Investigating Hessian-based inversion velocity analysis

Inversion velocity analysis (IVA) is an image domain method built upon the spatial scale separation of the model. Accordingly, the IVA method is performed with an iterative process composed of two minimization steps consisting of migration (inner loop) and tomography (outer loop), respectively, with each step accounting for its Hessian or not. The migration part provides the common image gathers (CIGs) with extension in the horizontal subsurface offset. Then, the differential semblance optimization (DSO) misfit measures the focusing of the events in the CIGs which indicates the quality of the velocity model. Commonly, the velocity updates are obtained from the DSO gradient. IVA is a modified version where the approximate inverse replaces the adjoint of the inner loop process: in that case, the migration Hessian is approximately diagonal in the high-frequency regime. In this work, we report the implementation of the tomographic Hessian (i.e., the second derivative of the DSO misfit with respect to the background model) for the estimation of the background velocity model. We apply the second-order adjoint-state method to obtain the application of the tomographic Hessian on a vector. Then, we use the truncated-Newton method to obtain the update directions by computing approximately the application of the inverse of the tomographic Hessian on the descent direction. We also make a theoretical comparison between the tomography in the IVA and full-waveform inversion contexts. Two numerical examples are used to compare, in terms of geophysical results and computational costs, the truncated-Newton method with different gradient-based optimization methods applied to IVA. A small model allows us to evaluate the eigenvalues of the tomographic Hessian which explains the large damping needed in the truncated-Newton case.