INTRODUCTION
This document describes recent work designed to address some of the shortcomings of the PSFs used in prophot photometry. The principal problem which has been identified so far is that of systematic distortions resulting from undersampling and subsquent sinc interpolation. Although the optimal cure for this problem is simply to calculate the PSF on a Nyquist-sampled grid, the computational burden makes this approach impractical with currently available resources. An alternate approach, involving a correction to the PSF, obtained by comparing the distorted initial estimate with the original data, has been outlined in a previous document.
Briefly, the correction is obtained by spatially binning the residuals between the initial PSF estimate and the data, and taking the average value of the PSF correction within each bin. To be more precise, a sliding Gaussian weighting kernel is used rather than a set of fixed bins. The weighting kernel corresponds to a spatial averaging, and therefore the choice of the kernel size is important: it must be large enough to collect sufficiently good statistics, but must be small enough so as not to produce undue smoothing of the correction.
For a given kernel size, the accuracy of the correction can be improved by increasing the spatial density of residuals. In the context of psfmake, this can be accomplished quite straighforwardly by processing several scans at once, rather than a single scan as is currently done. This would also make the subsquent PSF combination step unnecessary.
With the above considerations in mind, the goal of the present development work is:
(1) Incorporate the residual-based correction to the PSF
(2) Process several scans at once (corresponding to all scans within a given seeing-shape bin)
(3) Determine, from theoretical considerations, the optimal kernel size.
IMPLEMENTATION OF RESIDUAL-BASED CORRECTION
The algorithm has been coded and tested on a set of 5 scans, corresponding
to J shape bin 0.915 in the south:
981028s s078
981101s s073
981223s s112
990413s s018
Figure 1 shows images of (a) the estimated PSF,
(b) the variance map, and (c) the correction. All images are on a linear
intensity scale with a field of view corresponding to 5 x 5 focal-plane pixels.
The intensity scale of the correction map (panel c) covers the range -4% to
+4% of the peak value of the PSF.
Plots of the delta-mag (PSF - aperture) and chi squared as a function of magnitude are shown in Figure 2. These plots are somewhat incestuous, since they were constructed from photometry of the same stars used in the estimation of the PSF itself.
These results were repeated after implementing an additional modification in psfmake, namely trimming of the residuals. Specifically, the residuals used for the PSF correction and variance calculations were trimmed at the 3-sigma level. The resulting plots of delta-mag and chi squared are shown in Figure 3.
Unexpectedly, the trimming has perturbed the linearity. The problem was occurring in the PSF correction routine, and apparently the 3-sigma trimming was too severe, resulting in a pixel-value-dependent bias of its own. Since it would theoretically be desirable to have some trimming in order to weed out occasional large deviations, the trimming level was increased to 10-sigma. This produced results indistinguishable from the "no trimming" case above.
The resulting algorithm has been used
to generate the 5 PSFs used by prophot on the survey scan 991024s, s044.
The PSF names (labelled by seeing-shape), the number of scans
used to calculate them, and the corresponding plots (delta-mag and
chi squared), are as follows:
Name Nscans Plot
j09155 19 Figure 4a
h09325 20 Figure 4b
h09115 20 Figure 4c
k09355 17 Figure 4d
k09355 20 Figure 4e
Greyscale representations of the PSF, variance map, and correction map
in each case are shown in columns 1, 2, and 3, respectively, of
Figure 5
These PSFs were used in a prophot run on 991024s, s044, and linearity plots produced, as shown in Figure 6a, Figure 6b, Figure 6c, for J, H, and K, respectively. The top plot of each figure shows the current results, while the middle and lower plots show the version 2 and version 3 results for comparison.
Sigma plots are as follows:
J-band:
H-band:
K-band:
Overall assessment: H-band linearity is disappointing, and needs thinking
about. The sigmas for the new results seem good, and are a definite
improvement over version 3.
SOME THOUGHTS ON PSF-FLUX SIGMAS AND LINEARITY
We have been assuming that the elevated sigmas and nonlinearity are both symptoms of the undersampling-related PSF distortion. While this is very likely to be the case for the nonlinearity problems, there is an additional effect which may be involved in the elevated sigmas. Specifically, we may be paying a price for neglecting to take into account the spatial correlation of the seeing fluctuations. In prophot, the parameter estimation is accomplished using inverse-variance weighting, assuming that the errors in pixel values are uncorrelated. The original justification for this was that the PSF is not too much larger than a pixel, and this approximation considerably reduces the computation involved. However, the optimal solution involves weighting by the inverse of the full covariance matrix, and we need to consider whether some of the recent problems are due to the neglect of the off-diagonal terms.
The effect of the off-diagonal terms of the inverse covariance matrix can be thought of as a "spatial smearing" of the pixel values used in the maximum likelihood solution, and it is conceivable that at least some of the recently-investigated empirical "floor" to the variance maps can be regarded as a crude substitute for the proper application of these off-diagonal terms.
A possible approach to the investigation and correction of this effect, within the current time constraints is as follows:
(1) Have psfmake calculate the full covariance matrix of the PSF residuals (rather than simply the variance values, which are the diagonal terms).
