diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzmovr" "b/data_all_eng_slimpj/shuffled/split2/finalzzmovr"
new file mode 100644--- /dev/null
+++ "b/data_all_eng_slimpj/shuffled/split2/finalzzmovr"
@@ -0,0 +1,5 @@
+{"text":"\\section{Introduction}\n\n\nThere is no analytic method to determine the spectrum of the Hamiltonian\nof the gauge theory of the strong interaction. There is a long-standing conjecture that there is a gap in this spectrum between the ground-state and first-excited-state energies. One strategy to illuminate this problem is to study the physical space of configurations \n\\cite{singer}. These configurations are not choices of gauge field; rather they are gauge orbits. In this paper, we find coordinates on this space, eliminating gauge-fixing ambiguities \\cite{gribov}. This is more difficult in three or more space-time dimensions than in two\n\\cite{1+1}. \n\nThe space of gauge orbits is not a manifold, but an orbifold \\cite{FSS}. The Hamiltonian of the gauge theory is a linear combination of the Laplace-Beltrami operator and a certain potential function \non orbit space.\n\nDetermining some geometric quantities on orbit space (such as the Laplace-Beltrami operator, the Ricci curvature or the scalar curvature) require the \nevaluation of a trace. Such a trace does not exist without a regularization. Singer proposed \nzeta-function regularization for this purpose \\cite{singer}. In this paper we will regularize with a lattice. In particular, we use Kogut and Susskind's  gauge theory, defined on a spatial lattice, but with continuous time \\cite{k&s} (see Reference  \\cite{creutz} for a derivation of the Kogut-Susskind formalism from Euclidean path-integral lattice gauge theory \\cite{wilson}, with the \ntransfer matrix). \n\nTo make our treatment of gauge fixing simple, we break with the practice of using a toroidal lattice, using instead an open rectangular lattice.\n\n\n\nThe metric tensor on the space of gauge orbits can be understood as the projection operator\nwhich vanishes on gauge transformations \\cite{singer}, \\cite{bab-via}. Another point of view is to regard orbit space as a metric space; the same metric \ntensor \narises naturally in such a context \\cite{orl-metric}. This projection operator \nis singular by definition; it acts on functionals of the gauge field, not the physical wave functionals of  gauge \norbits. This is why it is desirable to find coordinates on orbit space. The metric between points of orbit space on the lattice was discussed in Reference \\cite{kud-mor-orl}. Several papers have partly reduced the number of orbit-space coordinates in an axial gauge \\cite{orl5}, but here all redundancies are completely eliminated. There are conically-singular points in the orbit-space orbifold, which arise from\ngauge configurations invariant under a subgroup of the gauge group \\cite{FSS}.\n\n\n\n\nOur approach to finding coordinates and the metric on lattice orbit space makes it possible to find the Riemann, Ricci and scalar curvatures, at least in principle. A lower bound on the Ricci curvature implies a gap in the spectrum of the kinetic term of the Hamiltonian \\cite{BochLichn}. The problem of calculating the curvature with the lattice regularization is more difficult than might be expected. This is because the metric tensor and inverse metric tensor need to be explicitly calculated. The lattice metric constructed in \\cite{kud-mor-orl} was derived by taking an infimum over distances between elements of two orbits. This can then be coordinatized to give a metric tensor. Finding the inverse metric tensor is nontrivial, but possible. The inverse metric tensor is contained in the Laplace-Beltrami operator, so can be extracted once coordinates have been chosen. Determining the inverse metric tensor is the focus of this paper.\n\nWe should mention that there is another way of studying the space of configurations of \n$2+1$-dimensional gauge theories using holomorphic coordinates which appear very useful \\cite{KJM}.  Some \nresults similar to those in Reference \\cite{KJM} were obtained in a simple formalism\n\\cite{giv}. \n\nThis paper is organized as follows. Some \ndefinitions are given in Section \n\\ref{sec:prelim}. Gauss' law and the definition of orbit space are given in Section \\ref{sec:orbit}. In\nparticular, we discuss how gauge-equivalent gauge configurations are eliminated. Section \n\\ref{subsec:last} is the heart of the paper, where the last step in the gauge fixing is \ndone. To place the ideas in context, we review the metric on orbit space in \nSection \\ref{sec:metric}. To make the coordinates on this space explicit, we introduce Euler angles for gauge fields in Section \\ref{sec:coord}. In Section \\ref{sec:imt}, we describe the form of the inverse metric tensor on a finite, $2$-dimensional rectangular lattice for gauge group ${\\rm SU}(2)$. Finally, we summarize our results and discuss some avenues for further work in Section \\ref{sec:concl}.\n\n\n\n\\section{Preliminaries}\\label{sec:prelim}\n\n\\begin{definition}\nThe $D$-dimensional \\emph{lattice} is the graph whose set of vertices is a subset of \n$\\mathbb{Z}^D$, and whose edges connect each vertex to its nearest neighbors.\\end{definition}\n\n\nWe will work with finite rectangular lattices. An example of such a lattice with $D=2$ is \nshown in Figure \\ref{lattice}. \n\nThe vertices of the lattice\nare denoted by $\\mathbf{x}\\equiv(x_1,x_2,\\dots,x_D)$. The numbers $x_{1}, \\dots, x_{D}$ are integer multiples of the lattice spacing $a$, specifically $x_{j}=0,a,2a,\\dots, L_{j}$, for $j=1,\\dots,D$.\nLet ${\\hat 1}, \\dots, {\\hat D}$ be unit vectors in the positive $1-,\\dots, D-$ directions, respectively. We denote the edge adjacent to the two vertices $\\mathbf{x}$ and $\\mathbf{x}+{\\hat{\\jmath}}$ \nby $(\\mathbf{x},j)$, for each $j=1,\\dots,D$. An element of ${\\rm SU}(n)$ is assigned to each edge of the lattice. The ${\\rm SU}(n)$ element\nat the edge $(\\mathbf{x},j)$ is denoted by $U_j(\\mathbf{x})$, Henceforth we shall take $n=2$, for simplicity. In the lattice-gauge literature, the \nvertices are called \\emph{sites} and the edges are called \\emph{links}.\n\n\n\n\n\n\\begin{figure}\n  \\caption{The finite rectangular lattice in 2 dimensions.}\n  \\begin{center}\n   \\includegraphics[height=150mm]{Lattice.jpg}\n     \\end{center}\n     \\label{lattice}\n\\end{figure}\n\n\n\\begin{definition}\nA \\emph{wave function} is a complex-valued function of all of the variables $U_j(\\mathbf{x})$ on all the edges. \n\\end{definition}\n\\begin{definition}\n\\emph{Gauge state space} is the Hilbert space of square-integrable wave functions, with the inner product \n\\beq\n\\langle\\Psi\\vert\\Phi\\rangle=\\int\\overline{\\Psi(U_j(\\mathbf{x}))}\\Phi(U_j(\\mathbf{x}))\\prod_{\\mathbf{x},j} \ndU_{j}(\\mathbf{x}),\n\\nonumber\n\\eeq\nwhere the integration measure on each edge is the Haar measure.\n\\end{definition}\n\n\nWe remark that only $n=2$ will be considered in any detail. We denote    \nthe basis vectors of $\\mathfrak{su}(2)$ by $t_{1}$, $t_{2}$ and $t_{3}$, normalized \nby ${\\rm Tr}\\,t_{a}t_{b}=\\delta_{ab}$.\n\n\n\n\nA column vector $l_{j}(\\mathbf{x})$ of the three differential operators, $[l_j(\\mathbf{x})]_1$, $[l_j(\\mathbf{x})]_2$\nand $[l_j(\\mathbf{x})]_3$,\nis assigned to each edge $(\\mathbf{x},j)$ of the lattice. We call these the {\\em electric-field operators}.\nThey are defined by the \ncommutation relations:\n\\begin{eqnarray}\n[l_{j}(x)_{b} , l_{k}(y)_{c} ]=\n{\\rm i}{\\sqrt 2}\\delta_{x\\,y}\\delta_{j\\,k} \\;\\epsilon^{bcd}\n\\;l_{j}(x)_{d} , \\nonumber \n\\end{eqnarray}\n\\begin{eqnarray}\n[l_{j}(x)_{b}, U_{k}(y)]        =\n-\\delta_{x\\,y}\\delta_{j\\,k}\\; t_{b}\\;U_{j}(x),\n\\nonumber\n\\end{eqnarray}\nwith all other commutators zero.\n\n \\begin{definition}\n The \\emph{Laplace-Beltrami operator} is\n\\beq\n -\\Delta\\equiv\\sum_{\\mathbb{Z}^D}\\sum_{j=1}^D\\sum_{j=1}^{n^2-1}{[l_j(\\mathbf{x})]}_b^2.\n \\nonumber\n \\eeq\nIts spectrum is unbounded and discrete. In $2$ dimensions this operator is:\n\\beq\n-\\Delta=\\sum_{x_1=0}^{L_1-1}\\sum_{x_2=0}^{L_2-1}\\sum_{j=1}^2\\sum_{b=1}^{3}{[l_j(x_1,x_2)]}_b^2.\n\\nonumber\n\\eeq\n \\end{definition}\n\n\n\\begin{definition}\nThe \\emph{covariant derivative} of $l_j(\\mathbf{x})$ is \n\\beq\n\\mathcal{D}_{j}l_j(\\mathbf{x})\\equiv\\mathcal{D}_j(\\mathbf{x})\\cdot l_j(\\mathbf{x})\\equiv l_j(\\mathbf{x})-(1-\\delta_{0}^{x_j})U_j(\\mathbf{x}-\\hat{\\jmath})l_j(\\mathbf{x}-\\hat{\\jmath})U_j(\\mathbf{x}-\\hat{\\jmath})^{-1}.\n\\label{CD1}\n\\eeq\n\\end{definition}\n\n\\noindent\nThe factor $(1-\\delta_{0}^{x_j})$ is needed  because the lattice is finite and rectangular.\n\nAn element of the adjoint representation $\\mathcal{R}_j(\\mathbf{x})$, is assigned to \n$U_j(\\mathbf{x})$ by\n\\beq\nUt_bU^{-1}\\equiv\\mathcal{R}t_b,\n\\nonumber\n\\eeq\nwhere the arguments denoting the edge are implicit. Notice that $\\mathcal{R}_j(\\mathbf{x})$\nlies in $SO(3)$. Hence \\eqref{CD1} may be written\n\\beq\n\\mathcal{D}_{j}l_j(\\mathbf{x})\\equiv l_j(\\mathbf{x})-(1-\\delta_{0}^{x_j})\\mathcal{R}_j(\\mathbf{x}-\\hat{\\jmath})l_j(\\mathbf{x}-\\hat{\\jmath}).\n\\label{CD2}\n\\eeq\n\n\\section{Orbit Space}\\label{sec:orbit}\n\n\n\n\\begin{definition}\n\\emph{Gauss' law}  is \n\\beq\n\\sum_j^D \\mathcal{D}_{j}l_j(\\mathbf{x})=0. \\nonumber\n\\nonumber\n\\eeq\nGauss' law is imposed at every vertex.\n\\end{definition}\n\n\n\nWe denote by $\\{U\\}$\nthe collection of $U_{j}({\\mathbf x})\\in {\\rm  SU}(2)$ for all the \nedges $({\\mathbf x},j)$. The equivalence relation \n$\\{U\\} \\simeq \\{V\\}$\nbetween two lattice-gauge configurations $\\{U\\}$\nand $\\{V\\}$\nmeans that there is gauge transformation $\\{K\\}$, i.e. \nsome collection $K({\\mathbf x})\\in {\\rm SU}(2)$ at sites $x$ such that\n\\begin{eqnarray}\nV_{j}({\\mathbf x})\\;=\\;K({\\mathbf x}+{\\hat\\jmath}a)^{-1}\\;U_{j}({\\mathbf x})\\;K({\\mathbf x})\\;.\n\\nonumber\n\\end{eqnarray}\nWe will sometimes use the obvious notation \n$\\{V\\}=\\{U\\}^{\\{K\\}}$ for this expression. \n\n\\begin{definition}\nA \\emph{gauge orbit} $u$ is \nan equivalence class of lattice-gauge configurations under the equivalence relation\n$\\simeq$, defined above. \n\\end{definition}\n\n\\noindent Gauss' law is the statement that wave \nfunctions depend on orbits rather than\ngauge configurations \\cite{k&s} \\cite{creutz}. To put coordinates on orbit space, we must first assign a unique element configuration for each equivalence class of \ngauge configurations. This is the procedure called {\\em gauge fixing}.\n\n\n\nA gauge transformation can easily be used to set the ${\\rm SU}(2)$ elements on edges in the \n$1$-direction to unity. A further gauge transformation can then used to set the ${\\rm SU}(2)$ elements \non the edges in the $2$-direction for which $x_1=0$, and so on. We have thereby fixed the gauge on a maximal tree:\n\\beq\nU_1(x_1, x_2, x_3, \\ldots)=\\mathbb{I}, \\nonumber \\\\\nU_2(0, x_2, x_3, \\ldots)=\\mathbb{I}, \\nonumber \\\\\nU_2(0, 0, x_3 \\ldots)=\\mathbb{I}, \\nonumber \\\\\n\\vdots\\;. \\nonumber\n\\eeq\nAs this is done, we use Gauss' law to write the electric-field operators on the fixed edges\nin terms of the \nelectric-field operators on the unfixed edges. In $2$ dimensions this is\n\\beq\nl_1(x_1,x_2)=-\\sum_{y_1=0}^{x_1}\\mathcal{D}_2 l_2(y_1,x_2), \\label{2dgauss1}\n\\eeq\n\\beq\nl_2(0,x_2)=-\\sum_{y_2=0}^{x_2}\\sum_{y_1=1}^{L_1}\\mathcal{D}_2 l_2(y_1,y_2) .\\label{2dgauss2}\n\\eeq\nThe procedure is similar for $D\\geq 3$.\n\nThe Laplace-Beltrami operator for $D=2$ may now be rewritten as\n\\beq\n\\nonumber\n-\\Delta&=&\\sum_{b=1}^3 \\Bigg\\lbrace \\sum_{x_1=2}^{L_1} \\sum_{x_2=0}^{L_2-1} [l_2(x_1,x_2)]^2 + \\sum_{x_2=1}^{L_2-1} [l_2(1,x_2)]^2  \\\\ \\nonumber\n&& -\\sum_{x_1=0}^{L_1-1} \\sum_{x_2=0}^{L_2} \\left[\\sum_{y_1=0}^{x_1} \\mathcal{D}_2 l_2(y_1,x_2)\\right]^2 \\label{LB} \\\\\n&& -\\sum_{x_2=0}^{L_2-1} \\left[\\sum_{y_2=0}^{x_2} \\sum_{y_1=1}^{L_1} \\mathcal{D}_2 l_2(y_1,y_2) \\right]^2  \\\\\n\\nonumber\n&& + \\ [l_2(1,0)]^2 \\Bigg\\rbrace .\\nonumber\n\\eeq\n\nThe gauge fixing in (\\ref{2dgauss1}) and (\\ref{2dgauss2})  is not yet complete. There are three remaining \nconditions to solve:\n\\beq\n\\sum_{y_2=0}^{L_2}\\sum_{y_1=1}^{L_1} \\mathcal{D}_2 {l}_2 (y_1,y_2)=0 .\n\\label{semifinalgauss}\n\\eeq\n\n\\subsection{Fixing the Last Edge}\\label{subsec:last}\n\nThe remaining global condition (\\ref{semifinalgauss}) can be solved by making a single\nelement of ${\\rm SU}(2)$ (at one edge) diagonal. No further gauge fixing is then possible. For $D=2$, we chose to diagonalize $U_2(1,0)$. As a result $R_2(1,0)$ will also be diagonal. For this purpose, we rewrite (\\ref{semifinalgauss}) as\n\\beq\n-[\\mathbb{I}-\\mathcal{R}_2 (1,0)]l_2(1,0)=l_2(1,1)+\\sum_{y_2=2}^{L_2}\\mathcal{D}_2l_2(1,y_2)+\\sum_{y_2=0}^{L_2}\\sum_{y_1=2}^{L_1}\\mathcal{D}_2l_2(y_1,y_2)\\equiv\\Xi.\n\\label{finalgauss}\n\\eeq\n\n\\section{The metric}\\label{sec:metric}\n\n\n\n\nThe metric distance $\\rho(u,v)$\nbetween two gauge orbits $v$ and \n$v$ on the lattice \nis given by \\cite{kud-mor-orl}\n\\begin{eqnarray}\n\\rho(u,v)^{2}\n&=&N-\\frac{1}{2}\\inf_{\\{K\\} }\\sum_{{\\mathbf x}}\\sum_{j=1}^{D} \\left[\n{\\rm Tr}\\; K({\\mathbf x})V_{j}({\\mathbf x})^{-1}K({\\mathbf x}+{\\hat\\jmath}a)^{-1} U_{j}({\\mathbf x}) \\right. \n\\nonumber \\\\\n&+&\\left. \n{\\rm Tr} \\;K({\\mathbf x}+{\\hat\\jmath}a)V_{j}({\\mathbf x})K({\\mathbf x})^{-1}U_{j}({\\mathbf x})^{-1} \\right]\n\\;,\\label{latmet}\n\\end{eqnarray}\nwhere $\\{U\\}$ is any element of $u$ and $\\{V\\}$ \nis any element \nof $v$. This function of two orbits is gauge invariant. Furthermore, it is a metric \\cite{kud-mor-orl}.\n\nThe partition function of a Wilson \nlattice gauge theory in $D+1$ dimensions, with discrete time $t$, is\n\\begin{eqnarray}\n\\prod_{{\\mathbf x},t,\\mu}\\int dU_{\\mu}({\\mathbf x})\\; e^{-S}\\;, \\label{wilson}\n\\end{eqnarray}\nwhere the index $\\mu$ runs from $0$ to \n$D$. The action $S$ may be split as \n\\begin{eqnarray}\nS=\\frac{a^{D-2}}{a_{\\rm t}g_{0}^{2}}\n\\sum_{t} {\\mathcal L}_{\\rm st}+\\frac{a_{\\rm t}a^{D-4}}{g_{0}^{2}}\n\\sum_{t}{\\mathcal L}_{\\rm ss}  \\;, \\nonumber\n\\end{eqnarray}\nwhere $a_{\\rm t}$ is the lattice spacing in the time direction, and\nwhere \n${\\mathcal L}_{\\rm st}$ is the contribution of a space-time plaquette \nand ${\\mathcal L}_{\\rm ss}$ is the contribution\nof a space-space plaquette. Explicitly \n\\begin{eqnarray}\n{\\mathcal L}_{\\rm st}\\!&\\!=\\!&\\!\\frac{N}{2}\n-\\frac{1}{2}\\sum_{{\\mathbf x}}\\sum_{j=1}^{D} \\left[\n{\\rm Tr}\\; U_{0}({\\mathbf x},t)U_{j}({\\mathbf x},t+a_{\\rm t})^{-1}\nU_{0}({\\mathbf x}+{\\hat\\jmath}a, t)^{-1} U_{j}({\\mathbf x},a) \\right. \\nonumber \\\\\n\\!&\\!+\\!&\\!\\left. {\\rm Tr} \\;U_{0}({\\mathbf x}+{\\hat\\jmath}a,t)\nU_{j}({\\mathbf x},t+a_{\\rm t})U_{0}({\\mathbf x},t)^{-1}U_{j}({\\mathbf x},t)^{-1} \n\\right]\\;,\\label{space-time}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n{\\mathcal L}_{\\rm ss}\\!&\\!=\\!&\\!\n\\frac{N}{4}-\\frac{1}{4}\\sum_{{\\mathbf x}}\\sum_{j\\neq k}\\left[\n{\\rm Tr}\\; U_{j}({\\mathbf x},t)U_{k}({\\mathbf x}+{\\hat\\jmath}a,t)\nU_{j}({\\mathbf x}+{\\hat k}a,t)^{-1}U_{k}({\\mathbf x},t)^{-1} \n\\right. \\nonumber \\\\\n\\!&\\!+\\!&\\! \\left. \n{\\rm Tr}\\; U_{k}({\\mathbf x},t)U_{j}({\\mathbf x}+{\\hat k}a,t)\nU_{k}({\\mathbf x}+{\\hat\\jmath}a,t)^{-1}U_{j}({\\mathbf x},t)^{-1} \n\\right]\\;.\\label{space-space}\n\\end{eqnarray}\nNote that the right-hand sides of (\\ref{latmet}) and \n(\\ref{space-time}) are very similar; if we\nsubstitute for each $x$ and $j$\n$U_{j}({\\mathbf x},t)\\rightarrow U_{j}({\\mathbf x})$, \n$U_{j}({\\mathbf x},t+a_{\\rm t})\\rightarrow V_{j}({\\mathbf x})$, and\n$U_{0}({\\mathbf x},t)\\rightarrow K({\\mathbf x})$, into the right-hand side of \n(\\ref{space-time}), and\ntake the infimum with respect to $K({\\mathbf x})$, we obtain the \nlattice metric. Thus, by an appropriate\ngauge fixing of the temporal gauge configuration $U_{0}({\\mathbf x},t)$, we \nmay replace ${\\mathcal L}_{\\rm st}$\nby $\\rho(u(t),u(t+a_{\\rm t}))$, where $u(t)$ is the gauge orbit \ncontaining $\\{U\\}$ at time $t$ and \n$u(t+a_{\\rm t})$ is the gauge orbit containing $\\{U\\}$ at \ntime $t+a_{\\rm t}$. Alternatively, if we\nsimply integrate out $U_{0}({\\mathbf x},t)$, the dominant contribution \nto (\\ref{wilson}) at weak coupling \nwill come from this choice of $U_{0}({\\mathbf x},t)$.\n\nTo see that (\\ref{latmet}) is a metric, we note\nthat for any two orbits $u$ and $v$, \n$\\rho(u,v)=\\rho(v,u) \\ge 0$, with\n$\\rho(u,v) = 0$ if and only if $u=v$. The only\nremaining property we need is the triangle inequality, proved in Reference\n\\cite{kud-mor-orl}. As the proof is not hard, we repeat it below.\n\nNotice that (\\ref{latmet}) is the same as\n\\begin{eqnarray}\n\\rho(u,v)\n&=&\\inf_{\\{K\\} } \\; I(\\{U\\},\\{V\\}^{\\{K\\}})\n=\\inf_{\\{K\\} } \\; I(\\{U\\}^{\\{K\\}},\\{V\\}) \\nonumber \\\\\n&=&\\inf_{\\{K\\},\\{L\\} } \\; I(\\{U\\}^{\\{K\\}},\\{V\\}^{\\{L\\}})\n\\;, \\label{latmet2}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nI(\\{U\\},\\{V\\})^{2}=\n\\frac{1}{2} \\sum_{{\\mathbf x}}\\sum_{j=1}^{D} \n{\\rm Tr}\\; \n\\left[ V_{j}({\\mathbf x})- U_{j}({\\mathbf x}) \\right]^{\\dagger}  \n\\left[ V_{j}({\\mathbf x})- U_{j}({\\mathbf x}) \\right] \\;. \\label{latmet3}\n\\end{eqnarray}\n\nNow for any three sets of matrices $\\{U\\}$ and $\\{V\\}$ $\\{W\\}$ we \nhave that\n\\begin{eqnarray}\nI(\\{U \\},\\{V\\})+I(\\{V \\},\\{ W\\}) \\ge I(\\{ U\\},\\{ W\\})\\;,\n\\nonumber \n\\end{eqnarray}\nwhich is a consequence of the triangle inequality of a vector \nspace over the complex field (this\nis formally true by (\\ref{latmet3}), even if we are not dealing \nwith special-unitary\nmatrices). Introducing gauge transformations $\\{K\\}$, $\\{L\\}$ \nand $\\{M\\}$, we have\n\\begin{eqnarray}\nI(\\{U \\}^{\\{K\\}},\\{V\\}^{\\{L\\}})+I(\\{V \\}^{\\{L\\}},\\{ W\\}^{\\{M\\}}) \n\\ge I(\\{ U\\}^{\\{K\\}},\\{ W\\}^{\\{M\\}})\n\\;,\n\\nonumber\n\\end{eqnarray}\nwhich implies that\n\\begin{eqnarray}\nI(\\{U \\}^{\\{K\\}},\\{V\\}^{\\{L\\}})+I(\\{V \\}^{\\{L\\}},\\{ W\\}^{\\{M\\}}) \n\\ge \\rho(u,w)\n\\;.\n\\nonumber\n\\end{eqnarray}\nTaking the infimum of the left-hand side of this equation \ngives the triangle inequality \n\\begin{eqnarray}\n\\rho(u,v)+\\rho(v,w)\\ge \\rho(u,w)\\;.\n\\label{triangle}\n\\end{eqnarray}\n\nWe next show that (\\ref{latmet}) \nprovides a Riemannian metric, except at conically-singular orbits \\cite{FSS}. \n\nLet us substitute $U_{j}({\\mathbf x})=e^{-i{\\mathcal A}_{j}({\\mathbf x})\\cdot t}$, \n$V_{j}({\\mathbf x})=e^{-i[\n{\\mathcal A}_{j}({\\mathbf x})\n+d{\\mathcal A}_{j}({\\mathbf x})\n]\\cdot t}$ and $K({\\mathbf x})=e^{d\\phi({\\mathbf x})\\cdot t}$ into (\\ref{latmet}) \nand expand to second order in\n$d{\\mathcal A}_{j}({\\mathbf x})$ and $d\\phi({\\mathbf x})$. The result is\n\\begin{eqnarray}\nd\\rho^{2}=\\rho(u,v)^{2}=\n\\inf_{d\\phi}\n\\sum_{{\\mathbf x},j} \\left\\{ e_{j}({\\mathbf x})_{\\alpha}^{\\;\\;\\;b}d{\\mathcal A}_{j}({\\mathbf x})^{\\alpha}\n+[-{\\mathcal D}_{j}^{\\dagger}d\\phi({\\mathbf x}+{\\hat\\jmath}a)]^{b} \\right\\}^{2} \n\\;. \\nonumber\n\\end{eqnarray}\nThe minimum of the sum on the right-hand side is unique. We \nfind that $d\\phi({\\mathbf x})$ is  \n\\begin{eqnarray}\nd\\phi({\\mathbf x})=\\sum_{{\\mathbf y},j}\n({-{\\mathcal D}^{\\dagger}\\cdot {\\mathcal D}})^{-1}\\delta_{\\mathbf {xy}}\\; \n{\\mathcal D}_{j}\\cdot e_{j}{\\mathcal A}_{j}({\\mathbf y})\n\\;, \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\ne_{\\alpha}^{\\;\\;\\;a} t_{a}=-i U^{-1}\\partial_{\\alpha} U\\;, \\nonumber\n\\end{eqnarray}\n(this is given explicitly by \n\\begin{eqnarray}\ne_{\\alpha}^{\\;\\;\\;a}  =\n-i\\left( \\frac{{\\mathbb I}- e^{i{\\mathcal A}\\cdot T}}{\n{\\mathcal A}\\cdot T}\\right)_{\\alpha}^{\\;\\;\\;a} \n\\;,\\nonumber\n\\end{eqnarray}\nin canonical coordinates, where $T_{1,2,3}$ constitutes a basis of the adjoint representation of the Lie algebra) and where the Green's function \n$({-{\\mathcal D}^{\\dagger}\\cdot {\\mathcal D}})^{-1}\\delta_{\\mathbf {xy}}$ is \nuniquely determined by\nthe boundary conditions. \nThis variational problem has the solution \\cite{kud-mor-orl}\n\\begin{eqnarray}\nd\\rho^{2}=\nG_{({\\mathbf x},j,\\alpha)({\\mathbf y},k,\\beta)}\nd{\\mathcal A}_{j}({\\mathbf x})^{\\alpha}\nd{\\mathcal A}_{k}({\\mathbf y})^{\\beta}\n\\;, \\label{riemann1}\n\\end{eqnarray}\nwhere we sum over lattice edges in our summation\nconvention and where the metric tensor is\n\\begin{eqnarray}\nG_{({\\mathbf x},j,\\alpha)({\\mathbf y},k,\\beta)}=\ne_{j}({\\mathbf x})_{\\alpha}^{\\;\\;\\;b}\n\\left\\{\\delta_{\\mathbf {xy}}\\delta_{jk}\\delta_{bc}-\n\\left[ (-{{\\mathcal D}_{j}}^{\\dagger})\\frac{1}{{-{\\mathcal D}^{\\dagger}\\cdot \n{\\mathcal D}}}{\\mathcal D}_{k}\\right]_{bc}\\delta_{{\\mathbf {xy}}}\n\\right\\}\ne_{k}({\\mathbf y})_{\\beta}^{\\;\\;\\;c}\n\\;. \\label{riemann2}\n\\end{eqnarray}\n\n\n\nNotice that the quantity in curled brackets in (\\ref{riemann2}) is \nidempotent, hence it is a \nprojection. In fact, the metric projects out\ngauge transformations in inner products. To remove \nthe zero eigenvalues, it is necessary to fix the gauge. The resulting induced metric is that on\nall of orbit space, except at conical singularities. The set of these singularities is of measure zero, but it is an open question whether they have consequences for the Yang-Mills spectrum \\cite{FSS}. \n\n\n\n\\section{Euler Angles}\\label{sec:coord}\n\nAs in our previous discussion, we specialize to gauge group SU($2$). W introduce Euler coordinates at each edge:\n\\beq\nU_{j}({\\mathbf x})=e^{i\\alpha_{j}({\\mathbf x}){\\sigma}_z}e^{i\\beta_{j}({\\mathbf x}){\\sigma}_x}e^{i\\theta_{j}({\\mathbf x}){\\sigma}_z},\n\\label{euler}\n\\eeq\nwhere ${\\sigma}_x, {\\sigma}_y, {\\sigma}_z$ are the Pauli matrices. The reason we use these coordinates (instead of the canonical coordinates of the previous section) is technical, not fundamental. It is easier to use Euler angles to perform the gauge fixing at the last edge.\n\nIn much of the discussion which follows, we denote the angles at the last edge $\\alpha_{2}(1,0)$,\n$\\beta_{2}(1,0)$ and $\\theta_{2}(1,0)$ by $\\alpha$, $\\beta$ and $\\theta$, respectively and $U_{j}(1,0)$ by $U$.\n\n\n\nA choice of basis vectors of $\\mathfrak{su}(2)$  is\n\\beq\n\\mathcal{M}_{\\gamma}^a \\sigma_a \\equiv -i \\partial_{\\gamma}U,  \\label{Mdef}\n\\eeq\nwhere $\\gamma$ denotes $\\alpha$, $\\beta$ or $\\theta$. From \\eqref{euler}, we can find $\\mathcal{M}_{\\gamma}^a$:\n\\beq\n\\mathcal{M}=\n\\begin{pmatrix} \n\\sin{2\\alpha}\\sin{2\\beta}& -\\cos{2\\alpha}\\sin{2\\beta}&\\cos{2\\beta} \\\\ \n\\\\\n\\cos{2\\alpha}&\\sin{2\\alpha}&0 \\\\\n\\\\\n0&0&1 \n\\end{pmatrix} . \\label{M}\n\\eeq\nThis result can be used to express the electric-field operators in terms of derivatives of the coordinates:\n\\newline\n\\beq\n\\begin{pmatrix} l_1 \\\\ l_2 \\\\ l_3  \\end{pmatrix}=\\mathcal{M}^{-1}\\begin{pmatrix}\\partial_{\\alpha}\\\\ \\partial_{\\beta} \\\\ \\partial_{\\theta} \\end{pmatrix}. \\nonumber\n\\eeq\nExplicitly:\n\\beq\n{[l_2(1,0)]}_1&=&\\frac{\\sin(2\\alpha)}{\\sin(2\\beta)}\\partial_{\\alpha}-\\cos(2\\alpha)\\partial_{\\beta}-\\frac{\\cos(2\\beta)\\sin(2\\alpha)}{\\sin(2\\beta)}\\partial_{\\theta} ,  \\label{l's} \\\\\n{[l_2(1,0)]}_2&=&\\frac{\\cos(2\\alpha)}{\\cos(2\\beta)}\\partial_{\\alpha}+\\sin(2\\alpha)\\partial_{\\beta}-\\sin(2\\alpha)\\partial_{\\theta} , \\nonumber \\\\\n{[l_2(1,0)]}_3&=&\\partial_{\\theta}  \\nonumber.\n\\eeq\nFrom \\eqref{euler}, and \\eqref{l's}, $\\mathcal{R}_j(x_1,x_2)$ can be explicitly calculated:\n\\beq\n\\;\\;\\mathcal{R}_j(x_1,x_2)=& \\nonumber \\\\\n\\;\\;&\\begin{pmatrix} \n \\cos(2\\beta)& -\\cos(2\\alpha)\\sin(2\\beta)&\\sin(2\\alpha) \\sin(2\\beta)\\\\ \n\\\\\n \\sin(2\\beta)\\cos(2\\theta)&\n\\begin{array}{c} -\\sin(2\\alpha)\\sin(2\\theta) \\\\ +\\cos(2\\alpha)\\cos(2\\beta)\\cos(2\\theta) \\end{array} &\n\\begin{array}{c} -\\cos(2\\alpha)\\sin(2\\theta) \\\\ -\\sin(2\\alpha)\\cos(2\\beta)\\cos(2\\theta) \\end{array}  \\\\\n\\\\\n\\sin(2\\beta)\\sin(2\\theta)& \n\\begin{array}{c} \\sin(2\\alpha)\\cos(2\\theta) \\\\ + \\cos(2\\alpha)\\cos(2\\beta)\\sin(2\\theta) \\end{array}&\n\\begin{array}{c} \\cos(2\\alpha)\\cos(2\\theta) \\\\  -\\sin(2\\alpha)\\cos(2\\beta)\\sin(2\\theta) \\end{array}  \n\\end{pmatrix}  . \\label{R}\n\\eeq\nDiagonalizing $\\mathcal{R}_2(1,0)$ by some $g\\in {\\rm SU}(3)$ yields the expression:\n\\beq\ng^{-1}\\mathcal{R}_2(1,0)g=\n\\begin{pmatrix}\n1 & 0 & 0 \\\\\n0 & k & 0 \\\\\n0 & 0 & \\frac{1}{k} \\\\\n\\end{pmatrix} , \\label{diag1}\n\\eeq\nwhere \n\\beq\nk&=& -\\frac{1}{2}\\left( (1-\\cos(2\\alpha-2\\theta)[1+\\cos(2\\beta)-\\cos(2\\beta)]  \\right.\\nonumber \\\\\n&+ &\n\\left.  \\{4+[(1-\\cos(2\\alpha-2\\theta)(1+\\cos(2\\beta)]-\\cos(2\\beta)\\}^2)^{\\frac{1}{2}}\\right).\n\\label{diag2}\n\\eeq\nSubstitution of (\\ref{diag1}) and (\\ref{diag2}) into \\eqref{finalgauss} gives two conditions, which \nallow the identification $\\beta=\\theta=\\frac{\\pi}{4}$. The gauge fixing is now complete.\nWe remove the ubiquitous factors of two in the remainder of this paper,  \nby redefining\n$\\alpha\\rightarrow 2\\alpha$, $\\beta\\rightarrow 2\\beta$, and $\\theta\\rightarrow 2\\theta$.\n\nTwo of the derivatives on the $(1,0)$ edge may be written in terms of the third derivative, in addition to some of the angles on other edges:\n\\beq\n\\partial_{\\theta}&=&\\frac{\\Xi_3}{1-\\frac{1}{k}}, \\label{theta} \\\\ \n\\partial_{\\beta}^{\\star}&=& \\frac{1}{\\sin{\\alpha}} \\left(\\sin{\\alpha}\\frac{\\Xi_3}{1-\\frac{1}{k}} +\\frac{\\Xi_2}{1-k} - \\frac{\\cos{\\alpha}}{\\cos{\\beta}} \\right) \\label{beta},\n\\eeq\nwhere $\\Xi$ is defined in \\eqref{finalgauss}. The asterisk on the derivative with respect to \n$\\beta$ means that derivatives with respect to $\\theta$ are replaced by the right-hand side of\n\\eqref{theta}.\n\nThe Laplace-Beltrami operator may now be written as:\n\\beq\n-\\Delta=-\\Delta_{1}-\\Delta_{2}-\\Delta_{3}-\\Delta_{4}-\\Delta_{5}-\\Delta_{6},\\label{lb}\n\\eeq\nwhere\n\\beq\n-\\Delta_{1}&=&\n\\sum_{x_2=1}^{L_2} [l_2(1,x_2)]^2 +\\sum_{x_1=2}^{L_1} \\sum_{x_2=0}^{L_2} [l_2(x_1,x_2)]^2,   \\nonumber \\\\\n-\\Delta_{2}&=& -\n\\sum_{x_1=0}^{L_1-1} \\sum_{x_2=0}^{L_2} \\left[\\sum_{y_1=0}^{x_1} \\mathcal{D}_2 l_2(y_1,x_2)\\right]^2, \\nonumber  \\\\\n-\\Delta_{3}&=& -\n\\sum_{x_2=0}^{L_2-1} \\left[\\sum_{y_2=0}^{x_2} \\sum_{y_1=1}^{L_1} \\mathcal{D}_2 l_2(y_1,y_2) \\right]^2,  \\nonumber \\\\\n-\\Delta_{4}&=& \\left(\\sqrt{2} \\sin{2\\alpha} \\ \\partial_{\\alpha} - \\cos{2\\alpha} \\ \\partial_{\\beta}^{\\star} - \\sin{2\\alpha} \\ \\frac{\\Xi_3}{1-\\frac{1}{k}}\\right)^2,\n\\nonumber \\\\\n-\\Delta_{5}&=& \\left(\\sqrt{2} \\cos{2\\alpha} \\ \\partial_{\\alpha} + \\sin{2\\alpha} \\ \\partial_{\\beta}^{\\star} - \\sin{2\\alpha}\\frac{\\Xi_3}{1-\\frac{1}{k}}\\right)^2, \\nonumber \\\\\n-\\Delta_{6}&=&\n\\left(\\frac{\\Xi_3}{1-\\frac{1}{k}}\\right)^2, \\nonumber  \n\\eeq\nwhere the angle $\\alpha\\equiv \\alpha_{2}(1,0)$ in $-\\Delta_{4}$, $-\\Delta_{5}$ and $-\\Delta_{6}$ is the sole remaining \ncoordinate specifying $U_{2}(1,0)$ (which is now diagonal).\n\nA comparison of  with the standard form of the Laplace-Beltrami operator:\n$-\\Delta \\equiv -\\frac{1}{\\sqrt{g}} \\partial_{\\mu} \\sqrt{g} \\ g^{\\mu \\nu} \\partial_{\\nu}, $\nyields the inverse metric tensor.\n\n\\section{The Inverse Metric Tensor}\\label{sec:imt}\nThe components of the inverse metric tensor may be read off by examining \n\\eqref{lb}. Fortunately, the determinant of the metric is not needed to find these components. Finding \nany given component (that is, $g^{\\mu \\nu}$) is done by selecting the function between two partial derivatives of the associated coordinates and multiplying by the function in front. This is because each term of the Laplace-Beltrami operator in equation \\eqref{lb} has the form \n$-\\frac{1}{h_{\\mu\\nu}} \\partial_{\\mu} h_{\\mu\\nu} g^{\\mu \\nu} \\partial_{\\nu}$ (the square root of the determinant of the metric $\\sqrt g$, automatically divides the\nproduct $\\prod_{\\mu\\nu}h_{\\mu\\nu}$).\n\nTo illustrate how the inverse metric tensor can be extracted, we give the example of the one-edge\nBeltrami-Laplace operator $-\\Delta_{\\rm one-edge}= l^{2}$. From the expressions\n\\eqref{l's} for the components of $l$, we find\n\\beq\ng^{\\alpha\\;\\alpha }\\!\\!&\\!\\!=\\!\\!&\\!\\! \\frac{1}{\\sin^2\\beta} ,\\nonumber \\\\\ng^{\\alpha\\;\\beta}\\!\\! &\\!\\! =\\!\\! & \\!\\! 0 , \\nonumber \\\\\ng^{\\alpha\\;\\theta}\\!\\! &\\!\\! =\\!\\! & \\frac{\\sin{\\alpha}(\\sin{(\\beta - \\alpha}))}{\\sin^2 \\beta} ,\n\\nonumber  \\\\\ng^{\\beta\\;\\beta}\\!\\!&\\!\\!=\\!\\!&\\!\\! 1 ,\\nonumber \\\\\ng^{\\beta\\;\\theta}\\!\\!&\\!\\!=\\!\\!&\\!\\! \\sin^2 \\alpha + \\frac{\\sin{\\alpha}\\cos{\\alpha}\\cos{\\beta}}{\\sin{\\beta}} ,\\nonumber \\\\\ng^{\\theta\\;\\theta}\\!\\!&\\!\\!=\\!\\!&\\!\\!\\frac{\\sin^2 \\alpha}{\\sin^2 \\beta} +1 , \\nonumber\n\\eeq\nNothing is new about this result, which is simply the inverse metric tensor of a three sphere.\n\n\nUsing \\eqref{CD2}, \\eqref{M}, and \\eqref{R}, the components of $[\\mathcal{D}_2 l_2(y_1,x_2)]_b$ \n(which are in $-\\Delta_{2}$ and $-\\Delta_{3}$) reduce to:\n\\beq\n[\\mathcal{D}_2 l_2(y_1,x_2)]_{1}&=&\n\\frac{\\sin\\alpha}{\\sin\\beta}\\partial_{\\alpha\n}-\\cos\\alpha\\partial_{\\beta\n}-\\frac{\\sin\\alpha\\cos\\beta}{\\sin\\beta}\\partial_{\\theta\n}  \\label{b1} \\\\ \n&+&\\left(\\cos^2\\alpha-\\frac{\\sin\\alpha\\cos\\beta}{\\sin\\beta}\\right)\\partial_{\\alpha_{2}(y_1,x_2-1)} \\nonumber \\\\\n&+&\\cos\\alpha(\\cos\\beta+\\sin\\alpha\\sin\\beta)\\partial_{\\beta_{2}(y_1,x_2-1)} \\nonumber \\\\\t\n&+&\\sin\\alpha\\left(\\frac{\\cos^2\\beta}{\\sin\\beta}+\\cos\\alpha\\sin\\beta-\\sin\\beta\\right)\n\\partial_{\\theta_{2}(y_1,x_2-1)} ,\\nonumber \n\\eeq\n\\beq\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\![\\mathcal{D}_2 l_2(y_1,x_2)]_{2}  \\label{b2} \\\\\n&=&\\frac{\\cos\\alpha}{\\sin\\beta}\\partial_{\\alpha\n}+\n\\sin\\alpha\\partial_{\\beta\n}+\\sin\\alpha\\partial_{\\theta\n}   \\nonumber \\\\\n&+&\\frac{\\sin\\alpha\\cos\\alpha\\sin\\theta-\\cos^2\\alpha\\cos\\beta\\cos\\theta-\\sin\\alpha\\cos\\theta\\sin\\beta}{\\sin\\beta}\\partial_{\\alpha_{2}(y_1,x_2-1)} \\nonumber \\\\\n&+&(\\sin^2\\alpha\\sin\\theta-\\sin\\alpha\\cos\\alpha\\cos\\beta\\cos\\theta+\\cos\\alpha\\sin\\beta\\cos\\theta)\\partial_{\\beta_{2}(y_1,x_2-1)} \\nonumber \\\\\n&+&(\\sin\\alpha\\cos\\beta\\cos\\theta+\\sin^2\\alpha\\sin\\theta+\\cos\\alpha\\sin\\theta \\nonumber \\\\\n &&  \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\   -\\sin\\alpha\\cos\\beta\\cos\\theta-\\sin\\alpha\\cos\\alpha\\cos\\beta\\cos\\theta)\\partial_{\\theta_{2}(y_1,x_2-1)}, \\nonumber \n\\eeq\n\\beq\n&&\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\![\\mathcal{D}_2 l_2(y_1,x_2)]_{3} \\label{b3}\\\\\n&=&\\partial_{\\theta\n} \n-\\left (\\sin\\alpha\\sin\\theta+\\frac{\\sin\\alpha\\cos\\alpha\\cos\\theta}{\\sin\\beta}+\\frac{\\cos^2\\alpha\\cos\\beta\\sin\\theta}{\\sin\\beta}\\right)\\partial_{\\alpha_{2}(y_1,x_2-1)} \\nonumber \\\\\n&+&(\\cos\\alpha\\sin\\beta\\sin\\theta-\\sin^2\\alpha\\cos\\theta-\\sin\\alpha\\cos\\alpha\\cos\\beta\\sin\\theta)\\partial_{\\beta_{2}(y_1,x_2-1)} \\nonumber \\\\\n&+&(\\sin\\alpha\\cos\\beta\\sin\\theta-\\sin^2\\alpha\\cos\\theta-\\sin\\alpha\\cos\\alpha\\cos\\beta\\sin\\theta \\nonumber \\\\\n&& \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \\ \n+\\cos\\alpha\\cos\\theta-\\sin\\alpha\\cos\\beta\\sin\\theta)\\partial_{\\theta_{2}(y_1,x_2-1)} ,\\nonumber\n\\eeq\nwhere the edge direction and adjacent vertex are only indicated explicitly for coordinates other than \n$\\alpha\\equiv\\alpha_{2}{(y_1,x_2)}$, $\\beta\\equiv\\beta_{2}{(y_1,x_2)}$, \n$\\theta\\equiv\\theta_{2}{(y_1,x_2)}$.\n\n\nIn $-\\Delta_{2}$ we sum over spatial dimensions after squaring, but only in the $1$-direction. These terms are merely a local term coupled with an adjoint term from the edge below. These can be constructed similarly by multiplying the associated pieces, and then summing.\n\nThe term $-\\Delta_{3}$ is somewhat more complicated, as the sums run in both directions. When constructing the contribution from this term for a given component of the metric tensor, it must be noted that there will be overlap from $\\mathcal{D}_2l_2(y_1,y_2+1)$ with $\\mathcal{D}_2l_2(y_1,y_2)$, except on the boundary. The form of these terms is $(l(x_{\\gamma})-\\mathcal{R} l(x_{\\gamma})) (l(y_{\\xi})-\\mathcal{R} l(y_{\\xi}))$, and can also be constructed similarly to the above.\n\nThe last three terms $-\\Delta_{4}$, $-\\Delta_{5}$ and $-\\Delta_{6}$ contain many pieces. They have many more combinations than than the above, because they each contain sums over most of the lattice. Only the third component of the vector is taken, however, so summing of components is not required in their construction. \n\nThis completely defines the inverse metric tensor for the finite rectangular lattice with $D=2$, working over ${\\rm SU}(2)$. Similar methods work for $D\\geq 3$. We believe it is possible to generalize to\nthe gauge group ${\\rm SU}(n)$. Angular coordinates are considerably more complicated for $n>2$, however.\n\n\\section{Conclusions}\\label{sec:concl}\n\nTo summarize, we have explicitly found the metric tensor for ${\\rm SU}(2)$ on the lattice with open boundary conditions, and determined the inverse metric tensor.  It is noteworthy that the gauge-fixing problem only becomes\ncomplicated when fixing the last edge.\n\nThe methodology used here can be generalized to construct results for higher-dimensional lattices and  and other gauge groups. There are no marked differences for $D\\ge 3$. Generalizing the results to gauge group ${\\rm SU}(n)$ is cumbersome, but the strategy is the same as for \n${\\rm SU}(2)$; this is under investigation.\n\nIt should be possible to study orbit-space geodesics in our coordinates. Any geodesic in the full space of lattice gauge fields is described by the real parameter $t$ through\n\\beq\nU_{j}({\\mathbf x}; t)=\\exp{\\rm i}\\,\\tau({\\mathbf x},j) \\,t , \\label{geod}\n\\eeq\nwhere $\\tau({\\mathbf x},j)$ is an arbitrary chosen element of the Lie algebra chosen for each edge $({\\mathbf x},j)$. The\ngeodesics in the completely-fixed axial gauge are obtained by gauge-fixing (\\ref{geod}) according to the prescription given in this paper.\n\nFinally, we believe that a detailed study of the set of conically-singular points \\cite{FSS} in the lattice formulation of gauge theory should be very fruitful. This set is of measure zero, but that \ndoes not mean it has no significance. Presumably the Riemann curvature diverges at \nsuch points (even on the lattice). An interesting question is whether this divergence\ncan be regularized in a sensible and physically-meaningful way.\n\n\\section{Acknowledgements}\nM.L. would like to thank David Stone and Tony Phillips for discussions. P.O. would like to thank both the Niels Bohr International Academy and the Kavli Institute for Theoretical Physics for their hospitality during which some of this work was done.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\n\nThe distance from an observer to a galaxy is nearly a prerequisite for\nany scientific study. Fundamental physical properties of galaxies\n(e.g., luminosity, star formation rate, and stellar masses) always include\nuncertainties from distance measurements. Ideally, precise redshifts\nare desired for all galaxies detected by imaging.  Although the\ncurrent technology cannot yet meet this demand, large efforts have\nbeen made to obtain spectroscopic data as efficiently as possible.\n\nLarge spectroscopic galaxy surveys have greatly enhanced our ability\nto explore the formation and evolution of galaxies by gathering a\nwealth of data sets of spectral features (emission lines, absorptions\nlines, and breaks), which in turn provide redshifts\n\\citep[e.g.,][]{York00, Lill07, Newm13, LeFe15}. Multi-object\nspectroscopy (MOS) is the most widely used technique for\nsimultaneously obtaining a large number of spectra over a wide area,\nbut this technique requires preselection of the targets from imaging\ndata and has difficulty with dense fields because of potential slit or\nfiber collisions (when only passing the field once).  A magnitude cut\nis a typical selection method for a target sample. This process can\nexclude faint galaxies, including quiescent absorption line galaxies\nor dust attenuated red galaxies.  Also, a relatively bright magnitude\ncut on the target sample and restrictions on slit positioning lead to\nsparse sky sampling and introduce low spectroscopic completeness, in\nparticular toward higher density regions.\n\nAnother approach is photometric redshift (photo-$z$), which is derived\nby fitting multiple galaxy templates as a function of redshift to the\nobserved photometry of a source.  Although photo-$z$ may be an\nalternative to redshifts from MOS observations, its error is typically\n$\\Delta z\/(1+z) \\sim 0.05$ \\citep[e.g.,][]{Bonn16,Beck17}. This\nprecision is inadequate for tasks such as finding (close) galaxy\npairs, identifying overdensity regions, and studying (local)\nenvironmental effects.\n\nAn integral field unit (IFU) with a wide field of view (FoV), high\nsensitivity, wide wavelength coverage, and high spectral resolution\ncan remedy some of these issues. The IFU yields spectroscopy for each\nindividual pixel over the entire FoV, which means that every detected\nobject in the field has a spectrum.  However, earlier generation IFUs\nhad FoV sizes that were too restrictive, insufficient spectral resolution, or\nspatial sampling that was too low for spectral surveys.\n\nThe Multi Unit Spectroscopic Explorer \\citep[MUSE;][]{Baco10}, is an\nIFU on the Very Large Telescope (VLT) Yepun (UT4) of the European\nSouthern Observatory (ESO).  It is unique in having a large FoV\n($1\\arcmin \\times 1\\arcmin$), wide simultaneous wavelength coverage\n($4650 - 9300$${\\rm \\AA}$\\xspace), relatively high spectral resolution\n($R \\sim 3000$), and high throughput (35\\% end-to-end).  The MUSE\ninstrument has a wide range of applications for observing objects such\nas H\\,\\textsc{ii} regions, globular clusters, nearby galaxies, lensing\ncluster fields, quasars, and cosmological deep fields.  Despite its\nuniversality, MUSE was originally designed and optimized for deep\nspectroscopic observations.  One of its advantages is that it can deal\nwith dense fields without making any target preselection, achieving a\nspatially homogeneous spectroscopic completeness and reducing the\nuncertainty in assigning a measured redshift to a photometric object\n\\citep[cf.][]{Baco15,subm_Brin17}.\n\nWe have conducted deep cosmological surveys in the {\\it Hubble} Ultra\nDeep Field \\citep[HUDF;][]{Beck06} with MUSE. This field has some of\nthe deepest multiwavelength observations on the sky, from X-ray to\nradio. However, the previously known secure spectroscopic redshifts\n(spec-$z$) in the HUDF are limited to about only $2\\%$ of the detected\ngalaxies \\citep[169 out of 9927 galaxies;][]{Rafe15}.  The\nthree-dimensional data of MUSE also facilitate blind identifications\nof previously unknown sources by searching for emission lines directly\nin the data cube \\citep[cf.][]{Baco15}. In addition to increasing the\nspectroscopic completeness in the HUDF with MUSE, the unique\ncapabilities of MUSE have permitted some of the first detailed\ninvestigations of photo-$z$ calibration of faint objects up to 30th\nmagnitude \\citep[][hereafter Paper~III]{subm_Brin17}, properties of a\nstatistical sample of C\\,\\textsc{iii}]\\xspace emitters \\citep{subm_Mase17}, spatially\nresolved stellar kinematics of intermediate redshift galaxies\n\\citep{subm_Guer17}, constraints on the faint-end slope of\nLyman-$\\alpha$ (Ly$\\alpha$\\xspace) luminosity functions and its evolution\n\\citep{subm_Drak17}, spatial extents of a large number of Ly$\\alpha$\\xspace haloes\n\\citep{subm_Lecl17}, redshift dependence of the Ly$\\alpha$\\xspace equivalent width\nand the UV continuum slope \\citep{subm_Hash17}, galactic winds in\nstar-forming galaxies \\citep{subm_Finl17}, evolution of the galaxy\nmerger rate \\citep{subm_Vent17}, and analyses of the cosmic web\n\\citep{subm_Gall17}.  In this paper, we report on the first set of\nredshift determinations in the two layered MUSE UDF fields with\n$\\sim 10$~hour and $\\sim 30$~hour integration times used for all of\nthe studies above.\n\nThe paper is organized as follows: In \\S\\ref{sec:obs}, we explain our\ndeep surveys, observations, and data reduction in brief.  