Quasi-Newton full-waveform inversion with a projected Hessian matrix

We present a method, in realistic-size full-waveform inversion (FWI), to explicitly construct a projected Hessian matrix and its inverse matrix, which we subsequently used to solve FWI with a quasi-Newton method. Newton's method is practically unfeasible in solving realistic-size FWI problems because of the prohibitive cost (computing time and memory consumption) of calculating the Hessian matrix and the inverse Hessian. Therefore, the Gauss-Newton method and various quasi-Newton methods are proposed to approximate a Hessian matrix. Particularly, current quasi-Newton FWI (QNFWI) commonly uses the limited-memory BFGS (L-BFGS) method, which, however, only implicitly approximates an inverse Hessian. We repose FWI as a sparse optimization problem in a sparse model space, which contains substantially fewer model parameters that are constrained by structures of the model. With respect to fewer parameters in the sparse model, we can avoid the "limited-memory" approximation and are able to explicitly compute and store a projected Hessian matrix that saves the computational time and required memory. We constructed such a projected Hessian matrix by adapting the classic BFGS method to a projected BFGS (P-BFGS) method in the sparse space. Using the projected Hessian matrix and its inverse, we can apply the P-BFGS method to solve FWI with a quasi-Newton method. In QNFWI with P-BFGS because we invert for a sparse model with much fewer parameters, the memory required to compute the projected Hessian is negligible compared to either forward modeling or gradient calculation. QNFWI with P-BFGS converges in fewer iterations than conjugate-gradient based methods and QNFWI with L-BFGS.