  We are concerned with high-fidelity subsurface imaging of the soil, which
commonly arises in geotechnical site characterization and geophysical
explorations. Specifically, we attempt to image the spatial distribution of the
Lame parameters in semi-infinite, three-dimensional, arbitrarily heterogeneous
formations, using surficial measurements of the soil's response to probing
elastic waves. We use the complete waveform response of the medium to derive
the inverse problem, by using a partial-differential-equation (PDE)-constrained
optimization approach, directly in the time-domain, to minimize the misfit
between the observed response of the medium at select measurement locations,
and a computed response corresponding to a trial distribution of the Lame
parameters. We discuss strategies that lend algorithmic robustness to our
proposed inversion scheme. To limit the computational domain to the size of
interest, we employ perfectly-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 Lame parameters for both smooth and sharp profiles.