Then we\ndescribe the source and spectral extraction methods, procedure of\nthe redshift determination, detected emission line flux measurements,\nand continuum flux measurements in \\S\\ref{sec:analysis}.  The\nresulting redshift distributions and global properties are shown in\n\\S\\ref{sec:result}, followed by \\S\\ref{sec:discussion} in which we\ndiscuss success rates of the redshift measurements, emission line\ndetected objects, comparisons with previous spec-$z$ and photo-$z$,\nand color selections of high-$z$ galaxies.  Finally, the summary and\nconclusion are given in \\S\\ref{sec:summary}. Along with this paper, we\nrelease the redshift catalogs (see Appendix~\\ref{app:cat}).  The\nmagnitudes are given in the AB system throughout the paper.\n\n\n\n\n\\section{Observations and data reduction}\\label{sec:obs}\n\nThe detailed survey strategy, data reduction, and data quality\nassessments are presented in \\citet[hereafter\nPaper~I]{subm_Baco17}. Here we provide a brief outline.  As part of\nthe MUSE Guaranteed Time Observing (GTO) program, we carried out deep\nsurveys in the HUDF.  There are two layers of different depths in\noverlapping areas: the $3\\arcmin \\times 3\\arcmin$ medium deep and\n$1\\arcmin \\times 1\\arcmin$ ultra deep fields. The medium deep field\nwas observed at a position angle (PA) of $-42 \\deg$ with a\n$3\\arcmin \\times 3\\arcmin$ mosaic (\\textsf{udf-01} to\n\\textsf{udf-09}), and thus it is named the \\textsf{mosaic}\\xspace. The ultra deep\nregion (named \\textsf{udf-10}\\xspace) is located inside the \\textsf{mosaic}\\xspace with a PA of\n$0 \\deg$.  We selected this field to maximize the overlap region with\nthe ASPECS\\,\\footnote{The ALMA SPECtroscopic Survey in the Hubble\n  Ultra Deep Field} project \\citep{Walt16}.\n\nThe observations were conducted between September 2014 and February\n2016. during eight GTO runs under clear nights, good seeing, and\nphotometric conditions.  The average seeing measured with the obtained\ndata has a full width at the half maximum (FWHM) of $\\approx 0.6''$ at\n$7750$${\\rm \\AA}$\\xspace. The total integration time for the \\textsf{mosaic}\\xspace is\n$\\approx 10$~hours for the entire field ($9.92$ sq. arcmin). The \\textsf{udf-10}\\xspace\nwas observed for $\\approx 21$~hours (at a PA of $0 \\deg$), but the final\nproduct was added together with the overlap region of the \\textsf{mosaic}\\xspace to\nachieve $\\approx 31$~hours.\n\nThe data reduction begins with the MUSE standard pipeline\n\\citep{Weil12}. For each exposure, it uses the corresponding\ncalibration files (flats, bias, arc lamps) to generate a {\\it\n  pixtable} (pixel table), which stores information of wavelength,\nphoton count, and its variance at each pixel location. After\nperforming astrometric and flux calibrations on the pixtable, a data\ncube is built. Then we implemented the following calibrations beyond\nthe standard pipeline.  The remaining low-level flat fielding was\nremoved by self-calibration, artifacts were masked, and the sky\nbackground was subtracted. Finally, we combined the 227 individual\ndata cubes to create the final data cube for the \\textsf{mosaic}\\xspace. For \\textsf{udf-10}\\xspace,\n51 of the PA $0 \\deg$ data cubes and 105 overlapping \\textsf{mosaic}\\xspace data\ncubes were used to make the final product.\n\nThe MUSE data cubes used for this work are version 0.42 for both the\n\\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace.  The estimated $1\\sigma$ surface brightness limits\nare $2.8$ and\n$5.5 \\times 10^{-20} \\, {\\rm\n  erg\\,s^{-1}\\,cm^{-2\\,}\\,\\AA^{-1}\\,arcsec^{-2}}$\nin the wavelength range of $7000-8500$${\\rm \\AA}$\\xspace (excluding regions of OH\nsky emission) for \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace, respectively. The $3\\sigma$\nemission line flux limits for a point-like source are $1.5$ and\n$3.1 \\times 10^{-19} \\, {\\rm erg\\,s^{-1}\\,cm^{-2\\,}}$, respectively,\nat $\\sim 7000$${\\rm \\AA}$\\xspace, where has no OH sky emission line.\n\n\n\n\n\\section{Analysis}\\label{sec:analysis}\n\n\\subsection{Source and spectral extractions}\n\nWe performed two different source extractions in the MUSE Ultra Deep\nField (UDF).  One uses the {\\it HST} detected sources from the UVUDF\ncatalog\\,\\footnote{The ultraviolet UDF (UVUDF) catalog includes the\n  photometries of 11 {\\it HST} broadbands from F225W to F160W.}\n\\citep{Rafe15} as the priors to extract continuum selected\nobjects. The other searches for emission lines blindly (without prior\ninformation) in the cube directly. The details of the source\ndetections in the MUSE UDF are discussed in Paper~I. Here we briefly\nsummarize the methods of the source detection and spectral extraction\nof the detected sources.\n\n\n\\subsubsection{Continuum selected objects}\\label{sec:analysis_cont}\n\nAmong all of the 9969 galaxies in the UVUDF catalog, $\\UDForgHSTpri$\nand $\\MOSorgHSTpri$ of these serve as the priors to extract objects in\nthe \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace survey fields, respectively. We performed the\nextraction at all of the {\\it HST} source positions regardless of\nwhether they are detected in the MUSE data cube.  When the {\\it\n  HST}-detected galaxies are located within $0.6''$, which cannot be\nspatially resolved by our observations, they were merged and treated\nas a single object.  We computed the new coordinates for a merged\nobject via the {\\it HST} F775W flux-weighted center of all objects\ncomposing the new merged source.  We neither used their photometric\nmeasurements in this work nor reported the photometric measurements in\nthe catalogs (see note (e) of Table~\\ref{tbl:cat}).  In total, we\nacquired $\\UDFHSTpri$ and $\\MOSHSTpri$ MUSE objects\\,\\footnote{These\n  numbers increase after the redshift analysis because we manage to\n  ``split'' some of these objects based on their emission line\n  narrowband images (see \\S\\ref{sec:result_cont}).} for the \\textsf{udf-10}\\xspace and\n\\textsf{mosaic}\\xspace fields, respectively.  Then we made a\n$5\\arcsec \\times 5\\arcsec$ subcube (or larger for extended objects)\nfor each of the extracted objects so that they are easy to handle.\nFor each object, we convolved its {\\it HST} segmentation map (provided\nby the UVUDF catalog) with the MUSE point spread function (PSF) to\nconstruct a mask image to mask out emission from nearby objects.\n\nFor the purpose of redshift determination, while we used all of the\nextracted objects in \\textsf{udf-10}\\xspace , we only inspected the objects with\n$\\rm F775W \\leq 27$~mag ($\\MOSHSTpriMAGcut$ MUSE objects) in the\n\\textsf{mosaic}\\xspace. This magnitude cut was selected after we completed the\nredshift analysis in \\textsf{udf-10}\\xspace. We discuss in detail how we decided\nto make the cut at this magnitude in Section~\\ref{subsec:mag_cut}.\n\n\n\n\\subsubsection{Emission line selected objects}\\label{subsec:analysis_origin}\n\nWe also searched for emission lines directly in the data cubes to\nidentify objects.  We adopted the software {\\tt ORIGIN} \\citep[detectiOn\nand extRactIon of Galaxy emIssion liNes;][]{prep_Mary17} to perform\nblind detections.\n\n{\\tt The ORIGIN} software uses a matched filter in three-dimensional\ndata to detect signals by correlating with a set of spectral\ntemplates.  It first uses a principal component analysis (PCA) to\nremove continuum emission.  A matched filter is then applied to the\ncontinuum removed data cube and the $P$-value test is used to assign a\ndetection probability to each emission line candidate.  When the test\ngives a high probability, a narrowband image is created using the raw\ndata cube to further test the significance of the line. The line is\nonly considered to be a real detected line when it survives the\nnarrowband image test.  Finally, the spatial position of each detected\nline is estimated by spatial deconvolution. For more details on the\nprocedures and parameters used for the detections in \\textsf{udf-10}\\xspace and the\n\\textsf{mosaic}\\xspace, we refer to Paper~I.  In \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace, $\\UDFORIGIN$\nand $\\MOSORIGIN$ emission line objects were detected with {\\tt\n  ORIGIN}, respectively.\n\nIn addition to {\\tt ORIGIN}, we also used the {\\tt MUSELET}\nsoftware\\,\\footnote{{\\tt MUSELET} is publicly available as a part of\n  the {\\tt MPDAF} software \\citep{subm_Piqu17}. More details at\n  \\url{http:\/\/mpdaf.readthedocs.io\/en\/latest\/muselet.html}} to test\nour blind search for line emitters in the Mosaic field. {\\tt The MUSELET}\nsoftware is a {\\tt SExtractor}-based detection tool, which identifies line\nemission sources in continuum-subtracted narrowband images of a\ncube. The narrowband images are produced by collapsing (weighted\n  average) the closest 5 wavelength planes (6.25${\\rm \\AA}$\\xspace) and subtracting\na median continuum from the closest 20 wavelength planes (25${\\rm \\AA}$\\xspace) on\neach of the blue and red sides of the narrowband regions. By\ncomparing the UVUDF sources and {\\tt ORIGIN} detections with the raw\n{\\tt MUSELET} catalogs of emission line sources in the \\textsf{mosaic}\\xspace, \n  {\\tt MUSELET} only detects 60\\% of the objects which {\\tt ORIGIN}\n  detects, but it finds 16 additional sources. The lower detection\n  rate and small number of additional detections of {\\tt MUSELET}\n  strengthens our choice of {\\tt ORIGIN} as the primary line\ndetection software for the analyses of these fields. All of the\nsources detected only by {\\tt MUSELET} are also included in the\nanalyses of this work and the released catalogs.\n\n\n\\subsubsection{Spectral extractions}\n\nWe adopted three different methods for spatially integrating the\ndata cube to create one-dimensional spectral extractions: unweighted\nsum, white-light weighted, and PSF weighted. The unweighted sum is a\nsimple summation of all of the flux in each wavelength slice over the\nmask region (the MUSE PSF convolved {\\it HST} segmentation map).\nThe white-light weighted spectrum is computed by giving a weight\n  based on the flux in each spatial element of the MUSE white-light\n  image, which is made by collapsing the MUSE data in the wavelength\n  direction.  For the PSF-weighted spectrum, the estimated PSF (as a\nfunction of wavelength) is used as the weight for the spectral\nextraction.  Local residual background emission is then subtracted\nfrom the extracted spectrum. The local background region is defined by\nthe area outside of the combined resampled {\\it HST} segmentation maps\nof all of the prior sources in the $5\\arcsec \\times 5\\arcsec$\nsubcube.  The global sky background emission has already\nbeen removed during the data reduction.\n\nThe weighted optimal extractions offer the advantage of reducing\ncontamination from neighboring objects that are not covered by the\nmask. Although in many cases the weighted extractions provide a higher\nsignal-to-noise ratio (S\/N) in the extracted spectrum, they can be\ndisadvantageous for objects whose emission line features are spatially\nextended or offset from the position of the white-light emission or\nthe PSF. This characteristic is often seen in Ly$\\alpha$\\xspace emitters\n\\citep[e.g.,][]{Wiso16, subm_Lecl17}.\n\nAs the default for the redshift determination process, because we\npreferentially require spectra with a good S\/N, we primarily use\nwhite-light and PSF weighted spectra, depending on the galaxy size\nfound by {\\tt SExtractor} \\citep{Bert96} in the {\\it HST} F775W image\nreported in the UVUDF catalog.  When the FWHM of an object is extended\nmore than the average seeing, 0.7\\arcsec, the primary spectrum to\ninspect is white-light weighted.  For the galaxies with FWHM\n$< 0.7$\\arcsec, the PSF-weighted spectrum is used. However, when\nnecessary, redshift investigators can use any of the extracted spectra\nor a user-defined spectrum to investigate spectral features in\ndepth. For some cases, a simple extraction in a small circular\naperture ($\\approx 0.8''$ in diameter) is useful to confirm\nobserved features.\n\n\n\n\\subsection{Redshift determination}\\label{subsec:z_meas}\n\nWe adopted a semi-automatic method for the redshift identification. For\nthe continuum selected galaxies, we used the redshift fitting software\n{\\tt MARZ} \\citep{Hint16}. The redshift determination of {\\tt MARZ} is\nbased on a modified version of the {\\tt AUTOZ} cross-correlation\nalgorithm \\citep{Bald14}.  From a user-defined input list of spectral\ntemplates, it automatically finds the best-fit template to the input\nspectrum and determines the redshift. If the resulting redshift is not\nideal, then the user can also update it interactively in the same\nwindow. We used the peak position of the most significant detected\nfeature to define redshift, including Ly$\\alpha$\\xspace.  The Ly$\\alpha$\\xspace peak\nemission may not represent the systemic redshift due to radiation\ntransfer effects of the resonant Ly$\\alpha$\\xspace line in bulk motion of neutral\nhydrogen gas \\citep{Hash13}.\n\n\nIn addition to its original capabilities, we customized {\\tt MARZ} to\nmatch our needs. We added a panel to display {\\it HST} images along\nwith the UVUDF ID numbers and MUSE white-light images.  The original\n{\\tt MARZ} already has a button for confidence level implemented\n(called the quality operator, QOP, in {\\tt MARZ}) and, in addition, we\nintegrated TYPE and DEFECT buttons (see below for the description of\nthese parameters).  Instead of only inputting a one-dimensional\nspectrum, we can feed the subcube such that it can create narrowband\nimages simultaneously when a redshift is selected or when certain\nemission\/absorption features are selected in the spectrum.  A\nscreenshot of the customized {\\tt MARZ} is presented in\nFigure~\\ref{fig:marz}.\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{figures\/marz.png}\n    \\caption{Redshift tool {\\tt MARZ} with some modifications for\n      our redshift analysis. This tool reports the most likely redshift and\n      the quality (QOP) by cross-correlating with input spectral\n      templates. The redshift identifiers verify the suggested\n      redshift or manually find a better redshift and provide its\n      quality (indicated as CONFID in the catalogs). We also implement\n      a feature to show images in the panel on the right.  }\n    \\label{fig:marz}\n  \\end{center} \n\\end{figure*}\n\nThe input spectral templates for {\\tt MARZ} are listed and explained\nin Appendix~\\ref{app:temples}. Some of the templates ($\\rm No.\\,6-19$)\nare made using the MUSE data. During the redshift identification\nprocess, no photometric redshift information is provided to the\ninspectors to avoid biases when we compare MUSE determined redshift\nagainst photometric redshift (Paper~III). Instead, we display all of\nthe existing {\\it HST} UV to near-infrared images as supplemental\ninformation to the MUSE spectra. Although it is not yet\n  implemented in our modified {\\tt MARZ}, we found that {\\it HST} color\n  images also help constrain spectroscopic redshifts and identify\n  the corresponding object that is associated with the determined\n  redshift.\n\nFor each of the determined redshifts, a confidence level (CONFID) from\n3 to 1 is assigned:\n\n\\begin{itemize}\n\\item[{\\bf 3}:] Secure redshift, determined by multiple features\n\\item[{\\bf 2}:] Secure redshift, determined by a single feature\n\\item[{\\bf 1}:] Possible redshift, determined by a single feature\n  whose spectral identification remains uncertain\n\\end{itemize}\n\nAlthough we identified emission and absorption features in the\none-dimensional spectra, we employed narrowband images of the features\nto confirm the significance of the detections. The CONFID is assigned\nonly when the identified features are seen in its narrowband\nimages. The redshifts with CONFID of 2 or 3 are reliable, but if one\nis interested in using the redshifts with CONFID of 1, it is highly\nrecommended to double check the spectrum and use it with care.\n\nWhile measuring redshifts, we kept records of which feature is\nmainly used for the identification. This parameter is saved as the\ninteger, TYPE, corresponding to\n\n\\begin{itemize}\n\\item[{\\bf 0}:] Star\n\\item[{\\bf 1}:] Nearby emission line object\n\\item[{\\bf 2}:] [O\\,\\textsc{ii}]\\xspace emitter\n\\item[{\\bf 3}:] Absorption line galaxy\n\\item[{\\bf 4}:] C\\,\\textsc{iii}]\\xspace emitter\n\\item[{\\bf 5}:] [O\\,\\textsc{iii}]\\xspace emitter\n\\item[{\\bf 6}:] Ly$\\alpha$\\xspace emitter\n\\item[{\\bf 7}:] Other types\n\\end{itemize}\n\nIn general, the most prominent feature is used to define the type.\nWe distinguished nearby galaxies (1) and [O\\,\\textsc{ii}]\\xspace emitters (2) by a\ndetection of H$\\alpha$\\xspace. Thanks to the relatively high spectral resolution of\nMUSE, it is possible to distinguish the [O\\,\\textsc{ii}]\\xspace$\\lambda\\lambda3726,3729$\nand C\\,\\textsc{iii}]\\xspace$\\lambda\\lambda1907,1909$ doublet features based on their\nintrinsic separations at a certain redshift. It also helps to resolve\nthe distinctive asymmetric profile of Ly$\\alpha$\\xspace.  We discuss this further in\n\\S\\ref{sec:result}, but the majority of redshifts are determined by\nidentifying the most prominent feature, the [O\\,\\textsc{ii}]\\xspace and Ly$\\alpha$\\xspace emission at\nlow redshift ($z < 1.5$), and high redshift ($z > 3$), respectively.\nGalaxies are only classified as [O\\,\\textsc{iii}]\\xspace emitters when [O\\,\\textsc{iii}]\\xspace is more\nprominent than [O\\,\\textsc{ii}]\\xspace in cases both are detected.\n\nWe also kept the information of the data quality with the parameter\nDEFECT. The default is $\\rm DEFECT= 0$, meaning no problem is found in\nthe data.  When $\\rm DEFECT = 1$, it indicates that there are some\nissues with the data, but the data are still usable. This flag is\nraised often owing to the object lying at the edge of the survey\narea. This may indicate that line fluxes of these sources are\nunderestimated because they are truncated in the data. As long as a\nspectral feature can be identified, we are able to determine a\nredshift, but we cannot recover the total line fluxes beyond the edge\nof the survey area.\n\nFor each continuum detected object, at least three investigators\nindependently assessed the redshift.  For \\textsf{udf-10}\\xspace, because it is our\nreference field, we had six investigators look at all of the\nobjects. For the \\textsf{mosaic}\\xspace, $\\MOSHSTpriMAGcut$ objects\n($\\rm F775W \\leq 27$~mag, see Section~\\ref{sec:analysis_cont}) are\ndivided equally for two groups with three investigators each. In the\ncase of the emission line detected objects, two investigators\nindividually determined redshift for all of these objects for both\n\\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace.  For both continuum and emission line detected\nobjects, among each of the groups, any disagreements in their\ndeterminations of redshift, CONFID, TYPE, and DEFECT were consolidated\nto provide a single set of redshift results. With the consolidated\nresults, we had at least one more person, who is not in the group,\ncheck all of the determined redshifts and related parameters to\nfinalize the redshift assessment.\n\nThen we refined the redshift by simultaneously fitting all of the\ndetected emission lines and measure line fluxes using a spectral\nfitting software. This process is discussed in the next subsection.\n\n\n\\subsection{Line flux measurements}\\label{subsec:line_flux}\n\nFor the redshift determination, the spectra utilized were primarily\nextracted based on either the white-light or PSF weighted extraction\nin order to optimize S\/N. For the purpose of cataloging line fluxes,\nwe alternatively used the unweighted summed spectra for all of the\nsources within each segmentation map\\,\\footnote{Line fluxes measured\n  in the segment may be less than the total flux because of aperture\n  effects.}.  This avoids possible biases introduced by the weighting\nscheme. It is also better when the spatial distribution of the line\nemission is more extended than the white-light emission or PSF. This\nis especially relevant for Ly$\\alpha$\\xspace emission \\citep[cf.][]{Wiso16,\n  subm_Lecl17}.  However, this results in an increased\nscatter for faint emission lines.\n\nFor the detected emission lines with $\\rm S\/N > 3$, the line flux and\nits uncertainty are provided along with the measured redshift in the\nreleased catalog (Appendix~\\ref{app:cat}). We used the spectral line\nfitting software {\\tt PLATEFIT} \\citep{Trem04, Brin08b} to\nsimultaneously fit all of the emission lines falling within the\nspectral range.  This tool first uses a set of model template spectra from\n\\cite{BC03} with stellar spectra from MILES \\citep{Sanc06} to fit the\ncontinuum of the observed spectrum at a predefined redshift with\nstrong emission lines masked.  After subtracting the fitted continuum\nfrom the observed spectrum, a single Gaussian profile is fitted to\neach expected emission line in velocity space.  The velocity offset\n(the velocity shift relative to the redshift of the galaxy) and\nvelocity dispersion of all of the emission lines are tied to be the\nsame.  See \\cite{Trem04} for a more detailed description of the\nsoftware.\n\nThe velocity offset is limited to vary within\n$\\pm 300 \\, {\\rm k m\\,s^{-1}}$. For emission line galaxies, we used the\nvelocity offset output from {\\tt PLATEFIT} to refine the original\n(input) redshift. The calculation of the velocity offset\nincludes Ly$\\alpha$\\xspace when it is detected. The typical velocity offset is\nabout $10$~km\\,s$^{-1}$\\xspace on average. When the original redshift is obtained\ndirectly from {\\tt MARZ}'s cross-correlation, we do not expect the\noffset to be significant, but it can be larger when the redshift\nidentifiers manually estimate the redshift.\n\nAs described above, {\\tt PLATEFIT} fits single Gaussian functions to\nthe emission lines. However, this is not suitable for Ly$\\alpha$\\xspace because it\nis often observed to have an asymmetric profile.\nThus, we added a feature to the software to fit the complicated profile\nof Ly$\\alpha$\\xspace. Instead of fitting Ly$\\alpha$\\xspace with a single Gaussian function, we\nused a combination of multiple Gaussian functions (up to 21, although\ntypically much fewer are used) at fixed relative positions to fit the\nLy$\\alpha$\\xspace profile.  We placed a central component Gaussian function at the\nposition of Ly$\\alpha$\\xspace as determined by the single Gaussian fitting carried out in\nthe previous step. We placed an additional 10 components on either side\nof this, at a constant spacing of $120\\,{\\rm km\\,s^{-1}}$ to be\nconsistent with the spectral resolution of the data. As such the\nfitting covers a velocity range of $\\pm 1200\\,{\\rm km\\,s^{-1}}$ from\nthe expected position of the line.\n\nWe then performed the minimization via a nonlinear least squares\nmethod.  During this process, the fluxes of each Gaussian component\nare allowed to vary independently (with most falling to 0 as they lie\nin the spectrum where Ly$\\alpha$\\xspace has no emission) until the aggregate of all\ncomponents best matches the observed spectrum.  The separation between\neach component is kept constant (in velocity) but the wavelength of\nthe central component is allowed to shift\n($\\pm 300 \\, {\\rm km\\,s^{-1}}$, consistent with fits for other lines).\nThis allows for the fitting to correct for small discrepancies in the\ncentral position determined during the multiple Gaussian component\nfitting so that it produces a more accurate fit to Ly$\\alpha$\\xspace.  The widths\nof individual Gaussian components are forced to be the same but the\nwidth itself is a free parameter. The  {\\tt PLATEFIT} tool places a limit of\n$< 500 \\, {\\rm km\\,s^{-1}}$ for each line width, but in general the\nGaussian components only vary slightly from the initial guess of\n$70 \\, {\\rm km\\,s^{-1}}$ and fall between $60$ and\n$120 \\, {\\rm km\\,s^{-1}}$.  Allowing the widths of the components to\nvary in this way produces a better fit to the data in a few cases,\ncompared to running the fitting with a single fixed width for all of\nthe components. However, the improvement is much less significant than\nthat obtained by allowing the central velocity of the components to\nshift.\n\nAfter the fitting, we studied the output spectrum of the complex\nfitted regions and identified individual complex ``lines'' (a\ncomposite of multiple Gaussian components) by searching the fit\nspectrum for local maxima. The routine then scans out from each maxima\non either side of the peak until it identifies a local minimum or the\nfitted spectrum returns to 0.  The flux of each complex line is then\ncalculated along with its associated error and fitted ``lines'' with a\nS\/N of less than 3 are dropped. This S\/N cut successfully avoids\noverfitting the noise and removes unphysical fits (such as more than\ntwo components or very broad components) in most cases.  The\nproperties of the remaining lines are then calculated directly from\nthe fitted spectrum using a model independent approach. Errors are\ndetermined using a Monte Carlo method by modifying the input spectrum\nwith random values within the associated errors at each point 100\ntimes. Each line property is then re-extracted for each realization\nand the standard deviation of these values is given as the error.  An\nexample of the complex fitting is presented in\nFigure~\\ref{fig:lya_complex_fit}.\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth\/2, trim=0 0 0 220]{figures\/lya_complex_fit.pdf}\n    \\caption{Demonstration of the complex fitting for a Ly$\\alpha$\\xspace\n      emission line presenting blue and red bumps. The fitting is performed\n      on a simulated spectrum (black line) to better illustrate\n      the procedure (the flux is in arbitrarily units). In the real\n      data, the final fit is usually comprised of fewer\n      components. The dashed red lines are the individual Gaussian\n      components and the solid blue line is the total of these\n      components (the final fit). The vertical green dashed\n      lines indicate the local minima. We\n        use the Ly$\\alpha$\\xspace peak position to measure the redshift (see\n        \\S~\\ref{subsec:z_meas}).}\n    \\label{fig:lya_complex_fit}\n  \\end{center} \n\\end{figure}\n\n\n\\subsection{Continuum flux measurements}\\label{subsec:NoiseChisel}\n\nThroughout the paper, we used the continuum fluxes from the {\\it HST}\nobservations provided in the UVUDF catalog.  In addition to that, we\nmeasured our own continuum fluxes for the objects that are not in the\nUVUDF catalog but are identified by directly detecting their emission\nline features with {\\tt ORIGIN} and {\\tt MUSELET} (see\n\\S\\ref{subsec:analysis_origin}). For these objects, we performed our own\nphotometric measurements on the {\\it HST} images to obtain\nsegmentation maps and continuum fluxes or upper limits. A full\nanalysis of why these objects are not in the UVUDF catalog and our\nmethod of extracting their broadband properties are described in\nPaper~I. Here we provide a brief summary.\n\nAmong $\\numnotinraf$ objects that are only found by {\\tt\n  ORIGIN\/MUSELET}, roughly one-quarter have very low or no continuum\nemission, and thus they are hardly detectable in any broadband\nimages.  The rest of the objects can be visually identified in the\nimages, but they are not detected by {\\tt SExtractor}\\,\\footnote{In\n  Section~7.3 of Paper~I, we discuss why they were not detected by\n  {\\tt SExtractor}.} \\citep{Bert96}.  In order to measure continuum\nfluxes (or upper limits) of these objects, we used {\\tt\n  NoiseChisel}\\,\\footnote{\\url{http:\/\/www.gnu.org\/software\/gnuastro\/manual\/html_node\/NoiseChisel.html}},\nwhich is an image analysis software employing a noise-based detection\nconcept \\citep{Akhl15}. It is run independently on each of the {\\it\n  HST} images and the largest (from all the filters) object closer\nthan $0.30\\arcsec$ to the position reported by {\\tt\n  ORIGIN\/MUSELET} is taken as representing the pixels associated with\nit to create a segmentation map for each object.  We are able to\nassociate a {\\tt NoiseChisel} segmentation map for $\\rafprobinnc\\%$ of\nthe objects only found by {\\tt ORIGIN\/MUSELET}. However, for the rest\nof the objects, no {\\tt NoiseChisel} segmentation map (in any of the\n{\\it HST} broadband images) can be found within\n$0.30\\arcsec$.  For these objects, a\n$0.50\\arcsec$ diameter circular aperture was placed\non the position reported by {\\tt ORIGIN\/MUSELET}.\n\nThe broadband magnitude is found by feeding the image of each filter\nand the final segmentation map into {\\tt MakeCatalog}, which takes\noutput of {\\tt NoiseChisel} as input to directly create a catalog.\nThe error in magnitude for each segment is derived from the S\/N\nrelation: $\\sigma_{M}=2.5\/(S\/N \\times \\ln{(10)})$ \\citep[see Eq. (3)\nin][]{Akhl15}.  {\\tt MakeCatalog} produces upper limit magnitudes for\neach object, by randomly positioning its segment in $200$\ndifferent blank positions over each broadband image and using \n  $\\mkcatupnsigma\\sigma$ of the standard deviation of the\n  resulting distribution. If the derived magnitude for an object in a\nfilter is fainter than this upper limit, the upper limit magnitude is\nused instead.\n\n\n\n\n\\section{Results}\\label{sec:result}\n\n\\subsection{Redshift determination in the MUSE Ultra Deep Field (\\textsf{udf-10}\\xspace)}\n\nIn this subsection, first we report the measured redshift and the\nparameters associated with it in our reference field, the MUSE Ultra\nDeep Field (\\textsf{udf-10}\\xspace). In addition to being the deepest spectroscopic\nfield so far observed with MUSE, all of the extracted MUSE spectra in\nthis field have been visually inspected. In the \\textsf{udf-10}\\xspace field, as\ndiscussed in \\S\\ref{sec:analysis}, we used two different procedures to\nextract the sources. Their redshifts were determined independently\nfirst, and then reconciled. Here we first present the basic\nproperties of the objects associated with their redshift measurements\nfor each extraction method independently. Based on our understanding\nof the differences and the relationship between the two source\nextraction methods and the redshift properties, we tried to find the\nbest combination of these two methods to efficiently maximize the\ndetection rate of redshifts in the even larger sample size of the MUSE\nDeep Field (the \\textsf{mosaic}\\xspace).\n\n\n\\subsubsection{Redshifts of the continuum selected objects}\\label{sec:result_cont}\n\nAmong $\\UDFHSTpri$ continuum selected objects in the MUSE Ultra Deep\nField (\\textsf{udf-10}\\xspace), we successfully measured the redshifts of $\\UDFzHSTCone$\nobjects in the redshift range from $z=0.21$ to $6.64$. The number of\nredshifts with confidence level $\\geq 2$ is $\\UDFzHSTCtwo$.\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{figures\/udf10_c042_e031_withz_iter6_muse-z_prop_HSTpri_ORIG_fade.pdf}\n    \\caption{MUSE redshift distributions of the objects extracted\n      with the {\\it HST} continuum-detected priors (filled bars) and\n      the direct line emission searches ({\\tt ORIGIN}; filled and\n      faded color bars on the right in each bin) in the MUSE Ultra\n      Deep Field (\\textsf{udf-10}\\xspace). All of the extracted sources are shown\n      without removing overlaps between the extraction methods. {\\bf\n        [Left]} The redshift histogram in bins of $\\Delta z = 0.35$.\n      The red, blue, and gray represent the confidence levels\n      3, 2, and 1, respectively, for the determined redshifts. The\n      filled dots show the percentage difference relative to the total\n      of {\\it HST} prior and {\\tt ORIGIN} MUSE-$z$ distributions.\n      {\\bf [Middle]} The same redshift distribution as the left\n      panel, but color coded by the classified type of the\n      objects. {\\bf [Right]} The histogram of the classified type of\n      the objects color coded by the redshift confidence levels. }\n    \\label{fig:u10_musez_hstpri_org}\n  \\end{center} \n\\end{figure*}\n\nIn the left two panels of Figure~\\ref{fig:u10_musez_hstpri_org}, we\nshow the distributions of redshifts found with the MUSE data.  \nBecause most of the objects at $0 < z < 1.5$ are usually identified by\nthe [O\\,\\textsc{ii}]\\xspace doublet, their CONFID is mostly 3 by definition when the\ndouble peaks are resolved and clearly seen.  There are six cases in which\nCONFID is 2 in this redshift range, whose spectra in general have\nlower S\/Ns but the features are obviously detected in narrowband\nimages.\n\nOn the other hand, galaxies at $z > 3$ are often found by a single\nfeature Ly$\\alpha$\\xspace, which gives CONFID of 2. For the case of $\\rm CONFID=3$\nat $z > 3$, the C\\,\\textsc{iii}]\\xspace emission or UV absorption features, and in some\nrare cases He\\,\\textsc{ii}, are also discerned. The number of\ndetermined redshifts drops significantly at\n$1.5 \\lesssim z \\lesssim 3.0$. This is a well-known ``redshift\ndesert'', where [O\\,\\textsc{ii}]\\xspace shifts out from the red end of the spectral\ncoverage, although Ly$\\alpha$\\xspace is still too blue to be detected. With the\ndeep data, we are able to recover some of the redshifts in this range\nby detecting C\\,\\textsc{iii}]\\xspace emission and some absorption features (e.g.,\nFe\\,\\textsc{ii}).\n\nThere are also 59 redshifts with $\\rm CONFID=1$.  Their redshifts are\ndifficult to determine with confidence, because in most cases, it is\nambiguous to use their line profile to specify the feature (e.g., [O\\,\\textsc{ii}]\\xspace\nversus Ly$\\alpha$\\xspace). For [O\\,\\textsc{ii}]\\xspace emitter at $0.25 < z < 0.85$, H$\\beta$\\xspace and [O\\,\\textsc{iii}]\\xspace\n  are in general also available, but they lie in a sky line crowded\n  region, which can sometimes prevent further constraint on the\n  measured redshift to give a higher CONFID.\n\n\n\\subsubsection{Redshifts of the emission line selected objects}\n\nOf the $\\UDFORIGIN$ emission line selected objects (objects detected\nwith {\\tt ORIGIN} regardless of whether they have UVUDF counterparts),\n$\\UDFzORGCtwo$ objects have secure redshift measurements with CONFID\nof 2 or 3 in the redshift range of $0.28-6.64$. When we include the\n$\\rm CONFID=1$ redshifts, the resulting number of measured redshifts\nis $\\UDFzORGCone$.  Since we have to actually detect the emission\nfeatures to select this sample, it is expected that the number\nfraction of $\\rm CONFID=1$ redshifts is much smaller for the emission\nline selected objects compared with the continuum selected objects.  A\ncrucial difference between the continuum and line emission selected\nobjects is that the former are extracted regardless of whether they\nare detected with MUSE or not, but the latter requires actual line\ndetections.\n\nThe redshift distributions of the emission line objects are shown as\nthe faded color bars (on the right in each bin) in\nFigure~\\ref{fig:u10_musez_hstpri_org}. Similar to the continuum\nselected objects, the majority of the redshifts at $z < 3$ have\n$\\rm CONFID=3$, while mostly $\\rm CONFID=2$ at $z > 3$, because of the\ndetectable spectral features. The main features that {\\tt ORIGIN}\ndetects to facilitate the redshift identifications are [O\\,\\textsc{ii}]\\xspace, Ly$\\alpha$\\xspace, and\nC\\,\\textsc{iii}]\\xspace. The lack of redshifts in this range is more significant for the\nemission line selected objects because by nature {\\tt ORIGIN} does not\nhave the ability to find any absorption galaxies.\n\n\n\n\\subsubsection{Redshift comparisons between the continuum and emission\n  line selected objects}\\label{sec:HSTpri_vs_ORIGIN}\n\nWhen we only consider the secure redshifts ($\\rm CONFID \\geq 2$), the\nderived redshifts of both the continuum and emission line selected\ngalaxies cover a similar range, from $z=0.2$ to $6.7$. A large\ndifference is the numbers of identified redshifts between these\ntwo data sets. The continuum selected galaxies have $\\UDFzHSTCtwo$\nsecure redshifts, while the emission line selected galaxies have\n$\\UDFzORGCtwo$. If we consider that of these $\\UDFzORGCtwo$ objects,\n$\\UDFzORGonlyCtwo$ do not have a continuum detection (not in the prior\nlist), the difference is even larger.\n\nIn the left panel of Figure~\\ref{fig:u10_musez_hstpri_org}, the\npercentage difference of the numbers of determined MUSE redshifts\nbetween the two extraction methods ({\\it HST} prior or {\\tt ORIGIN})\nis shown with the filled dots.  {\\tt The ORIGIN} method is optimized to detect\ncompact emission line objects with faint continuum, such that it is more\nsensitive to the detection of high-$z$ emission lines. This method misses\n$44\\%$ of the [O\\,\\textsc{ii}]\\xspace emitters identified in the continuum selected\ngalaxies. For the Ly$\\alpha$\\xspace emitters, $38\\%$ of the continuum detected\nsample are not found by {\\tt ORIGIN}.  There is no obvious trend of\nthe fraction of missed [O\\,\\textsc{ii}]\\xspace or Ly$\\alpha$\\xspace with redshift\n(Figure~\\ref{fig:u10_Fline_hst_org}). There is no trend found with\nline surface brightness either.\n\nAt $1.5 < z < 3$, the number of confirmed redshifts is smaller for\nthe emission line selected galaxies because in this redshift range we\nmostly rely on galaxies with absorption features to find their\nredshifts in addition to C\\,\\textsc{iii}]\\xspace.  {\\tt The ORIGIN} method is not designed to\ndetect absorption features, and thus it is not able to detect any of\nthese objects. While the other feature, C\\,\\textsc{iii}]\\xspace, is in emission, it is\nfairly weak, which makes it harder for {\\tt ORIGIN} to detect.  As\nshown in Figure~\\ref{fig:u10_Fline_hst_org}, line fluxes of the C\\,\\textsc{iii}]\\xspace\nemission not detected by {\\tt ORIGIN} are at the low end of the line\nfluxes.  Although it is possible to tune {\\tt ORIGIN} to detect\nfainter emission features in the cube, it also dramatically increases\nthe number of spurious detections.\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6_LineFlux_with_err_vs_muse-z.pdf}}\n  \\caption{ Line fluxes of [O\\,\\textsc{ii}]\\xspace$\\lambda\\lambda3726,3729$ (the red\n    circles), C\\,\\textsc{iii}]\\xspace$\\lambda\\lambda1907,1909$ (the blue squares), and\n    Ly$\\alpha$\\xspace (the green diamonds) of the redshift-identified galaxies in\n    \\textsf{udf-10}\\xspace ~(CONFID $\\geq 2$). The open and filled symbols are\n    the {\\it HST} continuum and emission line selected objects,\n    respectively. The filled symbols with the black edge indicate that\n    they are detected only by {\\tt ORIGIN}. }\n  \\label{fig:u10_Fline_hst_org}\n\\end{figure}\n\n\n\\subsubsection{Final redshifts for the MUSE Ultra Deep Field (\\textsf{udf-10}\\xspace)}\\label{sec:udf10_finalz}\n\nWe combined and compared the results from the independent measurements\nof the continuum and emission line selected objects to obtain the\nfinal redshifts. All of the determined redshifts are provided in the\ncatalog (Appendix~\\ref{app:cat}). We show some example spectra and\nnarrowband images of [O\\,\\textsc{ii}]\\xspace, C\\,\\textsc{iii}]\\xspace, and Ly$\\alpha$\\xspace emitters, and an\nabsorption line galaxy in Figures~\\ref{fig:eml_spec} and\n\\ref{fig:abs_spec}.\n\nWe successfully measured $\\UDFzALLCone$ and $\\UDFzALLCtwo$ redshifts\nwith $\\rm CONFID \\geq 1$ and $\\geq 2$, respectively, for the {\\it\n  unique} objects selected with the continuum or line emission\n(i.e., overlapping objects are removed).  The final MUSE redshift\ndistribution in \\textsf{udf-10}\\xspace is shown in Figure~\\ref{fig:u10_musez} and\nsummarized in Tables~\\ref{tbl:num_z} and \\ref{tbl:count}. The shape of\nthe histogram does not change compared with the individual redshift\nassessments (Figure~\\ref{fig:u10_musez_hstpri_org}): two peaks at\n$z \\approx 1$ and $z \\approx 3$. The number of $\\rm CONFID=2$\nredshifts at $z \\leq 1.5$ is $69$, at $1.5 < z \\leq 3$ (redshift\ndesert) it is $29$, and at $3 < z \\leq 6.7$ it is $155$.