Regularized extremal bounds analysis (REBA): An approach to quantifying uncertainty in nonlinear geophysical inverse problems

Abstract

[1] Geophysical measurements are band-limited in nature, contain noise, and bear a nonlinear relationship to the subterranean features being sampled. These result in uncertainty in data interpretation and necessitate a regularization approach. Although model uncertainty and non-uniqueness can be reduced by combining measurements of fundamentally different physical attributes of a subsurface target under investigation or by using available a priori information about the target, quantifying non-uniqueness remains a difficult task in nonlinear geophysical inversion. This paper develops a new theoretical framework for regularized extremal bounds analysis (REBA) of model uncertainty using the most-squares formalism. The nonlinear most-squares method allows for combining data constraints and their associated uncertainties in an objective manner to determine the model bounds. By making the assumption that the relevant bounding models must sample the same underlying geology, it is proposed that structural dissimilarity between the extreme models can be quantified using the cross-products of the gradients of their property fields and should serve as an objective measure of interpretational uncertainty in multidimensional inversion.

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

“…In the context of linearized inversion, the model Jacobian matrix is rank deficient, and the dimension of its null space accounts, locally, for the uncertainty around an inverse model. 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).…”

“…The estimation of inverse uncertainty is important, because it provides us with necessary information about the range of solutions that are informed equally well by our observations. As pointed out by Meju (2009), there are many approaches to the solution of both the linearized and nonlinear uncertainty problems. Perhaps the most apparent distinction is between stochastic and deterministic methods.…”

“…However, their uncertainty is still limited to small perturbations around a deterministic inverse solution. Because these deterministic methods commonly rely on linearized estimates of uncertainty about the last iteration of a nonlinear inversion, they tend to have limited relevance to actual nonlinear uncertainty (e.g., Meju, 1994;Alumbaugh, 2002).…”

“…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.…”

On the use of sensitivity tests in seismic tomography

“…Within a linear or weakly nonlinear framework, a number of different techniques have been tried, including multiple starting models (Vasco et al 1996), the so-called 'null space shuttle' (Deal & Nolet 1996;de Wit et al 2012), regularized extremal bounds analysis (Meju 2009), Lie group methods (Vasco 2007) and the dynamic objective function scheme (Rawlinson & Kennett 2008). However, it is in the realm of fully nonlinear sampling where the greatest strides are currently being made.…”