Recently developed hillslope storage dynamics theory can represent the essential physical behavior of a natural system by accounting explicitly for the plan shape of a hillslope in an elegant and simple way. As a result, this theory is promising for improving catchment-scale hydrologic modeling. In this study, grid digital elevation model (DEM) based algorithms for determination of hillslope geometric characteristics (e.g., hillslope units and width functions in hillslope storage dynamics models) are presented. This study further develops a method for hillslope partitioning, established by Fan and Bras (1998), by applying it on a grid network. On the basis of hillslope unit derivation, a flow distance transforms method (TD∞) is suggested in order to decrease the systematic error of grid DEM-based flow distance calculation caused by flow direction approximation to streamlines. Hillslope width transfer functions are then derived to convert the probability density functions of flow distance into hillslope width functions. These algorithms are applied and evaluated on five abstract hillslopes, and detailed tests and analyses are carried out by comparing the derivation results with theoretical width functions. The results demonstrate that the TD∞ improves estimations of the flow distance and thus hillslope width function. As the proposed procedures are further applied in a natural catchment, we find that the natural hillslope width function can be well fitted by the Gaussian function. This finding is very important for applying the newly developed hillslope storage dynamics models in a real catchment.
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 Hillslopes are regarded as one of the basic elements of catchments, which are made up of interconnected hillslopes and the channel network [Troch et al., 2007]. There have been considerable advances in our understanding and conceptualizations of hydrological processes at hillslope scale, and our knowledge of hillslope hydrology has improved the understanding of rainfall-runoff mechanisms and provided the theoretical basis for hydrological modeling [Sivapalan, 2003]. There have been some attempts to develop hillslope-based models by considering hillslope geometric characteristics. Models such as IHDM [Beven et al., 1987], KINEROS [Woolhiser et al., 1990], WEPP hillslope version [Flanagan and Nearing, 1995], CATFLOW [Maurer, 1997], and WASA [Güntner, 2002] are hydrological models that employ the representative hillslope scheme in a similar manner.
 Although modeling hydrological processes of natural hillslope units is considered important, it requires a cumbersome and sophisticated reprocessing of topographic data, and thus is a nontrivial problem [Bronstert, 1999; Bogaart and Troch, 2006]. For example, the exact procedures for deriving representative hillslopes are actually manually or semimanually determined through an on-screen digitizing technique [Francke, 2005]. Moreover, compared with regular grid hydrologic models, the hillslope-based models are difficult to build because of a lack of theories and methods to tailor the discretization scheme to the irregularly topographic characteristics.
 Recently developed hillslope storage dynamics models (HSDMs) [Fan and Bras, 1998; Troch et al., 2002, 2003] employ low-dimensional approximate equations to represent the essential physical behavior of a natural system. Compared with IHDM or CATFLOW, HSDMs account explicitly for plan shape of hillslopes in an elegant and simple way through using hillslope width functions. HSDMs, as implied by the name, use hillslope width function, w(x), defined perpendicular to x to represent soil moisture storage, S.
where h is the elevation of the groundwater table, ε is the drainable porosity. In the HSDMs, the flow rate is related to the storage S through kinematic wave approximation or a more general form of Darcy's equation. Hence the 3-D structure of soil mantle of a hillslope is reduced into a 1-D pore profile, and the 3-D flow problem is transformed into 1-D flow problem.
 So far, HSDMs have mainly been applied to theoretical cases using highly idealized hillslope geometries. For example, Troch et al. [2002, 2004] applied the hillslope storage kinematic wave (HSKW) and the hillslope storage Boussinesq (HSB) models on nine basic hillslope types with exponential hillslope width functions. The HSDMs which account for landscape geometric complexity have inspired many studies in the field of catchment hydrology. When applying to natural catchment, manually or semimanually determined methods through an on-screen digitizing technique were always used. For instance, Fan and Bras , first, applied the HSKW model at a catchment scale and an on-screen digitizing method for partitioning a catchment into the hillslopes was suggested. Matonse and Kroll,  further applied the HSKW and HSB models to a headwater portion of the Maimai watershed to simulate low streamflows according to the on-screen digitizing method. This on-screen digitizing method is time consuming and is difficult to derive the required hillslope geometric characteristics, which limits the application of HSDMs to the real-world scenarios.
