Statistical-analysis methods are generally derived under the assumption that forecast errors are strictly random and zero in the mean. If the short-term forecast, used as the background field in the statistical-analysis equation, is in fact biased, so will the resulting analysis be biased. The only way to account properly for bias in a statistical analysis is to do so explicitly, by estimating the forecast bias and then correcting the forecast prior to analysis.
We present a rigorous method for estimating forecast bias by means of data assimilation, based on an unbiased subset of the observing system. The result is a sequential bias estimation and correction algorithm, whose implementation involves existing components of operational statistical-analysis systems. The algorithm is designed to perform on-line, in the context of suboptimal data-assimilation methods which are based on approximate information about forecast- and observation-error covariances. The added computational cost of incorporating online bias estimation and correction into an operational system roughly amounts to one additional solution of the statistical-analysis equation, for a limited number of observations. Off-line forecast-bias estimates based on previously produced assimilated-data sets can be produced as well, using an existing analysis system.
We show that our sequential bias estimation algorithm fits into a broader theoretical framework provided by the separate-bias estimation approach of estimation theory. In this framework the bias parameters are defined rather generally and can be used to describe systematic model errors and observational bias as well. We illustrate the performance of the algorithm in a simulated data-assimilation experiment with a one-dimensional forced dissipative shallow-water model. A climate error is introduced into the forecast model via topographic forcing. while random errors are generated by stochastic forcing. In this simple setting our algorithm is well able to estimate and correct the forecast bias caused by this systematic error, and the climate error in the assimilated-data set is virtually eliminated as a result.