# Basic Basin Metrics

Contents
1.) Local Relief
2.) Basin Elongation & Orientation (Long Axis)
3.) Basin Relief Ratio
4.) Basin Midpoint
5.) Basin Centroid
6.) Basin Form Factor
7.) Basin Shape Factor
8.) Elevation Relief Ratio

1.) Local Relief
If relief is the difference between the highest and lowest elevations in a watershed, then local relief is the same thing only smaller (calculated by a roving window). There is no formal definition for “local” and there are several ways to calculate relief. All involve a neighborhood (roving window) usually sized hundreds of meters wide. Different study areas will require different size windows. Calculating local relief over large areas – like dozens of mountain catchments – will require a slightly different method than for a single watershed. Scale matters.

Local Relief Index (LAR)

LR = ∑i (Zmax – Zmin) / n

Zmax = Maximum elevation from DEM (highest pixel value)
Zmin = Minimum elevation from DEM (lowest pixel value)
n = number of pixels in catchment

Higher LR values may indicate the catchment has been subjected to more tectonic influence and/or incision.

Local Relief OPTION A:
– ArcToolbox > Spatial Analyst Tools > Neighborhood > Focal Statistics
– Input = DEM
– 15 x 15 cell window size
– Window shape = rectangular (or circular)
– Operator = Range

or

Local Relief OPTION B:
– ArcToolbox > Spatial Analyst Tools > Neighborhood > Focal Statistics
– Size window to slightly larger than half the perpendicular width of the catchment
– Window shape = rectangular or circular
– Operator = Range
* Expect some trial and error in determining this window size. Remember to convert meters to cells if you use Cell units.

* See Hammond Landforms lesson for a different calculation method. He called local relief the “profile parameter”.

* See Fonstad (2003, p. 84) for an alternative relief ratio equation applicable to studies where channel cross sections are made in the field. The equation is:
Subbasin Relief / Distance from Section Location to Subbasin Drainage Divide

* Optional: Properties > Symbology, change color ramp to red-to-blue (“Algorithmic”) and click Invert button so that blue corresponds with lower relief values.

2.) Elongation & Orientation
Automatically calculate the long axis and orientation of a watershed polygon (or set of watershed polygons), use Minimum Bounding Geometry tool (Data Management > Features > Minimum Bounding Geometry). The Convex Hull option will create several new fields in the attribute table of the new shapefile, including:

MBG_Length = Distance between the farthest 2 points on watershed boundary polygon (antipodal points)
MBG_Orientation = Orientation of the long axis

MBG Tool:
Input = watershed boundary polygon
Output = ws1_mbg
Geometry Type = CONVEX_HULL
Group = NONE (you want a convex hull created for each polygon, not one hull around all polygons)
Check the box for “Add geometry characteristics as attributes to output”
Run tool
Open attribute of output shapefile (polygon) to see long axis length and orientation info.

* The MBG_Length field in the attribute table of the output file will be reported in decimal degrees. If the value in the field MBG_APodX is in decimal degrees, then MBG_Length is also in decimal degrees (projected rasters will show distance in the projected distance units; if projected then ignore the rest of this). You need to convert decimal degrees to kilometers. With the attribute table open, use Go To XY tool to find the coordinates of the two points that define the long axis of the watershed boundary polygon. These are shown in the fields: MBG_APodX, MBG_APodY and MBG_BPodX, MBG_BPodY. Then use the Measure Tool in kilometers to find distance between points (or more rigorously by converting point graphics to features, then running Point Distance tool).
Example: MBG_Length = 0.111 DD –> 9.9 km (UTM_NAD83_Zone11N).

* Orientation field is returned in azimuth values (0-360) using the lower of two choices (i.e., 90 not 270). For watershed orientation, report the downstream azimuth (towards outlet).
Example: MBG_Orient = 68.3 –> add 180+68.3 –> 248.3 azimuth for a stream flowing to the southwest

Help
– Click the “Tool Help” button on the tool dialog window for more details.
– Search Desktop Help: Minimum Bounding Geometry

3.) Basin Relief Ratio
The ratio between total relief (max elev-min elev) and basin length (long axis length) is the Basin Length Ratio (Rr). This simple morphometric index can be used to compare a set of catchments in one mountain range to another.

Values may reflect differences in active tectonism, bedrock geology, geomorphic process, climate effects, etc. Use ArcMap Measure tool and summary statistics for basin DEM.

Rr = R / L

Rr = (m)
L = Basin length measured along long axis (m)

4.) Basin Midpoint
Erosion rate decreases down the access of a basin. The Basin Midpoint is the location at which the upstream drainage area (flow accumulation raster value of the trunk stream) approximates 1/2 the total basin area. The midpoint is measured in the down-basin direction starting at the dividing ridgeline above the primary headwater channel. The specific point is up to you to choose, similar to the “watershed divide point” in Hack’s SL index. Plot erosion rate (if you’ve got the data or can model it) against down-basin distance to the basin midpoint to determine if this relationship holds for your study area basin (A).

Basin Midpoint = Maximum FlowAcc / 2, or approximately 1/2 Area

FlowAcc = Flow accumulation raster value; maximum value is at the outlet
A = 2D area of basin

5.) Basin Centroid
The geometric center of the 2D basin polygon, which differs from Basin Midpoint. Find the centroid of a polygon using ArcToolbox > Data Management Tools > Features > Feature to Point (use Inside option). Or try the attribute table function Calculate Geometry in a new field. There are Python scripts, a statistical Central Feature method (Euclidean or Manhattan), and Field Calculator methods that accomplish the same thing. VBA is not supported in ArcGIS 10.

6.) Basin Form Factor
more to come…

F = A / L2

A = 2D area of watershed
L = Length of watershed; longest dimension measured parallel to mainstem channel (or approximately parallel)

7.) Basin Shape Factor
Basin shape can be described as circular, rectangular, triangular, or pear. The latter is most common. Shape can also be quantified using equations for Basin Shape Factor, sometimes called Shape Factor. Not to be confused with Form Factor.

Shape directly impacts the size of peak discharge and its arrival time at the basin outlet. Peak discharge for a circular basin will arrive sooner than that of an elongate basin of the same area because the tributary network in a circular basin is more compactly organized and tributary flows enter the mainstem at roughly the same time, thus more runoff is delivered to the outlet together, sooner (shorter duration, higher flood peak).

BSF = L2 / A = L / W
or
BSF = (L * Lca)0.3

L = Length of watershed
Lca = Line length of mainstem to center of watershed area
W = Width of watershed
A = Area of watershed

Refs:
Bedient (1992)
Gray (1970)
Grohmann & Riccomini (2009) Computers & Geosciences 35
Montgomery & Brandon (2002) Earth and Planetary Science Letters 201
Morisawa (1958)
Sarangi et al. (2003)
Sougnez & Vanacker (2011) Hydrology and Earth Systems Sciences 15
Wisler (1959)
Safran et al. (2005) ESPL 30, Fig. 7

8.) Elevation Relief Ratio
The Elevation-Relief Ratio provides hypsometric information about a watershed.

ERR = Zavg – Zmin / Zmax – Zmin

Refs:
Pike and Wilson (1971)