A skew-adjoint VTI elastic wave equation for full-waveform inversion

Abstract Full-waveform inversion (FWI) based on the full wave equation utilizes both travel-time and amplitude information from seismic records to recover high-resolution subsurface models. As seismic records are nonlinearly related to subsurface parameters, FWI is inherently a nonlinear inverse problem. Although global inversion methods can effectively handle such nonlinearity, they are computationally expensive. Therefore, local gradient methods are widely adopted. To avoid explicitly computing the Jacobian matrix, these approaches employ the adjoint-state method, which computes the gradient by cross-correlating the forward wavefield from the state equation with the adjoint wavefield from the adjoint equation. However, when FWI is based on the first-order VTI elastic wave equation, the adjoint-state equation is inconsistent with the state equation. This introduces two challenges: (1) The programming complexity increases because the forward and adjoint wavefields require separate simulation codes; (2) The adjoint-state equation cannot employ the same free-surface boundary conditions as the state equation. In FWI, the inability of the adjoint-state equation to correctly impose free-surface boundary conditions can adversely affect the final inversion results. To address these issues, we propose a skew-adjoint form of the first-order VTI elastic wave equation. In the skew-adjoint VTI elastic wave equation, the operator and its adjoint operator have the same form, differing only by a negative sign. Because it is mathematically equivalent to the first-order VTI elastic wave equation, the proposed skew-adjoint formulation allows both the state and adjoint-state equations to use consistent free-surface boundary conditions. Finally, we demonstrate the effectiveness of our method using two synthetic datasets.