Preconditioned non-linear conjugate gradient method for frequency domain full-waveform seismic inversion

We present preconditioned non-linear conjugate gradient algorithms as alternatives to the Gauss-Newton method for frequency domain full-waveform seismic inversion. We designed two preconditioning operators. For the first preconditioner, we introduce the inverse of an approximate sparse Hessian matrix. The approximate Hessian matrix, which is highly sparse, is constructed by judiciously truncating the Gauss-Newton Hessian matrix based on examining the auto-correlation and cross-correlation of the Jacobian matrix. As the second preconditioner, we employ the approximation of the inverse of the Gauss-Newton Hessian matrix. This preconditioner is constructed by terminating the iteration process of the conjugate gradient least-squares method, which is used for inverting the Hessian matrix before it converges. In our preconditioned non-linear conjugate gradient algorithms, the step-length along the search direction, which is a crucial factor for the convergence, is carefully chosen to maximize the reduction of the cost function after each iteration. The numerical simulation results show that by including a very limited number of non-zero elements in the approximate Hessian, the first preconditioned non-linear conjugate gradient algorithm is able to yield comparable inversion results to the Gauss-Newton method while maintaining the efficiency of the un-preconditioned non-linear conjugate gradient method. The only extra cost is the computation of the inverse of the approximate sparse Hessian matrix, which is less expensive than the computation of a forward simulation of one source at one frequency of operation. The second preconditioned non-linear conjugate gradient algorithm also significantly saves the computational expense in comparison with the Gauss-Newton method while maintaining the Gauss-Newton reconstruction quality. However, this second preconditioned non-linear conjugate gradient algorithm is more expensive than the first one.