\section{Introduction}\label{introduction}

Inverse boundary value problems are concerned with the determination of the physical properties (represented as various coefficients of a model equation) of an object from measurements taken on the boundary. The mathematical investigation of such problems already has a history spanning nearly four decades, going back to \cite{Ca}. In this context, uniqueness refers to the property certain equations might have that if two sets of coefficients produce identical sets of boundary data, then the coefficients must also be identical.

For second order elliptic equations, the usual boundary data used is the set of Dirichlet and Neumann boundary values of the  solutions of the equation. Physically, these may be interpreted as measurements of electric potential and current density in the case of the conductivity equation for the electric impedance tomography, or as measurements of displacement and traction in the case of the static elasticity equation for the nondestructive testing. One common way to encode the measurement data is as the Dirichlet-to-Neumann map, which associates to the one Dirichlet datum the corresponding Neumann one. It is also possible to consider ``local data'', i.e. Dirichlet data supported on a part of the boundary paired with the restriction of the corresponding Neumann data to that same part.

In the case of isotropic materials, many uniqueness results are known for these type of measurement data. See, for example, \cite{SU} for the case of isotropic conductivity, or \cite{NU-3} for isotropic elasticity.  For anisotropic materials uniqueness usually doesn't hold. In the case of the anisotroipc conductivity equation, for example, diffeomorphisms that leave the boundary fixed preserve the Dirichlet-to-Neumann map but change the interior conductivity. It is conjectured that this is the only obstruction. If one knows apriori that the coefficients belong to a more restricted class, which is not preserved by diffeomorphisms, then it can be expected that uniqueness may  hold. One such instance (for example for the anisotropic conductivity or anisotropic elasticity cases) would be the the class of piecewise constant coefficients. Uniqueness for the anisotropic conductivity case with this restriction was shown in \cite{AHG}. For the (static) anisotropic elasticity equation, it was shown in \cite{CHN}.

For second order hyperbolic equations, lateral Dirichlet and Neumann data may be used. That is, time dependent Dirichlet and Neumann data measured on the boundary of the object, for solutions with zero initial Cauchy data. Uniqueness has been proved for various second order hyperbolic equations (for example, see \cite{B}, \cite{B-3}, \cite{KKL}). These results are based on the so-called Boundary Control method. Computational aspects of this method have been studied recently in
\cite{BIKS}, \cite{hoop}, \cite{PBK}, see also \cite{BG} for the first computational
implementation of the method. Uniqueness up to diffeomorphism in anisotropic materials has been shown in \cite{B-3}, \cite{BK}. There is only one uniqueness result known for a time domain anisotropic elasticity equation with smooth coefficients. That is for the hexagonal elasticity equation with ellipsoidal slowness surfaces (see \cite{Mazzucato}). In \cite{HNZ} a uniqueness result is proved for the piecewise constant elastic coefficients in the generally anisotropic case (or piecewise analytic, but with greater symmetry assumptions).

The argument showing the uniqueness in \cite{HNZ}, as is typical for the above types of measurement data setups, consists of two steps. The first step is a ``boundary determination'' result. This is accomplished via a finite in time Laplace transform which relates the lateral Dirichlet-to-Neumann map of the hyperbolic problem to the symbol of the Dirichlet-to-Neumann map of an elliptic equation of elastic type. Using the techniques from \cite{NTU}, \cite{NU}, \cite{NU-2}, the coefficients at the boundary may be obtained from the given (local) data. The second step is an ``interior determination'' result. That is, for the above mentioned elliptic equation, it is shown that the local Dirichlet-to-Neumann on the boundary determines the local Dirichlet-to-Neumann map for the domain with the immediately adjacent domain of homogeneity removed, localized on a previously internal part of the boundary. This technique, inspired by \cite{I}, was used for interior determination in \cite{CHN}. It relies essentially on the Runge approximation property. Once these two steps are completed, the uniqueness result follows by induction.

In their ``boundary determination'' result, there is no restriction on the observation time. However, it is important to remark that the full elliptic Dirichlet-to-Neumann map is needed for the ``interior determination'' step, not just its symbol. This then introduces the restriction that the observation time has to be sufficiently long. For this type of method, this is to be expected since, due to the finite speed of propagation property of the equation, we expect to need to wait for a certain amount of time before the waves generated by the boundary data can travel to every part of the domain of interest and then return to the part of the boundary where observations are made. Only after this large amount of time has passed enough measurement data can be obtained which enable to do the determination of the coefficients in the whole domain. However, this is not the kind of result that should be expected for a hyperbolic equation. In the hyperbolic case, the boundary data gathered for any length of time should provide information on the material at least in the part of the domain from which the elastic waves have enough time to return to the surface on which the data is collected. One of the aims of this paper is to provide this type of result.

We will consider  the inverse boundary value problem for the  piecewise homogeneous, generally anisotropic, elasticity equation in the space-time domain and prove the above type of uniqueness result. We will use the boundary determination result of \cite{HNZ}. In Section \ref{boundary} we sketch the connection between the time domain equation and the elliptic one via the finite time Laplace transform and then quote the relevant boundary uniqueness result from \cite{CHN},\cite{HNZ},\cite{NTU}.

We will not use the elliptic equation for the interior determination part of our result. Instead, we use hyperbolic techniques inspired by \cite{KOP}. This allows us to obtain a uniqueness result that is more typical for hyperbolic equations, i.e. \emph{we obtain uniqueness closer to the part of the boundary where the measurements are taken for shorter observation times and  further away from that part of the boundary for longer times}.

We would like to remark on the importance of showing uniqueness in the inverse boundary value problem for the local Neumann-to-Dirichlet map. The vibroseis exploration technique in the reflection seismology is used to investigate the underground structure of the Earth. The measurements taken using this technique correspond almost precisely to this kind of map, and not to a local Dirichlet-to-Neumann map (see \cite{Bateman}). Furthermore, the geological structure of the Earth is isotropic or transversally isotropic layered in its shallow part, and in its further depth part, it is close to piecewise homogeneous and can contain regions with more complicated anisotropic elasticity than transversally isotropic elasticity.

The rest of this paper is organized as follows. In what remains of Section \ref{introduction}, we formulate our inverse  problem and give our main results on the unique identification of piecewise homogeneous density and elasticity tensor.
The first main results is the unique identification for the case when the geometry of the homogeneous pieces is known. The second main result is in  the case when the geometry of this pieces is unknown. In Section \ref{boundary} we briefly review a boundary determination result that has been obtained previously in \cite{CHN}, \cite{HNZ}. In Section \ref{inner} we discuss interior determination results for both Dirichlet-to-Neumann and Neumann-to-Dirichlet maps. Then in Section \ref{proof}, by combining these the results of the previous two sections, we complete the proofs of our main results, in the second case making use of the theory of subanalytic sets. Section A which is an appendix provides a brief summary of results on the theory of subanalytic sets which are used in the proof of the second main result.

\subsection{Preliminaries}

Let $\Omega\subset\mathbb{R}^3$ be an  open bounded connected domain, with Lipschitz boundary. For a $T>0$ we will denote $\Omega_T=(0,T)\times\Omega$ an $\Gamma=\partial\Omega$.

An \emph{elastic tensor} $\mathbf{C}=\mathbf{C}(x)=(C_{ijkl}(x))_{i,j,k,l=1,2,3}$, $x=(x_1,x_2,x_3)\in\overline\Omega$, is defined by real valued functions  $C_{ijkl}(x)$ which satisfy the symmetries
\begin{equation}
C_{ijkl}(x)=C_{ijlk}(x),\quad C_{ijkl}(x)=C_{klij}(x),\quad x\in\overline\Omega,\,\, i,j,k,l\in\{1,2,3\},
\end{equation}
and the strong convexity condition, i.e. there exists $\lambda>0$ such that for any symmetric matrix $\epsilon=(\epsilon_{ij})$,
\begin{equation}
\epsilon:(\mathbf{C}:\epsilon)=\sum_{i,j,k,l=1}^3 C_{ijkl}(x)\epsilon_{ij}\epsilon_{kl}\ge \lambda (\epsilon:\epsilon),\,\,x\in\overline\Omega,
\end{equation}
where $\epsilon:\eta$ is the inner product of matrices $\epsilon$ and $\eta=(\eta_{ij})$ defined by
$\epsilon:\eta=\sum_{i,j=1}^3\epsilon_{ij}\eta_{ij}$, and $\mathbf{C}:\eta$ is a matrix whose $(i,j)$ component $(\mathbf{C}:\eta)_{ij}$ is defined as
$(\mathbf{C}:\eta)_{ij}=\sum_{k,l=1}^3 C_{ijkl}(x)\eta_{kl}$. The density of mass function is given as a function $\rho(x)>\lambda$, $x\in\overline\Omega$, which will be simply called \emph{density}.

In this paper we will make the further assumption that there are a finite number of open, connected,  Lipschtiz subdomains $D_\alpha$, $\alpha\in A$, that is subdomains with Lipschitz boundaries, such that $\bar\Omega=\cup_{\alpha\in A}\bar D_\alpha$, $D_\alpha\cap D_\beta=\emptyset$ if $\alpha\neq\beta$, and $\mathbf{C}$, $\rho$  are homogeneous in each $D_\alpha$.

For a function $u:\Omega\to\mathbb{R}^3$ denoting the diplacement we write
\begin{equation}
(L_\mathbf{C} u)_i = \sum_{j,k,l=1}^3\partial_j \left( C_{ijkl} \partial_k u_l\right),
\end{equation}
where $\partial_j=\partial_{x_j}$
and for $u:\Omega_T\to \mathbb{R}^3$ we write
\begin{equation}
(P_{\rho,\mathbf{C}} u)_i = \rho\partial_t^2 u_i - (L_\mathbf{C} u)_i.
\end{equation}
Let $\nu$ be the unit outer normal to $\partial\Omega$. We will denote the traction at $\partial\Omega$ by
\begin{equation}
(\partial_\mathbf{C} u)_i:=[ (\mathbf{C}:Du)\nu]_i=\sum_{j,k,l=1}^3\nu_jC_{ijkl}\partial_ku_l.
\end{equation}

\subsection{The forward problem and local boundary data}

We want to consider in this paper the equations
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{in}\,\,\Omega_T,\\[5pt]
(u,\partial_t u)|_{t=0}=0,
\end{array}\right.
\end{equation}
with Dirichlet or Neumann boundary conditions. At one point in the argument we will however need a result with slightly more general boundary conditions, namely with both Dirichlet and Neumann data given on complementary parts of the boundary. This kind of problems have been considered in the case of piezoelectric equations in \cite{AN}. Here we will give the restriction of their result to the case of hyperbolic elasticity equations.