 In this study, the hillslope storage dynamics theories are used to guide the processing and deriving of hillslope geometric characteristics. The goal of this study is to derive hillslope width functions of HSDMs from flow distance field on the basis of grid digital elevation models (DEMs). In section 3, a modified, detailed algorithm based on grid DEMs is presented for automatic derivation of hillslopes, according to the illustrative procedures suggested by Fan and Bras . Then the algorithm based on flow distance transforms for derivation of hillslope width function, which is necessary for running the HSB or HSKW model, is described. Finally, the natural hillslope width functions are fitted by the Gaussian function curve.
2. Background: Definition and Derivation of Hillslope Width
 Hillslope width, according to Fan and Bras  and Paik and Kumar , was usually defined as contour lengths. As noted by Paik and Kumar , there are two assumptions underlying this definition: (1) the single direction of surface water flow paths is represented by streamlines, and (2) the streamlines can be identified as orthogonal to the geographic contours. As long as these assumptions are satisfied, contour length can be regarded as the equivalent hillslope width. It is convenient for on-screen or semimanual derivation of hillslope width in accordance with this definition for abstract or experimental hillslopes. However, it is much more difficult to derive natural hillslope width on the basis of contour lines manually.
 When dealing with a natural landscape, an automatic derivation method of hillslope width is required. This automatic derivation method can be developed on the basis of easily available geomorphologic data such as DEMs. For example, network width functions in the theory of geomorphological instantaneous unit hydrograph (GIUH) were defined as the number of junction nodes as a function of distance to outlet [Moussa, 2008]. The network width function was automatically derived from the topology of channel network and its morphometric properties. However, the role of geometric characteristics of individual hillslopes was ignored in the frame of GIUH, which may be applicable to large catchments where channel flow processes are dominant factors [Aryal et al. 2002].
Bogaart and Troch  suggested that the frequency of flow distances is equivalent to HSDMs' hillslope width function. It is an extremely important alternative for deriving the divergent or convergent characteristics of hillslopes using a probability density of flow distance. This definition is close to GIUH's width functions, and they all use frequency of area (or nodes) at a distance to represent hillslope (network) width functions. Both of these two definitions are indirect methods for defining width functions compared to the on-screen determination of contour length mentioned above, and can be readily implemented using grid DEMs.
 Although the probability density of flow distances provides an alternative way for deriving hillslope width function, more detailed procedures are still needed to supplement for practical applications. First, the probability density functions of flow distances need to be converted to hillslope width functions when applying HSDMs in a natural catchment. Second, there is always a trend error which tends to overestimate distances because of the inability to follow streamlines in all directions by regular flow direction assigning methods (e.g., D8 algorithm [O'Callaghan and Mark, 1984] that assigns a flow direction to a single neighboring pixel with the steepest downslope gradient, or D∞ algorithm [Tarboton, 1997] in which flow direction is defined as the direction of two downslope pixels on a facet). Though many other factors can cause different kinds of flow distance estimation errors, we focus on the former and procedures for reducing the trend errors will be discussed in section 3.
 The D8 algorithm provides a single direction to simulate the flow direction of streamlines. It is widely used in hydrologic models because it easily and simply specifies flow directions using GIS tools, and only one number is needed to be saved for each pixel to represent the flow field, thus saving computer storage. However, the D8 method has disadvantages arising from the discretization of flow into only one of eight possible directions separated by π/4 [Fairfield and Leymarie, 1991; Tarboton, 1997], which results in a loss of information about the real flow path and leads to biases of flow length. To minimize these errors, a distance transform method for correcting the D8 algorithm (named as TD8 in this study) was proposed by Butt and Maragos  and de Smith . In their method, flow distance in all eight possible directions, i.e., one unit length in orthogonal direction and unit length in diagonal direction, is multiplied by a coefficient of 0.96194. The transformed distance field represents an averaged approximation to the true one, without considering the specific bias between the direction of D8 and the streamline within individual raster.
 Since it is widely accepted that the D8 algorithm does a poor job when applied to divergent landscapes, many multidirectional flow algorithms have been proposed [Bogaart and Troch, 2006]. The D∞ algorithm produces a more reasonable flow direction field, which can be approximately in accordance with streamlines. However, the flow distance for the D∞ method, developed by Bogaart and Troch , is determined by proportioning flow between two downslope pixels. The systematic errors of length or flow distance calculations have not been diminished in this method. In this study, we suggest a flow distance transform procedure of the D∞ method for deriving hillslope width functions in comparison with D8, D∞, and TD8 methods.
