Local
topographic variables
are functions of first (p and q) and secondorder (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 (method 1, method 2). 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 secondorder polynomial
is fitted by the leastsquares approach to the
nine points of the 3x3 spheroidal equal angular window:
The orthogonal spheroidal
coordinates and elevations are known for the window points (‑c, e, z_{1}),
(0, e, z_{2}),
(c, e, z_{3}),
(‑b, 0, z_{4}),
(0, 0, z_{5}),
(b, 0, z_{6}),
(‑a, ‑d, z_{7}),
(0, ‑d, z_{8}),
and (a, ‑d, z_{9}). For the point (0, 0, z_{5}), 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
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 by formulae from the solution of the
inverse geodetic problem for short distances.
Reference
Florinsky, I.V., 1998.
Derivation of topographic variables from a digital elevation model given by
a spheroidal trapezoidal grid. International
Journal of Geographical Information Science, 12: 829–852. Article
at Taylor & Francis 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 9780128046326
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