Quantifying uncertainty in inverse scattering problems set in layered environments

TL;DR

The paper tackles inverse scattering in layered media with surface-wave data, aiming to quantify uncertainty in inclusions defined by a few parameters. It develops three complementary approaches: an automatic adaptive Levenberg–Marquardt–Fletcher scheme with adaptive finite elements and algorithmic differentiation, a fixed-mesh optimization method for efficiency, and affine-invariant MCMC to reveal multimodal posterior structures. Uncertainty is assessed via the Laplace approximation and through detailed MCMC studies, showing how prior information, noise, and layering shape the high-probability configurations, with frequency diversity reducing multimodality. The framework demonstrates the trade-offs between solver quality, adaptivity, and computational cost, and emphasizes the practical benefit of using multiple frequencies to improve convexification and uncertainty reduction in subsurface imaging.

Abstract

The attempt to solve inverse scattering problems often leads to optimization and sampling problems that require handling moderate to large amounts of partial differential equations acting as constraints. We focus here on determining inclusions in a layered medium from the measurement of wave fields on the surface, while quantifying uncertainty and addressing the effect of wave solver quality. Inclusions are characterized by a few parameters describing their material properties and shapes. We devise algorithms to estimate the most likely configurations by optimizing cost functionals with Bayesian regularizations and wave constraints. In particular, we design an automatic Levenberg-Marquardt-Fletcher type scheme based on the use of algorithmic differentiation and adaptive finite element meshes for time dependent wave equation constraints with changing inclusions. In synthetic tests with a single frequency, this scheme converges in few iterations for increasing noise levels. To attain a global view of other possible high probability configurations and asymmetry effects we resort to parallelizable affine invariant Markov Chain Monte Carlo methods, at the cost of solving a few million wave problems. This forces the use of prefixed meshes. While the optimal configurations remain similar, we encounter additional high probability inclusions influenced by the prior information, the noise level and the layered structure, effect that can be reduced by considering more frequencies. We analyze the effect on the calculations of working with adaptive and fixed meshes, under a simple choice of non-reflecting boundary conditions in truncated layered domains for which we establish wellposedness and convergence results.

Figures (15)

• Figure 1: Schematic representation of an imaging set-up. The emitters (red) generate waves which interact with the medium. The reflected waves are recorded at the receivers (blue). The velocities are in units of $10^3$m/s and the densities in units of kg/m$^3$. Units for $x$ and $y$ are km. Parameter values are typical of sandstone, shale, limestone, and salt for the inclusion.
• Figure 2: Types of meshes considered: (a) adapted to all the subdomains, (b) adapted to the stratified structure but not to the changing inclusion, (c) uniform.
• Figure 3: (a) Values recorded at the receivers with the adaptive mesh in Figure \ref{['fig2']}(a) for the parameter values and geometry in Figure \ref{['fig1']} with $\delta x = 0.04$, $\delta t = 1\text{e-}3$. Profiles are recorded at intervals of $0.01$. Errors when comparing the data in (a) with the observed values obtained with (b) the stratified mesh in Figure \ref{['fig2']}(b) and (c) the uniform mesh in Figure \ref{['fig2']}(c) and when comparing the data obtained with the latter two meshes between themselves (d).
• Figure 4: True inclusion (black curve) compared to the shapes obtained at successive iterations (red) for $10\%$ noise during adaptive constrained optimization.
• Figure 5: Evolution of (a) the cost, (b) $\rho$ and (c) $v_{\rm p}$ inside the inclusion along the iterations for the simulation in Figure \ref{['fig5']}.