 Our new algorithms for determination of hillslope width functions, based on grid DEMs, include three parts: (1) derivation of natural hillslope units, (2) flow distance transforms, and (3) derivation of hillslope width functions.
3.1. Derivation of Natural Hillslope Units
Fan and Bras  illustrated a method to partition a catchment into large hillslopes. Three procedures were included in their method: (1) definition of the channel network, (2) definition of the ridgeline network, and (3) delineation of the drainage area for each channel head. When the three layers together with the catchment boundary are overlaid, the natural hillslope units are derived. While this is an on-screen digitizing method, it is suitable for manually deriving the natural hillslope units on a contour map. Delineating a catchment into hillslopes is still a difficult task as reported by Bogaart and Troch , and more detailed procedures are needed.
 In the modified algorithm running on raster DEMs for this study, a portion of the Fan and Bras  procedures (including (1) and (3)) were retained, and then all the hillslopes were further divided into headwater and side slope hillslopes (hereafter referred to simply as side slopes) as defined by Bogaart and Troch .
 First, the channel network was defined. The channel nodes were numbered as 0 for the source nodes, 1 for the intermediate nodes, and 2 for the outlet node to distinguish the different types of nodes and channel segments (Figure 1a). Channel segments were defined as source channel segments, which originate from a source channel node, or intermediate channel segments, which are located between two intermediate nodes or one intermediate node and the outlet node. All the nodes and channel segments were represented by a grid network (Figure 1a). Then subcatchment that drains into each node of the channel segments, and the headwater hillslope that drains into source nodes were defined. It is important to note that the intermediate channel segments are always surrounded by two side slopes while three hillslopes drain to the source channel segments. So, each subcatchment was divided into two side slopes for intermediate channels or three hillslopes for source channel segments with one headwater and two side slopes (Figure 1b). The procedures for distinguishing headwater hillslopes from the subcatchment and for defining the left and right side slopes (Figure 1b) were given below.
 A flow vectors matrix on the basis of DEM was used to subdivide the subcatchment into different types of hillslopes. The source nodes were first evaluated to determine all the inflow pixels that belong to the headwater hillslope (Figure 1b). Then downstream pixels (e.g., (i, j) in Figure 2a) along the channel were identified to isolate adjoining pixels that drain directly to the channel pixels. Flow directions of all the eight surrounding pixels were judged to determine whether they belong to the side slopes. A procedure, which is identical to the Martz and Garbrecht  algorithm for defining left and right bank subwatershed, was used to determine left and right side slope pixels. As shown in Figure 2a, pixels highlighted in blue represent the channel segment, and two yellow pixels belong to left side slope and one orange pixel belong to right side slope. The angle (α), between the direction of upstream pixel (e.g., (i + 1, j)) and the study pixel (e.g., (i, j)), was determined by turning the direction of (i, j) clockwise, until the two directions overlap (Figure 2b). All pixels within this angle (α) belong to right side slope pixels; otherwise, they belong to the left side slope pixels. Once all surrounding pixels were assessed, the inflow pixels of these pixels were determined, and this process was repeated until all upslope areas were defined. The process for upslope area calculation was programmed using a recursive procedure developed by Tarboton . Figure 1b gives an example illustrating the result of headwater and left and right side slope definitions around a channel segment.
3.2. Flow Distance Transforms
 Within each hillslope, flow distance could then be calculated according to flow direction assigning methods, e.g., D8 and D∞ methods. Namely, distance calculations depended on the flow direction fields. For grid computation of flow distance in both D8 and D∞ methods, the smooth or straight streamlines had to be approximated by zigzag path over the grid network [Paz et al., 2008]. As shown in Figure 3a, the orthogonal (A9→I9) and diagonal (A9→I1) direction for streamlines could be perfectly approximated by the raster structure, while the nonorthogonal or nondiagonal directions (e.g., A9→E1), were represented by zigzag paths (e.g., A9→A8→C6→C4→E2→E1 for D8). If a raster length is set 1 unit, the real length for line segments A9→E1 is 8.944 units and the approximated value by zigzag paths is 9.657 units. There was a bias of about 8% between them.
