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###### Calculation of local morphometric variables on a spheroidal equal angular grid

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

Digital models of local morphometric variables are usually derived by methods based on the approximation of partial derivatives of elevation by finite differences. Such methods have been developed and intended to calculate partial derivatives from DEMs based on plane square grids (e.g., the Evans method and IF-2009 method). However, a plane square grid and a spheroidal equal angular grid have principally different geometry. Therefore, it is obvious that methods developed for plane square grids cannot be applied on spheroidal equal angular grids.

In our method, the second-order polynomial is fitted by the least-squares approach to the nine points of the 3 x 3 spheroidal equal angular window: The orthogonal spheroidal coordinates and elevations are known for the window points (‑cez1), (0ez2), (cez3), (‑b0z4), (00z5), (b0z6), (‑a‑dz7), (0‑dz8), and (a‑dz9). For the point (0, 0, z5), the polynomial coefficients (which are partial derivatives of elevation) are estimated by the following formulae: ,      , , ,      .

Moving the 3 x 3 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 spheroidal equal angular DEM, except for boundary rows and columns on each side of the DEM. If one processes a virtually closed, global spheroidal equal angular DEM, it is possible to estimate values of topographic variables in each point of such a DEM.

Values of a, b, c, d, and e vary depending on the latitude. Since geographic coordinates are known for every point of a spheroidal equal angular grid, so a, b, c, d, and e are easily calculated, for example, by well-known formulae from the solution of the inverse geodetic problem for short distances.

Reference

#### Florinsky, I.V., 2017. Spheroidal equal angular DEMs: The specificity of morphometric treatment. Transactions in GIS, 21: 1115–1129.  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