Scaling theory and scale problems are a hot topic in hydrology. The scale issue may be approached in two distinct ways. The first, an aggregation approach, can be expressed in terms of the question of how best can we use measurements and understanding at small scales to inform predictions of hydrological responses at larger (catchment) scales. For the second, a disaggregation approach, the question is how to use measurements and understanding at larger catchment scales to predict local scale responses. There have been numerous discussions in the literature, and at least four international workshops since 1981 (Caracas, Princeton, Robertson and Vienna) have been concerned specifically with scale problems. The papers arising from the Robertson, Australia workshop in 1995 made up four full issues of Hydrological Processes. Yet, despite all the paper and internet traffic expended on this topic, there seems to have been very little true progress. This should not really be surprising, in fact, since both aggregation and disaggregation problems are inherently impossible to resolve in hydrology, for similar reasons: the impossibility of knowing the details of both unsaturated and saturated flows in the subsurface at individual locations.
It may be, however, that both approaches might be able to give approximate extrapolations to other scales, based on some (perhaps restrictive) assumptions about the structure of the subsurface. Approximate also implies some uncertainty associated with the predictions. If we start to consider the nature of that uncertainty, there would appear to be a way of treating both aggregation and disaggregation approaches in a common framework of an uncertain (or fuzzy) mapping of the catchment landscape into an appropriate model space. Such an approach can deal with both scale dependent modelling, and scaling theories for inferring the properties at one scale from those at another.
The important feature of such an approach is that the models used should have sufficient flexibility in functionality to cover the range of hydrological responses seen in the real world at different scales. If this is so, then the model space will have the dimensions of the components, parameters or boundary conditions that are required to run the model and produce predictions. Within this space, the outputs are essentially known (even if the model is inherently stochastic). The requirement is then to map the catchment or units of the catchment into this space. This mapping process is essentially what happens in most modelling studies that require model calibration. If a set of model parameters is optimized against a catchment dataset, the mapping is to a single point in the model space. If uncertainty in the appropriate parameter sets (and perhaps model structure) is allowed for, then the mapping may be to a range of (possibly disjoint) locations in the model space.