Discharge-Area relations from Selected Drainages on
the Colorado Plateau:
A GIS Application
CEE 6440-GIS in Water Resources Term Project
Fall 2005
W. Scott Cragun
Geology Department
Introduction
Quantitative analyses of fluvial systems are important for understanding the fundamental relations between hydrologic parameters and the factors that control variations between different climatic or topographic regions. In the past, direct field or map measurements of parameters such as discharge, drainage area, slope, channel width, and channel depth were made in order to define empirical equations and predict of fluvial characteristics within basins where few measurements had been made (i.e. Leopold and Maddock, 1953; Howard and Kerby, 1983). While these studies have established valuable hydrologic relations that remain a solid foundation of many fluvial models, the ability to more precisely measure fluvial parameters has increased due to technological advances such as Geographic Information Systems (GIS).
The family of equations collectively known as the stream power laws is an example of an empirically derived fluvial model with parameters that can be precisely defined in a GIS. The stream power law is used to model aggradation and incision within a fluvial system and is generally written in the form:
(1) Z = kQmSn
where Z is incision, Q is discharge, S is channel slope, m and n are exponents whose exact values are debated (ratio of m/n = 0.3-1), and k is the bedrock resistance to erodibility. Validations of this model have been reported from several drainages in temperate climatic regimes, including the California Coast Range (Snyder et al., 2000) and the Cascade Mountains in Washington State (Stock and Montgomery, 1999). Field-tests in arid and semiarid environments are less common, although data reported from five drainages in New Mexico suggest that the stream power law does not always agree with observed rates of fluvial incision (Mitchell, 2000).
The substitution of contributing area (A) for discharge (Q) is a common assumption in many fluvial models, including the stream power laws. When contributing area (A) is substituted for discharge (Q) in the stream power law, it results in the new equation:
(2) Z = kAmSn.
It is commonly held that a power function defines the empirical relation between discharge and area in the form:
(3) Q = bAc
where b is a proportionality constant and c is an exponent that scales discharge to drainage area.
Appropriate values of c generally range from 0.65 to 1, although a linear or nearly linear relation (c = ~1) between discharge and area is most commonly used (Whipple, 2004). For example, a plot of mean annual discharge versus contributing drainage area for several basins in western Washington indicates an appropriate c value of 0.95, supporting the nearly linear relation between discharge and area (Figure 1).
Figure 1. Plot showing drainage area vs. mean annual discharge for several drainages in western Washington; regression line is a best-fit power function: Q = 0.08A0.95, r2 = 0.82 (from Mitchell, 2000).
The validity of assuming a linear or nearly linear relation between discharge and drainage area may not be valid in all environments, especially in semiarid landscapes where there is not a large influx of water from local catchments. This may be a common source of error in fluvial models attempting to extend hydrologic relations beyond the bounds in which they were previously defined. This study attempts to quantify the relation between discharge and area in the semiarid landscape of the Colorado Plateau using a GIS to compare contributing drainage area to observed discharge from several basins.
Objectives
The goal of this study is to test whether the discharge-area relation in the semiarid environment of the Colorado Plateau is linear as shown to be true in the temperate climatic zones of the Pacific Northwest. This study will attempt to answer 2 questions:
1) Is the simple substitution of contributing drainage area for discharge a valid assumption in semiarid fluvial systems?
2) Can runoff in these semiarid basins be predicted using a simple precipitation-runoff relation?
A quantitative analysis of this nature will provide a basis upon which further testing of the discharge-area relation in semiarid environments can completed and will ultimately provide us with a better understanding of the fluvial systems within the Colorado Plateau.
Study
Area
Six drainage basins were selected from the Colorado Plateau for analysis: the Fremont, Paria, Little Colorado, Upper Colorado, Gunnison and San Juan Rivers (Figure 2). These drainage basins were selected due to the availability of USGS gage station information and their nearly complete coverage of the plateau region. In addition, the six drainage basins chosen vary in size and will allow an analysis of relatively small and relatively large basins. The first three basins (the Fremont, Paria, and Little Colorado) have been designated in this study as the “southern basins” and the latter three (Upper Colorado, Gunnison, and San Juan) have been designated as the “northern basins”. This differentiation is made in order to distinguish between drainages with hydrographs (Figure 3) that are dominated by late fall monsoonal storms (southern basins) and drainages with hydrographs that are dominated by early spring runoff from snow melt (northern basins). This distinction will be important later on in the analysis.
Figure 2. Map showing the location of the six drainage basins examined in this study.
