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A multiscale data assimilation (MS-DA) scheme is formulated for fine-resolution models. A decomposition of the cost function is derived for a set of distinct spatial scales. The decomposed cost function allows for the background error covariance to be estimated separately for the distinct spatial scales, and multi-decorrelation scales to be explicitly incorporated in the background error covariance. MS-DA minimizes the partitioned cost functions sequentially from large to small scales. The multi-decorrelation length scale background error covariance enhances the spreading of sparse observations and prevents fine structures in high-resolution observations from being overly smoothed. The decomposition of the cost function also provides an avenue for mitigating the effects of scale aliasing and representativeness errors that inherently exist in a multiscale system, thus further improving the effectiveness of the assimilation of high-resolution observations. A set of one-dimensional experiments is performed to examine the properties of the MS-DA scheme. Emphasis is placed on the assimilation of patchy high-resolution observations representing radar and satellite measurements, alongside sparse observations representing those from conventional in situ platforms. The results illustrate how MS-DA improves the effectiveness of the assimilation of both these types of observations simultaneously.

An algorithm is proposed for the computation of streamfunction and velocity potential from given horizontal velocity vectors based on solving a minimization problem. To guarantee the uniqueness of the solution and computational reliability of the algorithm, a Tikhonov regularization is applied. The solution implies that the obtained streamfunction and velocity potential have minimal magnitude, while the given velocity vectors can be accurately reconstructed from the computed streamfunction and velocity potential. Because the formulation of the minimization problem allows for circumventing the explicit specification of separate boundary conditions on the streamfunction and velocity potential, the algorithm is easily applicable to irregular domains. By using an advanced minimization algorithm with the use of adjoint techniques, the method is computationally efficient and suitable for problems with large dimensions. An example is presented for coastal oceans to illustrate the practical application of the algorithm.

A set of four-dimensional variational data assimilation (4D-Var) experiments were conducted using both a standard method and an incremental method in an identical twin framework. The full physics adjoint model of the Florida State University global spectral model (FSUGSM) was used in the standard 4D-Var, while the adjoint of only a few selected physical parameterizations was used in the incremental method. The impact of physical processes on 4D-Var was examined in detail by comparing the results of these experiments. The inclusion of full physics turned out to be significantly beneficial in terms of assimilation error to the lower troposphere during the entire minimization process. The beneficial impact was found to be primarily related to boundary layer physics. The precipitation physics in the adjoint model also tended to have a beneficial impact after an intermediate number (50) of minimization iterations. Experiment results confirmed that the forecast from assimilation analyses with the full physics adjoint model displays a shorter precipitation spinup period. The beneficial impact on precipitation spinup did not result solely from the inclusion of the precipitation physics in the adjoint model, but rather from the combined impact of several physical processes. The inclusion of full physics in the adjoint model exhibited a detrimental impact on the rate of convergence at an early stage of the minimization process, but did not affect the final convergence. A truncated Newton-like incremental approach was introduced for examining the possibility of circumventing the detrimental aspects using the full physics in the adjoint model in 4D-Var but taking into account its positive aspects. This algorithm was based on the idea of the truncated Newton minimization method and the sequential cost function incremental method introduced by Courtier et al., consisting of an inner loop and an outer loop. The inner loop comprised the incremental method, while the outer loop consisted of the standard 4D-Var method using the full physics adjoint. The limited-memory quasi-Newton minimization method (L-BFGS) was used for both inner and outer loops, while information on the Hessian of the cost function was jointly updated at every iteration in both loops. In an experiment with a two-cycle truncated Newton-like incremental approach, the assimilation analyses turned out to be better than those obtained from either the standard 4D-Var or the incremental 4D-Var in all aspects examined. The CPU time required by this two-cycle approach was larger by 35% compared with that required by the incremental 4D-Var without almost any physics in the adjoint model, while the CPU time required by the standard 4D-Var with the full physics adjoint model was more than twice that required by the incremental 4D-Var. Finally, several hypotheses concerning the impact of using standard 4D-Var full physics on minimization convergence were advanced and discussed.