Bayesian Gaussian Mixture Linear Inversion for Geophysical Inverse Problems

A Bayesian linear inversion methodology based on Gaussian mixture models and its application to geophysical inverse problems are presented in this paper. The proposed inverse method is based on a Bayesian approach under the assumptions of a Gaussian mixture random field for the prior model and a Gaussian linear likelihood function. The model for the latent discrete variable is defined to be a stationary first-order Markov chain. In this approach, a recursive exact solution to an approximation of the posterior distribution of the inverse problem is proposed. A Markov chain Monte Carlo algorithm can be used to efficiently simulate realizations from the correct posterior model. Two inversion studies based on real well log data are presented, and the main results are the posterior distributions of the reservoir properties of interest, the corresponding predictions and prediction intervals, and a set of conditional realizations. The first application is a seismic inversion study for the prediction of lithological facies, P- and S-impedance, where an improvement of 30% in the root-mean-square error of the predictions compared to the traditional Gaussian inversion is obtained. The second application is a rock physics inversion study for the prediction of lithological facies, porosity, and clay volume, where predictions slightly improve compared to the Gaussian inversion approach.     

Conditioning Simulations of Gaussian Random Fields by Ordinary Kriging

Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.     

Kriging with Inequality Constraints

A Gaussian random field with an unknown linear trend for the mean is considered. Methods for obtaining the distribution of the trend coefficients given exact data and inequality constraints are established. Moreover, the conditional distribution for the random field at any location is calculated so that predictions using e.g. the expectation, the mode, or the median can be evaluated and prediction error estimates using quantiles or variance can be obtained. Conditional simulation techniques are also provided.     

Conditioning Channel Meanders to Well Observations

Assessment of uncertainty in the performance of fluvial reservoirs often requires the ability to generate realizations of channel sands that are conditional to well observations. For channels with low sinuosity this problem has been effectively solved. When the sinuosity is large, however, the standard stochastic models for fluvial reservoirs are not valid, because the deviation of the channel from a principal direction line is multivalued. In this paper, I show how the method of randomized maximum likelihood can be used to generate conditional realizations of channels with large sinuosity. In one example, a Gaussian random field model is used to generate an unconditional realization of a channel with large sinuosity, and this realization is then conditioned to well observations. Channels generated in the second approach are less realistic, but may be sufficient for modeling reservoir connectivity in a realistic way. In the second example, an unconditional realization of a channel is generated by a complex geologic model with random forcing. It is then adjusted in a meaningful way to honor well observations. The key feature in the solution is the use of channel direction instead of channel deviation as the characteristic random function describing the geometry of the channel.     

On the Likelihood and Fluctuations of Gaussian Realizations

The likelihood of Gaussian realizations, as generated by the Cholesky simulation method, is analyzed in terms of Mahalanobis distances and fluctuations in the variogram reproduction. For random sampling, the probability to observe a Gaussian realization vector can be expressed as a function of its Mahalanobis distance, and the maximum likelihood depends only on the vector size. The Mahalanobis distances are themselves distributed as a Chi-square distribution and they can be used to describe the likelihood of Gaussian realizations. Their expected value and variance are only determined by the size of the vector of independent random normal scores used to generate the realizations. When the vector size is small, the distribution of Mahalanobis distances is highly skewed and most realizations are close to the vector mean in agreement with the multi-Gaussian density model. As the vector size increases, the realizations sample a region increasingly far out on the tail of the multi-Gaussian distribution, due to the large increase in the size of the uncertainty space largely compensating for the low probability density. For a large vector size, realizations close to the vector mean are not observed anymore. Instead, Gaussian vectors with Mahalanobis distance in the neighborhood of the expected Mahalanobis distance have the maximum probability to be observed. The distribution of Mahalanobis distances becomes Gaussian shaped and the bulk of realizations appear more equiprobable. However, the ratio of their probabilities indicates that they still remain far from being equiprobable. On the other hand, it is observed that equiprobable realizations still display important fluctuations in their variogram reproduction. The variance level that is expected in the variogram reproduction, as well as the variance of the variogram fluctuations, is dependent on the Mahalanobis distance. Realizations with smaller Mahalanobis distances are, on average, smoother than realizations with larger Mahalanobis distances. Poor ergodic conditions tend to generate higher proportions of flatter variograms relative to the variogram model. Only equiprobable realizations with a Mahalanobis distance equal to the expected Mahalanobis distance have an expected variogram matching the variogram model. For large vector sizes, Cholesky simulated Gaussian vectors cannot be used to explore uncertainty in the neighborhood of the vector mean. Instead uncertainty is explored around the n-dimensional elliptical envelop corresponding to the expected Mahalanobis distance.     