 It is clear that flow distance determination largely depended on how to assign flow direction for each pixel. Here we assumed that direction of streamflow lines ( in Figure 3b) could be substituted by the flow direction of the D∞ method ( in Figure 3b). For each pixel, the overestimated flow distance of D8 could be transferred to its real value by projecting D8's flow segments on the directions of D∞. As the direction of any streamlines ( ) were approximated by D∞ method (e.g., and in Figure 3b), an angle of θ1 between the directions of D8 and D∞ (e.g., and in Figure 3b) could be used as included angle between D8 and the streamlines (e.g., and in Figure 3b). Since is approximately parallel to , θ2 is equal to θ3 in Figure 3b. Thus, the following flow distance transform equation was obtained:
where P1P3 is the real flow length of streamline, P1P2 and P2P3 are orthogonal and diagonal distances by the D8 method, P1M1 and P2M2 are projected distances of P1P2 and P2P3 along the direction defined by D∞.
 This flow distance transform method, named TD∞, is quite different from the flow distance algorithm suggested by Bogaart and Troch  on the basis of the D∞ method (abbreviated as D∞ herein). For computation of flow distance by D∞, the flow direction (e.g., in Figure 3b) on a facet is disintegrated along the cardinal and diagonal directions on the edge of the facet. So flow within each pixel has at least two flow paths to end up in the channel network, and for each flow path, it is zigzag just as D8 method. The estimated flow distance for each pixel is a weighted average value of the multiple paths. Therefore, for each path, flow distance estimated by D∞ is overestimated just like the D8 method, which leads to overestimation of flow distance for each pixel.
3.3. Derivation of Hillslope Width Functions
 As flow distance field was determined, a probability density function (PDF) of flow distance was determined. The PDF of flow distance was used to derive a real hillslope width function. The specific procedures are given below.
 For each pixel, flow distance could be calculated by tracing through a flow directions matrix. This made it possible to determine an equidistant belt (Figure 4a). Grid cells were sorted according to their distance from the channel, then the number of grid cells that falls onto the belt was tallied to derive the PDF, p(x) of flow distances (Figures 4a and 4b). The integration of p(x) equals 1, as given in the following formula:
where L is the entire length of the hillslope and x is the hillslope direction.
 If the hillslope width function (HWF) was given as w(x), hillslope area was obtained by its integration, via
where w [L] is the HWF, S [L2] is the area of the hillslope, n is the number of equidistant belts, and k is the order number of equidistant belts of .
 In order to make the PDFs of flow distance and HWFs comparable, a representative hillslope width, wr (L2 per unit length), was used to convert PDFs to HWFs.
where Lu (=1) is the unit length along x direction (Figure 4c). So, the physical meaning of wr, was explained as the hillslope area per unit length Lu, as the whole hillslope area was aligned in one row with its length equal to Lu. wr was defined as hillslope width transfer functions, HWTFs (Figure 4c).
 In addition, the general distribution pattern of w(x) requires some explanation. For headwater hillslopes, theoretically, the hillslope converges to one point, so the following equations are valid:
 For a natural hillslope, the single point (usually at mountain peak) of the maximum distance from the channel also exists for both headwater and side slopes, and the maximum distance was defined as the hillslope length, L. So, for any type of hillslopes, one assumed that
 Note, however, for side slope, w(x) was usually not equal to 0 at x = 0, as water from side slopes will drain divergently toward a channel segment instead of a single point, i.e., the channel source for a headwater hillslope.
 These algorithms, mentioned above, have been implemented in the software package DigitalHydro (J. T. Liu, Digital drainage network automatic extraction software, DigitalHydro V1.0, copyright 2009SR033528, China, 2009), which is a GIS tool and was originally developed for individual specific interests and is now used in numerous scientific research studies and engineering projects. DigitalHydro was first developed for defining the flow vectors matrix and digital channels. Many newly developed functions, e.g., rainfall or elevation interpolation, curvature calculation, determination of topographic index, etc., have been included in this software. While the definition of flow vectors was one of the basic functions of this software, using the D8 algorithm in the original version, later, several multiple direction methods, e.g., the Tarboton  D∞ method for computing the upstream area also were included in the software.
4. Performance Evaluation
4.1. Abstract Hillslopes
 We evaluated the above algorithms for five abstract hillslopes: (1) divergent-convex hillslope, (2) divergent-concave hillslope, (3) parallel-planar hillslope, (4) convergent-convex hillslope, and (5) convergent-concave hillslope (Figure 5). The shapes for these selected five hillslopes with concave, planar, and convex longitudinal profiles, and convergent, parallel, and divergent platforms represent some basic hillslope types in the real world. The theoretical hillslope elevations and width functions were listed in Table 1 by referring to Pan et al.  and Kanamaru . In addition, the average slope for all five hillslopes was set to be equivalent.
