interpolation, what splines are!

[Simon Cox]

Regarding Spline interpolation
(an attempt at a tutorial! - comments gratefully accepted):
=======================================

The principle of splines is easy to explain (though the mathematical
formulation may be less so!).

Back in the old days, before even flexi-curves were invented,
draftspeople who wanted to draw smooth curves through a set of
points achieved this by putting pins through the points, and
laying a flexible strip of (I think) bamboo against these.
This strip was called a "spline".  The minimum curvature was
controlled by the thickness (stiffness) of the spline, and if
a thick (stiff) spline was used, the draughtsperson may have to
choose to be a bit flexible (:-)) as to how close to the data
the spline actually passed.

The generalization of this for a surface is to replace the strip
with a plate or membrane, which is deformed to touch all the points
(no smoothing) or may only pass close to these (with smoothing):
again, the curvature is controlled by the thickness or stiffness
of the plate.  In addition, some "tension" may be added to the
plate, so it may be "stretched" as well as warped.  Of course,
no one does this physically, so the implementations of this
consist of mathematical approximations which solve the mechanics
equations!

Splines are one of a class of interpolators base on physical principles which
include the "minimum curvature" interpolators.  They are expected to give
good results where the surface being modelled is due to a low order physical
process (for example:  potential fields) where we KNOW that the surface
should NOT have discontinuities up to a few orders of spatial derivatives.
However, they are also found to give acceptable results in many other cases,
if the stiffness and tension parameters are adjusted suitably.  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.

Kriging, on the other hand, is a purely statistical method, based on
reducing an objective function allowing for the consideration of local
correlation lengths.  On its own terms, I think that Kriging, particularly
the more devloped forms (indicator, log-normal, etc) might be argued to
give a more honest result for cases where the processes are complex or
unknown, and it can also give some statistics about how good the estimate is!

Landscapes, for example, are the product of many processes, mostly non-linear,
etc,
so there is no good way of modelling elevations short of process simulations!
However, as we all have limited time and resources, splines will often serve us
well.

Overall, I think that an adequate toolkit for us all would include both
thin-plate-splines and kriging, since this would cover both the "integral"
and "statistical" approaches, each of which are useful in different
circumstances.
(so, thanks Helena, and please keep at it Chris!)

A couple of refs:

Smith W H F & Wessel P
Gridding with continuous curvature splines in tension
(Introduction, with a couple of applications to gravity and bathymetry data)
Geophysics, v55, p293-305  1990

Isaaks E H & Srivastava
An Introduction to Applied Geostatistics  (Basically a whole book on kriging!)
Oxford University Press 1989

And of course the chapter in Burrough's book on kriging.
=========================

If anyone thinks that I have really stepped out of line here, flame away, but
please try to generate at least as much light as heat!

Simon Cox

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__________________________________________________________________
				Dr Simon Cox
         __  L				
      ,~'  L_|\            	Department of Earth Sciences       
   ,-'         \         	Monash University    
   (            \		Clayton  Vic  3168  Australia
   \    ___     /	
    L,~'   "\_x/		Phone +61 3 565 5762
              u   		Fax   +61 3 565 5062
				simon at cerberus.earth.monash.edu.au
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