Formulations for Estimating Spatial Variations of Analysis Error Variance to Improve Multiscale and Multistep Variational Data Assimilation

Table 1

Entire-domain averaged RMS errors (in ms⁻¹) for the analysis increments obtained from SE, TEe, TEa, TEb, and TEc applied to the first set of innovations with periodic extension and consecutively increased , where is the number of iterations. All the RMS errors are evaluated with respect to the benchmark analysis increment. The relative error (RE) of the estimated analysis error covariance for updating the background error covariance in the second step of the two-step analysis is listed with the experiment name in the first column for each two-step experiment.

Experiment				Final
SE	0.671	0.365	0.187	0.013 at
TEe with RE() = 0.229	0.171	0.150	0.142	0.135 at
TEa with RE() = 0.156	0.169	0.142	0.144	0.144 at
TEb with RE() = 0.101	0.147	0.098		0.090 at
TEc with RE() = 0.042	0.145	0.063	0.062	0.032 at