WATER RESOURCES RESEARCH, VOL. 39, NO. 12, 1332, doi:10.1029/2003WR002576, 2003
A physically based model for calculating contributing area on hillslopes and along valley bottoms John B. Lindsay Department of Geography, University of Western Ontario, London, Ontario, Canada Received 9 August 2003; accepted 12 September 2003; published 3 December 2003.
[1] Most existing methods of calculating contributing area are unable to accurately
model the pattern of contributing area on hillslopes and along valley bottoms. This paper describes a new flow algorithm, the adjustable dispersion routing algorithm (ADRA). Rather than calculating contributing area using predetermined flow characteristics that are insensitive to location in the landscape, ADRA predicts channel location and adjusts flow characteristics accordingly. ADRA increases the degree of flow divergence downslope from divides until a channel head is detected. Channel head locations are estimated on the basis of a user-defined threshold of an area-slope function. Therefore the algorithm overcomes the problems of aggregated flow on hillslopes and divergent flow along valley bottoms. The pattern of catchment area produced by ADRA was compared with similar patterns calculated using a similar flow algorithm for a variety of natural landscapes. ADRA produced patterns of contributing area that were more consistent with the theory of channel INDEX TERMS: 1824 Hydrology: Geomorphology (1625); 1848 Hydrology: Networks; initiation. 1860 Hydrology: Runoff and streamflow; 1894 Hydrology: Instruments and techniques; KEYWORDS: flow routing, contributing area, digital elevation models, stream network extraction, channel initiation Citation: Lindsay, J. B., A physically based model for calculating contributing area on hillslopes and along valley bottoms, Water Resour. Res., 39(12), 1332, doi:10.1029/2003WR002576, 2003.
1. Introduction [2] Contributing area is the area upslope of a location in a catchment from which runoff is captured. In practice, contributing area is calculated per unit contour length, referred to as the specific catchment area, a. The utility of a as a surrogate for runoff volume makes it an essential parameter for modeling hydrologic, geomorphic, and other environmental processes. Specific catchment area has been used to estimate soil wetness [Beven and Kirkby, 1979], soil erosion and deposition potential [Mitasova et al., 1996], and to extract stream networks and watersheds [Jenson and Domingue, 1988] from digital elevation models (DEMs). Although digital terrain data are now abundant and software packages for analyzing these data are common, a remains inherently difficult to estimate accurately [Gallant et al., 2000]. [3] Specific catchment area is calculated using flow routing algorithms that direct and accumulate runoff over DEMs. Flow routing algorithms differ in the way that they calculate flow direction and in the method used to divide flow between each downslope neighbor. Some authors have categorized flow algorithms based on whether they allow for divergence [e.g., Wolock and McCabe, 1995]. Hence algorithms that are unable to disperse flow are single-flowdirection (SFD) algorithms, and those that are capable of divergence are multiple-flow-direction (MFD) algorithms. The most common SFD algorithm is referred to as steepest descent or D8 [O’Callaghan and Mark, 1984]. Commonly used MFD algorithms include FD8 [Freeman, 1991; Quinn et al., 1991], DEMON [Costa-Cabral and Burges, 1994], and D1 [Tarboton, 1997].
[4] Most existing flow algorithms do not explicitly differentiate between catchment hillslopes and channels, which is an important distinction given the contrast in geomorphic and hydrological processes operating on both. SFD algorithms are suited to modeling incisive channelized flow along valley bottoms but are unable to simulate divergence on hillslopes. MFD algorithms yield more realistic diffusive flow patterns on divergent hillslopes, but often result in braiding along valley bottoms. Braiding is an undesirable artifact that can cause discontinuous stream networks [Gallant and Wilson, 2000] and inappropriately decrease a downstream [Quinn et al., 1995]. The problem with existing flow algorithms is that the amount of divergence is determined by local slope or curvature without consideration of whether divergent flow is appropriate given the relative position of each cell in the landscape. The challenge in overcoming this problem is that the flow algorithm must estimate channel network extent during processing. This paper describes a flow routing scheme that has been developed to address these issues.