We need to introduce a few function spaces. Suppose $\Sigma$ is an open subset of $\Gamma$. Let
\begin{equation}
    H^{\pm1/2}_{co}(\Sigma)=\left\{ f\in H^{\pm1/2}(\Gamma): \mbox{supp\;} g\subset\overline{\Sigma}\right\}.
\end{equation}
with the restriction of the norm of $H^{\pm1/2}(\Gamma)$, and
\begin{equation}
    H^{\pm1/2}(\Sigma)=\left\{f|_\Sigma:f\in H^{\pm1/2}(\Gamma)\right\},
\end{equation}
with the norm
\begin{equation}
    ||f||_{H^{\pm1/2}(\Sigma)}=\inf_{\tilde f,\,\tilde f|_{\Sigma}=f}||\tilde f||_{H^{\pm1/2}(\Gamma)}.
\end{equation}

Suppose $\Gamma_u,\Gamma_s\subset\Gamma$  are disjoint connected open sets such that $\partial\Gamma_u=\partial\Gamma_s$ is a Lipschitz curve. We write
\begin{equation}
H^1_{\Gamma_u}(\Omega)=\left\{u\in H^1(\Omega): u|_{\Gamma_u}=0\right\}\,\,\text{and}\,\,(H^1_{\Gamma_u}(\Omega))'\,\,\text{for its dual space}.
\end{equation}

Let $f\in H^3(0,T;H^{1/2}(\Gamma_u))$, $g\in H^1(0,T;H^{-1/2}(\Gamma_s))$ be boundary data, $F\in H^1(0,T;(H^1_{\Gamma_u}(\Omega))')$ be a source term, and $u_0\in H^1(\Omega)$, $u_1\in L^2(\Omega)$ be initial data satisfying the compatibility condition
\begin{equation}
f(0,\cdot)=u_0|_{\Gamma_u}.
\end{equation}
Then we have the following theorem.
\begin{thm}\label{thm-forward}
There exists a unique $u\in L^\infty(0,T;H^1(\Omega))$ with $\partial_t u\in L^\infty(0,T;L^2(\Omega))$, $\partial_t^2u\in L^\infty(0,T;(H^1_{\Gamma_u}(\Omega))')$ such that
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=F\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
u=f\,\,\text{\rm on}\,\,(0,T)\times\Gamma_u,\\[5pt]
\partial_\mathbf{C} u=g\,\,\text{\rm on}\,\,(0,T)\times\Gamma_s\\[5pt]
u|_{t=0}=u_0,\;\partial_t u|_{t=0}=u_1,
\end{array}\right.
\end{equation}
and
\begin{multline}
||u||_{L^\infty(0,T;H^1(\Omega))}+||\partial_t u||_{L^\infty(0,T;L^2(\Omega))}+||\partial_t^2 u||_{L^\infty(0,T;(H^1_{\Gamma_u}(\Omega))')}\\[5pt]
\leq
C(||u_0||_{H^1(\Omega;\mathbb{R}^3)}+||u_1||_{L^2(\Omega)}
+||F||_{H^1(0,T;(H^1_{\Gamma_u}(\Omega))')}\\[5pt]
+||f||_{H^3(0,T;H^{1/2}(\Gamma_u))}
+||g||_{H^1(0,T;H^{-1/2}(\Gamma_s))}),
\end{multline}
where the constant $C$ depends on $\Omega$, $T$, $\lambda$, $||\mathbf{C}||_{L^\infty(\Omega)}$, $||\rho||_{L^\infty(\Omega)}$.
\end{thm}

\begin{remark}One reason why the source term $F$ has higher regularity in time than is usually assumed is the fact that it is more singular in space. We will in fact at one point in the proof need to solve the equation with a source term that would have exactly this regularity. The Dirichlet data is required to have three derivatives in time because, in the course of the proof of the result, it is necessary to convert the given equation to one with zero Dirichlet data, but with an additional source term, which has to have $H^1$ in time regularity.
\end{remark}

We can define the local Neumann-to-Dirichlet and Dirichlet-to-Neumann maps, which contain the information that may be collected by applying various tractions to $\Sigma$ and measuring displacements on $\Sigma$, or by producing displacements and measuring tractions on $\Sigma$ as follows. The local Neumann-to-Dirichlet map (abbreviated as ``ND map'')
\begin{equation}
\Phi^{T,\Sigma}_{\rho,\mathbf{C}}: H^1(0,T;H^{-1/2}_{co}(\Sigma))\to H^1(0,T;H^{1/2}(\Sigma))
\end{equation}
is defined by
\begin{equation}
\Phi^{T,\Sigma}_{\rho,\mathbf{C}}g=u|_{{(0,T)\times\Sigma}},\,\,\text{\rm with}\,\,{(0,T)\times\Sigma}=(0,T)\times\Sigma,
\end{equation}
where $u$ solves
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
\partial_\mathbf{C} u|_{{(0,T)\times\Sigma}}=g,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
The local Dirichlet-to-Neumann map (abbreviated as ``DN map'')
\begin{equation}
\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}: H^3(0,T;H^{1/2}_{co}(\Sigma))\to H^1(0,T;H^{-1/2}(\Sigma))
\end{equation}
is defined by
\begin{equation}
\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}f=\partial_\mathbf{C} u|_{{(0,T)\times\Sigma}},
\end{equation}
where $u$ solves
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
u|_{{(0,T)\times\Sigma}}=f,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}

\subsection{Main results}

If  $\Sigma\subset\partial D_\alpha$ is open, we will say that $\Sigma$ is {\it curved} if it is $C^1$ and $\{\nu(x):x\in\Sigma\}\subset S^2$ contains the image of a non-constant continuous curve.

Suppose $\mathbf{C}^{(I)}$, $\rho^{(I)}$ $I=1,2$, are two elastic tensors and two densities on $\Omega$, all of which are homogeneous in common Lipschitz subdomains $D_\alpha$. Since each boundary $\partial D_\alpha$ of $D_\alpha$ can have the discontinuity of the density and elastic tensor, we also call each $\partial D_\alpha$ interface. We will consider a chain $D_{\alpha_i}$, $i=1,\ldots,N$ of these subdomains (which we will abbreviate as $D_i$) and nonempty open surfaces $\Sigma_i\subset\partial D_i$ such that $\Sigma=\Sigma_1\subset\Gamma\cap\partial D_1$, and $\Sigma_{i+1}\subset\bar D_i\cap\bar D_{i+1}$, $i=1,\ldots,N-1$.

\begin{thm}\label{main-thm}
Let $\mathbf{C}^{(1)}$, $\mathbf{C}^{(2)}$, $\rho^{(1)}$, $\rho^{(2)}$ be as above, assume that each $\Sigma_i$, $i=1,\ldots,N-1$ is curved in the sense given above, and $\partial(\partial\Omega\cap\partial D_1)$, $\partial(\partial D_i\cap\partial D_{i+1})$, $i=1,\ldots,N-1$ are Lipschitz curves, then there exist times $0<T_1<\cdots<T_N<\infty$, such that if
\begin{equation}
\Lambda^{T_k,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Lambda^{T_k,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}},
\end{equation}
or if
\begin{equation}
\Phi^{T_k,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Phi^{T_k,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}},
\end{equation}
then
\begin{equation}
\rho^{(1)}|_{D_i}=\rho^{(2)}|_{D_i},\;\mathbf{C}^{(1)}|_{D_i}=\mathbf{C}^{(2)}|_{D_i},\quad i=1,\ldots,k.
\end{equation}
\end{thm}
\begin{remark}
The condition that $\partial(\partial\Omega\cap\partial D_1)$, $\partial(\partial D_i\cap\partial D_{i+1})$, $i=1,\ldots,N-1$ are Lipschitz curves is in fact not necessary. Given a chain of domains that satisfies the other assumptions, we can pick a smooth curve that connects a point in $\Sigma$ to a point inside $D_N$ and crosses each interface $\partial D_i$ transversely. The intersection of the original chain of domains with an appropriately chosen tubular neighborhood of this curve would satisfy this extra condition.
\end{remark}
\begin{remark}
Note that in the case when $\rho|_{D_i}$, $\mathbf{C}|_{D_i}$ are known for some $i$, then we do not need to assume that $\Gamma_i$ is curved. This would permit our result to apply, for example, to measurements of elastic waves taken on the surface of the Earth, which is locally very close to flat, as long as the properties of the top layer of the ground are known by other means. If the underground regions whose elastic properties are unknown can be reached by passing through a number of interfaces that do have curved portions, then the result still holds.
\end{remark}

\bigskip
Suppose $\mathbf{C}^{(I)}$, $\rho^{(I)}$ $I=1,2$, are two elastic tensors and two densities on $\Omega$, all of which are homogeneous on Lipschitz subdomains $D_\alpha^{(I)}$. Suppose further that there is  a  ``region of interest'' $R\subset\Omega$, $\Sigma\subset\partial R$, such  that any $D_\alpha^{(I)}\cap R$ is sub-analytic.\footnote{For a definition and summary of properties of sub-analytic sets, see Appendix \ref{sas}}

\begin{thm}\label{thm-interest}
Let $\mathbf{C}^{(1)}$, $\mathbf{C}^{(2)}$, $\rho^{(1)}$, $\rho^{(2)}$, be as above, and assume that $\Sigma$ is curved and that any boundary $(\partial D_\alpha^{(I)}\cap R)\setminus\partial R$  is curved on all its smooth components. Then there exists a time $0<T<\infty$ so that if
\begin{equation}
\Lambda^{T,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Lambda^{T,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}},
\end{equation}
or if
\begin{equation}
\Phi^{T,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Phi^{T,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}},
\end{equation}
then
\begin{equation}
\rho^{(1)}|_{R}=\rho^{(2)}|_{R},\;\mathbf{C}^{(1)}|_{R}=\mathbf{C}^{(2)}|_{R}.
\end{equation}
\end{thm}

\section{Boundary determination}\label{boundary}

In this section we follow \cite{HNZ} to show that the DN map $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$, or the ND map $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$, determines the values of $\rho|_\Sigma$ and $C|_\Sigma$. More precisely we will show the following.
\begin{prop}[see {\cite[Theorem 5.4]{HNZ}}]\label{boundary-prop}
If $\Sigma$ is curved, $0<T<\infty$, and $\Lambda^{T,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Lambda^{T,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}}$ or $\Phi^{T,\Sigma}_{\rho^{(1)},\mathbf{C}^{(1)}}=\Phi^{T,\Sigma}_{\rho^{(2)},\mathbf{C}^{(2)}}$, then $\rho^{(1)}|_{D_1}=\rho^{(2)}|_{D_1}$ and $\mathbf{C}^{(1)}|_{D_1}=\mathbf{C}^{(2)}|_{D_1}$.
\end{prop}

\begin{proof}
We will sketch the argument in the DN map and the ND map cases separately.

\paragraph{The DN map case:}
This is proved in \cite{HNZ}. Here we will repeat enough of the argument to give the reader an idea of how it works, but we will not reproduce it in full. The main idea is to use a finite-time Laplace transform in order to convert the hyperbolic elasticity boundary value problem to an elliptic boundary value problem. The boundary determination will then follow from the results on the elliptic case proven in \cite{CHN}, \cite{HNZ}.

Suppose $\mathbf{C}$ and $\rho$ are of the kind we are considering. Let $\psi\inH^{1/2}_{co}(\Sigma)$ and let  $u$  be a solution of
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}} u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
 u|_{(0,T)\times\Gamma}=t^2\psi,\\[5pt]
  (u, \partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
Also for $\phi\inH^{1/2}_{co}(\Sigma)$, we will consider the elliptic boundary value problem
\begin{equation}\label{elliptic}
\left\{\begin{array}{l}
\rho v-h^2L_\mathbf{C} v=0\,\,\text{\rm in}\,\,\Omega,\\[5pt]
v|_{\Gamma}=\phi
\end{array}\right.
\end{equation}
depending on a parameter $h>0$.
Let
\begin{equation}
\tilde\Lambda^{h,\Sigma}_{\rho,\mathbf{C}}( \phi)=h\partial_\mathbf{C} v|_{\Sigma}
\end{equation}
be the associated DN map.

