Interrelation between Laplace constants and the gradient distortion effect in Laplace-domain waveform inversion

Abstract

Laplace-domain waveform inversion (WI) is generally used to generate smooth initial velocity models for frequency-or time-domain full-waveform inversion. However, in the inversion results of Laplace-domain WI, anomalies such as salt domes are sometimes shifted. We evaluate the "gradient-distortion effect" that causes undesirable changes in parameter updates and found that this is caused by the relationship between the partial derivatives of Laplace wavefields with respect to two different parameters. By analyzing the gradient of the Laplace-domain misfit function, we found that the gradient distortion effect increases as the Laplace constants used in the Laplace-domain WI decrease. The velocity model inverted in the Laplace domain is generally blurred from shallower parameters to deeper parameters because the partial derivatives of the Laplace wavefields with respect to shallower parameters tend to be larger than those of deeper parameters. We found two solutions for suppressing the gradient distortion effect. The first one is the sequentially ordered Laplace constant approach with multiple Laplace constants. We discover that a dense, broad set of Laplace constants should be sequentially used in this approach. The second solution is the Gauss-Newton method, in which the Hessian matrix is considered. Numerical tests performed using a four-layer model and the BP benchmark model show the gradient distortion effect appeared in the inversion results and the effectiveness of the sequentially ordered Laplace constant approach. In addition, tests using an inverted BP benchmark model determine that the inversion results can be improved by applying a broad, dense set of Laplace constants to synthetic data. Finally, we verify the effectiveness of the GaussNewton method at suppressing the gradient distortion effect using the BP benchmark model.