2. Background [5] The FD8 algorithm is unique among existing flow routing algorithms in that it is possible to adjust the overall degree of divergence. The Freeman [1991] implementation of FD8 calculates the fraction (F ) of a apportioned to each downslope neighbor, i, by, Fi ¼
maxð0; tan Sip Þ 8 P maxð0; tan Sjp Þ
ð1Þ
j¼1
where S is the slope between cells. Summing Fiak for each upslope neighbor k yields ai. A larger value of p results in a
Copyright 2003 by the American Geophysical Union. 0043-1397/03/2003WR002576$09.00
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Figure 1. Fraction of a received by two downslope neighbors of varying slopes using the Freeman [1991] concentration parameter p.
greater fraction of a being apportioned to the lowest neighbor. As p ! 1, F ! 1 for the neighbor of steepest descent and F ! 0 for all other neighbors (Figure 1). [6] Many implementations of FD8 use a value for p between 1.1 – 6 but this practice results in a compromise in terms of modeling flow divergence. That is, single p values attempt to fit a catchment-wide average of flow divergence that results in flow on hillslopes appearing too convergent and braiding along valley bottoms. The same problem of hard-wired flow properties, which are indifferent to a cell’s location relative to the stream network, exists for the DEMON and D1 algorithms. Some researchers have suggested using hybrid flow algorithms to compensate for these problems. For example, equation (1) can be used on hillslopes until convergent topography is detected [Freeman, 1991] or a channel initiation threshold (CIT) catchment area is reached [Quinn et al., 1991], beyond which a is calculated using the D8 SFD flow algorithm. One criticism with this approach is that irregularities, or abrupt changes, occur at the transition to channelized flow [Gallant and Wilson, 2000]. Quinn et al. [1995] overcame this problem by increasing p continuously downslope until the CIT was reached, creating a downslope feedback between the degree of flow divergence and a. An adjustable power term determined how rapidly flow became convergent. This pattern of increasingly convergent flow toward channel heads is consistent with field observations [Quinn et al., 1995]. [7] There are several problems, however, with using p to adjust the degree of divergence. First, F is a nonlinear function of p, and therefore changing p an equivalent amount results in an unequal increase in the degree of flow convergence (Figure 1). Furthermore, when the slopes to neighboring cells are similar, substantial partitioning can occur even when p is large. When two or more cells have equal slopes, equation (1) will partition flow even if p equals infinity (Figure 1). This problem is further compounded by the practical limitation that large exponents can cause overflow errors in most computers.
[8] In addition to these technical issues, the threshold a used in the hybrid and Quinn et al. [1995] implementations of FD8 may not be physically realistic. Montgomery and Foufoula-Georgiou [1993] concluded that models with constant critical support areas for channel maintenance are theoretically and empirically less appropriate than slope-dependent critical support areas. This is because smaller contributing areas are needed to initiate channels on steeper slopes. If overland flow is assumed to be the mechanism for channel initiation, channel heads occur when the basal shear stress (tb) exceeds the critical shear stress of the ground surface (tcr). On the basis of a laminar flow model, Montgomery and Foufoula-Georgiou [1993] showed that the critical specific catchment area (acr) required for tb > tcr is, acr ¼
C ðtan qÞ2
C / t3cr ; q1 r
ð2Þ
where q is the local slope. C is a constant with units of length, which is proportional to tcr3 and inversely proportional to the steady state rainfall intensity, qr . [9] On the basis of equation (2), the transition from unchannelized to channelized flow occurs where aðtan qÞ2 C
ð3Þ
Montgomery and Dietrich [1992] and Montgomery and Foufoula-Georgiou [1993] found that this model could predict channel head locations. Therefore equation (3) is an empirical method for determining the channel network extent, although scatter in the relation can result from the spatial variation in tcr [Istanbulluoglu et al., 2002] and the assumption that the channel network is in a state of longterm equilibrium. Equation (3) is only necessarily valid at channel heads, i.e., a(tanq)2 can decrease downslope. [10] On the basis of field observations, Montgomery and Dietrich [1992] reported values of C between 25 m and
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Figure 2. Fraction of a received by two downslope neighbors of varying slopes using the ADRA concentration parameter n.
