st: Spline interpolation of spatial data

[Gordon Hughes <G.A.Hughes@ed.ac.uk>]

Dear Statalist,

I would be grateful for suggestions about whether there are any 
routines in Stata - or other software - to carry out a rather 
specific form of spline surface interpolation.  The context is fairly 
common with GIS raster data:  I have multiple sets of spatial data at 
different grid resolutions which I want to combine to form weighted 
averages.  Assuming a uniform distribution over the coarser grid 
units may introduce errors of unknown magnitude that I would like to examine.

As a concrete example, I have average temperatures for 1 deg grid 
cells covering the continental US.  In addition, I have estimates of 
total population by 30 arc-second grid cells for the same area.  I 
want to calculate estimates of population-weighted temperature 
exposure by state and/or county.  If population density and/or 
temperature distribution are not uniform within each 1 deg grid cell, 
the simple procedure of summing the population in each 1 deg cell and 
then computing population-weighted average temperatures by state 
fails to allow for the non-uniform distribution of population and/or 
temperature.

A better approach would be to convert each 1 deg grid cell to a 12 x 
12 (5 arc-min) mesh and use cubic or some other spline surfaces to 
interpolate temperatures over this mesh subject to a constraint on 
the average temperature for the whole grid cell and on knots at the 
boundary points.  This is a non-trivial exercise and I cannot locate 
any Stata routines that do anything like this.  There are monographs 
in mathematics and computational graphics that cover the general 
topic - notably a monograph by Paul Dierckx titled 'Curve and surface 
fitting with splines' (Clarendon Press, 1993).  In addition, there 
are specialised algorithms that are used in 3D graphical software, 
though generally these focus on interpolation of points rather than 
averages.  Some of Dierckx's algorithms - originally in Fortran and 
called FitPack - have been translated into R and Python, but these 
are not easy to convert to Stata or Mata.

Does anyone have any suggestions of routines in either Stata or 
Matlab or some other matrix language that might provide a starting 
point for spatial interpolation of this kind?

Gordon Hughes
g.a.hughes@ed.ac.uk 

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