We are interested in comparing $v(\cdot,h)$ to the finite-time Laplace transform
\begin{equation}
(\mathcal{L}_T u)(\cdot,h) := \int_0^T u(t,\cdot)e^{-\frac{t}{h}}\,\text{d} t,
\end{equation}
 where the Dirichlet data for $v$ is chosen so that
\begin{equation}\label{def-phi}
\phi=\psi\int_0^T t^2e^{-\frac{t}{h}}\,\text{d} t.
\end{equation}

Let $u_0\in H^1(\Omega)$ be the solution of
\begin{equation}
\left\{\begin{array}{l}
L_\mathbf{C} u_0=0\,\,\text{\rm in}\,\,\Omega,\\[5pt]
u_0|_{\Gamma}=\psi,
\end{array}\right.
\end{equation}
and define $u_1$ so that
\begin{equation}
u(t,x)=t^2u_0(x)+u_1(t,x),\quad (t,x)\in \Omega_T.
\end{equation}
Then $u_1$ should satisfy
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u_1=-2\rho u_0\,\,\text{\rm in}\,\,\Omega_T\\[5pt]
 u_1|_{(0,T)\times\Gamma}=0,\\[5pt]
  (u_1, \partial_t u_1)|_{t=0}=0.
\end{array}\right.
\end{equation}
It is known that by the Korn inequality, there exists a unique solution such that
\begin{equation}
u_1\in C([0,T];H^1(\Omega))\cap C^1([0,T]; L^2(\Omega))
\end{equation}
and
\begin{multline}
||u_1(t)||_{H^1(\Omega)}+||\partial_t u_1(t)||_{L^2(\Omega)}\\[5pt]\leq C||2\rho u_0||_{L^2(\Omega_T)}
\leq C||\psi||_{H^{1/2}(\Omega)},\,\,t\in[0,T]
\end{multline}
(see \cite{Wloka}).

\begin{comment}
Define
\begin{equation}
w(\cdot,h)=(\mathcal{L}_T u_1)(\cdot,h)=\int_0^Tu_1(\cdot,t)e^{-\frac{t}{h}}\,\text{d} t
\end{equation}
and notice that
\begin{equation}
\int_0^T\partial_t^2 u_1(\cdot,t)e^{-\frac{t}{h}}\,\text{d} t=h^{-2} w(\cdot, \tau)+ e^{-\frac{T}{h}}\left(\partial_tu_1(\cdot,T)+h^{-1} u_1(\cdot, T)\right).
\end{equation}
Let
\begin{multline}
F(\cdot,h)=-2h^3\rho(\cdot)u_0(\cdot)(1-e^{-\frac{T}{h}})\\-
h^2e^{-\frac{T}{h}}\left(\partial_tu_1(\cdot,T)+h^{-1} u_1(\cdot, T)\right)\in H^{-1}(\Omega).
\end{multline}
Then $w$ is a solution of
\begin{equation}
\left\{\begin{array}{l}
\rho w-h^2L_\mathbf{C} w=F\,\,\text{\rm in}\,\,\Omega,\\[5pt]
w|_{\partial\Omega}=0.
\end{array}\right.
\end{equation}
\end{comment}

Let
\begin{equation}
r(\cdot,h)=v(\cdot,h)-\int_0^Tu(t,\cdot)e^{-\frac{t}{h}}\,\text{d} t.
\end{equation}
\begin{comment}
If we also introduce
\begin{equation}
z(\cdot,h)=v(\cdot,h)-u_0(\cdot)\int_0^Tt^2e^{-\frac{t}{h}}\,\text{d} t,
\end{equation}
then
\begin{equation}
r(\cdot,h)=z(\cdot,h)-w(\cdot,h).
\end{equation}
Clearly, $z$ satisfies
\begin{equation}
\left\{\begin{array}{l}
\rho z-h^2L_\mathbf{C} z = -\rho u_0\int_0^Tt^2e^{-\frac{t}{h}}\,\text{d} t\,\,\text{\rm in}\,\,\Omega,\\[5pt]
z|_{\partial\Omega}=0.
\end{array}\right.
\end{equation}
Noting that
\begin{equation}
\int_0^Tt^2e^{-\frac{t}{h}}\,\text{d} t=\left[ 2h^3(1-e^{-\frac{T}{h}})-(T^2h+2Th^2)e^{-\frac{T}{h}}  \right],
\end{equation}
\end{comment}
An elementary computation shows that $r$ satisfies
\begin{equation}
\left\{
\begin{array}{l}
h^{-2}\rho r -L_\mathbf{C} r = e^{-\frac{T}{h}}\left[\partial_t u_1(T)+h^{-1} u_1(T)
+\rho u_0(T^2h^{-1}+2T)\right]\,\,\text{\rm in}\,\,\Omega,\\[5pt]
r|_{\Gamma}=0.
\end{array}
\right.
\end{equation}
Let $\tilde T=\max(1,T)$.
By the standard elliptic estimates it follows that
\begin{equation}
||r||_{H^1(\Omega)}\leq Ch^{-1}\tilde T^2e^{-\frac{T}{h}}||\psi||_{H^{1/2}_{co}(\Sigma)},
\end{equation}
for $0<h<1$, and with a constant $C>0$ independent on $T$ or $h$.

From \eqref{def-phi} it follows that $t^2\phi=\chi\psi$, where
\begin{equation}
\chi(t,T,h)=t^2\left(\int_0^T s^2 e^{-\frac{s}{h}}\,\text{d} s\right)^{-1}.
\end{equation}
It is easy to see that there exists $h_0>0$ so that if $0<h<h_0$, then
\begin{equation}
\chi(t,T,h)<CT^2h^{-3},\quad 0 \leq t\leq T,
\end{equation}
where the constant $C>0$ again is independent of $T$ or $h$.
We can conclude that
\begin{equation}
||\tilde\Lambda^{h,\Sigma}_{\rho,\mathbf{C}} \phi-h\mathcal{L}_T\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}(\chi\phi)||_{H^{-1/2}(\Sigma)}\leq
C\tilde T^4 h^{-3} e^{-\frac{T}{h}}||\phi||_{H^{1/2}_{co}(\Sigma)},
\end{equation}
or
\begin{equation}
||\tilde\Lambda^{h,\Sigma}_{\rho,\mathbf{C}} -h\mathcal{L}_T\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}\chi||_{H^{1/2}_{co}(\Sigma)\to H^{-1/2}(\Sigma)}\leq
C\tilde T^4 h^{-3} e^{-\frac{T}{h}},
\end{equation}
where the constant $C>0$ is independent of $T$ or $h$.

Considering $\tilde\Lambda^{h,\Sigma}_{\rho,C}$ as a semiclassical pseudodifferential operator with the small parameter $h$, it follows that its full symbol can be obtained from $\Lambda^{T,\Sigma}_{\rho,C}$ (but not necessarily the operator itself).

It is shown in \cite{HNZ} (see their Theorem 4.2) that the principal symbol of $\tilde\Lambda^{h,\Sigma}_{\rho,C}$ determines $\Gamma(x,h),\,x\perp\nu$, where $\Gamma(x,h)$ is the fundamental solution of $\rho-h^2L_C$ associated to the pair $\rho|_{D_1}$ and $C|_{D_1}$ whose Fourier transform with respect to $x$ with $x\perp\nu$ is bounded as $h\rightarrow 0$. See also \cite{NTU}, \cite{T}, for similar results. Once having this, it can be shown that by using $\Sigma$ is curved, we can recover $\rho|_{D_1}$ and $C|_{D_1}$. This is shown in \cite[Apendix B]{HNZ}.

\paragraph{The ND map case:}
The method of proof is almost entirely analogous to the DN map case. We will give only a brief sketch of its argument.

Let $\psi\inH^{-1/2}_{co}(\Sigma)$ and let  $u$  be a solution of
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}} u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
 \partial_\mathbf{C} u|_{(0,T)\times\Gamma}=t^2\psi,\\[5pt]
  (u, \partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
For $\phi\inH^{-1/2}_{co}(\Sigma)$ consider the elliptic boundary value problem
\begin{equation}\label{elliptic-N}
\left\{\begin{array}{l}
\rho v-h^2L_\mathbf{C} v=0\,\,\text{\rm in}\,\,\Omega,\\[5pt]
\partial_\mathbf{C} v|_{\Gamma}=\phi
\end{array}\right.
\end{equation}
depending on a parameter $h>0$.
Let
\begin{equation}
\tilde\Phi^{h,\Sigma}_{\rho,\mathbf{C}} \phi=h^{-1}v|_{\Sigma}
\end{equation}
be the associated ND map.

We will choose the Neumann data $\phi$ as
\begin{equation}
\phi=\psi\int_0^T t^2e^{-\frac{t}{h}}\,\text{d} t
\end{equation}
and let $u_0\in H^1(\Omega)$ be the solution of
\begin{equation}
\left\{\begin{array}{l}
L_\mathbf{C} u_0=0\,\,\text{\rm in }\,\,\Omega,\\[5pt]
\partial_\mathbf{C} u_0|_{\Gamma}=\psi.
\end{array}\right.
\end{equation}
Likewise before for the DN map case, we define $u_1$ so that
\begin{equation}
u(t,x)=t^2u_0(x)+u_1(t,x),\quad (t,x)\in \Omega_T.
\end{equation}
Then $u_1$ should satisfy
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u_1=-2\rho u_0\,\,\text{\rm in }\,\,\Omega_T\\[5pt]
 \partial_\mathbf{C} u_1|_{(0,T)\times\Gamma}=0,\\[5pt]
  (u_1, \partial_t u_1)|_{t=0}=0.
\end{array}\right.
\end{equation}
By the Korn inequality, this equation has a unique solution such that
\begin{equation}
u_1\in C([0,T];H^1(\Omega))\cap C^1([0,T]; L^2(\Omega))
\end{equation}
and
\begin{multline}
||u_1(t)||_{H^1(\Omega)}+||\partial_t u_1(t)||_{L^2(\Omega)}\\[5pt]
\leq C||2\rho u_0||_{L^2(\Omega_T)}\leq C||\psi||_{H^{-1/2}(\Omega)},\,\,t\in[0,T]
\end{multline}
(see \cite{Wloka}).

\begin{comment}
Define
\begin{equation}
w(\cdot,h)=\mathcal{L}_Tu_1=\int_0^Tu_1(\cdot,t)e^{-\frac{t}{h}}\,\text{d} t
\end{equation}
and let
\begin{multline}
F(\cdot,h)=-2h^3\rho(\cdot)u_0(\cdot)(1-e^{\frac{T}{h}})\\-
h^2e^{-\frac{T}{h}}\left(\partial_tu_1(\cdot,T)+h^{-1} u_1(\cdot, T)\right)\in H^{-1}(\Omega).
\end{multline}
Then $w$ is a solution of
\begin{equation}
\left\{\begin{array}{l}
\rho w-h^2L_\mathbf{C} w=F,\\[5pt]
\partial_\mathbf{C} w|_{\partial\Omega}=0.
\end{array}\right.
\end{equation}
\end{comment}

Now let
\begin{equation}
r(\cdot,h)=v(\cdot,h)-\int_0^Tu(t,\cdot)e^{-\frac{t}{h}}\,\text{d} t.
\end{equation}
\begin{comment}
and
\begin{equation}
z(\cdot,h)=v(\cdot,h)-u_0(\cdot)\int_0^Tt^2e^{-\frac{t}{h}}\,\text{d} t.
\end{equation}
Then
\begin{equation}
r(\cdot,h)=z(\cdot,h)-w(\cdot,h).
\end{equation}
Since $z$ satisfies
\begin{equation}
\left\{\begin{array}{l}
\rho z-h^2L_\mathbf{C} z = -\rho u_0\int_0^Tt^2e^{-\frac{t}{h}}\,\text{d} t\,\,\text{\rm in}\,\,\Omega,\\[5pt]
\partial_\mathbf{C} z|_{\partial\Omega}=0,
\end{array}\right.
\end{equation}
\end{comment}
It satisfies
\begin{equation}
\left\{\begin{array}{l}
h^{-2}\rho r -L_\mathbf{C} z = e^{-\frac{T}{h}}\left[\partial_tu_1(T)+h^{-1} u_1(T)
+\rho u_0(T^2h^{-1}+2T)\right]\,\,\text{\rm in}\,\,\Omega,\\[5pt]
\partial_\mathbf{C} r|_{\Gamma}=0.
\end{array}\right.
\end{equation}
Let $\tilde T=\max(1,T)$.
By the standard elliptic estimates it follows that
\begin{equation}
||r||_{H^1(\Omega)}\leq Ch^{-1}\tilde T^2e^{-\frac{T}{h}}||\psi||_{H^{-1/2}_{co}(\Sigma)},
\end{equation}
for $0<h<1$ and with a constant $C$ independent on $T$ or $h$.

