Correlation functions (CFs) associated with the inverse background-error correlations (iBECs) represented by polynomials of the diffusion operator D are obtained analytically for the binomial approximations of the Gaussian BEC and in the general case of a quadratic polynomial of D. The respective analytical expressions for one-, two- and three-dimensional cases have two tuning parameters, which provide enough freedom in adjusting the CFs' shape to experimental data. The polynomial coefficients of the corresponding iBEC operator are obtained in terms of these tuning parameters and may be useful in the design of the BEC models for variational data assimilation. Published in 2011 by John Wiley & Sons Ltd.

High-resolution (swath) altimeter missions scheduled to monitor the ocean surface in the near future have observation-error covariances (OECs) with slowly decaying off-diagonal elements. This property presents a challenge for the majority of the data assimilation algorithms which were designed under the assumption of the diagonal OECs being easily inverted. In this note, we present a method of approximating the inverse of a dense OEC by a sparse matrix represented by the polynomial of spatially inhomogeneous differential operators, whose coefficients are optimized to fit the target OEC by minimizing a quadratic cost function. Explicit expressions for the cost function gradient and the Hessian are derived. The method is tested with an OEC model generated by the SWOT simulator.

Correlation functions associated with the inverse covariances represented by polynomials of the homogeneous diffusion operator D are obtained analytically for an arbitrary polynomial of D, constrained by the positive-definiteness condition of the covariance operator. Copyright © 2012 Royal Meteorological Society

Numerical experiments with a regional configuration of the CICE model (Los‐Alamos Community sea Ice CodE) in the Beaufort Sea assimilating Special Sensor Microwave Imager/Sounder ice concentration (IC) and CryoSat ice thickness (IT) data acquired for September–December of 2015 are presented. We explore sensitivity of the 24‐hr IT/IC forecast skill to the system updates, which include introduction of the IT assimilation capability, flow‐dependent correlations, and Gaussianization of IC innovations. Experiments with IC data assimilation have shown that the flow‐dependent correlations provide 5–7% improvement of the forecast skill during the freezing period (10 September to 10 November) while Gaussianization contributes an additional improvement of 3–4% in most of the cases. In winter (11 November to 31 December) IC assimilation did not produce any statistically significant improvement of the skill due to the loss of dynamical information in the IC fields associated saturation of the ice cover. In contrast, IT assimilation provides larger improvement in December compared to October–November due to the better coverage of the Beaufort Sea by observations and their higher relative accuracy in winter. Comparison of the IT forecast fields with independent in situ observations by two upward looking sonars demonstrates statistically insignificant improvements. Much better improvement (15–25%) is observed when comparing monthly mean IT assimilation runs against independent Advanced Microwave Scanning Radiometer for Earth observing system (AMSR‐E) data. Introduction of the heuristic in situ IC/IT correlations into the background covariance model did not produce any improvements of the forecast skill.

Improving the performance of ensemble filters applied to models with many state variables requires regularization of the covariance estimates by localizing the impact of observations on state variables. A covariance localization technique based on modeling of the sample covariance with polynomial functions of the diffusion operator (DL method) is presented. Performance of the technique is compared with the nonadaptive (NAL) and adaptive (AL) ensemble localization schemes in the framework of numerical experiments with synthetic covariance matrices in a realistically inhomogeneous setting. It is shown that the DL approach is comparable in accuracy with the AL method when the ensemble size is less than 100. With larger ensembles, the accuracy of the DL approach is limited by the local homogeneity assumption underlying the technique. Computationally, the DL method is comparable with the NAL technique if the ratio of the local decorrelation scale to the grid step is not too large.