Multi-scale uncertainty quantification in geostatistical seismic inversion

Geostatistical seismic inversion is commonly used to infer the spatial distribution of the subsurface petro-elastic properties by perturbing the model parameter space through iterative stochastic sequential simulations/co-simulations. The spatial uncertainty of the inferred petro-elastic properties is represented with the updated a posteriori variance from an ensemble of the simulated realizations. Within this setting, the large-scale geological (metaparameters) used to generate the petro-elastic realizations, such as the spatial correlation model and the global a priori distribution of the properties of interest, are assumed to be known and stationary for the entire inversion domain. This assumption leads to underestimation of the uncertainty associated with the inverted models. We propose a practical framework to quantify uncertainty of the large-scale geological parameters in seismic inversion. The framework couples geostatistical seismic inversion with a stochastic adaptive sampling and Bayesian inference of the metaparameters to provide a more accurate and realistic prediction of uncertainty not restricted by heavy assumptions on large-scale geological parameters. The proposed framework is illustrated with both synthetic and real case studies. The results show the ability retrieve more reliable acoustic impedance models with a more adequate uncertainty spread when compared with conventional geostatistical seismic inversion techniques. The proposed approach separately account for geological uncertainty at large-scale (metaparameters) and local scale (trace-by-trace inversion)

A Sparse Bayesian Estimation Framework for Conditioning Prior Geologic Models to Nonlinear Flow Measurements

We present a Bayesian framework for reconstruction of subsurface hydraulic properties from nonlinear dynamic flow data by imposing sparsity on the distribution of the solution coefficients in a compression transform domain

A Practical Method to Estimate Information Content in the Context of 4D-Var Data Assimilation. I: Methodology

Data assimilation obtains improved estimates of the state of a physical system by combining imperfect model results with sparse and noisy observations of reality. Not all observations used in data assimilation are equally valuable. The ability to characterize the usefulness of different data points is important for analyzing the effectiveness of the assimilation system, for data pruning, and for the design of future sensor systems. This paper focuses on the four dimensional variational (4D-Var) data assimilation framework. Metrics from information theory are used to quantify the contribution of observations to decreasing the uncertainty with which the system state is known. We establish an interesting relationship between different information-theoretic metrics and the variational cost function/gradient under Gaussian linear assumptions. Based on this insight we derive an ensemble-based computational procedure to estimate the information content of various observations in the context of 4D-Var. The approach is illustrated on linear and nonlinear test problems. In the companion paper [Singh et al.(2011)] the methodology is applied to a global chemical data assimilation problem