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

Abstract

We have developed a new uncertainty estimation method that accounts for nonlinearity inherent in most geophysical problems, allows for the explicit search of model posterior space, is scalable, and maintains computational efficiencies on the order of deterministic inverse solutions. We accomplish this by combining an efficient parameter reduction technique, a parameter constraint mapping routine, a sparse geometric sampling scheme, and an efficient forward solver. In order to reduce our model domain and determine an independent basis, we implement both a typical principal component analysis, which factorizes the model covariance matrix, and an alternative compression method, based on singularvalue decomposition, which acts on training models, directly, and is storage efficient. Once we have a reduced base, we map parameter constraints, from our original model domain, to this reduced domain to define a bounded geometric region of feasible model space. We utilize an optimal scheme to sample this reduced model space that uses Smolyak sparse grids and minimizes the number of forward solves by finding the sparsest sampling required to produce convergent uncertainty measures. The result is an ensemble of equivalent models, consistent with our inverse solution structure, which is used to infer inverse uncertainty. We tested our method with a 1D synthetic example, a comparison with a published Metropolis-Hastings sampling example, and an extension to 2D problems using a field data inversion.

Accuracy and effectiveness of three-dimensional controlled source electromagnetic data inversions

“…Resistivity volumes that are slowly varying, such as those in Figure 2, can be accurately represented with a small number of principal components (Tompkins et al, 2011). We can construct new resistivity volumes by perturbing the first few principal components of the volumes produced by inversion.…”

Integration of well data into geostatistical seismic amplitude variation with angle inversion for facies estimation

“…However, independent of the inverse methodology, due to the intrinsic characteristics of seismic inversion problems, there is always a variable degree of uncertainty related to the retrieved subsurface elastic models. For more reliable reservoir modeling and characterization and decision making, this uncertainty should be propagated during the entire inversion procedure and should be an integral part of the inverse solution (Grana and Della Rossa, 2010;Tompkins et al, 2011). Casting the seismic inversion problem in a probabilistic framework is a natural way to account for this uncertainty.…”

“…In particular, seismic inversion problems try to model the spatial distribution of the subsurface acoustic and/or elastic properties (e.g., density and Pwave and S-wave velocities) recurring to available seismic reflection data. Seismic inversion problems are nonlinear, nonunique, and ill-posed due to measurement errors and the intrinsic properties of the seismic method: the limited bandwidth and resolution, noise and modeling imperfections (Tarantola, 2005;Bosch et al, 2010;Tompkins et al, 2011), and they can generally be mathematically summarized by m ¼ F −1 ðd obs Þ þ e;…”

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

“…Examples that represent this type of approach include Metropolis-Hastings, simulated annealing, neighborhood algorithm, and genetic algorithm. While random sampling methods avoid burdensome inversions and can account for nonlinearity, they come at the high cost of inefficiency (e.g., Haario et al, 2001;Tompkins et al, 2011bTompkins et al, , 2012 and thus are limited to modest-sized problems (10 s of unknowns). Recently, a new method has emerged that solves the same posterior sampling problem, but uses sparse-grid interpolation with orthogonal polynomials as opposed to random sampling (i.e., Tompkins et al, 2011aTompkins et al, , 2011b.…”