Seismic spectral sparse reflectivity inversion based on SBL-EM: experimental analysis and application

Abstract

In this paper, we propose a new method of seismic spectral sparse reflectivity inversion that, for the first time, introduces Expectation-Maximization-based sparse Bayesian learning (SBL-EM) to enhance the accuracy of stratal reflectivity estimation based on the frequency spectrum of seismic reflection data. Compared with the widely applied sequential algorithm-based sparse Bayesian learning (SBL-SA), SBL-EM is more robust to data noise and, generally, can not only find a sparse solution with higher precision, but also yield a better lateral continuity along the final profile. To investigate the potential of SBL-EM in a seismic spectral sparse reflectivity inversion, we evaluate the inversion results by comparing them with those of a SBL-SA-based approach in multiple aspects, including the sensitivity to different frequency bands, the robustness to data noise, the lateral continuity of the final profiles and so on. Furthermore, we apply the mean square error (MSE), residual variance (RV) of seismograms and residual energy (RE) between the frequency spectra of the true and inverted reflectivity model to highlight the advantages of the proposed method over the SBL-SA-based approach in terms of spectral sparse reflectivity inversion within a sparse Bayesian learning framework. Multiple examples, including both numerical and field experiments, are carried out to validate the proposed method.

DuRIN: A Deep-unfolded Sparse Seismic Reflectivity Inversion Network

“…performance of DuRIN-1 and DuRIN-2 in comparison with the benchmark techniques, namely, basis-pursuit inversion (BPI) [13], [12], fast iterative shrinkage-thresholding algorithm (FISTA) [15], [16], and expectation-maximization-based sparse Bayesian learning (SBL-EM) [27], [8]. To quantify the performance, we employ the objective metrics listed in the following section.…”

“…However, the finite length and measurement noise, and the loss of low and high-frequency information due to convolution of the reflectivity with a bandlimited wavelet [1], [6], [7], lead to non-unique solutions to the above problem [8]. This ill-posed inverse problem is tackled by employing a sparsity prior on the solution [9].…”

“…The sparsity of seismic data is difficult to estimate in practice and the adequacy of the 1 norm for the reflectivity inversion problem has not been established [8], [19]. Also, 1 -norm regularization results in a biased estimate of x [20], [21], [22].…”

“…2) Data-driven Methods: The application of sparse Bayesian learning (SBL) for reflectivity inversion has also been explored [26], [8], where the sparse reflection coefficients are recovered by maximizing the marginal likelihood. This is done through a sequential algorithm-based approach (SBL-SA) to update the sparsity-controlling hyperparameters [26], or through the expectation-maximization algorithm (SBL-EM) [8], [27]. The application of neural networks to inversion problems in geophysics [28], particularly reflectivity inversion, is a recent development [3], [29].…”

“…The limitations of 1 regularization [20], [21], [22], in addition to the challenge of estimating the sparsity of seismic reflection data [8], [19], can be overcome through nonconvex regularization [19]. However, the corresponding nonconvex cost suffers from local minima, and optimization becomes quite challenging.…”