\n\n\\begin{table*}[htp]\n  \\caption{Counts of determined redshifts for different source\n    extractions in the MUSE Ultra Deep (\\textsf{udf-10}\\xspace), Deep\n    field (\\textsf{mosaic}\\xspace), and the unique objects of \\textsf{udf-10}\\xspace $+$ the \\textsf{mosaic}\\xspace }\n\\begin{center}\n\\begin{tabular}{cccccccccc}\n\\hline \\hline\n                & \\multicolumn{3}{c}{\\textsf{udf-10}\\xspace ($1\\arcmin \\times 1\\arcmin$)}\n                & \\multicolumn{3}{c}{\\textsf{mosaic}\\xspace ($3\\arcmin \\times 3\\arcmin$)}\n                & \\multicolumn{3}{c}{combined~~\\tablefootmark{a}} \\\\\nConfidence level (CONFID)    & $\\geq 3$ & $\\geq 2$ & $\\geq 1$ \n                             & $\\geq 3$ & $\\geq 2$ & $\\geq 1$\n                             & $\\geq 3$ & $\\geq 2$ & $\\geq 1$ \\\\\n\\hline\n{\\it HST} continuum selected~~\\tablefootmark{b} \n                             & \\multirow{2}{*}{$\\UDFzHSTCthree$}  & \\multirow{2}{*}{$\\UDFzHSTCtwo$}  & \\multirow{2}{*}{$\\UDFzHSTCone$}\n                             & $\\MOSzHSTCthree$                   & $\\MOSzHSTCtwo$                   & $\\MOSzHSTCone$\n                             & \\multirow{2}{*}{$\\COMBzHSTCthree$} & \\multirow{2}{*}{$\\COMBzHSTCtwo$} & \\multirow{2}{*}{$\\COMBzHSTCone$} \\\\\nWith UVUDF counterparts~~\\tablefootmark{b}\n                             &                       &                     &           \n                             & $\\MOSzHSTallCthree$   & $\\MOSzHSTallCtwo$   & $\\MOSzHSTallCone$\n                             &                       &                     &                 \\\\\nEmission line selected       & $\\UDFzORGCthree$      & $\\UDFzORGCtwo$      & $\\UDFzORGCone$\n                             & $\\MOSzORGCthree$      & $\\MOSzORGCtwo$      & $\\MOSzORGCone$\n                             & $\\COMBzORGCthree$     & $\\COMBzORGCtwo$     & $\\COMBzORGCone$ \\\\\nEmission line only~~\\tablefootmark{c} & $\\UDFzORGonlyCthree$  & $\\UDFzORGonlyCtwo$  & $\\UDFzORGonlyCone$\n                             & $\\MOSzORGonlyCthree$  & $\\MOSzORGonlyCtwo$  & $\\MOSzORGonlyCone$\n                             & $\\COMBzORGonlyCthree$ & $\\COMBzORGonlyCtwo$ & $\\COMBzORGonlyCone$ \\\\\n\\hline\nTotal unique objects         & $\\UDFzALLCthree$  & $\\UDFzALLCtwo$  & $\\UDFzALLCone$\n                             & $\\MOSzALLCthree$  & $\\MOSzALLCtwo$  & $\\MOSzALLCone$\n                             & $\\COMBzALLCthree$ & $\\COMBzALLCtwo$ & $\\COMBzALLCone$ \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\tablefoot{ \n\\tablefoottext{a}{Number of unique objects in \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace.}\n\\tablefoottext{b}{For the \\textsf{mosaic}\\xspace, the {\\it HST} continuum selected\n  galaxies to inspect are limited to $\\rm F775W \\leq 27$~mag (\\S\\ref{subsec:mag_cut}). The rest\n  of the galaxies are selected by direct detection of emission lines\n  in the data cube. Among the emission line selected galaxies, some galaxies have counterparts\n  in the UVUDF catalog with $\\rm F775W > 27$~mag. They are listed in the row labeled ``With UVUDF\n  counterparts'' in the table.}\n\\tablefoottext{c}{The objects selected by emission lines that have no\n  counterpart in the UVUDF catalog \\citep{Rafe15}.} }\n\\label{tbl:num_z}\n\\end{table*}%\n\n\n\\begin{table*}[htp]\n\\caption{Census of the objects in the MUSE Ultra Deep (\\textsf{udf-10}\\xspace) and Deep\n  field (the \\textsf{mosaic}\\xspace) sorted by categories}\n\\begin{center}\n\\begin{tabular}{lrccrccr}\n\\hline \\hline\n                & \\multicolumn{3}{c}{\\textsf{udf-10}\\xspace ($1\\arcmin \\times 1\\arcmin$)} \n                & \\multicolumn{3}{c}{\\textsf{mosaic}\\xspace ($3\\arcmin \\times 3\\arcmin$)}\n                & combined \\\\\nCategory\/Type   & Counts~~\\tablefootmark{a} & Redshift~~\\tablefootmark{b} & F775W mag~~\\tablefootmark{b} \n                & Counts~~\\tablefootmark{a} & Redshift~~\\tablefootmark{b} & F775W mag~~\\tablefootmark{b}\n                & Counts~~\\tablefootmark{a} \\\\\n\\hline\n0. Stars                    &   0   (0) &      ...      &      ...       \n                            &   9   (1) &      ...      & $19.0 - 24.8$\n                            &   9   (1) \\\\ \n1. Nearby galaxies          &   5   (1) & $0.21 - 0.31$ & $22.6 - 30.0$  \n                            &  47   (2) & $0.10 - 0.42$ & $18.6 - 27.1$\n                            &  47   (3) \\\\ \n2. [O\\,\\textsc{ii}]\\xspace emitters            &  61  (27) & $0.33 - 1.45$ & $20.4 - 28.8$  \n                            & 465  (49) & $0.28 - 1.49$ & $19.4 - 28.3$\n                            & 473  (73) \\\\ \n3. Absorption line galaxies &  12   (4) & $0.95 - 3.00$ & $21.9 - 26.1$  \n                            &  57  (22) & $0.60 - 2.95$ & $21.0 - 26.2$\n                            &  63  (23) \\\\ \n4. C\\,\\textsc{iii}]\\xspace emitters           &  16   (2) & $1.55 - 2.54$ & $23.8 - 29.8$  \n                            &  41  (18) & $1.55 - 2.86$ & $23.4 - 27.0$\n                            &  50  (18) \\\\ \n5. [O\\,\\textsc{iii}]\\xspace emitters           &   1   (0) & $0.71$        &    $27.3$  \n                            &   2   (1) & $0.42 - 0.71$ & $27.0 - 27.3$\n                            &   2   (1) \\\\ \n6. Ly$\\alpha$\\xspace  emitters           & 158  (26) & $2.94 - 6.64$ & $25.5 - 31.1+$~~\\tablefootmark{c} \n                            & 624  (97) & $2.91 - 6.63$ & $24.4 - 31.2+$~~\\tablefootmark{c}\n                            & 692 (115) \\\\ \n7. Others                   &    0  (1) &      ...      &      ...       \n                            &    2  (2) & $1.22 - 3.19$ & $21.0 - 24.6$\n                            &    2  (2) \\\\ \n\\hline\n\\end{tabular}\n\\end{center}\n\\tablefoot{ \n\\tablefoottext{a}{Counts of $\\rm CONFID \\geq 2$ redshifts. The\n  numbers in parentheses indicate redshifts with $\\rm CONFID = 1$.}\n\\tablefoottext{b}{The ranges are for the secure redshifts ($\\rm CONFID \\geq 2$).}\n\\tablefoottext{c}{For some Ly$\\alpha$\\xspace emitters, the F775W continuum emission\n  is not detected ($> 31.2$~mag).} }\n\\label{tbl:count}\n\\end{table*}%\n\n\nIn Figure~\\ref{fig:u10_musez}, it is also immediately noticeable that\nbeyond $z = 3$, the fraction of confirmed redshifts of the objects\ndetected {\\it only} by {\\tt ORIGIN} (the faded color regions)\nincreases.  By $z \\approx 6$, it reaches $\\approx 50\\%$.  According to\nFigure~\\ref{fig:u10_Fline_hst_org}, the {\\tt ORIGIN} detections do not\nseem to favor any specific redshift ranges or line fluxes (except of\ncourse when the line flux is below its detection limit).  Two [OII]\nemitters ($\\rm CONFID \\geq 2$) were only detected by {\\tt ORIGIN}:\nMUSE ID 6314 and 6315.  The disturbed morphology of ID 6314 in the\n{\\it HST} imaging likely accounts for the no detection with {\\tt\n  SExtractor}. The other object, ID 6315, is blended with a nearby\nbright source and are difficult to separate using solely imaging data.\nFor the 28 Ly$\\alpha$\\xspace emitters identified only by {\\tt ORIGIN}, most of\nthese sources have surface brightness continuum emission that is too\nlow to be detected with {\\tt SExtractor} (we measured $\\approx 30$~mag\nwith {\\tt NoiseChisel}) or are not visible by eye\n(Figure~\\ref{fig:ORIonly_Lya_spec}), but there are a small number of\ncases that are missed in the UVUDF catalog because of blending or\nbecause they are lying close to nearby bright sources (ID 6313) and\ndisturbed\/complex morphology in the {\\it HST} images (ID 6324).  We\nemphasize that these 28 objects represent $\\sim 20\\%$ of Ly$\\alpha$\\xspace emitters\nfound in this work and they are in a small $1\\arcmin \\times 1\\arcmin$\narea of the entire sky where the deepest {\\it HST} data exist.\n\n\n\\subsection{Redshift determination in the MUSE Deep Field (the \\textsf{mosaic}\\xspace)}\n\n\\subsubsection{Strategy based on the results in \\textsf{udf-10}\\xspace}\\label{subsec:mag_cut}\n\nBased on the redshift analysis in our deepest survey field \\textsf{udf-10}\\xspace, we\ntried to find the best combination of the two extraction methods to\nmaximize the efficiency of redshift identifications. As shown in\nFigure~\\ref{fig:u10_musez_mag}, for the objects with redshift\ndetermined, 85 out of 98 objects at $z < 3$ ($\\rm CONFID \\geq 2$),\nmostly [O\\,\\textsc{ii}]\\xspace emitters, have $\\rm F775W \\leq 27$~mag (see also\nFigure~\\ref{fig:u10_accum_type}). While the majority at $z > 3$ are\nfainter than 27~mag, $106$ out of $155$ objects ($70\\%$) in this\nredshift range are detected by line emission directly in the data cube\nwith {\\tt ORIGIN}. In addition, all of the absorption galaxies, which\n{\\tt ORIGIN} cannot detect, are brighter than $26.1$~mag.\n\nWe simulated expected [O\\,\\textsc{ii}]\\xspace line fluxes to investigate the sudden\nreduction of [O\\,\\textsc{ii}]\\xspace emitters fainter than 27~mag visible in\nFigure~\\ref{fig:u10_accum_type}. With the spectral energy distribution\n(SED) fitting code {\\tt FAST}\\,\\footnote{Fitting and Assessment of\n  Synthetic Templates. We assume the dust extinction curve of\n  $\\tau \\approx \\lambda^{-1.3}$ from \\cite{Char00} together with the\n  median value of the measured Balmer decrements of the MUSE sources\n  that have two or more Balmer lines detected.}  \\citep{Krie09}, we\nused all of the galaxies in the \\textsf{mosaic}\\xspace with photo-$z$ (BPZ) of\n$0 - 1.5$ provided in the UVUDF catalog to predict their [O\\,\\textsc{ii}]\\xspace fluxes.\nWith the simple assumption that all of the [O\\,\\textsc{ii}]\\xspace emission originates\nfrom star formation, we can directly translate the star formation\nrates obtained from SED fits of the {\\it HST} photometry into expected\n[O\\,\\textsc{ii}]\\xspace fluxes.  Then we correct the [O\\,\\textsc{ii}]\\xspace fluxes for dust extinction\nusing the median dust correction factor,\nwhich is derived for the galaxies that are detected with MUSE using the\nBalmer decrement.  The modeled [O\\,\\textsc{ii}]\\xspace fluxes decrease with the F775W\nmagnitude.  From the $3\\sigma$ line flux detection limit of the\n\\textsf{mosaic}\\xspace, $\\approx 3 \\times 10^{-19} \\, {\\rm erg\\,s^{-1}\\,cm^{-2}}$,\nthe fraction of detectable [O\\,\\textsc{ii}]\\xspace emitters drastically decreases at\n$\\rm F775W \\gtrsim 27$~mag. At 27~mag, we expect $\\approx 80\\%$ of the [O\\,\\textsc{ii}]\\xspace\nemitters to be detected, but it abruptly decreases to\n$\\approx 20-50\\%$ between $27-28$~mag.  These numbers are\nlikely to be upper limits because, for simplicity of the model, all\ngalaxies are assumed to be star forming.\n\nThus, we make a cut at $\\rm F775W \\leq 27$~mag to limit the number of\nthe continuum selected objects based on the {\\it HST} priors for which\nwe need to perform the visual inspection on redshift\ndetermination. This reduces the total number of $\\MOSHSTpri$ continuum\nselected objects to $\\MOSHSTpriMAGcut$ to be inspected.  For the\nemission line selected objects, we do not apply any preselection and\nexamine all of the $\\MOSORIGIN$ spectra of the emission line detected\nobjects.\n\nIn the near future, we plan to extend the redshift determination\ntoward galaxies with $\\rm F775W > 27$~mag. At present, we only\nrelease the redshift measured in the $\\rm F775W \\leq 27$~mag continuum\nselected sample and all of the emission line selected sample as the\nfirst version of the MUSE UDF redshift catalog\n(Appendix~\\ref{app:cat}).\n\n\n\\subsubsection{Redshifts in the \\textsf{mosaic}\\xspace}\n\nThe process of the redshift evaluation in the \\textsf{mosaic}\\xspace follows the same\nprocedure as \\textsf{udf-10}\\xspace. We inspected the continuum detected and emission\nline detected objects individually, and then reconciled these two objects to obtain\nthe final redshifts.\n\nThe summary and distributions of the determined redshifts are shown in\nTables~\\ref{tbl:num_z} and \\ref{tbl:count} and\nFigure~\\ref{fig:mos_musez}. Out of the $\\MOSHSTpriMAGcut$ continuum\nselected objects ($\\rm F775W \\leq 27$~mag), redshifts of\n$\\MOSzHSTCone$ and $\\MOSzHSTCtwo$ objects are determined with\nconfidence levels of $\\geq 1$ and $\\geq 2$, respectively. When we\ninclude the emission line selected objects, these numbers increase to\n$\\MOSzALLCone$ and $\\MOSzALLCtwo$. Among these, $\\MOSzHSTallCone$ and\n$\\MOSzHSTallCtwo$ objects, respectively, have detectable UVUDF\ncounterparts.\n\nThe fractions of the identified redshifts below and above $z = 3$ are\ndifferent compared with those in \\textsf{udf-10}\\xspace. Out of $\\MOSzALLCtwo$\nredshifts with $\\rm CONFID \\geq 2$ in the \\textsf{mosaic}\\xspace, $52\\%$ ($650$) are\nat $z \\leq 3$ and $48\\%$ ($597$) are at $3 < z \\leq 6.7$.  Whereas, in\n\\textsf{udf-10}\\xspace, there are $39\\%$ ($98$) and $61\\%$ ($155$), respectively, out\nof $\\UDFzALLCtwo$ redshifts with $\\rm CONFID \\geq 2$. A larger\npercentage of Ly$\\alpha$\\xspace emitters are detected in the deeper data of \\textsf{udf-10}\\xspace\nthan in the \\textsf{mosaic}\\xspace, whose integration times are about three times\ndifferent.  This trend stays the same, even when we only account for\nthe continuum detected objects (i.e., excluding the objects detected\n{\\it only} by {\\tt ORIGIN} and {\\tt MUSELET}): $56\\%$ ($631$) at\n$z \\leq 3$ and $44\\%$ ($502$) at $z > 3$ for the \\textsf{mosaic}\\xspace, and $43\\%$\n($96$) and $57\\%$ ($127$), respectively, for \\textsf{udf-10}\\xspace.  In addition, a\nsmaller fraction of redshifts ($10\\%$) is found in the ``redshift\ndesert'' at $1.5 < z \\leq 3.0$ compared with \\textsf{udf-10}\\xspace ($13\\%$). These\ndifferences between the \\textsf{mosaic}\\xspace and \\textsf{udf-10}\\xspace, due to the depth of the\ndata, are also implied by the higher fraction of $\\rm CONFID=2$\nredshifts at $z < 3$ in the \\textsf{mosaic}\\xspace.  Interestingly, the deeper data\nof the \\textsf{udf-10}\\xspace increase the number fraction of $\\rm CONFID=1$ redshifts\nin the [O\\,\\textsc{ii}]\\xspace emitters compared to the \\textsf{mosaic}\\xspace.  Among all of the\ndetermined [O\\,\\textsc{ii}]\\xspace and Ly$\\alpha$\\xspace emitters, the $\\rm CONFID=1$ redshift is 11\\%\nand 16\\% of the $\\rm CONFID \\geq 2$ redshift, respectively, for the\n\\textsf{mosaic}\\xspace, while it is 44\\% and 16\\%, respectively, for \\textsf{udf-10}\\xspace.  This is\npossibly because with higher S\/N spectra, it is easier to find more\nfeatures.  The depth of the data also affects the detection rate of\nthe {\\tt ORIGIN\/MUSELET}-only objects at $z > 3$. We do not see the\nsame tendency as in \\textsf{udf-10}\\xspace, where the fraction of the determined\nredshifts of the {\\tt ORIGIN\/MUSELET}-only objects to the continuum\nextracted objects increases with redshift.  One nearby galaxy, 14 [O\\,\\textsc{ii}]\\xspace\nemitters, and 99 Ly$\\alpha$\\xspace emitters are found only by {\\tt ORIGIN\/MUSELET}\nwith $\\rm CONFID \\geq 2$ in the \\textsf{mosaic}\\xspace.\n\nThanks to the larger survey area of the \\textsf{mosaic}\\xspace, we find more rare\nobjects than in \\textsf{udf-10}\\xspace. As an example, the one-dimensional spectrum of\na quasar at $z = 1.22$ (MUSE ID 872 or UVUDF ID 23796) with prominent\nMg\\,\\textsc{ii} emission is shown in\nFigure~\\ref{fig:qso_spec}\\,\\footnote{The optical spectrum of this\n  object was taken before with {\\it HST\/ACS} slitless grism\n  spectroscopy and Mg\\,\\textsc{ii} is detected \\citep{Stra08}.}. \nThis spectrum also exhibits Fe\\,\\textsc{ii}\\xspace and Mg\\,\\textsc{ii}\\xspace absorption features. These\nfeatures only appear in the spectrum when it is extracted within the\nquasar PSF and are likely to be due to an intervening absorber at\n$z=0.98$ \\citep[cf.][]{Rigb02}.\n\n\\begin{figure\n  \\includegraphics[width=\\textwidth\/2]{figures\/udf10_c042_e031_withz_iter6__OII3727_spec.pdf}\n  \\includegraphics[width=\\textwidth\/2]{figures\/udf10_c042_e031_withz_iter6__CIII1909_spec.pdf}\n  \\includegraphics[width=\\textwidth\/2]{figures\/udf10_c042_e031_withz_iter6__LYALPHA_spec.pdf}\n  \\caption{ Full MUSE spectrum and a zoomed-in portion of the\n    [O\\,\\textsc{ii}]\\xspace, C\\,\\textsc{iii}]\\xspace, and Ly$\\alpha$\\xspace detected wavelength region at the top of\n    each panel, and the {\\it HST} F775W, MUSE white-light, and\n    narrowband images of each emission line in the bottom of each\n    panel. The yellow contours in the images are the boundary of the\n    UVUDF catalog segmentation for the object, but those in the\n    MUSE white-light image are convolved with the MUSE beam size.  The\n    white boundaries indicate masked objects excluded from local sky\n    residual estimates. The green cross indicates the central position\n    of the extracted object.  The ID numbers and measured redshift are\n    indicated at the top of each panel. All of these redshifts are\n    secure ($\\rm CONFID = 2 \\, or \\, 3)$.}\n  \\label{fig:eml_spec}\n\\end{figure}\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6__abs_spec.pdf}}\n  \\caption{ Full MUSE spectrum and a zoomed-in portion of the\n    SiIV, SiII and CIV, and FeII absorption features. The {\\it HST}\n    F775W and the MUSE white-light images are shown on the right. The\n    yellow contours in the images are the boundary of the UVUDF\n    catalog segmentation for the object, but those in the MUSE\n    white-light image are convolved with the MUSE beam size.  The white\n    boundaries indicate masked objects excluded from local sky\n    residual estimates. The green cross indicates the central position\n    of the extracted object. The ID number and measured redshift are\n    indicated at the top.}\n  \\label{fig:abs_spec}\n\\end{figure}\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{figures\/udf10_c042_e031_withz_iter6_muse-z_prop_fade.pdf}\n    \\caption{Final MUSE redshift distribution of the {\\it unique}\n      objects (i.e., overlapping objects are removed) combine both the\n      continuum and emission line detected sources in the MUSE Ultra\n      Deep Field (\\textsf{udf-10}\\xspace). {\\bf [Left]} The redshift histogram in bins\n      of $\\Delta z = 0.35$.  The red, blue, and gray colors represent\n      the confidence levels 3, 2, and 1, respectively, for the\n      determined redshift. The objects {\\it only} found by {\\tt\n        ORIGIN} are indicated by the faded colors, whereas in\n      Fig.~\\ref{fig:u10_musez_hstpri_org} {\\it all} of the {\\tt\n        ORIGIN} detected objects are shown. {\\bf [Middle]} The same\n      redshift distribution as the left planel, but color coded by\n      classified type of the objects. {\\bf [Right]} The histogram of\n      the classified type of the objects color coded by the redshift\n      confidence levels. }\n    \\label{fig:u10_musez}\n  \\end{center} \n\\end{figure*}\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6__ORIGIN_only_Lya_spec.pdf}}\n  \\caption{ Example of an object that is not detected by\n    continumm emission but found by {\\tt ORIGIN}.  The full MUSE\n    spectrum and zoomed-in part of the Ly$\\alpha$\\xspace detected wavelength region are\n    presented at the top. The {\\it HST} F775W, MUSE white-light,\n    and Ly$\\alpha$\\xspace narrowband images are at the bottom.  }\n  \\label{fig:ORIonly_Lya_spec}\n\\end{figure}\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6_muse-z_vs_mag_with_err.pdf}}\n  \\caption{ Magnitudes of {\\it HST} F775W plotted against determined\n    redshift ($\\rm CONFID \\geq 2$) for \\textsf{udf-10}\\xspace. The open and filled\n    symbols represent the continuum ({\\it HST} prior) and emission\n    line ({\\tt ORIGIN} or {\\tt MUSELET}) extracted objects. The filled\n    squares are the objects detected only with {\\tt ORIGIN} or {\\tt\n      MUSELET}, and thus their F775W mags are upper limits. The\n    horizontal dashed line indicates 27~mag where we make the cut to\n    the continuum selected galaxies to perform the redshift\n    determination in the \\textsf{mosaic}\\xspace field (see \\S\\ref{subsec:mag_cut}).}\n  \\label{fig:u10_musez_mag}\n\\end{figure}\n\n\\begin{figure}\n  \\includegraphics[width=\\textwidth\/2]{figures\/udf10_c042_e031_withz_iter6_muse-z_cumul_vs_mag__type.pdf}\n  \\caption{ Cumulative counts of secure redshifts in \\textsf{udf-10}\\xspace. The colors\n    indicate different types with the same scheme as in\n    Figure~\\ref{fig:u10_musez_hstpri_org}.}\n  \\label{fig:u10_accum_type}\n\\end{figure}\n\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{figures\/mosaic_c042_e030_withz_iter8_muse-z_prop_fade.pdf}\n    \\caption{Same histograms as Figure~\\ref{fig:u10_musez} but for\n      the MUSE Deep Field (the \\textsf{mosaic}\\xspace). }\n    \\label{fig:mos_musez}\n  \\end{center} \n\\end{figure*}\n\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6__qso_spec.pdf}}\n  \\caption{ Full MUSE spectrum of a quasar at $z=1.22$ detected with a\n    broad, prominent Mg\\,\\textsc{ii}\\xspace emission feature. An intervening absorber\n    (Fe\\,\\textsc{ii}\\xspace and Mg\\,\\textsc{ii}\\xspace) at $z=0.98$ is also detected. The cutout\n    ($7.6\\arcsec \\times 7.6\\arcsec$) for this object is larger than\n    the standard $5\\arcsec \\times 5\\arcsec$.}\n  \\label{fig:qso_spec}\n\\end{figure}\n\n\n\\subsection{Redshift comparisons between the overlap region in MUSE\n  Ultra Deep (\\textsf{udf-10}\\xspace) \\& Deep Field (the \\textsf{mosaic}\\xspace)}\\label{subsec:comp_udf10_mosaic}\n\nThe \\textsf{mosaic}\\xspace redshift determination is performed independently from\n\\textsf{udf-10}\\xspace. In this subsection, we show direct comparisons of the measured\nredshifts and the associated parameters in the overlap region of these\ntwo fields.\n\nIn the overlapping part of the \\textsf{mosaic}\\xspace and \\textsf{udf-10}\\xspace fields, all of the\nredshifts (regardless of CONFID) identified in the \\textsf{mosaic}\\xspace are also\nidentified in \\textsf{udf-10}\\xspace with the same CONFID or higher, except MUSE ID 275\nand four objects detected only by {\\tt ORIGIN} or {\\tt MUSELET}\n(MUSE IDs 6432, 6447, 6865, and 7396). The redshift of ID 275 is\nmeasured to be $z=2.9$ in both \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace, but CONFID are 1\nand 2, respectively. This is because the object lies at the edge of\nthe \\textsf{udf-10}\\xspace field, which gives less confidence. The four objects\ndetected only by emission lines found in the \\textsf{mosaic}\\xspace data cube (with\n$\\rm CONFID \\geq 2$) are missed by the {\\tt ORIGIN} run in \\textsf{udf-10}\\xspace.\n\nAmong $\\UDFzALLCone$ redshifts identified in \\textsf{udf-10}\\xspace, there are 13\nredshifts that disagree with the redshifts identified in the \\textsf{mosaic}\\xspace\nby $| \\Delta z | > 0.01$. For the object discussed above, MUSE ID 275,\nthe redshift is measured to be $z=2.899$ and $2.931$ in \\textsf{udf-10}\\xspace and the\n\\textsf{mosaic}\\xspace, respectively. Apart from this object, one (MUSE ID 6684) has\nthe same CONFID of 2 and four (MUSE IDs 44, 49, 90, 127) have the same\nCONFID of 1 in both \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace, but all of the rest (MUSE\nIDs 64, 103, 718, 6335, 6339, 6676, 6686) have the high\nCONFID of 2 or 3 in the \\textsf{udf-10}\\xspace whereas CONFID of 1 in the\n\\textsf{mosaic}\\xspace. Below, we analyze why the obtained redshifts have differences\n(CONFID is given in the parentheses):\n\n\\begin{description}\n\n\\item[\\underline{With the same CONFID}]\\mbox{}\\\\\n\n\\item[{\\bf 6684}   $z_{\\rm \\textsf{udf-10}\\xspace}=4.740$ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=0.871$ (2)]\\mbox{}\\\\\n  A clear emission feature is detected in both \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace,\n  but it is identified as Ly$\\alpha$\\xspace in \\textsf{udf-10}\\xspace and as [O\\,\\textsc{ii}]\\xspace in the\n  \\textsf{mosaic}\\xspace. This is because the line profile in the \\textsf{mosaic}\\xspace spectrum\n  shows double peaks (due to lower S\/N) whose separation matches that\n  of [O\\,\\textsc{ii}]\\xspace at $z=0.871$. However, because this object is not detected\n  in the UV imaging, it is more likely at higher redshift. Thus, we\n  conclude that the feature is Ly$\\alpha$\\xspace and its redshift should be $4.740$\n  as determined in \\textsf{udf-10}\\xspace.\n  \\\\\n\n\\item[{\\bf 44}   $z_{\\rm \\textsf{udf-10}\\xspace}=1.610 $ (1), $z_{\\rm \\textsf{mosaic}\\xspace}=1.436 $ (1)]\\mbox{}\\\\\n  Both of the redshifts are from absorption features, but a set of\n  multiple Fe\\,\\textsc{ii}\\xspace absorption features are found in different wavelength\n  regions. The CONFID for both of these redshifts are low.\n  \\\\\n\n\\item[{\\bf 49}   $z_{\\rm \\textsf{udf-10}\\xspace}=1.864 $ (1), $z_{\\rm \\textsf{mosaic}\\xspace}=1.578 $ (1)]\\mbox{}\\\\\n  The Al\\,\\textsc{iii}\\xspace and Fe\\,\\textsc{ii}\\xspace absorptions are identified in \\textsf{udf-10}\\xspace, but a\n  different set of absorptions are possibly seen in the \\textsf{mosaic}\\xspace. Owing\n  to low S\/N in the continua, they are both not certain.\n  \\\\\n\n\\item[{\\bf 90}   $z_{\\rm \\textsf{udf-10}\\xspace}=0.734 $ (1), $z_{\\rm \\textsf{mosaic}\\xspace}=2.389 $ (1)]\\mbox{}\\\\\n  The same emission feature detected at $6465$${\\rm \\AA}$\\xspace in \\textsf{udf-10}\\xspace and the\n  \\textsf{mosaic}\\xspace is identified as [O\\,\\textsc{ii}]\\xspace and C\\,\\textsc{iii}]\\xspace, respectively. With the low\n  S\/N data, it is difficult to confidently distinguish whether it is\n  [O\\,\\textsc{ii}]\\xspace or C\\,\\textsc{iii}]\\xspace.\n  \\\\\n\n\\item[{\\bf 127}   $z_{\\rm \\textsf{udf-10}\\xspace}=0.616$ (1), $z_{\\rm \\textsf{mosaic}\\xspace}=4.035$ (1)]\\mbox{}\\\\\n  This is an interesting case in which different sets of emission\n  features are spotted in the \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace spectra. In \\textsf{udf-10}\\xspace,\n  emission features at $6025$${\\rm \\AA}$\\xspace, $7855$${\\rm \\AA}$\\xspace, and $8090$${\\rm \\AA}$\\xspace are\n  identified as [O\\,\\textsc{ii}]\\xspace, H$\\beta$\\xspace, and \\oiii5007 at $z=0.616$. In contrast,\n  none of these features are identified in the \\textsf{mosaic}\\xspace, but an\n  emission line is identified at $6120$${\\rm \\AA}$\\xspace. This feature is\n  attributed to Ly$\\alpha$\\xspace. Reviewing the \\textsf{udf-10}\\xspace spectrum, this feature is\n  clearly detected.  Thus, we think that there are probably two\n  objects at $z=0.616$ and $z=4.035$ lying along the sightline.  In\n  the combined \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace catalog, we use $z=0.616$ for ID~127\n  and add a new object with ID~7582 for the Ly$\\alpha$\\xspace emitter at $z=4.035$.\n  \\\\\n\n\\item[\\underline{Higher CONFID in \\textsf{udf-10}\\xspace}]\\mbox{}\\\\\n\n\\item[{\\bf 64}   $z_{\\rm \\textsf{udf-10}\\xspace}=1.847 $ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=1.566 $ (1)]\\mbox{}\\\\\n  The C\\,\\textsc{iii}]\\xspace doublet is well detected at $5430$${\\rm \\AA}$\\xspace in \\textsf{udf-10}\\xspace. The\n  \\textsf{mosaic}\\xspace redshift was measured by tentative absorption\n  features. However, with a closer look, the same C\\,\\textsc{iii}]\\xspace seen in \\textsf{udf-10}\\xspace\n  is weakly detected in the \\textsf{mosaic}\\xspace. Thus, the redshift of this object\n  is likely to be $1.847$ as measured in \\textsf{udf-10}\\xspace.\n  \\\\\n\n\\item[{\\bf 103}   $z_{\\rm \\textsf{udf-10}\\xspace}=3.002$ (3), $z_{\\rm \\textsf{mosaic}\\xspace}=2.986$ (1)]\\mbox{}\\\\\n  A clear Ly$\\alpha$\\xspace absorption detected in both \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace\n  spectra. In \\textsf{udf-10}\\xspace, there are also multiple UV absorption\n  lines. Taking into account that the features used to determine the\n  redshift are absorption and very broad, $| \\Delta z |$ of 0.016 is\n  not a significant difference. We use the \\textsf{udf-10}\\xspace redshift as the\n  final redshift.\n  \\\\\n\n\\item[{\\bf 718}   $z_{\\rm \\textsf{udf-10}\\xspace}=4.524$ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=0.801$ (1)]\\mbox{}\\\\\n  Similar to ID 6684, the spectrum of the \\textsf{mosaic}\\xspace is noisier than\n  \\textsf{udf-10}\\xspace, which causes spurious double peaks of the identified emission\n  feature. In the \\textsf{udf-10}\\xspace spectrum, the feature shows a clear asymmetric\n  profile with a red wing, indicating Ly$\\alpha$\\xspace. Thus, the redshift should\n  be $4.524$ as in \\textsf{udf-10}\\xspace.\n  \\\\\n\n\\item[{\\bf 6335}   $z_{\\rm \\textsf{udf-10}\\xspace}=4.370 $ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=1.098$ (1)]\\mbox{}\\\\\n  A clear emission feature is detected in \\textsf{udf-10}\\xspace at $6525$${\\rm \\AA}$\\xspace whose\n  profile indicates Ly$\\alpha$\\xspace, while the identified feature ([O\\,\\textsc{ii}]\\xspace) in the\n  \\textsf{mosaic}\\xspace is unclear. We do not see any obvious features around\n  $6525$${\\rm \\AA}$\\xspace in the \\textsf{mosaic}\\xspace spectrum.  This is not surprising because\n  the line flux is\n  $\\approx 1 \\times 10^{-18} \\, {\\rm erg\\,s^{-1}\\,cm^{-2}}$, just\n  around the $3\\sigma$ detection limit of the \\textsf{mosaic}\\xspace.\n  \\\\\n\n\\item[{\\bf 6339}   $z_{\\rm \\textsf{udf-10}\\xspace}=5.131 $ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=5.121$ (1)]\\mbox{}\\\\\n  The same emission feature is recognized as Ly$\\alpha$\\xspace. It looks as if the\n  detected Ly$\\alpha$\\xspace has a blue bump (which is likely to be sky residual)\n  in the \\textsf{mosaic}\\xspace. The blue bump is used to determine redshift, causing\n  a redshift difference of 0.01.  However, this blue bump is not\n  visible in the deeper data of \\textsf{udf-10}\\xspace. Using $z=5.131$ from \\textsf{udf-10}\\xspace is\n  reasonable.\n  \\\\\n\n\\item[{\\bf 6676}   $z_{\\rm \\textsf{udf-10}\\xspace}=3.723$ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=0.541$ (1)]\\mbox{}\\\\\n  The emission line clearly looks like Ly$\\alpha$\\xspace in the \\textsf{udf-10}\\xspace spectrum and\n  the UV continuum emission is not detected. However, in the \\textsf{mosaic}\\xspace\n  spectrum, although the same feature is identified, it is classified\n  as [O\\,\\textsc{ii}]\\xspace. Thus, the \\textsf{udf-10}\\xspace redshift should be the correct one.\n  \\\\\n\n\\item[{\\bf 6686}   $z_{\\rm \\textsf{udf-10}\\xspace}=0.307$ (2), $z_{\\rm \\textsf{mosaic}\\xspace}=4.383$ (1)]\\mbox{}\\\\\n  In the \\textsf{udf-10}\\xspace spectrum, two emission features at $4875$${\\rm \\AA}$\\xspace and\n  $6545$${\\rm \\AA}$\\xspace correspond to [O\\,\\textsc{ii}]\\xspace and \\oiii5007, which gives $z=0.307$.\n  The former feature, in addition to not being convincing in \\textsf{udf-10}\\xspace, is\n  not seen in the \\textsf{mosaic}\\xspace.  The latter feature in the \\textsf{mosaic}\\xspace is\n  recognized as Ly$\\alpha$\\xspace because of its distinctive profile. This object\n  is not detected up to the F435W band, which indicates that this\n  object is likely to be at high redshift. Thus, the correct redshift\n  is probably $z=4.383$ measured in the \\textsf{mosaic}\\xspace.\n\n\\end{description}\n\nThese results suggest that it is not necessary true that the success\nrate of CONFID in the shallower \\textsf{mosaic}\\xspace region is lower when comparing\nthe same CONFID. We only found one case (ID 6684) in which the redshift in\nthe \\textsf{mosaic}\\xspace field is not correct when both of the redshifts are\n$\\rm CONFID=2$.  When both redshifts are $\\rm CONFID=1$ (IDs 44, 49\n90, 127), their redshifts are too uncertain to make a judgement for\nthe quality.\n\nBecause of the shallower depth of the \\textsf{mosaic}\\xspace, not all of the\nredshifts measured in \\textsf{udf-10}\\xspace are found in the \\textsf{mosaic}\\xspace. Excluding the\nobjects only found by {\\tt ORIGIN\/MUSELET}, the \\textsf{mosaic}\\xspace misses 118 out\nof $\\UDFzHSTCone$ redshifts with $\\rm CONFID \\geq 1$ and 64 out of\n$\\UDFzHSTCtwo$ with $\\rm CONFID \\geq 2$ (night [O\\,\\textsc{ii}]\\xspace emitters, six\nabsorption line galaxies, six C\\,\\textsc{iii}]\\xspace emitters, and 43 Ly$\\alpha$\\xspace\nemitters). We only visually inspected the spectra of galaxies with\n$\\rm F775W \\leq 27$~mag and for the rest we relied on {\\tt ORIGIN}\n(and {\\tt MUSELET}). Thus, a subset of these missing redshifts may be\nrecovered if we actually look at the spectra or change the tuning of\n{\\tt ORIGIN} (and {\\tt MUSELET}).  If we include the objects that have\nno UVUDF counterpart (i.e., detected only by {\\tt ORIGIN\/MUSELET}),\nthen these numbers increase to 133 and 78 for $\\rm CONFID \\geq 1$ and\n$\\geq 2$, respectively.  The smaller number of {\\tt ORIGIN\/MUSELET}\ndetections in the overlap region in the \\textsf{mosaic}\\xspace can also be attributed\nto the depth of the data.\n\nTogether with this paper, we release both the \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace\nredshift catalogs (Appendix~\\ref{app:cat}). As explained above, the ID\nnumbers of the objects in the overlapping region are the same in \\textsf{udf-10}\\xspace\nand the \\textsf{mosaic}\\xspace. In addition, we compiled a final redshift catalog\ncombining the \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace results.  For the objects that have\nmeasurements from both of the catalogs, the information is from \\textsf{udf-10}\\xspace,\nunless there is a disagreement with the \\textsf{mosaic}\\xspace. For those cases, we\nadopted the redshifts discussed above.  For the discussion below, we\nused the redshifts and the associated parameters from this combined\ncatalog.\n\n\n\\subsection{Final redshifts in the entire MUSE UDF survey region}\n\nIn the entire MUSE UDF survey region (\\textsf{udf-10}\\xspace $+$ the \\textsf{mosaic}\\xspace), there\nare $\\COMBHSTpri$ UVUDF sources that we used as priors to extract the\ncontinuum selected objects. As shown in Table~\\ref{tbl:num_z}, for\n$\\COMBzHSTCtwo$ (15\\%) of them, we successfully obtained secure\nredshifts ($\\rm CONFID \\geq 2$). As discussed above, some of the\nobjects were merged because of the lower spatial resolution of MUSE\nthan {\\it HST} and we did not investigate a subset\n($\\rm F775W \\leq 27$~mag in the \\textsf{mosaic}\\xspace where not overlapping with\n\\textsf{udf-10}\\xspace) of their spectra to determine redshifts.  The direct searches\nof emission line objects enabled us to find $\\COMBzORGonlyCtwo$ more\nredshifts in addition.  Thus, in total, we obtained $\\COMBzALLCtwo$\nunique redshifts with high confidence ($\\rm CONFID \\geq 2$). The final\nredshift distribution is presented in Figure~\\ref{fig:comb_musez}. As\na simple test, we compared the broadband (F775W) luminosity against\nthe measured MUSE redshifts ($\\rm CONFID \\geq 2$) to check whether\nthere are any catastrophic measured redshifts. This test does not show\nany extreme outliers.\n\nWe used the redshifts determined in both \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace to\nderive the distribution of redshift differences between these two\ncatalogs.  The standard deviation of the $\\Delta z\/(1+z)$ distribution\nis 0.00017. Assuming that the errors are similar for the \\textsf{udf-10}\\xspace and\n\\textsf{mosaic}\\xspace redshifts (i.e., assuming that the depth of the data is not a\ndominant factor for redshift errors), then we obtain a global estimate\nof the redshift\/velocity uncertainty to be $\\sigma_z = 0.00012 (1+z)$\nor $\\sigma_v \\approx 40 \\, {\\rm km\\,s^{-1}}$.\n\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=\\textwidth]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_muse-z_prop_fade.pdf}\n    \\caption{Same histograms as Figures~\\ref{fig:u10_musez} and\n      \\ref{fig:mos_musez}, but for the combined redshifts of the unique\n      objects from the MUSE Ultra Deep Field (\\textsf{udf-10}\\xspace) and MUSE Deep\n      Field (the \\textsf{mosaic}\\xspace). }\n    \\label{fig:comb_musez}\n  \\end{center} \n\\end{figure*}\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\n\n\\subsection{Redshift measurement success rates}\n\nAssuming all of the determined redshifts are correct for\n$\\rm CONFID \\geq 2$, we calculated the success rates of the redshift\nmeasurements. Here we define the success rate as the number fraction\nof the obtained secure redshifts over the sample size we inspected. We\nonly used the sample extracted with the {\\it HST} priors for this\ncalculation (i.e., the objects detected only by {\\tt ORIGIN\/MUSELET}\nare excluded).  In the \\textsf{mosaic}\\xspace, all galaxies with\n$\\rm F775W > 27$~mag are extracted by {\\tt ORIGIN\/MUSELET} (or due to\nsplitting; see \\S\\ref{subsec:split}). In other words, the success\nrates at $\\rm F775W > 27$~mag for the \\textsf{mosaic}\\xspace should be considered as\nlower limits.\n\n\\begin{figure}\n  \\begin{center} \n    \\resizebox{\\hsize}{!}\n    {\\includegraphics[width=0.48\\hsize]{figures\/udf10_c042_e031_withz_iter6_z_success_rates.pdf}}\n    \\caption{ Success rates of obtained secure redshifts are shown in\n      the top panel for \\textsf{udf-10}\\xspace (red) and \\textsf{mosaic}\\xspace (blue). The horizontal\n      dashed line indicates $50\\%$ completeness. We only use the\n      objects with {\\it HST} counterparts in the UVUDF catalog in this\n      plot (i.e., the objects detected only by {\\tt ORIGIN\/MUSELET}\n      are excluded). The middle and bottom panels show the counts of\n      the total number of the {\\it HST} objects (solid bars) and the\n      MUSE redshift determined objects (hatched bars). The vertical\n      dashed line in the bottom panel indicates the magnitude cut\n      where we perform the redshift investigation on the continuum\n      detected objects. }\n    \\label{fig:z_success_rate}\n  \\end{center} \n\\end{figure}\n\nIn Figure~\\ref{fig:z_success_rate}, the success rates of \\textsf{udf-10}\\xspace and\n\\textsf{mosaic}\\xspace are plotted against the {\\it HST} F775W magnitude.  In \\textsf{udf-10}\\xspace,\nwe successfully measured secure redshifts ($\\rm CONFID \\geq 2$) for\nall of the galaxies brighter than $25$~mag. At the same magnitude, the\nsuccess rate for the \\textsf{mosaic}\\xspace is $87\\%$.  The $100\\%$ success rate for\nthe \\textsf{mosaic}\\xspace is at $< 22.5$~mag (or $23.5$~mag if we ignore one object\nMUSE ID 6934 without MUSE redshift in the $22.5-23.0$~mag bin). The\n50\\% completeness with respect to the {\\it HST} F775W magnitude is\nreached at $26.5$~mag and $25.5$~mag, in \\textsf{udf-10}\\xspace and the \\textsf{mosaic}\\xspace,\nrespectively.  The success rates decrease with the F775W magnitudes,\nin particular there is a sudden drop at $\\approx 25$~mag for both of\nthe fields. However, the success rate remains greater than\n$\\approx 20\\%$ at $< 28-29$~mag in \\textsf{udf-10}\\xspace and $\\lesssim 27$~mag in the\n\\textsf{mosaic}\\xspace.  Except in the $25.0-25.5$~mag bin, at all magnitudes, the\n\\textsf{udf-10}\\xspace success rate is higher than the \\textsf{mosaic}\\xspace.\n\nAnother MUSE deep survey in the {\\it Hubble} Deep Field South\n\\citep[HDFS;][]{Baco15}, whose coverage is also a single MUSE field of\nview as our \\textsf{udf-10}\\xspace and has a similar depth ($27$~hours), reaches the\n$50\\%$ completeness at $26$~mag in the {\\it HST} F814W band.  The\nachieved slightly higher completeness in \\textsf{udf-10}\\xspace is a natural consequence of the\nbetter line flux detection limit, which results from the improved data\nreduction and analysis compared to the HDFS data cubes \\citep{Baco15}.\n\nThe spectroscopic completeness and comparisons with photometric\nredshifts are discussed in Paper~III.\n\n\n\\subsection{Objects only detected by {\\tt ORIGIN} or {\\tt MUSELET}}\n\nOwing to its wide FoV IFU with high sensitivity, MUSE\\ enables direct\ndetection of emission line objects with very faint continuum emission,\nwhich are improbable to target with slit spectroscopy and inefficient\nto find with narrowband imaging.  In the entire\n$3\\arcmin \\times 3\\arcmin$ survey area, we discovered\n$\\COMBzORGonlyCtwo$ emission line detected objects with confident\nredshifts which are not on the {\\it HST} prior list.  In the deepest\nregion ($1\\arcmin \\times 1\\arcmin$ area of \\textsf{udf-10}\\xspace), this number is\n$\\UDFzORGonlyCtwo$. Among these $\\COMBzORGonlyCtwo$ emission line\nobjects, the majority are Ly$\\alpha$\\xspace emitters (117). The rest are [O\\,\\textsc{ii}]\\xspace\nemitters (14) and a nearby galaxy (1).  Some of these objects in fact\ncan be visually identified in the images but are\nblended\\,\\footnote{Object MUSE ID 6449 actually has UVUDF ID 4293\n  (CANDELS ID 18674) in the UVUDF segmentation map. However, UVUDF ID\n  4293 does not exist in the UVUDF catalog. Since the actual\n  measurement is not provided in the UVUDF catalog, here we consider\n  that it was not successfully extracted in UVUDF.} (see\n\\S\\ref{sec:udf10_finalz}). However, especially for the Ly$\\alpha$\\xspace emitters,\neven when they can be visually identified, they have very low surface\nbrightness or are not detected in the continuum. As mentioned in\n\\S\\ref{subsec:NoiseChisel}, we performed our own flux or upper limit\nmeasurements for these galaxies, but here we look into the most recent\n{\\it HST} catalogs in more detail to investigate their properties.