  Elastic full-waveform inversion (EFWI) is a process used to estimate
subsurface properties by fitting seismic data while satisfying wave propagation
physics. The problem is formulated as a least-squares data fitting minimization
problem with two sets of constraints: Partial-differential equation (PDE)
constraints governing elastic wave propagation and physical model constraints
implementing prior information. The alternating direction method of multipliers
is used to solve the problem, resulting in an iterative algorithm with
well-conditioned subproblems. Although wavefield reconstruction is the most
challenging part of the iteration, sparse linear algebra techniques can be used
for moderate-sized problems and frequency domain formulations. The Hessian
matrix is blocky with diagonal blocks, making model updates fast. Gradient
ascent is used to update Lagrange multipliers by summing PDE violations.
Various numerical examples are used to investigate algorithmic components,
including model parameterizations, physical model constraints, the role of the
Hessian matrix in suppressing interparameter cross-talk, computational
efficiency with the source sketching method, and the effect of noise and
near-surface effects.
