# Compound Regularization of Full-waveform Inversion for Imaging Piecewise Media

## TL;DR

This paper introduces compound regularization techniques combining Tikhonov and total variation functionals within wavefield reconstruction inversion to improve seismic imaging of piecewise media, demonstrating superior reconstruction accuracy.

## Contribution

It develops and implements hybrid regularizers using infimal convolution and convex combination in WRI, enhancing model reconstruction for piecewise media in seismic imaging.

## Key findings

- IC-based regularization yields lowest reconstruction error.
- Compound regularization outperforms individual Tikhonov or TV methods.
- WRI with compound regularization improves imaging accuracy.

## Abstract

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.