Anisotropic Waveform Inversion with Well Constraints

The term, "Waveform inversion" (WFI), refers to a collection of techniques that use the information from seismic data to derive high-fidelity earth models for seismic imaging. The attractiveness of WFI lies mainly in its lack of approximations, at least in a theoretical sense, in contrast to other model determination techniques such as semblance or tomography. However, a whole raft of approximations must be made to make the technique viable with today's computing technology and restrictions of seismic acquisition. These are collectively referred to as "waveform inversion strategies" and in this paper we mainly discuss regularization and preconditioning strategies. Because the wavefields need to be accurately modeled to represent the kinematics of all the waves during WFI iterations, the effects of anisotropy often help to improve WFI results. In this paper, forward modeling and its adjoint computation are based on acoustic wave equations in vertical transversely isotropic (VTI) media. We introduce a multi-parameter inversion for P-wave velocity and anisotropy parameters. WFI is a highly nonlinear, ill-posed problem. We introduce additional information and turn the unconstrained optimization problem into a constrained optimization problem in order to reduce the ill-posedness. The geophysics of the problem leads to appropriate constraints, such as restriction of model parameters, or information from well logs. In this paper, we use well logs as constraints and solve the problem using the augmented Lagrangian method (ALM), a mathematical method that replaces a constrained optimization problem by a series of unconstrained problems. The ALM with well constraints aims at preserving velocity characteristics from well logs and providing us with more reliable velocity updates. This paper presents the acoustic anisotropic WFI implementation using ALM with well constraints. It also discusses practical strategies for regularization and preconditioning and their influences on the models that are obtained from WFI. We illustrate these approaches on a 2D synthetic Marmousi example and another application to 3D VSO OBC data from the Green Canyon area of the Gulf of Mexico. From the results, we show that multi-parameter VTI WFI with ALM provides us with more useful and reliable model updates. To further evaluate our WFI results, we also compare offset gathers and RTM images and illustrate their significant improvements using updated models generated from WFI.