Inversion for non‐smooth models with physical bounds

Abstract

SummaryGeological processes produce structures at multiple scales. A discontinuity in the subsurface can occur due to layering, tectonic activities such as faulting, folding and fractures. Traditional approaches to invert geophysical data employ smoothness constraints. Such methods produce smooth models and thefore sharp contrasts in the medium such as lithological boundaries are not easily discernible. The methods that are able to produce non-smooth models, can help interpret the geological discontinuity. In this paper we examine various approaches to obtain non-smooth models from a finite set of noisy data. Broadly they can be categorized into approaches: (1) imposing non-smooth regularization in the inverse problem and (2) solve the inverse problem in a domain that provides multi-scale resolution, such as wavelet domain. In addition to applying non-smooth constraints, we further constrain the inverse problem to obtain models within prescribed physical bounds. The optimization with non-smooth regularization and physical bounds is solved using an interior point method. We demonstrate the applicability and usefulness of these methods with realistic synthetic examples and provide a field example from crosswell radar data.

Regularization of linear and non-linear geophysical ill-posed problems with joint sparsity constraints

“…Hence, when attempting to penalize the small coefficients due to the noise, the large coefficients which are associated with the signal are more strongly penalized, leading to the distortion of the recovered solution (Youzwishen & Sacchi 2006; Gholami & Siahkoohi 2009a). The sparsity regularizer can be used to less penalize the outlier coefficients and hence capturing small‐scale details in the solution (Bertete‐Aguirre et al 2002; Routh et al 2007; Loris et al 2007; Herrmann et al 2008b). It can yield a solution which is sparse under the regularization operator L .…”

“…Non‐smooth inversion is a suitable approach to overcome such difficulty with more computational complexities (Sacchi & Ulrych 1995; Ajo‐Franklin et al 2007; Loris et al 2007; Routh et al 2007). Edge‐preserving inversion is sensitive to presence of the noise in data (Charbonnier et al 1997; Routh et al 2007), but well preserves sharp discontinuities in model parameters. CHA‐based sparsity inversion is stable against noise (Donoho 1995a; Donoho & Johnstone 1998; Daubechies et al 2004; Loris et al 2007; Routh et al 2007).…”

“…Edge‐preserving inversion is sensitive to presence of the noise in data (Charbonnier et al 1997; Routh et al 2007), but well preserves sharp discontinuities in model parameters. CHA‐based sparsity inversion is stable against noise (Donoho 1995a; Donoho & Johnstone 1998; Daubechies et al 2004; Loris et al 2007; Routh et al 2007). The method is computationally efficient (Kim et al 2007; Goldstein & Osher 2008) and enables one to apply regularization separately at different scales (Li et al 1996; Loris et al 2007; Gholami & Siahkoohi 2009a).…”

Rayleigh wave dispersion curve inversion: Occam versus the L1‐norm

“…For example, non-negativity is a special type of bound constraint. Routh et al (2007) examined various approaches to obtain non-smooth models and models within prescribed physical bounds in addition to non-smoothness. We denote these constraints as β ∈ C where C is a closed convex set.…”

“…In contrast, the algorithms for solving a constrained TV-regularized convex minimization problem (23) are gradually appearing. Routh et al (2007) solved the optimization with non-smooth regularization and physical bounds using an interior point method. Krishnan et al (2009) developed a primaldual active-set algorithm for bound constrained TV deblurring problems.…”