SUMMARYFull waveform inversion (FWI) is a nonlinear waveform matching procedure, which suffers from cycle skipping when the initial model is not kinematically accurate enough. To mitigate cycle skipping, wavefield reconstruction inversion (WRI) extends the inversion search space by computing wavefields with a relaxation of the wave equation in order to fit the data from the first iteration. Then, the subsurface parameters are updated by minimizing the source residuals the relaxation generated. Capitalizing on the wave-equation bilinearity, performing wavefield reconstruction and parameter estimation in alternating mode decomposes WRI into two linear subproblems, which can be solved efficiently with the alternating-direction method of multiplier (ADMM), leading to the so-called iteratively refined WRI (IR–WRI). Moreover, ADMM provides a suitable framework to implement bound constraints and different types of regularizations and their mixture in IR–WRI. Here, IR–WRI is extended to multiparameter reconstruction for vertical transverse isotropic (VTI) acoustic media. To achieve this goal, we first propose different forms of bilinear VTI acoustic wave equation. We develop more specifically IR–WRI for the one that relies on a parametrization involving vertical wave speed and Thomsen’s parameters δ and ϵ. With a toy numerical example, we first show that the radiation patterns of the virtual sources generate similar wavenumber filtering and parameter cross-talks in classical FWI and IR–WRI. Bound constraints and TV regularization in IR–WRI fully remove these undesired effects for an idealized piecewise constant target. We show with a more realistic long-offset case study representative of the North Sea that anisotropic IR–WRI successfully reconstruct the vertical wave speed starting from a laterally homogeneous model and update the long wavelengths of the starting ϵ model, while a smooth δ model is used as a passive background model. VTI acoustic IR–WRI can be alternatively performed with subsurface parametrizations involving stiffness or compliance coefficients or normal moveout velocities and η parameter (or horizontal velocity).
Reconstruction of the wave speed from transmission eigenvalues for the spherically symmetric variable-speed wave equation
