Implementation of a reduced rank square-root smoother for high resolution ocean data assimilation

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Abstract

Optimal smoothers enable the use of future observations to estimate the state of a dynamical system. In this paper, a square-root smoother algorithm is presented, extended from the Singular Evolutive Extended Kalman (SEEK) filter, a square-root Kalman filter routinely used for ocean data assimilation. With this filter algorithm, the smoother extension appears almost cost-free. A modified algorithm implementing a particular parameterization of model error is also described. The smoother is applied with an ocean circulation model in a double-gyre, 1/4° configuration, able to represent mid-latitude mesoscale dynamics. Twin experiments are performed: the true fields are drawn from a simulation at a 1/6° resolution, and noised. Then, altimetric satellite tracks and sparse vertical profiles of temperature are extracted to form the observations.

The smoother is efficient in reducing errors, particularly in the regions poorly covered by the observations at the filter analysis time. It results in a significant reduction of the global error: the Root Mean Square Error in Sea Surface Height from the filter is further reduced by 20% by the smoother. The actual smoothing of the global error through time is also verified. Three essential issues are then investigated: (i) the time distance within which observations may be favourably used to correct the state estimates is found to be 8 days with our system. (ii) The impact of the model error parameterization is stressed. When this parameterization is spuriously neglected, the smoother can deteriorate the state estimates. (iii) Iterations of the smoother over a fixed time interval are tested. Although this procedure improves the state estimates over the assimilation window, it also makes the subsequent forecast worse than the filter in our experiment.

Introduction

The development of data assimilation in geophysics has been triggered by the need of an accurate representation of the current atmospheric state to initialize a numerical weather forecast. In the framework of estimation theory, this issue is a filtering problem of estimating a dynamical state given a model (numerical), past and present observations (the only available data for a forecast). The spearhead of dynamical estimation methods is the Kalman filter (Kalman, 1960). The Kalman filter has received much interest in geophysics. This is due to its solid roots in estimation theory on the one hand, and to its possible implementation – provided some simplifications are made – with large numerical models on the other hand [e.g. Parrish and Cohn, 1985, Todling and Cohn, 1994, Evensen, 1994, Fukumori and Malanotte-Rizzoli, 1995, Houtekamer and Mitchell, 1998]. However, many oceanographic applications now expect more from data assimilation than only initialize a prediction. The study of climate variability and evolution, the targeted case studies of ocean dynamics, or biogeochemistry, require reanalyses of the ocean circulation: complete, consistent, and accurate datasets of the variables that describe the ocean, over a continuous time period in the past. The estimation of past ocean states implies that posterior observations exist (up to present), and may clearly be used retrospectively to improve the data assimilation outputs. Such estimation problem is tackled by smoothers.

Optimal linear smoothers stemming from estimation theory can be considered as an extension of the Kalman filter that takes future observations into account. Actually, all optimal linear smoother algorithms involve the Kalman filter. For a detailed description of the various types of smoothers based on Kalman’s theory, and their algorithms, we refer the reader to textbooks such as Anderson and Moore (1979) or Simon (2006). In this paper, we are concerned with the sequential approach of smoothing, as exposed by Cohn et al. (1994) or Evensen and van Leeuwen (2000): the Kalman filter analysis is followed by retrospective analyses, i.e. corrections of the past state estimates using the Kalman filter innovation.

Smoother algorithms involve the Kalman filter, however, it is well known that the Kalman filter cannot be implemented in its canonical form with realistic models of the ocean. First, models are generally nonlinear. Linearity is an essential hypothesis for the Kalman filter to be optimal. Various strategies can be adopted to extend the Kalman filter to nonlinear systems. The Extended Kalman filter is the straightforward, first order generalization of the Kalman filter. Higher order approaches exist, such as the unscented Kalman filter (Julier and Uhlmann, 1997). Evensen (1994) proposed an ensemble approach to nonlinear Kalman filtering. The second hindrance to the application of the standard Kalman filter is due to computer storage and CPU requirements. The Kalman filter is computationally untractable, and approximations have to be made for implementation. The most common strategy in oceanography consists in reducing the dimension of the error space. The Ensemble Kalman Filter (EnKF) of Evensen (1994) does it implicitly, by handling an ensemble of states far smaller than the state vector dimension. The Reduced-Rank SQuare-RooT filter (Verlaan and Heemink, 1997), the Error Subspace Statistical Estimation (ESSE) algorithm (Lermusiaux and Robinson, 1999), and the Singular Evolutive Extended Kalman (SEEK) filter (Pham et al., 1998, Brasseur and Verron, 2006) are explicitly founded on the order reduction.

