Lower-dimensional features in climate models and their fuzzy modeling

Following common practice in data assimilation schemes, most diagnostic tools and metrics for intercomparison of reanalyses, correct a climate model forecast (hincast) or background field of continuous variables based on optimal minimization of the model variable with respect to observed values, summed over some or all grid points in a discretization. Evaluated properly, this procedure allows for effective utilization of innovation, increments, and residuals to improve parameterizations and physical understanding. The least squares difference is often used as a basic measure of accuracy that is then normalized to form an agreement index/metric and to quantify the correction needed. These metrics are most appropriate for continuous fields where the observed and model variables are commensurate (i.e., measured with the same units). They are, however, flawed when used in the presence of sharp gradients and discontinuities and when used to evaluate a model's success in predicting or reproducing smaller scale lower dimensional features contained within a bulk simulation. These features, occur frequently in geophysical climate applications and often represent discontinuities that are associated with important climate physical and dynamic processes.