WaterSubject

Kalman filter (KF), a sequential data assimilation technique, and its extensions such as extended Kalman filter (EKF) and ensemble Kalman filter (EnKF), have been extensively used in hydrological modeling. This includes but not limited to soil moisture and temperature retrieval, aquifer parameter identification, and simultaneous state and parameter estimation. The common purpose of these applications is to quantify and manipulate uncertainties and errors involved in model structure, parameter and observation, given that the statistical properties of both model and observation errors are known in advance. If both model and observation errors are zero-mean (i.e. unbiased) and uncorrelated in time, KF is the best (i.e. minimum variance) linear unbiased estimator (BLUE) among any linear combinations of model forecasts and measurements. Additionally, if the errors are Gaussian, KF is the minimum variance unbiased estimator. From the Bayesian view, KF is both the minimum mean squared error (MMSE) estimation (i.e. conditional mean) and the maximum a posteriori (MAP) probability estimation (i.e. conditional mode), provided that all the assumptions on errors as stated above are valid.

However, for hydrological and environmental systems, accurate simulation models are not readily available, and the statistical nature of the uncertainties impinged in those systems is rarely understood completely. Thus, the limitations of applying KF and its extensions come out around the nature of errors. For example, if the distribution of the model or observation error is biased uniform such as U[0, 1], or even unbiased-uniform such as U[?1, 1], KF and its extensions will no longer be a BLUE (or MAP). The model error usually violates the zero-mean Gaussian assumption because the inputs and parameters of a hydrological model, such as rainfall and transmissivity, are not exactly known. Those uncertainties in inputs and parameters can cause biased errors. Moreover, if instant human interference is neglected in a simulation model, a large bias of state prediction may occur at the time points when human interference is imposed. In particular, when heavy human interferences exist, the unknown time dependent human inputs may cause non-stationary, unpredictable and biased error to the model forecast. Observation error of streamflow can be non-stationary when river flow is affected by storage regulation.

Another limitation of KF is that the statistics of errors must be known in advance, but the difficulty in obtaining such measurement and quantifying model error statistics is always a challenge, as argued by Reichle et al. “Any data assimilation approach that provides for model error faces the serious challenge of determining the true model error covariance matrix in operational applications.”

The ultimate purpose of data assimilation is to balance information from different sources – modeled outputs and system observations. Effective hydrologic data assimilation in operational settings needs to be based on realistic descriptions of uncertainties, thus, research may need to pay attention to the way in which uncertainties are actually represented. This is related to the robustness of data assimilation technique to error nature. Several studies have investigated the robustness issue of KF. These studies showed that the KF was quite robust with respect to state estimation, but the specification of error statistics had a severe impact on the covariance estimate. To deal with biased error, Drécourt et al. discussed the “separate bias” KF method, which used two KF schemes for the state vector and the bias vector, respectively. This method, like the primary KF, needs to know in advance both the error covariance of random errors and the bias.

This paper will examine the KF estimation performance when the assumptions are violated and compare KF to another filter, called H-infinity filter (H_? or HF), which has a different way of error representation and does not need the assumptions with KF regarding the nature of errors. HF has so far been mainly applied in electrical engineering system modeling. The design criterion of HF is based on the worst-case disturbance; therefore, the filter is less sensitive to uncertainty in the exogenous input statistics and simulation model structure. According to our knowledge, HF has not been applied in hydrological and environmental systems, although it has potential in resolving some limitations existing with KF and its extensions.

In the remainder of this paper, the formulations of KF and HF are presented and compared in Section 2, with special emphasis on their objective function, in order to show the essential difference between the two filters. To compare their robustness, the two filters are applied to a one-dimensional coupled soil moisture and temperature simulation model. In Section 3, the one-dimensional coupled soil moisture and temperature simulation model is briefly described. In Section 4, the performances of KF and HF are compared under the following different cases: (1) unknown or not fully known error statistics, (2) non-Gaussian errors, (3) biased errors, and (4) biased model errors caused by unknown instant human interferences, taking a hypothesized input of irrigation as example. These conditions are easily encountered in environmental and hydrological systems due to the complexity and the existence of human interferences. The merits and limitations of HF compared to KF are discussed in the conclusion section.