\n\nOur prior list is based on the UVUDF catalog created by running {\\tt\n  ColorPro} \\citep[a wrapper of {\\tt SExtractor};][]{Coe06} with the\ndetection image obtained by averaging four optical (F435W; F606W,\nF775W, F850LP) and four near-infrared (F105W, F125W, F140W, F160W)\nimages \\citep{Rafe15}. Except for the blended galaxies, the continuum\nof the MUSE emission line objects was still too faint to be originally\ndetected in the combination of this deepest detection image.\n\nWe also checked if any of the emission line only objects are in the\nCANDELS\\,\\footnote{Cosmic Near-IR Deep Extragalactic Legacy Survey}\nGOODS-S multiwavelength catalog \\citep{Guo13}, and identified 52\nobjects (one nearby object, eight [O\\,\\textsc{ii}]\\xspace emitters, and 43 Ly$\\alpha$\\xspace emitters)\nthat have potential counterparts within $1''$. This does not\nimmediately mean that they are the actual corresponding objects\nbecause some {\\tt ORIGIN\/MUSELET} detected objects are completely\nblended with known objects or happen to lie on the same sightline. We\ninspected these objects one by one and found one nearby galaxy, seven\n[O\\,\\textsc{ii}]\\xspace emitters, and one Ly$\\alpha$\\xspace emitter that exist in the CANDELS catalog.\nFor this Ly$\\alpha$\\xspace emitter (MUSE ID 6343 or CANDELS ID 15913), based on the\nprospective Ly$\\alpha$\\xspace in its MUSE spectrum, its redshift is determined to\nbe $5.5$. However, its $U$ magnitude is 28.5~mag in the CANDELS\ncatalog, which is not probable for a high-$z$ galaxy because of the\nLyman break at 912\\AA.  It is more likely that this Ly$\\alpha$\\xspace emitter\nhappens to lie on the same line of sight as this galaxy.  The\ntemplate-fitting technique, {\\tt TFIT} \\citep{Laid07}, is utilized for\nthe CANDELS catalog. The {\\tt TFIT} tool uses spatial positions and\nmorphologies in a high-resolution image as the priors to fit objects\nin lower resolution images. Until objects cannot be resolved in the\nhigh-resolution image, it can further perform deblending in the lower\nresolution images even when object separations are $\\lesssim 1.5$\ntimes the PSF FWHM, which is the limit for {\\tt SExtractor}.  Thus,\nsome of these closely separated objects could not be deblended in the\nUVUDF catalog but were in CANDELS.  In fact, the one [O\\,\\textsc{ii}]\\xspace emitter\n(MUSE ID 6315) that is not in the CANDELS catalog is not resolved even\nwith {\\it HST,} and the 42 Ly$\\alpha$\\xspace emitters are barely detected or not at\nall.\n\n\n\\subsection{Blend and split of merged objects}\\label{subsec:split}\n\nDuring the redshift evaluation process, we found some cases in which we\nwere able to split the merged MUSE objects (see \\S\\ref{sec:analysis_cont}).\nComparisons between a detected emission line narrowband image and the\n{\\it HST} image provide the spatial information of the corresponding\n{\\it HST} counterpart of the emission line object. An example is shown\nin Figure~\\ref{fig:split}. The object shown in the MUSE white\nimage (the middle panel) is originally extracted as a merged object,\nwhich corresponds to two {\\it HST}-detected objects with UVUDF IDs of 8147\nand 8118. While these two objects are clearly resolved in the {\\it\n  HST} F775W image (the left panel), they are not resolved in the MUSE\nwhite image (the middle panel). We managed to associate the determined\nredshift (based on Ly$\\alpha$\\xspace) with one of the {\\it HST} objects, UVUDF ID\n8147, by creating a narrowband image of the detected Ly$\\alpha$\\xspace at $z=3.67$\n(the right panel). Although we cannot fully deblend the MUSE emission\nbecause of the limited spatial resolution, we can assign individual MUSE\nID numbers for these objects using their UVUDF coordinates\n(``split'').  This Ly$\\alpha$\\xspace emitter is given MUSE ID 6290 and the other is\n6749.  This process does not increase or decrease the number\nof objects whose redshifts are successfully reported as identified\nin the earlier sections, unless both (or multiple) of the split\nobjects show clear features that can be used to determine redshift.\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/udf10_c042_e031_withz_iter6_split_object.pdf}}\n  \\caption{ Stamp impage of $5\\arcsec \\times 5\\arcsec$ centered at\n    $\\rm R.A.=53.169458$ and $\\rm Dec=-27.778191$. The {\\it HST} F775W\n    image (white-light, left panel), the MUSE $\\lambda$-collapsed\n    image (middle), and the continuum subtracted narrowband image\n    created at $\\lambda=5676$${\\rm \\AA}$\\xspace with a width of $10$${\\rm \\AA}$\\xspace (right).\n    While it is not possible to resolve the two nearby {\\it HST}\n    objects (UVUDF IDs of 8147 and 8118) with the MUSE white-light\n    image, the combination of the {\\it HST} image with the narrowband\n    image of the detected emission line (Ly$\\alpha$\\xspace in this case) makes it\n    possible to identify the origin of the emission (UVUDF ID 8147\n    corresponds to MUSE ID 6290 with $z=3.67$).}\n  \\label{fig:split}\n\\end{figure}\n\nAlthough small in number, there are also cases in which multiple\nredshifts are found in a single merged object.\nThe MUSE IDs of 6877 and 6878 are originally merged and treated as a\nsingle MUSE extracted object. However, clear detections of the [O\\,\\textsc{ii}]\\xspace\ndoublet, [O\\,\\textsc{iii}]\\xspace$\\lambda\\lambda4959,5007$, and several Balmer lines at\n$z=0.734$ for ID 6877 and Ly$\\alpha$\\xspace at $z=3.609$ for ID 6878 allow us to\nassign confident redshifts for both of the objects in addition to\nsplitting them (Figure~\\ref{fig:split_bothz}). \n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/mosaic_c042_e030_withz_iter8_split_object_bothz.pdf}}\n  \\caption{ Spectra (extracted by PSF weighting) of MUSE IDs 6877\n    ($z=0.734$) and 6878 ($z=3.609$) at the top. The images shown at\n    the bottom are {\\it HST} F775W, MUSE white-light, and narrowband\n    images of [O\\,\\textsc{ii}]\\xspace at $z=0.734$ and Ly$\\alpha$\\xspace at $z=3.609$, from left to\n    right. Because these two objects are barely resolved with MUSE,\n    they were originally treated as a single ``merged''\n    object. However, their spectra and a comparison of the narrowband\n    images with the {\\it HST} image help to ``split'' and associate\n    the MUSE redshifts with the corresponding {\\it HST} counterparts.}\n  \\label{fig:split_bothz}\n\\end{figure}\n\nThere are also some rare cases in which even though two or more of the\nmerged objects have more than one redshift identified, they cannot be\nassociated with any of the known objects owing to complex morphology,\ntoo close separation even for the {\\it HST}, or no continuum\ndetection. As shown in Figure~\\ref{fig:split_bothz_noID}, in the\none-dimensional spectrum of an absorption galaxy at $z=2.575$, MUSE ID\n942, multiple emission lines are visible at $7805$${\\rm \\AA}$\\xspace, $8100$${\\rm \\AA}$\\xspace,\n$8590$${\\rm \\AA}$\\xspace, and $9090$${\\rm \\AA}$\\xspace. These features do not correspond to any of\nthe features that can be seen at $z=2.575$ of ID 942, but are\nconsistent with [O\\,\\textsc{ii}]\\xspace, [Ne\\,\\textsc{iii}]\\xspace$3869$, H$\\delta$\\xspace, and H$\\gamma$\\xspace at $z=1.094$ (MUSE ID\n7382), respectively. We use the center of the [O\\,\\textsc{ii}]\\xspace narrowband image as\nthe coordinates of this object. This particular case was found by\nhand, but {\\tt ORIGIN} has played an important role in finding these\nkinds of blended objects that are hard to find by humans. For example,\nmultiple pronounced emission features seen in MUSE ID 945 immediately\nreveal its redshift to be $z=0.605$. It is very easy to neglect a\nweaker emission feature detected at $7230$${\\rm \\AA}$\\xspace in the same spectrum,\nwhich is not associated with ID 945. {\\tt ORIGIN} successfully found\nthis feature and we identify it as Ly$\\alpha$\\xspace at $z=4.947$ (MUSE ID 6470, no\nUVUDF counterpart).\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/mosaic_c042_e030_withz_iter8_split_object_bothz_noID.pdf}}\n  \\caption{ Similar to Figure~\\ref{fig:split_bothz}, but for MUSE IDs\n    942 ($z=2.575$) and 7382 ($z=1.094$). The extracted spectra\n    clearly show that there are at least two objects lying on the same\n    sight line. A small offset is also seen between the narrowband\n    images of the C\\,\\textsc{iv} absorption feature at $z=2.575$ and\n    [O\\,\\textsc{ii}]\\xspace at $z=1.094$. However, there is no obvious {\\it HST}\n    object to provide the counterpart for the [O\\,\\textsc{ii}]\\xspace emitter. }\n  \\label{fig:split_bothz_noID}\n\\end{figure}\n\nIn the end, we split about 150 of the merged objects.  Among\n$\\COMBzALLCone$ objects with MUSE-$z$ ($\\rm CONFID \\geq 1$), there are\n79 that remain as merged objects.  We stress that we do not use or\nprovide the photometric measurements for the merged objects.  When\nmerged objects have been successfully split during the redshift\ndetermination, the split objects have been associated with the\ncorresponding photometries from the UVUDF catalog.  If we find\nmultiple redshifts in an object that is not merged (the case of\nFigure~\\ref{fig:split_bothz_noID}), only the object with the centroid\nin the narrowband images of the detected features closest to the UVUDF\ncoordinates or the object with the reasonable {\\it HST} color is\nassigned to have the UVUDF photometries.\n\n\n\\subsection{Comparisons with previous spectroscopic redshifts}\n\nThere has been a large effort to measure spectroscopic redshifts\n(spec-$z$) in the UDF and the surrounding regions, such as GOODS,\nVVDS, and VUDS.  According to the list of spec-$z$ compiled in the\nUVUDF catalog by \\cite{Rafe15}\\,\\footnote{The spectroscopic redshifts\n  provided in the UVUDF catalog are gathered from the following\n  surveys: VVDS \\citep{LeFe04}, Szokoly \\citep{Szok04}, K20\n  \\citep{Mign05}, GRAPES \\citep{Dadd05}, Vanzella GOODS\n  \\citep{Vanz05,Vanz06,Vanz08,Vanz09}, Popesso GOODS \\citep{Pope09},\n  Balestra GOODS \\citep{Bale10}, GMASS \\citep{Kurk13}, and 3D-HST\n  \\citep{Morr15}.}, there are $\\COMBprevzUVUDF$ previously known {\\it\n  secure} spec-$z$ in our survey region ($3' \\times 3'$). The total\nnumber of published redshifts in the UVUDF catalog is 169 over the\nentire HUDF area.  In addition, within a $0.5''$ search radius, we\nfind that there are $\\COMBprevzVUDS$ MUSE objects\\,\\footnote{MUSE ID\n  6945 has a previously known spec-$z$ in both of the UVUDF and VUDS\n  catalogs (1.098 and 1.096, respectively). These redshifts agree, and\n  thus we do not count this as an additional spec-$z$ and only use the\n  spec-$z$ from UVUDF.}  that have previously known spec-$z$ measured\nby VUDS with high reliability \\citep[redshift flags of 4, 3, or\n2;][]{Tasc17}. Thus, in total, we compile $\\COMBprevz$ previously\nknown reliable spec-$z$ in our survey field. It is noteworthy that\nMUSE delivers a drastic improvement not only in the number of\nspec-$z$, but also in the range of redshift and the limiting magnitude\nof the objects with confident spec-$z$ (Figure~\\ref{fig:prev_specz}).\n\n\\begin{figure}\n  \\begin{center} \n    \\includegraphics[width=\\textwidth\/2]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_muse-z_vs_mag_with_err_specz.pdf}\n    \\caption{ Magnitude vs. redshift space of secure MUSE-$z$\n      ($\\rm CONFID \\geq 2$; red circles) and previous reliable\n      spec-$z$ for the unique objects in the entire MUSE deep survey\n      region (\\textsf{udf-10}\\xspace $+$ \\textsf{mosaic}\\xspace).  The previously known spec-$z$ taken\n      from the UVUDF and VUDS catalogs are indicated with blue and\n      cyan squares, respectively.  The horizontal dashed line\n      indicates 27~mag where we make the cut to the continuum selected\n      galaxies to perform the redshift determination in the \\textsf{mosaic}\\xspace\n      field (see \\S\\ref{subsec:mag_cut}).}\n    \\label{fig:prev_specz}\n  \\end{center} \n\\end{figure}\n\nThe majority (all for \\textsf{udf-10}\\xspace) of the published spec-$z$ are in the\nredshift range of $0 < z < 3$ and F775W magnitude of $< 25$. The three\nobjects at $z > 3.5$ in the UVUDF catalog are from the spectroscopic\nobservations of Lyman break galaxies \\citep{Vanz09}\\,\\footnote{Their\n  object IDs are J033236.83-274558.0, J033240.01-274815.0, and\n  J033239.06-274538.7}.\n\nA simple check of MUSE redshifts is to compare them with the published\nspectroscopic redshifts.  In Fig.~\\ref{fig:u10_comp_specz}, our\nMUSE spectroscopic redshifts are compared to these previously known\nspectroscopic redshifts.  When we only use the robust redshifts\n(CONFID $\\geq 2$), we find six objects with $|\\Delta z| > 0.01$ (MUSE\nIDs 29, 949, 957, 997, and 1048 from UVUDF and MUSE ID 6891 from\nVUDS), but all except IDs 997 and 6891 have $0.01 < |\\Delta z| < 0.02,$\nwhich is expected to happen owing to observations with different\nspectroscopic resolutions.  We investigate the two objects with\ncatastrophic differences below (CONFID of MUSE-$z$ is given in the\nparentheses):\n\n\\begin{description}\n\n\\item[{\\bf 997} $z_{\\rm MUSE}=1.041$ (3), $z_{\\rm known}=1.603$ \\citep{Morr15}]\\mbox{}\\\\\n  The previous spec-$z$ is measured from WFC3 grism (G141) data by\n  identifying H$\\beta$\\xspace and [O\\,\\textsc{iii}]\\xspace (with a good quality, 3 out of 4), while\n  MUSE-$z$ is determined by a well-detected [O\\,\\textsc{ii}]\\xspace doublet,\n  [Ne\\,\\textsc{iii}]\\xspace$3869$, H$\\delta$\\xspace, and H$\\gamma$\\xspace (with CONFID of 3, the\n  highest). Reinspecting the WFC3 grism spectrum verifies that\n  $z=1.603$ is not as convincing as originally determined. In\n  addition, there is an extra complication in extracting the grism\n  slitless spectrum due to multiple nearby objects. Thus, we conclude\n  that our MUSE redshift is likely to be correct for MUSE ID 997\n  (UVUDF ID 8592).\n  \\\\\n\n\\item[{\\bf 6891} $z_{\\rm MUSE}=0.227$ (3), $z_{\\rm known}=3.647$ \\citep{Tasc17}]\\mbox{}\\\\\n  Although the redshift of ID 6891 (UVUDF ID 4864) is identified by\n  good detections of H$\\beta$\\xspace , [O\\,\\textsc{iii}]\\xspace, and H$\\alpha$\\xspace, its spectrum is blended\n  with ID 6892 (UVUDF ID 4863), which shows a Ly$\\alpha$\\xspace emission line at\n  $z_{\\rm MUSE} = 3.648$ ($\\rm CONFID = 2$).  The separation of these\n  two objects is $0.3''$. The previously measured spec-$z$ corresponds\n  to MUSE ID 6892, but the pointing coordinates are closer to MUSE ID\n  6891. Also, these two objects were not successfully\n  deblended in the CANDELS catalog (CANDELS ID 13745 for both of the\n  objects). Thus, we think that our MUSE redshift measurement for MUSE\n  ID 6891 (and 6892) has no problem.\n\n\\end{description}\n\n\\begin{figure}\n  \\begin{center} \n    \\includegraphics[width=\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_spec-z_vs_muse-z.pdf}\n    \\caption{ Comparisons of MUSE-$z$ to the previous spectroscopic\n      redshifts from the other ground- and space-based telescopes\n      compiled in the UVUDF catalog. The high and low confidence\n      MUSE-$z$ are shown as red and gray symbols, respectively.\n      MUSE ID 997 indicated in the figure is the only object with a\n      catastrophic difference in the spec-$z$ comparisons. See the\n      text for more details.}\n    \\label{fig:u10_comp_specz}\n  \\end{center} \n\\end{figure}\n\nWhen including $\\rm CONFID=1$ redshifts, one MUSE object (MUSE ID\n1143) has previously known spectroscopic redshifts but disagrees with\nour MUSE redshifts \\citep[$z_{\\rm MUSE}=1.262$ and\n$z_{\\rm known}=1.332$ from][]{Morr15}.  The previously known redshift\n1.332 is measured in the near-infrared spectrum obtained with HST\/WFC3\ngrism (G141) spectroscopy. \\cite{Morr15} identified H$\\beta$\\xspace and [O\\,\\textsc{iii}]\\xspace. In\nour MUSE spectrum, we identified [O\\,\\textsc{ii}]\\xspace$\\lambda3729$ at $8435$${\\rm \\AA}$\\xspace, which\ngives $z=1.262$. Its CONFID is 1 because [O\\,\\textsc{ii}]\\xspace$\\lambda3726$ is not\ndetected probably because it overlies a sky line. The difference in\nredshift is $\\delta z = 0.07,$ which cannot be explained by the low\nspectral resolution ($R \\sim 130$) of the grism spectroscopy.  We found\nthat the 3D-HST survey \\citep{Bram12} also published a redshift for\nthis object of $z=1.299$ \\citep[by identifying H$\\alpha$\\xspace;][]{Momc16}, which is\nconsistent with our measurement. The slight discrepancy\n($|\\Delta z| = 0.037$) can be attributed to the fact that the object\nis spatially extended in the dispersion direction for the grism. Thus,\nwe assessed that the MUSE redshift for this object is more feasible.\n\nThere are 10 objects whose spec-$z$ are reported in the UVUDF (seven\nobjects) and VUDS (three objects) catalogs but are missed by our\nredshift determination. These objects are listed below with MUSE ID,\nUVUDF ID in the parentheses, previously measured spec-$z$, and the\nreference:\n\n\\begin{description}\n\n\\item[{\\bf 55 (24380)}: $z = 1.39 \\pm 0.01$ \\citep{Dadd05}]\\mbox{}\\\\\n  The spectrum was obtained with {\\it HST}\/ACS grism low-resolution\n  spectroscopy (G800L, $R \\approx 100$ at $8000$${\\rm \\AA}$\\xspace for a point\n  source), which detected a broad Mg\\,\\textsc{ii}\\xspace absorption line.  This feature\n  is not seen in our MUSE spectrum even with smoothing probably due to\n  much higher noise level in the continuum. This object is located in\n  the overlapping region of the \\textsf{mosaic}\\xspace and \\textsf{udf-10}\\xspace, but we cannot find\n  a redshift from the spectrum extracted from the deeper \\textsf{udf-10}\\xspace data.\n  \\\\\n\n\\item[{\\bf 1199 (3482)}: $z = 1.91 \\pm 0.01$ \\citep{Dadd05}]\\mbox{}\\\\\n  \\cite{Dadd05} found a broad blended Mg\\,\\textsc{ii}\\xspace$\\lambda\\lambda2796,2803$\n  doublet absorption feature (and likely the Fe\\,\\textsc{ii}\\xspace multiplet\n  absorption) in their spectrum taken with HST\/ACS grism\n  spectroscopy. This feature can be identified in our MUSE spectrum\n  too, but only after knowing the redshift or heavily smoothing the\n  spectrum.\n  \\\\\n\n\\item[{\\bf 1308 (10544)}: $z = 1.313$ \\citep{Morr15}]\\mbox{}\\\\\n  The known redshift was determined using the spectrum taken with {\\it\n    HST}\/WFC3 grism spectroscopy (G141, $R \\approx 130$ at\n  $1.1 \\leq \\lambda \\, {\\rm [\\mu m]} \\leq 1.7$). The emission features\n  H$\\beta$\\xspace, [O\\,\\textsc{iii}]\\xspace, and H$\\alpha$\\xspace were identified to obtain $z=1.313$.  With this\n  redshift, we confirm the [O\\,\\textsc{ii}]\\xspace emission at $8630$${\\rm \\AA}$\\xspace in our MUSE\n  spectrum. It is located in the wavelength region where sky emission\n  dominates, which made it easily missed during our redshift\n  determination process. This redshift is added into the combined\n  catalog by hand ($z_{\\rm MUSE}=1.315$ with $\\rm CONFID = 3$).\n  \\\\\n\n\\item[{\\bf 1312 (21773)}: $z = 1.76 \\pm 0.02$ \\citep{Dadd05}]\\mbox{}\\\\\n  The {\\it HST}\/ACS grism spectrum detected a broad Mg\\,\\textsc{ii}\\xspace absorption\n  line. With the previously determined redshift, we can confirm\n  the same feature in the MUSE data. However, it lies in a wavelength\n  region crowded with sky lines, which makes it even harder to identify\n  the feature without the a priori information.\n  \\\\\n\n\\item[{\\bf 1351 (4562)}: $z = 2.145$ \\citep{Bale10}]\\mbox{}\\\\\n  The spectrum was taken with VLT\/VIMOS.  The previously known\n  redshift was determined based on the O\\,\\textsc{i} ($1302.20$${\\rm \\AA}$\\xspace)\n  and C\\,\\textsc{ii} ($1335.10$${\\rm \\AA}$\\xspace) absorption features detected at\n  $4095$${\\rm \\AA}$\\xspace and $4200$${\\rm \\AA}$\\xspace, respectively.  These features are both\n  outside the MUSE wavelength coverage, preventing us from recovering\n  its redshift with MUSE. At this redshift, C\\,\\textsc{iii}]\\xspace and some absorption\n  features are covered by the MUSE wavelength range, but none of these\n  features are visible.\n  \\\\\n\n\\item[{\\bf 1364 (8292)}: $z = 2.067$ \\citep{Morr15}]\\mbox{}\\\\\n  The redshift was measured in the spectrum taken with the {\\it\n    HST}\/WFC3 near-infrared grism. These authors detected H$\\beta$\\xspace and\n  [O\\,\\textsc{iii}]\\xspace. We can indeed recognize the Mg\\,\\textsc{ii}\\xspace$\\lambda\\lambda2796,2803$\n  absorption lines (and possibly Fe\\,\\textsc{ii}\\xspace$\\lambda2344$ and\n  Fe\\,\\textsc{ii}\\xspace$\\lambda\\lambda2374,2382$) in the MUSE data with this\n  redshift. It is difficult to notice these features in our data\n  because at this redshift, the Mg\\,\\textsc{ii}\\xspace features lie in the low S\/N\n  region due to many sky lines.\n  \\\\\n\n\\item[{\\bf 1520 (21730)}: $z = 1.98 \\pm 0.02$ \\citep{Dadd05}]\\mbox{}\\\\\n  A broad Mg\\,\\textsc{ii}\\xspace absorption feature (and possibly the Fe\\,\\textsc{ii}\\xspace multiplet\n  absorption) was detected in their {\\it HST}\/ACS grism spectrum.\n  This feature might be visible in the MUSE data, but is not obvious\n  because of crowded sky lines. It is difficult to use our MUSE data\n  solely to determine this redshift.\n  \\\\\n\n\\item[{\\bf 4070 (635)}: $z = 4.498$ ($z_{\\rm flag}=3$) \\citep{Tasc17}]\\mbox{}\\\\\n  The VUDS redshift (VUDS ID 532000222) may have been determined by a\n  tentative Ly$\\alpha$\\xspace emission at 6685${\\rm \\AA}$\\xspace.  In our MUSE spectrum we also\n  discern an emission line at 6685${\\rm \\AA}$\\xspace, but this is contamination from\n  MUSE ID 1185 (Ly$\\alpha$\\xspace at $z_{\\rm MUSE}=4.50$), which is $2.38''$\n  away. It is possible that the VUDS slit also covered the extended\n  Ly$\\alpha$\\xspace emission from ID 1185, which led to $z = 4.498$.\n  \\\\\n\n\\item[{\\bf 5216 (8384)}: $z = 2.095$ ($z_{\\rm flag}=2$) \\citep{Tasc17}]\\mbox{}\\\\\n  It looks as though the VUDS redshift (VUDS ID 539990803) was\n  derived from a good cross-correlation signal. However, our\n  cross-correlation with {\\tt MARZ} does not provide a good signal and\n  we are not able to find the expected absorption features at this\n  redshift in our MUSE spectrum.\n  \\\\\n\n\\item[{\\bf 5494 (7285,7374)}: $z = 0.768$ ($z_{\\rm flag}=3$) \\citep{Tasc17}]\\mbox{}\\\\\n  At the VUDS redshift (VUDS ID 532000256), we might expect to detect\n  Fe\\,\\textsc{ii}\\xspace, Mg\\,\\textsc{ii}\\xspace, H$\\beta$\\xspace, and [O\\,\\textsc{iii}]\\xspace in the MUSE spectrum, but none of these\n  features are seen. The VUDS redshift is likely obtained through a\n  good cross-correlation signal, but we are not able to do so with the\n  {\\tt MARZ} cross-correlation.\n  \\\\\n\n\\end{description}\n\nOf these 10 redshifts that were missed by our redshift determination process,\nfour were obtained via ground-based spectroscopy.  However,\nfor one of these redshifts (MUSE ID 1351), the detected features are not\naccessible with MUSE, and for the other three, we cannot confirm the\npreviously measured redshifts in the MUSE spectra (IDs 4070, 5216,\nand 5494). The rest were taken with sky emission free space-based\nspectroscopy. The four optical spectra ({\\it HST}\/ACS) provide much\nhigher sensitivity, especially in the continua, which facilitates\ndetections of absorption features (Mg\\,\\textsc{ii}\\xspace in this case). An extremely\nbroad feature is also easily washed out in our higher resolution\nspectra without smoothing.  The remaining two redshifts were measured\nwith the near-infrared spectra from space ({\\it HST}\/WFC3). We can\nconfirm the expected features in the MUSE spectra with the known\nredshift, although they are difficult to identify because their\nwavelengths are in the region of many strong sky emission lines.\n\n\n\\subsection{Comparisons with photometric redshifts}\n\nWe compared our MUSE determined spec-$z$ against the published\nphotometric redshifts (photo-$z$) from \\cite{Rafe15} in\nFigure~\\ref{fig:u10_comp_photz}. The merged objects\n(extracted with {\\it HST} priors that cannot be resolved by MUSE) are\nnot included in the plots. \\cite{Rafe15} provide two sets of\nphoto-$z$, based on the Bayesian photometric redshift (BPZ) estimation\n\\citep{Beni00} and the EAZY software \\citep{Bram08}.\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_bpz_vs_muse-z.pdf}\n    \\includegraphics{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_eazy_vs_muse-z.pdf}}\n  \\caption{ Comparisons of redshifts measured from the MUSE data in\n    \\textsf{udf-10}\\xspace $+$ \\textsf{mosaic}\\xspace compared with photometric redshifts, computed\n    with BPZ (left) and EAZY (right). The redshifts with\n    $\\rm CONFID \\geq 2$ and $\\rm CONFID = 1$ are plotted with the red\n    and gray circles, respectively. }\n  \\label{fig:u10_comp_photz}\n\\end{figure}\n\nAlthough MUSE spec-$z$ and photo-$z$ agree well with both BPZ and EAZY\nup to $z \\approx 3$, a systematic difference appears at $z \\gtrsim 3$.\nBoth of the photo-$z$ measurements tend to underestimate the redshift.\nAll of the MUSE redshifts at $z > 3$ are identified by Ly$\\alpha$\\xspace, which is\nknown to cause the apparent offset toward higher redshift \\citep[of a\nfew hundred $\\rm km\\,s^{-1}$;][]{Shap03,Stei10,Hash13} attributed to\ngalactic-scale outflows or absorption by the intergalactic\nmedium. However, this Ly$\\alpha$\\xspace apparent offset is not the major cause of\nthe photo-$z$ offset because the median of the measured offsets\n($\\Delta z$) between MUSE-$z$ ($\\rm CONFID \\geq 2$) and photo-$z$ is\nsignificantly larger than the expected Ly$\\alpha$\\xspace velocity offset\n($\\approx -0.13$ and $\\approx -0.32$ for BPZ and EAZY, respectively).\nIn Figure~\\ref{fig:u10_comp_photz_diff}, the redshift difference is\nplotted against the {\\it HST} F775W magnitude. It is clear that the\nphoto-$z$ measurements are biased at the faint end.\n\nFurther analyses and discussions including the redshift completeness\ncan be found in Paper~III. In this paper, we checked the\n  catastrophic redshift outliers and systematic biases.  We\n  found that changes in model of treatments of inter-galactic medium\n  (IGM) absorption for Ly$\\alpha$\\xspace-forest and Lyman continuum absorption can\n  reduce the photo-$z$ and MUSE-$z$ discrepancy. We also adopted\nphotometric redshifts from the {\\tt BEAGLE} software \\citep{Chev16}.\n\n\\begin{figure}\n  \\resizebox{\\hsize}{!}\n  {\\includegraphics{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_dz_bpz_muse-z_vs_mag.pdf}\n    \\includegraphics{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_dz_eazy_muse-z_vs_mag.pdf}}\n  \\caption{ Difference in the photometric redshift and MUSE\n    redshift (\\textsf{udf-10}\\xspace $+$ \\textsf{mosaic}\\xspace) against the F775W magnitude. The\n    redshifts with $\\rm CONFID = 1$ are plotted with gray\n    circles. The secure redshifts ($\\rm CONFID \\geq 2$) are separated\n    into two groups: $z < 3$ (red circles) and $z \\geq 3$ (blue\n    squares).}\n  \\label{fig:u10_comp_photz_diff}\n\\end{figure}\n\n\n\n\n\\subsection{Color selections of high-z galaxies}\n\nThe presence of the 912${\\rm \\AA}$\\xspace Lyman break enables selection of\nhigh-redshift galaxies using their colors. Here we compared MUSE-$z$\nto some commonly used color-color selection diagrams to discuss the\nfraction of successful color selections. The combination of the MUSE\nwavelength coverage and the {\\it HST} filter sets observed in our deep\nfield facilitates some tests on the color selection technique of\n$2 \\lesssim z \\lesssim 6$ candidate galaxies. Known stars are excluded\nfrom all of these plots.\n\n\\begin{figure*}\n  \\begin{center}\n    \\resizebox{\\hsize}{!}\n    {\\includegraphics[width=0.98\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_hst_dropouts_F336W_model_err_nohiz.pdf}\n      \\includegraphics[width=0.75\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_hst_dropouts_F336W_hist.pdf}}\n    \\caption{ {\\bf [Left]} Color-color diagram of $F336W - F435W$\n      vs.  $F435W - F606W$ to select $z \\sim 2.7$ galaxies (F336W\n      dropouts). Only the MUSE spectroscopic redshifts with\n        $\\rm CONFID \\geq 2$ are plotted. The galaxies in the targeted\n      redshift range ($2.3 \\leq z \\leq 3.0$) are shown with red\n      open squares. The violet, blue, green, gold, orange, magenta,\n      and red filled circles, with sizes from small to large,\n      indicate galaxies in $0 \\leq z < 4$ with steps\n      of $\\Delta z=1$, respectively.  The model tracks of the\n        starburst template \\citep[SB1;][]{Kinn96} and the CWW\n        templates \\citep[Scd and E,][]{Cole80} are overlaid. The\n        crosses on the model tracks indicate redshifts from $z=0$ in\n        increments of $\\Delta z=1$.  {\\bf [Right]} The counts of\n      galaxies in each redshift bin (only the galaxies detected in all\n      of the F336W, F435W, and F606W bands are used here). The empty\n      bars are all of the galaxies with MUSE redshifts and the filled\n      bars are a subset of the MUSE redshifts lying within the F336W dropout\n      selection boundary (the left panel).  The percentages shown\n      above each bar are the fractions of the galaxies within the\n      color-color boundary for each bin.  The vertical dashed blue\n      lines indicate the targeted redshift range,\n      $2.3 \\leq z \\leq 3.0$. The percentage indicated in between these\n      lines in blue is the fraction of the galaxies within the\n      color-color boundary for the targeted redshift range. }\n    \\label{fig:F336W_dropout}\n  \\end{center} \n\\end{figure*}\n\nIn Figure~\\ref{fig:F336W_dropout} (left panel), we overplot all of the\nMUSE-$z$ on the F336W dropout selection diagram ($\\rm F336W - F435W$\nversus  $\\rm F435W - F606W$). The bandwidth of F336W favors Lyman break\ngalaxies (LBGs) at $2.3 \\lesssim z \\lesssim 3.0$.  Although\n\\cite{Hath10} and \\cite{Tepl13} also show the F225W and F275W dropout\nselections, we exclude these from this paper because they\npreferentially select $z \\sim 1.7$ and $z \\sim 2.1$, respectively,\nwhich is a range in which MUSE does not recover a large number of\nredshifts.  We follow the selection criteria of \\cite{Hath10}, i.e.,\n\\\\\n\n\\noindent\nF336W dropouts:\n\\[\n  \\begin{cases}\n    & {\\rm F336W - F435W > 0.8} \\\\\n    & {\\rm F435W - F606W < 1.2} \\\\\n    & {\\rm F435W - F606W > -0.2} \\\\\n    & {\\rm F336W - F435W > 0.35 + [1.3 \\times (F435W - F606W)]}\n  \\end{cases}\n\\]\n\nThe original selection conditions also apply a magnitude cut of\n${\\rm F435W \\leq 26.5}$~mag and S\/N limits of ${\\rm S\/N(F435W) > 3}$,\n${\\rm S\/N(F336W) < 3}$, ${\\rm S\/N(F275W) < 1}$, and\n${\\rm S\/N(F225W) < 1}$ to securely find high-$z$ candidates. We do not\nadopt these limits because our purpose here is to demonstrate where\nour galaxies with measured redshift lie in the diagrams. However, we\ndo require solid detections, and thus no limit is shown in the\ndiagrams.\n\nThe fraction of galaxies that meet these selection criteria in each\nredshift bin is shown in the right panel of\nFigure~\\ref{fig:F336W_dropout}. This plot indeed identifies galaxies\nat $2.3 \\lesssim z \\lesssim 3.0$ among the galaxies with secure\nredshift ($81\\%$). If we limit to $2.5 < z < 3.0$, $93\\%$ of the\ngalaxies in this redshift range are classified as F336W dropouts.\nThose missed (the filled green circles in the left panel of\nFigure~\\ref{fig:F336W_dropout}) lie very close to the selection\nboundary where lower-$z$ galaxies start to get intermingled, in\nparticular around $\\rm 0.5 \\lesssim F336W - F435W \\lesssim 0.8$ and\n$\\rm -0.2 \\lesssim F435W - F606W \\lesssim 0.4$.  There is only one\ninterloper from galaxies at $0 \\leq z < 1$ at\n$\\rm F336W - F435W = 0.93$ and $\\rm F435W - F606W = 0.27$, almost on\nthe borderline.  This object is MUSE ID 2537 ($z=0.717$) whose\nredshift is identified by H$\\delta$\\xspace and [O\\,\\textsc{iii}]\\xspace$\\lambda\\lambda4959,5007$.\n\nFor even higher redshifts, we utilize the F435W, F606W, and F775W\ndropout techniques to inspect $z \\sim 3.5$, $z \\sim 5.0$, and\n$z \\sim 6.0$ galaxy selections, respectively \\citep{Star09,Star10}, as\nshown in Figure~\\ref{fig:hiz_dropout}. The selection boundaries are\nlisted below.  Again, here we do not include any S\/N cuts on the\ndropout conditions.\n\\\\\n\n\\noindent\nF435W dropouts:\n\\[\n  \\begin{cases}\n    & {\\rm F435W - F606W > 1.1} \\\\\n    & {\\rm F606W - F850LP < 1.6} \\\\\n    & {\\rm F435W - F606W > 1.1 + F606W - F850LP}\n  \\end{cases}\n\\]\n\n\\noindent\nF606W dropouts:\n\\[\n  \\begin{cases}\n    & {\\rm F606W - F775W > 1.2} \\\\\n    & {\\rm F775W - F850LP < 1.3} \\\\\n    & {\\rm F606W - F775W > 1.47 + (F775W - F850LP) ~ or ~ 2.0}\n  \\end{cases}\n\\]\n\n\\noindent\nF775W dropouts:\n\\[\n  \\begin{cases}\n    & {\\rm F775W - F850LP > 1.3}\n  \\end{cases}\n\\]\n\nIn all of these dropout selections, the targeted dropout galaxies are\nwell captured. No low-$z$ interloper is found in any of these\nselections.  The fraction of galaxies that lie within the boundaries\nis much smaller compared the F336W selection\n(Figure~\\ref{fig:hiz_dropout_hist}). This fraction also decreases from\nthe lower-$z$ to higher-$z$ dropout selections ($41\\%$, $34\\%$, and\n$22\\%$ for the F435W, F606, and F775W dropouts, respectively). If we\nlimit the redshift bin to $3.5 < z < 4.0$, which is the most sensitive\nredshift range for the F435W filter to detect the dropouts, then\n$81\\%$ of galaxies in this redshift bin meet the dropout\ncriteria. Similarly, for selecting F606W and F775W dropouts, the\nfractions are highest ($67\\%$ for both) in the redshift bins of\n$5.0 < z < 5.5$ and $6.0 < z < 6.5$. Although error bars are not shown\nin the diagrams (in order to make them readable), the errors of the\nobjects outside of the selection boundaries are almost always smaller\nthan those in the boundaries. Their positions are reliable in the\ncolor-color planes.\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=0.98\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_hst_dropouts_hiz_model_err__new.pdf}\n    \\caption{ Similar to the left panel of\n      Figure~\\ref{fig:F336W_dropout} but selecting $z \\sim 3.5$\n      (F435W dropouts), $z \\sim 5.0$ (F606W dropouts), and\n      $z \\sim 6.0$ (F775W dropouts), shown in the left, middle, and\n      right panels, respectively. For the F435W and\n          F606W dropouts, we do not plot the objects at $z > 5.0$\n          and $z > 6.0$, respectively. The galaxies\n      indicated with the red squares are not in the full targeted\n      redshift range, but are restricted to narrower ranges (see also\n      Figure~\\ref{fig:hiz_dropout_hist}).  The solid and dashed black\n      lines are the boundaries of the selection criteria from\n      \\cite{Star09,Star10}. The dashed black lines are shown because\n      they are at the outside of the plotting regions in\n      \\cite{Star09,Star10}.  The solid red lines are our empirically\n      redefined new selection boundaries to gain more galaxies in the\n      targeted redshift ranges. The model tracks for the F850LP\n        magnitude in the F775W dropout diagram is computed for an\n        absolute magnitude of $-22$~mag. The 5$\\sigma$ limit of the\n        F850LP magnitude is 28.9 mag \\citep{Rafe15}.}\n  \\label{fig:hiz_dropout}\n  \\end{center} \n\\end{figure*}\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=0.98\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_hst_dropouts_hiz_hist.pdf}\n    \\caption{ Similar to the right panel of\n      Figure~\\ref{fig:F336W_dropout} but for investigating the\n      fractions of galaxies that meet the dropout conditions at\n      $z \\sim 3.5$, $z \\sim 5.0$, and $z \\sim 6.0$, shown in the\n      left, middle, and right panels, respectively.  }\n  \\label{fig:hiz_dropout_hist}\n  \\end{center} \n\\end{figure*}\n\nFrom these results, we can empirically adjust the selection boundary\nto increase the fraction of the candidate galaxies at the targeted\nredshift ranges. Based on visual modifications, we relax the selection\nconditions based on our MUSE redshifts as follows:\n\\\\\n\n\\noindent\nRedefined F435W dropouts from MUSE-$z$:\n\\[\n  \\begin{cases}\n    & {\\rm F435W - F606W > 0.7} \\\\\n    & {\\rm F606W - F850LP < 1.6} \\\\\n    & {\\rm F435W - F606W > 0.7 + F606W - F850LP} \\\\\n  \\end{cases}\n\\]\n\n\\noindent\nRedefined F606W dropouts from MUSE-$z$:\n\\[\n  \\begin{cases}\n    & {\\rm F606W - F775W > 1.0 + 0.89 \\times (F775W - F850LP) ~ or ~ 2.0} \\\\\n  \\end{cases}\n\\]\n\n\\noindent\nRedefined F775W dropouts from MUSE-$z$:\n\\[\n  \\begin{cases}\n    & {\\rm F775W - F850LP > -0.17 \\times F850LP + 5.5 ~ or ~ 1.3} \\\\\n  \\end{cases}\n\\]\n\nThese updated boundaries are shown with the solid red lines in\nFigure~\\ref{fig:hiz_dropout}. The histograms of the selected galaxies\nwith these new criteria are shown in\nFigure~\\ref{fig:hiz_dropout_hist_new}. We are trying this experiment,\nbut not all of our sample galaxies in the diagrams (in particular\nthose lying between the original and new boundaries) necessarily show\na clear Lyman break because we only use Ly$\\alpha$\\xspace emission to determine\ntheir redshifts.\n\n\\begin{figure*}\n  \\begin{center}\n    \\includegraphics[width=0.98\\hsize]{figures\/combined_udf10_c042_e031_withz_iter6_mosaic_c042_e030_withz_iter8_hst_dropouts_hiz_hist__new.pdf}\n    \\caption{ Same plots as Figure~\\ref{fig:hiz_dropout_hist}, but\n      for our empirically redefined new selection boundaries shown in\n      Figure~\\ref{fig:hiz_dropout} with the red lines.}\n  \\label{fig:hiz_dropout_hist_new}\n  \\end{center} \n\\end{figure*}\n\nWe allow a slightly bluer $\\rm F435W - F606W$ color for selecting\n$z \\sim 3$ galaxies (F435W dropouts) because there is a crowd of\n$3 \\leq z < 4$ galaxies lying right below the original boundary. The\nnew boundary significantly improves the $z \\sim 3$ galaxy selection\nwithout picking up contaminants. In particular, now it finds\n$\\approx 40\\%$ more galaxies at $3.0 < z < 3.5$.\n\nFor the F606W dropouts, the targeted $z \\sim 5$ galaxies also extend\ntoward bluer $\\rm F606W - F775W$ colors. Thus, we lower the color\nlimit of $\\rm F606W - F775W$. In addition, we remove the border at\n$\\rm F775W - F850LP = 1.3$ because some of galaxies at $5.5 < z < 6.0$\nhave redder colors. These new conditions successfully increase the\nfraction of $z \\sim 5$ galaxies lying within the boundary from $34\\%$\nto $59\\%$ and now capture $88\\%$ of the galaxies in the\n$5.0 < z < 5.5$ bin and $66\\%$ in the $4.5 < z < 5.0$ bin.  This also\nadds one low-$z$ interloper and two galaxies at\n$3.0 < z < 4.0$\\,\\footnote{The low-$z$ interloper is MUSE ID 873\n  ($z=0.66$). The other galaxies are MUSE IDs 493 ($z=3.18$) and 7377\n  ($z=3.42$).}.\n\nIt is more difficult to enhance the $z \\sim 6$ galaxy selection. We\ndouble the number of galaxies at $5 \\lesssim z \\lesssim 6.5$ to meet\nthe new conditions, but this also increases the number of galaxies at\nlower redshifts. With the newly suggested criteria, we cover galaxies\nat $6.0 < z < 6.5$ well ($83\\%$) and improve a lot for galaxies\nat $5.5 < z < 6.0$ (from $29\\%$ to $63\\%$). There are more\ngalaxies below $z=5$ in the new selection box, but only one is below\n$z=1$ (MUSE ID 2478 $z=0.73$). The remaining eight lower-$z$ galaxies\nhave redshifts between $3$ and $5$.  In total, we successfully select\n29 galaxies at $5.0 < z < 6.5$. It is challenging to purely select\ngalaxies at $5.0 < z < 5.5$ with this color-magnitude diagram because\ngalaxies at $3 < z < 5$ have similar colors and magnitudes.  We only\ncollect two galaxies ($5.3\\%$) in this bin using the new diagram\n(0 in the original one).\n\nThere are still significant numbers of high-$z$ galaxies lying outside\nof the selection boxes. We have checked these galaxies individually to\ndetermine whether their bluer colors are due to blending or\nphotometric issues. In the F435W dropout diagram, we find 10 objects\nwith $3.5 \\leq z \\leq 4.0$ for $\\rm F435W-F606W < 0.9$.  Among these\nobjects, one may be blended with a nearby object, one is a dusty\ngalaxy (ID 6672, see Figure~\\ref{fig:dropout_outliers}) and one may\nhave an ambiguous Ly$\\alpha$\\xspace feature. In the remaining seven objects, two\nhave good photometries lying near the revised selection box, and five\nhave $S\/N {\\rm (F435W)} < 5$ but the error budgets of their\nphotometries make them agree with the revised selection box. For the\nF606W dropout diagram, we checked 12 galaxies with\n$4.5 \\leq z \\leq 5.5$ for $\\rm 0 \\lesssim F606W-F775W \\lesssim 0.4$.