Jungmin Kwon

Laplace domain waveform inversion (WI) is one of the most effective algorithms to generate an initial velocity model. Because of its bandwidth independence with respect to the source wavelet, this method can yield reasonable initial models without low-frequency components in the seismic data. However, the conventional Laplace domain WI algorithm has an accuracy problem from its simultaneous consideration of the first arrival traveltime and apparent amplitude in the Laplace domain wavefield. This simultaneous consideration creates undesirable cross-correlation terms between the residual of the traveltime and the partial derivatives of the apparent amplitude in the gradient directions and between the residual of the apparent amplitude and the partial derivatives of the traveltime in the gradient directions. In this paper, we introduce a new objective function that uses a shifted Laplace domain wavefield to solve the problem of Laplace domain WI. Information that is associated with the traveltime and apparent amplitude can be separately inverted by using this shifted Laplace domain WI. This separation of the information can suppress the undesirable cross-correlation terms between the residual of the traveltime and the partial derivatives of apparent amplitude and between the residual of the apparent amplitude and the partial derivatives of the traveltime in the gradient directions. We can effectively perform shifted Laplace domain modelling by using the damped monochromatic wave equation. We verify the accuracy of this shifted Laplace domain modelling scheme by comparing the shifted Laplace-transformed result from a synthetic seismogram to a wavefield that is modelled in the shifted Laplace domain. We perform a contribution analysis to demonstrate that the shifted Laplace domain wavefield is essential to improve the accuracy of the inverted results. Finally, we confirm the robustness of the shifted Laplace domain WI algorithm by testing it against a BP model.

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 Gauss-Newton method at suppressing the gradient distortion effect using the BP benchmark model.