One common application for curvature data is hydrology where this data helps identify where streams will erode or deposit sediment. it can also be used to enhance terrain rendering by providing shadow.
Some reference information can be found here:
https://li360.pitneybowes.com/s/question/0D58000003rDQA3CAO/what-are-surface-variables-and-why-are-they-important
In addition to this, I have prepared the following notes about Slope, Aspect and Curvature that may assist clients trying to use and understand this tool.
Slope
In each cell we record the maximum downhill slope across that cell, or mark it as invalid. The slope is always positive.
This can be recorded in degrees or as a percentage (computed as change in height over distance).
Note that
- a slope of 26 degrees is equivalent to a percentage of 50%
- a slope of 45 degrees is equivalent to a percentage of 100%
- a slope of 63 degrees is equivalent to a percentage of 200%
We assume the vertical unit of the raster is equivalent to the horizontal unit. If not, you can compensate by specifying a vertical scale. This is used to pre-multiply the height values prior to the analysis. For example, if the raster horizontal unit is yards and the vertical unit is feet, you will need to scale the height by 1/3 to equalise the units.
To compute slope for a valid cell, we require all 8 surrounding cells to be valid. The analysis compares the height values at the centre of the 8 cells to the height value in the central cell.
Aspect
In each cell we record the direction, measured in degrees east of north, of the maximum downward slope. It is the direction that water would flow if rain fell in the middle of that cell.
Aspect will change direction by 180 degrees as you cross a ridge line. There is a mathematical discontinuity at north, so where the slope direction is approximately north you may see aspect values switch rapidly between above zero and below 360. This can make visual representations of aspect difficult to interpret. It can be useful to design a color table that wraps around on itself so that there is no discontinuity in color at north.
To compute aspect for a valid cell, we require all 8 surrounding cells to be valid.
Curvature
We compute three different measures of curvature at each cell - the omnidirectional ‘surface’ curvature, ‘profile’ curvature parallel to the aspect and ‘plan’ or ‘planform’ curvature perpendicular to the aspect. ‘Surface’ curvature is computed by summing the change in slope in the X and Y directions so, in this case, the four diagonal cells surrounding the central cell are not used.
‘Profile’ curvature gives an indication of how fluid flow might accelerate or decelerate down a slope. A negative value indicates the surface is upwardly convex, zero indicates it is linear and positive indicates it is downwardly concave.
‘Plan’ curvature gives an indication of how fluid flow might converge or diverge down a slope. A negative value indicates the surface is sidewardly concave and flow will converge, zero indicates it is linear and positive indicates the surface is sidewardly convex and flow will diverge.
Whereas slope is equivalent to the first horizontal derivative (the change in height across the cell), curvature is equivalent to the second horizontal derivative (the rate of change in height across the cell). The second derivative always enhances high frequency data and noise, so it can be advantageous to smooth the source raster prior to computing curvature. Also, if you are only interested in the amplitude of the curvature you can post-process the result with the calculator to obtain the absolute value of the curvature.
Note that curvature data will be compromised if the DTM contains other surface contamination data like trees and houses.
To compute curvature for a valid cell, we require all 8 surrounding cells to be valid.
In general, curvature is computed by solving a fourth-order polynomial for 9 cells. We follow the same approach taken by ESRI, and the following is taken from their documentation. The polynomial equation is of the form:
Z = Ax²y² + Bx²y + Cxy² + Dx² + Ey² + Fxy + Gx + Hy + I
Where –
A = [(Z1 + Z3 + Z7 + Z9) / 4 - (Z2 + Z4 + Z6 + Z8) / 2 + Z5] / L^4
B = [(Z1 + Z3 - Z7 - Z9) /4 - (Z2 - Z8) /2] / L^3
C = [(-Z1 + Z3 - Z7 + Z9) /4 + (Z4 - Z6)] /2] / L^3
D = [(Z4 + Z6) /2 - Z5] / L^2
E = [(Z2 + Z8) /2 - Z5] / L^2
F = (-Z1 + Z3 + Z7 - Z9) / 4L^2
G = (-Z4 + Z6) / 2L
H = (Z2 - Z8) / 2L
I = Z5
‘Surface’ curvature is computed as: -2(D + E)
![9Cell]()