Covariance regularization in inverse space

Ueno, G.; Tsuchiya, Takashi

doi:10.1002/qj.445

Cited by 22 publications

(28 citation statements)

References 62 publications

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“…Some interesting general remarks may be found in (Breiman, 2001) (Burnham & Anderson, 2002) (Burrill, 1900) (Cantu-Paz, 2004 (Claeskens & Hjort, 2008) (Hand, 2006) (Ueno & Tsuchiya, 2009). …”

Section: Appendix a Linear Regressionmentioning

confidence: 87%

Bridging the Gap Between Deterministic and Probabilistic Uncertainty Quantification Using Advanced Proxy Based Methods

Goodwin

2015

SPE Reservoir Simulation Symposium

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The use of proxy models for optimisation of expensive functions has demonstrated its value since the 1990's in many industries. Within reservoir engineering, similar techniques have been used for over a decade for history matching, both in commercial tools and in-house software.In addition to efficient history matching, proxy models have a distinct advantage when performing uncertainty quantification of probabilistic forecasts. Markov Chain Monte Carlo (MCMC) methods cannot realistically be applied directly with reservoir simulations, and even fast proxy models can fail dramatically to adequately represent the range of uncertainty if implemented without due care.A pitfall of the use of proxy models is that they are considered 'black box' and their quality is difficult to measure. Engineers prefer to deal with deterministic simulation models which they can evaluate and understand.The main pitfall of simple random walk MCMC techniques, which have begun to appear within reservoir engineering workflows, is a focus on theoretical properties which are not observed in practical implementations. This gives rise to potential gross errors, which are not generally appreciated by practitioners. Advances in recent years within the field of Bayesian statistics have significantly improved this situation, but have not yet been disseminated within the oil and gas industry. This paper describes the limitations of random walk MCMC techniques which are currently used for reservoir prediction studies, and shows how Hamiltonian MCMC techniques, together with an efficient implementation of proxy models, can lead to a more reliable and validated probabilistic uncertainty quantification, whilst also generating a suitable ensemble of deterministic reservoir models. Scientific comparison studies are performed for both an analytical case and a realistic reservoir simulation case to demonstrate the validity of the approach.The benefit of this methodology is to allow asset teams to effectively manage reservoir decisions using a robust and validated understanding of uncertainty. It lays the scientific foundations for the next generation of uncertainty tools and workflows.

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Section: Appendix a Linear Regressionmentioning

confidence: 87%

Bridging the Gap Between Deterministic and Probabilistic Uncertainty Quantification Using Advanced Proxy Based Methods

Goodwin

2015

SPE Reservoir Simulation Symposium

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“…To be concrete, R t is assumed to be either a scalar multiple of a fixed matrix or diagonal ( R t = α t Σ or R t =diag r t ). As the fixed matrix Σ , we choose a regularized sample covariance matrix of detrended SSH observations (Ueno and Tsuchiya, ). Estimation of an R t that has no specific structure is addressed in section 7.…”

Section: Application Part I: Rt = αT σ and Rt = Diag Rtmentioning

confidence: 99%

Bayesian estimation of the observation‐error covariance matrix in ensemble‐based filters

Ueno

Nakamura

2016

Quart J Royal Meteoro Soc

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We develop a Bayesian technique for estimating the parameters in the observation-noise covariance matrix R t for ensemble data assimilation. We design a posterior distribution by using the ensemble-approximated likelihood and a Wishart prior distribution and present an iterative algorithm for parameter estimation. The temporal smoothness of R t can be controlled by an adequate choice of two parameters of the prior distribution, the covariance matrix S and the number of degrees of freedom ν. The ν parameter can be estimated by maximizing the marginal likelihood. The present formalism can handle cases in which the number of data points or data positions varies with time, the former of which is exemplified in the experiments. We present an application to a coupled atmosphere-ocean model under each of the following assumptions: R t is a scalar multiple of a fixed matrix (R t = α t , where α t is the scalar parameter and is the fixed matrix), R t is diagonal, R t has fixed eigenvectors or R t has no specific structure. We verify that the proposed algorithm works well and that only a limited number of iterations are necessary. When R t has one of the structures mentioned above, by assuming S to be the previous estimate we obtain a Bayesian estimate of R t that varies smoothly in time compared with the maximum-likelihood estimate. When R t has no specific structure, we need to regularize S to maintain the positive-definiteness. Through twin experiments, we find that the best estimate of R t is, in general, obtained by a combination of structure-free R t and tapered S using decorrelation lengths of half the size of the model ocean basin. From experiments using real observations, we find that the estimates of the structured R t lead to overfitting of the data compared with the structure-free R t .

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“…If we assumed R = αS rather than R = α , the singularity of S prevents us from evaluating the log-likelihood function (18) because its determinant |R t | becomes zero and the inverse R −1 t does not exist. We convert S to a regularized matrix using a Gaussian graphical model of 12 neighbours (Ueno and Tsuchiya, 2009). The obtained covariance has identical values for the diagonal elements (variances) and for the nondiagonal elements for variables inside the 12 neighbours.…”

Section: Observation Noisementioning

confidence: 99%

Maximum likelihood estimation of error covariances in ensemble‐based filters and its application to a coupled atmosphere–ocean model

Ueno¹,

Higuchi²,

Kagimoto³

et al. 2010

Quart J Royal Meteoro Soc

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We propose a method for estimating optimal error covariances in the context of sequential assimilation, including the case where both the system equation and the observation equation are nonlinear. When the system equation is nonlinear, ensemble-based filtering methods such as the ensemble Kalman filter (EnKF) are widely used to deal directly with the nonlinearity. The present approach for covariance optimization is a maximum likelihood estimation carried out by approximating the likelihood with the ensemble mean. Specifically, the likelihood is approximated as the sample mean of the likelihood of each member of the ensemble. To evaluate the sampling error of the proposed ensemble-approximated likelihood, we construct a method for examining the statistical significance using the bootstrap method without extra ensemble computation. We apply the proposed methods to an EnKF experiment where TOPEX/POSEIDON altimetry observations are assimilated into an intermediate coupled model, which is nonlinear, and estimate the optimal parameters that specify the covariances of the system noise and observation noise. Using these optimal covariance parameters, we examine the estimates by the EnKF and the ensemble Kalman smoother (EnKS). The effect of smoothing decreases by 1/e approximately one year after the filtering step. One of the properties of the smoothed estimate is that westerly wind anomalies over the western Pacific are not reproduced around the period of an El Niño event, while those over the central Pacific are strengthened. From additional experiments, we find that (1) the westerly winds in the western Pacific are phenomena outside of the coupled model and are not necessary to model El Niño, (2) the model El Niño is maintained by the westerlies over the central Pacific, and (3) the modelled evolution process of the sea-surface temperature (SST) requires improvement to reproduce the westerly winds over the western Pacific.

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Covariance regularization in inverse space

Cited by 22 publications

References 62 publications

Bridging the Gap Between Deterministic and Probabilistic Uncertainty Quantification Using Advanced Proxy Based Methods

Bridging the Gap Between Deterministic and Probabilistic Uncertainty Quantification Using Advanced Proxy Based Methods

Bayesian estimation of the observation‐error covariance matrix in ensemble‐based filters

Maximum likelihood estimation of error covariances in ensemble‐based filters and its application to a coupled atmosphere–ocean model

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