Similarly to the DN case, we can conclude that
\begin{equation}
||\tilde\Phi^{h,\Sigma}_{\rho,\mathbf{C}} \phi-h^{-1}\mathcal{L}_T\Phi^{T,\Sigma}_{\rho,\mathbf{C}}(\chi\phi)||_{H^{1/2}(\Sigma)}\leq
C\tilde T^4 h^{-5} e^{-\frac{T}{h}}||\phi||_{H^{-1/2}_{co}(\Sigma)},
\end{equation}
or
\begin{equation}
||\tilde\Phi^{h,\Sigma}_{\rho,\mathbf{C}} -h^{-1}\mathcal{L}_T\Phi^{T,\Sigma}_{\rho,\mathbf{C}}\chi||_{H^{-1/2}_{co}(\Sigma)\to H^{1/2}(\Sigma)}\leq
C\tilde T^4 h^{-5} e^{-\frac{T}{h}},
\end{equation}
where the constant $C$ is independent of $T$ or $h$.

As above, we can obtain the symbol of $\tilde\Phi^{h,\Sigma}_{\rho,\mathbf{C}}$ from $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$. The principal symbol of $\tilde\Lambda^{h,\Sigma}_{\rho,\mathbf{C}}$ is the inverse of the symbol of $\tilde\Phi^{h,\Sigma}_{\rho,\mathbf{C}}$. We can therefore conclude as above that the local ND map determines the elastic tensor and density at the boundary.
\end{proof}

\section{Interior determination}\label{inner}

For the purposes of this section, $\Omega$ will be a domain in $\mathbb{R}^3$, $D\subset\Omega$ a subdomain, $\Sigma\subset \partial D\cap\partial\Omega\neq\emptyset$. Let $\Omega_2=\Omega\setminus \overline D$ and $\Sigma_2=\partial\Omega_2\setminus\partial\Omega$. Suppose $\mathbf{C}$ and $\rho$ are homogeneous in $D$ and let $\Lambda^T_{\Sigma_2}$ be the DN map for the domain $\Omega_2$ with data on $\Sigma_2$ and $\Phi^T_{\Sigma_2}$ be the similarly defined ND map. We will prove the following proposition.
\begin{prop}\label{interior-prop}
There exists some $0<\delta<\infty$ depending on $D$, $\mathbf{C}|_{D}$, $\rho|_{D}$ such that
\begin{itemize}
    \item[(i)] $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$ determines $\Lambda^{T-2\delta}_{\Sigma_2}$, \item[(ii)] $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$ determines $\Phi^{T-2\delta}_{\Sigma_2}$.
\end{itemize}
\end{prop}

We need the following  result from \cite{ET}:
\begin{prop}\label{ucp}
Suppose $\mathbf{C}$, $\rho$ are homogeneous in $D$. There is a (non-Riemannian) metric $N$ on $\mathcal{T} D$, determined by $\mathbf{C}|_D$ and $\rho|_D$, such that if $P_{\rho,\mathbf{C}}w=0$ in $D\times(0,T)$, $(w, \partial_\mathbf{C} w)|_{(0,T)\times\Sigma}=0$, then $w(T/2,x)=0$ for any $x\in D$ such that $d_N(x,\Sigma)<T/2$.
\end{prop}
Here $N$ is a family of norms $N_x$ on $\mathbb{R}^3\equiv\mathcal{T}_x D$, $x\in\Omega$, which induces a distance on $D$ by
\begin{equation}
d(x,y)=\inf_{\gamma}\int_0^1 N_{\gamma(t)}(\gamma'(t))\,\text{d} t,
\end{equation}
where the infimum is taken over curves $\gamma\in C^1([0,1];D)$ such that $\gamma(0)=x$ and $\gamma(1)=y$. The distance to the boundary is defined in the usual way as
\begin{equation}
d_N(x,\Sigma)=\inf_{y\in\Sigma}d_N(x,y).
\end{equation}

Now let $H^{-1}(\Omega)$ be the dual space of $H^1_0(\Omega)$, and for $F\in H^1((0,T); H^{-1}(\Omega))$ let $u$ satisfy
\begin{equation}\label{inner-data}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=F\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
u|_{(0,T)\times\Gamma}=0,\\[5pt]
(u, \partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
Then $u$ has the estimate
\begin{equation}
||u||_{X_D}\leq C||F||_{H^1((0,T);H^{-1}(\Omega))},
\end{equation}
with
\begin{equation}
 ||u||_{X_D}=||u||_{L^\infty((0,T);H^1_0(\Omega))}+||\partial_t u||_{L^\infty((0,T); L^2(\Omega))}.
\end{equation}
We write
\begin{equation}
G^D_{\rho,\mathbf{C}}(F)=u, \quad G^D_{\rho,\mathbf{C}}:H^1((0,T);H^{-1}(\Omega))\to X_D.
\end{equation}

Also, if $u$ satisfies
\begin{equation}\label{inner-data-N}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=F\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
\partial_\mathbf{C} u|_{(0,T)\times\Gamma}=0,\\[5pt]
(u, \partial_t u)|_{t=0}=0,
\end{array}\right.
\end{equation}
then
\begin{equation}
||u||_{X_N}\leq C||F||_{H^1((0,T);H^{-1}(\Omega))},
\end{equation}
with
\begin{equation}
 ||u||_{X_N}=||u||_{L^\infty((0,T);H^1(\Omega))}+||\partial_t u||_{L^\infty((0,T); L^2(\Omega))}.
\end{equation}
We write
\begin{equation}
G^N_{\rho,\mathbf{C}}(F)=u, \quad G^N_{\rho,\mathbf{C}}:H^1((0,T);H^{-1}(\Omega))\to X_N.
\end{equation}

\bigskip
Now let $\delta=\sup_{x\in D} d_N(x,\Sigma)$. Then we have the following lemma.
\begin{lem}\label{green-determination}
For any $F\in C^\infty(\Omega_T)$, $\mbox{supp\;} F\subset (0, T-\delta)\times D$, the DN map $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$ determine $G^D_{\rho,\mathbf{C}}(F)|_{ (0, T-2\delta)\times D}$.
\end{lem}
\begin{proof}
For $f\in C_0^\infty((0,T)\times\Sigma)$ we define $S^T_{\Sigma}(f)=u|_{(0,T)\times D}$, where $u$ satisfies
\begin{equation}\label{boundary-data}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
u|_{{(0,T)\times\Gamma}}=f,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
 Similarly, if $F\in C_0^\infty((0,T)\times D)$ we define $S^T(F)=u|_{(0,T)\times D}$, where $u$ satisfies \eqref{inner-data}.\footnote{We use this two notations as defined here only within the proof of this lemma.}

The first step in the proof is to show that $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$ determine $S^{T-\delta}_\Sigma$. Let $f\in C_0^\infty((0,T)\times\Sigma)$ and  $u$ that satisfies \eqref{boundary-data}. Extend $u$ by $0$ to $(-\infty,0)\times\Omega$. Let
\begin{multline}
\mathcal{S}_{f_1,f_2}^T=\{ v\in C^\infty((-\infty,T)\times D): P_{\rho,\mathbf{C}} v=0, \\
v|_{(-\infty,T)\times \Sigma}=f_1, \partial_\mathbf{C} v|_{(-\infty,T)\times \Sigma}=f_2\},
\end{multline}
which is determined by $f_1$, $f_2$, $D$, $\Sigma$, $\rho|_D$, and $\mathbf{C}|_D$. Let $u'\in\mathcal{S}_{f,\Lambda^{T,\Sigma}_{\rho,\mathbf{C}} f}^T$, and set $w=u-u'$. Then $P_{\rho,\mathbf{C}}w=0$ on $D$,  and $(w, \partial_\mathbf{C} w)|_{(0,T)\times\Sigma}=0$. For $t<T-\delta$ we can apply Proposition \ref{ucp} with the time interval $(t-\delta, t+\delta)$ to conclude that $w(t,x)=0$ for all $x\in D$. So we can now assume $S^{T-\delta}_\Sigma$ is known.

 Let $\mathfrak{r}$ be the time reversal operator on $(0,T-\delta)$. That is $(\mathfrak{r}\,\ell)(t)=\ell(T-\delta-t),\,t\in(0,T-\delta)$ for any function $\ell$ over $(0,T-\delta)$. If we write
\begin{equation}
u^*=\mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r}f,
\end{equation}
$u^*$ satisfies
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u^*=0\,\,\text{\rm in}\,\,(0, T-\delta)\times D,\\[5pt]
 u^*|_{(0,T)\times\Gamma}= f, (u^*, \partial_t u^*)|_{t=T-\delta}=0.
\end{array}\right.
\end{equation}
We want to identify the adjoint in $L^2((0,T-\delta)\times D)$ of $\mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r}$. Let $F\in C_0^\infty((0,T)\times D)$ and denote $v=G^D_{\rho,\mathbf{C}}(F)$. Then, using integration by parts,
\begin{multline}
\langle F, \mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r} f \rangle=
\int_0^{T-\delta}\int_D F(t,x) u^*(t,x) \,\text{d} x\,\text{d} t=
\int_0^{T-\delta}\int_\Omega P_{\rho,\mathbf{C}}(v) u^* \,\text{d} x\,\text{d} t\\=
-\int_0^{T-\delta}\int_\Sigma \partial_\mathbf{C} v f.
\end{multline}
We may then take the map $F\to \partial_\mathbf{C} v|_\Sigma$ to be known. Extend $v$ as $0$ to $(-\infty,0)\times\Omega$ and let $v'\in \mathcal{S}^{T-\delta}_{0, \partial_\mathbf{C} v|_\Sigma}$. Define $w=v-v'$. As above we can conclude that $w(t,x)=0$ for all $t<T-2\delta$ and $x\in D$. This proves that $S^{T-2\delta}(F)$ is determined by the  knowledge of $D$, the DN map $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$ and $\mathbf{C}|_D$.
\end{proof}

\begin{lem}\label{green-determination-N}
For any $F\in C^\infty(\Omega_T)$, $\mbox{supp\;} F\subset (0, T-\delta)\times D$, the ND map $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$ determine $G^N_{\rho,\mathbf{C}}(F)|_{ (0, T-2\delta)\times D}$.
\end{lem}
\begin{proof}
For $g\in C_0^\infty((0,T)\times\Sigma)$ we define $S^T_{\Sigma}(g)=u|_{(0,T)\times D}$, where $u$ satisfies
\begin{equation}\label{boundary-data-N}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
\partial_\mathbf{C} u|_{{(0,T)\times\Gamma}}=g,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
 Similarly, if $F\in C_0^\infty((0,T)\times D)$ we define $S^T(F)=u|_{(0,T)\times D}$, where $u$ satisfies \eqref{inner-data-N}.

The first step in the proof is to show that $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$ determine $S^{T-\delta}_\Sigma$. Let $g\in C_0^\infty((0,T)\times\Sigma)$ and  $u$ that satisfies \eqref{boundary-data-N}. Extend $u$ by $0$ to $(-\infty,0)\times\Omega$. Let $u'\in\mathcal{S}_{\Phi^{T,\Sigma}_{\rho,\mathbf{C}} g, g}^T$, and set $w=u-u'$. Then $P_{\rho,\mathbf{C}}w=0$ on $D$,  and $(w, \partial_\mathbf{C} w)|_{(0,T)\times\Sigma}=0$. For $t<T-\delta$ we can apply Proposition \ref{ucp} with the time interval $(t-\delta, t+\delta)$ to conclude that $w(t,x)=0$ for all $x\in D$. So we may now assume $S^{T-\delta}_\Sigma$ is known.

