The ultimate goal of my Ph.D. was to generalise the EnKF and to use this gen-eralised scheme to develop an ensemble-based filtering method, or essentially an ensemble updating method, for high-dimensional, categorical state vectors. The reason for wanting to generalise the EnKF is that many studies show that the filter provides quite good results even in non-linear, non-Gaussian situations, so therefore it would be interesting to investigate whether some of the underlying properties of the filter that contribute to this appealing behaviour can be trans-ferred to a categorical sample space. Throughout the thesis, we focus on state vectors with a one-dimensional spatial arrangement, meaning that the vector is spatially arranged along a line. Extending the proposed methods to two, and possibly three, dimensions is an interesting area for future research. The remains of the thesis consists of four papers, all closely related. The papers can be read independently, but we recommend reading paper I before the others, especially before papers II and III. Below, we briefly summarise each paper.
Paper I
The first paper, ”Ensemble updating of binary state vectors by maximising the expected number of unchanged components”, describes our first e↵ort on
(a)
(b)
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Figure 3: EnKF simulation example: (a) True state vectorxt(black dots) and observed vector yt (red stars) at timet = 40. (b) True state vector xt (black dots) and filtering ensemble members (coloured lines) at time t = 40 obtained using the standard stochastic EnKF. (c) Corresponding output as in (b), obtained using stochastic EnKF with covariance localisation, where⇢is defined by Eq. (29) withL= 10. (d) Corresponding output as in (b) and (c), obtained using a modified stochastic EnKF where the true Kalman filter forecast covariance matrices are used in the ensemble update.
developing a new ensemble updating method for categorical state vectors. As implied by the title, the paper restricts the focus to vectors where each element is a binary variable. The paper starts with the description of a general ensem-ble updating framework based on a generalisation of the statistical properties of the EnKF. In the EnKF, a Gaussian approximation to the forecast distribution is (implicitly) constructed. Combined with the assumption of a linear-Gaussian likelihood model, a corresponding Gaussian approximation to the filtering dis-tribution can be computed according to Bayes’ rule. Given that the Gaussian approximation to the forecast distribution is correct, the EnKF linear update corresponds to conditional simulation from a Gaussian distribution such that the marginal distribution of each updated sample is equal to the Gaussian approxi-mation to the filtering distribution. More generally, one can imagine to proceed in a similar fashion, but pursue another parametric model than the Gaussian. That is, instead of assuming that the forecast model is Gaussian and that the likeli-hood model is linear-Gaussian, other models can be chosen. Moreover, instead of linearly shifting the forecast samples, the posterior samples can be obtained by conditional simulation from some distribution such that, under the assump-tion that the forecast samples are distributed according to the assumed forecast model, the marginal distribution of each updated sample is equal to the corre-sponding assumed filtering distribution. Generally, an infinite number of such conditional distributions may exist. To choose a solution, one could for example seek a solution which is optimal with respect to some chosen optimality criterion.
To update a vector of binary variables, we propose to construct a first-order Markov chain approximation to the forecast distribution and assume that the elements of the observation vector are conditionally independent. This choice of forecast and likelihood models constitutes an HMM and returns a correspond-ing tractable first-order Markov chain approximation to the filtercorrespond-ing distribution.
Based on the assumed HMM, the next task is to construct an appropriate condi-tional distribution from which the posterior samples can be simulated. Because of the discrete context, this conditional distribution is a transition matrix, not a density as in the EnKF. A simple yet na¨ıve option is to set the transition ma-trix equal to the already established first-order Markov chain approximation to the filtering distribution. However, this na¨ıve approach entails that the posterior samples are simulated independently of the forecast samples and may result in that important information about thetrue forecast and filtering models, possibly not captured with the assumed Markov chain models, is lost. To capture more
information from the forecast samples, we propose to construct an optimal tran-sition matrix with respect to the optimality criterion of maximising the expected number of components of a forecast realisation that remain unchanged. A dy-namic programming algorithm for recursively constructing the optimal solution is presented, and the proposed procedure is demonstrated in a geophysics-inspired simulation example.
