Adaptive Regularization of the Reference Model in an Inverse Problem
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Pure and Applied Geophysics
Adaptive Regularization of the Reference Model in an Inverse Problem MEIJIAN AN1 Abstract—The solution to an inverse problem is often resolved by inverting the perturbation to a reference model of physical parameters and using regularizations. However, the most commonly used higher-order Tikhonov regularizations, which are unrelated to the reference model, are generally unable to correct false variations in the reference model, since these regularizations tend to minimize the model variations in the inverted perturbations. A viable approach to overcome this shortcoming is to adapt the regularization for the reference model such that a sharp variation around a given position in the reference model (regardless of whether the variation is true or false) receives a smaller weighting in the regularizations. Linear and nonlinear inversion tests show that this new adaptive regularization can improve the inversion results at or around positions with either badly constructed or true variations in the reference model. Keywords: Tikhonov regularization, linear or linearized inversion, tomography, adaptive regularization.
1. Introduction Inverse problems can be resolved via two different inversion schemes: Scheme 1, with inversions to determine the best-fit model of physical parameters, and Scheme 2, with inversions investigating perturbations relative to a reference model. Scheme 2 is termed the creeping strategy (Shaw and Orcutt 1985; Parker 1994) because the inverted solution is the model perturbation, and the model solution is the summation of the reference model and inverted perturbation. This scheme is effectively driven by the fact that it is easier to discover truth by building on previous discoveries (Keith et al. 2016) or by standing on the shoulders of giants (I. Newton, Letter to R. Electronic supplementary material The online version of this article (doi:https://doi.org/10.1007/s00024-020-02530-z) contains supplementary material, which is available to authorized users. 1
Chinese Academy of Geological Sciences, Beijing 100037, China. E-mail: [email protected]; [email protected]
Hooke, 5 Feb. 1676). All previous discoveries can be used to construct the reference model, such that Scheme 2 is widely applied in practical studies. A reference model allows many nonlinear inverse problems to be resolved via linearization using the first-order Taylor series approximation (e.g., Backus and Gilbert 1967, 1968; Aster et al. 2005). Many modern physical and geophysical tomographic studies, such as electrical resistance (e.g., Xu et al. 2016), impedance (e.g., Vauhkonen et al. 1998; Babaeizadeh and Brooks 2007), optical emission (e.g., Darne et al. 2013), and seismic body-wave (Nolet 2008) studies, define the reference model as a crucial and necessary component of the inversion process. Both inversion schemes should yield the same model solution if the inverse problem is linear and well-posed. However, most practical inverse problems (e.g., Vauhkonen et al. 1998; Aster et al. 2005; Xu et al. 2
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