A Multi-scale Three-Dimensional Variational Scheme Based on Diffusion Filter

Traditional three-dimensional variational (3DVAR) schemes are inefficient in capturing the multi-scale information resolved by observations. This study first indicates that the root cause lies in the spatial incoherence of the gradient of cost function caused by the irregular distribution of observations. Then an improved 3DVAR scheme based on diffusion filter is proposed to solve this problem. By employing diffusion filters to smooth out the erroneous signals implied in the gradient, the new scheme can incorporate scales into a gradient-based minimization algorithm and correct the observational information over different scales. For illustration, a two-dimensional idealized SST assimilation experiment is carried out to demonstrate the performance of the new scheme in extracting multi-scale signals. Results show that the proposed scheme has an obvious advantage over the traditional 3DVAR schemes and the multi-scale observational information, from longer to shorter wavelengths, can be extracted successively as expected, which suggests that enough considerations should be given to the scales information but not only to the rate of convergence when a minimization algorithm is employed in 3DVAR.