Table 1. Elevation Functions, Theoretical Width Functions, and Other Hillslope Geometry for Five Abstract Hillslopes
Here x and y are projection coordinates in longitude and latitude directions.
R and r are radius corresponding to outlet arc, and t is the distance to outlet arc in radial direction.
 The computation domain was defined through projections of the elevation surface (Figure 5). In order to generate grid DEMs for algorithms evaluation, continuous hillslope surfaces were discretized into nrows × ncols DEMs (nrows and ncols are the numbers of rows and columns, respectively) with horizontal resolution hr. The DEMs for all these five hillslopes were the same size. The coordinates for each pixel in the DEM were defined as
where i and j are row and column number for the pixel (i, j) and x and y are projection coordinates in longitude and latitude directions, respectively.
 Algorithms for derivation of HWFs were evaluated by comparing theoretical width functions with derived ones according to D8 and D∞ algorithms with and without flow distance transforms (D8, D∞, TD8 and TD∞). Tests were performed on the abstract hillslopes with a fixed DEMs' resolution (hr = 2 m); nrows and ncols were set equal to 201 for these 2m DEMs. The results were evaluated by the root-mean-square error (RMSE) of the derived hillslope width functions as follows:
where n is the number of data series, is theoretical hillslope width function, and is derived hillslope width function.
 The derived and theoretical hillslope lengths and correspondent relative errors for the five abstract hillslopes by different flow distance computation methods were listed in Table 2. For D8 and D∞ methods, flow distances were significantly overestimated. TD8 method improves flow distance computation for convergent hillslopes (hillslopes 4 and 5 in Table 2), but does not improve flow distance computation for divergent and planar hillslopes (hillslopes 1, 2, and 3 in Table 2). TD∞ method significantly improved the derived hillslope lengths, compared with those from D∞ method. Relative errors between derived and theoretical hillslope lengths, Re, are 7.5% by D∞ method and were reduced to be −0.45% and −0.43% by TD∞ method for hillslopes 1 and 2, Re are 8.3% by D∞ method and were reduced to be 4.2% for hillslopes 4 and 5 in Table 2.
Table 2. Derived and Theoretical Hillslope Lengths (Dl and Tl) and Correspondent Relative Errors (Re) for the Five Abstract Hillslopes by Different Flow Distance Computation Methods
 For convergent hillslopes (hillslopes 4 and 5 in Table 2), although hillslopes length by TD8 was relatively more accurate than that of TD∞, the derived hillslope width functions by TD8 in Figure 6 presents a zigzag formation, which were obviously unreasonable. Moreover, as TD8 method was used in areas with large part of parallel-planar hillslope (e.g., hillslope 3 in Table 2), it will inevitably underestimate the true flow distance. This is because it cannot tell whether or not raster flow directions coincide with the direction of streamlines, and thereby TD8 method transforms flow distance in all flow directions, and never considers whether or not the flow direction within a pixel is matching the streamlines' direction. For example, for the hillslope 3, the relative errors Re were 0 for D8, D∞ and TD∞ but Re was −4.8% for TD8. Therefore, our developed TD∞ method performed satisfactorily for deriving hillslopes length for all types of hillslopes.
Figure 6 showed the derived hillslope width functions versus the theoretical ones for 2 m DEMs, and the corresponding RMSEs between the derived and theoretical width functions were listed in Table 3. Great differences were among different types of hillslopes. For instance, RMSEs were very close to zero for parallel-planar hillslopes except for TD8 method, and the result was the same as flow distance derivations. RMSEs were moderate for divergent hillslopes (hillslopes 1 and 2 in Table 3), while greatest biases existed between the theoretical and the derived functions ones for convergent hillslopes (e.g., hillslopes 4 and 5 in Table 3). As shown in Figures 6d and 6e, for all the convergent hillslopes, the derived hillslope length, i.e., distance to channel (DC), has been enlarged for all the used methods. As hillslope area keeps constant, and according to equation (4), the derived hillslope width functions would be a little smaller than the theoretical ones (Figure 6) to compensate the lengthen part of hillslopes. For further revealing the large biases between the derived and theoretical hillslope widths, abstract hillslope 5 in Figure 5 was evaluated in Figure 7. It was obvious that biases were greatly increased while it deviated from the centerline (e.g., l-l' in Figures 7a and 7b). On the contrary, the errors were very small and the values closed to zero when it happened to be on the centerline except for the TD8 method because that area near line l-l' was relatively planar. The TD8 method tended to lead to a larger bias at such regions because of a larger error of the derived hillslope length (Table 2). As shown in Table 3 and Figure 6, our developed TD∞ significantly decreased the error between the derived and theoretical width functions. Although the TD8 method reduced computation errors of hillslope lengths for hillslopes 1, 2, 4, and 5 in Table 2, it did not reduce the errors of derived width functions because of the wave-like curves shown in Figure 6. In addition, the derived hillslope width functions were always with sharp decreasing tails (Figure 6) because the lengthen part of the hillslopes became relatively small and the area decreased significantly as the flow distance surpassed the normal length of hillslopes (e.g., 400 m contour line in Figure 7).
