# Channel Concavity

Channel concavity (θ) is the longitudinal change in the slope of a reach. Most channel profiles are concave due to detachment-limited erosion of bedrock in headwater areas and the usual downstream relationships: increase in discharge, decrease in bed grainsize, and increase in channel width.

Concavity is quantifiable at the reach scale (reach length choice is up to you). The same concavity can exist for channels having different gradients, but the difference in gradient will produce very different channels forms.

There are several ways to represent concavity for channels. Here are 3 ways.

METHOD A: Concavity instructions
Example watershed metrics work by former Boise State University students:
Concavity Instructions_JWade and MBrucker 2014.docx
Watershed Metrics Poster_Lange and Vega_Spring2015_20 in x 20 in.pdf

`Concavity Index (θ): Theta (θ) is the slope of a line regressed through a log-log plot of Channel Slope (percent rise) vs. Drainage Area (km2).`

Step #1: Extract Trunk Stream
Use Toolbars > Draw tools to create a point graphic and place the point somewhere along the trunk stream (must be in the headwaters of the watershed).

Convert the graphic to a feature layer and save the shapefile as Converted_Graphics.

Next, snap the Converted_Graphics to the nearest pixel using Spatial Analyst Tools > Hydrology > Snap Pour Point tool…
– Input raster or feature pour point data = Converted_Graphics
– Input accumulation raster = FlowAcc
– Output raster = Downpoint

Isolate the trunk stream using the Cost Path tool. Imagine dropping a ball into the headwaters of the watershed and trace the balls path of least resistance along the trunk stream. Use Spatial Analyst Tools > Distance > Cost Path tool…
– Point Source = Downpoint
– Cost Distance = FlowAcc
– Cost Backlink = FlowDir
– Output = Trunk_stream

Clip Trunk_stream to the bounds of watershed with Raster > Raster Processing > Clip tool…
– Use clip tool using the outline of the watershed as the constraining boundary.
– Output = WS_trunk

Step #2: Extract the properties of the WS_trunk raster using the filled DEM
Spatial Analyst > Extraction > Extract by Mask tool…
– Input raster = DEM_Filled
– Feature Mask Data = WS_trunk
– Output = Trunk_DEM
– Note: If the unit of the DEM is in feet instead of in meters, convert to meters using Raster Calculator. Enter the formula: “DEM”/3.2808

Step #3: Obtain slope values for each pixel along the trunk stream channel
Spatial Analyst > Surface > Slope tool…
– Input raster = Trunk_DEM
– Change Output Measurement to percent rise (not degrees).
– Output = Trunk_slope

Step #4: Extract flow accumulation data along the trunk stream
Spatial Analyst > Extraction > Extract by Mask tool…
– Input raster = FlowAcc
– Feature Mask Data = WS_trunk
– Output = Trunk_flow

Step #5: Convert ws_trunk into points
Conversion Tools > From Raster > Raster to Point tool…
– Input raster = WS_trunk
– Output point features = Trunk_points

Step #6:  Create a data table to be used in Excel to create the concavity graph
Spatial Analyst > Extraction > Sample tool…
– Input Rasters = Trunk_flow, Trunk_slope
– Input Location Data = Trunk_points
– Output = Concavity_data.dbf
– Note: Add file extension .dbf to your output so the table can be opened in MS excel.

Step #7: Open concavity_data.dbf in an excel spreadsheet
– Open Microsoft Excel and create a new, blank spreadsheet
– Open the concavity_data.dbf file that you created using ArcMap …or…Open the attribute table and copy/paste directly into an excel spreadsheet.
– Scan your data and delete all zero values from the slope data
– The Power trendline (see chart below) cannot be computed if there are zero values in the data. Deleting these values will have a negligible impact on the graphical data.
– Note: for our dataset of >1900 points, less that 1% contained zero values.

Create a new column and name it “Drainage_Area”.
– In this step, you will convert the data (pixel count) in Trunk_flow into upstream drainage area in km² using the following formula:
=(F2*(Pixel Size * Pixel Size))/1,000,000
– Note: the size of the pixel may vary depending on the resolution of the original DEM (e.g., 30m DEM, 10m DEM).

Select the Trunk_slope and drainage_area columns and plot the data in a scatter plot.
– Convert the x and y axis to logarithmic scale and add a Power-type trend line.

The Concavity Index is the exponent from the power trendline equation.

METHOD B.) Area-normalized Stream Concavity Index (SCI) is based on an integral area between the channel profile (a curve) and a line connecting channel endpoints (a line). The curve and line may cross for some channels. SCI covaries with θ (Zaprowski et al., 2005). See example dataset farther down in this post.

METHOD C.) Incision Values method. See figure below. Two values, “big Z” and “little z”, are measures of channel incision determined from the channel profile and a line connecting its endpoints (channel head and outlet). Big Z is maximum channel incision with respect to local base level. Little z is the metric for channel incision. By convention, both Z and z are measured at the median value position for Downstream Distance (x-axis), where both axes are normalized (Elev and Dist). That is, the minimum and maximum values must be rescaled from 0 to 1.

