Waveform inversion in the Laplace domain

For the last 30 yr, since Tarantola's pioneering theoretical study of waveform inversion, geophysicists and applied mathematicians have utilized a waveform inversion to delineate the earth's structures. However, successful applications of waveform inversion to real data are nominal. The failures are mainly caused by the high non-linearity of the waveform inversion and the real data containing insufficient low-frequency components. We propose a waveform inversion algorithm that is robust and is not sensitive to the initial model by exploiting the wavefield in the Laplace domain and the adjoint property of the wave equation. The wavefield in the Laplace domain is equivalent to the zero frequency component of the damped wavefield. Therefore, the inversion of Poisson's equation in electrical prospecting can be viewed as a waveform inversion problem, exploiting the zero frequency component of an undamped wavefield. Since our inversion algorithm in the Laplace domain is strongly associated with the nature of DC inversion for Poisson's equation in electrical prospecting, our algorithm can generate a velocity model that is equivalent to a long-wavelength velocity model (a smooth velocity model). Through numerical tests, we found that the Laplace-transformed wavefields for optimally low Laplace damping constants are critically needed in our algorithm as the low-frequency components are necessary in the waveform inversion of the frequency domain. We note that the objective function by l(2) norm of the logarithmic wavefield in the Laplace domain behaves as if it has no local minimum points in low and high Laplace damping constants. Moreover, we note that the forward modelling in the Laplace domain could be accurately calculated by using a coarser grid of several hundreds of metres than that of the frequency domain. Numerical tests of the salt dome model with high-velocity contrast demonstrate the robustness of our algorithm to both synthetic and real data. To apply our algorithm to real data more successfully, we need to improve the accuracy of the Laplace transformation for the wavefields in the time domain. However, our waveform inversion in the Laplace domain can be successfully applied to real data since our algorithm has more advantages in forward modelling and inversion than those in the frequency domain.