If we write
\begin{equation}
u^*=\mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r}g,
\end{equation}
$u^*$ satisfies
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u^*=0\,\,\text{\rm in}\,\,(0, T-\delta)\times D,\\[5pt]
\partial_\mathbf{C}  u^*|_{(0,T)\times\Gamma}= g,\\[5pt]
 (u^*, \partial_t u^*)|_{t=T-\delta}=0.
\end{array}\right.
\end{equation}
We want to identify the adjoint in $L^2((0,T-\delta)\times D)$ of $\mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r}$. Let $F\in C_0^\infty((0,T)\times D)$ and denote $v=G^N_{\rho,\mathbf{C}}(F)$. Then, using integration by parts,
\begin{multline}
\langle F, \mathfrak{r}S^{T-\delta}_\Sigma\mathfrak{r} f \rangle=
\int_0^{T-\delta}\int_D F(t,x) u^*(t,x) \,\text{d} x\,\text{d} t=
\int_0^{T-\delta}\int_\Omega P_{\rho,\mathbf{C}}(v) u^* \,\text{d} x\,\text{d} t\\=
\int_0^{T-\delta}\int_\Sigma v\cdot g.
\end{multline}
We can then take the map $F\to  v|_\Sigma$ to be known. Extend $v$ as $0$ to $(-\infty,0)\times\Omega$ and let $v'\in \mathcal{S}^{T-\delta}_{v|_\Sigma,0}$. Define $w=v-v'$. As above we can conclude that $w(t,x)=0$ for all $t<T-2\delta$ and $x\in D$. This proves that $S^{T-2\delta}(F)$ is determined by the knowledge of $D$, the ND map $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$ and $\mathbf{C}|_D$.
\end{proof}

Let  $f\in C^1((0,T); H^{-1/2}(\Sigma_2))$, and define $T_f\in C^1((0,T); H^{-1}(\Omega))$ by
\begin{equation}
\langle T_f,\phi\rangle=\int_0^T\langle f,\phi|_{\Sigma_2}\rangle,\quad \phi\in H^1_0(\Omega).
\end{equation}
Define the operator  $\mathscr{L}^T_D$ by  $\mathscr{L}^T_D(f)=u|_{\Sigma_2}$, where
\begin{equation}\label{definition-LD}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=T_f\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
 u|_{(0,T)\times\Gamma}=0,\\[5pt]
(u, \partial_t u)|_{t=0}=0,
\end{array}\right.
\end{equation}
and the operator $\mathscr{L}^T_N$ by $\mathscr{L}^T_N(f)=u|_{\Sigma_2}$, where
\begin{equation}\label{definition-LN}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=T_f\,\,\text{\rm in}\,\,\Omega_T,\\[5pt]
 \partial_\mathbf{C} u|_{(0,T)\times\Gamma}=0,\\[5pt]
(u, \partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}

\begin{comment}
These definitions require a discussion of regularity. Let $F\in C_0^\infty(Q)$. Set  $u=G^D_{\rho,C}(F)$ or $u=G^N_{\rho,C}(F)$ and define
\begin{equation}
E(t)=\frac{1}{2}\int_\Omega\left(\rho|\partial_t u|^2+Du:\mathbf{C}:Du\right)\,\text{d} x.
\end{equation}
Then
\begin{equation}
\frac{\,\text{d}}{\,\text{d} t}E(t)= \int_\Omega \partial_t u  F\,\text{d} x,
\end{equation}
and, since $E(0)=0$,
\begin{multline}
E(t)=\int_0^t \int_\Omega \partial_t u(s,x)  F(s,x)\,\text{d} x\,\text{d} s\\= \int_\Omega u(t,x) F(t,x)\,\text{d} x- \int_0^t\int_\Omega u(s,x) \partial_t  F(s,x)\,\text{d} x\,\text{d} s\\
\leq C\left(  \sqrt{E(t)}|| F(t,\cdot)||_{H^{-1}(\Omega)}  +\int_0^t\sqrt{E(s)}||\partial_t  F(s,\cdot)||_{H^{-1}(\Omega)}\,\text{d} s  \right),
\end{multline}
Let $\zeta(t)=\max(\sqrt{E(t)}, 1)$. Then
\begin{equation}
\zeta(t)\leq 1+ C\left( || F(t,\cdot)||_{H^{-1}(\Omega)}+\int_0^t\zeta(s)||\partial_t  F(s,\cdot)||_{H^{-1}(\Omega)}\,\text{d} s   \right).
\end{equation}
By Gronwall's inequality
\begin{multline}
\zeta(t)\leq 1+C [ || F(t,\cdot)||_{H^{-1}(\Omega)}\\
+\int_0^t || F(s,\cdot)||_{H^{-1}(\Omega)}||\partial_t  F(s,\cdot)||_{H^{-1}(\Omega)}
\exp\left( \int_s^t ||\partial_t  F(r,\cdot)||_{H^{-1}(\Omega)}\,\text{d} r  \right)\,\text{d} s   ].
\end{multline}
Let $Y$ be the space $C_0^1 ((0,T); H^{-1}(\Omega))$ with the norm
\begin{equation}
||F||_Y=\sup_{t\in[0,T]}\left(|| F(t,\cdot)||_{H^{-1}(\Omega)} +||\partial_t  F(t,\cdot)||_{H^{-1}(\Omega)}  \right).
\end{equation}
We have already shown that $G^D_{\rho,\mathbf{C}}:Y\to X_D$, $G^N_{\rho,\mathbf{C}}:Y\to X_N$. Suppose $||F_n||_Y\to 0$. Let $E_n(t)$ and $\zeta_n(t)$ be the corresponding functionals defined as above. Clearly $||\zeta_n||_{L^\infty(0,T)}$ is bounded and
\begin{equation}
E_n(t)\leq C ||\zeta_n||_{L^\infty(0,T)} ||F_n||_Y\to0.
\end{equation}
Therefore, $G^D_{\rho,\mathbf{C}}$ and $G^N_{\rho,\mathbf{C}}$ are continuous from $Y$ to $X_D$ and $X_N$, respectively.
\end{comment}

\begin{lem}\label{lemma33}
$\mathscr{L}^{T-2\delta}_D$ is determined by the  knowledge of $D$, the DN map $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$.
$\mathscr{L}^{T-2\delta}_N$ is determined by the  knowledge of $D$, the ND map $\Phi^{T,\Sigma}_{\rho,\mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$.
\end{lem}
\begin{proof}
Let  $f\in C^2_0((0,T);C_0^\infty(\Sigma_2))$. In local (in space) coordinates we can arrange that $\Sigma_2$ is $\{x_3=0\}$ and $D$ is  $\{x_3>0\}$. Suppose the spatial support of $f$ is entirely in this coordinate patch. For $\epsilon>0$ define $F_\epsilon\in C^1((0,T);H^{-1}(\Omega))$ by
\begin{equation}
 F_\epsilon (\phi)(t)= \int_{x_3=0} f(t,x')\phi(x',\epsilon)\,\text{d} x', \quad \phi\in C_0^\infty(D).
\end{equation}
Then
\begin{multline}
|F_\epsilon(\phi)(t)-T_f(\phi)(t)|= \left|\int_{0\leq x_3\leq\epsilon} f(t,x')\partial_3\phi(x',x_3)  \right|\\
\leq C||\phi||_{H^1_0(\Omega)}||f||_{L^\infty((0,T)\times\Sigma_2)}\epsilon^{1/2},
\end{multline}
and similarly
\begin{equation}
|\partial_t F_\epsilon(\phi)(t)-\partial_tT_f(\phi)(t)| \leq C||\phi||_{H^1_0(\Omega)}||\partial_t f||_{L^\infty((0,T)\times\Sigma_2)}\epsilon^{1/2}.
\end{equation}
Using a partition of unity argument, we can therefore construct a sequence $F_n\in C^1((0,T);H^{-1}(\Omega))$, $\mbox{supp\;} F_n\subset (0,T)\times D$, such that $F_n\to T_f$ in $C^1((0,T);H^{-1}(\Omega))$.

Suppose now that we take two functions $f,h \in C^2_0((0,T-2\delta);C_0(\Sigma_2))$ and construct sequences $F_n$, $H_n$ as above. By Lemma \ref{green-determination}, $H_n(G^D_{\rho,C}(F_n))$ is determined by the  knowledge of $D$, the DN map $\Lambda^{T,\Sigma}_{\rho, \mathbf{C}}$, $\rho|_D$, and $\mathbf{C}|_D$. Passing to the limit we see that so is $T_h( G^D_{\rho,\mathbf{C}}(T_f))$ and therefore so is $\mathscr{L}^{T-2\delta}_D$. The same is true for $\mathscr{L}^{T-2\delta}_N$.
\end{proof}

 Let  $\Lambda^{T,+}_{\Sigma_2}$ be the DN map for the domain $D$ with data on $\Sigma_2$. Also let $\Phi^{T,+}_{\Sigma_2}$ be the ND map for the domain $D$ with data on $\Sigma_2$.

\begin{lem}
If $(\Lambda^T_{\Sigma_2}-\Lambda^{T,+}_{\Sigma_2})f=0$   for $f\in C^\infty_0((0,T);H^{1/2}_{co}(\Sigma_2))$, then $f=0$. If $(\Phi^T_{\Sigma_2}-\Phi^{T,+}_{\Sigma_2})g=0$ for $g\in C^\infty_0((0,T);H^{-1/2}_{co}(\Sigma_2))$, then $g=0$.
\end{lem}
\begin{proof}
Suppose $(\Lambda^T_{\Sigma_2}-\Lambda^{T,+}_{\Sigma_2})f=0$. Let $u$ and $u^+$ be the solutions in $\Omega_2$ and $D$ respectively with Dirichlet data $f$ on $\Sigma_2$ and zero on the rest of their respective boundaries. Define $\tilde u= u\chi_{\Omega_2}+u_+\chi_D$. By the assumption, since both Dirichlet and Neumann data accross $\Sigma_2$ match, we have that $P_{\rho,\mathbf{C}} \tilde u=0$ in $\Omega$. Since $\tilde u$ has zero initial Cauchy and lateral Dirichlet data, it must be zero.

In the case when $(\Phi^T_{\Sigma_2}-\Phi^{T,+}_{\Sigma_2})f=0$, the argument is identical.
\end{proof}

\begin{lem}\label{lem-surjective-D}
For $f\in C^\infty_0((0,T)\times\Sigma_2)$,
\begin{equation}
(\Lambda^T_{\Sigma_2}-\Lambda^{T,+}_{\Sigma_2})\mathscr{L}^T_D f=f.
\end{equation}
\end{lem}
\begin{proof}
Let $u$ be as in \eqref{definition-LD} and $\phi\in C_0^\infty((0,T)\times\Omega)$. Then
\begin{equation}
\langle f,\phi|_{\Sigma_2}\rangle=\int_0^T\int_\Omega P_{\rho,\mathbf{C}}(u) \phi=
\int_0^T\int_\Omega \big(\partial_t^2 u \phi+Du:(\mathbf{C}:D\phi)\big)
\end{equation}
and
\begin{multline}
\int_0^T\int_\Omega Du:(\mathbf{C}:D\phi) =\int_0^T\int_{\Omega_2} Du:(\mathbf{C}:D\phi)+\int_0^T\int_D Du:(\mathbf{C}:D\phi)\\=
-\int_0^T\int_{\Omega_2}L_{\mathbf{C}}(u)\phi-\int_0^T\int_{D}L_{\mathbf{C}}(u)\phi\\
+\langle \Lambda^T_{\Sigma_2}\mathscr{L}^T_D f,\phi|_{\Sigma_2}\rangle-\langle \Lambda^{T,+}_{\Sigma_2}\mathscr{L}^T_D f,\phi|_{\Sigma_2}\rangle.
\end{multline}
\end{proof}

\begin{proof}[Proof of Proposition \ref{interior-prop} (i)]It follows that $\Lambda^{T-2\delta}_{\Sigma_2}-\Lambda^{T-2\delta,+}_{\Sigma_2}$ is the inverse of $\mathscr{L}^{T-2\delta}_D$, which is then determined by $\Lambda^{T,\Sigma}_{\rho,\mathbf{C}}$. The claim (i) follows as $\Lambda_{\Sigma_2}^{T-2\delta,+}$ is clearly determined by $D$, $C_D$
and $\rho|_D$.
\end{proof}

The argument for the ND map case is a little bit more involved. Before stating the lemma for ND maps that is analogous to Lemma \ref{lem-surjective-D} we introduce two more notations. If $u=G^N_{\rho,\mathbf{C}}(T_f)$, then $\partial_{\mathbf{C}_D}u|_{\Sigma_2^+}$ will denote the restriction of $\partial_\mathbf{C} u$ to $\Sigma_2$, taken from the $D$ side of $\Sigma_2$. Similarly, $\partial_{\mathbf{C}_{\Omega_2}}u|_{\Sigma_2}$ will be the restriction of the same function to $\Sigma_2$, taken from the $\Omega_2$ side of $\Sigma_2$. Note that here the unit normal vector in all cases points away from $\Omega_2$ and into $D$.

