Full waveform inversion in the Laplace and Laplace-Fourier domains

We present a review of Laplace and Laplace-Fourier domain waveform inversion. The wave equation in the Laplace and Laplace-Fourier domains can be solved by changing the real frequencies from the Fourier transform into imaginary frequencies. The initial model of Laplace-domain inversion can be a scratch such as a homogeneous velocity model. The inversion provides a long-wavelength velocity model that can be used as a starting velocity model for conventional waveform inversion, which uses the zero-frequency components of the damped wavefield. Laplace-Fourier domain inversion can recover long-, medium- and short-wavelength velocity models by adjusting the complex frequencies. Careful muting of noise should be applied before the first arrival because the damped wavefield is sensitive to random noise. Numerical experiments and real data examples show that full waveform inversion in the Laplace and Laplace-Fourier domains can provide an alternative for seismic velocity estimation.