\nThere is one (ID 6324, Figure~\\ref{fig:dropout_outliers}) whose\nphotometries may be contaminated by a neighboring object, but the rest\nof the objects have good photometries (seven with\n$S\/N {\\rm (F606W)} \\gtrsim 5$ and four with $S\/N {\\rm (F606W)} > 10$;\ne.g., ID 2727, Figure~\\ref{fig:dropout_outliers}). However, we find\nthat all of these objects have the Ly$\\alpha$\\xspace emission line lying in the\nF606W coverage, which is likely to produce bluer $\\rm F606W-F775W$\ncolors. This effect is visible in the diagrams in\nFigures~\\ref{fig:hiz_dropout_hist_new} and\n\\ref{fig:hiz_dropout_hist}. Allowing the boundary of $\\rm F606W-F775W$\nto be bluer increases more lower-z selections in the targeted redshift\nrange of $4.2 < z < 5.8$. In fact, this trend is also seen in the 13\noutlier galaxies with $5.5 \\leq z \\leq 6.5$ for\n$\\rm 0 \\lesssim F775W-F850LP \\lesssim 0.5$ in the F775W dropout\ndiagram (e.g., ID 3238, Figure~\\ref{fig:dropout_outliers}). All of\nthese galaxies have $S\/N {\\rm (F775W)} \\gtrsim 5$ and the detected\nLy$\\alpha$\\xspace line is lying in the F775W coverage; three of these galaxies are\nstarting to enter the F850LP coverage, but at the wavelength where the\nF850LP filter throughput is still $< 50\\%$. Similar to the outliers in\nthe F606W dropout diagram, the enhanced blue $\\rm F775W-F850LP$ colors\nof these objects can be ascribed to Ly$\\alpha$\\xspace emission contributing to the\nF775W flux.  The newly defined selection boundaries help to increase\nthe completeness of $3 \\lesssim z \\lesssim 7$ galaxy selections. It is\nnot surprising that we find outliers as color selection criteria are\nnot designed for comprehensive detection.  Although rest-frame UV\nspectra of high-$z$ galaxies are affected by IGM transmission, the\nhigh equivalent width of Ly$\\alpha$\\xspace emission also causes bluer colors, which\nmakes the apparent Ly$\\alpha$\\xspace break less significant (particularly in the\nF606W and F775W dropout selections).  In addition, color selections\nare known to exclude some high-$z$ passive and dust-obscured galaxies\n\\citep[e.g.,][]{Dadd04, Redd05, vanD06}.  The outliers in the dropout\ndiagrams are also found in the VVDS survey \\citep{LeFe05c, Palt07}.\n\nHere we limit to the comparisons of measured redshifts against the\ndropout selections. The galaxy populations plotted here are likely to\nbe different from the traditional dropout selected galaxies because\nthe majority of the redshifts are determined with emission lines.\nHowever, candidate galaxies lying between the original and new\nselection conditions may be able to serve as second highest priority\ntargets for multi-object spectroscopy to increase the efficiency in\nfinding high-$z$ galaxies.  Also, we do not adopt any S\/N or magnitude\ncuts here.  For future work, we will discuss in more detail why some\ngalaxies fail to meet the dropout conditions (with respect to their\nspectroscopic and photometric redshifts), the relationship between\nLy$\\alpha$\\xspace emitters and dropout galaxies and their physical properties, and\nthe evolution of the fraction of Ly$\\alpha$\\xspace emitters.\n\n\n\\begin{figure}\n  \\begin{center}\n    \\includegraphics[width=0.98\\hsize]{figures\/outside_dropout_seds__F435W_dropouts_06672.pdf}\n    \\includegraphics[width=0.98\\hsize]{figures\/outside_dropout_seds__F606W_dropouts_06324.pdf}\n    \\includegraphics[width=0.98\\hsize]{figures\/outside_dropout_seds__F606W_dropouts_02727.pdf}\n    \\includegraphics[width=0.98\\hsize]{figures\/outside_dropout_seds__F775W_dropouts_03238.pdf}\n    \\caption{ Examples of the outliers in the F435W\n          (top panel), F606W (two middle panels), and F775W (bottom\n          panel) dropout selections. In each panel, a spectral energy\n          distribution using the {\\it HST} photometries is shown at\n          the top left, a zoomed-in part of the Ly$\\alpha$\\xspace detected\n          wavelength region is presented at the top right, and\n          thumbnail {\\it HST} images of the object are shown at\n          the bottom. The object ID, MUSE spec-$z$, and the colors of\n          each galaxy can be found at the top of each panel. }\n  \\label{fig:dropout_outliers}\n  \\end{center} \n\\end{figure}\n\n\n\n\n\n\\section{Summary and conclusions}\\label{sec:summary}\n\nWe have conducted a deep spectroscopic survey in the HUDF with\nMUSE. The $3\\arcmin \\times 3\\arcmin$ deep survey ($\\approx 10$~hour\ndepth, the \\textsf{mosaic}\\xspace) region almost covers the entire HUDF with the\n$1\\arcmin \\times 1\\arcmin$ ultra deep survey ($\\approx 30$~hour depth,\n\\textsf{udf-10}\\xspace) region enclosed.  We used two different spectral extraction\nmethods for the redshift identification: the {\\it HST} prior continuum\nselected objects \\citep[based on the UVUDF catalog;][]{Rafe15} and the\nemission line objects selected directly in the cubes without prior\ninformation.  We visually inspected these spectra to determine\nredshifts via redshift analysis software.  We also measured the line\nfluxes for the objects with redshifts determined. Along with this\npaper, we release the redshift and line flux catalogs.  Here we\nsummarize our findings in the redshift assessments:\n\n\\begin{enumerate}\n\n\\item In \\textsf{udf-10}\\xspace, we obtained $\\UDFzALLCtwo$ secure redshifts\n  ($\\rm CONFID \\geq 2$) in the redshift range $0.21 \\leq z \\leq 6.64$.\n  Among these, $\\UDFzORGonlyCtwo$ are not in the {\\it HST} UVUDF\n  catalog that we used for the prior based source extraction. The\n  majority of these redshifts (28) are Ly$\\alpha$\\xspace emitters. When we only count {\\it\n    HST} prior extracted objects, we managed to retrieve $\\UDFzHSTCtwo$\n  redshifts out of $\\UDFHSTpri$ continuum selected objects\n  ($26\\%$). We reach a $50\\%$ completeness at $26.5$~mag (F775W), and\n  the completeness stays around $20\\%$ up to $28-29$~mag.\n\n\\item In the \\textsf{mosaic}\\xspace, in addition to investigating all of the emission\n  line galaxies, we performed visual inspections for those with\n  $\\rm F775W \\leq 27$~mag for the continuum selected galaxies. This\n  arbitrary cut is made based on our findings for \\textsf{udf-10}\\xspace that\n  $\\approx 90\\%$ of galaxies at $z < 3$ are brighter than this\n  magnitude. At $z > 3$, although most of the galaxies are fainter\n  than $27$~mag, we are able to detect $70\\%$ of these galaxies with\n  emission features directly in the data cube without any continuum\n  information.  Together with the direct search for emission line\n  objects in the cube, this helps to increase the efficiency of\n  redshift determination.\n\n\\item Out of the $\\MOSHSTpriMAGcut$ continuum selected objects with\n  $\\rm F775W \\leq 27$~mag in the \\textsf{mosaic}\\xspace field, we determined secure\n  redshift of $\\MOSzHSTCtwo$ objects. When we included the emission\n  line detected objects (regardless of magnitude), we obtained\n  $\\MOSzALLCtwo$ unique redshifts with confidence. The $50\\%$\n  completeness with respect to F775W mag is at $25.5$~mag.\n\n\\item In the overlapping region of \\textsf{udf-10}\\xspace and \\textsf{mosaic}\\xspace, all of the\n  redshifts measured in  \\textsf{mosaic}\\xspace were also measured in the \\textsf{udf-10}\\xspace\n  with at least the same confidence level or higher, except one object\n  lying at the edge of the \\textsf{udf-10}\\xspace field and four objects detected\n  only through emission lines in the \\textsf{mosaic}\\xspace cube.  On the other hand,\n  78 secure redshifts were missing in \\textsf{mosaic}\\xspace when compared with\n  \\textsf{udf-10}\\xspace. There are a few discrepancies but the measurements in the\n  \\textsf{udf-10}\\xspace, which has higher S\/N spectra, are usually more convincing.\n\n\\item Among $\\COMBHSTpri$ unique continuum selected galaxies in the\n  entire MUSE UDF survey region (\\textsf{udf-10}\\xspace $+$ the \\textsf{mosaic}\\xspace), we recovered\n  redshifts for $\\COMBzHSTCtwo$ (15\\%) objects. We also found\n  $\\COMBzORGonlyCtwo$ objects only via emission line searches\n  directly in the data cubes (no counterparts in the {\\it HST} UVUDF\n  catalog). Thus, in total, we obtained $\\COMBzALLCtwo$ unique\n  redshifts with high confidence.\n\n\\item Compared with the {\\it HST} beamsize, the MUSE data suffer from\n  source confusion. However, we were able to partly solve this issue\n  when an emission\/absorption feature is identified. The location of\n  the detected feature in its narrowband image can be associated with\n  the location of the corresponding source in the {\\it HST}\n  images. There are also a few cases where two (or more) galaxies are\n  completely blended even with the {\\it HST} resolution, but we\n  identified both (or all) of these cases with two (or more) different\n  sets of emission lines at different redshifts.  Finally, among\n  $\\COMBzALLCone$ objects with MUSE-$z$ ($\\rm CONFID \\geq 1$), there\n  are 79 that remain as merged objects.\n\n\\item Of the previously known $\\COMBprevz$ spectroscopic redshifts in\n  our survey field, we recovered all except 10 objects. Of these, four\n  were measured in spectra taken with ground-based\n  spectroscopy. However, for one of these redshifts, the detected\n  features are at wavelengths shorter than the blue end of MUSE, and\n  for the other three, we cannot confirm the previously measured\n  redshifts based on the expected detectable features. The other six\n  spectra were taken with {\\it HST} grism spectroscopy. The features\n  used for determining their redshifts either were a very broad\n  absorption feature or lay on the red side of MUSE spectra where sky\n  emission dominates.\n\n\\item The comparison of MUSE-$z$ and photometric redshifts revealed\n  that they agree well up to $z \\approx 3$. At higher redshift, an\n  offset appears. Although most of the $z > 3$ redshifts are\n  determined with Ly$\\alpha$\\xspace, the median offset between MUSE-$z$ and\n  photo-$z$ is much larger than the expected Ly$\\alpha$\\xspace velocity offset to\n  the systematic redshift due to gas motion. We observed a trend that\n  the photo-$z$ offset increases with fainter continuum emission in\n  {\\it HST}\/F775W.\n\n\\item We investigated all of the galaxies with secure MUSE-$z$ in some\n  common color selection (dropout) diagrams of high-$z$ galaxies. With\n  the F336W dropout selection criteria, $81\\%$ of targeted\n  $z \\sim 2.7$ galaxies are captured. This fraction decreases for\n  higher-$z$ selections: $41\\%$ for F435W dropouts ($z \\sim 3$),\n  $34\\%$ for F606 dropouts ($z \\sim 5$), and $22\\%$ for F775W dropouts\n  ($z \\sim 6$). We empirically redefined the selection boundaries to\n  increase the fractions to $68\\%$, $59\\%$, and $45\\%$, but our galaxy\n  populations are likely to be different from the traditional dropout\n  selected galaxies (continuum selected) because the majority of the\n  redshifts were determined using emission lines.\n\n\\end{enumerate}\n\nWith deep MUSE spectroscopic observations in HUDF, we dramatically\nimprove the redshift completeness.  The improvements are not only the\nincrease in number from $\\COMBprevz$ to $\\COMBzALLCtwo$, but also the\ncoverage of the redshift range well beyond $z > 3$ and depths up to\nthe 30th magnitude (F775W).  Together with existing large telescopes\nand planned future observatories such as {\\it JWST} and 30 m class\ntelescopes, it opens new horizons for exploring the early universe.\n\nIn the near future, we plan further advances in the analysis, such as\ninvestigating spectra extracted at the positions of continuum selected\nobjects fainter than $27$~mag in the \\textsf{mosaic}\\xspace, fine tuning {\\tt ORIGIN}\n(and {\\tt MUSELET}) to further improve the direct detection of\nemission lines without enhancing spurious sources, upgrading {\\tt\n  MARZ} to increase the accuracy of automated redshift determinations\nwith newly built spectral templates, and developing deblending\ntechniques based on prior information.\n\n\n\n\\begin{acknowledgements} \n\n  The authors are grateful for useful suggestions by the referee that\n  improved the manuscript.  We thank Laure Piqueras for her support\n  with analysis and observation software.  This work is supported by\n  the ERC advanced grant 339659-MUSICOS (R. Bacon).  JB acknowledges\n  support by Funda\\c{c}\\~ao para a Ci\\^encia e a Tecnologia (FCT)\n  through national funds (UID\/FIS\/04434\/2013) and by FEDER through\n  COMPETE2020 (POCI-01-0145-FEDER-007672) and was in part supported by\n  FCT through Investigador FCT contract IF\/01654\/2014\/CP1215\/CT0003.\n  JR and BC acknowledge support from the ERC starting grant\n  336736-CALENDS. TC acknowledges support of the ANR FOGHAR\n  (ANR-13-BS05-0010-02), the OCEVU Labex (ANR-11-LABX-0060), and the\n  A*MIDEX project (ANR-11-IDEX-0001-02) funded by the\n  ``Investissements d'avenir'' French government program managed by\n  the ANR. JS acknowledges support of the ERC Grant agreement\n  278594-GasAroundGalaxies. LW acknowledges funding by the Competitive\n  Fund of the Leibniz Association through grant SAW-2015-AIP-2. RAM\n  acknowledges support by the Swiss National Science Foundation.\n\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\nThe multiple signal classification (MUSIC) algorithm \\cite{Schmidt}, \\cite{Viberg} is a well established technique for estimating direction-of-arrivals (DOAs) of signals impinging on an array of sensors. It operates by finding DOAs with corresponding array steering vectors that have minimal projections onto the empirical noise subspace, whose spanning vectors are obtained via eigendecomposition of the sample covariance matrix (SCM) of the array outputs.\n\nIn the presence of outliers, possibly caused by heavy-tailed impulsive noise, the SCM poorly estimates the covariance of the array outputs, resulting in unreliable DOAs estimates. In order to overcome this limitation, several MUSIC generalizations have been proposed in the literature that replace the SCM with robust association or scatter matrix estimators, for which the empirical noise subspace can be determined from their eigendecomposition. \n\nUnder the assumption that the signal and noise components are jointly $\\alpha$-stable processes \\cite{AlphaStable}, it was proposed in \\cite{ROCMUSIC} to replace the SCM with empirical covariation matrices that involve fractional lower-order statistics. Although $\\alpha$-stable processes are appropriate for modelling impulsive noise  \\cite{AlphaStable2}, the assumption that the signal and noise components are jointly $\\alpha$-stable is restrictive. In \\cite{LIU}, a less restrictive approach considering circular signals contaminated by additive $\\alpha$-stable noise was developed that replaces the SCM with matrices comprised of empirical fractional-lower-order-moments. Although this approach is less restrictive than the one proposed in \\cite{ROCMUSIC}, violation of the signal circularity assumption, e.g., in the case of BPSK signals, results in poor DOA estimation performance \\cite{LIU}.  \n\nIn \\cite{swami1997}-\\cite{swami2002} it was proposed to apply MUSIC after passing the data through a zero-memory non-linear (ZMNL) function that suppresses outliers by clipping the amplitude of the received signals. The ZMNL approach has simple implementation having low computational complexity, and unlike the methods proposed in \\cite{ROCMUSIC}, \\cite{LIU} it does not require restrictive assumptions on the signal and noise probability distributions. Although the ZMNL preprocessing may result in more accurate DOA estimation than the methods in \\cite{ROCMUSIC}, \\cite{LIU}, it may not preserve the noise subspace which can lead to performance degradation \\cite{Sadler}. \n\nUnder the assumption of normally distributed signals in heavy tailed noise, a similar approach was proposed in \\cite{Zoubir} that is based on successive outlier trimming until the remaining data is Gaussian. Normality  of the data is tested using the Shapiro-Wilk's test. Similarly to the ZMNL preprocessing, the noise subspace may not be preserved after the trimming procedure. Moreover, the key assumption that the signals are Gaussian may not be satisfied in some practical scenarios. \n\nIn \\cite{Visuri}, a different MUSIC generalization was proposed that replaces the SCM with empirical sign or rank covariances. Using only the assumption of spherically distributed noise, it was shown that convergent estimates of the noise subspace can be obtained from their eigendecomposition.  The influence functions \\cite{Hampel} of the empirical sign and rank covariance matrices, that measure their sensitivity to an outlier, are bounded \\cite{croux2002sign}. In other words, these estimators are B-robust \\cite{Hampel}. However, it can be shown that the Frobenius norms of their matrix valued influence functions do not approach zero as the magnitude of the outlier approaches infinity, i.e., they do not reject large outliers. Indeed, the empirical sign and rank covariance matrices have influence functions with constant Frobenius norms for spherically symmetric distributions. \n\nIn \\cite{Visa}, robust M-estimators of scatter \\cite{Marrona}, \\cite{Huber}, such as the maximum-likelihood, Huber's \\cite{Huber}, and Tyler's \\cite{Tyler} M-estimators, extended to complex elliptically symmetric (CES) distributions, were proposed as alternatives to the SCM. Under the class of CES distributions having finite second-order moments, these estimators provide consistent estimation of the covariance up to a positive scalar, resulting in consistent estimation of the noise subspace. Although this approach can provide robustness against outliers with negligible loss in efficiency when the observations are normally distributed, it may suffer from the following drawbacks. First, when the observations are not elliptically distributed, M-estimators may lose asymptotic consistency \\cite{Hallin}, which may lead to poor estimation of the noise subspace. Second, M-estimators of scatter are often computed using an iterative fixed-point algorithm that converges to a unique solution under some regularity conditions. Each iteration involves matrix inversion which may be computationally demanding in high dimensions, or unstable when the scatter matrix is close to singular. Moreover, although the influence functions of M-estimators may be bounded, they may not behave well for large norm outliers that can negatively affect estimation performance. Indeed, similarly to the method of \\cite{Visuri}, Tyler's scatter M-estimator does not reject large outliers and its matrix valued influence function \\cite{Visa} has constant Frobenius norm for spherically symmetric distributions.\n\nIn \\cite{LP_MUSIC}, a robust MUSIC generalization called $l_{p}$-MUSIC was proposed that estimates the noise subspace by minimizing the $l_{p}$-norm ($1<p<2$) of the residual between the data matrix and its low-rank representation. The $l_{p}$-norm with $p<2$ is less sensitive to outliers than the $l_{2}$-norm. Therefore, $l_{p}$-MUSIC is more robust against impulsive noise as compared to MUSIC that is based on $l_{2}$-norm minimization of the data fitting error matrix \\cite{LP_MUSIC}. However, unlike MUSIC and other robust generalizations, in $l_{p}$-MUSIC \nthe empirical noise subspace is not determined  by solving a simple eigendecomposition problem. Indeed, in \\cite{LP_MUSIC} the non-convex $l_{p}$-norm minimization is performed by alternating convex optimization scheme that may converge to undesired local minima.  \n\nIn this paper, we introduce a new framework for robust MUSIC. The proposed framework, called robust measure-transformed MUSIC (MT-MUSIC), is inspired by a measure transformation approach that was recently applied to canonical correlation analysis \\cite{MTCCA} and independent component analysis \\cite{MTICA}. Robust MT-MUSIC is based on applying a transform to the probability distribution of the data. The proposed transform is defined by a non-negative function, called the MT-function, and maps the probability distribution into a set of new probability measures on the observation space. By modifying the MT-function, classes of measure transformations can be obtained that have useful properties. Under the proposed transform we define the measure-transformed (MT) covariance and derive its strongly consistent estimate, which is also shown to be Fisher consistent \\cite{Cox}. Robustness of the empirical MT-covariance is established in terms of boundedness of its influence function. A sufficient condition on the MT-function that guarantees B-robustness of the empirical MT-covariance is also obtained. \n\nIn robust MT-MUSIC, the SCM is replaced by the empirical MT-covariance. The MT-function is selected such that the resulting empirical MT-covariance is B-robust, and the noise subspace can be determined from the eigendecomposition of the MT-covariance. By modifying the MT-function such that these conditions are satisfied a class of robust MT-MUSIC algorithms can be obtained. \n\nSelection of the MT-function under the family of zero-centered Gaussian functions, parameterized by a scale parameter, results in a new algorithm called Gaussian MT-MUSIC. We show that the empirical Gaussian MT-covariance is B-robust with influence function that approaches zero as the outlier magnitude approaches infinity. Under the additional assumption that the noise component has a spherically contoured distribution, we show that the noise subspace can be determined from the eigendecomposition of Gaussian MT-covariance. Note that this spherically contoured noise distribution assumption is weaker than the standard i.i.d. Gaussian noise assumption. We propose a data-driven procedure for selecting the scale parameter of the Gaussian MT-function. This procedure has the property that it prevents significant transform-domain Fisher-information loss when the observations are normally distributed. \n\nIn this paper, a robust estimate of the number of signals is proposed that is based on minimization of a measure-transformed version of the minimum description length (MDL) criterion \\cite{Wax}. This criterion, called MT-MDL, is obtained by replacing the eigenvalues of the SCM with the eigenvalues of the empirical MT-covariance. We show that under some mild conditions, minimization of the MT-MDL criterion results in strongly consistent estimation of the number of signals regardless the underlying distribution of the data. These conditions are satisfied when the Gaussian MT-function is implemented and the noise component has a spherically contoured distribution. The MT-MDL criterion with the Gaussian MT-function is called the Gaussian MT-MDL.\n\nThe proposed Gaussian MT-MUSIC algorithm is extended to the case of coherent signals impinging on a uniform linearly spaced array (ULA) \\cite{VanTrees}. This extension is carried out through forward-backward spatial smoothing of the empirical Gaussian MT-covariance matrix.  \n\nThe Gaussian MT-MUSIC algorithm and the Gaussian MT-MDL criterion are evaluated by simulations to illustrate their advantages relative to other robust MUSIC and MDL generalizations. We examine scenarios of non-coherent and coherent signals contaminated by several types of spherically contoured noise distributions arising from the compound Gaussian (CG) family. This family encompasses common heavy-tailed distributions, such as the $t$-distribution, the $K$-distribution, and the CG-distribution with inverse Gaussian texture, and have been widely adopted for modeling radar clutter \\cite{CG1}-\\cite{CG4}. \n\nThe paper is organized as follows. In Section \\ref{MUSIC_ALG}, the robust MT-MUSIC framework is presented. In Section \\ref{GMTMUSIC}, the Gaussian MT-MUSIC algorithm is derived. In Section \\ref{MTMDLEST}, we propose a measure-transformed generalization of the MDL criterion for estimating the number of signals.  In Section \\ref{CMTMUSIC}, a spatially smoothed version of the Gaussian MT-MUSIC algorithm for coherent signals is developed. The proposed approach is illustrated by simulation in Section \\ref{Examples}. In Section \\ref{Conclusions}, the main points of this contribution are summarized. The proofs of the propositions and theorems stated throughout the paper are given in the Appendix.\n\\section{Robust measure-transformed MUSIC}\n\\label{MUSIC_ALG}\nIn this section, the robust MT-MUSIC procedure is presented. First, the sensor array model is introduced. Second, a general transformation on probability measures is established. Under the proposed transform, we define the MT-covariance matrix and derive its strongly consistent estimate. Robustness of the empirical MT-covariance is studied by analyzing its influence function. Finally, based on the assumed array model, we propose a robust MT-MUSIC procedure that replaces the SCM with the empirical MT-covariance of the received signals. \n\\subsection{Array model}\nConsider an array of $p$ sensors that receive signals generated by $q<p$ narrowband incoherent far-field point sources with distinct azimuthal DOAs $\\theta_{1},\\ldots,\\theta_{q}$. Under this model, the array output satisfies \\cite{Viberg}:\n\\begin{equation}  \n\\label{ArrayModel}\n{\\bf{X}}\\left(n\\right)={\\bf{A}}{\\bf{S}}\\left(n\\right)+{\\bf{W}}\\left(n\\right),\n\\end{equation}\nwhere $n\\in\\mathbb{N}$ is a discrete time index, ${\\bf{X}}\\left(n\\right)\\in{\\mathbb{C}}^{p}$ is the vector of received signals, ${\\bf{S}}\\left(n\\right)\\in{\\mathbb{C}}^{q}$ is a zero-mean latent random vector, comprised of the emitted signals, with non-singular covariance, and ${\\bf{W}}\\left(n\\right)\\in{\\mathbb{C}}^{p}$ is an additive spatially white noise with zero location parameter. The matrix ${\\bf{A}}\\triangleq\\left[{\\bf{a}}\\left(\\theta_{1}\\right),\\ldots,{\\bf{a}}\\left(\\theta_{q}\\right)\\right]\\in{\\mathbb{C}}^{{p}\\times{q}}$ is the array steering matrix, where ${\\bf{a}}\\left(\\theta\\right)$ is the steering vector of the array toward direction $\\theta$. We assume that the array is unambiguous, i.e., any collection of $p$ steering vectors corresponding to distinct DOAs forms a linearly independent set. Therefore, ${\\bf{A}}$ has full column rank, and identification of its column vectors is equivalent to the problem of identifying the DOAs. We also assume that ${\\bf{S}}\\left(n\\right)$ and ${\\bf{W}}\\left(n\\right)$ are statistically independent and first-order stationary. To simplify notation, the time index $n$ will be omitted in the sequel except where noted.\n\\subsection{Probability measure transform}\nWe define the measure space $\\left({\\mbox{$\\mathcal{X}$}},\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}},P_{\\Xmatsc}\\right)$, where ${\\mbox{$\\mathcal{X}$}}$ is the observation space of a random vector ${\\bf{X}}\\in{\\mathbb{C}}^{p}$, $\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$ is a $\\sigma$-algebra over ${\\mbox{$\\mathcal{X}$}}$, and $P_{\\Xmatsc}$ is a probability measure on $\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$. Let $g:{\\mbox{$\\mathcal{X}$}}\\rightarrow{\\mathbb{C}}$ denote an integrable scalar function on ${\\mbox{$\\mathcal{X}$}}$. The expectation of $g\\left({\\bf{X}}\\right)$ under $P_{\\Xmatsc}$ is defined as\n\\begin{equation}\n\\label{ExpDef}\n{\\rm{E}}\\left[g\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]\\triangleq\\int\\limits_{{\\mbox{\\tiny $\\mathcal{X}$}}}g\\left({\\bf{x}}\\right)dP_{\\Xmatsc}\\left({\\bf{x}}\\right),\n\\end{equation}\nwhere ${\\bf{x}}\\in{\\mbox{$\\mathcal{X}$}}$. The empirical probability measure $\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ given a sequence of samples ${\\bf{X}}\\left(n\\right)$, $n=1,\\ldots,N$ from $P_{\\Xmatsc}$ is specified by\n\\begin{equation}\n\\label{EmpProbMes}\n\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(A\\right)=\\frac{1}{N}\\sum\\limits_{n=1}^{N}\\delta_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}(n)}\\left(A\\right),\n\\end{equation}\nwhere $A\\in\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$, and $\\delta_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}(n)}\\left(\\cdot\\right)$ is the Dirac probability measure at ${\\bf{X}}\\left(n\\right)$ \\cite{Folland}.\n\\begin{Definition}\n\\label{Def1}\nGiven a non-negative function $u:{\\mathbb{C}}^{p}\\rightarrow{\\mathbb{R}}_{+}$ satisfying \n\\begin{equation} \n\\label{Cond}\n0<{{{\\rm{E}}}\\left[u\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]}<\\infty,\n\\end{equation}\na transform on $P_{\\Xmatsc}$ is defined via the relation:\n\\begin{equation}\n\\label{MeasureTransform} \nQ^{\\left(u\\right)}_{\\Xmatsc}\\left(A\\right)\\triangleq{\\rm{T}}_{u}\\left[P_{\\Xmatsc}\\right]\\left(A\\right)=\\int\\limits_{A}\\varphi_{u}\\left({\\bf{x}}\\right)dP_{\\Xmatsc}\\left({\\bf{x}}\\right),\n\\end{equation}\nwhere $A\\in\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$, ${\\bf{x}}\\in{\\mbox{$\\mathcal{X}$}}$, and\n\\begin{equation}\n\\label{VarPhiDef} \n\\varphi_{u}\\left({\\bf{x}}\\right)\\triangleq\\frac{u\\left({\\bf{x}}\\right)}{{{\\rm{E}}}\\left[u\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]}.\n\\end{equation}\nThe function $u\\left(\\cdot\\right)$, associated with the transform ${\\rm{T}}_{u}\\left[\\cdot\\right]$, is called the MT-function.\n\\end{Definition}  \n\\begin{Proposition}[Properties of the transform]\n\\label{Prop1}\nLet $Q^{\\left(u\\right)}_{\\Xmatsc}$ be defined by relation (\\ref{MeasureTransform}). \nThen\n\\begin{enumerate}\n\\item\n\\label{P1}\n$Q^{\\left(u\\right)}_{\\Xmatsc}$ is a probability measure on $\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$.\n\\item\n\\label{P2}\n$Q^{\\left(u\\right)}_{\\Xmatsc}$ is absolutely continuous w.r.t. $P_{\\Xmatsc}$, with Radon-Nikodym derivative \\cite{Folland}:\n\\begin{equation}\n\\label{MeasureTransformRadNik}     \n\\frac{dQ^{\\left(u\\right)}_{\\Xmatsc}\\left({\\bf{x}}\\right)}{dP_{\\Xmatsc}\\left({\\bf{x}}\\right)}=\\varphi_{u}\\left({\\bf{x}}\\right).\n\\end{equation}\n\\item \n\\label{P3} \nAssume that the MT-function $u\\left(\\cdot\\right)$ is strictly positive, and let ${\\bf{g}}:{\\mbox{$\\mathcal{X}$}}\\rightarrow{\\mathbb{C}}^{m}$ denote an integrable function over ${\\mbox{$\\mathcal{X}$}}$.\nIf the covariance of ${\\bf{g}}\\left({\\bf{X}}\\right)$ under $P_{\\Xmatsc}$ is non-singular, then it is non-singular under the transformed probability measure $Q^{\\left(u\\right)}_{\\Xmatsc}$. \n\\end{enumerate} \n[A proof is given in Appendix \\ref{Prop1Proof}]\n\\end{Proposition}\nThe probability measure $Q^{\\left(u\\right)}_{\\Xmatsc}$ is said to be generated by the MT-function $u\\left(\\cdot\\right)$. By modifying $u\\left(\\cdot\\right)$, such that the condition (\\ref{Cond}) is satisfied, virtually any probability measure on $\\mathcal{S}_{{\\mbox{\\tiny $\\mathcal{X}$}}}$ can be obtained. \n\\subsection{The measure-transformed covariance}\nAccording to (\\ref{MeasureTransformRadNik}) the covariance of ${\\bf{X}}$ under $Q^{\\left(u\\right)}_{\\Xmatsc}$ is given by  \n\\begin{equation} \n\\label{MTCovZ}   \n{\\mbox{\\boldmath $\\Sigma$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\triangleq{\\rm{E}}\\left[\\left({\\bf{X}}-{\\mbox{\\boldmath $\\mu$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)\n\\left({\\bf{X}}-{\\mbox{\\boldmath $\\mu$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)^H\\varphi_{u}\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right],\n\\end{equation}\nwhere \n\\begin{equation} \n\\label{MTMean} \n{\\mbox{\\boldmath $\\mu$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\triangleq{\\rm{E}}\\left[{\\bf{X}}\\varphi_{u}\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]\n\\end{equation}\nis the expectation of ${\\bf{X}}$ under $Q^{\\left(u\\right)}_{\\Xmatsc}$. Equation (\\ref{MTCovZ}) implies that ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is a weighted covariance matrix of ${\\bf{X}}$ under $P_{\\Xmatsc}$, with weighting function $\\varphi_{u}\\left(\\cdot\\right)$. Hence, ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ can be estimated using only samples from the distribution $P_{\\Xmatsc}$. By modifying the MT-function $u\\left(\\cdot\\right)$, such that the condition (\\ref{Cond}) in definition \\ref{Def1} is satisfied,  the MT-covariance matrix under $Q^{\\left(u\\right)}_{\\Xmatsc}$ is modified. In particular, by choosing $u\\left({\\bf{x}}\\right)\\equiv{1}$, we have $Q^{\\left(u\\right)}_{\\Xmatsc}=P_{\\Xmatsc}$, for which the standard covariance matrix ${\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$  is obtained. \n\nIn the following proposition, a strongly consistent estimate of ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is constructed, based on $N$ i.i.d. samples of ${\\bf{X}}$.\nUnlike the empirical MT-covariance proposed in \\cite{MTCCA}, the construction is based on complex observations and its almost sure convergence conditions are different.\n\\begin{Proposition}[Strongly consistent estimate of the MT-covariance]\n\\label{ConsistentEst}\nLet ${\\bf{X}}\\left(n\\right)$, $n=1,\\ldots,N$ denote a sequence of i.i.d. samples from $P_{\\Xmatsc}$, and define the empirical covariance estimate\n\\begin{equation}   \n\\label{Rx_u_Est}   \n\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\triangleq\\sum\\limits_{n=1}^{N}\\left({\\bf{X}}\\left(n\\right)-\\hat{{\\mbox{\\boldmath $\\mu$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)\\left({\\bf{X}}\\left(n\\right)-\\hat{{\\mbox{\\boldmath $\\mu$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)^{H}\\hat{\\varphi}_{u}\\left({\\bf{X}}\\left(n\\right)\\right)\n\\end{equation}\nwhere \n\\begin{equation}   \n\\label{Mu_u_Est} \n\\hat{{\\mbox{\\boldmath $\\mu$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\triangleq\\sum_{n=1}^{N}{\\bf{X}}\\left(n\\right)\\hat{\\varphi}_{u}\\left({\\bf{X}}\\left(n\\right)\\right)\n\\end{equation}\nand\n\\begin{equation}\n\\label{hat_varphi}  \n\\hat{\\varphi}_{u}\\left({\\bf{X}}\\left(n\\right)\\right)\\triangleq\\frac{u\\left({\\bf{X}}\\left(n\\right)\\right)}{\\sum\\limits_{n=1}^{N}u\\left({\\bf{X}}\\left(n\\right)\\right)}.\n\\end{equation}\nIf \n\\begin{equation}\n\\label{Cond11}\n{\\rm{E}}\\left[\\left\\|{\\bf{X}}\\right\\|^{2}_{2}u\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]<\\infty,\n\\end{equation}\nwhere ${\\|\\cdot\\|}_{2}$ denotes the $l_{2}$-norm, then $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\rightarrow{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ almost surely (a.s.) as $N\\rightarrow\\infty$. \n[A proof is given in Appendix \\ref{Prop2Proof}]\n\\end{Proposition}\nNote that for $u({\\bf{X}})\\equiv{1}$ the estimator $\\frac{N}{N-1}\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ reduces to the standard unbiased SCM. Also notice that $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ can be written as a statistical functional ${\\mbox{\\boldmath $\\Psi$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[\\cdot]$ of the empirical probability measure $\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ defined in (\\ref{EmpProbMes}), i.e.,\n\\begin{equation}\n\\label{StatFunc} \n\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}=\\frac{{\\rm{E}}[({\\bf{X}}-{\\mbox{\\boldmath $\\eta$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}])({\\bf{X}}-{\\mbox{\\boldmath $\\eta$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}])^{H};\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]}{{\\rm{E}}[{u}\\left({\\bf{X}}\\right);\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]}\n\\triangleq{\\mbox{\\boldmath $\\Psi$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}],\n\\end{equation} \nwhere \n\\begin{equation}\n\\label{StatFuncMean}\n{\\mbox{\\boldmath $\\eta$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]\\triangleq\\frac{{\\rm{E}}[{\\bf{X}}{u}\\left({\\bf{X}}\\right);\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]}{{\\rm{E}}[{u}\\left({\\bf{X}}\\right);\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]}. \n\\end{equation}\nBy (\\ref{VarPhiDef}), (\\ref{MTCovZ}) and  (\\ref{StatFunc}), when $\\hat{P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is replaced by ${P}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ we have ${\\mbox{\\boldmath $\\Psi$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}^{(u)}[P_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]={\\mbox{\\boldmath $\\Sigma$}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$, which implies that $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is Fisher consistent \\cite{Cox}. \n\\subsection{Robustness of the empirical MT-covariance}\nHere, we study the robustness of the empirical MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$  using its matrix valued influence function \\cite{Hampel}.\nDefine the probability measure $P_{\\epsilon}\\triangleq(1-\\epsilon)P_{\\Xmatsc}+\\epsilon\\delta_{{\\bf{y}}}$, where $0\\leq\\epsilon\\leq1$, ${\\bf{y}}\\in{\\mathbb{C}}^{p}$, and\n$\\delta_{{\\bf{y}}}$ is the Dirac probability measure at ${\\bf{y}}$. The influence function of a Fisher consistent estimator with statistical functional $\\textrm{H}[\\cdot]$ at probability distribution $P_{\\Xmatsc}$ is defined as \\cite{Hampel}:\n\\begin{equation}\n\\label{IFDef}\n{\\rm{IF}}_{\\textrm{H}}\\left({\\bf{y}};P_{\\Xmatsc}\\right)\\triangleq\\lim\\limits_{\\epsilon\\rightarrow{0}}\\frac{\\textrm{H}\\left[P_{\\epsilon}\\right]-\\textrm{H}\\left[P_{\\Xmatsc}\\right]}{\\epsilon}\n=\\left.\\frac{\\partial{\\textrm{H}}\\left[{P}_{\\epsilon}\\right]}{\\partial{\\epsilon}}\\right|_{\\epsilon=0}.\n\\end{equation}\nThe influence function describes the effect on the estimator of an infinitesimal contamination at the point ${\\bf{y}}$. An estimator is said to be B-robust if its influence function is bounded \\cite{Hampel}. Using (\\ref{StatFunc}) and (\\ref{IFDef}) one can verify that the influence function of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is given by\n\\begin{equation}\n\\label{MT_COV_INF}\n{\\rm{IF}}_{{\\tiny{\\mbox{\\boldmath $\\Psi$}}}_{{\\bf{x}}}^{(u)}}\\left({\\bf{y}};P_{\\Xmatsc}\\right)\n=\\frac{{u\\left({\\bf{y}}\\right)}\n[({\\bf{y}}-{\\mbox{\\boldmath $\\mu$}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}})({\\bf{y}}-{\\mbox{\\boldmath $\\mu$}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}})^{H} -  {{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}]}\n{{\\rm{E}}[u({\\bf{X}});P_{\\Xmatsc}]}.\n\\end{equation}\nThe following proposition states a sufficient condition for boundedness of (\\ref{MT_COV_INF}).  This condition is satisfied for the Gaussian MT-function proposed in Section \\ref{GMTMUSIC}.\n\\begin{Proposition}\n\\label{RobustnessConditions}\nThe influence function (\\ref{MT_COV_INF}) is bounded if the MT-function $u({\\bf{y}})$ and the product $u({\\bf{y}})\\|{\\bf{y}}\\|^{2}_{2}$ are bounded over ${\\mathbb{C}}^p$. [A proof is given in Appendix \\ref{InfBoundProof}]\n\\end{Proposition}\n\\subsection{The robust MT-MUSIC procedure}\n\\label{RobMTMUS}\nIn robust MT-MUSIC the measure transformation (\\ref{MeasureTransform}) is applied to the probability distribution $P_{\\Xmatsc}$ of the array output ${\\bf{X}}\\left(n\\right)$ (\\ref{ArrayModel}).  \nThe MT-function $u\\left(\\cdot\\right)$ is selected such that the following conditions are satisfied:\n\\begin{enumerate}\n\\item\n\\label{Cond1} \nThe resulting empirical MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is B-robust.\n\\item\n\\label{Cond2}\nLet $\\lambda^{\\left(u\\right)}_{1}\\geq\\cdots\\geq\\lambda^{\\left(u\\right)}_{p}$ denote the eigenvalues of  ${{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$. The $p-q$ smallest eigenvalues of ${{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ satisfy:\n\\begin{equation}\n\\label{EigCond}\n\\lambda_{q}^{(u)}>\\lambda^{\\left(u\\right)}_{q+1}=\\cdots=\\lambda^{\\left(u\\right)}_{p},\n\\end{equation}\nand their corresponding eigenvectors span the null-space of ${\\bf{A}}^{H}$, also called the noise subspace.\n\\end{enumerate} \n\nLet $\\hat{{\\bf{V}}}^{(u)}\\in{\\mathbb{C}}^{p\\times{(p-q)}}$ denote the matrix comprised of $p-q$ eigenvectors of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ corresponding to its smallest eigenvalues. The DOAs are estimated by finding the $q$ highest maxima of the measure-transformed pseudo-spectrum:\n\\begin{equation}\n\\label{MTEmpSpectrum}\n\\hat{P}^{(u)}(\\theta)\\triangleq{\\left\\|\\hat{{\\bf{V}}}^{(u)H}{\\bf{a}}(\\theta)\\right\\|}^{-2}_{2}.\n\\end{equation}\nBy modifying the MT-function $u(\\cdot)$ such that the stated conditions \\ref{Cond1} and \\ref{Cond2}  are satisfied a family of robust MT-MUSIC algorithms can be obtained. \nIn particular, by choosing $u\\left({\\bf{x}}\\right)\\propto\\left\\|{\\bf{x}}\\right\\|^{-2}_{2}$ one can verify using (\\ref{MTCovZ}) that for zero-centered symmetric distributions the resulting MT-covariance is proportional to the sign-covariance, proposed for the robust MUSIC generalization in \\cite{Visuri}. Another particular choice of MT-function leading to the Gaussian MT-MUSIC algorithm is discussed in the following section.