Table 3. Root-Mean-Square Error (RMSE) of the Derived and Theoretical Hillslope Width Functions for the Five Abstract Hillslopes by Different Flow Distance Computation Methodsa
Units are meters.
5. Real-World Application
5.1. Study Catchment Description and Materials
 The 1.35 km2 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 Hydrology-Water Resources and Hydraulic Engineering (Hydro-Lab) 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.
 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.
 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.
 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 km2 area was divided into 0.47 km2 left bank side slope, 0.37 km2 right bank side slope, and 0.51 km2 headwater hillslope. For each hillslope, the hillslope length and average slope were also computed (Table 4).
Table 4. Hillslope Geometry of the Hemuqiao Catchment
The first number in this column represents the channel number, and it is associated with the second number, 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.
Average contour curvature.
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.
 The algorithms of HWTFs were used to determine the HWFs for each hillslope. According to equation (8), the value of wr 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.
 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.
Troch et al.  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. Curve-Fitted Results for Natural Hillslopes of the Hemuqiao Catchment
 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 parallel-planar 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.
 A procedure for automatic derivation of natural hillslope unit based on Fan and Bras's  methodology was presented in this study, and the algorithm implemented Fan and Bras's method on a grid network. The procedures of our deriving hillslope width function based on grid DEMs can be summarized as follows: first, the whole catchment was divided into headwater hillslopes and side slope hillslopes, and each side slope was further divided into right and left side of hillslopes. Second, On the basis of hillslope unit derivation, a flow distance transforms method, TD∞ based on grid DEMs was proposed. Then the equidistant belts and the probability density functions (PDFs) of flow distance were determined by tracing the flow distance field. Last, on the basis of the transformed flow distance field, the hillslope width transfer functions (HWTFs) were derived to convert probability density of flow distance into a hillslope width function.
 The results in the five theoretical hillslopes demonstrated that flow distance by grid DEM-based algorithms of D8 and D∞ were always overestimated and thus may result in errors of derivation of hillslope width functions. The TD8 method, a transformed D8 method proposed by Butt and Maragos  and de Smith , can be used to decrease errors in flow distance for the divergent and convergent hillslopes, but this method increases computation errors for the parallel-planar hillslopes and hillslope width function computation. In this study, we developed the TD∞ method which transforms the zigzag flow distance by multiplying a variable coefficient that equals to the cosine value of the included angle between the direction of D8 and D∞ within individual pixels. The TD∞ method improved the flow distance estimation and was proven to be more accurate than D8 and D∞, and TD8 methods in computation of HWFs.
 Mathematic expression of HWFs is very important for deriving analytical solution of HSDMs. DEM-derived HWFs in a natural catchment (Hemuqiao Catchment) demonstrated that the derived natural hillslopes by DEMs were a composition of different types of simple hillslopes, e.g., convergent, parallel, and divergent hillslopes. HWFs for the derived natural hillslopes cannot be simply expressed by the exponential function, but can be expressed by the Gaussian function with different parameter values.
 By introducing the hillslope width function, the HSDMs could be explicitly expressed to describe subsurface flow and saturation along complex hillslopes. Our study offers efficient and accurate methods for dividing a catchment into hillslope units and for deriving hillslope width functions through flow distance transforms algorithm, TD∞, which provides general improvements for derivation of flow pathways needed in many hydrologic research fields.
 This work was supported by the National Natural Science Foundation of China (grants 40801013, 51190091, 41030636, and 51079038), the National Natural Science Foundation of Jiangsu Province (BK2010516), the China Postdoctoral Science Foundation funded project (201003572), and the Special Fund of State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering (2010585612). We thank Rachael Hoagland (U.S. Geological Survey) for her reviewing the English language for this paper.