Recent diagnostic work within 3D-Var and 4D-Var (Desroziers and Ivanov, 2001, Chapnik et al, 2004, 2006, Talagrand 1999, Cardinali et al. 2004, Rabier et al. 2002, Navascues et al. 2006 at HIRLAM, and others) suggest that innovation and other statistics can be used to tune observation and background errors. Miyoshi (2005) reported the use of the innovation statistics to estimate the background error inflation factor online within the LETKF. Although the results were satisfactory they did not take into account the fact that discrepancy between estimated and diagnosed total errors can be also due to observational errors.
Here we propose to estimate observational (for each type of instrument) errors and the inflation coefficient for the background error simultaneously within the Local Ensemble Transform Kalman Filter (LETKF). Since the Kalman Filter equations are solved exactly within a local domain, it is possible to compute on the fly statistics such as the Degrees of Freedom of the Signal DFS=Trace (KH)=(I-A/B), for each type of observation which is equal to the number of model dof of each type reduced by A/B, where A, B are the analysis and background error covariance respectively (Cardinale et al, 2004).
Following Miyoshi's (2005) approach, we will estimate and correct online within the LETKF the error variance of different instruments and the optimal inflation factor, using statistics of diagnosed (observed) <[y-h(xa)]'[y-h(xb)]>~Tr(R)=px(ob error variance), and <[h(xa)-h(xb)]'[y-h(xb)] >~Tr(HBH')= px(background error variance) to estimate and correct the variance parameters. This can be done using a simple Kalman Filter (Kalnay, 2003, appendix C), and persistence as their forecast, or by augmenting the state vector in the LETKF. We will test whether we can recover the correct error statistics with simulated observations and whether the analysis is improved within a simple model, and with a global model with real observations.