Abstract. We present a detailed description of TOPAZ4, the latest version of TOPAZ -a coupled ocean-sea ice data assimilation system for the North Atlantic Ocean and Arctic. It is the only operational, large-scale ocean data assimilation system that uses the ensemble Kalman filter. This means that TOPAZ features a time-evolving, state-dependent estimate of the state error covariance. Based on results from the pilot MyOcean reanalysis for 2003-2008, we demonstrate that TOPAZ4 produces a realistic estimate of the ocean circulation in the North Atlantic and the sea-ice variability in the Arctic. We find that the ensemble spread for temperature and sea-level remains fairly constant throughout the reanalysis demonstrating that the data assimilation system is robust to ensemble collapse. Moreover, the ensemble spread for ice concentration is well correlated with the actual errors. This indicates that the ensemble statistics provide reliable state-dependent error estimates -a feature that is unique to ensemble-based data assimilation systems. We demonstrate that the quality of the reanalysis changes when different sea surface temperature products are assimilated, or when in-situ profiles below the ice in the Arctic Ocean are assimilated. We find that data assimilation improves the match to independent observations compared to a free model. Improvements are particularly noticeable for ice thickness, salinity in the Arctic, and temperature in the Fram Strait, but not for transport estimates or underwater temperature. At the same time, the pilot reanalysis has revealed several flaws in the system that have degraded its performance. Finally, we show that a simple bias estimation scheme can effectively detect the seasonal or constant bias in temperature and sea-level.

This study investigates the relation between two common localisation methods in ensemble Kalman filter (EnKF) systems: covariance localisation and local analysis. Both methods are popular in large-scale applications with the EnKF. The case of local observations with non-correlated errors is considered. Both methods are formulated in terms of tapering of ensemble anomalies, which provides a framework for their comparison. Based on analytical considerations and experimental evidence, we conclude that in practice the two methods should yield very similar results, so that the choice between them should be based on other criteria, such as numerical effectiveness and scalability.

A B S T R A C TThe use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple modification to the traditional EnKF that results in matching the analysed error covariance given by Kalman filter in cases when the correction is small; without perturbed observations. The proposed filter is based on the recognition that in the case of small corrections to the forecast the traditional EnKF without perturbed observations reduces the forecast error covariance by an amount that is nearly twice as large as that is needed to match Kalman filter. The analysis scheme works as follows: update the ensemble mean and the ensemble anomalies separately; update the mean using the standard analysis equation; update the anomalies with the same equation but half the Kalman gain. The proposed filter is shown to be a linear approximation to the ensemble square root filter (ESRF). Because of its deterministic character and its similarity to the traditional EnKF we call it the 'deterministic EnKF', or the DEnKF. A number of numerical experiments to compare the performance of the DEnKF with both the EnKF and an ESRF using three small models are conducted. We show that the DEnKF performs almost as well as the ESRF and is a significant improvement over the EnKF. Therefore, the DEnKF combines the numerical effectiveness, simplicity and versatility of the EnKF with the performance of the ESRFs. Importantly, the DEnKF readily permits the use of the traditional Schur product-based localization schemes.

A B S T R A C TThe use of perturbed observations in the traditional ensemble Kalman filter (EnKF) results in a suboptimal filter behaviour, particularly for small ensembles. In this work, we propose a simple modification to the traditional EnKF that results in matching the analysed error covariance given by Kalman filter in cases when the correction is small; without perturbed observations. The proposed filter is based on the recognition that in the case of small corrections to the forecast the traditional EnKF without perturbed observations reduces the forecast error covariance by an amount that is nearly twice as large as that is needed to match Kalman filter. The analysis scheme works as follows: update the ensemble mean and the ensemble anomalies separately; update the mean using the standard analysis equation; update the anomalies with the same equation but half the Kalman gain. The proposed filter is shown to be a linear approximation to the ensemble square root filter (ESRF). Because of its deterministic character and its similarity to the traditional EnKF we call it the 'deterministic EnKF', or the DEnKF. A number of numerical experiments to compare the performance of the DEnKF with both the EnKF and an ESRF using three small models are conducted. We show that the DEnKF performs almost as well as the ESRF and is a significant improvement over the EnKF. Therefore, the DEnKF combines the numerical effectiveness, simplicity and versatility of the EnKF with the performance of the ESRFs. Importantly, the DEnKF readily permits the use of the traditional Schur product-based localization schemes.

