Optimal Seismic Reflectivity Inversion: Data-Driven $\ell_p$ -Loss-$\ell_q$ -Regularization Sparse Regression

Abstract

Seismic reflectivity inversion is widely applied to improve the seismic resolution to obtain detailed underground understandings. Based on the convolution model, seismic inversion removes the wavelet effect by solving an optimization problem. Taking advantage of the sparsity property, the ℓ 1 norm is commonly adopted in the regularization terms to overcome the noise/interference vulnerability observed in the ℓ p-losses minimization. However, no one has provided a deterministic conclusion that ℓ 1 norm regularization is the best choice for seismic reflectivity inversion. Instead of using an unproved fixed regularization norm, we propose an optimal seismic reflectivity inversion approach. Our method adaptively adopts a ℓ p-loss-ℓ q-regularization (i.e. ℓ p,q regularization) for p = 2, 0 < q < 1 to estimate a more accurate and detailed reflectivity profile. In addition, we employ a K fold cross-validation based approach to obtain the optimal damping factor λ to further improve the seismic inversion results. The letter starts with the introduction of non-convex constraint for seismic inversion, and the necessity of the ℓ q norm regularization. Then the majorization-minimization and cross validation algorithms are briefly described. The performance of the proposed seismic inversion approach is evaluated through synthetic examples and a field example from the Bohai Bay Basin, China.

Inverse spectral decomposition using an lp-norm constraint for the detection of close geological anomalies

“…In this paper, the selection of the regularization parameter is mainly based on the integrity and clarity of geological bodies in the spectral amplitude slice of horizon as a quality-control standard. How to pick a reasonable p and the regularization parameter adaptively (e.g., Li et al 2019) is also worth studying in the future.…”

DuRIN: A Deep-unfolded Sparse Seismic Reflectivity Inversion Network

“…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].…”

“…The advantages of adopting nonconvex regularizers over 1 have been demonstrated in sparse recovery problems [22], [24], [25]. Particularly pertinent to this discussion is the data-driven q -norm regularization (0 < q < 1) for seismic reflectivity inversion proposed by [19], wherein the optimal q was chosen based on the input data.…”

“…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.…”

NuSPAN: A Proximal Average Network for Nonuniform Sparse Model -- Application to Seismic Reflectivity Inversion

“…The fast iterative shrinkage-thresholding algorithm (FISTA) [11] has been employed for reflectivity inversion [12] along with an amplitude recovery boost through debiasing steps of least-squares inversion [13] and "adding back the residual" [14]. The 1 -norm, although convex, is not the best sparsity constraint for reflectivity inversion, and the accurate estimation of the sparsity of seismic reflections is challenging [15], [16]. Further, 1 -norm regularization underestimates the high-amplitude components and introduces a bias in the estimate of the sparse code x [17], [18], [19].…”