Dealing with Unknown and Biased Errors in Hydrological Modeling - A Comparison of Kalman and H-infinity Filters

In hydrological modeling, model and observation errors are often non-Gaussian and/or biased; and the statistical properties of the errors are often unknown or not fully known. Thus determining the true error covariance matrices is a challenge for data assimilation approaches such as the most widely used Kalman filter (KF) and its extensions. This paper compares KF to the H-infinite filter (HF), which is based on worst-case disturbance and less sensitive to uncertainty in the exogenous input statistics and simulation model structure. KF needs the assumptions of zero mean Gaussian errors but HF does not need such assumptions; KF is an optimal approach to the minimum mean square error estimation (MMSE), and it performs better than HF only under unbiased Gaussian model and observation errors whose statistics are known in advance. To compare their performance when the KF's assumptions are violated, the two filters are applied to a one-dimensional coupled soil moisture and temperature simulation model under the two error cases: biased Gaussian model error and non-stationary model errors caused by unknown, instant human interferences. HF is found to be more robust to model errors than KF. In particular, the HF estimation recovers to the true state faster than the KF right after a human interference that causes a non-stationary, unknown model error. Thus, the HF might be more appropriate for hydrological models, which need to account the impact of human interferences that are usually uncertain, unknown or estimated with biased errors compared to natural inputs such as precipitation.