200 m for the Tennessee Valley area of northern California. Dietrich et al. [1993] and Montgomery and FoufoulaGeorgiou [1993] present several methods for estimating C for different regions. One of the most practical of these estimation methods is based on evidence that channel head locations define the limit to convergent topography [Montgomery and Dietrich, 1992]. Montgomery and Foufoula-Georgiou [1993] suggest that an appropriate value of C reflects the smallest value that does not result in the extension of the channel network onto the surrounding planar or divergent topography. Thus C is the smallest value that does not result in ‘‘feathering’’ of the channel network along headwater channels.
3. Adjustable Dispersion Routing Algorithm (ADRA) [11] The adjustable dispersion routing algorithm (ADRA) was developed to address some of the problems with existing algorithms. ADRA calculates Fi as,
Fi ¼
B maxð0; tan Si Þ maxð0; tan Si Þ C C B þ nBki 8 C 8 A @ P P maxð0; tan Sj Þ maxð0; tan Sj Þ j¼1
n¼
t aðtan qÞ2 < C 1 aðtan qÞ2 C
ð5Þ
where the transition function, t, is a curve that rises monotonically upward such that t(0) = 0 and t(C) = 1. Although other models may be suitable (e.g., a sigmoidal curve), a linear transition function was used in this paper, of the form: t¼
aðtan qÞ2 ; C
aðtan qÞ2 < C
ð6Þ
[13] Using the recursive drainage accumulation approach [Mark, 1988; Freeman, 1991], a is calculated for a cell only after each upslope neighbor has been solved. Thus, for any cell i, a(tanq)2 and u are known for each upslope neighbor k and ai = Fiak.
4. Flow Algorithm Comparison
1
0
tered, flow divergence is undesirable and all subsequent downslope cells are treated as channel cells. Thus
ð4Þ
j¼1
where the concentration parameter, u, ranges from zero to one and k equals one for the first cell detected with the maximum downward slope and zero for all other cells. For the cell with the maximum downward slope, F(u) is a line from S/S to one, and for all other cells with downward slopes, F(u) is a line from S/S to zero (Figure 2). [12] ADRA has been designed to vary u with the areaslope function a(tanq)2. If a cell has a low a(tanq)2 value (i.e., a(tanq)2 < C) then it is above the channel head and a low u should be used to divide a between downslope neighbors. Similarly, a high a(tanq)2 value indicates channelized flow and u should equal one. Once C is encoun-
[14] ADRA was compared with the Quinn et al. [1995] variable-p implementation of FD8, because this algorithm produces the most logically consistent pattern of a (i.e., divergent flow on hillslopes and convergent flow in valleys). Several researchers have used artificial surfaces (e.g., planes and cones) to evaluate flow algorithms because the actual contributing area is known for all locations [e.g., Freeman, 1991; Costa-Cabral and Burges, 1993; Tarboton, 1997]. However, this approach was inappropriate for evaluating ADRA and the Quinn et al. [1995] algorithm because artificial surfaces lack stream channels. The only alternative therefore was to compare the visual patterns of a generated by both flow algorithms in a variety of natural settings. Nevertheless, this approach offered valuable insight into the applicability of ADRA because theory indicates, at least qualitatively, what the spatial pattern of a should look like.
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Figure 3. Patterns of catchment area calculated using ADRA and the Quinn et al. [1995] algorithm for (a) a mountainous catchment in coastal Washington, (b) a moderately steep catchment in northern Vermont, (c) a low-relief catchment in Texas, and (d) a low-relief catchment in Oklahoma.
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The three data points in Figure 4 with slopes greater than 60 resulted from missing data in the SRTM-1 data, which occur on mountainsides. These missing data (given the value of 32768) were filled during preprocessing, yielding spurious ‘‘cliff faces’’ on the upslope side of the pit. These sites are recognized as channel heads by ADRA despite their small catchment areas because of their erroneously high slopes. The Quinn et al. [1995] algorithm does not include local slope in its estimation of channel extent and therefore is not similarly susceptible to this type of error. ADRA would not identify channel heads in these areas if missing data were removed using an interpolation scheme that approximates the original hillslope rather than using pit filling.
