We are concerned with high-
fidelity subsur
face imaging o
f the soil, which commonly arises in geotechnical site characterization and geophysical explorations. Speci
fically, we attempt to image the spatial distribution o
f the Lamé parameters in semi-in
finite, three-dimensional, arbitrarily heterogeneous
formations, using sur
ficial measurements o
f the soil’s response to probing elastic waves. We use the complete wave
form response o
f the medium to drive the inverse problem, by using a partial-di
fferential-equation (PDE)-constrained optimization approach, directly in the time-domain, to minimize the mis
fit between the observed response o
f the medium at select measurement locations, and a computed response corresponding to a trial distribution o
f the Lamé parameters. We discuss strategies that lend algorithmic robustness to our proposed inversion scheme. To limit the computational domain to the size o
f interest, we employ per
fectly-matched-layers (PMLs).
In order to resolve the forward problem, we use a recently developed hybrid finite element approach, where a displacement–stress formulation for the PML is coupled to a standard displacement-only formulation for the interior domain, thus leading to a computationally cost-efficient scheme. Time-integration is accomplished by using an explicit Runge–Kutta scheme, which is well-suited for large-scale problems on parallel computers.
We verify the accuracy of the material gradients obtained via our proposed scheme, and report numerical results demonstrating successful reconstruction of the two Lamé parameters for both smooth and sharp profiles.