\n\\section{The Gaussian MT-MUSIC}\n\\label{GMTMUSIC}\nIn this section, we parameterize the MT-function $u\\left(\\cdot;\\tau\\right)$, with scale parameter $\\tau\\in{\\mathbb{R}}_{++}$\nunder the Gaussian family of functions centered at the origin. This results in a B-robust empirical MT-covariance matrix that rejects large outliers. Under the assumption of spherically contoured noise distribution, we show that the noise subspace can be determined from the eigendecomposition of the MT-covariance. Choice of the scale parameter $\\tau$ is also discussed.\n\\subsection{The Gaussian MT-function}\n\\label{GaussMTFunc}\nWe define the Gaussian MT-function $u_{\\rm{G}}\\left(\\cdot;\\cdot\\right)$ as \n\\begin{eqnarray}\n\\label{GaussKernel}  \nu_{\\Gausssc}\\left({\\bf{x}};\\tau\\right)\\triangleq\\left(\\pi\\tau^{2}\\right)^{-p}\\exp\\left(-{\\left\\|{\\bf{x}}\\right\\|^{2}_{2}}\/{\\tau^{2}}\\right),\\hspace{0.2cm}\\tau\\in{\\mathbb{R}}_{++}.\n\\end{eqnarray}\nUsing (\\ref{VarPhiDef})-(\\ref{MTCovZ}) and (\\ref{GaussKernel}) one can verify that the resulting Gaussian MT-covariance always takes finite values. \nAdditionally, notice that the Gaussian MT-function satisfies the condition (\\ref{Cond11}) in Proposition \\ref{ConsistentEst}, and therefore, the empirical Gaussian MT-covariance, based on i.i.d. samples from any probability distribution $P_{\\Xmatsc}$,  is strongly consistent. For any fixed scale parameter $\\tau$, the Gaussian MT-function also satisfies the condition in proposition \\ref{RobustnessConditions}, resulting in a B-robust empirical Gaussian MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$. The following proposition, which follows directly from (\\ref{MT_COV_INF}) and (\\ref{GaussKernel}), states that the Frobenius norm of the corresponding influence function approaches zero as the contamination norm approaches infinity.\n\\begin{Proposition}\n\\label{InfFuncLim}\nFor any fixed scale parameter $\\tau$ of the Gaussian MT-function (\\ref{GaussKernel}), the influence function of the resulting empirical Gaussian MT-covariance satisfies\n\\begin{equation}\n\\label{IFLim}\n{\\left\\|{\\rm{IF}}_{{\\tiny{\\mbox{\\boldmath $\\Psi$}}}_{{\\bf{x}}}^{(u_{\\rm{G}})}}\\left({\\bf{y}};P_{\\Xmatsc}\\right)\\right\\|}_{\\rm{Fro}}\\rightarrow{0}\\hspace{0.2cm}\\textrm{as}\\hspace{0.2cm}{\\left\\|{\\bf{y}}\\right\\|}_{2}\\rightarrow\\infty, \n\\end{equation}\nwhere $\\left\\|\\cdot\\right\\|_{\\rm{Fro}}$ denotes the Frobenius norm. [A proof is given in Appendix \\ref{InfFuncLimProof}]\n\\end{Proposition}\nThus, unlike the SCM and other robust covariance approaches, the empirical Gaussian MT-covariance rejects large outliers. This property is illustrated in Fig. \\ref{INF_FRO} for a standard bivariate complex normal distribution, as compared to  the empirical sign-covariance, Tyler's scatter M-estimator, and the SCM.\n\\begin{figure}[htbp!]\n  \\begin{center}\n    {\\includegraphics[scale=0.35]{INF_FUNC_FRO_NORM.eps}}\n  \\end{center}\n  \\caption{Frobenius norms of the influence functions associated with the empirical Gaussian MT-covariance for $\\tau=1$, $\\tau=1.5$ and $\\tau=2$, Tyler's scatter M-estimator, the empirical sign-covariance, and the SCM, versus the contamination norm, for a bivariate standard complex normal distribution. Notice that the influence function approaches zero for large $\\|\\mathbf y\\|$ only for the proposed Gaussian MT-covariance estimator, indicating enhanced robustness to outliers.}\n\\label{INF_FRO}\n\\end{figure}\n\nNotice that as the scale parameter $\\tau$ of the Gaussian MT-function (\\ref{GaussKernel}) approaches infinity, the corresponding empirical Gaussian MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ approaches the non-robust standard SCM $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$, whose influence function is unbounded. On the other hand, as $\\tau$ decreases it can be shown using the upper bound in (\\ref{InfFuncGauss}) that the influence function of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ has a faster asymptotic decay, as illustrated in Fig. \\ref{INF_FRO}, i.e., the empirical Gaussian MT-covariance becomes more resilient to large outliers. However, we note that this may come at the expense of information loss. The trade-off between robustness and information loss is discussed in Subsection \\ref{TauChoice}.\n\\subsection{The Gaussian MT-covariance for spherically distributed noise}\n\\label{CGNoise}\nWe assume that the noise component in (\\ref{ArrayModel}) has a complex spherically contoured distribution, also known as a spherical distribution \\cite{Visa} having stochastic representation:\n\\begin{equation}\n\\label{CompGauss}\n{\\bf{W}}\\left(n\\right)=\\nu\\left(n\\right){\\mbox{\\boldmath $\\zeta$}}\\left(n\\right),\n\\end{equation}\nwhere $\\nu\\left(n\\right)\\triangleq\\rho\\left(n\\right)\/\\left\\|{\\mbox{\\boldmath $\\zeta$}}\\left(n\\right)\\right\\|_{2}$, $\\rho\\left(n\\right)\\in{\\mathbb{R}}_{++}$ is a first-order stationary process, and ${\\mbox{\\boldmath $\\zeta$}}\\left(n\\right)\\in{\\mathbb{C}}^{p}$ is a proper-complex wide-sense stationary Gaussian process with zero-mean and unit covariance, which is statistically independent of $\\rho\\left(n\\right)$. The stochastic representation (\\ref{CompGauss}) is a direct consequence of the following properties \\cite{Visa}:\n\\begin{inparaenum}\n\\item\nAny spherically distributed complex random vector ${\\bf{W}}$ can be represented as ${\\bf{W}}=\\rho{\\bf{U}}$, where $\\rho$ is a strictly positive random variable, and ${\\bf{U}}$  is uniformly distributed on the complex unit sphere and statistically independent of $\\rho$.\n\\item\nAny random vector ${\\bf{U}}$ that is uniformly distributed on the complex unit sphere can be represented as ${\\bf{U}}={\\mbox{\\boldmath $\\zeta$}}\/\\left\\|{\\mbox{\\boldmath $\\zeta$}}\\right\\|_{2}$, where ${\\mbox{\\boldmath $\\zeta$}}$ is a complex random vector with zero-mean spherically contoured distribution, for example a complex Gaussian random vector with zero-mean and unit covariance.\n\\end{inparaenum}\n\nThe structure of the resulting Gaussian MT-covariance of the array output is given in the following theorem. \n\\begin{Theorem}\n\\label{CompGaussStruct}   \nUnder the array model (\\ref{ArrayModel}) and the spherical noise assumption (\\ref{CompGauss}), the Gaussian MT-covariance of ${\\bf{X}}\\left(n\\right)$  takes the form:\n\\begin{equation}\n\\label{GaussMTCov}\n{\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)={\\bf{A}}{\\mbox{\\boldmath $\\Sigma$}}^{\\left(g\\right)}_{\\alpha^{2}{\\mbox{\\boldmath \\tiny $\\Smat$}}}\\left(\\tau\\right){\\bf{A}}^{H}+\\sigma^{2\\left(h\\right)}_{\\alpha{\\mbox{\\boldmath \\tiny $\\Wmat$}}}\\left(\\tau\\right){\\bf{I}},\n\\end{equation} \nwhere ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(g\\right)}_{\\alpha^{2}{\\mbox{\\boldmath \\tiny $\\Smat$}}}\\left(\\tau\\right)$ is a non-singular covariance matrix of the scaled signal component $\\alpha^{2}\\left(n\\right){\\bf{S}}\\left(n\\right)$, $\\alpha\\left(n\\right)\\triangleq\\sqrt{\\frac{\\tau^{2}}{\\tau^{2}+\\nu^{2}\\left(n\\right)}}$, under the transformed joint probability measure $Q^{\\left(g\\right)}_{\\alpha,{\\mbox{\\boldmath \\tiny $\\Smat$}}}$ with the MT-function  \n$g\\left(\\alpha,{\\bf{S}};\\tau\\right)\\triangleq{(\\frac{\\pi\\tau^{2}}{\\alpha^{2}})}^{-p}\\exp(-{\\alpha^{2}\\|{\\bf{A}}{\\bf{S}}\\|^{2}_{2}}\/{\\tau^{2}})$. The term \n$\\sigma^{2\\left(h\\right)}_{\\alpha{\\mbox{\\boldmath \\tiny $\\Wmat$}}}\\left(\\tau\\right)$, multiplying the identity matrix ${\\bf{I}}$, is the variance of the scaled noise component $\\alpha\\left(n\\right){\\bf{W}}\\left(n\\right)$ under the transformed joint probability measure $Q^{\\left(h\\right)}_{\\alpha,{\\bf{W}}}$ with the MT-function $h\\left(\\alpha;\\tau\\right)\\triangleq{\\rm{E}}\\left[g\\left(\\alpha,{\\bf{S}};\\tau\\right);P_{{\\mbox{\\boldmath \\tiny $\\Smat$}}}\\right]$. [A proof is given in Appendix \\ref{CompGaussStructProof}]\n\\end{Theorem}\nThus, by the structure (\\ref{GaussMTCov}) and the facts that the steering matrix ${\\bf{A}}$ has full column rank and the MT-covariance ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(g\\right)}_{\\alpha^{2}{\\mbox{\\boldmath \\tiny $\\Smat$}}}\\left(\\tau\\right)$ is non-singular, we conclude that Condition \\ref{Cond2}\nin Subsection \\ref{RobMTMUS} is satisfied.\n\\subsection{The Gaussian MT-MUSIC algorithm}\nThe empirical Gaussian MT-covariance is B-robust, and, under the spherical noise assumption (\\ref{CompGauss}), the noise subspace can be determined from the eigendecomposition of the  Gaussian MT-covariance. The Gaussian MT-MUSIC algorithm is implemented by replacing the MT-function in (\\ref{MTEmpSpectrum}) with the Gaussian MT-function (\\ref{GaussKernel}). \n\\subsection{Choosing the scale parameter of the Gaussian MT-function}\n\\label{TauChoice}\nWhile de-emphasizing non-informative outliers, e.g., caused by heavy-tailed distributions, the empirical Gaussian MT-covariance is less informative than the standard sample-covariance when the observations are normally distributed. This is seen in the following theorem that follows from the Gaussian Fisher information formula \\cite{Schreier} and elementary trace inequalities \\cite{TraceIneq}.\n\\begin{Theorem}\n\\label{TauProp}\nAssume that the probability distribution $P_{\\Xmatsc}$ of the array outputs (\\ref{ArrayModel}) is proper complex normal. The ratio between the Fisher information for estimating $\\theta_{k}\\in\\{\\theta_{1},\\ldots,\\theta_{q}\\}$ under the transformed probability measure $Q^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ (with the MT-function (\\ref{GaussKernel})) and the corresponding Fisher information under $P_{\\Xmatsc}$ satisfy:\n\\begin{equation}\n\\label{FIMRatio}\n\\frac{\\tau^{4}}{({\\lambda_{\\rm{max}}}\\left({\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)+\\tau^{2})^{2}}\\leq\\frac{{F}(\\theta_{k};{Q^{\\left(u_{\\Gausssc}\\right)}_{{\\bf{x}}}})}{F\\left(\\theta_{k};{P_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}}\\right)}  \n\\leq\\frac{\\tau^{4}}{(\\lambda_{\\rm{min}}\\left({\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)+\\tau^{2})^{2}},\n\\end{equation}\nwhere $\\lambda_{\\rm{min}}\\left(\\cdot\\right)$ and $\\lambda_{\\rm{max}}\\left(\\cdot\\right)$ are the minimum and maximum eigenvalues, respectively.\n[A proof is given in Appendix \\ref{TauPropProof}]\n\\end{Theorem}\nTherefore, in order to prevent a significant transform-domain Fisher information loss when the observations are normally distributed, we propose to choose the following safe-guard scale parameter:\n\\begin{equation}\n\\label{TAUSAFE}  \n\\tau=\\sqrt{c{\\lambda_{\\rm{max}}}\\left({\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\right)}, \n\\end{equation}\nwhere $c$ is some positive constant that guarantees that the Fisher information ratio (\\ref{FIMRatio}) is greater than $(c\/(1+c))^{2}$. Since in practice ${\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ is unknown, it is replaced by the following empirical robust estimate that is based on its relation (\\ref{GaussCovGauss}) to the Gaussian MT-covariance for normally distributed observations:\n\\begin{equation}\n\\label{RobCovEst}\n\\hat{{\\mbox{\\boldmath $\\Sigma$}}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}=\\tau^{2}\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u_{\\Gausssc})}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\left(\\tau^{2}{\\bf{I}}-\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u_{\\Gausssc})}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\right)^{-1},\n\\end{equation}\nwhere the empirical Gaussian MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u_{\\Gausssc})}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ is obtained using (\\ref{Rx_u_Est}), and $\\tau$ must be greater than ${\\lambda_{\\rm{max}}}\\left(\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u_{\\Gausssc})}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\right)$ in order to guarantee positive definiteness of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$. Therefore, substitution of (\\ref{RobCovEst}) into (\\ref{TAUSAFE}) results in the following data-driven selection rule:\n\\begin{equation}\n\\label{TAUSAFE_EMP}\n\\tau=\\sqrt{\\left(c+1\\right){\\lambda_{\\rm{max}}}\\left(\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{(u_{\\Gausssc})}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\right)},\n\\end{equation}\nwhich can be solved numerically, e.g., using fixed-point iteration \\cite{FXP}.\n\nIn the general case, when the observations are not necessarily Gaussian, the selection rule (\\ref{TAUSAFE}) controls the amount of second-order statistical information loss caused by the measure transformation. Increasing the constant $c$ increases the scale parameter $\\tau$ and reduces the information loss, while on the other hand, makes the estimator more sensitive to large-norm outliers, as illustrated in Fig. \\ref{INF_FRO}.\n\\section{Estimation of the number of signals}\n\\label{MTMDLEST}\nWe estimate the number of signals using a measure-transformed version of the minimum description length (MDL) criterion \\cite{Wax}, called MT-MDL, that is obtained by replacing the eigenvalues of the SCM with the eigenvalues of the empirical MT-covariance. The MT-MDL criterion takes the form:\n\\begin{eqnarray}\n\\label{MTMDL}\n{\\rm{MDL}}^{\\left(u\\right)}\\left(k\\right)&=&-\\log\\left(\\frac{\\left(\\prod\\limits_{m=k+1}^{p}\\hat{\\lambda}^{\\left(u\\right)}_{m}\\right)^{\\frac{1}{p-k}}}{\\frac{1}{p-k}\\sum\\limits_{m=k+1}^{p}\\hat{\\lambda}^{\\left(u\\right)}_{m}}\\right)^{({p-k}){N}}\n\\\\\\nonumber&+&\\frac{1}{2}k\\left(2p-k\\right)\\log{N},\n\\end{eqnarray}\nwhere $\\hat{\\lambda}^{\\left(u\\right)}_{1}\\geq\\ldots\\geq\\hat{\\lambda}^{\\left(u\\right)}_{p}$ denote the eigenvalues of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ and $N$ is the number of observations (snapshots). The estimated number of signals, $\\hat{q}$, is obtained by minimizing (\\ref{MTMDL}) over $k\\in\\left\\{0,\\ldots,p-1\\right\\}$. \n\nUnder the conditions that the eigenvalues of the SCM are strongly consistent with asymptotic convergence rate of $O\\left(\\sqrt{N^{-1}\\log\\log{N}}\\right)$ and that the $p-q$ smallest eigenvalues of the covariance matrix are equal and separated from its $q$ largest eigenvalues, it has been shown in \\cite{Zhao} that minimization of the MDL criterion leads to strongly consistent estimates of the number of signals for any underlying probability distribution of the data. Thus, when the eigenvalues of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ and ${{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$  satisfy these conditions, namely $\\hat{\\lambda}^{\\left(u\\right)}_{k}$ converges almost surely to ${\\lambda}^{\\left(u\\right)}_{k}$ for all $k=1,\\ldots,p$ with the same asymptotic convergence rate as the eigenvalues of the SCM, and the eigenvalues of ${{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ satisfy (\\ref{EigCond}), the resulting MT-MDL based estimator, $\\hat{q}$, must be strongly consistent. This rationale is used for proving the following Theorem that states a sufficient condition for strong consistency of the estimator $\\hat{q}$.\n\\begin{Theorem}\n\\label{Th2}\nLet ${\\bf{X}}\\left(n\\right)$, $n=1,\\ldots,N$ denote a sequence of i.i.d. samples from the probability distribution $P_{\\Xmatsc}$ of the array output (\\ref{ArrayModel}), with MT-covariance ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$ whose eigenvalues satisfy (\\ref{EigCond}). If \n\\begin{equation}\n\\label{Cond3}\n{\\rm{E}}\\left[u^{2}\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]<\\infty \\hspace{0.2cm} {\\rm{and}}\\hspace{0.2cm}{\\rm{E}}\\left[\\left\\|{\\bf{X}}\\right\\|^{4}_{2}u^{2}\\left({\\bf{X}}\\right);P_{\\Xmatsc}\\right]<\\infty,\n\\end{equation}\nthen $\\hat{q}\\rightarrow{q}$ a.s. as $N\\rightarrow\\infty$. [A proof is given in Appendix \\ref{Th2Proof}]\n\\end{Theorem}\n\nNotice that the Gaussian MT-function (\\ref{GaussKernel}) always satisfies the condition (\\ref{Cond3}). Furthermore, as shown in subsection \\ref{CGNoise}, the Gaussian MT-covariance ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ satisfies Condition \\ref{Cond2} in Subsection \\ref{RobMTMUS} for spherically distributed noise. Therefore, in this case, minimization of the MT-MDL criterion with the Gaussian MT-function (Gaussian MT-MDL) results in robust and strongly consistent estimate of the number of signals. We propose to choose the scale parameter $\\tau$ of the Gaussian MT-function using the same selection rule (\\ref{TAUSAFE_EMP}) that prevents significant information loss for estimation of the DOAs (model parameters), and does not require any knowledge about the number of signals (model order). The idea of estimating the DOAs and the number of signals using the same scale parameter $\\tau$, i.e., under the same transformed probability measure, arises from the intuition that if there is no significant information loss for estimating the model parameters, then there will be no significant information loss for estimating the model order.   \n\\section{The spatially smoothed Gaussian MT-MUSIC for coherent signals}\n\\label{CMTMUSIC}\nIn this section, we consider the case of coherent signals contaminated by spherically distributed noise. In this scenario, the components of the latent vector ${\\bf{S}}\\left(n\\right)$ in (\\ref{ArrayModel}) are phase-delayed amplitude-weighted replicas of a single first-order stationary random signal $s(n)$, i.e.,  \n\\begin{equation}\n\\label{CoherentModel}\n{\\bf{S}}\\left(n\\right)={\\mbox{\\boldmath $\\xi$}}{s}\\left(n\\right),\n\\end{equation}\nwhere ${\\mbox{\\boldmath $\\xi$}}\\in{\\mathbb{C}}^{q}$ is a vector of deterministic complex attenuation coefficients. Similarly to the standard covariance ${\\mbox{\\boldmath $\\Sigma$}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}$, the noise subspace cannot be determined from the eigendecomposition of  the Gaussian MT-covariance ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$, and therefore, the Gaussian MT-MUSIC will fail in estimating the DOAs. Fortunately, similarly to \\cite{Pillai},  we show that for uniform linearly spaced array (ULA) \\cite{VanTrees} the DOAs can be determined using a spatially smoothed version of the Gaussian MT-covariance. \n\nWe partition ${\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ into $L=p-r+1$ overlapping forward and backward square sub-matrices of dimension $r<q$. The entries of the $l$-th forward sub-matrix are given by\n\\begin{equation}\n\\label{FOR}\n\\left[{\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f,l}\\left(\\tau\\right)\\right]_{j,k}\\triangleq\\left[{\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\right]_{l+j-1,l+k-1},\n\\end{equation}\n$j,k=1,\\ldots,r$, and the entries of the $l$-th backward sub-matrix are given by\n\\begin{equation}\n\\label{BCK}   \n\\left[{\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{b,l}\\left(\\tau\\right)\\right]_{j,k}\\triangleq\\left[{\\mbox{\\boldmath $\\Sigma$}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)\\right]^{*}_{p-l-j+2,p-l-k+2},\n\\end{equation}\n$j,k=1,\\ldots,r$, where $\\left(\\cdot\\right)^{*}$ denotes the complex conjugate. These matrices correspond to overlapping forward and backward subarrays of size $r$, respectively. The forward spatially smoothed Gaussian MT-covariance is defined as the average over the $L$ forward sub-matrices:\n\\begin{equation}\n\\label{FGMTC}\n{\\bf{C}}^{(u_{\\Gausssc})}_{f}\\left(\\tau\\right)\\triangleq\\frac{1}{L}\\sum\\limits_{l=1}^{L}{\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f,l}\\left(\\tau\\right).\n\\end{equation}\nSimilarly, the backward spatially smoothed Gaussian MT-covariance is defined as the average over the $L$ backward sub-matrices:\n\\begin{equation}\n\\label{BGMTC} \n{\\bf{C}}^{(u_{\\Gausssc})}_{b}\\left(\\tau\\right)\\triangleq\\frac{1}{L}\\sum\\limits_{l=1}^{L}{\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{b,l}\\left(\\tau\\right).\n\\end{equation}\nThe forward\/backward spatially smoothed Gaussian MT-covariance matrix is given by\n\\begin{equation}\n\\label{FBGMTC}  \n{\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)\\triangleq\\frac{1}{2}\\left({{\\bf{C}}^{(u_{\\Gausssc})}_{f}\\left(\\tau\\right) + {\\bf{C}}^{(u_{\\Gausssc})}_{b}\\left(\\tau\\right)}\\right).\n\\end{equation}\n\nWe define ${\\bf{B}}\\triangleq\\left[{\\bf{b}}\\left(\\theta_{1}\\right),\\ldots,{\\bf{b}}\\left(\\theta_{q}\\right)\\right]\\in{\\mathbb{C}}^{r\\times{q}}$ as the steering matrix of a ULA with $r<p$ sensors, where ${\\bf{b}}\\left(\\theta\\right)\\triangleq\\left[1,e^{-i2\\pi{(d\/\\lambda)}\\sin\\left(\\theta\\right)},\\ldots,e^{-i2\\pi(r-1){(d\/\\lambda)}\\sin\\left(\\theta\\right)}\\right]^{T}$ is the array steering vector toward direction $\\theta$, and $d$ is the sensors spacing, i.e., ${\\bf{B}}$ is a sub-matrix of the steering matrix ${\\bf{A}}$ in (\\ref{ArrayModel}) comprised of its first $r$ rows.\nThe following Proposition states sufficient conditions under which the null-space of ${\\bf{B}}^{H}$ can be determined from the eigendecomposition of ${\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$ (\\ref{FBGMTC}). The same conditions were proved in  \\cite{Pillai} for the spatially smoothed SCM.\n\\begin{Proposition}\n\\label{FBSTh}\nDefine ${\\bf{H}}\\triangleq{\\bf{G}}^{-1}_{f}{\\bf{G}}_{b}$, where ${\\bf{G}}_{f}\\triangleq{\\rm{diag}}\\left({\\mbox{\\boldmath $\\xi$}}\\right)$, ${\\bf{G}}_{b}\\triangleq{\\rm{diag}}\\left({\\mbox{\\boldmath  $\\delta$}}\\right)$, ${\\mbox{\\boldmath  $\\delta$}}\\triangleq\\left({\\bf{D}}^{p-1}{\\mbox{\\boldmath $\\xi$}}\\right)^{*}$, and ${\\bf{D}}^{l}$ is the $l$-th power of the diagonal matrix \n${\\bf{D}}\\triangleq{\\rm{diag}}\\left(\\left[v\\left(\\theta_{1}\\right),\\ldots,v\\left(\\theta_{q}\\right)\\right]\\right),\\hspace{0.2cm}v\\left(\\theta\\right)\\triangleq{\\exp}\\left({-i2\\pi{(d\/\\lambda)}\\sin\\left(\\theta\\right)}\\right)$.\nAdditionally, let $\\lambda^{\\left(u_{\\Gausssc}\\right)}_{1}\\geq\\cdots\\geq\\lambda^{\\left(u_{\\Gausssc}\\right)}_{r}$ denote the eigenvalues of  ${\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$. If the dimension $r$ of the forward and backward sub-matrices (\\ref{FOR}) and (\\ref{BCK}) is chosen such that the resulting number of sub-matrices in each direction (forward or backward) $L$ satisfies: \n\\begin{enumerate}\n\\item\n$L\\geq{q}$, or\n\\item\n$2L\\geq{q}$ and  the largest subset of equal diagonal entries of ${\\bf{H}}$ is at most of size $L$,\n\\end{enumerate}\nthen the $r-q$ smallest eigenvalues of ${\\bf{C}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$ satisfy $\\lambda^{\\left(u_{\\Gausssc}\\right)}_{q}>\\lambda^{\\left(u_{\\Gausssc}\\right)}_{q+1}=\\cdots=\\lambda^{\\left(u_{\\Gausssc}\\right)}_{r}$, and their corresponding eigenvectors span the null-space of ${\\bf{B}}^{H}$.  [A proof is given in Appendix \\ref{FBSThProof}]\n\\end{Proposition}\n\nHence, by proper choice of $r$, such that either one of the stated conditions above is satisfied, the spatially smoothed Gaussian MT-MUSIC is obtained by replacing the empirical Gaussian MT-covariance $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$ with its spatially smoothed version $\\hat{{\\bf{C}}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$. \n\nThe number of signals is estimated using a measure-transformed version of the modified MDL (MMDL) criterion used in \\cite{Xu} for cases where forward\/backward spatial smoothing is performed. Similarly to (\\ref{MTMDL}), this criterion, called here Gaussian MT-MMDL, is obtained by replacing the eigenvalues of the spatially smoothed SCM with the eigenvalues of $\\hat{{\\bf{C}}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$. The Gaussian MT-MMDL criterion takes the form: \n\\begin{eqnarray}\n\\label{GMTMDL}\n{\\rm{MDL}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(k\\right)&=&-\\log\\left(\\frac{\\left(\\prod\\limits_{m=k+1}^{r}\\hat{\\lambda}^{\\left(u_{\\Gausssc}\\right)}_{m}\\right)^{\\frac{1}{r-k}}}{\\frac{1}{r-k}\\sum\\limits_{m=k+1}^{r}\\hat{\\lambda}^{\\left(u_{\\Gausssc}\\right)}_{m}}\\right)^{({r-k}){N}}\n\\\\\\nonumber&+&\\frac{1}{4}k\\left(2r-k+1\\right)\\log{N},\n\\end{eqnarray}\nwhere $\\hat{\\lambda}^{\\left(u_{\\Gausssc}\\right)}_{1}\\geq\\ldots\\geq\\hat{\\lambda}^{\\left(u_{\\Gausssc}\\right)}_{r}$ denote the eigenvalues of $\\hat{{\\bf{C}}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$ and $N$ is the number of observations (snapshots). The estimated number of signals, $\\hat{q}$, is obtained by minimizing (\\ref{GMTMDL}) over $k\\in\\left\\{0,\\ldots,r-1\\right\\}$.\n\nFinally, we choose the scale parameter $\\tau$ of the Gaussian MT-function using the selection rule (\\ref{TAUSAFE_EMP}) with $\\hat{{\\bf{C}}}^{\\left(u_{\\Gausssc}\\right)}_{f\/b}\\left(\\tau\\right)$ instead of $\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\left(u_{\\Gausssc}\\right)}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}}}\\left(\\tau\\right)$. \n\\section{Numerical examples}\n\\label{Examples}\nWe evaluate and compare the performance of the proposed MT-MUSIC DOA estimator and the MT-MDL order estimator. What follows is a  summary of these comparisons.\nThe DOAs estimation performances are evaluated under the assumption that the number of signals is known. We perform a separate evaluation of the  proposed MT-MDL estimator of the number of signals.  We examine scenarios of non-coherent and coherent signals. For non-coherent signals, the Gaussian MT-MUSIC algorithm is compared to the standard SCM-based MUSIC (SCM-MUSIC) \\cite{Schmidt} and to its robust generalizations based on the ZMNL preprocessing (ZMNL-MUSIC) \\cite{swami2002}, the empirical sign-covariance (SGN-MUSIC) \\cite{Visuri}, Tyler's scatter M-estimator (TYLER-MUSIC) \\cite{Visa}, \\cite{Tyler} and \nthe maximum-likelihood (ML) estimators of scatter corresponding to each of the considered non-Gaussian noise distributions (ML-MUSIC) \\cite{Visa}, \\cite{MestMDL}. The estimation performance of the number of signals using the MT-MDL criterion (\\ref{MTMDL}) with the Gaussian MT-function is compared to estimators using the standard MDL criterion \\cite{Wax} based on the standard SCM (SCM-MDL), and the MDL variants based on the SCM of the pre-processed data with the ZMNL function (ZMNL-MDL) \\cite{Sadler}, the empirical sign-covariance (SGN-MDL) \\cite{Visuri}, Tyler's scatter M-estimator (TYLER-MDL) \\cite{MestMDL} and the ML estimators of scatter corresponding to each of the considered non-Gaussian noise distributions (ML-MDL) \\cite{Visa}, \\cite{MestMDL}.  For coherent signals, the spatially smoothed (SS) Gaussian MT-MUSIC algorithm, discussed in Section \\ref{CMTMUSIC}, is compared to the spatially smoothed versions of the SCM-MUSIC \\cite{Pillai}, ZMNL-MUSIC,  SGN-MUSIC \\cite{Visuri}, TYLER-MUSIC and ML-MUSIC. Estimation performance of the number of signals using the Gaussian MT-MMDL criterion (\\ref{GMTMDL}) is compared to those obtained by the modified MDL criterion for coherent signals \\cite{Xu} based on the forward\/backward spatially smoothed versions of the standard SCM, the SCM of the preprocessed data with the ZMNL function \\cite{swami2002}, the empirical sign-covariance \\cite{Visuri}, Tyler's scatter M-estimator and the ML estimators of scatter corresponding to each non-Gaussian noise distribution considered in the simulation examples. \n\nWe consider the following $p$-variate complex spherical compound Gaussian noise distributions with zero location parameter and isotropic dispersion $\\sigma^{2}_{{\\mbox{\\boldmath \\tiny $\\Wmat$}}}{\\bf{I}}$:  \nGaussian, Cauchy, $K$-distribution with shape parameter $\\nu=0.75$, and compound Gaussian distribution with inverse Gaussian texture and shape parameter $\\lambda=0.1$. Notice that unlike the Gaussian distribution, the other noise distributions are heavy tailed. Random sampling from the considered noise distributions and their applicability for modelling radar clutter are discussed in detail in \\cite{Visa} and \\cite{CG4}. Let $\\sigma^{2}_{S_{k}}$, $k=1,\\ldots,q$ denote the variances of the received signals, the generalized signal-to-noise-ratio (GSNR) is defined as $\\textrm{GSNR}\\triangleq{10}\\log_{10}{\\frac{1}{q}\\sum_{k=1}^{q}\\sigma^{2}_{S_{k}}}\/{\\sigma^{2}_{{\\mbox{\\boldmath \\tiny $\\Wmat$}}}}$ and is used to index the estimation performance.\n\nFollowing the approach proposed in subsection \\ref{TauChoice}, for non-coherent signals, we select the scale parameter $\\tau$ of the Gaussian MT-function (\\ref{GaussKernel}) as the solution of (\\ref{TAUSAFE_EMP}) with $c=5$. This choice of the constant $c$ guarantees relative transform-domain Fisher information loss of no more than $\\approx30\\%$. The solution of (\\ref{TAUSAFE_EMP}) is obtained using fixed-point iteration with initial condition $\\tau_{0}=5\\sqrt{\\frac{1}{p}\\sum_{k=1}^{p}\\hat{\\sigma}^{2}_{X_{k}}}$, where $\\hat{\\sigma}^{2}_{X_{k}} = \\gamma^{2}[(\\textrm{MAD}(\\{\\textrm{Re}(X_{k,n})\\}_{n=1}^{N}))^{2} + (\\textrm{MAD}(\\{\\textrm{Im}(X_{k,n})\\}_{n=1}^{N}))^{2}]$, $\\gamma\\triangleq{1}\/\\textrm{erf}^{-1}(3\/4)$, is a robust median absolute deviation (MAD) estimate of variance \\cite{Huber}. The maximum number of iterations and the stopping criterion were set to 100 and ${\\left|\\tau_{l}-\\tau_{l-1}\\right|}\/{\\tau_{l-1}}<10^{-6}$, respectively, where $l$ is an iteration index. For coherent signals, we replaced the empirical Gaussian MT-covariance in (\\ref{TAUSAFE_EMP}) with its spatially smoothed version and applied the same selection procedure for $\\tau$. \n\nThe maximum number of iterations and the stopping criterion in Tyler's scatter M-estimator and the ML estimators of scatter were set to 100 and \n$${\\|\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\textrm{Tyler\/ML}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}},l}-\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\textrm{Tyler\/ML}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}},l-1}\\|_{\\rm{Fro}}}\/{\\|\\hat{{\\mbox{\\boldmath $\\Sigma$}}}^{\\textrm{Tyler\/ML}}_{{\\mbox{\\boldmath \\tiny $\\Xmat$}},l-1}\\|_{\\rm{Fro}}}<10^{-6},$$ respectively. \n \nIn all examples, the performances versus GSNR were evaluated for $N=1000$ i.i.d. snapshots.\nThe performances versus the number of snapshots were evaluated at the threshold GSNR point obtained by the Gaussian MT-MUSIC algorithm for $N=1000$ i.i.d. snapshots. The parameter space $\\Theta\\triangleq\\left[-90^{\\circ},90^{\\circ}\\right)$ was sampled uniformly with sampling interval \n$\\Delta=0.0018^{\\circ}$. All performance measures were averaged over $10^{4}$ Monte-Carlo simulations.\n\\subsection{Non-coherent signals}\n\\label{NonCoherentSig} \nIn this example, we considered five independent 4-QAM signals with equal power $\\sigma^{2}_{{\\mbox{\\boldmath \\tiny $\\Smat$}}}$ impinging on a $16$-element uniform linear array with $\\lambda\/2$ spacing from DOAs $\\theta_{1}=-10^{\\circ}$, $\\theta_{2}=0^{\\circ}$, $\\theta_{3}=5^{\\circ}$, $\\theta_{4}=15^{\\circ}$, and $\\theta_{5}=35^{\\circ}$. The average RMSEs for estimating the DOAs and the error rates for estimating the number of signals versus GSNR and the number of snapshots \nare depicted in Figs. \\ref{GAUSS_NON_COHERENT}-\\ref{IG_NON_COHERENT} for each noise distribution. Notice that for the Gaussian noise case, there is no significant performance gap between the compared methods. \nFor the other noise distributions, the proposed Gaussian MT-MUSIC and the Gaussian MT-MDL based estimation of the number of signals outperform all other robust MUSIC and MDL generalizations in the low GSNR and low sample size regimes, with significantly lower threshold regions. This performance advantage may be  attributed to the fact that unlike the empirical sign-covariance, Tyler's scatter M-estimator, and the ML estimators of scatter corresponding to each non-Gaussian noise distribution, the influence function of the empirical Gaussian MT-covariance is negligible for large norm outliers (as illustrated in Fig. \\ref{INF_FRO}), which are likely in low GSNRs and become more defective when the sample size decreases. Furthermore, unlike the ZMNL preprocessing based technique, the proposed measure-transformation approach preserves the noise subspace and effectively suppresses outliers without significant information loss for estimating the DOAs and the number of signals.\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_GAUSS_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_GAUSS_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_GAUSS_NON_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_GAUSS_NON_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n  \\caption{\\textbf{Non-coherent signals in Gaussian noise:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots. \n   (d) Probability of error for estimating the number of signals versus the number of snapshots.\n   The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of snapshots were evaluated for ${\\rm{GSNR}}=-11$ [dB].\n   Notice that all algorithms perform similarly.}\n\\label{GAUSS_NON_COHERENT}\n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_CAUCHY_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_CAUCHY_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_CAUCHY_NON_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_CAUCHY_NON_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n    \\caption{\\textbf{Non-coherent signals in Cauchy noise:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-11$ [dB].\n   Notice that the Gaussian MT-MUSIC outperforms all other compared algorithms in the low GSNR and low sample size regimes.  Also notice that the Gaussian MT-MDL criterion leads to significantly lower error rates for estimating the number of signals.}\n\\label{CAUCHY_NON_COHERENT} \n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_K_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_K_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_K_NON_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_K_NON_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n \\caption{{\\textbf{Non-coherent signals in K-distributed noise with shape parameter $\\nu=0.75$:}}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-19$ [dB]. Notice that the Gaussian MT-MUSIC outperforms all other compared algorithms in the low GSNR and low sample size regimes.  Also notice that the Gaussian MT-MDL criterion leads to significantly lower error rates for estimating the number of signals.}\n\\label{K_NON_COHERENT}\n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_IG_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_IG_NON_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_IG_NON_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_IG_NON_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n \\caption{\\textbf{Non-coherent signals in spherical compound Gaussian noise with inverse-Gaussian texture and shape parameter $\\lambda=0.1$:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-22$ [dB]. Notice that the Gaussian MT-MUSIC outperforms all other compared algorithms in the low GSNR and low sample size regimes.  Also notice that the Gaussian MT-MDL criterion leads to significantly lower error rates for estimating the number of signals.}\n\\label{IG_NON_COHERENT}  \n\\end{figure}\n\\subsection{Coherent signals}\nIn this example, we considered five coherent signals impinging on a $22$-element uniform linear array with $\\lambda\/2$ spacing from DOAs $\\theta_{1}=-17^{\\circ}$, $\\theta_{2}=-3^{\\circ}$, $\\theta_{3}=2^{\\circ}$, $\\theta_{4}=13^{\\circ}$ and $\\theta_{5}=20^{\\circ}$. The signals were generated according to the model (\\ref{CoherentModel}), where $s\\left(n\\right)$ is a 4-QAM signal with power $\\sigma^{2}_{S}$. The attenuation coefficients were set to $\\eta_{1}=0.8\\exp(i\\pi\/3)$, $\\eta_{2}=1$, $\\eta_{3}=0.9\\exp(i\\pi\/4)$, $\\eta_{4}=0.7\\exp(i\\pi\/5)$ and $\\eta_{5}=0.6\\exp(i\\pi\/6)$. The dimension of the spatially smoothed covariance was set to $r=16$. The average RMSEs for estimating the DOAs and the error rates for estimating the number of signals versus GSNR are depicted in Figs. \\ref{GAUSS_COHERENT}-\\ref{IG_COHERENT} for each noise distribution. \nNotice that for the Gaussian noise case, there is no significant performance gap between the compared MUSIC algorithms. Regarding the estimation of the number of signals, the sign-covariance based modified MDL criterion results in better estimation performance. This may be attributed to the fact that in the sign-covariance based modified MDL criterion \\cite{Visuri} the eigenvalues are estimated in a more stable manner. For the other noise distributions, the spatially smoothed Gaussian MT-MUSIC and the Gaussian modified MT-MDL based estimation of the number of signals outperform all other robust MUSIC and MDL generalizations in the low GSNR and low sample size regimes, with significantly lower breakdown thresholds. Again, as in the non-coherent case, this performance advantage may be attributed to the following facts. First, unlike the empirical sign-covariance, Tyler's scatter M-estimator and the ML-estimators of scatter corresponding to each non-Gaussian noise distribution, the influence function of the empirical Gaussian MT-covariance is very small for large norm outliers. Such outliers are likely in low GSNRs and become more frequent when the sample size is small. Second, unlike the ZMNL preprocessing based technique, the proposed measure-transformation approach preserves the noise subspace and effectively suppresses outliers without significant performance loss in estimating the DOAs and the number of signals.\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_GAUSS_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_GAUSS_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_GAUSS_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_GAUSS_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n  \\caption{\\textbf{Coherent signals in Gaussian noise:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. \n   The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of snapshots were evaluated for ${\\rm{GSNR}}=-12$ [dB].\n    Notice that there is no significant performance gap between the compared MUSIC algorithms. The sign-covariance based modified MDL criterion results in better estimation of the number of signals. This may be attributed to the fact that the sign-covariance based modified MDL criterion \\cite{Visuri} involves more stable eigenvalues estimation.}\n\\label{GAUSS_COHERENT}\n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_CAUCHY_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_CAUCHY_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_CAUCHY_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_CAUCHY_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n  \\caption{\\textbf{Coherent signals in Cauchy noise:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-14$ [dB]. Note that similarly to the non-coherent case, the Gaussian MT-MUSIC estimator has significantly lower GSNR and sample size threshold regions than the other methods.  \n  Also notice that the Gaussian MT-MMDL estimator of the number of signals outperforms all other MDL based estimators with significantly lower probability of error at the low GSNR and low sample size regimes.}\n\\label{CAUCHY_COHERENT}\n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_K_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_K_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_K_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_K_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n  \\caption{\\textbf{Coherent signals in K-distributed noise with shape parameter $\\nu=0.75$:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-25$ [dB]. Note that similarly to the non-coherent case, the Gaussian MT-MUSIC estimator has significantly lower GSNR and sample size threshold regions than the other methods.  \n  Also notice that the Gaussian MT-MMDL estimator of the number of signals outperforms all other MDL based estimators.}\n\\label{K_COHERENT}\n\\end{figure}\n\\begin{figure}[htp]\n  \\begin{center}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_IG_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_IG_COHERENT.