Computation of Hessian Vector Product by Spectral-Element and Adjoint Methods Using Wavefield Storage Scheme

Abstract:

To further improve the accuracy of the model update and model convergence, we show how to compute the Hessian vector product Hδm based upon the spectral-element and adjoint methods using the wavefield storage scheme (WSS). There are four simulations required to construct the Hessian vector product by the spectral-element and adjoint methods when the WSS is applied. The first is a forward simulation computing and saving four fields, that is, s(m₁), s(m₂), a(m₁), and a(m₂), where s denotes the displacement field and a the acceleration field. The two models are m₁ and m₂, where m₂=m₁+vδm. The second simulation includes two adjoint simulations. The first adjoint simulation is designed to compute and store the adjoint field (s_s)^†(m₁) due to the perturbation of the adjoint source. The second adjoint simulation is to compute and store the adjoint filed (s)^†(m₂) due to the perturbation of the model. The last simulation is a simultaneous adjoint simulation and the computation of the Hessian vector product, where the adjoint simulation is to compute the adjoint field s^†(m₁) on the fly for each time step, and the Hessian vector product is constructed simultaneously with the adjoint simulation. In the last simulation, the Fréchet kernels and the Hessian vector product are simultaneously constructed by these wavefields for each time step or subsampled time steps. We presented the computation workflow and the implementation strategies using 2D synthetic models and the Specfem2D package as examples. By using the present-day computational facility, the WSS for computing the Hessian vector product is suitable for 2D and small-scale 3D simulations, such as applications in local exploration seismology, non-destructive testing, and medical imaging. For a large-scale or the global-scale simulation, it could be computationally practical when storing the wavefields with advanced compression or interpolation techniques.