A method for improving the computational efficiency of a Laplace–Fourier domain waveform inversion based on depth estimation

• The effective inversion depth in Laplace–Fourier domain is frequently less than model depth.
• The effective inversion depth depends on many factors.
• We proposed a method for estimating the effective inversion depth quantitatively.
• We used the estimated depth to improve the Laplace–Fourier domain inversion efficiency.
• Numerical experiments demonstrated the decrease of more than 50% computation cost.

The Laplace–Fourier domain full waveform inversion can simultaneously restore both the long and intermediate short-wavelength information of velocity models because of its unique characteristics of complex frequencies. This approach solves the problem of conventional frequency-domain waveform inversion in which the inversion result is excessively dependent on the initial model due to the lack of low frequency information in seismic data. Nevertheless, the Laplace–Fourier domain waveform inversion requires substantial computational resources and long computation time because the inversion must be implemented on different combinations of multiple damping constants and multiple frequencies, namely, the complex frequencies, which are much more numerous than the Fourier frequencies. However, if the entire target model is computed on every complex frequency for the Laplace–Fourier domain inversion (as in the conventional frequency domain inversion), excessively redundant computation will occur. In the Laplace–Fourier domain waveform inversion, the maximum depth penetrated by the seismic wave decreases greatly due to the application of exponential damping to the seismic record, especially with use of a larger damping constant. Thus, the depth of the area effectively inverted on a complex frequency tends to be much less than the model depth. In this paper, we propose a method for quantitative estimation of the effective inversion depth in the Laplace–Fourier domain inversion based on the principle of seismic wave propagation and mathematical analysis. According to the estimated effective inversion depth, we can invert and update only the model area above the effective depth for every complex frequency without loss of accuracy in the final inversion result. Thus, redundant computation is eliminated, and the efficiency of the Laplace–Fourier domain waveform inversion can be improved. The proposed method was tested in numerical experiments. The experimental results show that the Laplace–Fourier domain waveform inversion with variable depth can effectively reduce the calculation time to less than 50% without loss of quality in the inversion results.