Hillslopes are dominated by divergent, diffusive flow and sediment transport processes (soil creep, landslides, rainsplash, sheetwash). Channels are characterized by convergent, often linear transport (debris flows, rilling, channelized flow). Slope-Area plots are created for watersheds from DEM data. They are useful in delineating geomorphic process domains, the transitions between them, and the location of the hillslope-channel transition (channel heads). Mapping the spatial distribution of Slope-Area data has applicability to soil functional properties and digital soil mapping (Cohen et al., 2008).
Flint (1974) described the power law relationship between slope and stream magnitude.
S = cAα
S = Slope (local slope)
c = Slope scaling constant
α = Exponent coefficient
The log-log scatter plots used in Slope-Area analyses have a distinctive, interpretable shape; you can look at a curve and distinguish 2 primary regions that have meaning, assuming you are dealing with a transport-limited (soil covered) landscape. Diffusive “hillslope” points cluster on the left side, are loosely organized, correspond with low area values, convex landforms, and generally have a positive slope. Fluvial “channel” points cluster on the right side, correspond with high area values, concave landforms, and have a negatively sloping line. Between the hillslope cluster and channel cluster lies the inflection, or “transition” point (actually a zone more than a point). The channel initiation area, the flow accumulation threshold, is usually placed at the downstream end of the curve’s turnover, though opinions differ (Willgoose et al., 1991; Tarboton et al., 1992; Montgomery and Foufoula-Georgiou, 1993). By manually picking a particular threshold area on a plot (hillslope cluster to the left, channel cluster to the right), you are identifying the location where most channel heads will occur.
The Slope-Area relationship for the fluvial regime:
S = (U / β1 β3m1)1 / n1 A1 – m1m3 / n1 ∝ Aα
The Slope-Area relationship for the hillslope regime:
S = (U / D) Asα
S = Slope
U = Tectonic uplift
As = Flow accumulation at a pixel (specific area)
D = Creep (hillslope diffusivity)
α = Area exponent
β1 = …, constant in space and time
β3 = …, constant in space and time
m1 = …, constant in space and time
m3 = …, constant in space and time
n1 = …, constant in space and time
Slope is the hillslope gradient in percent rise for each cell in the DEM. Percent slope values are scale in log space.
Area is the upstream contributing area, also called flow accumulation. Every cell in a flowacc raster contains the total count of cells that flow into it. Pixels on summits or ridges have flowacc values near zero. Trunk streams have flowacc values in the 10′s to 100′s of thousands (higher for large watersheds). The highest flowacc value in a watershed is at the outlet. Area values are binned and plotted in log space.
We create a Slope-Area Plot of a single basin in this example. You need to first create Flow Accumulation and Slope rasters from your filled DEM. An optional Smoothed Slope raster is an alternative to Slope – it can make for a cleaner plot in certain cases.
CREATE INPUT RASTERS
Flow Accumulation: Spatial Analyst > Hydrology (create Flow Direction raster first)
Slope: Spatial Analyst > Surface > Slope (percent rise)
Smoothed Slope: Spatial Analyst > Neighborhood > Focal Statistics
* If ArcGIS gives you problems when these creating rasters, try xToolsPro or DEM Surface Tools extensions. These rarely fail and both are free. See ‘Extensions, Freeware & Links’ post.
EXTRACT RASTER VALUES
There are several ways to do this in ArcGIS. One is to convert the FlowAcc raster to points using Conversion Tools > Raster to Points tool. Now you have a shapefile. Next, use Spatial Analyst > Extraction > Extract Multiple Values to Points tool to append the values from other rasters (slope, smoothed slope) into new fields in the point attribute table. This tends to take awhile to process. There are certainly other database methods out there.
SET-UP SPREADSHEET: Open Office
Row 1 contains column headings.
Column A contains unordered list of flow accumulation values (integer) in cells A2 to A13058.
Column B contains unordered list of slope values (percent rise, float).
Column C contains unordered list of smoothed slope values (3×3, percent rise, float).
Column D is empty.
Column E contains list of bin break values (100, 200, 300…13100), cells E2 to E132.
Column F will contain counts of values for each bin.
For example, Bin #1 contains count of values 1-100; Bin #2:101-200; Bin #3: 201-300.
Column G is empty.
Column H contains 1% rise bins (1 through 100) in cells H2 to H101.
Column I will contain slope bin counts.
Column J is blank.
Column K contains smoothed slope bins (K2-K101).
Column L will contain smoothed slope bin counts.
* Excel formulas differ slightly from those in Open Office Spreadsheet, used here
* Float = floating point format (decimal values)
* Integer = whole number format (no decimals)
1.) In cell F2, type: =FREQUENCY(A2:A13058;E2:E133)
Make sure you add one row more to the second part of the formula (E133 not E132).
2.) Drag-copy formula down through row 132. This is the count of flow accumulation values in each bin, where bins contain increasing blocks of 100 values.
3.) In cell I2, type: =FREQUENCY(B2:B13058;H2:H102)
Make sure you add one row more to the second part of the formula (H102 not H101).
4.) Drag-copy formula down through row 101. This is the count of slope values in each bin, where slope values are binned by 1-percent rise increments.
SMOOTHED SLOPE 3×3 (Optional)
5.) In cell L2, type: =FREQUENCY(C2:C13058;K2:K102)
6.) Make sure you add one row more to the second part of the formula (K102 not K101).
7.) Drag-copy formula down through row 101. This is the count of smoothed slope values in each bin, where slope values are binned by 1-percent rise increments.
Create a log-log Slope (y) vs. Area (x) plot for the basin….spreadsheet screenshots on the way…
Tarboton et al. (1992) Water Resources Research 33
Brocklehurst and Whipple (2004) Earth Surface Processes and Landforms 29
Cohen et al. (2008) Journal of Geophysical Research 113
Seidl and Dietrich (1992)
Walcott and Summerfield (2007) Geomorphology 96
Montgomery and Foufoula-Georgiou (1993)
Snyder et al. (2000)
Kirby et al. (2003)
Lague and Davy (2003)
Stock and Dietrich (2003)
Duvall et al. (2004)
Tarolli and Tarboton (2006)
Wobus et al. (2006) GSA Special Paper 398 Tectonics, Climate, and Landscape Evolution
Tarolli and Fontana (2009)
Willgoose et al. (1991) Water Resources Research 27
Flint (1974) Water Resources Research 10