Re: suggestions?

• To: mathgroup at christensen.cybernetics.net
• Subject: [mg554] Re: [mg549] suggestions?
• From: el at qua.crl.melco.co.jp (E. Lange)
• Date: Thu, 16 Mar 95 14:22:53 JST

```> There is a rectangular plot of land, several acres in size, which is
> contaminated by a single hazardous waste. We have some chemical
> measurements of the concentation of the waste in the soils at N randomly
> spaced points -- these points are NOT on a grid.
>
> From other information about the deposition of the waste, we know that the
> function
>
>         concentration( x, y )
>
> is well behaved and smoothly varying.....
>
> So, I want to use a 2D InterpolationFunction in Mma to model the

You might try a simple interpolation using radial basis functions:

data = Table[Random[], {10}, {3}]
c = 5

Transpose[data][[3]] . # / Plus @@ # & [
E^(-c^2((x-#[[1]])^2 + (y-#[[2]])^2))& /@ data ]
Plot3D[%, {x, 0, 1}, {y, 0, 1}]

The parameter c adjusts the `stiffness' of the interpolation.
For c = 0, the interpolation result is the average concentration of the
waste. For c-->Infinity, the interpolation result is the concentration
at the closest measurement point. You can also try to use other basis
functions, such as 1/d^c instead of E^(-c^2 d), or incorporate
knowledge about the physics of the problem.

InterpolationRBF[data_, c_][x_, y_] :=
(x - data[[i, 1]])^2 + (y - data[[i, 2]])^2) ),
{i, Length[data]} ];