Quasi‐static estimation of background‐error covariances for variational data assimilation

Variational data assimilation produces an analysis (estimated atmospheric state) that depends on the background‐error covariance matrix ( B matrix), which can be estimated from a sample of training data following the same statistical distribution. A quasi‐static iterative estimation procedure is proposed that can be started without prior knowledge of the background‐error characteristics. The initial iteration limits non‐stationary error growth by only using a small number of assimilation/forecast cycles. Increased confidence in the updated estimate of the B matrix then allows the number of cycles to be increased in the next iteration, continuing the process until the desired number of cycles has been reached. The iterative procedure then continues using the full set of cycles until the convergence criteria have been satisfied. The use of a robust Gauss‐Newton algorithm allows ill‐conditioned minimizations (resulting from poor B matrix estimates in early quasi‐static iterations) to be completed successfully, allowing the iterative process to continue until a sensible B matrix has been generated. This quasi‐static technique is particularly useful for idealized data assimilation experiments, which can be prone to non‐stationarity when there is no prior knowledge of the background‐error characteristics or existing training data. It could also reduce the computational cost of retuning the B matrix of an operational forecast/assimilation system following a significant model or observing‐system change. The technique is illustrated using the three‐body gravitational model with an observational configuration in which the global average influence of the observations on the analysis matches that of an operational weather forecasting system.