Paper II
The second paper, ”Geophysics-based fluid-facies predictions based on ensem-ble updating of binary state vectors”, is much more applied than the remaining papers. Basically, the paper presents a synthetic geophysical filtering problem where the method proposed in Paper I is applied. The problem considered is a two-phase fluid flow problem originating from water injection in a petroleum reservoir. Based on noisy measurements of a geophysical property called resistiv-ity recorded at various times and at di↵erent locations in the reservoir, the goal is to monitor the oil displacement. Simulation examples with a two-dimensional reservoir model are presented. Here, to deal with the two-dimensional context, the updating of the variables associated with each column of the grid is done independently of the remaining variables.
Paper III
The third paper, ”A generalised and fully Bayesian ensemble updating frame-work”, is an extension of the work presented in Paper I. The contribution of the paper is three-fold. Firstly, the general updating framework proposed in Paper I is modified to a Bayesian context where the parameters of the assumed forecast dis-tribution are also treated as random variables. Secondly, the proposed Bayesian framework is investigated under the assumption of a linear-Gaussian model. An important result of this part of the paper is the proof that a particular EnSRF scheme is optimal with respect to the optimality criterion of minimising the ex-pected Mahalanobis distance between a prior and posterior ensemble member.
Thirdly, the framework is examined under the assumption of a binary HMM.
Simulation examples for both the linear-Gaussian model and the binary HMM are presented.
A consequence of the proposed Bayesian setup is that, prior to the updating of each forecast ensemble member, a corresponding parameter vector needs to be
simulated, and this simulation is to be done conditionally on both the incoming observation and all the forecast samples except the forecast sample which is to be updated. This is di↵erent from existing fully Bayesian approaches, such as the hierarchical EnKF (HEnKF) of Myrseth and Omre (2010), where the parameters are simulated conditionally on all the forecast samples (including the sample which is to be updated), but not the data. In the simulation example with the linear-Gaussian model, we observe that the exclusion of the forecast sample which is to be updated can have a major impact on the results when the ensemble size is small. In particular, compared to the traditional EnKF and the HEnKF, we observe that the proposed Bayesian EnSRF scheme provides more reliable results and gives a much better representation of the uncertainty.
Currently, although referred to as a paper in the thesis, Paper III is not really a paper, but a technical report. However, it will be submitted to a journal in the future, after some revisions.
Paper IV
The fourth paper, ”Ensemble updating of categorical state vectors”, is another extension of work presented in Paper I. The paper follows in the same Bayesian spirit as Paper III, but focuses entirely on categorical state vectors. A slightly modified version of the general Bayesian framework proposed in Paper III is pre-sented and an improved version of the updating method proposed in Paper I is described. Two important limitations about the algorithm proposed in Paper I are that it works for binary variables only and that the assumed forecast distribution is restricted to be a first-order Markov chain. In Paper IV, we address these two issues and present an improved method which is computationally feasible also for situations with more than two classes and which allows for a higher-order Markov chain as the assumed forecast distribution. While the algorithm proposed in Pa-per I is based on a certain directed acyclic graph for the dependencies between the variables of a prior and posterior ensemble member, the algorithm proposed in Paper IV instead makes use of an undirected graph. The chosen structure of this undirected graph makes it possible to efficiently construct the optimal transition matrix by solving a linear program. A simulation example where each variable of the state vector can take three di↵erent values is presented. (J. L. Anderson, 2007, 2009)
References
Anderson, J. L. (2001). An ensemble adjustment Kalman filter for data assimila-tion.Monthly Weather Review,129, 2884–2903.
Anderson, J. L. (2007). An adaptive covariance inflation error correction algorithm for ensemble filters.Tellus A: Dynamic Meteorology and Oceanography,59, 210–224.
Anderson, J. L. (2009). Spatially and temporally varying adaptive covariance in-flation for ensemble filters. Tellus A: Dynamic Meteorology and Oceanog-raphy,61, 72–83.
Anderson, J. L., & Anderson, S. L. (1999). A Monte Carlo implementation of the nonlinear filtering problem to produce ensemble assimilations and fore-casts.Monthly Weather Review,127, 2741–2758.
Bishop, C. H., Etherton, B. J., & Majumdar, S. J. (2001). Adaptive sampling with the ensemble transform Kalman filter. Part 1: Theoretical aspects.
Monthly Weather Review,129, 420–436.
Burgers, G., van Leeuwen, P. J., & Evensen, G. (1998). Analysis scheme in the ensemble Kalman filter.Monthly Weather Review,126, 1719–1724.
