## 1. Introduction

[2] TOPMODEL is a topography-based concept for watershed hydrology modeling. Since the TOPMODEL was first proposed in 1979 [*Beven and Kirkby*, 1979], it has been widely used to study the effects of topography on hydraulic processes including flood frequency, streamflow generation, flow paths, geomorphic characteristics, and water quality [*Wolock and McCabe*, 1995]. In addition to the success of the TOPMODEL concept in traditional hydrologic modeling, this concept has also been successfully incorporated into several ecosystem-atmosphere models including the Regional Hydro-Ecological Simulation System (RHESSys) [e.g., *Ford et al.*, 1994]; the Land Ecosystem-Atmosphere Feedback model (LEAF-2) [*Walko et al.*, 2000]; the TOPMODEL-based Land Atmosphere Transfer Scheme (TOPLATS) [*Famiglietti and Wood*, 1994; *Peters-Lidard et al.*, 1997]; the Catchment Model [*Koster et al.*, 2000]; and the Common Land Model (CLM) [*Dai et al.*, 2003].

[3] To apply the TOPMODEL, a modeled catchment is partitioned with a regular grid or lattice. The so-called ‘topographic index’ is then calculated for each cell in the catchment. The topographic index, ln(*a*/tan β), is the natural logarithm of the ratio of the specific flow accumulation area *a* to the ground surface slope tanβ. The surface slope can be evaluated from digital elevation model (DEM) data. The specific flow accumulation area is the total flow accumulation area (or upslope area) *A* through a unit contour length *L*. To compute the total flow accumulation area *A*, flow directions are tracked upslope, starting from the cell of interest to the upstream divide of the watershed, and then tracked downslope accumulating cells contributing to the drainage area of the cell of interest. Here we note an uncertainty associated with the definition of the flow accumulation area *A*. From the presentations of *Beven and Wood* [1983, Figure 2] and *Kirkby* [1997, Figure 1], one can reasonably conclude that the flow accumulation area is defined along the ground surface. However, using DEMs in geographical information systems (GIS), the computed flow accumulation area is generally the area projected to x-y plane, and this calculation of *A* has become standard practice. The difference between these two areas is negligible if the slope is less than 0.5 (m/m), and most of the slopes in the watersheds to which the TOPMODEL is applied are less than 0.5 (m/m) [e.g., *Montgomery and Dietrich*, 1992]. Therefore, for consistency with standard practice, we adopt the convention of calculating the flow accumulation area or upslope area as the area projected to x-y plane.

[4] The slope term (tan β) in the topographic index arises from the assumption that the surface of the water table is parallel to the ground surface. Thus the local hydraulic gradient is assumed to be equal to the slope of the ground. Because flow direction depends on hydraulic gradient, or ground surface slope, flow direction and the calculation of the upslope accumulation area should be consistent with the local slope value that is used to compute ln(*a*/tan β)*.* Thus the computed topographic index is dependent upon the calculation of both slope and flow direction.

[5] Although many researchers have investigated algorithms for calculation of slope and flow direction, those studies tend to focus on either slope [e.g., *Jones*, 1998; *Zhang et al.*, 1999] or flow accumulation area [e.g., *Tarboton*, 1997; *Rieger*, 1998]. Few studies have examined the combined effects of slope and flow direction algorithms on the topographic index. The works of *Quinn et al.* [1991], *Wolock and McCabe* [1995], and *Mendicino and Sole* [1997] are exceptions. However, these studies only investigated the difference in the statistical moments or distributions of the computed topographic index; no comparisons of errors between the “true,” i.e., analytically solved, and the numerically computed topographic indices were carried out. Comparing against “truth” is difficult because an analytical expression for the real terrain does not normally exist. Therefore one cannot usually determine which topographic index algorithm is more accurate. This paper seeks to resolve that issue and provide objective evaluation of the appropriate numerical algorithm for calculating the topographic index.