\begin{lem}\label{lem-surjective-N}
For $f\in C^\infty_0((0,T)\times\Sigma_2)$,
\begin{equation}
(\Phi^T_{\Sigma_2}-\Phi^{T,+}_{\Sigma_2})(\partial_{\mathbf{C}|_D}u|_{\Sigma_2^+}+f)=-\Phi^{T,+}_{\Sigma_2}(f).
\end{equation}
\end{lem}
\begin{proof}
As in the proof of Lemma \ref{lem-surjective-D}, for $\phi\in C_0^\infty((0,T)\times\Omega)$
\begin{equation}
\langle f,\phi|_{\Sigma_2}\rangle=\int_0^T\int_\Omega P_{\rho,\mathbf{C}}(u) \phi=
\int_0^T\int_\Omega\big( \partial_t^2 u \phi+Du:(\mathbf{C}:D\phi)\big),
\end{equation}
and
\begin{multline}
\int_0^T\int_\Omega Du:(\mathbf{C}:D\phi) =\int_0^T\int_{\Omega_2} Du:(\mathbf{C}|_{\Omega_2}:D\phi)+\int_0^T\int_D Du:(\mathbf{C}|_D:D\phi)\\=
-\int_0^T\int_{\Omega_2}L_{\mathbf{C}}(u)\phi-\int_0^T\int_{D}L_{\mathbf{C}}(u)\phi\\
+\langle \partial_{\mathbf{C}_{\Omega_2}}u|_{\Sigma_2},\phi|_{\Sigma_2}\rangle-\langle \partial_{\mathbf{C}_D}u|_{\Sigma_2^+},\phi|_{\Sigma_2}\rangle.
\end{multline}
It follows that
\begin{equation}
\partial_{\mathbf{C}_{\Omega_2}}u|_{\Sigma_2}=\partial_{\mathbf{C}_D}u|_{\Sigma_2^+}+f.
\end{equation}
Since $u(t,\cdot)\in H^1(\Omega)$, its Dirichlet data on each side of $\Sigma_2$ coincide. Therefore
\begin{equation}
\Phi^T_{\Sigma_2}(\partial_{\mathbf{C}_{\Omega_2}}u|_{\Sigma_2})=\Phi^{T,+}_{\Sigma_2}(\partial_{\mathbf{C}_{D}}u|_{\Sigma_2^+}).
\end{equation}
The conclusion follows immediately.
\end{proof}

The surjectivity of the difference $\Phi^T_{\Sigma_2}-\Phi^{T,+}_{\Sigma_2}$ follows from the next lemma.

\begin{lem}\label{phi-surjective}
The local ND map
\begin{equation}
\Phi^{T,+}_{\Sigma_2}: C_0^\infty((0,T];H^{-1/2}_{co}(\Sigma_2))\to C_0^\infty((0,T];H^{1/2}(\Sigma_2))
\end{equation}
 is surjective.
\end{lem}

First note that since the coefficients of the equation are time independent, taking time derivatives of all orders and applying Theorem \ref{thm-forward}, we have the following corollary with the same notations as in Theorem \ref{thm-forward}.
\begin{cor}
If $f\in C_0^\infty((0,T];H^{1/2}(\Gamma_u))$, $g\in C_0^\infty((0,T];H^{-1/2}(\Gamma_s))$,
then there exists a unique $u\in C_0^\infty((0,T];H^1(\Omega))$ such that
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,\Omega,\\[5pt]
u|_{(0,T)\times\Gamma_u}=f,\\[5pt]
\partial_\mathbf{C} u|_{(0,T)\times\Gamma_s}=g,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
\end{cor}
In particular, taking $\Gamma_u = \emptyset$ and $\Gamma_s = \Gamma$, this justifies the spaces in between which $\Phi^{T,+}_{\Sigma_2}$ maps in the statement of the lemma.

\begin{proof}[Proof of Lemma \ref{phi-surjective}]
To prove the surjectivity, let $f\in C_0^\infty((0,T];H^{1/2}(\Sigma_2))$. We choose $\Gamma_u=\Sigma_2$, $\Gamma_s=\partial D\setminus\overline{\Sigma_2}$. Let $u\in C_0^\infty((0,T];H^1(\Omega))$ be such that
\begin{equation}
\left\{\begin{array}{l}
P_{\rho,\mathbf{C}}u=0\,\,\text{\rm in}\,\,D,\\[5pt]
u|_{(0,T)\times\Gamma_u}= f,\\[5pt]
\partial_\mathbf{C} u|_{(0,T)\times\Gamma_s}=0,\\[5pt]
(u,\partial_t u)|_{t=0}=0.
\end{array}\right.
\end{equation}
Since $\Phi^{T,+}_{\Sigma_2}(\partial_{\mathbf{C}|_D} u|_{\Sigma_2^+})=f$, we have our desired conclusion.
\end{proof}

\begin{proof}[Proof of Proposition \ref{interior-prop} (ii)]With the same notation used in Lemma \ref{lem-surjective-N}, for $f\in C^\infty_0((0,T);H^{-1/2}(\Sigma_2))$ define the operator
\begin{equation}
\begin{array}{c}
\mathscr{K}:C^\infty_0((0,T);H^{-1/2}_{co}(\Sigma_2))\to C^\infty_0((0,T);H^{-1/2}_{co}(\Sigma_2)),\\[5pt]
\mathscr{K}(f)=\partial_{\mathbf{C}|_D}u|_{\Sigma_2^+}+f.
\end{array}
\end{equation}
By Lemmas \ref{lem-surjective-N} and \ref{phi-surjective}, this operator is surjective. It follows from the proof of Lemma \ref{lemma33} that $\mathscr{K}$ is  determined by the knowledge of $\Omega$, $D$, $\mathbf{C}|_D$, $\rho|_D$, and $\Phi^{T,\Sigma}_{\rho, \mathbf{C}}$. Its right inverse, which we will denote by $\kappa$, is then determined by the same quantities.
We have now that $\Phi^T_{\Sigma_2}-\Phi^{T,+}_{\Sigma_2}=-\Phi^{T,+}_{\Sigma_2}\circ\kappa$ is determined by the knowledge of $\Omega$, $D$, $\mathbf{C}|_D$, $\rho|_D$, and $\Phi^{T,\Sigma}_{\rho, \mathbf{C}}$. This ends the proof.
\end{proof}

\section{Proofs of Theorem \ref{main-thm} and Theorem \ref{thm-interest}}\label{proof}

\begin{proof}[Proof of Theorem \ref{main-thm}]The time $T_1$ can be chosen arbitrarily small. By Proposition \ref{boundary-prop}, if $\Lambda^{T_1,\Sigma}_{\rho^{(1)},C^{(1)}}=\Lambda^{T_1,\Sigma}_{\rho^{(2)},C^{(2)}}$ or $\Phi^{T_1,\Sigma}_{\rho^{(1)},C^{(1)}}=\Phi^{T_1,\Sigma}_{\rho^{(2)},C^{(2)}}$, then $\rho^{(1)}|_{D_1}=\rho^{(2)}|_{D_1}$ and $C^{(1)}|_{D_1}=C^{(2)}|_{D_1}$. Now let $\delta$ be as in Proposition \ref{interior-prop}, with $D=D_1$. We can choose $T_2=T_1+2\delta$. If $\Lambda^{T_2,\Sigma}_{\rho^{(1)},C^{(1)}}
=\Lambda^{T_2,\Sigma}_{\rho^{(2)},C^{(2)}}$ or $\Phi^{T_2,\Sigma}_{\rho^{(1)},C^{(1)}}
=\Phi^{T_2,\Sigma}_{\rho^{(2)},C^{(2)}}$, then $\Lambda^{T_1,\Gamma_2}_{\rho^{(1)},C^{(1)}}=\Lambda^{T_1,\Gamma_2}_{\rho^{(2)},C^{(2)}}$ or $\Phi^{T_1,\Gamma_2}_{\rho^{(1)},C^{(1)}}=\Phi^{T_1,\Gamma_2}_{\rho^{(2)},C^{(2)}}$, respectively, where these DN and ND maps are taken relative to $\Omega\setminus \overline{D_1}$. By Proposition \ref{boundary-prop} it follows that $\rho^{(1)}|_{D_2}=\rho^{(2)}|_{D_2}$ and $C^{(1)}|_{D_2}=C^{(2)}|_{D_2}$. It is clear that we may continue in this way to inductively construct all the times $T_k$.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm-interest}]
It is clear that we only need to prove the result in the case $R=\Omega$. Notice that $\rho^{(I)}$, $C^{(I)}$ are all constant on all the elements of the common partition
\begin{equation}
\{\tilde D_\gamma\}=\{D_\alpha^{(1)}\cap D_\beta^{(2)}\}.
\end{equation}
 Let $P\in R$ and pick a smooth curve $\omega:[0,1]\to R$ so that $\omega(0)\in\Sigma$, $\omega(1)=P$, and which intersects the boundaries of the subdomains $\tilde D_\gamma$ only at smooth points and transversally. Let $V_\epsilon$ be the tubular neighborhood of $\omega([0,1])$ of radius $\epsilon>0$. We can choose $\epsilon$ small enough that $V_\epsilon$ only intersects the smooth components of the boundaries of the subdomanins $\tilde D_\gamma$ and $\partial V_\epsilon$ is transversal to all of them. It then follows that each $D'_\gamma=\tilde D_\gamma\cap V_\epsilon$ is a Lipschitz set.

 We can label by $D'_1$, \ldots, $D'_N$ the chain of non-empty sets in $\{D'_\gamma\}$, in the order in which the curve $\omega$ intersects them. This chain satisfies the conditions of Theorem \ref{main-thm} and we may conclude that there is a time $0<T_P<\infty$ such that if $\Lambda^{T_P,\Sigma}_{\rho^{(1)},C^{(1)}}= \Lambda^{T_P,\Sigma}_{\rho^{(2)},C^{(2)}}$ or $\Phi^{T_P,\Sigma}_{\rho^{(1)},C^{(1)}}= \Phi^{T_P,\Sigma}_{\rho^{(2)},C^{(2)}}$, then $\rho^{(1)}(P)=\rho^{(2)}(P)$ and $C^{(1)}(P)=C^{(2)}(P)$. Choosing $T=\max_{P\in R}T_P<\infty$, we have the result.
\end{proof}

\noindent
\bigskip{\bf\large Acknowledgement}${}$
\newline
This project began while the first author was employed at the Hong Kong University of Science and Technology, Jockey Club Institute for Advanced Study, and concluded while he was visiting Hokkaido University, partially supported by Sichuan University.
The second author was partially supported by Grant-in-Aid for Scientific Research (15K21766, 15H05740) of the Japan Society for  the  Promotion  of  Science  doing  the  research  of  this  paper. The third author was supported by EPSRC grants EP/P01593X/1 and EP/R002207/1.


\section{Introduction}
Outcome-based Learning is gaining prominence in learner-centric educational services\cite{lo_ed}. Rather than focusing on the the approach to learning, outcome-based learning sets the goals on what a learner can accomplish. It is no coincidence that learning outcomes (LOs) are typically defined using action oriented verbs, largely concerned with academic achievements\cite{lo_def}. This pivoting of measuring LOs allows for self-discovery and has the potential to move away from one-size-fits-all learning solutions. However, such a paradigm shift can put undue stress on already over stretched human resources in the education sector. But, thanks to deeper mobile penetration and faster technology adoption rates, online learning platforms can, not only supplement, but also amplify the reach and potential of every stakeholder in the system, especially teachers and students\cite{lo_dhar}. In fact, the opportunity is so huge that we can even revisit the scope of LOs. Studies have shown that besides skill and competency, psychological traits such as a grit and resilience also contribute to success in general, and in academics in particular \cite{flow,psyq_grit}. This leads to the following important question - how do we measure LOs. There are both direct and indirect ways to measure them\cite{lo_price}. Test scores such as GMAT, GRE, JEE, are the most widely used form of direct measurements. Psychometric tests are available to indirectly measure non-cognitive, psychological constructs\cite{psyq_flow,psyq_res}. In this work, we consider Test Scores as a form of an LO measurement except that the LO is now composite in nature. Our thesis is that, an assortment of skills are required to \textit{perform} well on a test and \textit{succeed}. Online learning platforms have the benefit of capturing fine-grained learner interactions. Depending on the depth of instrumentation, they can track various pedagogic and behavioural aspects of the learning process. We set forth to a framework to leverage such fine-grained signals to positively drive LOs, and do this in a way that the student can be an active participant in the decision process.