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{DOA_EST_IG_COHERENT_SNAP.eps}}}}\n    {{\\subfigure[]{\\includegraphics[scale = 0.235]{SIGNALS_EST_IG_COHERENT_SNAP.eps}}}}\n      \\end{center}  \n  \\caption{\\textbf{Coherent signals in spherical compound Gaussian noise with inverse-Gaussian texture and shape parameter $\\lambda=0.1$:}\n   (a) Average RMSE versus GSNR. (b) Probability of error for estimating the number of signals versus GSNR. \n   (c) Average RMSE versus the number of snapshots.  \n   (d) Probability of error for estimating the number of signals versus the number of snapshots. The performance measures versus GSNR were evaluated for $N=1000$ i.i.d. snapshots. The performance measures versus the number of     snapshots were evaluated for ${\\rm{GSNR}}=-24$ [dB]. Note that similarly to the non-coherent case, the Gaussian MT-MUSIC estimator has significantly lower GSNR and sample size threshold regions than the other methods.  \n  Also notice that the Gaussian MT-MMDL estimator of the number of signals outperforms all other MDL based estimators with significantly lower probability of error at the low GSNR and low sample size regimes.}\n\\label{IG_COHERENT}  \n\\end{figure}\n\\section{Conclusion}\n\\label{Conclusions}\nIn this paper, a new framework for robust MUSIC was proposed that applies a transform to the probability distribution of the data prior to forming the sample covariance. \nUnder the assumption of spherically contoured noise distribution, a new robust MUSIC algorithm, called Gaussian MT-MUSIC, was presented based on a Gaussian shaped measure transform (MT) function. Furthermore, a new robust generalization of the MDL criterion for estimating the number of signals, called MT-MDL, was derived that is based on replacing the eigenvalues of the SCM with those of the empirical MT-covariance. The proposed Gaussian MT-MUSIC algorithm was extended to the case of coherent signals by applying spatial smoothing to the empirical Gaussian MT-covariance. Exploration of other classes of MT-functions may result in additional robust MUSIC algorithms that have different useful properties. Furthermore, extending the MT-MDL criterion to sample-starved scenarios \\cite{R1} or to cases where there is additional information on the sample eigenvalues distribution \\cite{R2} are worthwhile topics for future research.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conclusions}\n\\label{sec:conclusions}\n\nIn this work, we have introduced a framework to model the different attacks of the statistical disclosure family, showing how better results can be achieved when performing more complex operations with the observations from the system. We have formalized two new variants of the SDA, which we called $\\SDAenhone$ and $\\SDAenhtwo$, and showed that they significantly improve the state-of-the-art SDA in threshold mixes, $\\SDAenhzero$. Furthermore, we have shown that the LSDA, introduced in \\cite{petsLSDA}, can be seen as an upgraded version of statistical disclosure that solves the problem jointly for all users.\n\nWe have also improved the previous theoretical analysis on LSDA and derived for the first time an expression which accurately approximates the error of $\\SDAenhtwo$. Our experiments confirm these theoretical results.\n\n\n\n\n\n\\section{Evaluation}\n\\label{sec:eval}\n\nWe evaluate the performance of the attacks in Sect.~\\ref{sec:revision} in terms of $\\MSEi$, simulating a threshold mix system as described in Sect.~\\ref{sec:sysmodel}.\\footnote{The simulator, written in Matlab, will be available upon request.}\nWe exclude $\\SDA$ from this evaluation and use its generalization $\\SDAenhzero$ instead.\n\nWe vary the number of users in the population $\\nusers$, the threshold $\\thre$, the sending frequencies $\\sendfreq{i}$, the number of rounds observed by the attacker $\\rho$ and the uniformity of the sending profiles $\\uniformi{i}$.\n\n\\subsection{Performance with respect to the uniformity $\\uniformi{i}$}\n\nAs we have shown in Sect.~\\ref{sec:mse}, the uniformity of the sender profiles is a key parameter to show the difference in performance between $\\SDAenhtwo$ and $\\LSDA$. For simplicity, we assume that each user $i$ has $\\numberfriends$ friends to whom she sends messages uniformly, which are users $\\mbox{mod}\\left( i+k, \\nusers\\right)$ for $k=0, ..., \\numberfriends-1$. This allows us to vary the uniformity of the sender profile of each user with a single parameter: $\\uniformi{i}=\\frac{1-\\numberfriends}{\\numberfriends}$. We choose the number of friends $\\numberfriends$ from $\\{10, 25, 50, 100\\}$ and, for each value, perform $100$ repetitions of the experiment.\n\nFigure~\\ref{fig:boxplot} shows a box-and-whiskers plot of the average MSE per sender profile, $\\MSEim$. On the boxes, the central mark is the mean and the edges are the 25th and 75th percentiles. The black circles $\\bullet$ represent the theoretical asymptotic values of the $\\MSEim$, from (\\ref{eq:MSE_lsda}) and (\\ref{eq:MSE_sda2}). Since $\\rho$ is finite, $\\MSEim$ does not coincide exactly with its theoretical value, although (\\ref{eq:MSE_lsda}) and (\\ref{eq:MSE_sda2}) reliably describe the accuracy of the attacks. As expected, when the uniformity of the sender profiles is low and the background uniformity $\\uniformi{\\backg{}}$ is large, $\\LSDA$ outperforms the other estimators, but as the uniformity of each user increases and therefore becomes closer to the background uniformity, the advantage of $\\LSDA$ decreases.\nAlso, note that the proposed estimators $\\SDAenhone$ and $\\SDAenhtwo$ outperform $\\SDAenhzero$.\n\n\\begin{figure}[!t]\n\\centering\n\\includegraphics[width=0.45\\textwidth]{prueba}\n\\caption{Average $\\mbox{MSE}$ for the different attacks, as a function of the number of friends $\\numberfriends$ of each user ($\\rho=20000$, $\\nusers=100$, $\\sendfreq{i}=1\/\\nusers$, $t=10$).}\n\\label{fig:boxplot}\n\\end{figure}\n\n\\subsection{Performance with respect to the other parameters}\n\nDue to space limitations, we are not able to plot the results obtained when varying all the other parameters. We summarize the basic results next and refer to \\cite{petsLSDA} for further information about LSDA. First, the $\\MSEi$ decreases with $1\/\\rho$ in each of these attacks, as in (\\ref{eq:MSE_lsda}) and (\\ref{eq:MSE_sda2}). Also, in every attack, the $\\MSEi$ is approximately proportional to the inverse of the sending frequency $\\sendfreq{i}^{-1}$, due to the increasing difficulty of estimating the sender profile of a user when she rarely participates in the system. The threshold $\\thre$ has little influence on the $\\MSEi$ of $\\SDAenhtwo$ and $\\LSDA$ but does, however, decrease the number of rounds that can be used to estimate the background (\\ref{eq:back_est1}) in $\\SDAenhzero$ and $\\SDAenhone$, thus increasing the $\\MSEi$ in these estimators. Finally, we note that increasing $\\nusers$ adds an extra error in $\\LSDA$ which is not predicted by (\\ref{eq:MSE_lsda}) and that stems from the matrix inversion \nin (\\ref{eq:LSDA}). This error can be reduced by increasing the number of rounds observed. This can be seen in Fig.~\\ref{fig:boxplot}, where the mean values of $\\MSEim$ obtained for $\\LSDA$ are slightly above their asymptotic value.\n\nThe improvements in performance achieved by the more sophisticated versions of statistical disclosure come at the price of an increase in the computational cost. While $\\SDAenhzero$ adds the observations where the user whose profile is being estimated has participated, $\\SDAenhone$ needs to perform an additional multiplication for each of these rounds. $\\SDAenhtwo$ has a higher computational cost since it requires solving a system of two equations for each user, and $\\LSDA$ requires solving a linear system of $\\nusers$ equations with $\\nusers$ unknowns.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nMixes constitute the basic building block of high-latency anonymous communication systems \\cite{mixes}. They act as a channel that hides the correspondence between incoming and outgoing messages, thus preventing a potential adversary from unveiling users' communication patterns (e.g. friendships, frequency). \n\nThere exist a wide variety of attacks that compromise the anonymity provided by mixes. In this paper, we revisit a particularly efficient family of attacks which is based on the Statistical Disclosure Attack (SDA) \\cite{originalSDA} and propose a framework that allows us to easily compare the attacks when performed on threshold mixes. We revisit Mathewson and Dingledine's generalization of the SDA and propose two new variants that outperform previous work. We also illustrate the relation between the SDA and the Least Squares Disclosure Attack (LSDA).\n\nAdditionally, we improve the theoretical analysis of the LSDA in \\cite{petsLSDA} and extend it to one of the proposed variants of the SDA, which helps us understand the tradeoffs in performance versus complexity when attacking mixes.\n\nThe rest of the paper is organized as follows: we start with a brief overview of the current attacks on threshold mixes in Sect.~\\ref{sec:relwork}. In Sect.~\\ref{sec:sysmodel}, we introduce our system model and notation and then proceed with our revision of statistical disclosure attacks in Sect.~\\ref{sec:revision}. We perform a theoretical analysis of the attacks in Sect.~\\ref{sec:mse} and validate our results in Sect.~\\ref{sec:eval}. Finally, we conclude in Sect.~\\ref{sec:conclusions}.\n\n\n\n\\section*{\\small ACKNOWLEDGEMENTS}\n\\small\nThis research was supported by the European Union under\nproject LIFTGATE (Grant Agreement Number 285901), the European\nRegional Development Fund (ERDF) and the Spanish Government\nunder projects DYNACS (TEC2010-21245-C02-02\/TCM) and\nCOMONSENS (CONSOLIDER-INGENIO 2010 CSD2008-00010),\nand the Galician Regional Government under projects \"Consolidation\nof Research Units\" 2009\/62, 2010\/85. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n\n\\section{Performance Analysis}\n\\label{sec:mse}\n\nIn this section, we aim at deriving a theoretical expression for the \\emph{Mean Squared Error of sender profile $i$}, which we define as $\\MSEi\\doteq||\\sendprof{i}-\\sendprofest{i}||^2=\\sum_{j=1}^{\\nusers} \\left(\\prob{j}{i}-\\probest{j}{i}\\right)^2$, for the described estimators. Due to space limitations, we reduce our analysis to  $\\SDAenhtwo$ and $\\LSDA$.\n\nWe start by deriving an expression of $\\mbox{MSE}_i$ in $\\LSDA$. In order to do so, we first show, by using the law of total expectation together with $\\mbox{E}\\left\\{ \\recvv{j}{}| \\sendm{} \\right\\}=\\sendm{} \\cdot \\recvprof{j}$, that this estimator is unbiased, since\n\\begin{equation}\n \\mbox{E}\\{ \\recvprofest{j} \\}=\\mbox{E}\\left\\{ \\mbox{E}\\left\\{ \\recvprofest{j} | \\sendm{} \\right\\} \\right\\}= \\mbox{E}\\left\\{ \\left( \\sendm{T} \\sendm{} \\right)^{-1} \\sendm{T} \\mbox{E}\\left\\{ \\recvv{j}{} | \\sendm{}\\right\\} \\right\\}=\\recvprof{j}\n\\end{equation}\n\nUsing this fact, along with the law of total variance, we can write the covariance matrix of $\\recvprof{j}$ as\n\\begin{equation} \\label{eq:sigmaycond}\n \\mathbf{\\Sigma}_{\\recvprof{j}}=\\mbox{E}\\left\\{ \\mathbf{\\Sigma}_{\\recvprof{j} | \\sendm{} } \\right\\}=\\mbox{E}\\left\\{ \\left(\\sendm{T} \\sendm{} \\right)^{-1} \\sendm{T} \\mathbf{\\Sigma}_{\\recvv{j}{}|\\sendm{}}\\sendm{}\\left(\\sendm{T}\\sendm{}\\right)^{-1} \\right\\}\n\\end{equation}\n\nWe model $\\{\\send{1}{r}, ..., \\send{\\nusers}{r}\\}$ together as a multinomial distribution with $t$ trials and probabilities $\\{\\sendfreq{1}, ..., \\sendfreq{\\nusers}\\}$. In order to compute (\\ref{eq:sigmaycond}), we first assume that the number of observations is large enough, so that we can approximate $\\sendm{T}\\sendm{}\\approx\\mbox{E}\\{\\sendm{T}\\sendm{}\\}=\\autocorr \\cdot \\rho$, where $\\autocorr$ is the autocorrelation matrix of the input process,\n\\begin{equation}\n \\autocorr=\\thre\\left[\\sendfreqm + \\left(\\thre-1\\right)\\sendfreqm \\mathbf{1}_{\\nusers\\times \\nusers} \\sendfreqm\\right]\n\\end{equation}\nwhere $\\sendfreqm=\\mbox{diag}\\{ \\sendfreq{1}, ..., \\sendfreq{\\nusers}\\}$.\n\nApplying the matrix inversion lemma, we can write the inverse of this autocorrelation matrix as\n\\begin{equation} \\label{eq:invrx}\n \\autocorr^{-1}=\\frac{1}{t}\\left[\\sendfreqm^{-1}-\\left(1-\\frac{1}{\\thre}\\right)\\mathbf{1}_{\\nusers\\times \\nusers} \\right]\\,.\n\\end{equation}\n\nNow that, using $\\sendm{T}\\sendm{} \\approx \\autocorr \\cdot \\rho$, the only term remaining inside $\\mbox{E}\\{\\cdot\\}$ in (\\ref{eq:sigmaycond}) is $\\mbox{E}\\{ \\sendm{T}\\mathbf{\\Sigma}_{\\recvv{j}{}|\\sendm{}} \\sendm{}\\}$. We model $\\recv{j}{r}|\\sendm{}$ as the sum of $\\nusers$ binomial processes with $\\send{i}{r}$ trials and probabilities $\\prob{j}{i}$, for $i=1, 2, ..., \\nusers$. Let $\\binvar{k}=\\prob{j}{k}\\cdot(1-\\prob{j}{k})$ and $\\binvarm=\\mbox{diag}\\{\\binvar{1}, ..., \\binvar{\\nusers}\\}$. Then, $\\mathbf{\\Sigma}_{\\recvv{j}{}|\\sendm{}}$ is a diagonal matrix whose $(r,r)$-th element is $\\left(\\mathbf{\\Sigma}_{\\recvv{j}{}|\\sendm{}}\\right)_{r,r}=\\sum_{k=1}^{\\nusers} \\send{k}{r} \\binvar{k}$. Operating,\n\\begin{equation} \\label{eq:midterm}\n\\begin{array}{l}\n \\mbox{E}\\{ \\sendm{T}\\mathbf{\\Sigma}_{\\recvv{j}{}|\\sendm{}} \\sendm{}\\}=\\\\\n \\rho \\left\\{ \\sendfreqm \\left( \\eta_j \\thre^{(3)} \\mathbf{1}_{\\nusers\\times\\nusers} + \\binvarm \\mathbf{1}_{\\nusers\\times\\nusers} \\thre^{(2)} \n + \\mathbf{1}_{\\nusers\\times\\nusers} \\binvarm \\thre^{(2)} \\right) \\sendfreqm \\right\\}\\\\\n + \\ \\rho \\left\\{\\left(\\eta_j \\thre^{(2)} \\mathbf{I}_{\\nusers\\times\\nusers} + \\thre \\binvarm \\right) \\sendfreqm \\right\\}\n \\end{array}\n\\end{equation}\nwhere $\\eta_j=\\sum_{k=1}^{\\nusers} \\sendfreq{k} \\binvar{k}$ and $\\thre^{(n)}=\\thre\\cdot(\\thre-1)\\cdot ... \\cdot(\\thre-n+1)$.\n\nPlugging (\\ref{eq:invrx}) and (\\ref{eq:midterm}) into (\\ref{eq:sigmaycond}) we get an approximation of $\\mathbf{\\Sigma}_{\\recvprof{j}}$. Now, taking each of the diagonal elements of this matrix, which are $\\mbox{Var}\\{\\probest{j}{i}\\}$ for $i=1, ..., \\nusers$ and adding them along $j$ to obtain $\\MSEi=\\sum_{j=1}^{\\nusers} \\mbox{Var}\\{ \\probest{j}{i} \\}$, we finally get\n\\begin{equation} \\label{eq:MSE_lsda}\n \\MSEi^{\\LSDA}\\approx \\frac{1}{\\rho} \\left\\{ \\left(\\sendfreq{i}^{-1}-1\\right)\\left(1-\\frac{1}{\\thre}\\right)\\meanuniformi_{\\LSDA} + \\frac{\\sendfreq{i}^{-1}}{\\thre} \\cdot \\uniformi{i}\\right\\}\n\\end{equation}\nwhere $\\meanuniformi_{\\LSDA}=\\sum_{k=1}^{\\nusers} \\sendfreq{k} \\uniformi{k}$ is the average uniformity of the sender profiles.\n\n\nFollowing a similar approach, it can be shown for $\\SDAenhtwo$ that, when the number of observed rounds is large enough,\n\\begin{equation} \\label{eq:MSE_sda2}\n \\MSEi^{\\SDAenhtwo}\\approx \\frac{1}{\\rho}  \\left\\{ \\left(\\sendfreq{i}^{-1}-1\\right)\\left(1-\\frac{1}{\\thre}\\right)\\meanuniformi_{\\SDAenhtwo} + \\frac{\\sendfreq{i}^{-1}}{\\thre} \\cdot \\uniformi{i}\\right\\}\n\\end{equation}\nwhere $\\meanuniformi_{\\SDAenhtwo}=\\sendfreq{i}\\uniformi{i} + (1-\\sendfreq{i})\\uniformi{\\backg{i}}$ is the average uniformity considering that there are only two users in the system: the user $i$ and her background. \n\nNote that the only approximations made to derive (\\ref{eq:MSE_lsda}) and (\\ref{eq:MSE_sda2}) were $\\sendm{T}\\sendm{}\\approx\\mbox{E}\\{\\sendm{T}\\sendm{}\\}=\\autocorr \\cdot \\rho$ and its equivalent with matrix $\\sendmcomp{i}$. Therefore, these MSE estimators are more accurate as the number of observed rounds is large.\n\nGiven the definition of the background sending profile in Sect.~\\ref{sec:sysmodel}, it is easy to see that $\\meanuniformi_{\\SDAenhtwo}\\geq \\meanuniformi_{\\LSDA}$, and therefore $\\MSEi^{\\SDAenhtwo}\\geq \\MSEi^{\\LSDA}$, where the equality holds only when all users have the same sending profile. This proves that $\\LSDA$ will eventually outperform $\\SDAenhtwo$ in terms of MSE when the attacker observes the system indefinitely.\n\n\n\\section{Previous work}\n\\label{sec:relwork}\n\nThe Disclosure Attack \\cite{agrawal03} relies on Graph Theory to reveal the exact set of friends of a user (Alice), seeking for mutually disjoint sets of receivers. This attack is known to be NP-complete but there exist other implementations that speed up the search \\cite{kesdoganHS}.\n\nDanezis proposed the Statistical Disclosure Attack (SDA) \\cite{originalSDA} as a faster alternative to the Disclosure Attack, which is based on the idea that it is possible to statistically isolate Alice's sending behavior after observing a large amount of her message's sets of receivers. The original SDA is limited to a specific scenario and was extended later to a more general user model and more complex mixing algorithms \\cite{mathewsonSDA}.\n\nThe Least Squares Disclosure Attack (LSDA) \\cite{petsLSDA} models profiling as a least squares problem, minimizing the error between the actual number of output messages and a prediction based on the input messages.\n\nIn this work, we present an analysis of the family of statistical disclosure attacks \\cite{originalSDA,mathewsonSDA} and the LSDA \\cite{petsLSDA}, which share the goal of estimating the sending behavior of the users by combining the set of observations in an appropriate way. Other approaches that we leave out of our work are the Two-Sided SDA (TS-SDA) \\cite{tsSDA} and the Reversed SDA (RSDA) \\cite{RSDA}, which assume that users reply to messages; the Perfect Matching Disclosure Attack (PMDA) and the Normalized Statistical Disclosure Attack (NSDA) \\cite{PMDA}, which exploit that the relationship between sent and received messages is one-to-one; and the Bayesian inference-based approach, Vida \\cite{Vida}.\n\n\n\\section{Revisiting the Family of Disclosure Attacks}\n\\label{sec:revision}\n\n\\subsection{The Original Statistical Disclosure Attack}\n\nDanezis introduced the original Statistical Disclosure Attack ($\\SDA$) in \\cite{originalSDA}, which provides an estimator of $\\prob{j}{i}$ under the assumptions that the user $i$ does not send more than one message each round and the background traffic for that user is uniform, i.e. $\\prob{j}{\\backg{i}}=\\frac{1}{N}$ for $j=1,2,...,\\nusers$.\n\nDanezis claims that, by using the Law of Large Numbers, the mean of the observations $\\recv{j}{r}$ in the rounds where $i$ has sent at least one message can be written as\n\\begin{equation} \\label{eq:OSDA_y_first}\n \\frac{\\sendvplus{i}{T} \\recvv{j}{}}{\\sendvplus{i}{T} \\mathbf{1}_{\\rho}} \\approx  \\prob{j}{i} + (t-1) \\cdot \\prob{j}{\\backg{i}}\\,,\n\\end{equation}\nand therefore an estimator for $\\prob{j}{i}$ is\n\\begin{equation}\n \\probest{j}{i}^{\\SDA} = \\frac{\\sendvplus{i}{T} \\recvv{j}{}}{\\sendvplus{i}{T} \\mathbf{1}_{\\rho}}- (t-1) \\cdot \\probest{j}{\\backg{i}}\\,\n        \\mbox{, with } \\probest{j}{\\backg{i}}=\\frac{1}{N}\\,.\n\\end{equation}\n\nIn order to compare $\\SDA$ with its variants, note that we can write (\\ref{eq:OSDA_y_first}) as\n\\begin{equation} \\label{eq:OSDA_y}\n \\sendvplus{i}{T} \\recvv{j}{} \\approx \\sendvplus{i}{T} \\mathbf{1}_{\\rho} \\cdot \\prob{j}{i} \n                              + \\sendvplus{i}{T} \\left( \\mathbf{1}_{\\rho} \\cdot t - \\mathbf{1}_{\\rho} \\right) \\cdot \\prob{j}{\\backg{i}}\\,.\n\\end{equation}\n\n\n\n\\subsection{Generalized Statistical Disclosure Attack}\n\nMathewson and Dingledine extended Danezis' attack in \\cite{mathewsonSDA}, allowing user $i$ to send multiple messages in a round and estimating the background from the observations.\n\nUsing this extension, (\\ref{eq:OSDA_y}) becomes\n\\begin{equation} \\label{eq:SDAenh0_y}\n \\sendvplus{i}{T} \\recvv{j}{} \\approx \\sendvplus{i}{T} \\sendv{i}{} \\cdot \\prob{j}{i} \n                              + \\sendvplus{i}{T} \\sendv{b}{} \\cdot \\prob{j}{\\backg{i}}\\,,\n\\end{equation}\nwhere we have just replaced the $\\mathbf{1}_\\rho$s which referred to the number of messages sent by user $i$ in each round in (\\ref{eq:OSDA_y}) with the actual number of messages sent by $i$, $\\sendv{i}{}$, and $\\mathbf{1}_{\\rho}\\cdot t-\\sendv{i}{}=\\sendv{b}{}$. \n\nThe background profile is estimated by computing the average number of messages received by $j$ in the rounds where $i$ does not participate and dividing by the total number of messages exiting the mix each round ($\\thre$),\n\\begin{equation} \\label{eq:back_est1}\n \\probest{j}{\\backg{i}}=\\frac{1}{t} \\cdot \\frac{\\left(\\mathbf{1}_{\\rho} - \\sendvplus{i}{}\\right)^{T} \\recvv{j}{} }{\\left(\\mathbf{1}_{\\rho} - \\sendvplus{i}{}\\right)^{T} \\mathbf{1}_{\\rho}}\\,.\n\\end{equation}\n\nWe denote this attack by $\\SDAenhzero$, whose estimator is\n\\begin{equation} \\label{SDAenh0}\n \\probest{j}{i}^{\\SDAenhzero} = \\frac{\\sendvplus{i}{T} \\recvv{j}{}}{\\sendvplus{i}{T} \\sendv{i}{}}\n                  - \\frac{\\sendvplus{i}{T}\\sendv{b}{}}{\\sendvplus{i}{T} \\sendv{i}{}} \\cdot \\probest{j}{\\backg{i}}\\,.\n\\end{equation}\n\n\n\\subsection{Improvements in the Generalized SDA}\n\nThe attack described in the previous section performs an average of the outputs in those rounds where user $i$ sends at least one message in order to compute $\\probest{j}{i}^{\\SDAenhzero}$, giving the same value to those outputs regardless of the actual participation of user $i$. We propose a new estimator, which we denote $\\SDAenhone$, that counts the outputs once for every message sent by user $i$, therefore giving more weight to those rounds where the number of messages sent by $i$ is larger.\n\n\nUsing this approach, (\\ref{eq:SDAenh0_y}) becomes\n\\begin{equation} \\label{eq:SDAenh1_y}\n \\sendv{i}{T} \\recvv{j}{} \\approx \\sendv{i}{T} \\sendv{i}{} \\cdot \\prob{j}{i} \n                              + \\sendv{i}{T} \\sendv{b}{} \\cdot \\prob{j}{\\backg{i}}\\,,\n\\end{equation}\nwhere we have replaced the vector we used to select the rounds we were taking into account, $\\sendvplus{i}{}$, by the vector with the actual number of messages sent by $i$ in each round, $\\sendv{i}{}$.\n\nFrom (\\ref{eq:SDAenh1_y}), we get the following estimator,\n\\begin{equation} \\label{SDAenh1}\n \\probest{j}{i}^{\\SDAenhone} = \\frac{\\sendv{i}{T} \\recvv{j}{}}{\\sendv{i}{T} \\sendv{i}{}}\n                  - \\frac{\\sendv{i}{T}\\sendv{b}{}}{\\sendv{i}{T} \\sendv{i}{}} \\cdot \\probest{j}{\\backg{i}}\\,,\n\\end{equation}\nwhere $\\probest{j}{\\backg{i}}$ is estimated as in (\\ref{eq:back_est1}).\n\nNote that the idea behind this estimator appears in \\cite{mathewsonSDA} applied to other mixing algorithms. The analysis of SDA in \\cite{mathewsonSDA} also features the idea of exploiting observations from rounds where user $i$ appears as a sender in order to compute $\\probest{j}{\\backg{i}}$.\n\n\nThe latter idea inspires our second variant, denoted $\\SDAenhtwo$, which uses the observations from \\emph{all rounds} to get the background estimation. Following (\\ref{eq:SDAenh1_y}), we can write\n\\begin{equation} \\label{eq:SDAenh2_y}\n\\begin{cases} \n \\sendv{i}{T} \\recvv{j}{} = \\sendv{i}{T} \\sendv{i}{} \\cdot \\probest{j}{i} \n                              + \\sendv{i}{T} \\sendv{\\backg{i}}{} \\cdot \\probest{j}{\\backg{i}}\\\\\n \\sendv{\\backg{i}}{T} \\recvv{j}{} = \\sendv{\\backg{i}}{T} \\sendv{i}{} \\cdot \\probest{j}{i} \n                              + \\sendv{\\backg{i}}{T}  \\sendv{\\backg{i}}{} \\cdot \\probest{j}{\\backg{i}}\\,.\n\\end{cases}\n\\end{equation}\n\nIf we define the $\\rho \\times 2$ matrix $\\sendmcomp{i}=\\left( \\sendv{i}{},  \\sendv{\\backg{i}}{} \\right)$, the new estimator $\\probest{j}{i}^{\\SDAenhtwo}$ can be obtained by solving\n\\begin{equation} \\label{eq:SDAenh2}\n \\left( \\begin{array}{c}\n \\probest{j}{i}^{\\SDAenhtwo} \\\\\n \\probest{j}{\\backg{i}}  \\end{array} \\right) =\n \\left( \\sendmcomp{i}^T  \\sendmcomp{i} \\right)^{-1}  \\sendmcomp{i}^T  \\recvv{j}{}\\,.\n\\end{equation}\n\n\n\\subsection{The Least Squares Disclosure Attack}\n\nThe estimator in (\\ref{eq:SDAenh2}) uses the information from all outputs when estimating both $\\prob{j}{i}$ and $\\prob{j}{\\backg{i}}$. However, users' profiles are solved independently, compressing information in matrices $\\sendmcomp{i}$. We can extend the idea in (\\ref{eq:SDAenh2_y}) considering that, when computing the sender profile of $i$, the background is formed by all the users but $i$. In that case, we would have $\\nusers$ equations with $\\nusers$ unknowns, which are\n\\begin{equation}\n \\sendv{i}{T} \\recvv{j}{} = \\sendv{i}{T} \\sum_{k=1}^{\\nusers} \\left(\\sendv{k}{} \\cdot \\probest{j}{k}\\right)\\,,\\,\\mbox{for}\\, i=1,...,\\nusers\\,.\n\\end{equation}\n\nPresenting this system in matricial form, we have\n\\begin{equation}\n \\sendm{T} \\recvv{j}{} = \\sendm{T} \\sendm{} \\recvprofest{j}\\,.\n\\end{equation}\n\nTherefore, if $\\sendm{T} \\sendm{}$ is not singular, we obtain the Least Squares Disclosure Attack ($\\LSDA$) estimator in \\cite{petsLSDA}, \n\\begin{equation} \\label{eq:LSDA}\n  \\recvprofest{j}^{\\LSDA} = \\left(\\sendm{T} \\sendm{}\\right)^{-1} \\sendm{T} \\recvv{j}{}\\,.\n\\end{equation}\n\n\n\n\n\n\n\\section{System Model and Notation}\n\\label{sec:sysmodel}\n\n\nThroughout the text, we will represent vectors using boldface lowercase characters and matrices using boldface capital letters. We will also use $\\mathbf{1}_N$ to refer to the column vector whose $N$ elements are equal to 1, and $\\mathbf{1}_{N\\times M}$ to the all-ones matrix of size $N\\times M$. The superscript $T$ will denote the transposing operation.\n\n\n\\paragraph{System Model}\n\nOur system consists of a population of $\\nusers$ users, designated by index $i \\in \\{1, 2, ... \\nusers\\}$, which communicate using a threshold mix. The system works as follows: every time a user $i$ in our population wants to send a message to another user $j$, she encrypts the message and sends it to the mix. The mix receives and stores the messages until it has gathered $\\thre$ of them. Then, it transforms the messages cryptographically to change their appearance and outputs them in a random order; hence hiding the correspondence between incoming and outgoing messages. We call this process a \\emph{round} of mixing, and $\\thre$ is the \\emph{threshold} of the mix.\n\nWe denote the number of messages user $i$ sends in round $r$ by $\\send{i}{r}$. We define the column vector containing all the messages sent by user $i$ up to round $\\rho$ as $\\sendv{i}{}=[ \\send{i}{1}, \\send{i}{2}, ..., \\send{i}{\\rho}]^T$, and the matrix of all observed inputs to the mix as $\\sendm{}=\\left( \\sendv{1}{}, \\sendv{2}{}, ..., \\sendv{\\nusers}{} \\right)$. Likewise, we denote the number of messages user $j$ receives in round $r$ by $\\recv{j}{r}$ and define $\\recvv{j}{}=[ \\recv{j}{1}, \\recv{j}{2}, ... , \\recv{j}{\\rho}]^T$ and $\\recvm{}=\\left( \\recvv{1}{}, \\recvv{2}{}, ..., \\recvv{\\nusers}{}\\right)$. Additionally, we define $\\sendplus{i}{r}$ as a binary representation of $\\send{i}{r}$, denoting whether there is at least one message sent by user $i$ in round $r$ ($\\sendplus{i}{r}=1$) or not ($\\sendplus{i}{r}=0$).\nWe also define, $\\sendvplus{i}{}=[\\sendplus{i}{1}, \\sendplus{i}{2}, ... , \\sendplus{i}{\\rho}]^T$. \n\nUser $i$ sends messages to their recipients according to her \\emph{sender profile} and her \\emph{sender frequency}. We define the sender profile of user $i$ as $\\sendprof{i}=[ \\prob{1}{i}, \\prob{2}{i}, ..., \\prob{\\nusers}{i}]^T$, where $\\prob{j}{i}$ models the probability that user $i$ sends a message to user $j$. The sender frequency $\\sendfreq{i}$ models the probability that a message arriving to the mix comes from user $i$ ($\\sendfreq{i}\\geq 0$ for $i=1,2, ..., \\nusers$ and $\\sum_{i=1}^{\\nusers} \\sendfreq{i}=1$). We also define the vector $\\recvprof{j}=[\\prob{j}{1}, \\prob{j}{2}, ..., \\prob{j}{\\nusers}]^T$ which shall come in handy later. \nWe make no assumptions on the distribution of each sender profile, other than $\\prob{j}{i}\\geq 0$ for $i,j=1,2,...,\\nusers$ and $\\sum_{j=1}^{\\nusers} \\prob{j}{i}=1$ for $i=1,2,...,\\nusers$.\n\nWe define the \\emph{uniformity} of the sender profile of user $i$ as $\\uniformi{i}=1-\\sum_{j=1}^{\\nusers} \\prob{j}{i}^2$. The uniformity $\\uniformi{i}$ ranges from $0$, when user $i$ always sends messages to the same contact (i.e. $\\prob{k}{i}=1$, $\\prob{j}{i}=0$ for $k \\in \\{1, ..., \\nusers\\}$ and $j\\neq k$, $j=1,...,\\nusers$), to $\\frac{\\nusers-1}{\\nusers}$, when she sends messages to all the other users equiprobably.\n\n\nFinally, we define the \\emph{background traffic} of a user $i$ as an aggregate of the traffic generated by all users but $i$. This way, vector $\\sendv{\\backg{i}}{}$ contains the messages sent by all users but $i$, $\\sendv{\\backg{i}}{}=\\sum_{\\substack{k=1\\\\k\\neq i}}^{\\nusers} \\sendv{k}{}=\\mathbf{1}_\\rho \\cdot t - \\sendv{i}{}$. The \\emph{background profile} is $\\sendprof{\\backg{i}}=[\\prob{1}{\\backg{i}}, \\prob{2}{\\backg{i}}, ... , \\prob{\\nusers}{\\backg{i}}]^T$ where  $\\prob{j}{\\backg{i}}=\\sum_{\\substack{k=1\\\\k\\neq i}}^{\\nusers}  \\frac{\\sendfreq{k}}{1-\\sendfreq{i}} \\cdot \\prob{j}{k}$ and the uniformity of this sender profile is denoted by $\\uniformi{\\backg{i}}$. In all cases, user $i$ will be clear from the context.\n\n\n\\paragraph{Adversary Model}\n\nWe consider a global passive adversary that observes the system during $\\rho$ rounds. The adversary observes the identity of the users communicating through the mix and knows all the parameters of the system. We also assume that the adversary is not able to link any messages by their content, i.e. the cryptographic transformations do not leak information.\n\nThe goal of the adversary is to infer the sending behavior of the users in the system from the observations, i.e. to obtain an estimator $\\probest{j}{i}$ of $\\prob{j}{i}$ given the input and output observations $\\sendm{}$ and $\\recvm{}$. \n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{S:1}\nIt is believed that a virus that causes the novel COVID-19 disease spreads mainly from having a close interaction or contact with the person already being affected with the virus, and still carrying the attributes of the virus. Since December 2019, work has already begun on the development of a potential vaccine for the cure, however, until the development of vaccines for masses, the only possible way of protection is to limit the interaction with the people, isolate the people through imposing full or smart lock-down. The efficient smart lockdown could be imposed by employing the track and trace method that only isolates the infected people and their contacts.  For this purpose, several smartphone apps for android, iOS, and Windows operating systems have been developed for tracing, tracking, and informing citizens about whether they have recently come in close contact with the person showing confirmed attributes of COVID-19. These apps can be either private or government-owned.\n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=15cm,height=8cm]{images\/contacttracing.png}\n\\caption{Tracing and Isolating through Smartphone based Contact-tracing \\cite{Ferrettieabb6936}}\n\\label{fig:tracing}\n\\end{figure*}\n\n\\color{blue} The Contact tracing operates by identifying positive cases of COVID-19 and asking them for their close contacts manually or identifying their close contacts in an automated way.  Figure \\ref{fig:tracing} presents the building block and working mechanism of the contact tracing apps \\cite{Ferrettieabb6936}.\nThe contact-tracing apps exchange information when the phones of two persons are close enough to each other. These people who came socially close to one another will be informed if their counterparts during the social interaction were officially infected with the COVID-19. These apps can only serve as the anchor point to inform citizens and provide suggestions on whether they should go to isolation or not. These apps have already shown efficiency in controlling the spread of the virus in South Korea and Singapore and flatten the spread curve through \"Test\" and then  \"Trace\" mechanism \\cite{southkorea,southkore1}.\n\\color{black}\n\nThe contact-tracing apps are broadly developed by the national or country lead health regulators. To provide a reliable and efficient decision, the developed apps utilize the information from various smartphone sensors (GPS, Bluetooth) along with names, addresses, gender, age, contact details, calling log history, and contact history, etc. to make the decision. These apps either interact automatically with the national health data system for the test results of the citizens or citizens could manually provide test results to the health organization. For instance, upon downloading, Pakistan's track \\& trace app \\footnote{https:\/\/tinyurl.com\/y4lt634d} requests permission to use the device's location and user personal details such as name, phone number, and email address, and that stored data will be shared with a third party. The current version of this app (as of June-2020) then uses the device's location and the location of users with positive COVID-19 test results to render a map showing high-level \\textit{hotspots} for COVID-19 infections. Similarly, Google and Apple jointly developed an API that enables app developers to use the Bluetooth beacon messages for exchanging information between two persons who are in close contact with each other and showing virus symptoms. Some countries use the call detailed records to trace the close contact of the infected person and isolate them as well. \n\nThe use of these apps is normally voluntary and is considered as the support to control the spread of the virus. The developed app requests users for specific permission e.g. contact details, call history, web searches, camera permissions, access to call records, messages, and mobile media (videos and photos). This information could pose serious privacy and security risks to the users and limit users to use these apps. The privacy of users may be protected through the use of different mechanism e.g. data anonymization, differential privacy, and decentralized app development \\cite{Troncoso2020}. However, it is already identified that anonymization systems are not providing effective privacy-preservation \\cite{ga1,ga2,ga3} and decentralized app development is still at the early stage and progressing very slowly. \n\n\nIn this paper, we provide a first look at the permission, security, and privacy analysis of the contact tracing apps available at the Android and iOS app stores. We have studied app stores to define the nature of the privacy risks these apps have, what types of permissions these apps are requesting for their functioning, which permissions are unnecessarily required, and how they store and process the user data. Currently, only a small number of certified apps (developed by the country regulator) are available at the play stores, so we performed an exhaustive manual analysis on the available apps. Our analysis shows that the majority of the track and trace apps collect personal information such as name, device ID, and location, however, some apps require access to further resources such as SMS, microphone, camera, and storage memory of the device. Access to such resources is not required for the accurate function of such apps, and, therefore should not be requested by the developers. Furthermore, a number of apps disclose sharing data with third parties, however, a small number of these acquire the permission of the user before sharing this data with the third parties. As track and trace apps are voluntary and rely on the public's trust to achieve their function effectively \\cite{IND2020}, addressing concerns with regards to data collection and sharing are paramount to their success to combat COVID-19.  \n\nThe rest of the paper is organized as follows: Section \\ref{sec:related} provides a discussion on privacy and permission analysis of mobile apps and work performed towards the development of track and trace apps. Section \\ref{sec:problem} defines the background on the trace and track smart applications. The section also provides the working mechanism and important features of developed apps. Section \\ref{sec:analysis} critically analyses the security and privacy of different apps. Section \\ref{sec:conclusion} provides recommendations for design chose secure development and concludes the paper. \n\n\\section{Related Work}\n\\label{sec:related}\n\nA large number of works have been presented that analyse the functionality of smartphone apps and the leakage of the sensitive information of their users \\cite{S1,S2,S3}. Many contact tracing applications involve tracing users using GPS, Bluetooth, and wireless technologies \\cite{ct12, ct13, ct14, ct15, ct16, ct17}. These approaches usually provide users with two options. Either the user has to self-report themselves, or the application takes the help of a wireless technology \\cite{ct11}. A large number of people are currently downloading and using contact-tracing apps, and hence the privacy aspects of these apps have become paramount for the research and development of unanimous privacy regulation. The regulators such as the Federal Trade Commission, the US National Telecommunication and information administrations, the European Union Commission \\cite{eu1,eu2}, Information Commissioner's Office \\cite{ICO} are analyzing and providing these guidelines to the app developers, content creators, website operators to improve the development of their products in terms of security and privacy. In this section, we summarize works related to the security and privacy of smartphone apps. \n\nSmartphone apps normally get access to user data and other information through the use of permission \\cite{appper} that the users provide to the smartphone app at the time of app installation. For example, an app might ask to grant the to see the location of the user, the messages stored on the mobile phone, the search history, etc. The user can still control the permission after the installation but it might affect the functionality of apps operations. Providing permission to various private information would expose the private information of users to the advertisers, insurance companies \\cite{per1,per4}, and publicly expose personal data of the users without the user's consent \\cite{per2,per3}. A large number of smartphones app also ask unnecessary permission that is not required for the functionality of the app, these apps might pose a serious threat to privacy and security of the users \\cite{un1,un2}. Muhammad et al. \\cite{muhammad,muhammad1} analyzed the security and privacy of smartphone apps designed for blocking the advertisement and providing mobile VPN clients. Ilaria et al. \\cite{Liccardi} analyzed the permissions requested by the smartphone apps and assigned a sensitivity score to the app if the app asked to read the personal information of the users. They concluded that around 56\\% of the app asks users to provide permission to sensitive parts of the user's data. Barrera et al. \\cite{berral} investigated the relationship between free android apps and the most popular 1100 Android apps by deploying machine learning methods. It is also concluded that people are willing to use the paid version of smartphone apps if apps are not asking for unnecessary permissions. Enck et al. \\cite{enck} proposed a lightweight certification mechanism to identify Android apps that are asking for suspicious permissions. Pern et al. \\cite{pern} studied the user-consent permission systems by using the user-centric data from the Facebook apps, chrome browser extensions, and Android smartphone apps. It is very important to develop tools or applications that inform users about the privacy indicator of the apps they are using for specific purposes. To address this issue, Max et al. \\cite{max} developed a prototype that provides users with privacy indicators of the app. The prototype also identifies previously exposed hidden information flows out of the apps.\n\nContact tracing with smartphones can be employed to restrict the transmission of a pandemic disease.  Utilizing computing technologies to avert and control the pandemic seems to be an obvious choice.  