Northern
Drainages
Southern
Drainages
Figure 3. Hydrographs
from the six drainage basins selected from the Colorado Plateau. The distinction between basins with
hydrographs dominated by early spring snow runoff and basins dominated by late
fall monsoonal bursts is made.
Methods
In order to address the two questions posed in the objectives of this project, the methods and results will be split into two parts. Part I will address the discharge-area relation of the six basins selected from the Colorado Plateau; part II will address the prediction of runoff from precipitation values.
Part
I:
1) Thirty-meter DEM’s from each of the six drainage basins were downloaded from the USGS seamless data website (http://seamless.usgs.gov/) and re-projected in UTM zone 12 coordinates. Each DEM was re-sampled at 100-meter cell sizes in order to reduce the size of the files (Figure 4).
Paria Little
Figure 4. Raw DEM's from the six basins used in this study.
2) The Terrain Analysis Using Digital Elevation Models (TauDEM) tools were downloaded from Dr. David Tarboton’s website at Utah State University (http://hydrology.neng.usu.edu/taudem/) and were added to the ArcMap toolbar.
3) TauDEM tools were then used on each of the drainage basin DEM’s in order to define flow directions and ultimately determine flow accumulation at each grid cell (Figure 5). As a side note, due to the large size and complexity of the terrain, the first step in the TauDEM tools (pit filling) took up to 12 hours. The largest rasters had approximately 4000 columns and 3000 rows.
Figure 5. Flow accumulation gird of the San Juan drainage produced using the Dinf flow directions of TauDEM. USGS stream gage locations shown in red dots.
4) Once TauDEM had been run on each of the rasters, USGS stream gage data were
obtained from the USGS national daily streamflow database at (http://waterdata.usgs.gov).
5) Latitude and longitude of each USGS stream gage were recorded in a .dbf table and imported into ArcMap using the “add xy data” operation. A point shapefile was then created and re-projected in UTM zone 12 coordinates in order to display the stream gage locations in each drainage basin (Figure 5).
6) Zooming in on the location of each USGS gaging station allowed the contributing drainage area (number of cells x 0.1 km x 0.1 km = km2) at each cell to be manually extracted (Figure 6) and recorded in a Microsoft Excel spreadsheet. The exact location of USGS stream gages was often directly on the flow accumulation grid; however, when it was not, the nearest major stream network cell was selected. There were never any gages that were more than ~4 cells away from the major flow network.
Figure 6. Showing the USGS stream gage on the flow accumulation raster, used to extract contributing drainage area.
7) Mean annual streamflow was calculated for each USGS stream gage. Record lengths varied from 3 to 93 years with an average record of 48 years.
8) Contributing drainage area as determined in ArcMap using the TauDEM tools was also compared to the drainage area reported by each USGS stream gage. In general, the two values agree quite well; however, some large discrepancies are observed between calculated and reported drainage areas (Table 1). These differences warrant the use of the GIS calculated contributing area in subsequent analyses.
|
Drainage Basin |
GIS Contrb. Area (km2) |
USGS Contrb. Area (km2) |
Difference |
|
Upper Colorado 1 |
167 |
138 |
29 |
|
Upper Colorado 2 |
834 |
837 |
3 |
|
Upper Colorado 3 |
2,044 |
2,044 |
0 |
|
Upper Colorado 4 |
6,170 |
6,169 |
1 |
|
Upper Colorado 5 |
11,398 |
11,380 |
18 |
|
Upper Colorado 6 |
15,607 |
15,574 |
33 |
|
Upper Colorado 7 |
20,714 |
20,849 |
135 |
|
Gunnison 1 |
1,345 |
1,362 |
17 |
|
Gunnison 2 |
1,933 |
1,919 |
14 |
|
Gunnison 3 |
2,396 |
2,388 |
8 |
|
Gunnison 4 |
2,623 |
2,621 |
2 |
|
Gunnison 5 |
10,306 |
10,269 |
37 |
|
Gunnison 6 |
14,518 |
14,576 |
58 |
|
Gunnison 7 |
19,113 |
20,533 |
|
|
San Juan 1 |
725 |
772 |
47 |
|
San Juan 2 |
3,256 |
3,186 |
70 |
|
San Juan 3 |
8,394 |
8,443 |
49 |
|
San Juan 4 |
18,674 |
18,752 |
78 |
|
San Juan 5 |
33,316 |
33,411 |
95 |
|
San Juan 6 |
37,679 |
37,814 |
135 |
|
San Juan 7 |
50,949 |
59,570 |
|
|
Paria 1 |
521 |
513 |
8 |
|
Paria 2 |
1,682 |
1,676 |
6 |
|
Paria 3 |
3,432 |
3,652 |
220 |
|
Fremont 1 |
1,947 |
1,945 |
2 |
|
Fremont 2 |
3,134 |
3,129 |
6 |
|
Fremont 3 |
10,304 |
10,772 |
468 |
|
Little Colorado 1 |
1,959 |
1,829 |
130 |
|
Little Colorado 2 |
2,235 |
2,189 |
46 |
|
Little Colorado 3 |
2,603 |
2,608 |
5 |
|
Little Colorado 4 |
21,009 |
20,906 |
103 |
|
Little Colorado 5 |
29,321 |
29,686 |
365 |
|
Little Colorado 6 |
31,725 |
32,074 |
349 |
|
Little Colorado 7 |
41,911 |
41,699 |
212 |
|
Little Colorado 8 |
59,740 |
68,528 |
|
Table 1. Contributing drainage area comparison between
GIS calculated and USGS reported values.