(2) After having taken the inverse of this covariance matrix (within psfmake),
use the results to generate a "pseudo-variance map" which, when applied in
the usual way by prophot, will simulate the spatial smearing which would
have been produced by the off-diagonal terms. A mathematical basis for
such an "effective variance map" has been derived, and will be detailed in
a subsequent document.
The 5 psfs used by prophot in scan 991024s/s044 (j09155, h09115, h09325,
k09355, k09545) were regenerated after making a fix in psfmake_multiscan.
The fix involved deferring the optimal solution for the PSF correction
until after all of the bad stars have been weeded out. Previously, bad
stars had been severely perturbing the variance map, which, in turn, had
perturbed the psf correction, resulting in the weeding out of good stars
(preferentially the brightest ones) early on in the iterative process.
Prophot was then re-run on 991024s/s044 using the new psfs. The
resulting magnitude bias plots are shown in
Figure 8.
Unfortunately, the new fix has not improved the linearity at H, which is
just as bad (possibly worse) than before. In light of this, it is worthwhile
to look at the plots for the photometry results produced by psfmake_multiscan.
These are shown in:
It is apparent that for the stars used to make the PSFs (about 800 for
each PSF) the photometry was very linear indeed, including at H band
where the problems occurred with 991024s/s044. This is mysterious.
It would be instructive to run prophot on the scans used to generate
the H-band psfs.
The following plots represent the results of psfmake photometry during the
production of the new south PSFs over the weekend of July 21-22, 2001:
These may be compared with the results of running prophot on the same
scans used by psfmake:
Note that when prophot was run on these scans, it did not always pick the
PSF corresponding to the nominal seeing bin. Presumably this is due
to seeing variations within a cal scan, but will be investigated further.
In the plots above, the sources for which prophot used the "wrong" PSF
have been filtered out, and hence the prophot plots contain somewhat less
data that the psfmake plots for the magnitude range of the psf stars. Taking
this into account, I draw the following conclusions from the comparison of
the psfmake and prophot plots:
For the prophot runs of the 5 sets of cal scans listed above, the actual
PSFs selected by prophot, and the percentage of the scan for which the
PSFs were used are as follows:
It is reassuring that the peak fraction of sources occurs for the PSF
corresponding to the nominal seeing bin, and that the range of PSFs
selected by prophot is consistent with the Delta-shape for that seeing
bin. So prophot is apparently
making the correct selection, but there is a significant range of
seeing shape variation within a scan.
In order to assess the effect of the seeing variation on the photometric
linearity, we can compare the "filtered" prophot results of Figure 11
with the following "unfiltered" results, which include all sources for
the scans in a given seeing bin, regardless of whether the PSF ID
selected by prophot corresponded to the nominal seeing bin:
Comparing the Figure 12 plots with their counterparts of Figure 11, it
is apparent that the seeing variations have a large effect on photometric
linearity. The difference is particularly apparent at J-band -- over the
magnitude range of the psf stars, the photometry based on the "correct" PSF
(Figure 11a) shows much better linearity than the unfiltered results of
Figure 12a.
Based on these results, it is likely that some or all of the linearity problems
that we've been experiencing are due to seeing variations during
the scans used for PSF generation, the effect of which is that the PSFs do
not fully characterize the spatial response for the particular shape bin.
So the moral of the story appears to be: select PSF scans with the smallest
possible value of delta-shape.
The above modifications have been incorporated into the new psfmake.
The results so far have not been successful. An example of the estimated
PSF, pseudo-variance map, and correction are shown in
Figure 7. It is apparent that the
pseudo-variance map has rather erratic behavior, which may be due to
the way in which the covariance matrix of the PSF is currently being
estimated. The process is susceptible to small errors in the off-diagnoal
terms; it may be that smooth basis functions should be used instead.
Depending on the results of using the PSF correction with the simple variance
maps, the covariance information may not actually be required after all.
This aspect of the investigation is continuing.
UPDATES
JULY 20: Re-run 991024s/s044 test.
Figure 9a: j09155
Figure 9b: h09115
Figure 9c: h09325
Figure 9d: k09355
Figure 9e: k09545
JUL 24: Comparison of prophot and psfmake photometry
Figure 10a: j09155
Figure 10b: h09115
Figure 10c: h09325
Figure 10d: k09355
Figure 10e: k09545
Figure 11a: j09155
Figure 11b: h09115
Figure 11c: h09325
Figure 11d: k09355
Figure 11e: k09545
(1) Prophot produced linear photometry on the psf stars.
(2) There is no significant difference between the photometry results of
psfmake and prophot.
JULY 25: The effect of seeing variations on photometric linearity.
Band PSF bin Delta- PSFs used Fraction
shape by prophot of sources
J 09155 0.024 09155 52%
09315 48%
H 09115 0.026 09115 71%
09325 29%
09325 0.025 09115 4%
09325 63%
09505 32%
09715 1%
K 09355 0.036 09355 57%
09545 41%
09705 2%
09545 0.035 09355 7%
09545 51%
09705 41%
09905 1%
Figure 12a: j09155
Figure 12b: h09115
Figure 12c: h09325
Figure 12d: k09355
Figure 12e: k09545