p-Kernel Stein Variational Gradient Descent for Data Assimilation and History Matching

A Bayesian method of inference known as “Stein variational gradient descent” was recently implemented for data assimilation problems, under the heading of “mapping particle filter”. In this manuscript, the algorithm is applied to another type of geoscientific inversion problems, namely history matching of petroleum reservoirs. In order to combat the curse of dimensionality, the commonly used Gaussian kernel, which defines the solution space, is replaced by a p-kernel. In addition, the ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons. Our experimental results in data assimilation are rather disappointing. However, the results from the subsurface inverse problem show more promise, especially as regards the use of p-kernels.publishedVersio

Improvements to ensemble methods for data assimilation in the geosciences

Data assimilation considers the problem of using a variety of data to calibrate model-based estimates of dynamic variables and static parameters. Geoscientific examples include (i) satellite observations and atmospheric models for weather forecasting, and (ii) well-log data and reservoir flow simulators for oil production optimization. Approximate solutions are provided by the set of techniques deriving from the ensemble Kalman filter (EnKF), which combines a Monte Carlo approach with assumptions of linearity and Gaussianity. This thesis proposes some improvements to the accuracy and understanding of such ensemble methods. Firstly, a new scheme is developed to account for model noise in the forecast step of the EnKF. The main aim is to eliminate the sampling errors of additive, simulated noise. The scheme is based on the previously developed "square root" schemes for the analysis step, but requires further consideration due to the limited subspace spanned by the ensemble. The properties of the square root scheme in general are surveyed. Secondly, the "finite size" ensemble Kalman filter (EnKF-N) is reviewed. The EnKF-N explicitly considers the uncertainty in the forecast moments (mean and covariance), thereby not requiring the multiplicative inflation commonly used to compensate for an intrinsic bias of the analysis step of the standard EnKF. Thus, in the perfect model setting, it avoids the process of tuning the inflation factor. This presentation consolidates the earlier literature on the EnKF-N, substantiates the scalar inflation perspective, and rectifies a deficiency. Thirdly, two ensemble "smoothers" expressed by different recursions, used in different applications, and hitherto thought to yield different results, are shown to be equivalent. The theory is revisited under practical considerations, where equivalence is broken due to inflation and localization, but the methods remain equally capable. In each case, the theory is tested and the accuracy performance is benchmarked against standard methods using numerical twin experiments.</p

Improvements to ensemble methods for data assimilation in the geosciences

Data assimilation considers the problem of using a variety of data to calibrate model-based estimates of dynamic variables and static parameters. Geoscientific examples include (i) satellite observations and atmospheric models for weather forecasting, and (ii) well-log data and reservoir flow simulators for oil production optimization. Approximate solutions are provided by the set of techniques deriving from the ensemble Kalman filter (EnKF), which combines a Monte Carlo approach with assumptions of linearity and Gaussianity. This thesis proposes some improvements to the accuracy and understanding of such ensemble methods. Firstly, a new scheme is developed to account for model noise in the forecast step of the EnKF. The main aim is to eliminate the sampling errors of additive, simulated noise. The scheme is based on the previously developed "square root" schemes for the analysis step, but requires further consideration due to the limited subspace spanned by the ensemble. The properties of the square root scheme in general are surveyed. Secondly, the "finite size" ensemble Kalman filter (EnKF-N) is reviewed. The EnKF-N explicitly considers the uncertainty in the forecast moments (mean and covariance), thereby not requiring the multiplicative inflation commonly used to compensate for an intrinsic bias of the analysis step of the standard EnKF. Thus, in the perfect model setting, it avoids the process of tuning the inflation factor. This presentation consolidates the earlier literature on the EnKF-N, substantiates the scalar inflation perspective, and rectifies a deficiency. Thirdly, two ensemble "smoothers" expressed by different recursions, used in different applications, and hitherto thought to yield different results, are shown to be equivalent. The theory is revisited under practical considerations, where equivalence is broken due to inflation and localization, but the methods remain equally capable. In each case, the theory is tested and the accuracy performance is benchmarked against standard methods using numerical twin experiments.</p

p-Kernel Stein Variational Gradient Descent for Data Assimilation and History Matching

A Bayesian method of inference known as “Stein variational gradient descent” was recently implemented for data assimilation problems, under the heading of “mapping particle filter”. In this manuscript, the algorithm is applied to another type of geoscientific inversion problems, namely history matching of petroleum reservoirs. In order to combat the curse of dimensionality, the commonly used Gaussian kernel, which defines the solution space, is replaced by a p-kernel. In addition, the ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons. Our experimental results in data assimilation are rather disappointing. However, the results from the subsurface inverse problem show more promise, especially as regards the use of p-kernels

p-Kernel Stein Variational Gradient Descent for Data Assimilation and History Matching

A Bayesian method of inference known as “Stein variational gradient descent” was recently implemented for data assimilation problems, under the heading of “mapping particle filter”. In this manuscript, the algorithm is applied to another type of geoscientific inversion problems, namely history matching of petroleum reservoirs. In order to combat the curse of dimensionality, the commonly used Gaussian kernel, which defines the solution space, is replaced by a p-kernel. In addition, the ensemble gradient approximation used in the mapping particle filter is rectified, and the data assimilation experiments are re-run with more relevant settings and comparisons. Our experimental results in data assimilation are rather disappointing. However, the results from the subsurface inverse problem show more promise, especially as regards the use of p-kernels

Adaptive covariance inflation in the ensemble Kalman filter by Gaussian scale mixtures

This paper studies multiplicative inflation: the complementary scaling of the state covariance in the ensemble Kalman filter (EnKF). Firstly, error sources in the EnKF are catalogued and discussed in relation to inflation; nonlinearity is given particular attention as a source of sampling error. In response, the “finite‐size” refinement known as the EnKF‐N is re‐derived via a Gaussian scale mixture, again demonstrating how it yields adaptive inflation. Existing methods for adaptive inflation estimation are reviewed, and several insights are gained from a comparative analysis. One such adaptive inflation method is selected to complement the EnKF‐N to make a hybrid that is suitable for contexts where model error is present and imperfectly parametrized. Benchmarks are obtained from experiments with the two‐scale Lorenz model and its slow‐scale truncation. The proposed hybrid EnKF‐N method of adaptive inflation is found to yield systematic accuracy improvements in comparison with the existing methods, albeit to a moderate degree