Bold values show differences of greater than 100 km2 and
circled values show differences of over 1000 km2.
9) Discharge could then be plotted against contributing drainage area for further analysis.
Part
II:
1) Average 30-year precipitation data for the western United States were downloaded from the Natural Resources Conservation Services (NRCS) PRISM website (http://www.wcc.nrcs.usda.gov/climate/prism.html), re-projected in UTM zone 12 coordinates, and re-sampled through cubic interpolation to 1000-meter grid cells (Figure 7).
Figure 7. Precipitation raster of the western US region.
2) The regional precipitation raster was then evaluated in “Raster Calculator” in order to clip out a separate precipitation raster for each of the six drainage basin DEM’s and to create a raster of equal extent and cell size for further analysis (Figure 8).
Figure 8. Precipitation raster for the San Juan drainage in mm/yr, showing the location of USGS gaging stations in red dots.
3) “Raster Calculator” was then used to convert the original precipitation units of 100ths of mm/year to mm/year.
4) In order to calculate the runoff from each grid cell, runoff coefficients had to be estimated. This was initially done for each drainage using the table in Figure 9 as a guide, adding the most appropriate value from each of the four rows to get a total runoff coefficient. This was a crude approximation and resulted in runoff overestimations in each drainage.
Figure 9. Table for estimating runoff coefficients in undeveloped drainages, taken from an online engineering textbook at: http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp0810.pdf.
5) “Raster Calculator” was then used to calculate the runoff from each grid cell in mm/year (Figure 10).
Figure 10. Raster Calculator window showing the calculation of ppt in mm/year to runoff in mm/year based on a runoff coefficient of 0.17 in the San Juan basin.
6) Runoff in mm/year was then converted to m3/year and then to m3/second using “Raster Calculator” (Figure 11).
Figure 11. Raster Calculator window showing the calculation of runoff in mm/year to m3/second.
7) TauDEM tools were then used to calculate flow (runoff) accumulation in each drainage basin using the precipitation raster (m3/sec) as the weighted grid (Figure 12).
Figure 12. TauDEM contributing area using the runoff raster as a weighted grid in order to determine flow accumulation.
8) Comparison was made between the observed mean annual discharge and predicted mean annual runoff values. Adjustments were made to the original runoff coefficient by calculating the percent difference between the observed and predicted values and returning to step 4 in order to select a more appropriate value.
Results
Past
I:
An initial plot (Figure 13) of the discharge-area relation for the six drainage basins selected in the Colorado Plateau indicates a large scatter in the data (r2 = 0.26), especially when compared to the tight cluster observed from the streams in western Washington data (r2 = 0.82).
Figure 13. Discharge-area relation plotted for all six drainage basin selected from the Colorado Plateau; dark line represents best-fit power function.
However, closer examination of the data reveals that each drainage basin has a unique downstream-discharge pattern. The Gunnison River loses discharge initially, followed by consistent increases downstream; discharge in the Upper Colorado River increases with greater contributing area; the San Juan River initially gains discharge, followed by a plateau of generally equal discharge along the remainder of its length; discharge along the Little Colorado River is complex, displaying several gains and losses; the Paria River displays a downstream increase in discharge that only increases from 0.2 to 0.8 m3/sec; and the Fremont River initially loses discharge, followed by a slight gain. In addition, the northern three drainages appear to cluster together, whereas the southern three drainages do not group tightly, yet appear to widely cluster below the northern drainages (Figure 14).
Figure 14.
Individual discharge-area plots for each of the six basins; dark lines
represent best-fit power functions for individual basins.
