5.1. Study Catchment Description and Materials
[35] The 1.35 km^{2} Hemuqiao catchment (119°48′E, 30°35′N) located upstream of Taihu Basin, China (Figure 8) was chosen for this study. The average annual precipitation is about 1580 mm, and the average annual evaporation and temperature are 805 mm and 14.6°C, respectively. This experimental catchment, operated by the State Key Laboratory of HydrologyWater Resources and Hydraulic Engineering (HydroLab) and Zhejiang Provincial Hydrology Bureau, possessed some typical hydrologic characteristics of humid climates. Hillslope hydrological processes here were dominated by lateral subsurface flow and saturation excess overland flow.
[36] The Hemuqiao catchment was characterized by mountains and steep forest slopes of 25°–45°. The surface elevation is as high as 500–600 m asl in the southwest elevated region, and decreased to 150 m asl at the outlet of the Hemuqiao hydrological station. The vegetation in the Hemuqiao was heavily dominated by bamboo forests, about 95% of the entire area, with the remainder in rural and crop land.
[37] Contour lines were digitized using the TOPOGRID function in ArcInfo (ESRI Inc.) from topography maps (scale 1:10,000) with 5 m vertical intervals. A DEM of 10 m resolution was generated for derivation of hillslope geometric characteristics.
5.2. Results
[38] By defining the channel network according to field observations of the actual river, the 8 digital channel nodes, e.g., 4 source channel nodes, 3 intermediate nodes and 1 outlet node, were assigned and 7 channel segments were derived for delineation of the natural hillslopes (Figure 9). The whole catchment was divided into 18 hillslopes including 4 headwater hillslopes and 7 pairs of side slopes according to the 7 channel segments (Table 4). In Figure 9 and Table 4, every channel segment corresponded to at least two side slopes, and one headwater hillslope drained to each of the source channels (e.g., channels 2, 4, 6, and 7 listed in Table 4). In Table 4, the first number of the hillslope number is the channel number and it is associated with the second numbers, which represents the type of hillslope: 1 represents a side slope on the left bank of a channel segment, 2 represents a side slope on the right bank of a channel segment, and 3 represents a headwater hillslope on the source of a source channel segment. In this catchment, the entire 1.35 km^{2} area was divided into 0.47 km^{2} left bank side slope, 0.37 km^{2} right bank side slope, and 0.51 km^{2} headwater hillslope. For each hillslope, the hillslope length and average slope were also computed (Table 4).
Table 4. Hillslope Geometry of the Hemuqiao CatchmentChannel  Hillslope Number^{a}  Area (km^{2})  Length L (m)  Average Slope  Average C_{c}^{b} (×10^{−3}) 


1  11  0.0519  361.4  0.40  1.47 
 12  0.0052  119.0  0.48  14.62 
2  21  0.0294  304.6  0.52  0.48 
 22  0.0445  179.0  0.56  5.44 
 23  0.1009  458.7  0.60  0.00 
3  31  0.1788  523.8  0.54  2.98 
 32  0.1430  498.7  0.56  1.64 
4  41  0.0398  375.6  0.48  3.49 
 42  0.0508  370.4  0.53  0.91 
 43  0.1322  607.7  0.47  −1.31 
5  51  0.0100  185.6  0.49  8.65 
 52  0.0272  302.8  0.48  1.58 
6  61  0.0308  250.0  0.65  3.82 
 62  0.0232  189.0  0.63  1.37 
 63  0.1051  592.8  0.55  −1.36 
7  71  0.1162  433.8  0.58  1.99 
 72  0.0680  388.0  0.65  2.78 
 73  0.1513  568.7  0.60  −1.63 
[39] Figure 10 shows the flow distance field for the 18 hillslopes in the Hemuqiao catchment. For each hillslope, the cumulative frequency curves of flow distances were computed (Figure 11) by ranking all the pixels within a hillslope according to their distance to the channel segment. The PDFs as a function of distance to the channel segment for all the hillslopes were derived (Figure 12) in terms of derivatives of these hillslope cumulative frequency curves of flow distances in Figure 11. For the headwater hillslopes (Figure 12a), PDFs curves increased from nearly zero near the top to a peak first and then they decreased to zero at the outlet. Peak values for these four headwater hillslopes ranged from 2.92 × 10^{−3} to 4.20 × 10^{−3}. For the side slopes (Figure 12b), there were two PDFs distribution patterns: for most side slopes (e.g., 11, 21, 52, etc.), the PDFs were high with a rapidly increase near the channel and then decreased; for other side slopes, the PDFs near the channel were similar to those located around hill top (e.g., 42). The peak values of PDFs for these 14 side slopes ranged from 3.06 × 10^{−3} to 1.79 × 10^{−2} with the averaged peak value 7.05 × 10^{−3}. Results indicated that the cumulative curves were fully aggregated beyond the 45° line for the side slopes. In addition, they generally appeared to be a symmetrically distributed around the 45° line for the headwater hillslopes. This means that the location of the peak for these probability distributions of hillslope width functions was near the channel segments for side slopes. For headwater hillslopes, the mean values were relatively far away from the channel source points.
[40] The algorithms of HWTFs were used to determine the HWFs for each hillslope. According to equation (8), the value of w_{r} could be defined through hillslope area, S. So the hillslope width functions as a function of distance to channel (DC) for the 18 hillslopes could be derived as shown in Figure 13. Results showed that although HWFs differ from each other, nevertheless, there were some general properties. For instance, all the curves of headwater hillslopes approximately conformed to the relationship of equations (10) and (11), and the HWFs appeared to be generally a normal distribution, as found in Figure 13. That is, headwater hillslopes always originated from a hill top and then converged at one point.
[41] Generally, the HWFs curves for side slopes decreased, and the overall shape was divergent contrary to headwater hillslopes, though there was always a small increase near the channel segments. This was because that most of the side slopes were composed of several parallel gullies, water near the channel segment more likely drained to the gully outlet instead of into the channel segment, so fewer cells had water flowing directly into channel, and hence the frequency or the hillslope width adjacent to the channel was lower compared with the middle part of the side slopes. All the HWFs of side slopes approximately conformed to equation (11), and the curves approximately belong to a partial normal distribution as can be seen from Figure 13.
[42] Troch et al. [2004] used exponential functions to express HWFs for deriving analytical solution of HSB equations for ideal hillslopes. As the natural hillslopes are composed by different types of simple hillslopes, e.g., convergent, parallel or divergent hillslopes and thereby the monotonically increasing or decreasing of exponential function cannot accurately describe the complex structure of natural hillslopes. Figure 13 demonstrates that HWFs in natural catchment display exponential decrease function (e.g., 12, 22, 51) and normal distribution (e.g., 23, 43). To express the HWFs functions accurately, we used Gaussian function for differentiating the shape characteristics of HWFs among the 18 natural hillslopes. The results of the HWFs curve fitted by Gaussian function are listed in Table 5 and shown in Figure 13.
Table 5. CurveFitted Results for Natural Hillslopes of the Hemuqiao CatchmentHillslope  Exponential Functions^{a}  Gaussian Functions^{b} 

