3 - Method for data merging
The method for extracting a single solution from the dataset is described in detail in previous papers: Gaillard (1992) for the stationary case and Gaillard et al.(1997) for the introduction of the time evolution. We intent to reconstruct a three dimensional field of temperature (T) and horizontal velocity (u,v), consistent with the datasets collected during Thetis 1. Salinity is not part of the reconstructed parameters because the relevant data are too scarse to expect a reliable estimation. For each type of measurement we write the relation between the data and the parameter. Complementary imformation extracted from our theoretical and statistical knowledge of the ocean is introduced either as statistical imformation or as additional relations between parameters. All the data-parameter relations are linear (or linearized around a reference state).
The estimation produced at different times are not obtained independently. The previous estimation and associated statistical error is used as the starting point of the current estimation. Time evolution is introduced and represented by a prediction model, a prediction error takes into account possible deviation from this model. In our case a simple persistence with limited memory is applied. This combination of updating the field with data, then carrying information in time, defines a Kalman filter. This method requires computing the full covariance matrix at each time step. This is problematic with large state vectors (vector of unknowns) as is the case with high resolution numerical models. In our case the prediction model is extremely simplified and the time step and grid scale is set up by the data resolution instead of being imposed by numerical requirements. Since we will manipulate a state vector with O(103) unknown components, full computation of the covariance matrix is feasible. A-priori information are introduced by specifyirlg the relative weight of the different vertical modes, based on the CTD statistics.
