3D acoustic waveform inversion in the Laplace domain using an iterative solver

In this paper we propose a 3D acoustic full waveform inversion algorithm in the Laplace domain. The partial differential equation for the 3D acoustic wave equation in the Laplace domain is reformulated as a linear system of algebraic equations using the finite element method and the resulting linear system is solved by a preconditioned conjugate gradient method. The numerical solutions obtained by our modelling algorithm are verified through a comparison with the corresponding analytical solutions and the appropriate dispersion analysis. In the Laplace-domain waveform inversion, the logarithm of the Laplace transformed wavefields mainly contains long-wavelength information about the underlying velocity model. As a result, the algorithm smoothes a small-scale structure but roughly identifies large-scale features within a certain depth determined by the range of offsets and Laplace damping constants employed. Our algorithm thus provides a useful complementary process to time- or frequency-domain waveform inversion, which cannot recover a large-scale structure when low-frequency signals are weak or absent. The algorithm is demonstrated on a synthetic example: the SEG/EAGE 3D salt-dome model. The numerical test is limited to a Laplace-domain synthetic data set for the inversion. In order to verify the usefulness of the inverted velocity model, we perform the 3D reverse time migration. The migration results show that our inversion results can be used as an initial model for the subsequent high-resolution waveform inversion. Further studies are needed to perform the inversion using time-domain synthetic data with noise or real data, thereby investigating robustness to noise.