\section{Related Work}
In \cite{score_Er}, predictive models were used to classify learners based on their engagement, which instructors can act upon \textit{in situ}. Behavioural patterns associated with performance were mined using Matrix Factorization techniques in \cite{score_meh}. Predicting student test scores based on activity was considered in \cite{score_michael} using a linear regression model with features such as \textit{tasks completed} and \textit{number of attempts}, among others. Using gradient-boosted trees, \cite{score_khe} found that factors such as \textit{scheduling} and \textit{content} are important predictors of students success in MOOCs. A critical review of the state of the art papers in predicting student scores (in MOOCs) was undertaken in \cite{score_josh}. One striking observation is that, in the context of MOOCs, there can be multiple definitions of success (or outcomes) measured in terms of \textit{dropout rate},\textit{certification}, and \textit{ final exam grade}. The importance of feature engineering and the need for providing actionable predictions was emphasized. In the next Section, we discuss the details of our approach.

\section{Modeling Framework}
We consider an extensible, end-to-end modeling framework, where multiple models work in concert, feeding forward their results to downstream models and applications. We lay emphasis on feature engineering pipelines, incorporating uncertainty quantification via prediction intervals, and an optimization framework to produce actionable feedback based on the predictive model.

\subsection{ Network of Models architecture}
A functional view of the \emph{Network of Models} architecture is shown in Fig.~\ref{fig_arch_01}. It is inspired by the modality of construction of deep learning networks, where intermediate networks can be seen as feature engineering models. Unlike in traditional deep learning networks, the modules in this architecture can be black box or hand-crafted.
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.35]{images/ESQ_Diagrams_Arch.png}
 \caption{Network of Models}
 \label{fig_arch_01}
 \end{figure}
For example,  we used a Bayesian Knowledge Tracing (\texttt{BKT})\cite{bkt_corbett} model to summarize temporal learner interactions into interpretable features, which are consumed by a Concept Mastery Model. We used Deep Factorization model \texttt{deepFM} \cite{deepfm_guo} to predict academic competency of student across 1,242 concepts. Note that \texttt{Embibe} platform has more than 11,000 concepts in its Knowledge Graph, and for this paper we merged granular concepts into broader concepts (eg: \textit{Differential equation dilution problems} and \textit{Differential Equation Growth and Decay Problems} are mapped to \textit{Application of Differential Equations}) to arrive at the 1,242 concept set. \texttt{deepFM} has the ability to learn complex learner-concept interactions, and has a highly configurable model architecture. Along with the aforementioned academic features, a set of behavioural and test-taking features drive the Embibe Score Quotient model (\texttt{ESQ}) Model. \texttt{ESQ} attempts to model the scoring potential of a learner in a test yet-to-be-taken based on all observations available prior to taking the test\cite{patent_keyur}. The nudges model, using the learnt \texttt{ESQ} as an Oracle, predicts the changes in the learner state (described in terms of the observed features) to achieve  the desired change in the LO. Subsequently we focus on ESQ, and briefly discuss the nudges model.

\subsection{Data Preparation}
\texttt{Embibe} is an online learning platform offering learning solutions for learners through K12 and beyond. It has a collection of immersive content as well as a repository of curated and AI generated assessments and tests. Tens of thousands of practice questions and hundreds of mock tests were made available for learners to prepare and practice \cite{edm_soma}. For the purpose of developing \texttt{ESQ}, we consider data from 175,441 users on the \texttt{Embibe} Platform between January 2017 and December 2019. The raw interactions amount to more than 110 million data points. Data generated from bots and system testers are removed. We also consider only tests where at least 10\% of total questions available should have been attempted, and 10\% of total time available should have been spent.

\subsection{Feature Engineering}
An extensive list of more than 200 features was discussed in \cite{kalyan} useful in MOOC and online platform settings. Pyschological traits such as flow and resilience can also be measured based on survey type questions \cite{psyq_flow,psyq_grit}. On similar lines, we have considered academic, behavioral, effort and test-taking attributes (denoted as AQ, BQ, EQ, TQ respectively), as features to estimate the test score\cite{patent_keyur}. This notion is depicted in Fig~.2.

\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.35]{images/05_esq_split_eq.png}
 \caption{Hypothesized feature categories driving test score}
 \label{fig_feature_02}
 \end{figure}

\textbf{AQ features:}  We use individualized \texttt{BKT}\cite{bkt_gordon}, to summarize the time series attempt level data. Whether a student is in a learned state in a concept/skill, whether the student has a tendency to guess a question against a concept are some of the features given by the \texttt{BKT} model. However, \texttt{BKT} is skill specific and it can not exploit student or concept similarities. In order to leverage this information,  we fit \texttt{deepFM}\cite{deepfm_guo} model to produce student competency across the 1,242 concept set. We reduce the dimension to 50 by random projections\cite{kdd_rp}. A subset of other features considered are \textit{mean accuracy in last three tests}, \textit{score on last test}.

\textbf{TQ features:} Fine-grained attempt-level events are meticulously captured while taking a test on \texttt{Embibe} platform. These events include looking at a question, choosing an answer option, changing an answer option, marking a question for review, attempting a question, swapping subjects, and many others. Using the timestamp for each event, we infer how a student has attempted the test paper at the event level. Each question is also tagged with its ideal time, which is defined as the expected time taken by an idealized student. Based on this we can infer what kinds of questions are attempted too quickly or too slowly. Each question is also tagged as whether \textit{first-look} or not. \textit{First-look} implies that an attempt decision on that question was taken by the student when the student viewed it for the first time, and has not altered that decision in any subsequent visit to that question. Features in TQ bucket summarize question answering behaviour while taking a test. All features are normalized to lie between 0 and 1.

\textbf{BQ features:} In addition to what a learner "knows", scoring on a test also depends on "how" the learner attempts any given test. Traits such as carelessness and over-confidence can affect the final test score. BQ features stand as a proxy for these traits. Features such as \textit{ratio of careless mistakes to all attempts}, \textit{ratio of time spent on non attempts to total time spent}, fall under this bucket.

\textbf{EQ features:} These features measure how much effort was spent by the student between two tests. On the \texttt{Embibe} platform, students can search, browse, watch content, ask questions and perform many other actions. We summarize how students engage based on time spent, and how the student is navigating from one environment to another. Quantification of effort based on what transpired between two events is a useful feature to include\cite{amar}.

\subsection{Model Building}
Let $y_{t} \in \mathbb{R}$ and $x_{t} \in \mathbb{R}^{p}$  be the response variable (score), and the $p-$dimensional feature vector at time $t$, respectively. Then, we are interested in finding the mapping: $$f: x_{0:t},y_{0:t} \xrightarrow{}  y_{t+1}$$
Data flow is depicted in Fig.~\ref{fig_dataflow_03}.
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.35]{images/03_ESQ_Model_SetUp.png}
 \caption{Data flow for Model Setup}
 \label{fig_dataflow_03}
 \end{figure}
We consider Random Forests (\texttt{RF})as a baseline model, with 500 trees with depth 5. We only consider $(x_{t}, y_{t})$ to predict $y_{t+1}$ as \texttt{RF} does not naturally accommodate time-series data. We also consider a Recurrent Neural Net (RNN) with stacked LSTM layers. We set the size of the LSTM's hidden state at 20. No hyper parameter tuning was done.
\subsection{Interpretability}
A model is an approximation of reality. An interpretation can be considered as a way to assert that reality either in quantitative terms or in natural language. It is urged to use interpretable models, where possible\cite{Rudin}. There is a surge in providing interpretations to even seemingly opaque deep learning models. A recently introduced \textit{Shapley values}\cite{shapely} based model explanation technique unifies many existing feature attribution methods such as a \texttt{LIME}, \texttt{DeepLift}, and \texttt{TreeInterpreter}. Shapley possesses some nice theoretical properties such as 1) local accuracy 2) missingness and 3) consistency. In addition, thanks to the additivity property,  feature attributions of any individual instance adds up to the predicted value. In our context, we can break up the test score into AQ, BQ, TQ, EQ components, and provide it as part of engaging feedback. We use \texttt{TreeExplainer} optimized for tree models such as a RFs, and \texttt{DeepExplainer} for RNNs, available in \cite{shapely}.

\subsection{Uncertainty Quantification}
Generally speaking, Machine Learning (ML) algorithms are concerned with predictions. Quantifying uncertainty, either in terms of confidence intervals (CIs) and/or prediction intervals (PIs) is usually an afterthought. The problem is exasperated in the Deep Learning space\cite{uq_nguyen}. Such information is particularly useful when deploying ML models in production settings. For example, the product owner may decide not to use the prediction when it is very vague. Generating CIs for models that use third-party tools are very hard. On the other hand, generating PIs is somewhat straightforward, even though not widely known. It is only recently that Quantile Regression (QR)\cite{Koenker2005} has gained popularity among ML practitioners\cite{takeuchi2006nonparametric,qr_rodrigues}. We use \texttt{sk-garden} to fit Quantile Random Forest, and implement Check Loss, defined below, to fit quantiles with RNNs. The Check Loss  is given as:
    $$ \rho_{\tau}(e) = (\tau - I(e<0))e $$
The non-parametric PIs can be obtained by reporting $(\tau, 1-\tau)^{th}$ quantiles at every observed set of predictors. Not only are PIs easier to implement with third party tools, they are also relatively easier to explain than CIs.

\subsection{From predictions to actions}
A nudge can be defined as an action taken by an intelligent agent to positively impact LOs. Damgaard et al. \cite{nudges_damgaard} studied the effectiveness of nudges in the education domain, taking examples of nudges tried in different countries to help students study better. Some of their suggested nudges include displaying peer performance, use of relative grading rather than absolute, re-framing incentives as loss, and showing learning videos on the importance of grit and hard work. A schematic of the \texttt{nudges} model is shown in Fig.~\ref{fig_nudges_schema}.
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.5]{images/04_esq_nudges2.png}
 \caption{A nudge model}
 \label{fig_nudges_schema}
 \end{figure}

Providing feedback, a kind of nudge, is also found to be useful \cite{nudges_david}. Bramucci et al \cite{nudges_sherpa} suggested the use of a recommendation engine to help students pick courses that would maximize their success. We can see nudges in the much broader context of Reinforcement Learning (RL). Shayan et al. \cite{ril_shayan} reviewed RL as a generic paradigm for planting nudges into the overall learning framework. In this work, we develop \texttt{ESQ} that can act as an Oracle, which for a given student's state, returns the reward function. This allows us to simulate how students learn, and design optimal policies to provide nudges. That is, we study the inverse problem: \textit{What state should a learner move to, from a given state in order to obtain a particular reward?}. It is akin to back propagating the output gradient (change in reward) to find the input gradient (change in input state). If indeed the causal factors are also encoded in the state, then by solving this inverse problem, we can also suggest which nudge must be given to the student, for the desirable change in the reward (improvement in the test score). We pose this as an optimization problem. Recall that, \texttt{SBT} learns the function $f(.)$, parameterized by $\theta$:  $y \leftarrow f(x;\theta) $. We treat $\theta$ as the unknown while learning the function(training phase), and  while solving for nudges, we treat $\delta x$ as the unknown in the following minimization problem:
$$\min_{\delta x} \ell (y + \delta y, f(x+\delta x; \theta))$$
Here, $\delta y$ is the desired improvement in the score from $y$, $\delta x$ is the anticipated change needed in the student's state from $x$, $\ell(.)$ is a suitable loss function that handles the domain constraints. It is out of scope of this paper to discuss the details, but we demonstrate its utility in the next section.

