are functions of partial derivatives of elevation , , , , . There are methods for computing r, t, s, p, and q from digital elevation models (DEMs) based on plane square
grids. These methods use an approximation of partial derivatives by finite
differences using the 3x3 plane square-gridded moving window.
Evans–Young method, the second-order polynomial
is fitted by the least-squares approach to the
nine points of the 3x3 square-spaced window with a grid spacing of w:
The Cartesian coordinates and
elevations of the topographic surface are known for the window points (-w, w, z1), (0, w, z2),
(w, w, z3), (-w, 0, z4),
(0, 0, z5), (w, 0, z6), (-w,
-w, z7), (0, -w, z8) and
(w, -w, z9). For the point (0, 0, z5), the polynomial coefficients (which are
partial derivatives of elevation) are estimated by the following formulae:
Moving the 3x3 window along a DEM, one can
calculate values of r, t, s, p, and q (and so values of local
morphometric variables) for all points of the plane square-gridded DEM,
except for boundary rows and columns.
I.S., 1979. Statistical Characterization of Altitude Matrices by
Computer. An Integrated System of Terrain Analysis and Slope
Mapping. The Final Report on Grant DA-ERO-591-73-G0040. Department of
Geography, University of Durham, Durham, 192 p.
M., 1978. Statistical Characterization of Altitude Matrices by Computer.
Terrain Analysis: Program Documentation. Report 5 on Grant
DA-ERO-591-73-G0040. Department of Geography, University of Durham,
Durham, 18 p.
details and examples, see:
DIGITAL TERRAIN ANALYSIS
IN SOIL SCIENCE AND GEOLOGY
2nd revised edition
Elsevier / Academic Press, 2016
Amsterdam, 486 p.