1710 MONTHLY WEATHER REVIEW VOLUME 133

Maximum Likelihood Ensemble Filter: Theoretical Aspects

MILIJA ZUPANSKI Cooperative Institute for Research in the Atmosphere, Colorado State University, Fort Collins, Colorado

(Manuscript received 6 May 2003, in final form 29 October 2004)

ABSTRACT A new ensemble-based method, named the maximum likelihood ensemble filter (MLEF), is presented. The analysis solution maximizes the likelihood of the posterior probability distri- bution, obtained by minimization of a cost function that depends on a general nonlinear observation operator. The MLEF belongs to the class of deterministic ensemble filters, since no perturbed observations are employed. As in variational and ensemble data assimilation methods, the cost function is derived using a Gaussian probability density function framework. Like other ensemble data assimilation algorithms, the MLEF produces an estimate of the analysis uncertainty (e.g., analysis error covariance). In addition to the common use of ensembles in calculation of the forecast error covariance, the ensembles in MLEF are exploited to efficiently calculate the Hessian preconditioning and the gradient of the cost function. A sufficient number of iterative minimization steps is 2–3, because of superior Hessian preconditioning. The MLEF method is well suited for use with highly nonlinear observation operators, for a small additional computational cost of minimization. The consistent treatment of nonlinear observation operators through optimization is an advantage of the MLEF over other ensemble data assimilation algorithms. The cost of MLEF is comparable to the cost of existing ensemble algorithms. The method is directly applicable to most complex forecast models and observation operators. In this paper, the MLEF method is applied to data assimilation with the one-dimensional Korteweg–de Vries–Burgers equation. The tested observation operator is quadratic, in order to make the assimilation problem more challenging. The results illustrate the stability of the MLEF performance, as well as the benefit of the cost function minimization. The improvement is noted in terms of the rms error, as well as the analysis error covariance. The statistics of innovation vectors (observation minus forecast) also indicate a stable performance of the MLEF algo- rithm. Additional experiments suggest the amplified benefit of targeted observations in ensemble data assimilation.

1. Introduction cause of a tremendous computational burden associ- ated with high dimensionality of realistic atmospheric Since early 1960s, data assimilation in atmospheric and oceanic data assimilation problems. So far, com- and oceanographic applications is based on the Kalman mon approaches to realistic data assimilation were to filtering theory (Kalman and Bucy 1961; Jazwinski approximate (e.g., model) error covariances, as well as 1970). Beginning with optimal interpolation (Gandin to avoid the calculation of posterior (e.g., analysis) er- 1963), and continuing with three-dimensional (Parrish ror covariance. These approximations have a common and Derber 1992; Rabier et al. 1998; Cohn et al. 1998; problem of not being able to use fully cycled error co- Daley and Barker 2001) and four-dimensional varia- variance information, as the theory suggests. The con- tional data assimilation (Navon et al. 1992; Zupanski sequence is not only that the produced analysis is of 1993; Zou et al. 1995, 2001; Courtier et al. 1994; Rabier reduced quality, but also that no reliable estimates of et al. 2000; Zupanski et al. 2002), data assimilation the uncertainties of the produced analysis are available. methodologies operationally used in atmospheric and For the first time, a novel approach to data assimila- oceanic applications can be viewed as an effort to ap- tion in oceanography and meteorology pursued in re- proximate the Kalman filter/smoother theoretical cent years (Evensen 1994; Houtekamer and Mitchell framework (Cohn 1997). The approximations are nec- 1998; Pham et al. 1998; Lermusiaux and Robinson 1999; essary because of the lack of knowledge of statistical Brasseur et al. 1999; Hamill and Snyder 2000; Evensen properties of models and observations, as well as be- and van Leeuwen 2000; Keppenne 2000; Bishop et al. 2001; Anderson 2001; van Leeuwen 2001; Haugen and Evensen 2002; Reichle et al. 2002b; Whitaker and Corresponding author address: Milija Zupanski, Cooperative Institute for Research in the Atmosphere, Colorado State Uni- Hamill 2002; Anderson 2003; Ott et al. 2004), based on versity, Foothills Campus, Fort Collins, CO 80523-1375. the use of in nonlinear Kalman E-mail: [email protected] filtering, offers the to consistently estimate the

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MWR2946 JUNE 2005 ZUPANSKI 1711 analysis uncertainties. The price to pay is the reduced uncertainty (e.g., analysis error covariance). The idea dimension of the analysis subspace (defined by en- behind this development is to produce a method ca- semble forecasts); thus there is a concern of not being pable of optimally exploiting the experience gathered sufficient to adequately represent all important dy- in operational data assimilation and the advancements namical features and instabilities. Preliminary results in ensemble data assimilation, eventually producing a show, however, that this may not always be a problem qualitatively new system. The practical goal is to de- (e.g., Houtekamer and Mitchell 2001; Keppenne and velop a single data assimilation system easily applicable Rienecker 2002). On the other hand, it is anticipated to the simplest, as well as to the most complex nonlin- that the ensemble size will need to be increased as more ear models and observation operators. realistic and higher-resolution models and observations While the maximum likelihood estimate has a unique are used. This, however, may be feasible even on cur- solution for unimodal probability density functions rently available computers. With the advancement in (PDFs), there is a possibility for a nonunique solution computer technology, and multiple processing in par- in the case of multimodal PDFs. This issue will be given ticular, which is ideally suited for ensemble framework, more attention in the future. the future looks promising for continuing development The method is explained in section 2, algorithmic and realistic applications of ensemble data assimilation details are given in section 3, experimental design is methodology. presented in section 4, results in section 5, and conclu- In achieving that goal, however, there are still few sions are drawn in section 6. unresolved methodological and practical issues that will be pursued in this paper. Current ensemble data assimi- lation methodologies are broadly grouped in stochastic 2. MLEF methodology and deterministic approaches (Tippett et al. 2003). A From variational methods it is known that a maxi- common starting point to these algorithms is the use of mum likelihood estimate, adopted in MLEF, is a suit- the solution form of the (e.g., able approach in applications to realistic data assimila- Evensen 2003), obtained assuming linearized dynamics tion in meteorology and oceanography. From opera- and observation operators, with Gaussian assumption tional applications of data assimilation methods, it is regarding the measurements and control variables (e.g., also known that a Gaussian PDF assumption, used in initial conditions). We refer to this as a linearized solu- derivation of the cost function (e.g., Lorenc 1986), is tion form. Since realistic observation operators are gen- generally accepted and widely used. Although the erally nonlinear, a common approach to nonlinearity in model and observ