Summary
We present an algorithm that provides improved Earth model reconstruction over Gaussian, or Tikhonov, regulated inversions. By utilizing a dynamic noise estimation, linear error propagation theory and a Bayesian construction this algorithm can: objectively indicate the amount of recoverable independent information, provide a framework on how to weight prior information, and to communicate the uncertainty of the estimates.
Introduction
The distribution of Earth reflectivity has been shown to depart significantly from a Gaussian distribution (Walden and Hosken, 1986, Velis, 2003, Painter et al., 1995). The findings of Painter et al. (1995) are that seismic reflectivity is best represented by a Levy-stable distribution with slowly decaying tails, and Walden and Hoskens (1986) found that mixtures of Laplace distributions are most efficient description of the distribution of reflectivity. These distributions motivate the use of priors other than an L2, or Gaussian, distributions including an L1, or Laplacian, and mixtures of L1 and L2 prior distributions on the Earth model within this paper. Inspired by these insights, a common objective is to inverse model, or deconvolve, seismic data to recover the discontinuities of the Earth, including using sparse-spike inversion (e.g. Debeye and Van Reil, 1990, and Barrodale and Roberts, 1973, Taylor, et al., 1979, Perez et al, 2013, Sacchi, 1997) and related basis pursuit methods (Zhang, and Castagna, 2011). The need for a sparsity constraint within the minimization function of this inverse problem makes it a nonlinear optimization problem, which leads to problems in converging to the global minima that require considerable constraints to converge to an acceptable solution. We are extending an iterative soft-thresholding algorithm (ISTA), which can be understood as a case of the expectation maximization (EM) algorithm, for solving linear inverse problem with a sparsity constraint (Daubechies et. al. 2003 and Figueiredo and Nowak 2003). This method had first been applied to seismic data by Dossal and Mallat (2005), and an implementation to 2-term prestack seismic inversion was introduced by Perez et al. (2013) using the fast iterative soft-thresholding algorithm (FISTA).