PSF Shape Tests

The Effect of PSF Shape Errors on Point Source Photometry

I. Introduction

In PROPHOT, the assumed form of the point spread function (PSF) is obtained from table lookup with respect to the seeing shape factor, a quantity which is updated at time intervals corresponding to a coadd. Seeing variations during this time interval could result in a mismatch between the assumed PSF and its true instantaneous value. The purpose of the present study is to investigate the effect of such mismatches on the accuracy of point source photometry. Preliminary studies of this question have been made by Light (1998) using 2MASS data, and by Wheaton (2001) using theoretical expressions based on uniform weighting of the PSF.

II. Procedure

Our test procedure involved taking a set of survey scans and comparing the photometric results from PROPHOT with those obtained by deliberately using the PSFs corresponding to the wrong seeing shape factors. The scans selected for this purpose covered all hardware periods in both hemispheres (4 north and 1 south), and covered the dominant range of seeing shape factors (nominally 0.95-1.25 in the north, and 0.90-1.20 in the south).

For each hardware period, scans with 5 nominal values of base shape were chosen (where base shape represents the dominant value of seeing shape factor assigned by the 2MAPPS pipeline to that scan). For each scan, PROPHOT was run using a set of shape offsets, spaced at intervals of 3 bins of the PSF lookup table (where the latter has a nominal sampling interval of 0.02 in shape factor). Both positive and negative offsets were used, designed to cover the full range of shape factors in the PSF lookup table for each base shape.

For each source in the scan, the analysis procedure involved taking the difference between the magnitude estimated using the base-shape PSF and the delta-shape PSF. For each base shape and delta-shape, the mean and standard deviation of the magnitude difference was calculated for all sources in a set of representative magnitude bins. The estimated magnitudes were also compared with those from aperture photometry. The analysis included all unsaturated point sources with S/N > 10 and reduced chi squared < 1.8.

Three plots were made for each base shape:

(1) Bias plot: Mean magnitude difference as a function of delta shape.

(2) Sigma plot: Standard deviation and standard error of the bias estimate as a function of delta shape.

(3) Linearity plot: (Fitted - Aperture) magnitude difference as a function of magnitude.

The bias values were plotted using dual sets of error bars, such that the larger (outer) error bars represent the standard deviation of an individual source, while the smaller (inner) error bars represent the standard deviation of the mean (i.e. the standard error).

III. Results

The complete set of bias and linearity plots for the southern hemisphere is presented in Tables 1a, 1b, and 1c for J, H, and K, respectively. In interpreting the data in these tables, please note the following:

(1) The magnitude values are "PROPHOT magnitudes"; to get the approximate true magnitude, add 2.5*log(4), or approximately 1.5.

(2) The quoted magnitude bias corresponds to the effect of delta(shape) on the PROPHOT magnitude -- it does not include the profile-fit normalization factor (expected to be a small fraction of the bias itself).

(3) The aperture magnitudes used in the (fitted-aperture) differences do not include the aperture correction, i.e., they correspond to the standard aperture, and have not been adjusted to correspond to an infinite aperture.

(4) The (fitted-aperture) differences in the linearity plots are subject to the same offset of 1.5 as discussed above, i.e. a zero difference of true magnitudes would correspond to 1.5 on the linearity plots.

(5) For the purpose of the sigma plots, the data were divided into two magnitude ranges, namely m < m0 and m > m0, where m0 = 14, 13, and 12, for J, H, and K, respectively (in PROPHOT magnitudes). For convenience, however, the "bright source" sigma plots have been placed in the first magnitude column in the tables, even though their magnitude ranges actually span the first two columns.

Table 1a: Southern hemisphere at J

Scan	Base shape	Mag < 13	Mag = 13-14	Mag > 14	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
990918s/034	0.917	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
990911s/022	0.974	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/091	1.063	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/054	1.127	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990923s/113	1.182	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 1b: Southern hemisphere at H

Scan	Base shape	Mag < 12	Mag = 12-13	Mag > 13	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
990918s/034	0.908	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
990911s/022	0.953	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/091	1.024	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/054	1.070	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990923s/113	1.135	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 1c: Southern hemisphere at K

Scan	Base shape	Mag < 11	Mag = 11-12	Mag > 12	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
990918s/034	0.939	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
990911s/022	0.982	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/091	1.046	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990909s/054	1.095	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
990923s/113	1.139	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

The results for the 4 hardware periods of the northern hemisphere are presented in the next set of tables. Please note that the magnitude offset of 1.5 has been removed for the northern plots, i.e., the quoted magnitudes correspond approximately to the true magnitudes.

