Abstract
An advance that made Gaussian-based three- and four-dimensional variational data assimilation (3D- and 4DVAR, respectively) operationally viable for numerical weather prediction was the introduction of the incremental formulation. This reduces the computational costs of the variational methods by searching for a small increment to a background state whose evolution is approximately linear. In this paper, incremental formulations for 3D- and 4DVAR with lognormal and mixed lognormal–Gaussian-distributed background and observation errors are presented. As the lognormal distribution has geometric properties, a geometric version for the tangent linear model (TLM) is proven that enables the linearization of the observational component of the cost functions with respect to a geometric increment. This is combined with the additive TLM for the mixed distribution–based cost function. Results using the mixed incremental scheme with the Lorenz’63 model are presented for different observational error variances, observation set sizes, and assimilation window lengths. It is shown that for sparse accurate observations the scheme has a relative error of ±0.5% for an assimilation window of 100 time steps. This improves to ±0.3% with more frequent observations. The distributions of the analysis errors are presented that appear to approximate a lognormal distribution with a mode at 1, which, given that the background and observational errors are unbiased in Gaussian space, shows that the scheme is approximating a mode and not a median. The mixed approach is also compared against a Gaussian-only incremental scheme where it is shown that as the z-component observational errors become more lognormal, the mixed approach appears to be more accurate than the Gaussian approach.
