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###### Calculation of local morphometric variables on a plane square grid

Once elevations are given by . where x and y are Cartesian co-ordinates, local topographic variables are functions of first- (p and q), second- (r, t, and s), and third-order (g, h, k, and m) partial derivatives of elevation: , ; , , ; , , , .

To compute partial derivatives of elevation from DEMs based on plane square grids, one can use methods including an approximation of partial derivatives by finite differences with the 3 x 3 plane square-gridded moving windows (e.g., the Evans method).

In our method, the third-order polynomial is fitted by the least-squares approach to the 25 points of the 5 x 5 square-spaced window with a grid spacing of w: The Cartesian coordinates and elevations of the topographic surface are known for 25 points of this window: (‑2w, 2wz1), (‑w, 2wz2), (0, 2wz3), (w, 2wz4), (2w, 2wz5), (‑2wwz6), (‑wwz7), (0, wz8), (wwz9), (2wwz10), (‑2w, 0, z11), (‑w, 0, z12), (0, 0, z13), (w, 0, z14), (2w, 0, z15), (‑2w, ‑wz16), (‑w, ‑wz17), (0, ‑wz18), (w, ‑wz19), (2w, ‑wz20), (‑2w, ‑2wz21), (‑w, ‑2wz22), (0, ‑2wz23), (w, ‑2wz24), and (2w, ‑2wz25). For the point (0, 0, z13), the polynomial coefficients (which are partial derivatives of elevation) are estimated by the following formulae:  ,  ,  ,  ,  ,  ,  ,  ,  .

Moving the 5x5 window along a DEM, one can calculate values of g, h, k, m, r, t, s, p, and q (and so values of local morphometric variables) for all points of the plane square-gridded DEM, except for two boundary rows and two boundary columns on each side of the DEM.

References

Florinsky, I.V., 2009. Computation of the third-order partial derivatives from a digital elevation model. International Journal of Geographical Information Science, 23: 213–231.  doi  pdf

For details and examples, see:

 DIGITAL TERRAIN ANALYSIS IN SOIL SCIENCE AND GEOLOGY   2nd revised edition     I.V. Florinsky   Elsevier / Academic Press, 2016 Amsterdam, 486 p.   ISBN 978-0-12-804632-6