Waveform inversion in the Laplace-Fourier domain

Since the pioneering work of Tarantola, waveform inversion has emerged as a tool for estimating velocity models of the subsurface using pre-stack seismic data. The waveform inversions have usually been performed in the time or frequency domain, but this can make it difficult to recover long-wavelength components of the velocity model due to the high non-linearity of the objective function and the lack of low-frequency components in the field data. Instead, it has been recently suggested that Laplace-domain waveform inversion can circumvent these limitations. By using the zero-frequency component of the damped wavefield, the Laplace-domain waveform inversion can recover long-wavelength structures of the velocity model even if low-frequency components less than 5 Hz are unreliable or would be unusable in conventional inversions. The main drawback is that the penetration depth of the Laplace-domain inversion depends on the offset distance and the choice of Laplace damping constants. In this paper, we propose an improved Laplace-Fourier-domain waveform inversion to compensate for these weak points. This is accomplished by exploiting low frequency components (less than 5 Hz)of the damped wavefield. The success of this technique arises from the 'mirage-like' resurrection of low-frequency omponents less than 5 Hz and the uniqu characteristics of the complex logarithmic wavefield. The latter is capable of separating the wavefield into amplitude and phase components, allowing us to simultaneously generate both long-wavelength and medium-short-wavelength velocity models. We successfully applied the aplace-Fourier-domain waveform inversion to a synthetic data set of the BP model calculated using the time-domain finite difference method. This not only produced a more refined velocity model when compared to Laplace-domain inversion results, but it also improved the penetration depth of the inversion. Furthermore, when the velocity model produced by the Laplace-Fourier-domain waveform inversion was then used as an initial velocity model of a conventional frequency-domain inversion, we obtained an inverted velocity model containing almost every feature of the true BP model. We applied our two-step, Laplace-domain waveform inversion to field data and obtained a refined velocity model containing short- and long-wavelength components. To convince ourselves of the accuracy of the inversion results, we computed a synthetic model using the estimated sourcewavelet and our velocity model from the inversion, andwe obtained amigrated image and angle-domain common-image gathers at several points by a reverse-time pre-stack depth migration in the frequency domain. The reconstructed synthetic data were in good agreement with the field data and most parts of the reflections in the image gathers were flattened.