Information about some of the ways in which models are both parametrically and structurally inadequate can be inferred by dynamically altering model parameters using a data assimilation framework. Instead of (or in addition to) altering model states to minimize a model-data misfit, one alters model parameters during the minimization process. Structural model inadequacy implies that one should not search for a global ``best" set of parameter values, but rather allow the parameter values to change as a function of state; different parameter values will be needed to compensate for the state-dependent variations of realistic model inadequacy. Care must be taken when interpreting results. It is shown that when the system of interest is stochastic, it is {\it{impossible}} to uncover the correct form of stochasticity even when the model is taken to be stochastic. The interpretation errors are especially egregious when the model is deterministic. Of course, this need not imply that the resulting parameter estimatates are not useful. The parameter estimation approach can be used both to attempt to quantitatively identify model inadequacy (with an eye towards model improvement), and to develop strategies for accounting for model inadequacy (with an eye towards forecast improvement).
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