However, these contact tracing apps might invade privacy, collect personal data, and justify mass surveillance against users' wishes. There must be a protocol for contact tracing that observes commitment to privacy, as well as provides the consent mechanism where there is a need to share individual data. Contact tracing may collect personal data such as location which is not an effective privacy control when it comes to user's data \\cite{ct7}. The process of contact tracing usually involves collecting users' privacy information without informing them. Privacy-literate individuals might be reluctant to share their information which in turn hampers the process of contact tracing. Privacy-preserving approaches might encourage individuals to participate more in this process and increase their confidence in those applications \\cite{ct11}. \n\nProminent privacy researchers from across the world are arguing with the government agencies and vendors involved in developing the contact-tracing application about the privacy, and also highlighted the catastrophic consequences these apps would have on the citizen's private lives \\cite{ross2,ross1}. To ensure privacy, Berke et al. \\cite{Cho2020ContactTM} utilized the semantics of private set interaction for assessing the risk exposure of users using encrypted and anonymous GPS locations. Manish et al. \\cite{manish} analyzed the privacy preservation mechanism for various contact tracing applications and discussed the attributes which contact-tracing apps should have to ensure the privacy of users.\\color{blue} Michael et al \\cite{ethics} discussed the ethical consideration of contact-tracing apps for fighting against the COVID-19. Several contact-tracing application has been compared in \\cite{comp} in terms of data collection, retention of data, purpose, and sharing of collected data, what mechanisms the apps have deployed to ensure the privacy of users. Most recently, Carmela et al. \\cite{Troncoso2020} describe and analyse a decentralized system for secure and privacy-preserving proximity tracing to combat the spread of COVID-19. The system is solely based on the anonymous identifiers of positive users of the COVID-19 without providing the exact location information to the health authorities. Serge \\cite{cryptoeprint} analyse the security and privacy properties of the pan-European Decentralized Privacy-Preserving Proximity Tracing (DP3T) system. Health authorities or any other users would not be able to learn the private information of the users except a notification message when a person is exposed to COVID-19 affected person.Ruoxi et al. \\cite{sun2020vetting} analyze the security and privacy of contact tracing apps in three dimensions: a) evaluate the design choice (centralized or decentralized) used for privacy preservation, b) static analysis for the identification of potential vulnerabilities and c) (iii) evaluate the robustness of approaches used for privacy-preservation. The paper has not analyzed the permission analysis. Yaron et al. \\cite{Yaron} analysed the security and privacy properties of Bluetooth based specification by Apple and Google, concluding that the specifications may have some significant security and privacy risks.  The Centers for Disease Control and Prevention (CDS) have issued guidelines that define a set of features a contact tracing app should have to help health departments to overcome the COVID-19 pandemic \\cite{cdc}. \n\\color{black}\n\n\n\n\n \n\\section{CHARACTERIZING Contact Tracing Apps}\n\\label{sec:problem}\nContact tracing is an important tool for the community to prevent the outspread of novel pandemic diseases, such as COVID-19 \\cite{ct1}. In past contact-tracing tools have shown effectiveness against the spread of transmissible diseases such as STD, HIV, Ebola, and tuberculosis \\cite{ct9, ct10}.\nContact tracing is the process of identifying persons who are in close contact with the infected person so that exposed targets can be informed to have self-isolation and quarantine, thus breaking the chain of transmission \\cite{ct7}. The current outbreak of COVID-19 and its highly contagious feature motivates technology developers to develop smartphone apps for the effective tracing of the footprint of the disease. In this section, we provide the architectural setup of contact-tracing apps and their significance towards controlling the spread of disease.\n\n\\subsection{Centralized and Decentralized Architecture}\n\\color{blue} The design of contact-tracing apps is mainly using data from the users thus has some privacy concerns which motivate the developer to come up with privacy-preserving solutions. The privacy of users can be addressed using the centralized and decentralized system setup. The centralized and decentralized apps entirely have different architecture and properties shown in Figure \\ref{fig:cendcen} and explained below.  \n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=15cm,height=5cm]{images\/COVID19.png}\n\\caption{Architectural Setup of Contact Tracing Apps}\n\\label{fig:cendcen}\n\\vspace{-0.5cm}\n\\end{figure*}\n\n\n\\subsubsection{Centralised models} In the centralized setup, the smartphones of the users having specific contact-tracing apps send the random identifier to the centralized trusted system. The centralized system in this setup holds the information from all users of the app. If a person has tested positive for the COVID-19 virus, the identifier of other users who have exchanged identifiers in the past can be sent to the centralized server along with other information e.g. time data is sent, a time when identifiers are exchanged, etc. The centralized system decrypts the identifiers and automatically notifies the interacted phones suggesting or informing users to self-isolate or take other preventive measures. The centralized system can also utilize the available information for further analysis and policies for placing lockdown in hard-hit proximities.\n \n\\subsubsection{Decentralised models} In the decentralized setup there is not a trusted centralized system that exists for the handling of the user's data and matching of smartphone's identifier. If a person is diagnosed positive with the COVID-19, the identifier of his phone and test result is uploaded to the centralized system. Other smartphones having the app can access these reports and locally establish the truth whether he was close to an infected individual or not. If a smartphone comes across the identity that has COVID-19 then alert is the sent to the user of the smartphone for precaution and self-isolation. The location and proximity of the person are not known to the centralized system thus ensuring the privacy of the users using the app.  The health organizations or the government still used the shared data to understand the spread of the virus in the community but would not have detailed information about the users. \n\\color{black}\n\\color{blue}\n\\subsection{Significance of Contact Tracing for COVID-19}\nSince the outbreak of the COVID-19 pandemic in December 2019, as of June 2020, there exists no medicine or vaccine to fight against the rapidly spreading pandemic. Governments across the world are currently focusing on the ways that would have the least load on their health systems. This has been achieved through imposing travel restrictions or lockdown however, it is not only affecting the economy but there are also fears of the second wave of infection once the restrictions are relaxed. The governments are finding ways to identify the methods for contact tracing in order to quickly identify and isolate the infected persons. The manual contact tracing is not only slow and has a late response, but would also require resources for identifying infected persons and then asking for his contacts and then contacts of his contacts to track the flow of the disease. The technologies soon realized the importance of smartphones and used the inbuilt smartphone sensors for tracking in an automated and efficient way.  The use of digital technologies help the citizen at the early stage of the virus spread and inform people for isolation at the early stage. The use of smartphone apps for contact tracing has shown promising results in several countries to combat the spread of the virus \\cite{RePEc}, however, the performance efficiency depends on the number of people using the application \\cite{tech}. One thing that limits the usage of the app is the privacy because a large number of existing apps store data at the central trusted system, and in some circumstance, this data is made available to the third party systems for performing artificial Intelligence and data analysis.\n\\color{black}\n \\subsection{Vendor Support}\n\\color{blue}As the healthcare officers and medical entities are working together worldwide to fight the spread of this pandemic, Google and Apple have joined an effort and developed a privacy-preserving contact tracing API that uses Bluetooth signals \\cite{ct18} for exchanging information between people who are in close contact with each other. The apps using this API operate in a decentralized fashion, however, a centralized database is maintained.\n\\color{black}\nThis framework allows healthcare agencies to propose or develop smartphone apps that help in limiting the spread of the disease with the help of Bluetooth technology. This API will bring interoperability between iOS and Android devices while maintaining privacy, consent, and transparency \\cite{ct19}. A test project, PACT (Private Automated Contact Tracing), was built at MIT to harness the strength of Bluetooth-based, privacy-preserving, automated contact tracing API. This project detects proximity between contacts with the help of Bluetooth signals within a 6-foot radius. Instead of relying on the GPS, this system sends out random Bluetooth numbers, which can later be updated to a database with the user's consent \\cite{ct21}. The first large-scale pilot for this joint venture has been launched in Switzerland, known as SwissCovid. This application determines the close contact that lasted for more than 15 minutes and notifies the user with the procedure to follow \\cite{ct20}. \n\nApple also released a new application for COVID-19 based on CDC guidelines that provide COVID-19 information across the USA. In this application, the users have to answer some questions related to recent exposure and risk factors. In return, they get a CDC recommendation on what their next step should be. However, this application does not replace a healthcare worker in any way \\cite{ct29, ct30}. Another application, \"HEALTHLYNKED COVID-19 Tracker\", which became the most downloaded coronavirus tracker application for March. The application enables users to track local cases and chat with other users around the world. The most unique feature of this application is that it enables real-time chat with other users and share updates \\cite{ct40, ct41}. \n\n \n \n\n\n\n\n \n  \n   \n   \n   \n   \n   \n   \n  \n\n\n\n \n  \n   \n   \n   \n\n\n\n\n\n\n\n\n\\section{ANALYSIS OF Contact Tracing apps }\n\\label{sec:analysis}\nIn this section, we present our approach to studying current contact tracing apps for COVID-19. We have focused on smartphone apps for any platform (iOS, Android, Microsoft) and available anywhere in the world. Although there appears a concerted effort by governments across the globe to contain the pandemic, we identified through our analysis that many such apps have been developed by third-party individuals or organizations. Therefore, we have also included these in our analysis. Furthermore, as the contact tracing technology is still in its early stages (especially within the context of COVID-19), although many apps claim to perform track and trace function, their effectiveness in this respect is subjective.  \n\n\\begin{figure*}[h!]\n\\centering\n\\includegraphics[width=18cm, height=4.6cm]{images\/covid-timeline.pdf}\n\\caption{Timeline for release of COVID-19 track \\& trace apps}\n\\label{fig:timeline}\n\\vspace{-2.0em}\n\n\\end{figure*}\n\n\\subsection{Data collection}\n\\color{blue}\nIn order to achieve an in-depth analysis of current track \\& trace apps, we performed exhaustive search techniques to collect relevant apps across two major platforms (iOS and Android), irrespective of the country or the developing organization (government and private). We have chosen iOS and Android because these systems currently hold the most market share in the smartphone industry. Specifically, we used keywords such as \\textit{COVID-19}, \\textit{COVID track \\& trace}, and \\textit{Coronavirus track \\& trace}. We analyzed the results of our search queries to filter apps that did not relate to contact tracing to mitigate the spread of COVID-19. We used a manual analysis of the app description to conduct this filtering. Furthermore, as several countries have encountered difficulties to achieve effective contact tracing, we did not exclude apps performing the partial or limited function in this regard. Overall, we identified 26 smartphone apps that claim to perform contact tracing in their description belonging to 17 different countries. Details of these apps are presented in table \\ref{tab:analysis} with a brief description of some apps presented below. Furthermore, Fig \\ref{fig:timeline} presents a graphical representation of the timeline with respect to launching dates of prominent apps. \n\n\\color{black}\n\\begin{itemize}\n   \\item \\textit{COVID-19 Gov PK} is an app developed by the government of Pakistan. Initially, the application provided awareness to citizens about COVID-19, however, with the development of the new \\textit{radius alert} feature, this application provides information which are hot-spot areas that help country to impose smart lock-down \\cite{ct22}. \n   \\item \\textit{Health Canada} is developed by the government of Canada to provide a personalized recommendation to the users based on their risk factors. Personal data collected is only used by Health Canada and is not shared with any other application or agency \\cite{ct31}. \n    \\item The government of Vietnam has developed an application named \\textit{COVID-19} which includes features such as chatbot, consultation, and live updates on COVID-19. The application requires access to media, location, storage, device ID, and call logs. The application's privacy policies are updated in its native language \\cite{ct32}.\n    \\item  \\textit{COVID19 - DXB Smart App} is developed by the government of Dubai and provides general information on COVID-19 and also provides correct statistics. The application collects personal information voluntarily but does not share with the third-party applications unless required by the law \\cite{ct34}. \n     \\item \\textit{COVI} is a third-party COVID-19 informative app developed by Droobi, a Qatar based digital company. This application collects personal data such as contact information, age, health information, and unique identifiers, etc. \\cite{ct33}.\n    \\item \\textit{Corona360} is an app developed in South Korea which enables users to update their COVID-19 status as well as view the status of other people. For privacy reasons, the app does not collect any personal information such as ID, name, or phone number \\cite{ct35}. \n    \\item \\textit{CoronaCheck} is a third-party application that has been developed to enable its users to conduct self-assessment and provide accurate expert COVID-19 information to the users. This application does not collect any personal information and does not share data with third-party vendors \\cite{ct36}. \n    \\item \\textit{The Beat COVID Gibraltar} app is developed for the region of Gibralter and it utilizes Bluetooth technology in a decentralized way to track other phones who come in close contact with the person declared himself as the virus affected. \n\n\n\\item \\textit{BC COVID-19} Support is developed for the resident of British Columbia, Canada, to inform the people about the status of COVID-19 in British Columbia and guide them on what next action people should take. All the recommendations are personalized so it involves some level of contact tracing. \n\n\n\\item \\textit{COVID Symptom Study} has been designed for everyone to report their health status to the people who are developing policies to fight against the Virus. \n\n\\item \\textit{BeAware Bahrain} is a mobile application developed for the region of Bahrain that helps citizens to contain the spread of the virus by using the contact tracing efforts.\n\n\\item \\textit{Tawakkalna (Covid-19 KSA)} is the official app of kingdom of Saudi Arabia. It helps in controlling the spread of COVID-19 and suggest authorities where to impose the curfew and lockdown.\n\n    \n\\end{itemize}\n\n\nOur study of these apps consisted of analyzing publicly available information shared by the app developers and platform i.e. privacy policy, permissions requested, and user reviews. Furthermore, as some of the apps did not use SSL, we performed black-box testing of the apps using the Burp suite to analyze the network traffic during the app usage. The traffic analysis did indeed help us identify the information collected by these apps and shared with back-end servers which are liable to interception using network sniffing software.  \n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=9cm,height=5.8cm]{images\/permissions_analysis.pdf}\n\\caption{Distribution of apps and requested permissions}\n\\label{fig:permissions_analysis}\n\\vspace{-2.0em}\n\n\\end{figure}\n\n\\subsection{Permission analysis}\nPermissions required by a smartphone app are significant as they communicate with the user the resources required by the app to perform its function. Therefore, presenting the user with a list of permissions not only achieves transparency (providing the user an insight into the app operation) but it also serves to seek user consent. Within the context of our study, we have gathered information about the permissions required by the track and trace apps under study. Figure \\ref{fig:permissions_analysis} presents a graphical representation of the distribution of the apps with respect to permissions required by them. \n\nAs presented in Figure \\ref{fig:permissions_analysis}, we expected the majority of the apps to require access to location data of a device however our study also identified permissions requested by the apps which are not necessary to perform their function. For instance, we identified 06 apps that require access to SMS and call information of the device. In some scenarios access to phone numbers can be envisioned however access to SMS within a device is not essential to the function of a track and trace app. Similarly, we identified 04 apps that require access to the camera of the device which is of course an unexpected request by a track and trace app. \n\nFurthermore, 04 apps studied require access to the microphone of the device whereas 05 apps require access to media and storage of the device which are of course not critical to the app's function. Such apps are indeed causing concern regarding the privacy of their users and exemplify a lack of attention to the security and privacy of the users by app developers.\n\n\n\n\n\n\n\n\n\\subsection{Analysing privacy}\nTo understand the privacy considerations applied for the apps within our study, our analysis took into account the privacy policy published by the application developer as well as the use of basic privacy protection mechanisms such as SSL\/TLS to achieve encrypted data transmissions. \n\n\\color{blue}\nThrough the study of privacy policies of the apps as well as traffic analysis, we identified that most of the apps collect personal data such as location information, name, and phone number, etc. Although collecting such information is vital for effective track \\& trace, appropriate mechanisms should be applied to ensure secure sharing, processing, and storage of such data. Such details were not available for most of the apps analyzed in this study. Furthermore, an interesting observation we made was the type of information gathered by the apps. Specifically, our analysis identified that the track \\& trace app developed by Dubai collects personal information such as date of birth, name, email address, and caller ID. Such information is not required for effective track and trace and risks privacy of users as any malicious actor with access to such information can easily perform ID theft attacks. We believe data collection policies for such apps require immediate attention to minimize risk to individual user privacy. \n\nA significant challenge with respect to security and privacy within smartphone apps is sharing data with third parties to aid targeted advertisements. Smartphone apps using such strategies leverage advanced analytics techniques to identify user-behavior and profiles to achieve personalized advertisements. Within this context, the information collected by track \\& trace apps is highly personalised and if made accessible to third-parties can lead to sophisticated advertisement techniques breaching public trust and confidence in such apps. Our analysis of the apps concluded that all apps are free to use and do not include any in-app advertisements. However, some apps (as highlighted in table \\ref{tab:analysis}) do share data with third parties without precise information about who these third parties, what data is shared, and how this data will be used by the third parties. Therefore, focused efforts from the research community are required to enhance data sharing, processing, and storage practices within such scenarios.  \n\n\nAnother aspect concerning the privacy of information collected is how it is stored and shared by the apps. In this respect, our analysis revealed that 10 of the apps shared data with \\textit{third parties} however the nature and identity of such parties are not identified in the privacy policies. This is a cause of concern regarding individual privacy as the aims of sharing this data are not clear and therefore users are not aware of how their data may be used. For instance, a common apprehension among users is the sharing of data with advertisement agencies who may wish to use such data for targeted advertisement and adware. \n\n\\color{black}\nHaving said this, we also identified examples of good practice within our analysis. Specifically, some of the apps clearly state requirements for user consent before sharing data with third parties thereby assuring users with respect to how their data is shared. For instance, Corona360 (the app developed by the government of South Korea) collects personal and sensitive data of users but whenever the data is used, the user is notified for the reason of data usage. \n\nIn addition to the above, our analysis also uncovered security vulnerabilities within some apps. In particular, we identified five apps that were not using SSL\/TLS to ensure secure communication made through the app. Pursuing this direction of analysis, we conducted traffic monitoring of such apps and identified serious flaws in the app developed by the government of Pakistan. Details of these vulnerabilities have been reported to the relevant authorities however such vulnerabilities do put user privacy at risk especially where the app is collecting and utilizing personal user data. \n\n\\subsection{App review analysis}\nIn addition to the permissions requested, privacy policy, and traffic analysis of the worldwide COVID-19 track and trace apps, we have also studied user comments available in Google Play and App Store reviewing these apps. Although the majority of these comments are related to the usability and general function of the app, we found some comments to be insightful with respect to how the app collects and utilizes data. For instance, the app developed by the government of Israel was commented by a user to ask for users' permission when sharing data with third parties as well as to \\textit{guarantee not to send the data anywhere but compare it locally on user's device against downloaded \"Corona paths\"}. Another interesting observation was made for the Corona-Datenspende app, where user comments suggest that \\textit{one cannot use the app without connecting to a fitness account and hence completely breaking the point of anonymity.} As suggested by the user, connecting to the app via a fitness app such as Fitbit indeed does indicate sharing personal user data across different apps which is a risk to user privacy. \n\nThrough analysis of the user reviews of apps, we observed that although some of the users have included concerns about the privacy of information through their feedback, these are relatively minor proportions of users. For instance, for the \\textit{COVID-19 Gov PK} app, user review includes comments highlighting a lack of encryption and concerns about data traveling in plain-text. However, for apps such as \\textit{Corona-Datenspende} which requires a user to connect to a fitness app as a pre-requisite, there are no user comments with regards to how data is captured, analyzed, stored and processed between the third-party fitness app and the government app for CVOID-19. These observations reflect a lack of awareness among users with regards to measures to preserve the privacy of personal data collected, stored, and analyzed by computing systems, therefore, requiring efforts to raise awareness among users.\n\n\\section{Conclusions and Recommendations}\n\\label{sec:conclusion}\nAs Coronavirus is a contagious disease that spreads through close social interaction between humans, contact tracing is vital for containing its spread. Mobile devices present an ideal platform to introduce contact tracing software due to their ease of use, widespread ownership, and personalized usage. Therefore, several smartphone apps have been developed by governments, international agencies, and other parties to mitigate the virus spread.\nHowever, there is an increasing concern regarding the collection and use of data, and out-sourcing data to third-party systems. In this paper, we analyzed a large set of contact-tracing apps with respect to different security and privacy metrics. Specifically, we analyzed contact-tracing apps for permission analysis, privacy analysis, the security of the apps, and reviews of the users. \\color{blue} Our major findings are as follows:\n\\begin{enumerate}\n    \\item Although there have been significant technological advancements to aid COVID-19 response, contact tracing apps require further enhancements to achieve desired objectives in a privacy-aware manner.\n    \\item A number of track \\& trace apps request permissions which may not be required for the successful operation of the app's function. These include access to storage media, camera, and microphone which might result in in a breach of user's privacy.\n    \\item Several apps mention outsourcing data to third parties, however, it is unclear who are these third parties, what data is shared, and how it is processed by these parties.\n    \\item Some apps (used in developing countries) have not adopted appropriate security measures for the exchange of the data to and from the user to the data centers.\n    \\item Our analysis of the user reviews and the ratings for contact-tracing apps suggested that a large number of users are aware of privacy concerns of these apps.\n\\end{enumerate}\n\nThough digital technologies could play a prominent role in addressing the current pandemic challenges and the containment of the spread of the virus. However, the effectiveness and accuracy of these systems depend upon the working architecture of applications and user participation. The user participation could be improved if systems employed mechanism that ensures the security and privacy of users. To ensure the privacy, security, and secure development of contact tracing apps,  we recommend following design choices that should be followed for the development of contact tracing apps:\n\n\\begin{enumerate}\n\n\\item In order to ensure the privacy and security of the user data, the contact tracing systems have to consider the well established and state of the art encryption systems for storing data, enable personalized access control mechanisms and utilize secure communication mechanisms for the exchange of data between the users and the data center. Furthermore, developers should also consider the semantics of secure software development, strong authentication mechanism possibly two-factor authentication to minimize the risk of misuse.\n\\item \tThe contact-tracing apps should perform their operations in a completely decentralized way. i.e. the system performs the bulk of its operation at the user side.\n\\item The app's privacy policy should be mentioned in a way that a user could easily understand. The developer should also adopt the mechanisms that they could easily destroy user data once this pandemic is over.\n\\item The design system should not unnecessarily seek permissions for example access to videos, browsing history, or the images.\n\\item The developers should consider the measures that assign a unique pseudonymized identifier for the users which must not be linked to the user's real identity and could not be used to learn the private information of users through background knowledge. \n\\item To improve usability, the design should be simple and should have a user interface for interaction and personal tracking.\n\\item We also recommend developers and regulators to use the identity verification (telephone number authentication) or authentication system within their trace and track system so the information could be exchanged through the reliable voice call. \n\\end{enumerate}\n\nIt is very important to incorporate the techniques that ensure the privacy of citizens so that they can confidently participate in limiting the spread of the disease. The apps should not serve as the tool for mass surveillance tools so that people trust the system without having any concerns about their privacy and tracking of their private lives. As a part of our future work, we are looking to conduct a user study using qualitative measures focused on directly considering the feedback from users to further understand the usability and security concerns of users.\n\n\\color{black}\n\n\\clearpage\n\\onecolumn\n\n\\fontsize{6.5}{7}\\selectfont\n\n\\centering\n\\begin{longtable}{|p{1cm}|p{1cm}|p{2.8cm}|p{3.0cm}|p{0.8cm}|p{0.8cm}|p{0.5cm}|p{3.0cm}|p{0.7cm}|p{0.8cm}|}\n\\caption{Analysis of Smartphone apps designed to limit spread of COVID-19.}\n\\\\\\hline \n \\textbf{App} & \\textbf{Platform} & \\textbf{Permissions Requested} & \\textbf{ Privacy Policy} & \\textbf{Country}& \\textbf{No of Downloads}& \\textbf{TLS\/ SSL} &  \\textbf{App Reviews} & \\textbf{App version} &\\textbf{API version}\\\\ \\hline\n\nCOVID-19 Gov PK  & Android & Location (approximate and precise), full network access, prevent device from sleeping & Data to be shared with third party & Pakistan & 500,000+ & No & lack of encryption. Data might be traveling in plaintext. Radius alert is not accurate. Doesn't show patients infected with COVID-19 & 3.0.7 &\t5.0 and up\\\\\n\\hline\nCOVID Symptom Tracker &    Android \\& iOS    & Wifi connection information, full network access, audio settings, run at startup, prevent the device from sleeping &  Collects sensitive personal information such as DOB, name, gender, COVID-19 tests status, location, details of any treatment, email, phone number, IP address. Shared with universities, research centers, amazon web service, google analytic, etc.  & United Kingdom  & 500,000+    & N\/A &  basic information related to COVID-19 symptoms, helps people take precautionary measures to self-isolate &0.14 &\t5.0 and up \\\\\n\\hline\nBC COVID-19 Support &    Android \\& iOS & Location (approximate and precise), full network access, prevent the device from sleeping & Personal information collected for COVID-19 alerts and management, only used by Ministry of Health &    Canada    & 10,000+ & Yes &  Doesn't update on a regular basis with current stats. No graph of active cases. Inaccurate and outdated information & 1.20.0 &\t5.0 and up\\\\\n\\hline\nOpenWHO: Knowledge for Health Emergencies &    Android \\& iOS & Wifi connection, full network access, media\/files, and storage, run at startup, prevent the device from sleeping & Requires name and email to create an account used for communications and the announcement of changes to the openWHO platform &    United States &    500,000+ &    No &    Language issue to some people. Gives out a certificate for completing the course, increases public health knowledge & 3.4 &\t5.0 and up \\\\\n\\hline\nPakistan's National Action Plan for COVID-19 &    Android    & This application requires no special permissions to run    & No information being shared    &     Pakistan &    50,000+    & Yes    &    Shares information related to COVID-19 and SOPs that government has launched for the safety of people & 1.1\t& 5.0 and up\\\\\n\\hline\nHealth Monitoring PDMA    & Android &    Location (approximate and precise), receives data from internet and full network access    & Information will be accessed by Smart Asset Sindh Health, shared with third party &    Pakistan    & 1000+    & No &    Data information being sent in plaintext. No encryption or algorithm is used for data protection. Too many bugs. Doesn't work efficiently. & 1.4\t& 4.0.3 and up\\\\\n\\hline\n\nCanada COVID-19    & Android & Location (approximate and precise), full network access, prevent device from sleeping &    Personal data is collected by Health Canada only to support COVID-19.  &Canada    & 50,000+    & Yes & App doesn't take into account pre-existing conditions. Will be much more effective if user can see map with active cases. & 4.0.0\t& 5.0 and up\\\\\n\\hline\n\nCOVID-19  & Android &    Location, phone, media, storage, camera, microphone, wifi, device ID, call information, download files without notification, run at startup, prevent the device from sleeping  & May use personal information with third party &    Vietnam &    100,000+ &    N\/A    &    App is only available for Veitnamese and not available in English. Very narrow coverage overall. Provides basic information. Only accessible in Veitnam & 1\t& 4.4 and up\\\\\n\\hline\n\nCOVI    & Android &    Location, phone, wifi, device ID, call information, pair with Bluetooth devices, receive data from Internet, run at startup, prevent device from sleeping &    Information such as DOB, name, the account number is collected and shared with trusted third parties & Qatar &    10,000+ &    N\/A & Only restricted for the people living in Qatar.  Doesn't get updates. Provides basic information & 2.0.2.2\t& 5.1 and up \\\\\n\\hline\n\nCOVID19 - DXB Smart App &    Android    & Microphone, camera, location, storage, calendar, Wifi connection, media, receive data from Internet, pair with Bluetooth devices, full network access, prevent device from sleeping, change audio settings & Sends personal information such as ID, name, DOB, email, geographical location to a third party & Dubai    & 1000+ & N\/A &   Only restricted for the people living in Dubai. Some users reported experiencing network error whenever they open this app & 3.8 &\t5.0 and up\\\\\n\\hline\n\nCorona 360    & Android &    Location (approximate and precise), receives data from the internet, full network access, prevent the device from sleeping &    Collects personal and sensitive data of user but whenever the data is being used, the user is notified for the reason    & South Korea &    10+    Yes &    & Useful and multilingual solution for find Corona free locations & 2.2.2\t& 4.3 and up \\\\\n\\hline\nCoronaCheck    & Android \\&iOS &    Full network access     & Will not share any information &    Pakistan    & 10,000+     & Yes    &    Gives detailed information related to COVID-19 protection, symptoms. Translate English to Vocal language. & 1.1\t& 4.1 and up \\\\\n\\hline\nCoronavirus Australia    & Android \\&iOS    & Location (approximate and precise), receives data from internet and full network access &    Collects information but does not use it without asking from the user &    Australia    & 500,000+    &     N\/A & App opens in the web browser which is clunky, the infection status is updated less often than the press releases, and is out of date later in the day. & 1.4.5\t& 6.0 and up \\\\\n\\hline\nNHS App    & Android \\& iOS    & Location, phone, media, storage, camera, microphone, Wifi, device ID, call information, download files without notification, run at startup, prevent device from sleeping    & No specific information about sharing data with third parties &United Kingdom &    500,000+ &    N\/A &    Requires personal details such as photo, name, DOB, NHS number. Requires 12 hours for the initial setup. Misleading\/inaccurate information about compatible operating systems & 1.36.3 &\t5.0 and up\\\\\n\\hline\nAarogya Setu &    Android    & Location (approximate and precise), receives data from internet and full network access &    Cannot access its privacy policy    & India    & 50,000,000+ &            N\/A &  Location, network and Bluetooth visibility  required.   No proper tracking, no radius alert, bugs, doesn't update cases. Takes a new location every time when accessed & 1.4.1\t& 5.0 and up\\\\\n\\hline\n\n\nTraceTogether & Android \\& iOS & Media, storage, receive data from Internet, pair wth Bluetooth devices, full internet access, prevent device from sleeping & Mobile number and anonymous ID are shared in a secure server and not available to be shared with Public &  Singapore & 500,000+ & N\/A & Doesn't alert you to infected cases in your area. Drains battery pretty fast due to Bluetooth connection. & 2.2.0\t& 5.1 and up\\\\ \\hline\n\nHaMagen    & Android \\&iOS    & Device and app history, location, Wifi connection, full network access, prevent device from sleeping, change network connectivity & Cross-referencing location data with the corona patients & Israel & 1,000,000+  & No & Correlates overlaps only since installation. Should extract and use historical information. Data processed locally when a user opts against downloaded \"Corona paths\" & 2.2.6\t&  5.0 and up\\\\ \\hline\n\n\nHome Quarantine (Kwarantanna domowa) & Android \\& iOS & Location, phone, media, storage, camera, microphone, wifi, device ID, call information, download files without notification, run at startup, prevent device from sleeping &    Collected data may be shared with third party &    Poland &   100,000+ &    N\/A &GPS location is invalid. Cannot add a phone number as it gives away error. & 1.39.5\t& 6.0 and up\\\\ \\hline\n\n\nNHS 24 : COVID-19 &    Android \\& iOS &    Full network access, receives data from internet prevent device from sleeping &    Collect personal data and share with third party &    UK    & 1000+        &    N\/A &Not Compatible     Basic information only & 1.0.3\t& 4.1 and up\\\\ \\hline\n\nBeat Covid Gibraltar &   Android &   view Wi-Fi connection,pair with Bluetooth devices, full network access &    No personal data will be stored or used &    Gibraltar    & 10000+        &    N\/A &Developed only for Gibraltar, easy to use & 1.18\t& 6 and up\\\\ \\hline\n\nEHTERAZ &   Android &   location data, phone access for calls, Photos \/ Media \/ Files,  full network access &   personal data will be stored or used &    Qatar  & 1,000,000+ &  n\/A & additional authentication performed, some privacy flaws are identified & 9.02\t& 6 and up\\\\ \\hline\n\nBeAware Bahrain &   Android &  require access to apps running, read calendar information, require access to location, media files, and storage, pair Bluetooth devices &   personal data will be stored or used &    Bahrain  & 100,000+ &    N\/A &some privacy flaws are identified & 0.2.1\t& 4.4 and up\\\\ \\hline\n\nShlonik\n&   Android &  record audio, access to running apps, require access to location, media files, and storage, pair Bluetooth devices, precise location information, full network access & Collects data and location information&    Kuwait  & 100,000+ &    N\/A &some privacy flaws are identified & varies\t& 4.4 and up\\\\\n\\hline\n\nCOVID Radar\n&   Android\/iOS & phone access, access to media files and storage, pair Bluetooth devices & Data is provided by users manually&    Netherlands  & 50,000+ &    N\/A & N\/A& 1.1.2\t& 6 and up\\\\ \\hline\n\nTawakkalna (Covid-19 KSA)&   Android & GPS Location, read the storage data, take pictures and make video, pair via Bluetooth device, full network access & Data is provided by users manually&    Saudi Arabia & 1,000,000+&    N\/A & helps in imposing curfew & 1.7\t& 6 and up\\\\ \\hline\n\nMySejahtera &   Android & precise location (GPS and network-based), call access, media \\& storage access, camera access, full network access\n & Data is provided by users manually&   Malaysia & 1000K+ &    N\/A & user also need to register through their website & 1.0.24\t& 4 and up\\\\\n\\hline\n\n\\end{longtable}\n\\label{tab:analysis}\n\n\\twocolumn\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}