Linearized seismic waveform inversion using a multiple re-weighted least-squares method with QR preconditioning

Abstract

Linearized inversion methods such as Gauss-Newton and multiple re-weighted leastsquares are iterative processes in which an update in the current model is computed as a function of data misfit and the gradient of data with respect to model parameters. The main advantage of those methods is their ability to refine the model parameters although they have a high computational cost for seismic inversion. In the Gauss-Newton method a system of equations, corresponding to the sensitivity matrix, is solved in the least-squares sense at each iteration, while in the multiple re-weighted least-squares method many systems are solved using the same sensitivity matrix. The sensitivity matrix arising from these methods is usually not sparse, thus limiting the use of standard preconditioners in the solution of the linearized systems. For reduction of the computational cost of the linearized inversion methods, we propose the use of preconditioners based on a partial orthogonalization of the columns of the sensitivity matrix. The new approach collapses a band of co-diagonals of the normal equations matrix into the main diagonal, being equivalent to computing the least-squares solution starting from a partial solution of the linear system. The preconditioning is driven by a bandwidth L which can be interpreted as the distance for which the correlation between model parameters is relevant. To illustrate the benefit of the proposed approach to the reduction of the computational cost of the inversion we apply the multiple re-weighted least-squares method to the 2D acoustic seismic waveform inversion problem.We verify the reduction in the number of iterations in the conjugate’gradient algorithm as the bandwidth of the preconditioners increases. This effect reduces the total computational cost of inversion as well.