Model Structure Identification and Correction Through Data Assimilation

The physical laws governing water movement at small scales have been understood for decades. What we don't understand well is how to apply these physical laws to systems that are complex and heterogeneous on all scales. To date, most 'physically based' models of hydrologic systems are based on an implicit up-scaling premise that the behavior at the model scale can be described by the small scale governing equations by spatial averaging of the state variables and by use of 'effective' parameters. Of course, the up-scaling assumption may be wrong, and the effective large scale governing equations for a heterogeneous system may be different in form, not just different in parameters, from the equations derived from small-scale physics. We suppose that there is a conceptual model of a hydrologic system; i.e. the major processes and their interconnections have been identified. We wish to know if it is possible to construct the mathematical relationships in question (or correct them) via data assimilation, using measurements made on the system inputs and outputs. Our approach is based upon the construction of a 'posterior' joint probability density functions for the relationships in question, in such a way that data assimilation helps to correct 'prior' belief about the dependences. In regions where no data are available the 'prior' knowledge dominates. The approach permits a representation of, and discrimination between, all three sources of uncertainty: initial conditions, input and structure uncertainty, and is illustrated using case studies.