Bayesian Spatial Prediction for Discrete Closed Skew Gaussian Random Field

Most approaches in statistical spatial prediction assume that the spatial data are realizations of a Gaussian random field. However, this assumption is hard to justify for most applications. When the distribution of data is skewed but otherwise has similar properties to the normal distribution, a closed skew normal distribution can be used for modeling their skewness. Closed skew normal distribution is an extension of the multivariate skew normal distribution and has the advantage of being closed under marginalization and conditioning. In this paper, we generalize Bayesian prediction methods using closed skew normal distributions. A simulation study is performed to check the validity of the model and performance of the Bayesian spatial predictor. Finally, our prediction method is applied to Bayesian spatial prediction on the strain data near Semnan, Iran. The mean-square error of cross-validation is improved by the closed skew Gaussian model on the strain data.     

Closed Form Solutions of the Two-Dimensional Turning Bands Equation

The turning bands method generates realizations of isotropic Gaussian random fields by means of appropriately summed line processes. For two-dimensional simulations the relation between the isotropic correlation function of the random field and the correlation function to be simulated along the lines is given by an integral equation of Abel type. We present closed form solutions of this integral equation for almost all two-dimensional correlation models encountered in practice and discuss their numerical implementation. As an additional benefit, our tables and illustrations serve as a concise guide to correlation models useful in geostatistics.     

The FFT Moving Average (FFT-MA) Generator: An Efficient Numerical Method for Generating and Conditioning Gaussian Simulations

A fast Fourier transform (FFT) moving average (FFT-MA) method for generating Gaussian stochastic processes is derived. Using discrete Fourier transforms makes the calculations easy and fast so that large random fields can be produced. On the other hand, the basic moving average frame allows us to uncouple the random numbers from the structural parameters (mean, variance, correlation length, ... ), but also to draw the randomness components in spatial domain. Such features impart great flexibility to the FFT-MA generator. For instance, changing only the random numbers gives distinct realizations all having the same covariance function. Similarly, several realizations can be built from the same random number set, but from different structural parameters. Integrating the FFT-MA generator into an optimization procedure provides a tool theoretically capable to determine the random numbers identifying the Gaussian field as well as the structural parameters from dynamic data. Moreover, all or only some of the random numbers can be perturbed so that realizations produced using the FFT-MA generator can be locally updated through an optimization process.     

Non-parametric simulations-based conditional stochastic predictions of geologic heterogeneities and leakage potentials for hypothetical CO2 sequestration sites

The present study focuses on understanding the leakage potentials of the stored supercritical CO₂ plume through caprocks generated in geostatistically created heterogeneous media. For this purpose, two hypothetical cases with different geostatistical features were developed, and two conditional geostatistical simulation models (i.e., sequential indicator simulation or SISIM and generalized coupled Markov chain or GCMC) were applied for the stochastic characterizations of the heterogeneities. Then, predictive CO₂ plume migration simulations based on stochastic realizations were performed and summarized. In the geostatistical simulations, the results from the GCMC model showed better performance than those of the SISIM model for the strongly non-stationary case, while SISIM models showed reasonable performance for the weakly non-stationary case in terms of low-permeability lenses characterization. In the subsequent predictive simulations of CO₂ plume migration, the observations in the geostatistical simulations were confirmed and the GCMC-based predictions showed underestimations in CO₂ leakage in the stationary case, while the SISIM-based predictions showed considerable overestimations in the non-stationary case. The overall results suggest that: (1) proper characterization of low-permeability layering is significantly important in the prediction of CO₂ plume behavior, especially for the leakage potential of CO₂ and (2) appropriate geostatistical techniques must be selectively employed considering the degree of stationarity of the targeting fields to minimize the uncertainties in the predictions.     

