  The nonlinear and ill-posed nature of full waveform inversion (FWI) requires
us to use sophisticated regularization techniques to solve it. In most
applications, the model parameters may be described by physical properties
(e.g., wave speeds, density, attenuation, anisotropic parameters) which are
piecewise functions of space. Compound regularizations are thus necessary to
reconstruct properly such parameters by FWI. We consider different
implementations of compound regularizations in the wavefield reconstruction
inversion (WRI) method, a formulation of FWI that extends its search space and
prevent the so-called cycle skipping pathology. Our hybrid regularizations rely
on Tikhonov and total variation (TV) functionals, from which we build two
classes of hybrid regularizers: the first class is simply obtained by a convex
combination (CC) of the two functionals, while the second relies on their
infimal convolution (IC). In the former class, the model of parameters is
required to simultaneously satisfy different priors, while in the latter the
model is broken into its basic components, each satisfying a distinct prior
(e.g. smooth, piecewise constant, piecewise linear). We implement these types
of compound regularizations in the WRI optimization problem using the
alternating direction method of multipliers (ADMM). Then, we assess our
regularized WRI in the framework of seismic imaging applications. Using a wide
range of subsurface models, we conclude that compound regularizer based on IC
leads to the lowest error in the parameter reconstruction compared to that
obtained with the CC counterpart and the Tikhonov and TV regularizers when used
independently.
