Zachary Genta
Math 365
Spring 2015
Area-Normalized Stream Concavity Index
When analyzing a channel profile, its concavity is
the change in its longitudinal slope. Finding a channel’s
concavity can be useful in many ways. It can help find
anomalies, the erosive state of bedrock, rock uplift,
increases in drainage area, and various downstream
relationships.
There are multiple ways to measure the concavity of a channel. Here we will be focusing
on area-normalized concavity. This method allows us to compare a channel’s concavity with
other channels made up of different lengths or gradients.
To find the area-normalized concavity of a channel segment, first obtain 30-50 channel
points starting from the head of the channel, with each point about equidistant to another.
Measure the channel’s cumulative downstream distance (0 at its head) with its elevation. Be sure
all distance measurements are done following the stream (and not measured as the crow flies, or
geodesic distance). Once the data is collected, normalize the downstream distance by dividing
the field by its total distance. This will make the final point have a distance of 1. Next, normalize
the elevation by subtracting the field by its last and lowest elevation, and then dividing the field
by the newly updated elevation at its head (the highest elevation). This will make the first point
have an elevation of 1 and the last point an elevation of 0.
Zachary Genta
Math 365
Spring 2015
Now that both measurements are normalized, the process of finding a channel’s
concavity is figuring out the integral area between the channel profile curve and the line between
channel endpoints. There are various methods to integrate, or find the area of the needed
polynomial. The area is then divided by 0.5 to give the final stream concavity index.
This method uses the area formula
through Matlab to find the area of the
channel profile polygon, and then
subtracting that area from 0.5 to get our
target area (highlighted in gray to the
left). 0.5 is the area underneath the line
connecting the channel endpoints.
After uploading the channel data
into Matlab, and normalizing distance
and elevation, find the area of the
channel profile. This requires adding the
point (0,0) to the end of the channel data set, and then wrapping the vertices so that the first point
is now also the last point. The data set can now be inputted into the polygon area formula. In
Matlab, the area formula can be vectorized producing quicker, less-expensive results.
% Area of polygon formula
for i=1:length(u);
A(i)= (poly_y(i+1)-poly_y(i))*(poly_x(i+1)+poly_x(i))/2;
end
Area=abs(sum(A(:)));
Zachary Genta
Math 365
Spring 2015
% Vectorized
Area=abs(sum(diff(poly_y).*(diff(poly_x)+2.*poly_x(1:end-1))./2));
Once the area is obtained, it just needs be divided by 0.5 (or multiplied by 2) to obtain the
channel’s correct stream concavity index (SCI). A SCI greater than 0 signifies a concave profile
while a SCI less than 0 signifies a convex profile. A SCI near 0 simply means the channel is
neither concave nor convex, or goes through a little of both as it travels downstream.
Other numerical integration methods may also be used to find the integral area between
the two lines. First use a natural cubic spline to approximate the function of the normalized
channel profile. An optimality property of natural cubic splines minimizes curvature, thus
creating a function that oscillates the least and hopefully doesn’t run ‘uphill,’ just how a river
shouldn’t. Quadrature methods such as trapezoidal rule and Simpson’s can be applied to
integrate the functions area. The area is then subtracted from 0.5 and divided by 0.5 to find the
SCI.
Here are the results of a sample data set of 50 points of the Seti River in Nepal. The data
set’s SCI found from using the area formula method is 0.5197. With 50 subintervals, the
midpoint rule, trapezoidal rule, and Simpson’s rule in approximating the same target area result
in SCI’s of 0.5201, 0.5195, and 0.5199 respectively. The greatest difference between any of
these three approximations and the area formula is only 0.0004. Thus, we may conclude any of
the above methods produce relatively the same result, and that the polygon area formula is a
nice, simplistic method for finding a channel profile’s SCI.
Zachary Genta
Math 365
Spring 2015
Matlab code: Concavity.m
References
Cooley, Skye. "Channel Concavity." GIS 4 Geomorphology. N.p., 2011. Web.
http://gis4geomorphology.com/concavity-soon/ 28 Apr. 2015.
Zaprowski, B. J., F. J. Pazzaglia, and E. B. Evenson (2005), Climatic influences on profile
concavity and river incision, J. Geophys. Res., 110