Improved Full-Waveform Inversion for Seismic Data in the Presence of Noise Based on the K-Support Norm

Abstract

In the inversion part of the full-waveform inversion (FWI) that brings high resolution in finding a convergence point in the model space, a localized numerical optimization technique reduces the objective function employing the $\ell_{2}$ norm in a least-squares approach. Given the $\ell_{2}$ norm's vulnerability to outliers and noise, this strategy might frequently yield imprecise imaging outcomes. Consequently, it's essential to introduce a novel regulatory approach that incorporates a more applicable form of relaxation to address the issue of overfitting, specifically, the K-support norm, characterized by its more logical and stringent restrictions. Unlike the least-squares method that targets the $\ell_{2}$ norm reduction, the $\ell_{1}$ norm is celebrated for its ability to ensure sparsity, durability, and superior noise reduction capabilities; thus, we consider a new regularization form, the K-support norm, which combines the $\ell_{2}$ and the $\ell_{1}$ norms in minimization. Subsequently, a quadratic penalty approach is utilized to expedite the convergence process and prevent entrapment in local minima, linearizing the nonlinear issue to reduce computational demands. This study presents the K-support norm concept, incorporating it with the quadratic penalty approach to enhance both convergence speed and resilience to ambient noise. In the numerical demonstration, three synthetic models are evaluated to demonstrate the K-support norm's superiority over Tikhonov regularization using two distinct noise data sets. Experimental results indicate that the modified FWI improves inversion accuracy by enhancing lateral resolution in deeper parts even with data with a low signal-to-noise ratio.

Appendix A. Wavefield-Reconstruction Inversion (WRI)

Wave equation FWI algorithm based on the penalty function is proposed by van Leeuwen and Herrmann proposed.

The regularization term of PDE constrained form can be expressed as follows:

$$\mathop {\min }\limits_{{{\mathbf{M}},{\mathbf{X}}}} \left\| {{\mathbf{AX}} - {\mathbf{D}}} \right\|_{2}^{2} ,{\text{ s}}{\text{.t}}{. }{\mathbf{C}}({\mathbf{M}}){\mathbf{X}} = {\mathbf{S}},$$

(13)

where $\left\| { \, \, } \right\|_{2}^{2}$ is the $\ell_{2}$ norm, and the ${\mathbf{M}} \in {\mathbb{R}}^{{{\mathcal{N}} \times 1}}$ is the model parameters; ${\mathbb{R}}$ is the regularization function, contains preliminary information about model parameters, ${\mathbf{X}} \in {\mathbb{R}}^{{{\mathcal{N}} \times 1}}$ represents the wavefield, ${\mathbf{D}} \in {\mathbb{R}}^{{{\mathcal{M}} \times 1}}$ is the recorded seismic data, ${\mathbf{S}} \in {\mathbb{R}}^{{{\mathcal{N}} \times 1}}$ is the source term, and the linear observation operator ${\mathbf{A}} \in {\mathbb{R}}^{{{\mathcal{M}} \times {\mathcal{N}}}}$ sampling ${\mathbf{X}}$ at the receiver positions.

From the mathematical point of view, the most appropriate way to solve this type of constrained optimization problem is in the form of a Lagrangian function:

$$\mathop {\min }\limits_{{{\mathbf{M}}{,}{\mathbf{X}}}} \mathop {\max }\limits_{{\mathbf{V}}} {\text{F}} ({\mathbf{M}},{\mathbf{X}},{\mathbf{V}}) = \mathop {\min }\limits_{{{\mathbf{M}},{\mathbf{X}}}} \mathop {\max }\limits_{{\mathbf{V}}} \left\| {{\mathbf{AX}} - {\mathbf{D}}} \right\|_{2}^{2} + {\mathbf{V}}^{T} \left[ {{\mathbf{C}}({\mathbf{M}}){\mathbf{X}} - {\mathbf{S}}} \right],$$

(14)

where ${\mathbf{V}} = \left[ {{\mathbf{V}}_{1} ;{\mathbf{V}}_{2} ; \ldots } \right]$. The ${\mathbf{V}}$ represents the Lagrange multiplier. The advantage of using the Lagrange function is that after iterative optimization, new fitting data, forward results, and an adjoint matrix will be obtained simultaneously, which avoids the need for an explicit solution in the optimization process of conventional FWI, which means that the wavefield reconstruction inversion algorithm obtains a solution consistent with augmented wave equation term.

Since the model parameter ${\mathbf{C}}$ based on the PDE operator may converge to an approximate minimum value when the start model is not ideal, therefore, we can redefine the original constrained problem as a quadratic penalty problem (van Leeuwen & Herrmann, 2013):

$$\mathop {\min }\limits_{{{\mathbf{M}},{\mathbf{X}}}} {\text{F}} ({\mathbf{M}},{\mathbf{X}}) = \mathop {\min }\limits_{{{\mathbf{M}},{\mathbf{X}}}} \left\| {{\mathbf{AX}} - {\mathbf{D}}} \right\|_{2}^{2} + \lambda \left\| {{\mathbf{C}}({\mathbf{M}}){\mathbf{X}} - {\mathbf{S}}} \right\|_{2}^{2} .$$

(15)

With the form of a quadratic penalty term, the original full space-constrained form can be turned into a double penalty term form, reducing the complexity of the algorithm while improving the ability of FWI to converge to the more exact minimum value in the presence of an inaccurate initial model.