A correction model for conditional bias in selective mining operations

A nonlinear correction functionK(Z*) is proposed to transform any initial linear grade estimatorZ* into a conditional unbiased estimatorZ**=K(Z*) with reduced conditional estimation variance. Such a corrected estimator allows more accurate prediction of ore reserves at any level of selection performed during the mine lifetime. The correction is based upon an analytical or isofactorial representation of a bivariate distribution model of true gradeZ and its estimatorZ*. This correction model allows derivation of conditional estimation variances for both estimatorsZ* andZ** and provides a solution to the problem of change of support. A case study is presented and performance of the proposed correction model is evaluated in terms of actual conditional bias and mean squared errors. Results obtained stress the practical importance of the correction model in selective mining operations.     

History matching of approximated lithofacies models under uncertainty

In history matching of lithofacies reservoir model, we attempt to find multiple realizations of lithofacies configuration that are conditional to dynamic data and representative of the model uncertainty space. This problem can be formalized in the Bayesian framework. Given a truncated Gaussian model as a prior and the dynamic data with its associated measurement error, we want to sample from the conditional distribution of the facies given the data. A relevant way to generate conditioned realizations is to use Markov chains Monte Carlo (MCMC). However, the dimensions of the model and the computational cost of each iteration are two important pitfalls for the use of MCMC. Furthermore, classical MCMC algorithms mix slowly, that is, they will not explore the whole support of the posterior in the time of the simulation. In this paper, we extend the methodology already described in a previous work to the problem of history matching of a Gaussian-related lithofacies reservoir model. We first show how to drastically reduce the dimension of the problem by using a truncated Karhunen-Loève expansion of the Gaussian random field underlying the lithofacies model. Moreover, we propose an innovative criterion of the choice of the number of components based on the connexity function. Then, we show how we improve the mixing properties of classical single MCMC, without increasing the global computational cost, by the use of parallel interacting Markov chains. Applying the dimension reduction and this innovative sampling method drastically lowers the number of iterations needed to sample efficiently from the posterior. We show the encouraging results obtained when applying the methodology to a synthetic history-matching case.     

Conditional estimation variances: An empirical approach

One of the tasks routinely carried out by geostatisticians is the evaluation of global mining reserves corresponding to a given cutoff grade and size of selective mining units. A long with these recovery figures, the geostatistician generally provides an assessment of the global estimation variance, which represents the precision of the overall average grade estimate, when no cutoff is applied. Such a global estimation variance is of limited interest for evaluating mining projects; what is required is the reliability of the estimate of recovered reserves or, in other words, the conditional estimation variance. Unfortunately, classical linear geostatistical methods fail to provide an easy way to estimate this variance. Through the use of simulated deposits (representing various types of regionalization)the present paper reviews and discusses the effects of changes in cutoff grade and selective mining unit size on the conditional estimation variance. It is shown that, when the cutoff grade is applied to a pointsupport (sample-size)distribution, the conditional estimation variance appears to be readily accessible by classical formulas, once the conditional semivariogram is known. However, the evaluation of the conditional estimation variance seems to be less straightforward for the general case when a cutoff is applied to the average grade distribution of selective mining units. Empirical approximation formulas for the conditional estimation variance are tentatively proposed, and their performance in the case of the simulated deposits is shown. The limitations of these approximations are discussed, and possible ways of formalizing the problem suggested.     

Application of the EnKF and Localization to Automatic History Matching of Facies Distribution and Production Data

