[GRASS-user] Third partial derivatives

[Jan Pacina]

Hello to all,
for a project of Geomophologic information system I need to compute 
morphometrical parameters (characteristics) of the third order (change 
of curvature). For this computation I need to compute the 3rd 
directional partial derivatives on the DEM (GRID).
I have tried "normal" (or classical) numerical methods computing the 
third derivatives from the GRID data, but the result was not good at all 
(huge inacurancies in the output morphometrical parameters).
I have two ideas which may work - but I need some additional help.

1. I compute the DEM by v.surf.rst - there is a possibility to compute 
partial derivatives (1st and 2nd order) and these derivatives are 
computed very precisely.
And my question is - is it possible to compute the 3rd partial 
derivative directly from the spline function, as it does for the 1st and 
2nd?? I have tried to go trough the source code of v.surf.rst - but I 
got completely lost (I am not a C programmer). I was wondering, if the 
RST is computing for each segment a "parametrical form" - then it could 
be possible to compute the 3rd derivative or if there is only some 
numerical computation ...

2. In the GRASS Handbuch is described computation for r.slope.aspect as 
fitting a polynom onto the input points. This works well for the 1st and 
2nd derivative, I would like to derive a formula for the 3rd derivative 
- this means, that the computation would be on 5x5 surroundind points 
(for the 2nd is 3x3). There is no problem with the "theory" but in the 
GRASS Handbuch is as well desribed the weight used for the weighted 
least square: $w_k = d_k^(-2), d_k^2 = (x - x_k)^^ 2+ (y - y_k)^2$ - do 
you know, how is this weight derived??? So I can derive the right weight 
for the 3rd derivative?
Thank you very much for any suggestion.
Jan Pacina

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  Ing. Jan Pacina                    e-mail: packy at kma.zcu.cz
  University of West Bohemia         phone:  (+420) 37763-2692
  Department of Mathematics          fax:    (+420) 37763-2602
  Plzen, Czech Republic              address: Univerzitni 20, 306 14
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