Sensitivity analysis of an inverse problem for the wave equation with caustics (Q2922789)

Let \(\Omega\) be a bounded, strictly convex and smooth domain in \(\mathbb R^d\), \(d \geq 2\), with boundary \(\Gamma\). Let \(c(x)\) be a velocity field in \(\Omega\) and let \(T\) be a sufficiently large positive number. Consider the wave equation system NEWLINENEWLINE\[NEWLINE\begin{aligned} \frac{1}{c^2} u_{tt} - \Delta u&=0,\\ u(0,x)&= u_t(0,x)=0, \;x \in \Omega, \\ u(x,t)&=f(x,t), \;(x,t) \in \Gamma \times (0,T).\end{aligned}NEWLINE\]NEWLINENEWLINE For each \(f \in H^1_0 ([0, T] \times \Gamma)\), it is known that there exists a unique solution \(u \in C^1((0,T); L_2(\Omega))\bigcup C([0,T]; H^1(\Omega))\) and furthermore \(\partial u/\partial \nu \in L^2([0,T] \times \Gamma)\), where \(\nu\) is the unit outward normal to the boundary. Define the Dirichlet-to-Neumann map by NEWLINE\[NEWLINE\Lambda_c(f)= \left . \frac{\partial u }{\partial \nu} \right |_{[0,T] \times \Gamma} .NEWLINE\]NEWLINE The paper is concerned with the stability of the inverse problem of recovering the velocity field \(c\) from the DDtN map \(\Lambda_c\). Assuming that two velocity fields are non-trapping and are equal to constant near the boundary, it is shown that the two induced scattering fields must be indentical if their corresponding DDtN maps are sufficiently close. A geodesic X-ray transform operator with matrix-valued weight is introduced by linearizing the operator which associates each velocity field with its induced Hamiltonian flow. A selected set of geodesics is used to recover the singularity of the X-ray transformed function at the point; a local stability estimate is established in this case. Suppose that a background velocity field is fixed and every interior point has the above required set of geodesics (fold-regular). Assuming that another velocity field is sufficiently close to it and satisfies a certain ortogonality condition, it is shown that if two corresponding DDtN maps are sufficiently close, then they must be equal.