Interactions between model predictions, parameters and DTM scales for topmodel

Abstract

The scale-dependence of the popular rainfall-runoff model, TOPMODEL, is assessed for a small headwater catchment in the Nepal Middle Hills. Digital terrain analysis is used to calculate the frequency distributions of slope, tanβ, upslope contributing area, a, and the combined hydrological wetness index ln(a/tanβ) for a range of digital elevation model (DEM) grid sizes between 20 and 500 m. The resulting distributions of the topographic index are strongly sensitive to grid size, in which changing estimates of the upslope contributing area are identified as the first-order control. Sensitivity analysis reveals that model predictions are consequently grid-size dependent, although this effect can be modulated by recalibrating the saturated hydraulic conductivity parameter of the model as grid size changes. An analytical link between this parameter and the shape of the probability distribution of the index is tested and found to be reliable over a wide range of scales. A significant change in the model response to scale is identified between grid sizes of 100 and 200 m. This change in grid size is also marked by rapid deterioration of the topographic information contained in the DEM, measured in terms of the statistical entropy. It is suggested that this break in the scaling relationship corresponds to typical hillslope lengths in the dissected terrain, and this scale thus marks a fundamental natural threshold for DEM-based applications.

Introduction

Topography is now recognised as a first-order control on the hydrological response of a catchment to rainfall. This reflects the role that topography plays in determining the spatial distribution of catchment-scale flow pathways resulting from the downward force of gravity. By accounting for local variations in hillslope gradient and curvature, a wide variety of hydrological processes can be related to catchment topography, including the spatial distribution of soil moisture, the generation of runoff and basin response times (Dunne and Black, 1970; Anderson and Burt, 1978; Maidment and others, 1996)

TOPMODEL (Beven and Kirkby, 1979), TOPographic MODEL, is a catchment scale rainfall-runoff model which makes an explicit link between catchment topography and the generation of streamflow. The model is based on a spatially distributed topographic index which is used to predict local variations in water table depths. This topographic index is defined as ln(a/tanβ)_i, where a is the total upslope area which drains through point (pixel) i, and tanβ is the local downslope topographic gradient. The upslope area, a, reflects the tendency for subsurface water to drain to i, whereas tanβ can be considered to be an approximation of the hydraulic gradient forcing water downslope. It follows that drainage will tend to accumulate in areas with high values of the topographic index. A consequence of this formulation is that areas of the catchment with approximately equal values of the topographic index may be assumed to behave in a hydrologically consistent manner. The index can, therefore, be considered as a measure of hydrological similarity. The TOPMODEL approach has been applied to a wide range of topics, including flood-frequency estimation (Beven, 1986), scaling theory in hydrology (Wood and others, 1988), water table estimation (Merot and others, 1995; Moore and Thompson, 1996) and uncertainty analysis (Freer, Beven and Ambroise, 1996).

Considerable recent interest has focused on methods for the extraction of this index and related measures from digital representations of topography known as digital terrain models, DTMs. Popular forms of terrain model include digital line (contour) graphs, triangular irregular networks (TINs) and regular grid or raster digital elevation models, DEMs. This later form of terrain model has gained popularity primarily due to computational and mathematical simplicity of the regular data structure. Numerous algorithms have been developed for the extraction of terrain attributes from DEMs (reviewed by Desmet and Govers, 1996) and are now frequently incorporated into commercial Geographic Information Systems. Interest in this and related modelling strategies has been promoted by the widespread availability of high resolution pre-processed DEMs, such as the UK 50 m and the US 30 m models. Catchment-scale terrain models may also be readily derived by digitising existing hard-copy maps and from stereo-matching aerial photographs or satellite images.

There remains a preliminary assumption that DEMs used in such analysis are of high enough resolution to enable reliable description of hillslope flow routing. This requires that the DEM accurately captures the local scales of variability in hillslope morphology that play an important role in the definition of flow pathways. Previous studies have shown that the TOPMODEL index is sensitive to DEM resolution. Quinn and others (1991)found significant differences in probability distributions of the topographic index computed from 12.5 and 50 m grids. Zhang and Montgomery (1994)also determined grid size to be a significant control. For a range of scales between 4-90 m they found that the mean of the topographic index increased progressively with grid size. The effect of both the topographic map scale used to derive a DEM and the resolution of the DEM itself were analysed by Wolock and Price (1994), who again found the mean of the index to increase with grid size. In the situation of changing map scale they found that this was attributable to increases in the mean of the upslope contributing area and decreases in the mean slope gradient. By contrast, they found that the influence of DEM scale was most profound through its effect on the calculation of the contributing area. The effect of DEM resolution has also been investigated in terms of the TOPMODEL hydrological predictions. Zhang and Montgomery (1994)found that the shift or translation of the index distribution towards higher values increased the rate of predicted peak streamflow and decreased the depth to the water table as grid size increased. Wolock and Price (1994)reported similar results and found that predicted hydrographs became more skewed as the ratio of predicted overland to subsurface flow increased for coarser DEMs.

More recently, a number of studies have shown that the effects of DEM resolution may be compensated for by the effective model parameters. Bruneau and others (1995)applied TOPMODEL to a 12 km² catchment in Brittany, France and found that by optimising the model parameters, it was possible to obtain approximately equivalent simulations of streamflow over a range of DEM scales from 20-100 m. Similar results have been obtained by Franchini and others (1996)and Saulnier, Obled and Beven (1997), who both suggest a significant interaction between the saturated hydraulic conductivity parameter, Ko, used in TOPMODEL and the DEM resolution.

This paper examines the nature of TOPMODEL dependence on DEM resolution for a small (4.5 km²) research catchment, the Bore Khola, which is the focus of an investigation into the dynamics of rainfall-runoff in the Nepal Middle Hills. This subject is examined for a range of grid dimensions between 20-500 m and the precise nature of model dependence is examined through the following experiments. First, the effect of grid size upon the spatial and probability distributions of the topographic index and its constituent variables, tanβ and a, is evaluated. Second, the sensitivity of the model to grid size is investigated by applying the model at progressively coarser resolutions with a single parameter set and examining the propagated errors. Third, the interaction between the model parameters and DEM scale is examined by calibrating the model parameters at each resolution and comparing these with estimates of the parameter values derived analytically.

A guiding principle behind this study is to identify the range of appropriate scales for a model application. This reflects a potential danger that developments in computer power and storage may encourage the use of high resolution DEMs in a belief that these provide more accurate representations of flow pathways. However, high resolution DEMs derived by interpolation from relatively coarse scale map data add no more topographic information and incorporate spurious artifacts. Moreover, as Farajalla and Vieux (1995)point out, the view that “the smaller the better” may result in wasted computer storage and longer run-times. There is a need therefore, to identify the optimum scale of representation that captures the essential variability of the terrain appropriate for a TOPMODEL application. The identification of this scale is finally examined by reviewing model scale dependence in relation to the changing information content of the DEM as measured by the statistical entropy.