Coefficients  R^{2}^{c}  RMSE (m)  Coefficients  R^{2}^{c}  RMSE (m) 

a1  b1  a2  b2  c2 


11  211.1  −0.00180  0.18  67.2  245.8  147.9  128.2  0.92  20.8 
12  110.4  −0.01396  0.69  18.1  87.2  31.4  43.5  0.95  7.4 
21  155.8  −0.00267  0.33  38.4  158.3  114.4  120.3  0.93  12.2 
22  462.5  −0.00631  0.66  65.7  365.5  48.7  106.5  0.85  44.6 
23  194.0  0.00077  0.07  101.2  359.0  261.2  173.3  0.92  30.2 
31  575.7  −0.00181  0.49  106.1  514.7  170.4  255.8  0.89  50.0 
32  537.7  −0.00254  0.65  83.5  442.4  112.5  263.1  0.83  57.1 
41  224.4  −0.00409  0.78  26.4  179.3  29.5  239.2  0.86  21.1 
42  139.5  0.00013  0.003  40.5  180.7  195.3  203.0  0.66  23.9 
43  268.9  −0.00058  0.03  156.5  471.9  266.4  152.7  0.92  45.3 
51  144.6  −0.01042  0.81  17.1  108.3  32.9  74.4  0.95  8.9 
52  143.8  −0.00262  0.28  39.1  148.2  112.2  118.8  0.86  17.2 
61  207.3  −0.00376  0.51  37.2  184.0  87.3  121.9  0.93  14.3 
62  217.9  −0.00481  0.32  58.8  214.5  73.1  70.7  0.90  22.9 
63  176.8  0.00014  0.003  88.9  277.2  307.9  239.7  0.74  45.7 
71  566.4  −0.00353  0.70  88.6  461.8  112.6  183.2  0.98  22.0 
72  376.2  −0.00392  0.75  48.6  294.7  56.9  221.7  0.88  34.6 
73  220.6  0.00073  0.09  124.4  425.7  321.5  221.8  0.86  48.3 
[43] As shown in Table 5 and Figure 13, HWFs for most natural hillslopes were poorly expressed by exponential function but well fitted by the Gaussian function. Squared correlation coefficient between the simulated and DEM derived HWFs were larger than 0.8 for most hillslopes. By adjusting parameter values, the Gaussian function was suitable for simulating HWFs for any types of natural hillslopes. Parameter, b2, in Gaussian function reflected the peak location of HWFs. Results in Table 5 show that values of b2 for all the headwater hillslopes were much larger than those of the side slopes. Moreover, the shape of HWFs for headwater hillslopes (e.g., 23, 43, 63 and 73 in Figure 13) was more symmetric (b2 about half of the longest DC) than that of side slopes hillslopes (b2 less than half of the longest DC). This symmetric rise and decline of HWFs indicated that natural headwater hillslopes nearby the outlet were primarily dominated by convergent type of hillslopes (hillslope width decreased with the increase of DC), and they were in divergent plan shape nearby catchment boundary (hillslope width increased with the increase of DC). For the side slopes, however, HWFs consisted of a larger portion of divergent type than convergent type. Particularly for the hillslopes of 12, 22, and 51, b2 was much smaller than half of the longest DC, indicating these hillslopes mostly belonged to the divergent type. Another parameter of c2 in the Gaussian function controls the width of the curve “bell.” The wide and flat curve's peaks (e.g., 31, 42, 61, 62, and 63) indicated that these natural hillslopes included parallelplanar type of hillslopes. Therefore, the Gaussian function with different parameter values was able to describe the natural hillslopes composed by different types of simple hillslopes.