Some algorithms approximating the Kalman filter have been extended for smoothing and applied in problems connected to oceanography. It is the case of the EnKF (van Leeuwen and Evensen, 1996, Evensen and van Leeuwen, 2000), with applications with a 2-layer quasigeostrophic model (van Leeuwen, 1999, van Leeuwen, 2001). Lermusiaux and Robinson (1999) have derived a ESSE smoother. This algorithm has been run in real data assimilation experiments at high resolution by Lermusiaux, 1999a, Lermusiaux, 1999b, and Lermusiaux et al. (2002). Fukumori (2002) have tested a Kalman filter and an approximation of the RTS smoother (Rauch et al., 1965), based on a state-partitioning approach, with a one-dimensional shallow-water model. Todling and Cohn (1996) and Todling et al. (1998) have introduced various strategies to make the Kalman filter and optimal smoothers applicable with large systems and tested some of them with a linear shallow-water model. Gaspar and Wunsch (1989) have experimented the RTS smoother with an over-simplified equation of ocean dynamics. Recently, Ravela and McLaughlin (2007) have illustrated fast ensemble smoothing algorithms in identical twin experiments with the Lorenz-95 100-variable model.

The work presented in this paper basically aims at pioneering the implementation of a smoother algorithm, based on the SEEK filter algorithm, in a high resolution ocean data assimilation system imbedding a primitive equations ocean circulation model. The mid-term perspective is to make high quality, high resolution reanalyses of the ocean circulation. In this prospect, the chosen smoothing algorithm is sequential, in the spirit of Evensen and van Leeuwen (2000) or Cohn et al. (1994). The first part of the paper is dedicated to the theoretical derivation and the algorithmic aspects of the SEEK extension to smoothing, including a method to handle a common parameterization of the model error. It is shown that the implementation of this sequential smoother is straightforward, and results in negligible additional cost, when the SEEK filter is already in place. In the second part of the paper, the application of the SEEK smoother with a mesoscale ocean circulation model is demonstrated and evaluated. Three issues expected to be tricky or relevant in a real data assimilation context are examined.

Section 2 recapitulates the Kalman filter and the sequential smoothing ingredients and algorithm. Section 3 presents the SEEK filter and the SEEK smoother algorithm. The SEEK filter is well known (Pham et al., 1998, Rozier et al., 2007). In Section 3.1, we stress the aspects particularly relevant or problematical for the smoother extension. In Section 3.2, the equations of the new SEEK smoother algorithm are written down. For clarity’s sake, the perfect model and imperfect model cases are considered separately. Two practical aspects are examined in Section 4: the computational complexity and the numerical implementation. In Section 5, we present the set-up of a twin experiment carried out with the SEEK smoother and an ocean circulation model in a high resolution, idealized configuration. Results are examined in Section 6, and Section 7 reports three examinations and numerical experiments concerning key issues with the SEEK smoother. Section 8 concludes.

Section snippets

Optimal linear smoothers

The Kalman filter and the most common optimal linear smoothers are derived and described in many textbooks (Anderson and Moore, 1979, Simon, 2006, Evensen, 2007). Here we first recall the well-known Kalman filter algorithm to set down the notations that will be used next. Then we provide an intuitive view and the equations of the less-known optimal smoothers. Note that we are interested in the smoothers of the sequential type here.

Square-root transformation and order reduction

The square-root transformation and the order reduction are introduced in the smoother equations. This is done here with the Singular Evolutive Extended Kalman (SEEK) filter, but adaptation to any other square-root filter is similar [e.g. Ravela and McLaughlin, 2007].

Algorithm complexity

Considering that the expensive operations in the Kalman filter are the model integrations (Eqs. (1a), (1b)) and the inversion of the innovation error covariance (Eqs. (2a), (2c)), the Kalman filter algorithm complexity approaches:AC(KF)(n+1)N+s3,where n is the size of the state vector, N the number of operations in one model integration, and s the number of observations.

The full rank fixed-lag smoother is far more expensive than the Kalman filter, for it involves L×n extra model integrations

Twin experiment with a nonlinear, ocean circulation model: description

The smoother algorithm is implemented and applied with an ocean circulation model. In this section, the experimental set-up is described. Results are presented in the next section.

Results

Concerning the smoother, we focus on the 8-days retrospective analysis in this section. The reason will be given in Section 7.1, where the choice of the lag is discussed.

On the smoother lag

In the framework of the Kalman filter hypotheses, the largest lag theoretically provides the best error reduction and smoothing. But it generally makes sense to consider a limited number of retrospective analyses. In presence of unstable or dissipative dynamics in particular, most of the smoother improvements are due to the first few retrospective analyses, the others having a minor impact on the results (Cohn et al., 1994). High resolution ocean circulation models develop nonlinear, unstable

Conclusion

In theory, the Kalman filter is well designed to provide an initial state for a prediction. To build a re-analysis of the ocean circulation, optimal smoothers are more appropriate. In particular, the impacts of gaps in the observation network are efficiently smoothed out by smoothers. In this paper, a smoother algorithm has been derived based on a reduced rank square-root filter, the SEEK filter, and implemented with a high resolution ocean circulation model.

The theoretical derivation of the

Acknowledgements

We thank Jean-Marc Molines for sharing his skills with the NEMO model. This work was supported by CNRS/INSU through the LEFE/ASSIM program and by the ANR program. Partial support of the European Commission under Grant Agreement FP7-SPACE-2007-1-CT-218812-MYOCEAN is gratefully acknowledged. Calculations were performed using HPC resources from GENCI-IDRIS (Grant 2009-011279). The original manuscript has been well improved thanks to Pr Pierre Lermusiaux and two other anonymous referees.

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