Up-scaling short-term process-level understanding to longer timescales using a covariance-based approach

General experience in hydrologic modelling suggests that the parameterisation of a model changes over different time and space scales. As a result, hydrologists often re-parameterise their models whenever different temporal or spatial resolutions are required. Here, we investigate theoretical aspects of this issue in a search for the cause(s) of the need for re-parameterisations. Based on Taylor series expansion, we present a mathematical approach for temporal up-scaling that involves covariance-based corrections. We apply the theory using a unique database of half-hourly pan evaporation measurements (comprising 237 days) and examine how the model parameters change when integrating from half-hour to daily and then monthly integration periods. We show that the model parameters change over different integration periods because of changes in the covariance between the model variables. In our model system, we find that the covariance-based correction is highly variable from day to day but settles down to a reasonably constant value over periods longer than about 15 days. The 15 days timescale is likely to be specific to our model system, nonetheless the underlying principle that there is a characteristic timescale for the covariance-based scaling correction of a particular hydrologic process might be general. If that proved true it would open up the possibility of systematically searching for characteristic integration periods for the key covariance-based scaling terms in other key hydrologic processes. That would in turn enable the development of more generalised hydrologic closure scheme(s).