Carpenter, J., Cli↵ord, P., & Fearnhead, P. (1999). Improved particle filter for nonlinear problems.IEE Proceedings - Radar, Sonar and Navigation,146, 2–7.
Chopin, N., & Papaspiliopoulos, O. (2020).An introduction to sequential Monte Carlo. Springer-Verlag, Cham.
Doucet, A., de Freitas, N., & Gordon, N. (2001).Sequential Monte Carlo methods in practice. Springer-Verlag, New York.
Doucet, A., Godsill, S., & Andrieu, C. (2000). On sequential Monte Carlo sampling methods for Bayesian filtering. Statistics and Computing,10, 197–208.
Evensen, G. (2003). The ensemble Kalman filter: Theoretical formulation and practical implementation.Ocean Dynamics,53, 343–367.
Evensen, G. (2004). Sampling strategies and square root analysis schemes for the EnKF.Ocean Dynamics,54, 539–560.
Gaspari, G., & Cohn, S. E. (1999). Construction of correlation functions in two or three dimensions. Quarterly Journal of the Royal Meteorological Society, 125, 723–757.
Geweke, J. (1989). Bayesian inference in econometric models using Monte Carlo integration.Econometrica,57, 1317–1339.
Gordon, N. J., Salmond, D. J., & Smith, A. F. M. (1993). Novel approach to non-linear/non-Gaussian Bayesian state estimation. IEE-Proceedings-F, 140, 107–113.
Hamill, T. M., & Whitaker, J. S. (2001). Distance-dependent filtering of back-ground error covariance estimates in an ensemble Kalman filter.Monthly Weather Review,129, 2776–2790.
Houtekammer, P. L., & Mitchell, H. L. (2001). A sequential ensemble Kalman filter for atmospheric data assimilation. Monthly Weather Review, 129, 123–137.
Hunt, B. R., Kostelich, E. J., & Szunyogh, I. (2007). Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter.Physica D: Nonlinear Phenomena,230, 112–126.
Janjic, T., Nerger, L., Albertella, A., Schr¨oter, J., & Skachko, S. (2011). On domain localization in ensemble-based Kalman filter algorithms.Monthly Weather Review,139, 2046–2060.
Kalman, R. E. (1960). A new approach to linear filtering and prediction problems.
Transactions of the ASME - Journal of Basic Engineering,82, 35–45.
Kong, A., Liu, J. S., & Wong, W. H. (1994). Sequential imputation and Bayesian missing data problems. Journal of the American Statistical Association, 89, 278–288.
K¨unsch, H. R. (2000). State space and hidden Markov models. In O. E. Barndor↵-Nielsen, D. R. Cox, & C. Kl¨uppelberg (Eds.),Complex stochastic systems.
Chapman and Hall/CRC, Chap. 3, p. 109-174.
Lui, J. S., & Chen, R. (1998). Sequential Monte-Carlo methods for dynamical systems.Journal of the American Statistical Association,90, 567–576.
Myrseth, I., & Omre, H. (2010). Hierarchical ensemble Kalman filter.SPE Jour-nal,15, 569–580.
Ott, E., Hunt, B. R., Szunyogh, I., Zimin, A. V., Kostelich, E. J., Corazza, M., Kalnay, E., Patil, D. J., & Yorke, J. A. (2004). A local ensemble Kalman filter for atmospheric data assimilation. Tellus A: Dynamic Meteorology and Oceanography,56, 415–428.
Sakov, P., & Bertino, L. (2011). Relation between two common localisation meth-ods for the EnKF.Computational Geosciences,15, 225–237.
Snyder, C., Bengtsson, T., Bickel, P., & Anderson, J. (2008). Obstacles to high-dimensional particle filtering.Monthly Weather Review,136, 4629–4640.
van Leeuwen, P. J., K¨unsch, H. R., Nerger, L., Potthast, R., & Reich, S. (2019).
Particle filters for high-dimensional geoscience applications: A review.
Quarterly Journal of the Royal Meteorological Society,145, 2335–2365.
Wang, X., & Bishop, C. H. (2003). A comparison of breeding and ensemble trans-form Kalman filter ensemble forecast schemes.Journal of the Atmospheric Sciences,60, 1140–1158.
Whitaker, J. S., & Hamill, T. M. (2002). Ensemble data assimilation without perturbed observations. Monthly Weather Review,130, 1913–1924.