Making this distinction and plotting the northern and southern data separately reveals a good correlation between discharge and contributing area in the northern drainages, with an r2 value of 0.88 (Figure 15). The spread in the discharge-area data from the southern drainages is moderate, with an r2 value of 0.54. The c values calculated for the northern (0.70) and southern (0.65) drainages are significantly lower than the empirical value of near 1 as reported in western Washington and assumed in many fluvial models; however, they are still within the expected range of 0.65 to 1. It is also interesting to note that for the same contributing area, the southern drainages produce an order of magnitude less runoff than that produced for the same drainage area in the northern drainages.
Figure 15. Separate discharge-area plots from the northern and southern basins from the Colorado Plateau.
Part
II:
Initial runoff coefficient estimates were significantly too high. More appropriate values were found by readjusting the runoff coefficient based on the average percent over-estimation in each basin. In particular, the southern basins (the last three in Table 2) were found to have runoff coefficients that were one or two orders of magnitude less than initially estimated. The range of runoff coefficients in the six basins from the Colorado Plateau was found to be approximately 0.0084 to 0.3.
|
|
Basin |
Initial Estimate K |
Readjusted K |
|
|
Gunnison |
0.3 |
0.22 |
|
Northern |
Upper Colorado |
0.42 |
0.30 |
|
|
San Juan |
0.26 |
0.17 |
|
|
Fremont |
0.21 |
0.062 |
|
Southern |
Little Colorado |
0.14 |
0.0084 |
|
|
Paria |
0.24 |
0.031 |
Table 2. Showing
estimated and readjusted runoff coefficients for each drainage basin.
These readjusted runoff coefficients produced predicted runoff values that were, in general, very close to the observed mean annual runoffs for each of the 34 USGS gaging stations (Table 3).
|
Drainage Basin |
Mean Annual Q (cms) |
Calculated GIS Q (cms) |
Difference |
Ratio of Runoff |
|
Gunnison 1 |
13.000 |
6.286 |
6.714 |
1.068087814 |
|
Gunnison 2 |
6.430 |
8.638 |
2.208 |
-0.255614726 |
|
Gunnison 3 |
-- |
10.182 |
-- |
-- |
|
Gunnison 4 |
20.530 |
10.404 |
10.126 |
0.973279508 |
|
Gunnison 5 |
36.560 |
38.705 |
2.145 |
-0.055419196 |
|
Gunnison 6 |
57.200 |
54.060 |
3.140 |
0.058083611 |
|
Gunnison 7 |
73.400 |
70.374 |
3.026 |
0.042998835 |
|
Upper Colorado 1 |
1.792 |
1.132 |
0.660 |
0.583038869 |
|
Upper Colorado 2 |
7.787 |
5.534 |
2.253 |
0.407119624 |
|
Upper Colorado 3 |
7.504 |
12.617 |
5.113 |
-0.405246889 |
|
Upper Colorado 4 |
34.009 |
34.815 |
0.806 |
-0.023150941 |
|
Upper Colorado 5 |
59.097 |
63.382 |
4.285 |
-0.067605945 |
|
Upper Colorado 6 |
95.315 |
91.726 |
3.589 |
0.039127401 |
|
Upper Colorado 7 |
108.963 |
117.493 |
8.530 |
-0.07260007 |
|
San Juan 1 |
10.421 |
3.154 |
7.267 |
2.304058339 |
|
San Juan 2 |
17.302 |
10.610 |
6.692 |
0.63072573 |
|
San Juan 3 |
32.848 |
25.990 |
6.858 |
0.26387072 |
|
San Juan 4 |
58.191 |
48.512 |
9.679 |
0.199517645 |
|
San Juan 5 |
58.531 |
69.407 |
10.876 |
-0.156698892 |
|
San Juan 6 |
61.023 |
76.784 |
15.761 |
-0.205264118 |
|
San Juan 7 |
63.883 |
101.345 |
37.462 |
-0.369648231 |
|
Fremont 1 |
2.490 |
1.691 |
0.799 |
0.472501478 |
|
Fremont 2 |
2.120 |
2.388 |
0.268 |
-0.112227806 |
|
Fremont 3 |
2.760 |
5.677 |
2.917 |
-0.513827726 |
|
Little Colorado 1 |
0.614 |
0.268 |
0.346 |
1.291044776 |
|
Little Colorado 2 |
0.095 |
0.291 |
0.196 |
-0.673539519 |
|
Little Colorado 3 |
0.208 |
0.322 |
0.114 |
-0.354037267 |
|
Little Colorado 4 |
1.416 |
2.153 |
0.737 |
-0.342313052 |
|
Little Colorado 5 |
3.653 |
2.838 |
0.815 |
0.287174066 |
|
Little Colorado 6 |
-- |
3.011 |
-- |
-- |
|
Little Colorado 7 |
2.410 |
4.052 |
1.642 |
-0.405231984 |
|
Little Colorado 8 |
6.286 |
5.996 |
0.290 |
0.048365577 |
|
Paria 1 |
0.233 |
0.157 |
0.076 |
0.484076433 |
|
Paria 2 |
0.303 |
0.485 |
0.182 |
-0.375257732 |
|
Paria 3 |
0.801 |
0.984 |
0.183 |
-0.18597561 |
Table 3. Values
of observed USGS gage and predicted GIS runoff, showing the difference and
ratio of difference between the two.