Table 2a: North Period 1 at J

Scan	Base shape	Mag < 14.5	Mag = 14.5-15.5	Mag > 15.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980429n/063	0.980	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980429n/095	1.029	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980526n/052	1.111	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980428n/027	1.173	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980427n/021	1.239	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 2b: North Period 1 at H

Scan	Base shape	Mag < 13.5	Mag = 13.5-14.5	Mag > 14.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980429n/063	0.966	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980429n/095	1.009	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980526n/052	1.085	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980428n/027	1.140	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980427n/021	1.199	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 2c: North Period 1 at K

Scan	Base shape	Mag < 12.5	Mag = 12.5-13.5	Mag > 13.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980429n/063	0.985	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980429n/095	1.026	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980526n/052	1.098	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980428n/027	1.143	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980427n/021	1.166	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 3a: North Period 2 at J

Scan	Base shape	Mag < 14.5	Mag = 14.5-15.5	Mag > 15.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980618n/021	0.967	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980610n/054	1.026	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980617n/054	1.106	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980624n/096	1.147	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 3b: North Period 2 at H

Scan	Base shape	Mag < 13.5	Mag = 13.5-14.5	Mag > 14.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980618n/021	0.993	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980610n/054	1.018	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980617n/054	1.018	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980624n/096	1.111	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 3c: North Period 2 at K

Scan	Base shape	Mag < 12.5	Mag = 12.5-13.5	Mag > 13.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980618n/021	1.042	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980610n/054	1.057	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980617n/054	1.140	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980624n/096	1.186	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 4a: North Period 3 at J

Scan	Base shape	Mag < 14.5	Mag = 14.5-15.5	Mag > 15.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980929n/087	0.965	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980929n/114	1.024	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980927n/074	1.115	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/014	1.183	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/016	1.225	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 4b: North Period 3 at H

Scan	Base shape	Mag < 13.5	Mag = 13.5-14.5	Mag > 14.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980929n/087	0.953	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980929n/114	1.005	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980927n/074	1.101	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/014	1.149	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/016	1.184	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 4c: North Period 3 at K

Scan	Base shape	Mag < 12.5	Mag = 12.5-13.5	Mag > 13.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
980929n/087	0.943	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
980929n/114	1.005	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
980927n/074	1.061	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/014	1.099	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
981004n/016	1.133	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 5a: North Period 4 at J

Scan	Base shape	Mag < 14.5	Mag = 14.5-15.5	Mag > 15.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
991108n/010	0.954	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
991113n/147	1.025	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991104n/136	1.111	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991030n/102	1.170	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991203n/101	1.211	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 5b: North Period 4 at H

Scan	Base shape	Mag < 13.5	Mag = 13.5-14.5	Mag > 14.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
991108n/010	0.918	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
991113n/147	0.990	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991104n/136	1.075	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991030n/102	1.127	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991203n/101	1.154	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

Table 5c: North Period 4 at K

Scan	Base shape	Mag < 12.5	Mag = 12.5-13.5	Mag > 13.5	All mags	Delta(shape) <0	Delta(shape) =0	Delta(shape) >0
991108n/010	0.939	Bias Sigma	Bias	Bias Sigma	Bias	-	Linearity	Linearity
991113n/147	1.004	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991104n/136	1.076	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991030n/102	1.117	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity
991203n/101	1.152	Bias Sigma	Bias	Bias Sigma	Bias	Linearity	Linearity	Linearity

IV. Discussion

Some conclusions that may be drawn from the foregoing plots are:

(1) The slope of the magnitude bias v. delta(shape) is typically about -1 at all 3 bands. So, for example, a PSF mismatch of 1 bin (0.02 in shape factor) would cause typically a 2% bias in flux. Note that in order to use our estimated systematic biases to obtain the contribution of PSF mismatch to the PROPHOT error budget, we would need to fold in these results with the statistical distribution of seeing-shape error.

(2) The absolute value of the slope increases with magnitude, particularly between the first two magnitude ranges in the tables, indicating that the effect of shape mismatch is greater for the fainter sources. Since the latter are dominated by Poisson noise (rather than PSF error) which gives a more uniform spatial weighting in the photometric solution, this suggests that the flux bias due to PSF mismatch comes more from the central portion of the PSF rather than the wings.

(3) A consequence of the previous item is that PSF mismatches affect the photometric linearity, and this is quite evident from the linearity plots.

To better illustrate the effect of PSF shape mismatches on photometric linearity, plots have been made of the deviation from linearity as a function of base shape. For this purpose, we define "deviation from linearity" as the change in the (fitted-aperture) magnitude difference over a span of 5 magnitudes, and is obtained from the slopes of the linearity plots in the above tables. The deviation v. shape plots have been made for delta(shape) values corresponding to +3 and -3 shape bins, i.e., deviations of +/- 0.06 in shape factor. The results for the northern hemisphere data (combining all 4 hardware periods) are presented in Figure 1a (J-band), Figure 1b (H-band), and Figure 1c (K-band).

In these plots, open circles represent delta(shape) = 0, while the + and - symbols represent shape mismatches of +3 and -3 shape bins, respectively.

It is apparent that at a base shape of 1.1, a shape mismatch of 3 bins can induce a substantial degree of nonlinearity, corresponding to 6-8% in flux. By interpolation, even a 1-bin shape mismatch could produce nonlinearities of 2-3%. Thus it seems that the nonlinearity problems that we experienced in the development of PSFs could easily be explained in terms of PSF shape mismatches, which could have occurred both during the generation of the PSFs themselves as well as between the shape-update intervals in survey scans. Seeing variations therefore probably set a practical limit to photometric linearity regardless of the theoretical accuracy of the PSF estimation algorithm.

Last Update - 2001 November 27
K. A. Marsh & W. Wheaton - IPAC