Geomorphic Curvature

Curvature is the rate of change of slope, so it is the second derivative of the elevation surface.

Profile Plan Cross section Minimum Maximum

The curvature maps are all scaled to show the 5th to 95th percentiles, and they usually have distributions with very large tails.

Curvature (convexity)
  • Profile convexity (change of slope)
  • Plan convexity (change of aspect)
Point parameter, from 9 point neighborhood
  • According to Evans (1998), three "methods":
    • Evans 1980 deg/100 m
    • Z&T 1987
    • Eyton 1991
  • Heerdegen, R.C., and Beran, M.A., 1982
Second derivative of elevation

First derivative of slope

Surface Curvature.

If we wish to create a single measure of the second order derivatives we must derive that measure for an intersecting plane so as to reduce the expression to an ordinary differential one.  Thus we have several choices depending on the orientation of this intersecting plane. The plane can be defined uniquely by two vectors.
From Wood, J. (1996) The geomorphological characterisation of Digital Elevation Models, PhD Thesis, University of Leicester.

Curvature Equations Used in MICRODEM

Znw

Zn

Zne

Zw

Z

Ze

Zsw

Zs

Zse

z1 := znw - z;
z2 := zn - z;
z3 := zne - z;
z4 := zw - z;
z5 := 0;
z6 := ze - z;
z7 := zsw - z;
z8 := zs - z;
z9 := zse - z;     
A := ((z1 + z3 + z4 + z6 + z7 + z9) / 6 - (z2 + {z5 +} z8) / 3) / XSp / YSp;
B := ((Z1 + z2 + z3 + z7 + z8 + z9) / 6 - (z4 + {z5 +} z6) / 3) / XSp / YSp;
C := (z3 + z7 -z1 - z9) / 4  / XSp / YSp;
D := (z3 + z6 + z9 - z1 - z4 - z7) / 6 / XSp;
E := (z1 + z2 + z3 - z7 - z8 - z9) / 6 / YSp;
F := (2 * ( z2 + z4 + z6 - z8) - (z1 + z3 + z7 + z9) {+ 5 * z5} ) / 9;
SqABC := sqrt(sqr(A-B) + sqr(C));

 

MaxCurve := 20 * (ASp)*(-A - B + SqABC);
MinCurve := 20 * (ASp)*(-A - B - SqABC);
SqED := (sqr(E) + sqr(D));
if SqED > 0.000001 then begin
         SlopeCurvature {Profile convexity} := -200 * (A * sqr(D) + B * sqr(E) + C * D * E) / SqED / Math.Power(1 + SqED, 1.5);
         PlanCurvature {Plan convexity} := 200 * (B * sqr(D) + A * sqr(E) - C * D * E) / Math.Power(1 +SqED, 1.5);
         crossc := -20 * (ASp)*(B*D*D + A*E*E - C*D*E)/  SqED;
end
else begin
         SlopeCurvature := 0;
         PlanCurvature := 0;
         CrossC := 0;
end;

XSp and YSp are are the horizontal data spacings in the x and y directions, which can be different (for example SRTM, NED, or DTED), and can be a function of latitude within a single DEM.

 

References:

The algorithm option computes using four algorithms, with an option to use a larger computation region:


Last revised  1/8/2017