The performance of the ensemble Kalman filter (EnKF) for continuous updating of facies location and boundaries in a reservoir model based on production and facies data for a 3D synthetic problem is presented. The occurrence of the different facies types is treated as a random process and the initial distribution was obtained by truncating a bi-Gaussian random field. Because facies data are highly non-Gaussian, re-parameterization was necessary in order to use the EnKF algorithm for data assimilation; two Gaussian random fields are updated in lieu of the static facies parameters. The problem of history matching applied to facies is difficult due to (1) constraints to facies observations at wells are occasionally violated when productions data are assimilated; (2) excessive reduction of variance seems to be a bigger problem with facies than with Gaussian random permeability and porosity fields; and (3) the relationship between facies variables and data is so highly non-linear that the final facies field does not always honor early production data well. Consequently three issues are investigated in this work. Is it possible to iteratively enforce facies constraints when updates due to production data have caused them to be violated? Can localization of adjustments be used for facies to prevent collapse of the variance during the data-assimilation period? Is a forecast from the final state better than a forecast from time zero using the final parameter fields?To investigate these issues, a 3D reservoir simulation model is coupled with the EnKF technique for data assimilation. One approach to enforcing the facies constraint is continuous iteration on all available data, which may lead to inconsistent model states, incorrect weighting of the production data and incorrect adjustment of the state vector. A sequential EnKF where the dynamic and static data are assimilated sequentially is presented and this approach seems to have solved the highlighted problems above. When the ensemble size is small compared to the number of independent data, the localized adjustment of the state vector is a very important technique that may be used to mitigate loss of rank in the ensemble. Implementing a distance-based localization of the facies adjustment appears to mitigate the problem of variance deficiency in the ensembles by ensuring that sufficient variability in the ensemble is maintained throughout the data assimilation period. Finally, when data are assimilated without localization, the prediction results appear to be independent of the starting point. When localization is applied, it is better to predict from the start using the final parameter field rather than continue from the final state.     

Gradual Deformation and Iterative Calibration of Sequential Stochastic Simulations

Based on the algorithm for gradual deformation of Gaussian stochastic models, we propose, in this paper, an extension of this method to gradually deforming realizations generated by sequential, not necessarily Gaussian, simulation. As in the Gaussian case, gradual deformation of a sequential simulation preserves spatial variability of the stochastic model and yields in general a regular objective function that can be minimized by an efficient optimization algorithm (e.g., a gradient-based algorithm). Furthermore, we discuss the local gradual deformation and the gradual deformation with respect to the structural parameters (mean, variance, and variogram range, etc.) of realizations generated by sequential simulation. Local gradual deformation may significantly improve calibration speed in the case where observations are scattered in different zones of a field. Gradual deformation with respect to structural parameters is necessary when these parameters cannot be inferred a priori and need to be determined using an inverse procedure. A synthetic example inspired from a real oil field is presented to illustrate different aspects of this approach. Results from this case study demonstrate the efficiency of the gradual deformation approach for constraining facies models generated by sequential indicator simulation. They also show the potential applicability of the proposed approach to complex real cases.     

Using non-Gaussian distributions in geostatistical simulations

Parametric geostatistical simulations such as LU decomposition and sequential algorithms do not need Gaussian distributions. It is shown that variogram model reproduction is obtained when Uniform or Dipole distributions are used instead of Gaussian distributions for drawing i. i.d. random values in LU simulation, or for modeling the local conditional probability distributions in sequential simulation. Both algorithms yield simulated values with a marginal normal distribution no matter if Gaussian, Uniform, or Dipole distributions are used. The range of simulated values decreases as the entropy of the probability distribution decreases. Using Gaussian distributions provides a larger range of simulated normal score values than using Uniform or Dipole distributions. This feature has a negligible effect for reproduction of the normal scores variogram model but have a larger impact on the reproduction of the original values variogram. The Uniform or Dipole distributions also produce lesser fluctuations among the variograms of the simulated realizations.     

Modeling soil variability as a random field

The observed variability in the spatial distribution of soil properties suggests that it is natural to describe such distribution as a random field. One of the ways to study engineering problems in such a stochastic setting is by the use of the Monte-Carlo simulation procedure. Application of this technique requires the capability to generate a large number of realizations of a given random field. A numerical procedure for the generation of such realizations in two-dimensional space is introduced as a finite difference approximation of a stochastic differential equation. The equation used was that suggested by Heine (1955). The resulting procedure is essentially similar to other autoregressive procedures used for the same purpose (Whittle, 1954; Smith and Freeze, 1979). However, contrary to these procedures, the present one is defined in terms of physically significant parameters:r ₀, the autocorrelation distance;, the discretization size; and the standard deviation, . Formulating the simulation procedure in terms of the physically significant parameters (r ₀,, ) greatly simplifies the task of generating realizations that are compatable with a given soil deposit.     

