Waveform inversion in the shifted Laplace domain

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 domainWI 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.