\section{Results and Discussion}
We fit a \texttt{RandomForest} model on the entire dataset. However, we observed that variance was changing with observed score in the residuals, particularly in the low scoring buckets (students score less than 25\%), and the Median Absolute Error (MedAE) was above 10\% which was not acceptable for real world deployment. To address this issue, We segment the users into four buckets based on test scores seen on their previous three test attempts, and fit one \texttt{RandomForest} for each score bucket. We did not use any bucketing strategy to fit RNNs. A two dimensional histogram of observed vs. predicted responses is shown in Fig.~\ref{fig_residuals_RF_RNN}. The residuals of the RNN models have some segmentation, which we believe is due to user encoding we did to reflect the score buckets.
\begin{figure}[htb]
 \centering

  \subfloat[RF]{{\includegraphics[scale=0.2]{images/03_RandomForest_ESQ_wo_quantile_density_plot_observed_vs_predicted_score.png} }}%
  \subfloat[RNN]{{\includegraphics[scale=0.2]{images/03_RNN_density_plot_observed_vs_predicted_score.png} }}%
 \caption{Observed vs Predicted Density}
 \label{fig_residuals_RF_RNN}
 \end{figure}
 Between the two models, the RNN had the smaller Median Absolute Error (MedAE) at 4.53\%, and the Pearson correlation $\rho$ between the predicted and observed density was 0.9331 (see Table ~\ref{tab_model_summary_overall}). With bucketwise RFs, the MedAE is 6.15\% and $\rho$ is 0.8235. To see the effect of adding Concept Mastery vectors, we ran an experiment on one third of the dataset.
\begin{table*}[ht]
\begin{center}
\caption{Performance summary of RF and RNN models. n: number of samples; p: number of features; r: train/test split ratio. ()* represents Model with Concept Mastery vectors}

\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|}
    \hline
    \multirow {2} {*} {Model} & \multirow {2} {*} {p} &  \multirow {2} {*} {n} & \multirow {2} {*} {r} & \multicolumn{4}{c|}{Train} &  \multicolumn{4}{c|}{Test}  \\ \cline{5-12}
     ~ & ~ & ~ & ~ & RMSE & MAE & MedAE &  $\rho$ &  RMSE & MAE & MedAE &  $\rho$\\
    \hline
    RF & 67489	& 54 & 0.8	& 11.21 & 8.13 & 6.16	& 82.35 & 12.50 & 9.12 & 6.73 & 77.15\\
    \hline
    RF* & 20358	& 104 & 0.95	& 11.50 &	8.13	& 5.83 & 80.40 & 12.08 & 8.84 & 6.67 &	77.07\\
    \hline
    RNN	 & 67489 & 54 & 0.8 &  7.1	& 5.51 &	4.53	& 91.31 &	7.43	&5.69	&4.57	&92.05 \\
    \hline
    RNN*	& 20358  & 104 & 0.95 & 8.84	&6.72	&5.37	&88.60	&10.42	&7.54	&5.91	&88.92 \\
    \hline
\end{tabular}
\label{tab_model_summary_overall}
\end{center}
\end{table*}
\begin{comment}
\begin{table}[h]
\caption{Bucketwise Model Performance}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Model &  Score Bucket & RMSE & MAE & MedAE \\
\hline
\hline
 RF &  0-25 & 12.21	& 8.36 &	5.56 \\
\hline
 RF &  25-50 & 11.28 & 8.56 &	6.67 \\
\hline
RF &  55-75 & 13.75 & 10.44	& 8.61 \\
\hline
RF &  75-100 & 25.95	& 18.12 &	10.28 \\
\hline
 RNN & 0-25 & 7.53	& 5.60 &	4.38 \\
\hline
 RNN &  25-50 & 9.55 &	7.77 &	7.04 \\
\hline
RNN  & 55-75 & 10.10	& 7.79 &	6.28 \\
\hline
RNN  & 75-100 & 14.49 &	10.63 &	8.31 \\
\hline
\end{tabular}
\label{tab:model_summary_bucketwise_noCM}
\end{center}
\end{table}
\begin{table}[h]
\caption{Model Performance with Concept Mastery}
\begin{center}
\begin{tabular}{|c|c|c|c|c|}
\hline
Model &   Score Bucket & RMSE & MAE & MedAE \\
\hline
\hline
 RF & 0-25 & 12.68	& 9.28 &	6.94 \\
\hline
 RF &  25-50 & 9.752	& 7.43 &	5.97 \\
\hline
RF & 55-75 & 15.24 &	11.04 &	7.78 \\
\hline
RF & 75-100 & 13.88 &	10.77 &	8.05 \\
\hline
 RNN  & 0-25 & 8.89	& 6.53 &	4.98 \\
\hline
 RNN  & 25-50 & 9.31 &	7.41 &	6.58 \\
\hline
RNN &  55-75 & 10.64 &	7.67 &	5.96 \\
\hline
RNN & 75-100 & 17.32 & 	11.46 &	6.75 \\
\hline
\end{tabular}
\label{tab:model_summary_bucketwise_withCM}
\end{center}
\end{table}
\end{comment}
We obtain 90\% PIs by fitting a Quantile RF model at $\tau=0.05 \text{ and } 0.95$. We train the same RNN model with Check Loss to fit quantiles. For a particular user, PIs over a period of time are shown in Fig.~\ref{fig_PI_RF_RNN}.
\begin{comment}
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.3]{images/04_RandomForest_ESQ_prediction_interval_plot_90_percentile_for_a_user.png}
  \caption{Prediction Intervals for a user}
 \label{fig_PI_RF}
 \end{figure}
\end{comment}

\begin{figure}[htb]
 \centering

  \subfloat[RF]{{\includegraphics[scale=0.15]{images/04_RandomForest_ESQ_prediction_interval_plot_90_percentile_for_a_user.png} }}%
  \subfloat[RNN]{{\includegraphics[scale=0.15]{images/04_RNN_PIs.png} }}%
 \caption{90\% Prediction Intervals for a user}
 \label{fig_PI_RF_RNN}
 \end{figure}
 We can use the width of PI as a measure to assert confidence in the predictions -- predictions with wide PIs can be not shown. The Shapley model summary is shown in Fig.~\ref{fig_Shapley_RF_Model}. It turns out that, not surprisingly, \textit{mean accuracy} (coded as \texttt{aq\_16}) is the most discriminating feature. It is interesting to note that, what kind of test a student is taking also influences the test score(\texttt{bq}\textunderscore\texttt{17}). A student with a particular ability can perform differently based on the difficulty of the test - a viewpoint fundamental in Item Response Theory modeling. Overall, AQ features' contribution is 60.98\% to the mean predictions. Feature attributions at an individual level can also be computed based on Shapley values.
\begin{figure}[htb]
 \centering
 \begin{turn}{270}
  \includegraphics[scale=0.3]{images/03_Shapley_RandomForest_ESQ_25-50_wo_quantile_feature_correlation_wrt_esq_predicted_score_plot.png}
  \end{turn}
 \caption{Shapley Model Summary}
 \label{fig_Shapley_RF_Model}
 \end{figure}
For a particular user, we can show the contribution of each of the features to the score. A force-plot of individualized \texttt{Shapley values} for a user is shown in Fig.~\ref{fig_indy_shapley}. For example, we can see that,  \texttt{aq}\textunderscore\texttt{16} contributed 1.57 points to the predicted score, whereas \texttt{bq}\textunderscore\texttt{20}(\textit{number of test sessions}) reduced predicted score by 0.73 points, possibly because, the student should have practiced this test on par with others. These individualized feature attributions can be used to break up the score into explainable and actionable components.
\begin{figure*}[ht]
 \centering
  \includegraphics[scale=0.4]{images/04_UserIndividualShapleyScoreInterpretation.png}
 \caption{Individualized Shapley values for a user}
 \label{fig_indy_shapley}
\end{figure*}
The nudges model takes this idea further and  systematically perturbs the inputs which can potentially induce a desired a change in the score (score is a proxy for LO in this case). Using the \texttt{nudges} model introduced earlier, we vary one feature at time - much like how coordinate descent works. In fact, the analogy is precisely that. We are optimizing the learning process to improve the LOs. In Fig.~\ref{fig_nudges_eg}, we show affected features which improve the current score by more than 10 points.
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.2]{images/04_esq_nudges.png}
 \caption{Nudges in Action}
 \label{fig_nudges_eg}
 \end{figure}

Our approach draws parallels to recent work on providing explanations to black-box deep learning models via counterfactual examples\cite{dice}. While the \texttt{nudges} model can figure out what the state should be, it does not influence the state directly. On the \texttt{Embibe} platform this happens via a messaging capability - a user can be shown actionable feedback like \textit{You seem to be making careless mistakes. Revise your calculations before submissions}. Once a causal link is established between an action and the affecting feature, \texttt{nudges} model can either influence the learner actions directly or it can be used in a gaming mode, where the learner can play with different simulated actions and pick a suitable action. To see the effect of the feedback, we mined historical data and sampled 1,116 users who have taken at least 10 valid tests (defined earlier) on our platform. In Fig.~\ref{fig_test_on_test_change}, we show how undesirable behavior parameters, such as \textit{wasted attempts, unused time, overtime incorrects}, trend down as marks scored increases test-on-test. The direction of the results is encouraging but we exercise caution in interpreting them. We have not controlled for user attributes, and can not attribute the behavioral change only to the feedback mechanism. However, the analysis suggests that we can persist with the hypothesis that providing useful feedback helps in impacting the intended behavioural aspects. A controlled A/B experiment will test that hypothesis.
\begin{figure}[htb]
 \centering
  \includegraphics[scale=0.5]{images/combo.png}
 \caption{Test-on-test progress of marks scored and change in behavior parameters. Reading clockwise from top are plots for \textit{marks scored, wasted attempt ratio, unused time ratio and overtime incorrects ratio}}
 \label{fig_test_on_test_change}
 \end{figure}
\begin{comment}
\begin{table}[h]
\caption{Feedback type: X=1(0): Actionable(Informational). Effectiveness: Y=1(0): Effective(Not effective).}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
X/Y  & Y=1 & Y=0 \\
\hline
 X=1 & 11,758	& 5,671 \\
\hline
 X=0 & 8,833	& 12,100 \\
\hline
\end{tabular}
\label{tab:rankup_analysis}
\end{center}
\end{table}
\end{comment}
\section{Conclusions and Future Work}
In this work, we proposed a framework for predicting Learning Outcomes, defined in terms of test scores. Various features were considered that summarize learner interactions on the platform. We used individualized \texttt{BKT} parameters to summarize temporal attempt level data. Deep Factorization model, with dimensionality reduction, was used to model a user's mastery in 1242 concepts. We are testing our approach on larger datasets. We considered RandomForest model as a baseline to predict test scores. We also considered RNNs as they can naturally handle the sequential nature of the attempt data. While the RNN model gave best results, the residuals did not look smooth. Using \texttt{TreeExplainer}, we obtained individualized Shapley values. Based on this data, we observed that, AQ features contribute about 60.98\% of the scores. In order to fit Shapley values to an RNN, based on \texttt{DeepExplainer}, we had to drop the time-distributed layer, which reduced the RNN performance. We are working on architecture search and feature encodings to improve the performance and model diagnostics, while retaining interpretability. Individualized feature attributions can be presented to provide a breakup of the test score in terms of the features. If the features themselves are actionable, it is possible to directly avail this information to drive actions. We modelled this idea formally by posing nudge recommendation as an optimization problem. Since we are treating the \texttt{ESQ} as an Oracle, many simulations can be performed to identify the optimal policy. Our formalism can be easily incorporated into a Reinforcement Learning paradigm. We are working on the framework to provide optimal nudges and validate this hypothesis.

\section*{Acknowledgements}
The authors would like to thank Achint Thomas for very helpful discussions and Anwar Sheikh for pedagoical suggestions.
\bibliographystyle{abbrv}