Curvature

Geomorphic Curvature

Based on this refereence, I intend to relook at the curvature computations.

Jozef Minár, Ian S. Evans, Marián Jenčo, 2020, A comprehensive system of definitions of land surface (topographic) curvatures, with implications for their application in geoscience modelling and prediction, Earth-Science Reviews, Volume 211, 103414, ISSN 0012-8252, https://doi.org/10.1016/j.earscirev.2020.103414
Florinsky IV (2009) Computation of the third-order partial derivatives from a digital elevation model. International Journal of Geographical Information Science 23: 213–231.
Shary PA (1995) Land surface in gravity points classification by a complete system of curvatures. Mathematical Geology 27: 373–390.

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.

–profc or profile convexity or slope curvature (intersecting with the plane of the Z axis and aspect direction). Emphasizes terraces.
- A negative value is convex upward, and slope will be increasing moving downhill.
–planc or plan convexity (intersecting with the XY plane). Emphasizes ridges and valleys.
- A positve value will have diverging flow
- a negative value will have converging flow
–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

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.

Parametric isotropic smoothing: recommended by Hengl and Evans (2008, after Sharry and others, 2002) before computing curvature or other derivatives.

References:

Schmidt and others, 2003.

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

Evans, 1979, as defined in Woods 1996 PhD thesis
Heerdegen, R.C., and Beran, M.A., 1982, Quantifying source areas through land surface curvature and shape: Journal of Hydrology, vol.57, p.359-373; modified to (1) work with variable x and y spacing; (2) have the same sign for Plan curvature as the others; and (3) multiply both curvatures by 100;
After Shary, 1991, as given in Florinsky, 1998; multiply both curvatures by 100;
Zevenbergen and Thorne

Last revised 10/12/2021