Compound Regularization of Full-Waveform Inversion for Imaging Piecewise Media

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.

Efficient Wavefield Inversion With Outer Iterations and Total Variation Constraint

“…Total variation (TV) regularization was proposed to preserve sharp edges for salt body inversion [24]. To further improve the performance of regularization, compound regularization was used to combine the advantages of Tikhonov regularization and TV regularization [25].…”

Robust wavefield inversion via phase retrieval

“…By doing this, we come up with a least-squares problem with a closed-form expression to update m1 and m2, a proximity subproblem to update the auxiliary variables and a gradient ascent step to update dual variables. The reader is referred to Aghamiry et al (2019b) for the detailed derivation of the algorithm.…”

“…ADMM also provides a suitable framework to implement bound constraints and 1-based nonsmooth regularizations (such as total variation (TV) regularization) in IR-WRI (Aghamiry et al 2018c(Aghamiry et al , 2019c. IR-WRI with TV regularization was further improved by using more versatile TT regularization combining Tikhonov and Total-variation regularizers through infimal convolution (Aghamiry et al 2018a(Aghamiry et al , 2019b. TT regularization explicitly decomposes the model into two components of different statistical properties (a smooth one and a blocky one) such that each component can be recovered by a suitable regularization (Tikhonov and TV) (Gholami & Hosseini 2013).…”

“…TT+λ L(u k )m−|y(u k )|e j∠L(u k )m k 2 2 , (4.16) where C = {x ∈ R n×1 | m l ≤ x ≤ mu} is the set of all feasible models bounded by the lower bound m l and the upper bound mu and m TT is the TT regularization functional as defined byAghamiry et al (2018aAghamiry et al ( , 2019b m TT = minm=m 1 +m 2 m1 TV + α m2 Tikh,(4.17)with m TV = |∇xm| 2 + |∇zm| 2 m Tikh = |∇xxm| 2 + 2|∇xzm| 2 + |∇zzm| 2 (4.18)…”

An all‐at‐once approach to full waveform inversion in the viscoelastic regime

“…In the subsequent publication, 9 WRI has been generalized to other nondynamic (frequency domain) inverse problems. We refer to Aghamiry et al 10 for a recent study of several extensions of WRI (still in the frequency range). All these contributions are done on a discrete, matrix-based level in an optimization, data fitting framework which is in contrast to our infinite-dimensional setting.…”