Uncertainty and resolution analysis for anisotropic tomography using iterative eigendecomposition

Abstract

Tomographic velocity model building has become an industry standard for depth migration. Anisotropy of the Earth challenges tomography because the inverse problem becomes severely ill-posed. Singular value decomposition (SVD) of tomographic operators or, similarly, eigendecomposition of the corresponding normal equations, are well known as a useful framework for analysis of the most significant dependencies between model and data. However, application of this approach in velocity model building has been limited, primarily because of the perception that it is computationally prohibitively expensive, especially for the anisotropic case. In this paper, we extend our prior work (Osypov et al., 2008) to VTI tomography, modify the process of regularization optimization, and propose an updated way for uncertainty and resolution quantification using the apparatus of eigendecomposition. We demonstrate the simultaneous tomographic estimation of VTI parameters on a real dataset. Our approach provides extra capabilities for regularization optimization and uncertainty analysis in anisotropic model parameter space which can be further translated into the structural uncertainty within the image.

Scalable uncertainty estimation for nonlinear inverse problems using parameter reduction, constraint mapping, and geometric sampling: Marine controlled-source electromagnetic examples

“…Interestingly, some progress has been made in extending these linear methods either by extremizing parameters (Oldenburg, 1983;Meju, 2009), null-space projection with random sampling (Osypov et al, 2008), or prior sampling with deterministic inverse mapping (Materese, 1995;Alumbaugh and Newman, 2000;Alumbaugh, 2002). Osypov et al (2008), for example, present a null-space projection method, which incorporates stochastic sampling, to determine velocity uncertainties for the anisotropic seismic tomography problem. However, their uncertainty is still limited to small perturbations around a deterministic inverse solution.…”

“…Thus, many deterministic methods rely on the Jacobian (or related linear operators) to compute the posterior covariance or resolution matrices and define linearized uncertainties (e.g., Meju, 1994;Zhang and Thurber, 2007). Interestingly, some progress has been made in extending these linear methods either by extremizing parameters (Oldenburg, 1983;Meju, 2009), null-space projection with random sampling (Osypov et al, 2008), or prior sampling with deterministic inverse mapping (Materese, 1995;Alumbaugh and Newman, 2000;Alumbaugh, 2002). Osypov et al (2008), for example, present a null-space projection method, which incorporates stochastic sampling, to determine velocity uncertainties for the anisotropic seismic tomography problem.…”

Anisotropic Imaging in Complex Media: Step Change in Quality and Interpretability of the Results

“…Once a high-resolution model is produced, many equally probable models may also be generated using the uncertainty analysis introduced by Osypov et al (2008). This method is based on sampling the posterior distribution of the final anisotropic tomography operator with its embedded geological and rock-physics constraints.…”

Efficient estimation of nonlinear posterior model covariances using maximally sparse cubature rules

“…While the reasons for the occurrence of solution uncertainty are reasonably well understood (e.g., measurement error, solution nonuniqueness, data coverage and bandwidth limitation, and physical assumptions), efficient methods to estimate such uncertainty are not. The geophysics community has been reasonably successful at quantifying uncertainty associated with large linearized problems (e.g., Zhang and Thurber, 2007;Osypov et al, 2008), but solutions to the large-scale nonlinear uncertainty problem have been elusive. In the context of nonlinear inversion, this uncertainty problem is that of quantifying the range or variability in the model space supported by prior information, the data, and any errors.…”