The Impact of Background Error on Incomplete Observations for 4D-Var Data Assimilation with the FSU GSM

In the study of 4D Var data assimilation of atmospheric models, an important issue to address is the case of incomplete observations in either the space or time dimension. In an ideal setting 4-D Var data assimilation assumes the observation data field to be complete, and if there are gaps in the data, these are being taken into account in the process of data assimilation. To assess the impact of incomplete observations on the 4D Var data assimilation, we carried out some assimilation experiments with a model consisting of the dynamical core of FSU GSM by reducing the number of observations in both the space and time dimensions respectively. The impact of the Jb background error covariance term on problem of incomplete observations in either time or space direction has been investigated using the new FSU GSM consisting of a T126L14 global spectral model in a parallel environment using MPI version of its adjoint model. Numerical experiments aimed at assessing impact of incomplete observations on 4-D Var data assimilation were carried out as follows: first, a twin experiment with observations available at every model grid point (thereafter referred to complete observations) was carried out using the dynamic core of FSU GSM and its adjoint model to ensure that the assimilation system is well ∗Supported by N.S. F.Grant ATM-0201808 1 constructed. For such an experiment, one knows in advance the exact solution, and the minimum value of the cost function is zero. We then reduced the available observations to every 2, 4 or 8 spatial grid points, respectively. We also carried out another set of experiments with data holes where all the observations were missing, e.g. at ocean grid points locations. We then carried out experiments reducing the number of time instants where observations are available only every 2, 4 or 8 time steps in the window of assimilation . The results obtained show that spatial incomplete observations lead to a slow down in the cost functional minimization. Although the decrease rate of cost function with incomplete observations where observations are available only at every 2, 4 or 8 observation grids exhibited a similar pattern, the impact(degree) of reasonable retrieval initial data strongly depends on the density of observations. The impact of incomplete observations is even more pronounced for experiments where no observations were available over oceans, in which case the lack of fit between a control run and the aforementioned could not be reduced. In contrast to above results, experiments involving reduction of the number of time instants where observations are available in the assimilation window allowed a successful retrieval of the initial data. The results obtained were insensitive to whether observational data was available only every 2,4 or 8 time steps versus that of the full observation. To sum-up, the lack of the observations in grid space strongly affect the results of the minimization and retrieval of initial data, while that in time dimension or some variables will have no significant affection on the results. Impact of various scenarios of incomplete observations on ensuing forecasts, and root mean square error were investigated for 24-72h forecasts for cases when the cost functional included and/or excluded the background covariance term. To further investigate the issue of incomplete observations, we carried out another set of 2 experiments by adding a background term Jb to the cost function. Background state propagates information from observations at early times into the data holes. By considering an assimilation with a single observation, it can be shown that the background error covariance matrix controls the way in which information is spread from that observation to provide statistically consistent increments at the neighboring grid points.