next up previous [pdf]

Next: Theory Up: Xue et al.: FWI Previous: Xue et al.: FWI

Introduction

Full waveform inversion (FWI) is a data-fitting procedure used to construct high-resolution quantitative subsurface models by exploiting full information in observed data (Tarantola, 1984; Virieux and Operto, 2009; Lailly, 1983). However, the problem is inherently ill-posed, and suffers from artifacts that could be falsely interpreted as ``geological structures''. One way to mitigate this problem is by adding regularization or preconditioning to guide the inversion toward a model consistent with a priori geological or geophysical constraints. The choice among different regularization techniques depends on the specific problem, and can be governed by the need to preserve or emphasize particular desired features of the model (Loris et al., 2007). For example, if one is mostly interested in large-scale features, it is natural to introduce a regularizing constraint or penalty term involving spatial derivatives. On the other hand, if one seeks to find solutions that are sparse with respect to a given basis, this can be achieved by imposing a sparsity constraint involving appropriate transforms such as Fourier, wavelet or curvelet transforms (Loris et al., 2007). This regularization procedure finds solutions that are faithfully represented by a relatively small number of non-zero coefficients in the transformed domain.

We propose to impose sparsity regularization on the model in the seislet domain (Fomel and Liu, 2010) to improve the robustness of FWI. We refer to this regularization as seislet regularization. A model in the seislet domain is expressed using basis functions aligned along locally planar structures. Other researchers have previously used the sparsity of velocity model in other transform domains for velocity model building. For instance, Loris et al. (2007) applied $ l_1$ -norm regularization in a wavelet basis to solve global seismic tomography problems, allowing for the possibility of sharp discontinuities superimposed on a smoothly varying background model. Li et al. (2012) computed the model updates from random subsets of data and used curvelet domain sparsity promotion to suppress crosstalk between different sources. Curvelets are appropriate for seismic data because they provide a provably optimal decomposition of wave-propagation operators (Candès and Demanet, 2005). In this paper, the sparsity regularization for velocity model is implemented by imposing a soft thresholding on the updated model in the seislet domain. Compared to other transform domains, seislets exhibit a superior data compression and sparsity for events with dominant local slopes (Chen et al., 2016). The classic digital wavelet transform is equivalent to seislet transform with the erroneous zero slope (Fomel and Liu, 2010).

Seislet regularization allows us to build a model that both fits the data and has a strong tendency to be sparse in the seislet basis. Because seislet basis functions are aligned along locally planar structures, this helps to attenuate random noise and build geologically-consistent models. Different approaches to generating geologically sensible models for seismic inversion have been proposed before. Guitton et al. (2012) used directional Laplacian filter as model preconditioning operator in FWI to smooth gradients along geological dips. Ma et al. (2012) proposed to invert for a sparse velocity model in FWI, and connected the sparse and dense models through image-guided interpolation. Xue et al. (2016) incorporated linear shaping regularization (Fomel, 2007) into least-squares reverse-time migration (RTM) and used structure-enhancing filtering to mitigate artifacts caused by simultaneous-source or incomplete data. Compared to these methods, the proposed method offers an alternative formulation with a highly efficient implementation. It is formulated as the sparsity constraint on the model in the seislet domain, and is implemented by imposing a soft thresholding on the updated model at each iteration.

In this paper, we first introduce FWI with and without seislet regularization, and illustrate their implementation differences. Next, we use two numerical examples to verify the effectiveness of the proposed method in improving the robustness of FWI by suppressing artifacts caused by encoded data and random noise. Both examples are based on the 2D Marmousi synthetic model (Bourgeois et al., 1991).


next up previous [pdf]

Next: Theory Up: Xue et al.: FWI Previous: Xue et al.: FWI

2017-10-09