Laplace-domain inversions generate long-wavelength velocity models from synthetic and field data sets, unlike full-waveform inversions in the time or frequency domain. By examining the gradient directions of Laplace-domain inversions, we explain why they result in long-wavelength velocity models. The gradient direction of the inversion is calculated by multiplying the virtual source and the back-propagated wavefield. The virtual source has long-wavelength features because it is the product of the smooth forward-modeled wavefield and the partial derivative of the impedance matrix, which depends on the long-wavelength initial velocity used in the inversion. The back-propagated wavefield exhibits mild variations, except for near the receiver, in spite of the short-wavelength components in the residual. The smooth back-propagated wavefield results from the low-wavenumber pass-filtering effects of Laplace-domain Green’s function, which attenuates the high-wavenumber components of the residuals more rapidly than the low-wavenumber components. Accordingly, the gradient direction and the inversion results are smooth. Examples of inverting field data acquired in the Gulf of Mexico exhibit long-wavelength gradients and confirm the generation of long-wavelength velocity models by Laplace-domain inversion. The inversion of moving-average filtered data without short-wavelength features shows that the Laplace-domain inversion is not greatly affected by the high-wavenumber components in the field data.
A comparison between the behavior of objective functions for waveform inversion in the frequency and Laplace domains