Reducing scale dependence in TOPMODEL using a dimensionless topographic index

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 x_i=ln(a_i/S_i), where a_i is the specific contributing area per unit contour length and S_i is the topographic slope. In the SI unit system, a_i/S_i is in meters, and the unit of x_i is problematic. We propose a simple solution in the 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 y_i=x_i-lnC. Reformulating TOPMODEL equations to use y_i instead of x_i helps giving the units of all their terms and emphasizes the scale dependence of these equations via the explicit use of C outside from the topographic index, in what can be defined as the transmissivity at saturation per unit contour length T₀/C. The term lnC defines the numerical effect of DEM resolution, which contributes to shift the spatial mean x of the classical topographic index when the DEM cell size C varies. A key result is that both the spatial mean y of the dimensionless index and T_0/C are much more stable with respect to DEM resolution than their counterparts x and T₀ in the classical framework. This shows the importance of the numerical effect in the dependence of the classical topographic index to DEM resolution, and reduces the need to recalibrate TOPMODEL when changing DEM resolution.