Innovation-Based Methods for Online Estimates of Observation Error Variances within Ensemble Data Assimilation Cycles

Many data assimilation methods require knowledge of the first two moments of the background and observation errors to function optimally. To ensure the effective performance of such methods, it is often advantageous to estimate the second moment of the observation errors directly. We examine three different strategies for doing so, focusing specifically on the case of a single scalar observation error variance parameter r. The first method is the well-known Desroziers et al. “diagnostic check” iteration (DBCP). The second method, described in Karspeck, adapts the “spread–error” diagnostic–used for assessing ensemble reliability–to observations and generates a point estimate of r by taking the expectation of various observation-space statistics and using an ensemble to model background error statistics explicitly. The third method is an approximate Bayesian scheme that uses an inverse-gamma prior and a modified Gaussian likelihood. All three methods can recover the correct observation error variance when both the background and observation errors are Gaussian and the background error variance is well specified. We also demonstrate that it is often possible to estimate r even when the observation error is not Gaussian or when the forward operator mapping model states into observation space is nonlinear. The DBCP method is found to be most robust to these complications; however, the other two methods perform similarly well in most cases and have the added benefit that they can be used to estimate r before data assimilation. We conclude that further investigation is warranted into the latter two methods, specifically into how they perform when extended to the multivariate case.