Acoustic waveform inversion with the Lippmann-Schwinger equation constraint Rahul Sarkar and Biondo Biondi, Stanford University SUMMARY We develop the theory for performing acoustic waveform in- version in the frequency domain in both 2D and 3D, with the Lippmann-Schwinger equation as the constraint. The Lippmann-Schwinger equation that we consider has the spe- cial structure that the background velocity depends only on depth, in which case an efficient method exists to compute the forward and adjoint actions of the integral kernel. We treat the inversion as a joint optimization problem, where both the model to be inverted and the wavefields for each source and frequency are simultaneously treated as optimization vari- ables. Here we explore the penalty method formulation of the problem, and a two-step alternating minimization strategy to solve it is presented, where each step involves solving a linear least squares problem. We point out the similarities and differ- ences between the computational structure of this problem and the closely related wavefield reconstruction inversion problem, where the Helmholtz equation is used as the constraint instead of the Lippmann-Schwinger equation. Finally numerical ex- amples in 2D illustrating the inversion method is presented. INTRODUCTION Full waveform inversion in the frequency domain was pio- neered by Pratt (1999), and since then a lot of subsequent work has focused on tackling the non-convex nature of the optimiza- tion problem and improving its convergence, especially when the starting model is very far away from the true model. One well-known approach for tackling non-convexity is wavefield reconstruction inversion (WRI) (van Leeuwen and Herrmann, 2013, 2015), where one relaxes the optimization problem by extending the search space, so that the variables of the opti- mization problem involve both the model parameters and the wavefields for each source and frequency. The basic method was subsequently improved to be more computationally effi- cient in Aghamiry et al. (2019b), and strategies to incorpo- rate bound constraints and other regularization terms was car- ried out in Aghamiry et al. (2019a). However in the acous- tic case, in each of these works, the partial differential equa- tion (PDE) constraining the wavefields and the model param- eters is the Helmholtz equation. In this paper, we develop a method where the inversion is instead constrained by the Lippmann-Schwinger equation. In our setting, as explained in the next section, we seek to recover the perturbation in slow- ness squared with respect to some fixed background, as com- pared to recovering slowness squared as in the WRI case. In Sarkar and Biondi (2021), we developed a numerical scheme to solve the acoustic Lippmann-Schwinger equation for a lin- early varying background velocity, extending the truncated ker- nel method of Vico et al. (2016) for the constant velocity back- ground case. The method has been further extended to cover the case where the background velocity is a smooth function of a single coordinate (e.g. depth), which will be publiahed separately. We describe the salient features of the method very briefly in the next section. The goal of this paper is to leverage this computational technique to solve the waveform inversion problem with the Lippmann-Schwinger equation constraint. The Lippmann-Schwinger equation has been previously stud- ied in great detail in seismic imaging because of its connection to the forward and inverse scattering problems, for example in applications involving imaging and multiple removal with- out requiring a velocity model (Weglein et al., 2001, 2003; Zhang and Weglein, 2009; Zou et al., 2019). Another direc- tion of study has focused on designing convergent summation schemes for the Born-Neumann series, which is a series sum- mation technique for solving the Lippmann-Schwinger equa- tion (see for example the works of Wu and Zheng (2014); Lesage et al. (2014); Jakobsen and Wu (2015); Yao et al. (2015); Huang et al. (2019); Jakobsen et al. (2020)). Our method of solving the Lippmann-Schwinger equation instead relies on Krylov subspace methods and is thus convergent for arbitrar- ily large perturbations, provided the resulting linear system is invertible after discretization. The rest of the paper is structured as follows. In the next sec- tion, we begin by briefly discussing the specific form of the Lippmann-Schwinger equation that we will need. This is fol- lowed by the formulation of the acoustic case of the waveform inversion problem in extended search space, and an alternating minimization strategy to solve the resulting optimization prob- lem is presented. We then discuss the similarities and differ- ences of the computational structure of the proposed method with that of WRI. In the subsequent section, we present some numerical examples in 2D to illustrate the performance of the method. We finish by making some conclusions. THEORY AND METHOD The Lippmann-Schwinger equation The material here will be expanded in a separate publication; so we briefly discuss only what we need for this paper here. Let Ω =[− 1 2 , 1 2 ] d × [a, b], and consider the Helmholtz equation in R d+1 = {(x, z) : x ∈ R d , z ∈ R}, for d = 1, 2 (here x, z denotes horizontal and vertical coordinates respectively)  Δ + ω 2 c(x, z) 2  u = f , (1) where Δ = ∂ 2 ∂ z 2 + ∑ d i=1 ∂ 2 ∂ x 2 i , ω > 0, c(x, z)= v(z)+ w(x, z) > 0, and w ∈ C ∞ 0 (Ω) 1 . Furthermore assume that 0 < c 1 ≤ c ≤ c 2 for some constants c 1 and c 2 , and v ∈ C ∞ (R) with the prop- erty that v(z)= v 1 for all z ≤ a and v(z)= v 2 for all z ≥ b for constants v 1 and v 2 . If one imposes the Sommerfeld radia- tion boundary condition (Colton and Kress, 1998), and if f is a compactly supported distribution, then the solution u is unique. When f = δ (x − x 0 , z − z 0 ) and w = 0, the solution is called the “Green’s function”, which is translation invariant in the hori- zontal coordinates, and we denote it by G ω (x − x 0 , z, z 0 ). Now 1 The notation C ∞ 0 means smooth functions with compact support. 10.1190/image2022-3752039.1 Page 877 Second International Meeting for Applied Geoscience & Energy © 2022 Society of Exploration Geophysicists and the American Association of Petroleum Geologists Downloaded 08/17/22 to 100.24.209.23. Redistribution subject to SEG license or copyright; see Terms of Use at http://library.seg.org/page/policies/terms DOI:10.1190/image2022-3752039.1