Commun. Comput. Phys. Vol. 28, No. 1, pp. 228-248 doi: 10.4208/cicp.OA-2018-0087 July 2020 Subspace Methods in Multi-Parameter Seismic Full Waveform Inversion Yu Geng1,∗, Kristopher A. Innanen1 and Wenyong Pan2,3 1 Department of Geosciences, CREWES Project, University of Calgary, Alberta T2N 1N4, Canada. 2 Department of Geoscience, CREWES Project, University of Calgary, AB T2N 1N4, Canada. 3 Los Alamos National Laboratory, Geophysics Group, MS D452, Los Alamos, NM 87545, USA. Received 2 April 2018; Accepted 6 March 2019 Abstract. In full waveform inversion (FWI) high-resolution subsurface model param- eters are sought. FWI is normally treated as a nonlinear least-squares inverse problem, in which the minimum of the corresponding misfit function is found by updating the model parameters. When multiple elastic or acoustic properties are solved for, simple gradient methods tend to confuse parameter classes. This is referred to as parameter cross-talk; it leads to incorrect model solutions, poor convergence and strong depen- dence on the scaling of the different parameter types. Determining step lengths in a subspace domain, rather than directly in terms of gradients of different parameters, is a potentially valuable approach to address this problem. The particular subspace used can be defined over a span of different sets of data or different parameter classes, pro- vided it involves a small number of vectors compared to those contained in the whole model space. In a subspace method, the basis vectors are defined first, and a local min- imum is found in the space spanned by these. We examine the application of the sub- space method within acoustic FWI in determining simultaneously updates for velocity and density. We first discuss the choice of basis vectors to construct the spanned space, from linear updates by distinguishing only the contributions of different parameter classes towards nonlinear updates by adding the contributions of higher-order pertur- bations of each parameter class. The numerical character of FWI solutions generated via subspace methods involving different basis vectors is then analyzed and compared with traditional FWI methods. The subspace methods can provide better reconstruc- tions of the model, especially for the velocity, as well as improved convergence rates, while the computational costs are still comparable with the traditional FWI methods. AMS subject classifications: 35L05, 35R30, 86A22 Key words: Waveform inversion, inverse problem, subspace method. ∗Corresponding author. Email addresses: [email protected] (Y. Geng), [email protected] (K. A. Innanen), [email protected] (W. Pan) http://www.global-sci.com/cicp 228 c 2020 Global-Science Press Y. Geng, K. A. Innanen and W. Pan / Commun. Comput. Phys., 28 (2020), pp. 228-248 229 1 Introduction In full waveform inversion (FWI) [11, 25, 27], subsurface model parameters are deter- mined by minimizing an objective function measuring the difference between predicted data and recorded data, related through forward modelling. The forward modelling in- volves wave propagation physics which can range from scalar acoustic, to acoustic, up to viscoelastic anisotropic approximations, and beyond. Simultaneous inversion for dif- ferent parameter classes (see, e.g., [3, 6, 16, 17, 19–21]), including for instance P-wave and S-wave velocities, density, as well as the various attenuation and anisotropic parameters, etc., are critical for a wide application of FWI in reservoir characterization. Similar to the mono-parameter inversion under the scalar acoustic approximation, in which only P-wave velocity is considered, in multi-parameter FWI a misfit function is set up to de- scribe the distance between the recorded data and the predicted data, and FWI is treated as a nonlinear least squares problem, which can be solved by gradient-based methods or Newton-type methods. Multi-parameter inversion is more complicated than mono- parameter inversion, because the additional parameter classes increase the ill-posedness and the nonlinearity of the inverse problem. Different parameter classes can be more or less coupled, and it may be difficult to distinguish the contribution of each parameter class to changes in the data. Mitigating cross-talk between different parameter classes is a key issue. Studies have shown that the Hessian operator contains some information concerning the coupling between different parameter classes. Different ways of incorpo- rating the inverse of the Hessian operator, especially in multi-parameter inversion, have been proposed to better decouple different parameter classes in the inversion. These in- clude preconditioning the gradient using the pseudo Hessian matrix [24], quasi-Newton method, truncated Newton method [12–15,18] and so on. Hierarchical strategies can be applied to successively invert different parameter classes to mitigate the