interpolation

[Chris W Skelly]

I would like to thank Helena for her very informative posting and assure her
that she is _NOT_ taking up too much space!  If we could sustain dialogue on
the important topics, like interpolation, for even 10% of the time it would
be a signigicant improvement to this discussion group.  IMHO GRASS is about
spatial analysis and modelling and about providing the community development
of these tools that researchers are not likely to get in commercial products.

Helena, thanks for the tutorial it was much appreciated.  However, there is
one aspect of your reply to Simon that you try and clarify for me.  I have
chopped the following from your post:

On Fri, 9 Jul 1993, Helena Mitasova wrote:

> ----------------Simon writes:--------------------------------------------------
> >However, it
> >is probably worth noting that in most cases, splines are just a fancy form of
> >curve-fitting (which is what they were for the wizened old draftspeople) and
> >their use often cannot be rigorously justified in terms of the processes being
> >modelled.
> 
[stuff deleted...]

> Splines by itself do not model any process, splines create a curve, surface
> or volume function which reproduces the original phenomenon as closely as 
> the functional form enables. HOWEVER they can be and they are used as a TOOL
> for modeling processes and our and others experience is that they are
> very successful in this matter. For example, 


Now, I think I may have changed the context of your reply, but I am interested
in what I think Simon has said because it seems that there is almost a
philosophy-split between the "kriging camp" and the "spline camp". Can
you comment on this?

I have pulled the following quote from Cressie 1991 (_Statistics for
Spatial Data_ John Wiley & Sons, New York, pg 182) to emphasis this point:

"the spline technology is built for, and is justified by, surfaces that
are deterministic or deterministic plus white noise.
   There is a fundametal difference between 'krigers' and 'spliners',
which is manifest in how they report their results.  Only the spline...is
given ...without any local meausure of precision associated with each
Z^&(So).  In contrast, at the very foundation of kriging is the
minimization of local mean-squared prediction error, whose minimized
value...is given by...the kriging predictor...
  One exception to the essentially deterministic treatment given to
splines can be found in Wahba (1983)...Wahba's simulations appear to show
accurate intervals because she simulates with additive Gaussian white
noise..."

Cressie goes on to explain why kriging  is more generally suitable and
ends this section, entitled "Kriging and Splines", with,

"In conclusion, kriging and splines are formally alike, but
practically very different.  Both disciplines can benefit from
each other's knowledge base...Nevertheless, between the two methods, my
preference is with kriging's obligatory spatial-dependence assessment 
and its automatic calculation of mean-squared prediction errors (for
possibly different supports)."


I think Cressie does a fairly good job of objectively stating these
oposing interpolation philosophies,  _however_ he did manage to write
an entire reference book on "statistics of spatial data" without
using splines :-)

I think our collective "tool box" really does need splines, we all
_know_ they are bloody useful.  But, the geographer in me is 
hesitant in using them for anything but making nice contour maps...
Is this just mathematical ignorance on my part (not hard to imagine) or
amongst Cressis's equations is he stating some sound earth science
principle? 

Ooooooh too heavy, its time for a beer, so I'll leave it there.

Cheers,
Chris