The study considers an iterative formulation of the ensemble Kalman filter (EnKF) for strongly nonlinear systems in the perfect-model framework. In the first part, a scheme is introduced that is similar to the ensemble randomized maximal likelihood (EnRML) filter by Gu and Oliver. The two new elements in the scheme are the use of the ensemble square root filter instead of the traditional (perturbed observations) EnKF and rescaling of the ensemble anomalies with the ensemble transform matrix from the previous iteration instead of estimating sensitivities between the ensemble observations and ensemble anomalies at the start of the assimilation cycle by linear regression. A simple modification turns the scheme into an ensemble formulation of the iterative extended Kalman filter. The two versions of the algorithm are referred to as the iterative EnKF (IEnKF) and the iterative extended Kalman filter (IEKF). In the second part, the performance of the IEnKF and IEKF is tested in five numerical experiments: two with the 3-element Lorenz model and three with the 40-element Lorenz model. Both the IEnKF and IEKF show a considerable advantage over the EnKF in strongly nonlinear systems when the quality or density of observations are sufficient to constrain the model to the regime of mainly linear propagation of the ensemble anomalies as well as constraining the fast-growing modes, with a much smaller advantage otherwise. The IEnKF and IEKF can potentially be used with large-scale models, and can represent a robust and scalable alternative to particle filter (PF) and hybrid PF–EnKF schemes in strongly nonlinear systems.

This paper considers implications of different forms of the ensemble transformation in the ensemble square root filters (ESRFs) for the performance of ESRF-based data assimilation systems. It highlights the importance of using mean-preserving solutions for the ensemble transform matrix (ETM). The paper shows that an arbitrary mean-preserving ETM can be represented as a product of the symmetric solution and an orthonormal mean-preserving matrix. The paper also introduces a new flavor of ESRF, referred to as ESRF with mean-preserving random rotations. To investigate the performance of different solutions for the ETM in ESRFs, experiments with two small models are conducted. In these experiments, the performances of two mean-preserving solutions, two non-mean-preserving solutions, and a traditional ensemble Kalman filter with perturbed observations are compared. The experiments show a significantly better performance of the mean-preserving solutions for the ETM in ESRFs compared to non-mean-preserving solutions. They also show that applying the mean-preserving random rotations prevents the buildup of ensemble outliers in ESRF-based data assimilation systems.

International audienceThe iterative ensemble Kalman filter (IEnKF) was recently proposed in order to improve the performance of ensemble Kalman filtering with strongly nonlinear geophysical models. The IEnKF can be used as a lag-one smoother and extended to a fixed-lag smoother: the iterative ensemble Kalman smoother (IEnKS). The IEnKS is an ensemble variational method. It does not require the use of the tangent linear of the evolution and observation models, nor the adjoint of these models: the required sensitivities (gradient and Hessian) are obtained from the ensemble. Looking for optimal performance, out of the many possible extensions we consider a quasi-static algorithm. The IEnKS is explored for the Lorenz '95 model and for a two-dimensional turbulence model. As the logical extension of the IEnKF, the IEnKS significantly outperforms standard Kalman filters and smoothers in strongly nonlinear regimes. In mildly nonlinear regimes (typically synoptic-scale meteorology), its filtering performance is marginally but clearly better than the standard ensemble Kalman filter and it keeps improving as the length of the temporal data assimilation window is increased. For long windows, its smoothing performance outranks the standard smoothers very significantly, a result that is believed to stem from the variational but flow-dependent nature of the algorithm. For very long windows, the use of a multiple data assimilation variant of the scheme, where observations are assimilated several times, is advocated. This paves the way for finer reanalysis, freed from the static prior assumption of 4D-Var but also partially freed from the Gaussian assumptions that usually impede standard ensemble Kalman filtering and smoothing

We present a detailed description of TOPAZ4, the latest version of TOPAZ – a coupled ocean-sea ice data assimilation system for the North Atlantic Ocean and Arctic. It is the only operational, large-scale ocean data assimilation system that uses the ensemble Kalman filter. This means that TOPAZ features a time-evolving, state-dependent estimate of the state error covariance. Based on results from the pilot MyOcean reanalysis for 2003–2008, we demonstrate that TOPAZ4 produces a realistic estimate of the ocean circulation and the sea ice. We find that the ensemble spread for temperature and sea-level remains fairly constant throughout the reanalysis demonstrating that the data assimilation system is robust to ensemble collapse. Moreover, the ensemble spread for ice concentration is well correlated with the actual errors. This indicates that the ensemble statistics provide reliable state-dependent error estimates – a feature that is unique to ensemble-based data assimilation systems. We demonstrate that the quality of the reanalysis changes when different sea surface temperature products are assimilated, or when in situ profiles below the ice in the Arctic Ocean are assimilated. We find that data assimilation improves the match to independent observations compared to a free model. Improvements are particularly noticeable for ice thickness, salinity in the Arctic, and temperature in the Fram Strait, but not for transport estimates or underwater temperature. At the same time, the pilot reanalysis has revealed several flaws in the system that have degraded its performance. Finally, we show that a simple bias estimation scheme can effectively detect the seasonal or constant bias in temperature and sea-level