On the adjoint state method for the gradient computation in full waveform inversion: a complete mathematical derivation for the (visco-)elastodynamics approximation

SUMMARY High-resolution seismic imaging at all scales using full waveform inversion is now routinely used in the industry and in the academy. One key element for the success of this approach is a numerical method, named adjoint state method, originally designed for optimization problems constrained by partial differential equations, a category to which full waveform inversion belongs. This method provides an efficient way to compute the gradient of the full waveform inversion misfit function, which is the most computationally demanding task in the implementation of full waveform inversion. While well known, the complete and rigorous mathematical derivation of the adjoint state method for full waveform inversion remains missing in the scientific bibliography. The aim of this study is to remedy this lack. The derivation is performed in general settings, that is in the elastodynamics approximation, with and without considering viscosity. Through the calculus, the mechanism of the adjoint state strategy makes clear the connection between the incident and adjoint fields, especially regarding their initial and boundary conditions. The impact of introducing the viscosity is carefully analysed. The resulting gradient formulas are analysed and shown to be consistent with already published ones. The generic approach which is adopted also makes it possible to derive misfit function gradients with respect to other quantities than the subsurface mechanical parameters, for instance with respect to the initial or the boundary conditions, which could be of interest for specific applications where the reconstructed parameters are not only volumetric mechanical parameters but boundary parameters or initial field values.