Curvature

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.

–profc or profile convexity (intersecting with the plane of the Z axis and aspect direction);
–planc or plan convexity (intersecting with the XY plane);
–longc or longitudinal curvature (intersecting with the plane of the slope normal and aspect direction)
–crosc or cross-sectional curvature (intersecting with the plane of the slope normal and perpendicular aspect direction);
–maxic or maximum curvature (in any plane);
–minic or minimum curvature (in any plane);
–meanc or mean curvature (in any plane).

From Wood, J. (1996) The geomorphological characterisation of Digital Elevation Models, PhD Thesis, University of Leicester.

Profile and Plan Convexity follow suggestions of Evans

Znw	Zn	Zne
Zw	Z	Ze
Zsw	Zs	Zse

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;


Profile	Plan	Cross section	Minimum	Maximum