Much of the post can relate back to the many discussions about Steig et al here and at tAV, namely the physical basis for the eigenvectors used in Steig’s reconstruction.

Evan, kriging interpolation is new to me, so thanks for the suggestion. I worked out an analytical solution for the center station’s kriging weight. If there are N+1 stations, station 0 at the pole and stations [1..N] evenly distributed around the perimeter (thetai = 2×i×pi/N), then

lambda0 = (A-f(N))/(2-f(N))

where A = 32/9/pi + 1/3 ~ 1.4651
and f(N) = 2/N * SUM(sin(thetai/2))

For the case of 36 stations on the perimeter, this equation gives

lambda0 ~ 0.264

in agreement with Evan’s solution. For a large N, f(N) –> 4/pi

[This is probably of little interest unless you're a calculus junky like me ©¿© ]

THERMAP:
Ice Temperature Measurements of the Antarctic Ice Sheet

Interesting stuff, it seems that teams have been roaming Antarctica since 1957 drilling holes and taking the subsurface temperatures. The penguin tracks are from the last 5 treks. The yellow dots are I guess the sampling sites. Anyway they seem to have archived all the data, etc. Really clear and friendly site.

The grahpic I used seems to have lost its temperature scale but the one on this link is fine. It has many of the features of Steig 1. But then it would as does altitude, distance from coast line, etc.

Alex

Thanks much, Ryan, for the background information. I would be most interested in what you find if you do follow through with the “complete” process.

Alex, what are the yellow dots? Penguin migration routes?? GISS Urban areas??

More seriously, what’s the NSIDC URL where you found this interesting graphic?

No Steve it is not “only” a boundary problem .
However “spatial autocorrelation” is just a cumbersome and unnatural way to speak about wave propagation in a very general way .
Indeed spatial autocorrelation means that what happens at a point M is influenced by what happened at point N .
And a propagating wave means that what happens at a point M will arrive sooner or later to point N .
Now a propagating wave obeys the wave equation which is given by the Laplacian regardless if one talks about physical waves (drums , fluids) or abstract waves (psi in quantum mechanics) .
That’s why a wave , ANY wave , can be written as a combination of the eigenvectors of the Laplacian and so it doesn’t surprise me that the eigenvectors of a spatial autocorrelation matrix have the same shape as the eigenvectors of wave mechanics .
When you solve the Schrodinger equation of a simple system of 1 particle in a potential well (solve = find eigenvalues of the Laplacian) you find functions that have exactly the same shape as those presented on this thread .
You can even do the exercice the other way round – combine manually some eigenfunctions in order to reconstruct the original picture of Antarctica and this for any arbitrary picture .
.
It seems to me that what Steig “discovered” and what anyone applying PCA to spatially autocorrelated data would “discover” is that the underlying process is a (any) wave propagation process .
Indeed the shape of the eigenfunctions and their symmetries are constant and completely independent of the actual system and its temporal evolution be it a drum , a tsunami or the energy of a particle .
They are of course also influenced by the geometry of the boundary (square , disk , ellipse) but it is not a fundamental influence .
.
Everybody who wants to check that can play with the following applet :
This applet gives the eigenfunctions (quantum states) of a particle in a 2D potential box (potential is 0 inside and infinite outside the square) .