Moving average fields, macro-scale response measures, and homogenizing micro-scale variation

There are two critical issues when deriving a macro-scale prediction model starting from a more complete, underlying model. The first is the precise relationship of the fields predicted by the more complete model and the fields predicted by the macro-scale model. The second is the manner of solving a closure problem that is invariably encountered in all such derivations. The understanding that moving averages of the fields predicted by the more complete model are the fields predicted by the macro-scale model is challenged on the grounds that accomplishing a moving average does not eliminate micro-scale variation, it only appears to do so in one representation of the moving average field. The solution of a closure problem by assumption is challenged on the grounds that the most common assumptions are demonstrably invalid, even while leading to prediction models that can provide reasonable estimates of the macro-scale response in some scenarios. In presenting the challenges, it is further shown how a multiresolution analysis by an orthogonal wavelet system provides a framework for both precisely defining macro-scale response fields, i.e., fields from which all micro-scale variation has been eliminated, and presenting a formally exact solution for a precisely described closure problem.