Distance measures for hydrological data having a support

Covariance and correlation are usually functions of Euclidian distance between observation points in space. The distance is well defined for points but how to define a distance between drainage basin is not obvious. The traditional distance measure between two drainage basins in hydrology has been the distance between the centres of gravity of the respective basins. The alternative distance measure promoted in this technical note is the expected value of all possible distance between the respective basins. It is shown that this measure has a well founded theoretical background. Furthermore this distance measure may serve to simplify the numerical calculation of integral expression of covariance and correlation for data having a support. This distance measure can be evaluated analytically for lines and for simple geometries in space like rectangles. For more complex geometries in space Monte Carlo methods apply. The theoretical results are illustrated with examples from drainage basins in Norway, France and Costa Rica.