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Calculation of local morphometric variables on a plane square grid

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 an approximation of partial derivatives by finite differences with the 3x3 or 5x5 plane square-gridded moving windows.

In our method, the third-order polynomial

is fitted by the least-squares approach to the 25 points of the 5x5 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.  Article at Taylor & Francis  PDF

Florinsky, I.V., 2009. Accurate method for derivation of local topographic variables. Geodezia i Cartografia, No. 4: 19–22 (in Russian).  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

 

 

 

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