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.
Rocks can be anisotropic due to a variety of reasons. When estimating rock velocities from seismic data, failure to introduce anisotropy into earth models could generate distortions in the final images that can have enormous economic impact. To estimate anisotropic earth velocities by tomographic methods, it is necessary to trace rays or to solve the wave equation in models where anisotropy has been properly considered. Thus, in this work we present a 3-D generalized ellipsoidal travel time formulation that allow us to trace rays in an anisotropic medium. We propose to trace rays in anisotropic media by solving a set of nonlinear optimization problems, where the group velocities for P and S wave propagation modes are 3-D ellipsoidal approximations that have been recently obtained. Moreover, we prove that this 3-D ellipsoidal anisotropic ray tracing formulation is a convex nonlinear optimization problem, and therefore any solution of the problem is a global minimum. Each optimization problem is solved by the global spectral gradient method, which requires first order information and has low computation and low storage requirements. Our approach for tracing rays in anisotropic media is a generalization in the sense that handles titled axis of symmetry and, close to the axis of symmetry, it is an accurate formulation for 2-D transversely isotropic media and 3-D orthorhombic media, depending on the input parameters. Moreover, this formulation gives the exact ray trajectories in 2-D and 3-D homogeneous isotropic media. The simplicity of the formulation and the low computational cost of the optimization method allow us to Partially supported by Fonacit project UCV-97-003769 D. Cores ( ) 374 D. Cores, M.C. Loreto present a variety of numerical results that illustrate the behavior and computational advantages of the approach, and the difficulties when working in anisotropic media.
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