ill-posedness of FWI [1, 2, 9]. In most cases involving incorporation of Hessian information, significant increases in computational cost ensue. In both gradient-based methods and Newton-type methods (see, e.g., [7,12,13,18,22, 27, 31]), a line search scaling the descent direction tends to be necessary for convergence. One scalar is found for all parameter classes regardless of their contributions to the data. To combat cross-talk, distinguishing between the contributions of each parameter class during updating could be helpful in multi-parameter inversion. Application of subspace methods in large-scale inverse problems was first discussed by [10,23] as an approach to adjusting the update descent directions according to different parameter classes’ contributions. Baumstein [5] showed that using an extended subspace method in multi-parameter inversion can also help to mitigate cross-talk. In subspace FWI, basis vectors are determined first, and the optimization problem is then solved in this spanned space to minimize the quadratic approximation of the misfit function, with only a few coefficients to be determined as compared to the traditional gradient-based or Newton-type methods. Although projection of the full Hessian or Gauss-Newton Hes- sian onto the subspace is needed for each iteration, the calculation is much cheaper com- 230 Y. Geng, K. A. Innanen and W. Pan / Commun. Comput. Phys., 28 (2020), pp. 228-248 pared to Newton-type methods. In this study, we evaluate different basis vectors, constructed from the gradients of two different acoustic parameter classes, and related Hessian-vector products (see, e.g., [12, 18]), for their ability to construct better multi-parameter inversions. Our re- sults are restricted to the acoustic approximation (with varying density) both for forward modelling and inversion; however we expect our results to be by-and-large true for other parameterizations and elastic extensions. 2 Full waveform inversion and subspace methods: review We use the frequency-space domain acoustic wave equation to describe wave motion: ω2 1 u(x,x ,ω)+∇· ∇u(x,x ,ω) = f (ω)δ(x−x ), (2.1) ρ(x)v2(x) s ρ(x) s s s where ρ is the density and v is the velocity. Radiation patterns, or scattering patterns (see, e.g., [3,19,26,28]) of v and ρ are plotted in Fig. 1. Compared to the radiation pattern of v, which is isotropic, the radiation pattern of ρ decreases progressively from small scattering angles, which are related to backward scattering, to large scattering angles, which are related to forward scattering (see, e.g., [16, 27]). This is consistent with the fact that a change of velocity affects the waves at all scattering angles, while a change in density only has significant influence on the amplitudes of the precritical reflections at small scattering angles. This is indicative of the possible difficulties of the reconstruction of these two parameters, especially using only a small range of near-offset traces. 0° 330° 30° 300° 60° 1 0.6 0.8 0.2 0.4 270° 90 ° 240° 120° 210° 150° 180° v Figure 1: Radiation patterns of velocity and density in acoustic FWI. Y. Geng, K. A. Innanen and W. Pan / Commun. Comput. Phys., 28 (2020), pp. 228-248 231 Expressing model parameters of all types in one vector m, the source term as f(xs,ω) and the forward modelling operator, which is the discretized impedance matrix, as F(m,ω), a discretized form of the wave equation can be written in matrix form as F(m,ω)u(m,xs,ω)=f(xs,ω), (2.2) The forward modeling (2.2) describes a nonlinear relationship between the wavefield u(m,xs,ω) and the model parameters m. In FWI inversion is formulated as an optimization problem, in which a model m is sought which minimizes the misfit functional φ(m) 1 2 1 2 φ(m)= ∑∑∑ dobs(xs,xg,ω)−dsyn(m,xs,xg,ω) = ∑∑∑ ∆d(m,xs,xg,ω) , 2 xs xg ω 2 xs xg ω (2.3) where dobs(xs,xg,ω) are the observed data at receiver location xg for each source location xs for one frequency ω, and dsyn(m,xs,xg,ω)= Ru(m,xs,ω) are the synthetic data gener- ated by simulating wave propagation in the current model iterate m via (2.2), and sam- pling the wavefield with an operator R to the receiver locations. ∆d(m,xs,xg,ω) are the data residuals, which are defined as the difference between the observed and synthetic data. Expanding the misfit functional (2.3) up to second order around the vicinity of the model m, 1 φ(m+δm)= φ(m)+hg,δmi+ hHδm,δmi+O kδmk3 , (2.4) 2 where g and H are the gradient and the Hessian operator of the misfit function, respec- tively. Since the misfit functional is generally non-quadratic, Eq. (2.4) represents a local quadratic approximation, in which the model can be iteratively updated: mn+1 =mn +αnδm. (2.5) The perturbation can be determined by a descent direction, which is the negative of the gradient in gradient-based methods, or it can be the solution of a Newton system, respec- tively δm=−g, or (2.6) δm=−H−1g.