Concavity Instructions with MatLab_Zach Genta_2015.docx

Stream Concavity Index (SCI) is described in BLACK text (Zaprowski method). Integrate between the channel profile curve and the line between endpoints for area. A best-fit polynomial function, usually 2nd order, through the profile points works well for the curve (Excel trendline). Use Excel to create the endpoint line too. Triangles at bottom show how the shape of the channel profile corresponds to SCI values with respect to 0: Straight profiles (SCI ~ 0), Concave profiles (SCI > 0), Convex profiles (SCI < 0). The RED text describes another concavity metric. Find two channel incision values (big Z and little z) by simple vertical measurements at the median value of Dowstream Distance, where values have been normalized. Normalization rescales min and max values along an axis from 0 to 1. Click to enlarge image.

Charting the example channel profile. RED DOTS are scatter plot of Elevation vs. Downstream Distance for the channel. RED DASHED LINE is the trendline fit through the points (a 2nd order polynomial in this example). BLUE DASHED LINE is constructed between channel endpoints (channel head and outlet). The two equations will be entered into WolframAlpha to find the integral area between the line and curve (integrate with respect to x). Integrate over the entire channel length (0 to 500).

Wolfram Alpha Instructions
You’ll first find root values for integration (x limits), then you’ll integrate to find the area between line and curve.

#1: Open WolframAlpha website.

#2: Example function looks like this when plotted with Online Function Grapher –>

#3: Find the roots (limits for x) by entering the equations of your functions into the yellow bar in Wolfram. Arrange your functions in parentheses as shown below. Root values are the upper and lower limits of integration. Both are x values located where the two functions cross. The area to be found in next step is that between the GREEN line and BLUE curve.

Example function equations:
f(x) = -3.4625x+1731.3
g(x) = 0.0001x^2-0.5229x+483.41

#4: With your root values in hand (x = -29814 and x = 418), integrate to find the area between the line and the curve. Again, enter the function equations and add the root values into yellow bar, following this pattern for integrals (partially cut off):

#5: Area
Example area returned for above equation = 4.60571 x 10^8 = 460,571

– Higher values of m/n indicate greater concavity. Increasing concavity or gradient will tend to increase incision.

– Changes in profile concavity are often a channel’s response to changes in base level, onset of uplift, change in rock uplift rate, an abrupt climate shift, or a combination of these factors.

– Lines take the form of Y = mX + b, or in this case the line is y=ax^k. Where m = k = the slope of the line. Where b = log a = the intercept of the line with the log y-axis (where log x = 0). So, reversing the logs, a is the y value corresponding to x = 1. By plotting in log-log space, the relationship appears as straight lines, with the power (k) and constant (a) terms corresponding to slope and y-intercept of the line on the log-y axis, respectively.

– Since θ and Ks are strongly correlated, a Reference Concavity (θref) is required in order to be able to compare different streams. θref is calculated from the regional mean of observed θ values for a number of channel segments. Typical reference concavities range from 0.35 to 0.65 (and possibly from ~0.2 to 1.0 in certain regions). Assuming a value of 0.5 for m/n is not uncommon in models.

– See Stream Power & Erosion equations for additional information.

Refs
Anderson & Anderson (2010) Geomorphology: The mechanics and chemistry of landscapes, Cambridge Univ Press, 637 pgs.
Bull(1979)
Clark et al. (Sept 2005) GSA Today 15
Collins & Bras (2010) Water Resources Research 46
Craddock et al. (2007) Journal of Geophysical Research 112
Densmore et al. (2007) Transient landscapes at fault tips, Journal of Geophysical Research 112
Duvall et al. (2004) Journal of Geophysical Research 109
Howard & Kerby (1983) GSA Bulletin 94
Gasparini et al. (2006) Numerical modeling of non–steady-state…, p. 127-141, in GSA Special Paper 398
Jones, R. (2002) Algorithms for using a DEM for mapping catchment areas…, Computer & Geoscience 28, p. 1051-1060
Kirby et al. (2003) Journal of Geophysical Research 108
Kirby & Whipple (2001) Geology 29, p. 415-418
Korup, O. (2006) Geology 34, p. 45-48
May & Lisle (2012) Journal of Geophysical Research 117
Norton, K.P., Abbuhl, L.M., Schlunegger, F. (2010) Geology 38, p. 655-658
Norton et al. (2010) Data Repository #2012175, ftp://rock.geosociety.org/pub/reposit/2010/2010175.pdf)
Rehak et al. (2010) ESP&L 35
Rossi et al. (2009), http://adsabs.harvard.edu/abs/2009AGUFMEP51B0604R
RSG.TU (c. 2012) River Profile Analysis, http://www.rsg.tu-freiberg.de/twiki/bin/view/Main/RiverProfileAnalysis
Safran et al. (2005) ESPL 30
Schumm & Lichty (1965)
Seidl & Dietrich (1992) Catena Supplement 23
Snyder et al. (2000) GSA Bulletin 112
Stock et al. (2005)
Tan et al. (2007) Hydrological Processes 21
Tarboton et al. (1989)
Tinkler, K.J. (1998) Rivers Over Rock, AGU Publishers, 323 pgs.
Whipple (2001) American Journal of Science 301
Whipple (2003) Annual Review of Earth & Planetary Science 32, p. 151-185
Whipple & Tucker (1999) Journal of Geophysical Research 104, p. 17,661-17,674
Whittaker (2012) How do landscapes record tectonics and climate? Lithosphere 4
Wobus et al. (2006) p. 55-74, in Willett et al. (editors), GSA Special Paper 398
XiaoFei et al. (2010) Chinese Science Bulletin 55
Zaprowski et al. (2005) Journal of Geophysical Research 110