The Role of DEM Resolution and Entropy in the Parameterisation and Evaluation of Hydrologic Models

James Brasington1 and Keith Richards2
1Research Institute for Environmental Science and Management, School of Geography and Earth Resources, University of Hull, Hull HU6 7RX, United Kingdom
2Department of Geography, University of Cambridge, United Kingdom

Distributed hydrological and geomorphical modelling is increasingly facilitated by the widespread availability of digital elevation models (DEMs). The objective of the work reported here is to evaluate the role of the gridcell resolution in the parameterisation and evaluation of a terrain based rainfall-runoff model, TOPMODEL (Beven & Kirkby, 1979). High resolution digital elevation data from a small (c. 4.5 km2) catchment in the Nepal Middle Hills have been gridded at 10, 20, 40, 60, 80, 100, 200, 300, 400, and 500m scales. The upslope contributing area (a) and local slope angles (tan b) have been derived using the multiple flowdirection algorithm of Quinn (1991) and combined to yield the wetness index, ln(a/tan b) used to predict the location of runoff source areas in TOPMODEL. Frequency distributions of the derived terrain attributes show a strong scale dependence. The consequence of this scale dependence is explored systematically through the application of TOPMODEL to a one-month rainfall-runoff record sampled at half-hourly frequency during the 1992 Nepalese monsoon.

The sensitivity of the model to DEM scale is examined from two directions. Firstly the model is applied with parameters identified applying the ln(a/tan b) frequency distribution derived from the 20m DEM. Simulations using the distribution functions derived over a range of DEM scales are then realised whilst the model parameter set is held constant. A significant systematic relationship between increasing model error and gridcell size is identified. This relationship is found to be related to an overestimation of upslope area with increasing gridcell size, resulting in an overprediction of the saturated contributing area and a switch between the balance of predicted subsurface and saturation excess flow production mechanisms.

A second sensitivity analysis is reported in which a global optimisation procedure is used to identify parameter sets for each DEM scale. Equally successful parameterisations are found over the whole range of scales. This analysis clearly shows a strong topographic dependence in the optimised parameter sets in which the identified parameters were found to compensate for the changing forms of the topographic index distribution function. This evidence for scale dependent parameters in a physically based hydrologic model is profound.

Lastly the results of the sensitivity analyses are investigated in terms of the changing information content (entropy) of the DEMs with changing resolution. All terrain attributes show significant decreases in entropy with increasing gridcell size. Firstly, results from the former sensitivity experiments are investigated and a significant double-log relationship between the model error and relative entropy of the DEMs is identified. A clear step in this relationship between the 100m and 200m scales may be indicative of a critical resolution required for accurate representation of flowpaths at the catchment scale. Secondly, results from the parameter optimisation experiments show a highly significant non-linear relationship between the optimised saturated hydraulic conductivity (Ko) and the change in entropy with DEM scale. Again a clear step in the this relationship is found at the 100-200m scale, reinforcing the findings of the first sensitivity experiment, and enabling an estimation of the critical scale required for distributed modelling of the system.

The findings presented demonstrate a scale dependence in the application of TOPMODEL. The use of information theory provides an analytical tool for investigating the scaling relationships evident in model response and parameter compensation. Further analysis may show information theory and the related fractal dimension to be an effective measure for characterisation of fundamental critical spatial scales applied in distributed hydrological modelling.