An efficient retrospective optimal interpolation algorithm compared with the fixed-lag Kalman smoother by assuming a perfect model

We developed an efficient retrospective optimal interpolation (ROI) algorithm by which we can avoid the overlap of model integration periods, which appears in the procedure of evolving a previous analysis error covariance. If the fixed-lag Kalman smoother (FLKS) is used to determine the analysis state at the beginning of the fixed analysis window as is done for ROI, the FLKS can be considered a suboptimal version of the efficient ROI when the involved model is non-linear and has no errors. We confirm that the efficient ROI analyses are more accurate than the FLKS analyses with the increase of the analysis window size. Nevertheless, the computation costs for implementing efficient ROI are almost the same as those for FLKS. Additionally, the reduced-rank version of the efficient ROI is developed based on the accuracy-saturation property.

From the experiments using Lorenz 3-variable model, it is confirmed that the non-linearity of numerical model, which becomes stronger with the increase of the analysis window size, makes the analysis of efficient ROI more accurate than that of FLKS. Our Lorenz 40-variable experiments show that the average analysis error of the efficient ROI is smaller than that of the FLKS for an analysis window size of up to 4 d. However, the efficient ROI and the FLKS requires almost the same costs for computation. From the results of Lorenz 40-variable model experiments, it is suggested that, by using the reduced-rank formulation of the efficient ROI, we can obtain the suboptimal analysis more cost-effectively rather than FLKS.