A New Data-Space Inversion Procedure for Efficient Uncertainty Quantification in Subsurface Flow Problems

Uncertainty quantification for subsurface flow problems is typically accomplished through model-based inversion procedures in which multiple posterior (history-matched) geological models are generated and used for flow predictions. These procedures can be demanding computationally, however, and it is not always straightforward to maintain geological realism in the resulting history-matched models. In some applications, it is the flow predictions themselves (and the uncertainty associated with these predictions), rather than the posterior geological models, that are of primary interest. This is the motivation for the data-space inversion (DSI) procedure developed in this paper. In the DSI approach, an ensemble of prior model realizations, honoring prior geostatistical information and hard data at wells, are generated and then (flow) simulated. The resulting production data are assembled into data vectors that represent prior ‘realizations’ in the data space. Pattern-based mapping operations and principal component analysis are applied to transform non-Gaussian data variables into lower-dimensional variables that are closer to multivariate Gaussian. The data-space inversion is posed within a Bayesian framework, and a data-space randomized maximum likelihood method is introduced to sample the conditional distribution of data variables given observed data. Extensive numerical results are presented for two example cases involving oil–water flow in a bimodal channelized system and oil–water–gas flow in a Gaussian permeability system. For both cases, DSI results for uncertainty quantification (e.g., P10, P50, P90 posterior predictions) are compared with those obtained from a strict rejection sampling (RS) procedure. Close agreement between the DSI and RS results is consistently achieved, even when the (synthetic) true data to be matched fall near the edge of the prior distribution. Computational savings using DSI are very substantial in that RS requires \(O(10^5\)–\(10^6)\) flow simulations, in contrast to 500 for DSI, for the cases considered.     

Probabilistic assessment of soil liquefaction considering spatial variability of CPT measurements

The conventional liquefaction potential assessment methods (also known as simplified methods) profoundly rely on empirical correlations based on observations from case histories. A probabilistic framework is developed to incorporate uncertainties in the earthquake ground motion prediction, the cyclic resistance prediction, and the cyclic demand prediction. The results of a probabilistic seismic hazard assessment, site response analyses, and liquefaction potential analyses are convolved to derive a relationship for the annual probability and return period of liquefaction. The random field spatial model is employed to quantify the spatial uncertainty associated with the in-situ measurements of geotechnical material.     

Numerical evaluation of multicomponent cation exchange reactive transport in physically and geochemically heterogeneous porous media

Experimental evidence and stochastic studies strongly show that the transport of reactive solutes in porous media is significantly influenced by heterogeneities in hydraulic conductivity, porosity, and sorption parameters. In this paper, we present Monte Carlo numerical simulations of multicomponent reactive transport involving competitive cation exchange reactions in a two-dimensional vertical physically and geochemically heterogeneous medium. Log hydraulic conductivity, log K, and log cation exchange capacity (log CEC) are assumed to be random Gaussian functions with spherical semivariograms. Random realizations of log K and log CEC are used as input data for the numerical simulation of multicomponent reactive transport with CORE^2D, a general purpose reactive transport code. Longitudinal features of the fronts of reactive and conservative species are computed from the temporal and spatial moments of depth-averaged concentrations. Monte Carlo simulations show that: (1) the displacement of reactive fronts increases with increasing variance of log K, while it decreases with the variance of log CEC; (2) second-order spatial moments increase with increasing variances of log K and log CEC; (3) uncertainties in the mean arrival time are largest (smallest) for negatively (positively) correlated log K and Log CEC; (4) cations undergoing competitive cation exchange exhibit different apparent velocities and retardation factors due to both physical and geochemical heterogeneities; and (5) the correlation between log K and log CEC affects significantly apparent cation retardation factors in heterogeneous aquifers.     

Bidirectional Reflectance of Gaussian Random Surfaces and Its Scaling Properties

Using rigorous probabilistic techniques we define and compute the shadowing factor, the conditional mean slope and the conditional standard deviation of the slope for a random rough surface described as an isotropic Gaussian stochastic process with Gaussian autocorrelation function. We use these quantities in order to obtain the bidirectional reflectance distribution function for incoherent light scattering by rough surfaces. The calculated quantities depend on reduced, affine invariant parameters, such that changing roughness is equivalent to rescaling the slopes of incident and emergent direction. We discuss some possible applications of these scaling properties to remote sensing.