Pavel Sakov

A deterministic formulation of the ensemble Kalman filter: an alternative to ensemble square root filters

The 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.

Implications of the Form of the Ensemble Transformation in the Ensemble Square Root Filters

Abstract 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-preservi...

An Iterative EnKF for Strongly Nonlinear Systems

AbstractThe 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...

An iterative ensemble Kalman filter in the presence of additive model error

The iterative ensemble Kalman filter (IEnKF) in a deterministic framework was introduced in Sakov et al. (2012) to extend the ensemble Kalman filter (EnKF) and improve its performance in mildly up to strongly nonlinear cases. However, the IEnKF assumes that the model is perfect. This assumption simplified the update of the system at a time different from the observation time, which made it natural to apply the IEnKF for smoothing. In this study, we generalise the IEnKF to the case of imperfect model with additive model error. The new method called IEnKF-Q conducts a Gauss-Newton minimisation in ensemble space. It combines the propagated analysed ensemble anomalies from the previous cycle and model noise ensemble anomalies into a single ensemble of anomalies, and by doing so takes an algebraic form similar to that of the IEnKF. The performance of the IEnKF-Q is tested in a number of experiments with the Lorenz-96 model, which show that the method consistently outperforms both the EnKF and the IEnKF naively modified to accommodate additive model noise.

Ocean forecasting of mesoscale features can deteriorate by increasing model resolution towards the submesoscale

Submesoscale dynamics are ubiquitous in the ocean and important in the variability of physical, biological and chemical processes. Submesoscale resolving ocean models have been shown to improve representation of observed variability. We show through data assimilation experiments that a higher-resolution submesoscale permitting system does not match the skill of a lower resolution eddy resolving system in forecasting the mesoscale circulation. Predictability of the submesoscale is inherently lower and there is an inverse cascade in the kinetic energy spectrum that lowers the predictability of the mesoscale. A benefit of the higher-resolution system is the ability to include information content from observations to produce an analysis that can at times compare more favourably with remotely sensed satellite imagery. The implication of this work is that in practice, higher-resolution systems will provide analyses with enhanced spatial detail but will be less skilful at predicting the evolution of the mesoscale features.The degree to which increasing the resolution of ocean models to consider submesoscale dynamics will improve prediction of mesoscale features remains uncertain. Here, via data assimilation experiments, the authors show higher resolution models do not necessarily provide improved dynamical solutions.

Asynchronous data assimilation with the EnKF in presence of additive model error

Abstract The term ‘asynchronous data assimilation’ (ADA) refers to modifications of sequential data assimilation methods that take into consideration the observation time. In Sakov et al. [Tellus A, 62, 24–29 (2010)], a simple rule has been formulated for the ADA with the ensemble Kalman filter (EnKF). To assimilate scattered in time observations, one needs to calculate ensemble forecast observations using the forecast ensemble at observation time. Using then these ensemble observations in the EnKF update matches the optimal analysis in the linear perfect model case. In this note, we generalise this rule for the case of additive model error.

An adaptive quality control procedure for data assimilation

We describe a simple adaptive quality control procedure that limits the impact of individual observations likely to be inconsistent with the state of the data assimilation system. It smoothly increases the observation error variance depending on the projected increment, state error variance and so-called K-factor so that the resulting increment does not exceed the estimated state error times K. Because an estimate of the state error is readily available in the Kalman filter (KF), the method is particularly suitable for the KF, ensemble Kalman filter (EnKF), or ensemble optimal interpolation systems. The tests show that setting K to about 1.5–2 or above has no detrimental effect for performance of nearly optimal systems; at the same time it still makes it possible to make use of observations that might otherwise be discarded by the background check. The technique is successfully used in the EnKF codes TOPAZ and EnKF-C.

Validation of ice‐ocean models using acoustic measurements.

The Fram Strait is the main passage through which the ocean mass and heat exchange between the Atlantic and Arctic Oceans take place. On the eastern side of the strait the northbound West Spitzbergen Current transports Atlantic water to the Arctic Ocean, whereas on the western side the southbound East Greenland Current transports sea ice and polar water from the Arctic Ocean to the Nordic Seas and the Atlantic Ocean. The currents in the Fram Strait are characterized by significant recirculation and numerous propagating mesoscale eddies. Under the EU project DAMOCLES single‐track acoustic travel time measurements were carried out for a year. Acoustic travel time pattern is modeled by feeding oceanographic fields from ice‐ocean models into acoustic propagation models. By comparing modeled travel times to accurate acoustic observations we can validate the ocean models. Two different ice‐ocean models are considered: a single member high‐resolution model and a multi member ice‐ocean model system at coarser res...