5. Summary
Figure 4. Relation between slope and estimated catchment area at channel heads. [15] Four study catchments were identified in different physiographic regions. These catchments represented ranges in relief (i.e., 56 to 786 m) and climate (i.e., marine, humid sub-tropical, humid continental, and midlatitude dry). Shuttle Radar Topography Mission (SRTM-1) DEMs [Farr and Kobrick, 2000] were used as the necessary input to the flow algorithms. The original DEMs, with grid resolutions of 1 arc-second, were projected into UTM coordinates and resampled to 30 m grids. The vertical precision of SRTM-1 DEMs is 1 m. [16] ADRA and the Quinn et al. [1995] algorithm produced very similar patterns of a, with divergent flow on hillslopes and fully convergent flow along valley bottoms (Figure 3). Stream channels were evident in the flow accumulation maps as single-cell wide networks of high a-values. Braiding did not occur using either flow algorithm. Stream network extent was marked by gradually increasing flow convergence near channel heads in each study catchment. The differences between the flow accumulations maps produced by ADRA and the Quinn et al. [1995] algorithm were most pronounced near valley heads where the rate of downslope flow convergence and the positioning and number of channel heads differed. These dissimilarities were most apparent in Figure 3b because of the large hillslopes and low drainage density of the Vermont study catchment. Conversely, the differences between the visual patterns of a produced by the two flow algorithms were negligible for the low-relief catchment in Oklahoma (Figure 3d), because of the relatively high drainage density and little variation in slope. [17] The relation between q and the estimated value of a at channel heads is illustrated in Figure 4 for the mountainous catchment in Washington (Figure 3a). Figure 4 clearly demonstrates that a varied inversely with slope for ADRA while the Quinn et al. [1995] algorithm did not exhibit a similar relation. Thus the spatial pattern of a derived using ADRA was more consistent with the theory of channel initiation. However, Figure 4 also demonstrates that ADRA can be more susceptible to a particular type of DEM error.
[18] Most existing SFD and MFD routing algorithms are inadequate for modeling contributing area over entire catchments. Rather than basing the degree of divergence on local slope alone, ADRA divides flow using local slope and an estimate of the cell’s position relative to channel heads. Thus, by incorporating a model of channel initiation directly into the flow calculation, ADRA can model catchment area on hillslopes and along valley bottoms. The transition between hillslope and channel flow is based on a physically realistic model, capable of locating channel heads. Also, the proportion of the catchment area that a cell receives is a linear function of the concentration parameter, which has a range from zero to one. In contrast, exponent based concentration parameters divide flow nonlinearly and range from zero to infinity. A comparison between ADRA and the Quinn et al. [1995] flow algorithm, across a range of basin types, showed that the two methods produce visually similar spatial patterns of a, but differ in the representation of flow accumulation near valley heads. ADRA’s flow accumulation patterns were found to be more consistent with the theory of channel initiation. However, if DEMs contain missing data values, ADRA can yield artifact channels because of erroneous slopes. Thus care is needed to correctly handle missing data values. [19] Acknowledgments. The author thanks I. Creed and three anonymous reviewers for their invaluable comments and advice. This work was supported by the Ontario Graduate Scholarship.
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O’Callaghan, J. F., and D. M. Mark, The extraction of drainage networks from digital elevation data, Comput. Vision Graphics Image Processes, 28, 323 – 344, 1984. Quinn, P., K. Beven, P. Chevallier, and O. Planchon, The prediction of hillslope flow paths for distributed hydrological modelling using digital terrain models, Hydrol. Processes, 5, 59 – 79, 1991. Quinn, P. F., K. J. Beven, and R. Lamb, The ln(a/tanB) index: How to calculate it and how to use it within the TOPMODEL framework, Hydrol. Processes, 9, 161 – 182, 1995. Tarboton, D. G., A new method for the determination of flow directions and upslope areas in grid digital elevation models, Water Resour. Res., 33(2), 309 – 319, 1997. Wolock, D. M., and G. J. McCabe Jr., Comparison of single and multiple flow direction algorithms for computing topographic parameters in TOPMODEL, Water Resour. Res., 31(5), 1315 – 1324, 1995.
J. B. Lindsay, Department of Geography, University of Western Ontario, London, Ontario, Canada N6A 5C2. (
[email protected])
