Explicitation of an important scale dependence in TOPMODEL using a dimensionless topographic index

Abstract

Abstract. This paper stems from the fact that the topographic index used in TOPMODEL is not dimensionless. In each pixel i in a catchment, it is defined as xi=ln (ai /Si), where ai is the specific contributing area per unit contour length and Si is the topographic slope. In the SI unit system, ai /Si is in meters, and the unit of xi is problematic. Even if all the equations in TOPMODEL are homogeneous, it is confusing to use the logarithm of a non-dimensionless quantity as an index, and we propose a simple solution to this issue in the nowadays widespread cases where the topographic index is computed from a regular raster digital elevation model (DEM). The pixel length C being constant, we can define a dimensionless topographic index yi=xi−ln C. Reformulating TOPMODEL's equations to use yi instead of xi helps giving the units of all the terms in TOPMODEL's equations. Another advantage is to raise awareness about the scale dependence of these equations via the explicit use of the DEM cell size C outside from the topographic index, in what what can be defined as the transmissivity at saturation per unit contour length T0/C. We eventually demonstrate, based on various examples from the literature, that both T0/C and the spatial mean y of the proposed dimensionless topographic index are very stable with respect to DEM resolution. This markedly reduces the recalibration necessity when changing DEM resolution, thus offering an efficient rescaling framework.