Discussion:
Part
I:
There is a close relation between discharge and contributing drainage area in the northern drainages; however, the scaling exponent (c) is not nearly 1 (c = 0.70). This may suggest that streams in humid environments such as the Pacific Northwest are capable of generating greater discharges per unit area than in semiarid regions. This is observed when comparing the discharge-area plots of the Washington, northern drainage, and southern drainage data. In Washington, a contributing area of ~1000 km2 produces discharges of ~100 m3/sec (Figure 1). In contrast, for similar contributing areas, the northern and southern drainages on the Colorado Plateau produce discharges of only ~10 and ~1 m3/sec, respectively (Figure 15).
The larger scatter observed in the discharge-area data from the southern drainages may suggest that local geologic or climatic controls influence discharge generation in semiarid fluvial systems. Rivers running over porous material such as sandstones or unconsolidated sediments may result in discharge loss through increased groundwater penetration. In addition, high temperatures in desert environments may result in the loss of discharge through evaporation in the absence of a steady influx of water from local catchments.
In conclusion, streams in semiarid regions display varied discharge-area relations and careful consideration should be made when substituting contributing area for discharge in fluvial models beyond the scope for which they have been previously defined. The simple assumption that drainage area can be substituted for discharge may not always be true.
Part
II:
Estimated runoff coefficients vary widely between the six basins selected from the Colorado Plateau, 0.0084 to 0.30. This may suggest that local climate conditions such as monsoonal versus snow melt runoff influences the relation between precipitation and runoff and ultimately the runoff generation in each basin.
There does not appear to be a distinct pattern in runoff predictions that stands out in the data, suggesting that each basin has a unique relation between precipitation and runoff. In addition, the simplicity of characterizing an entire basin with a single runoff coefficient is not realistic. The complexities of land use, land cover, vegetation cover, soil type, geology, and individual storm intensity would control the relation between precipitation and runoff. At first glance, runoff can be easily predicted on the Colorado Plateau; however, all controlling factors would need to be considered in order to more accurately define the precipitation-runoff relation in individual catchments.
Potential
Sources of Error:
Large cell size (100 m) in the raster coverage may decrease the precision of estimating contributing area.
Sensitivity of streamflow record compared to precipitation record.
Acknowledgements:
Special thanks to Dr. David Tarboton for his help in skirting potential hazards and for the hours he spent helping resolve my questions.
References:
Howard, A.D., and Kerby, G., 1983, Channel change in badlands, Geological Society of America Bulletin, v. 94, pp. 739-752.
Leopold, L.B., and Maddock, T., 1953, The hydraulic geometry of stream channels and some physiographic implications, US Geological Survey Professional Paper 252, 57 p.
Mitchell, D.K., 2000, Stream power and incision of five mixed alluvial-bedrock streams, northern New Mexico, M.S. Thesis, University of New Mexico, 102 p.
Runoff coefficient estimation chart: http://www.dot.ca.gov/hq/oppd/hdm/pdf/chp0810.pdf
Stock, J.D., and Montgomery, D.R., 1999, Geologic constraints on bedrock river incision using the stream power law, Journal of Geophysical Research, v. 104, no. B3, pp. 4983-4993.
Snyder, N., Whipple, K.X., Tucker, G.E., and Merritts, D., 2000, Landscape response to tectonic forcing: digital elevation model analysis of stream profiles in the Mendocino triple junction region, northern California, Geological Society of America Bulletin, v. 112, pp. 1250-1263.
Tarboton, D., Terrain Analysis Using Digital Elevation Models (TauDEM) tools downloaded from: http://hydrology.neng.usu.edu/taudem/
Whipple, K.X., 2004, Bedrock rivers and the geomorphology of active orogens, Annual Reviews Earth and Planetary Sciences, v. 32, pp. 151-185.