Yet another PSF model

[Andrew Bennett <andrew.bennett@ns.sympatico.ca>]

I have spent the summer growing grapes and, from
time to time, contemplating the problems with my
PSF fitting MK IV data analysis.

The model I was using used the sum of elliptical
Gaussian components in an attempt to model the
core/halo PSF of the MK IV. The implementation
I was using eventually grew to 3 components. 
Experimentally to 4.

This suffered from a number of problems. One obvious
ambiguity is that the solution with components
123 is identical to 231, 312, 321 etc - a total of
six solutions arrived at by swapping components -
and the code proved to be entirely willing to swap
components pretty much at random. This makes it very
difficult to compare the fit for consecutive images.

Worse: it appears the code was quite capable of making
different assignments of components in different parts
of the same image, thus giving grossly different apparent
spatial variation in PSF from one image to another. That 
is, the fitted PSF can be seriously wrong in some parts of
some images.

This appears to be one of the reasons for the poor 
performance of my Ensemble photometric calibration
code. The results other people have been showing us this
year are better than I managed to get, in spite of the
clear superiority (in principle) of PSF fitting over
aperture photometry. [AB: personal opinion]

As a result of my summer's thought, I am now trying
out a model based on Type A probability distributions -
essentially a Gaussian multiplied by a polynomial.
This model was originally rejected because it does not do
a good job with the large halos of the original MK IV
lenses: modelling a profile seriously different from a
plain Gaussian requires a very large number of coefficients
with this type of model. And has convergence problems.

This type of model does not suffer from the ambiguities
of the multi-component model. The first run, using Data
Set 20, shows mostly smooth variation of coefficients
from image to image; most of the 35 coefficients are
significantly different from zero. Preliminary Ensemble
analysis suggests that the new model works better than
the old. Or, at least, less badly.

Current model
2nd order terms 2nd order spatial variation 18 coefficients
3rd order terms 1st order spatial variation 12 coefficients
4th order terms 5 coefficients
Total 35 coefficients

Old model
3 ellipses: size 2nd order spatial variation 18 coefficients
   ellipticities 1st order spatial variation 18 coefficients
2 separations 1st order spatial variation 12 coefficients
2 amplitudes 2 coefficients
Total 50 coefficients

The 50 coefficients give a better fit than the 35. Some
of the time! Adding one more order everywhere to the new
fit needs 75 coefficients which is rather a lot ... the
hope is that some will prove negligible. 

Andrew Bennett, Avondale Vineyard