Robust Elastic Full Waveform Inversion using Students t-distribution in the Frequency domain

Abstract

Seismic full waveform inversion (FWI) is a numerical technique that estimates subsurface parameters. FWI is usually based on a nonlinear least-squares optimization problem. However, it has been known that the least-squares objective function cannot properly estimate subsurface material properties when field data are contaminated with noise such as outliers. In this study, we propose a 2D elastic FWI algorithm based on Students t-distribution, which has an overdispersed density compared to the Normal distribution and can be useful for data with outliers. To apply t-distribution to the elastic FWI, the statistical techniques such as Maximum a posteriori (MAP) and Maximum likelihood (ML) are used. The inversion algorithm is based on the finite-element modeling and the adjoint state of the wave equation. To calculate gradient directions efficiently, the gradients were computed by the cross-correlation of back-propagated residuals and virtual sources and the pseudo-Hessian matrix was applied. Also, the conjugate gradient method was used to accelerate the convergence rate of inversion.

The elastic FWI using Students t-distribution is demonstrated for 2D synthetic data set for the Modified elastic Marmousi-2 model. For comparison, the l2- and l1-norm-based FWI have also been applied to the model. For noise-free data, all the inversion results obtained by the three objective functions are in good agreement with the true velocities. For data with 10 outliers, the magnitude of outliers is 150 % of the maximum amplitude of signal in each frequency. While the velocity model inverted by the l2-norm FWI is severely distorted by the outliers, the l1-norm and Students t misfit yield reliable results. When both outliers and random noises are applied, inversion results obtained by the l2-norm are much poorer than those obtained for the data with only outliers. It seems like that the l1-norm FWI is less influenced by random noise compared to the other methods. Although the RMS errors of Students t-distribution are lower than those of the other methods and yields better inversion results than the l2- and l1-norm objective functions, the distortions caused by random noise appear throughout the entire P-wave velocity model. From these results, we note that Students t misfit can decrease the influence of large outliers on inversion results, in particular for deep structures. We expect that other statistical distributions can be applied to seismic FWI.