Fast multi-trace impedance inversion using anisotropic total p-variation regularization in the frequency domain

Abstract

In this article, we find that the transformational matrix between the natural logarithm of the acoustic impedance and the reflection coefficient is a Toeplitz matrix. As a result, we can convert the matrix multiplication into convolution operation according to properties of the Toeplitz matrix. In this way, we calculate the convolution of matrices through a dot product operation in the frequency domain based on the convolution theorem. Thus a fast multi-trace impedance inversion method by using two dimensional fast Fourier transform in the frequency domain is proposed, which can greatly improve the inversion speed. Considering the non-convex Lp (0 ≤ p < 1) quasi-norm is more suitable for sparse optimization than the L1 norm, anisotropic total variation regularization based on the Lp quasi-norm is used to improve the inversion result. Synthetic seismic data and field data inversion results show that the proposed method improves the inversion speed more than ten times compared with the multi-trace inversion method with total variation (TV) regularization in the time domain. At the same time, the proposed method has the advantages over the inversion methods with TV regularization in the time domain of less inversion error and better depiction of thin layers.