Non‐monotone spectral projected gradient method applied to full waveform inversion

Abstract

A B S T R A C TThe seismic inversion problem is a highly non-linear problem that can be reduced to the minimization of the least-squares criterion between the observed and the modelled data. It has been solved using different classical optimization strategies that require a monotone descent of the objective function. We propose solving the full-waveform inversion problem using the non-monotone spectral projected gradient method: a low-cost and low-storage optimization technique that maintains the velocity values in a feasible convex region by frequently projecting them on this convex set. The new methodology uses the gradient direction with a particular spectral step length that allows the objective function to increase at some iterations, guarantees convergence to a stationary point starting from any initial iterate, and greatly speeds up the convergence of gradient methods. We combine the new optimization scheme as a solver of the full-waveform inversion with a multiscale approach and apply it to a modified version of the Marmousi data set. The results of this application show that the proposed method performs better than the classical gradient method by reducing the number of function evaluations and the residual values. I N T R O D U C T I O NTraveltime tomography (Dines and Lytle 1979;Bishop et al. 1985) and full-waveform inversion (Tarantola 1984a(Tarantola , 1986 have been used widely to estimate velocity fields. Tomography cannot resolve complex structures because of the highfrequency approximations and the poorly posed formulation of the problem (Snieder 1998). Full-waveform inversion is a highly non-linear problem with numerous local minima in the objective function. Linear (Tarantola 1984b) and non-linear (Gauthier et al. 1986;Tarantola 1987;Mora 1987;Sun and McMechan 1992) approaches have been proposed for solving the problem of minimizing the residual function. The first ap- * E-mails: zeevn@cantv.net, savastao@cantv.net, cores@cesma.usb.ve proach performs well when simple geological structures are involved in the model. In the non-linear approaches, the presence of different local minima could impede iterative optimization techniques reaching the global solution. Moreover, the convergence of these schemes, when starting from any initial model, is not guaranteed.Different approaches have been proposed in an attempt to find the minimum of the full-waveform inversion problem.

Spectral Projected Gradient Methods

“…This method is applicable to convex constrained problems in which the projection on the feasible set is easy to compute. Since its appearance, the method has been intensively used in applications [3,6,10,14,15,19,20,24,26,35,42,50,59,63,64,65,69]. Moreover, it has been the object of several spectral-parameter modifications, alternative nonmonotone strategies have been suggested, convergence and stability properties have been elucidated and it has been combined with other algorithms for different optimization problems.…”

Partial spectral projected gradient method with active-set strategy for linearly constrained optimization

“…It has been intensively used in many different applied problems [9,11,13,22,29,43,44,52,58,60,65]. Several interesting parameter modifications were proposed in the papers [24,23,26,27,33,41,55,64].…”

Estudo comparativo de passos espectrais e buscas lineares não monótonas

“…Desde o seu surgimento, o método SPG tem sido muito utilizado em aplicações de diversasáreas tais como física [4,24,51,59,60], geofísica [6,31,61], química [34], engenharia [23], entre outras [8,11,12,17,18,44,55]. Em [6], o SPG foi adaptado para resolver problemas de tomografia de reflexão sísmica e aplicado tanto a problemas com restrições de caixa quanto problemas sem restrições de um conjunto de testes baseado em dados sintéticos.…”

“…Verificouse nos experimentos que o método SPGé uma técnica robusta na solução desses problemas quandoé feita uma definição adequada da região viável convexa. Em [61], o método SPGé utilizado para resolver o problemas de inversão sísmica, que até então era resolvido utilizandose diferentes estratégias de otimização clássicas que necessitam de decréscimo monótono da função objetivo. Os resultados dos testes apontam que a aplicação com SPG apresentou desempenho melhor do que com o método de gradiente clássico pois reduziu o número de avaliações de função e valores residuais.…”