Multistep variational data assimilation: important issues and a spectral approach

In this paper, two important issues are raised for multistep variational data assimilation in which broadly distributed coarse-resolution observations are analysed in the first step, and then locally distributed high-resolution observations are analysed in the second step (and subsequent steps if any). The first one concerns how to objectively estimate or efficiently compute the analysis error covariance for the analysed field obtained in the first step and used to update the background field in the next step. To attack this issue, spectral formulations are derived for efficiently calculating the analysis error covariance functions. The calculated analysis error covariance functions are verified against their respective benchmarks for one- and two-dimensional cases and shown to be very (or fairly) good approximations for uniformly (or non-uniformly) distributed coarse-resolution observations. The second issue concerns whether and under what conditions the above calculated analysis error covariance can make the two-step analysis more accurate than the conventional single-step analysis. To answer this question, idealised numerical experiments are performed to compare the two-step analyses with their respective counterpart single-step analyses while the background error covariance is assumed to be exactly known in the first step but the number of iterations performed by the minimisation algorithm is limited (to mimic the computationally constrained situations in operational data assimilation). The results show that the two-step analysis is significantly more accurate than the single-step analysis until the iteration number becomes so large that the single-step analysis can reach the final convergence or nearly so. The two-step analysis converges much faster and thus is more efficient than the single-step analysis to reach the same accuracy. Its computational efficiency can be further enhanced by properly coarsening the grid resolution in the first step with the high-resolution grid used only over the nested domain in the second step.