Full Waveform Inversion Using Student's t Distribution: a Numerical Study for Elastic Waveform Inversion and Simultaneous-Source Method

摘要

Seismic full waveform inversion (FWI) has primarily been based on a least-squares optimization problem for data residuals. However, the least-squares objective function can suffer from its weakness and sensitivity to noise. There have been numerous studies to enhance the robustness of FWI by using robust objective functions, such as l (1)-norm-based objective functions. However, the l (1)-norm can suffer from a singularity problem when the residual wavefield is very close to zero. Recently, Student's t distribution has been applied to acoustic FWI to give reasonable results for noisy data. Student's t distribution has an overdispersed density function compared with the normal distribution, and is thus useful for data with outliers. In this study, we investigate the feasibility of Student's t distribution for elastic FWI by comparing its basic properties with those of the l (2)-norm and l (1)-norm objective functions and by applying the three methods to noisy data. Our experiments show that the l (2)-norm is sensitive to noise, whereas the l (1)-norm and Student's t distribution objective functions give relatively stable and reasonable results for noisy data. When noise patterns are complicated, i.e., due to a combination of missing traces, unexpected outliers, and random noise, FWI based on Student's t distribution gives better results than l (1)- and l (2)-norm FWI. We also examine the application of simultaneous-source methods to acoustic FWI based on Student's t distribution. Computing the expectation of the coefficients of gradient and crosstalk noise terms and plotting the signal-to-noise ratio with iteration, we were able to confirm that crosstalk noise is suppressed as the iteration progresses, even when simultaneous-source FWI is combined with Student's t distribution. From our experiments, we conclude that FWI based on Student's t distribution can retrieve subsurface material properties with less distortion from noise than l (1)- and l (2)-norm FWI, and the simultaneous-source method can be adopted to improve the computational efficiency of FWI based on Student's t distribution.