s.surf.tps-curvatures

[Helena Mitasova]

>By the way: can Helena tell us a bit more about the derived parameters
>pcurv, tcurv and mcurv that you get for free with s.surf.tps? I made some
>nice-looking maps with them, but I have some difficulty interpreting these
>'curvatures': how are they calculated and what is the difference between
>these three types of curvature?

You will find the exact derivation probably in August or September
issue of Mathematical Geology in a paper:

Mitasova H., and Hofierka, J. Interpolation by Regularized Spline with
Tension: II. Application to terrain modeling and surface geometry
analysis.

also, there is a short description in manual page and 1992 summer
issue of GRASSCLIPPINGS

In short, pcurv is profile curvature = normal curvature of 2d surface
measured in the direction of flow 
- convex areas (yellow through red)have accelerated flow,
 concave areas(cyan through blue) have flow with decreasing velocity.
Zero isoline of this curvature (where yellow changes to cyan - if wide
enough it should be light green) shows the local extremas(max or min.)
 of slope - and also the inflex points on flowlines.
tcurv is tangential curvature - normal curvature of 2d surface measured
in the direction of the tangent to the contour (perpendicular to the direction
of flow) - convex areas have dispersed water flow, concave areas
have convergent water flow. Zero isoline of this curvature
passes through the inflex points of contours.
mcurv is the mean curvature - it is the average of maximum and minimum
normal curvature in the given point (it is the average of so called
principal curvatures)

the best thing how to understand and interpret these curvatures,
is to drape them over the terrain  (using SG3d for those who have
Silicon Graphics or d.3d should help too)

spline used in s.surf.tps was specially designed to have
regular derivatives and we are computing the first and second
order partial derivatives of interpolation function simultaneosly
with interpolation. This gives the estimates of derivatives with
higher accuracy than with standard local polynomial using the 
3*3 neighborhod and the maps of curvature look better
 (and should be more accurate)
than what you can get from standard raster based methods.

I hope this helps,

Helena