Instrument Science Report WFPC2 98-02
The long vs. short anomaly
in WFPC2 images
S. Casertano and M. Mutchler
September 24, 1998
ABSTRACT
We present an improved characterization of the so-called “long vs. short” anomaly, a nonlinear behavior of the WFPC2 chips which results in a decrease of the measured count
rate with decreasing exposure time. We demonstrate that: 1) this non-linearity depends
strictly on total counts in a stellar image, and is independent of exposure time; 2) the effect
depends on position on the chip only through the well-known CTE effect, and is independent of position after the CTE correction is applied; 3) F555W and F814W appear similarly affected; and 4) there is a marginal decrease in non-linearity at high background
levels, but the effect is statistically insignificant in the overall description of the anomaly.
We present a simple formula that can be used to correct the observed counts in a variety of
conditions; this formula is recommended for use with most point-source photometry
obtained with WFPC2.
1. Introduction
The so-called “long vs. short” anomaly was discovered early on in the operational life
of WFPC2 by Stetson (1995) by comparing ground-based photometry with WFPC2 exposures of length ranging from over 1000s to a few tens of seconds. Early reports can be
found in Kelson et al (1996) and Saha et al (1996). The name derives from the initial interpretation of the anomaly as a zero point difference of about 5% between long and short
exposures. This effect has been independently confirmed in a number of studies (see the
WFPC2 Clearinghouse at http://www.stsci.edu/ftp/instrument_news/WFPC2/
Wfpc2_clear/wfpc2_clrhs.html for an up-to-date list of references). Casertano (1995)
used multiple exposures of a field in the globular cluster ω Cen to characterize this anomaly, finding that the count rate measured for a given object increases with increasing
exposure time. Similar results were obtained by Stiavelli (1995) using observations of the
globular cluster NGC 2419. There is also a report (Ratnatunga 1995) of consecutive
images with a pixel-to-pixel ratio about 3% smaller than the ratio of the exposure times,
1
indicating a possible non-linearity in the opposite sense; we have been unable to confirm a
pattern of such occurrences. For a short summary of the earlier results, see also Casertano
(1997).
On the basis of the evidence collected so far, the anomaly appears to be more properly
a function of total counts in a stellar image, rather than a direct function of exposure time.
The commonly used “long vs. short” name is thus somewhat of a misnomer.
The information obtained from archival data proved adequate to quantify the effect of
the anomaly under a specific set of circumstances, but was insufficient to obtain a correction suitable for general use. The WFPC2 group has therefore obtained an extensive set of
HST observations, under the calibration program CAL 7630, to help achieve a broader
characterization of the “long vs. short” anomaly. We present here a preliminary analysis of
this dataset, and give a correction formula that reduces the residual systematic effect of the
anomaly below the random error due to the photon noise in our observations.
Users interested solely in the correction formula can skip directly to Section 5, which
details the recommended correction procedure. Since we could not possibly reproduce all
realistic observing conditions in our tests, we encourage users to report to us any successful or unsuccessful applications of our correction formula by sending e-mail to
help@stsci.edu.
2. Description of CAL 7630
The observing program CAL 7630 was designed to obtain the most detailed characterization of the long vs. short anomaly possible within a limited investment in observing
time. A total of 14 orbits were used to carry out a large number of separate tests on a previously studied field including the Galactic globular cluster NGC 2419. The tests included
the following:
1. Basic measurement of the effect: obtain images with exposure time ranging from
10s to 1000s; this provides a test of the effect in extreme conditions;
2. Dependence on background: apply preflash levels between 5 and 1000 e/pixel
(previous tests used a maximum of 300 e/pixel);
3. Wavelength dependence: besides F814W, the filter used for the primary characterization of the effect, we took data with F555W and F300W as well;
4. Hysteresis: data were sequenced in both ascending and descending exposure times,
to test whether the order of the exposures affects the measurements;
5. Noiseless preflash: multiple exposures were taken about 35 minutes after flooding
the chip with charge that was then read out; it was hoped that this procedure would
fill any traps on the chip (possible causes of the anomaly) and thus greatly reduce
the non-linearity;
2
6. Position on the chip: one set of observations was taken with a large shift, thus testing the same stars in different positions.
In total, 52 images were taken at essentially the same pointing, and 5 more at the
shifted pointing. All exposures were taken at gain 7. The complete list of exposures, given
in Table 1, includes also an INTFLAT exposure used for the noiseless preflash. The heavy
horizontal lines in Table 1 mark the orbit boundaries.
Table 1: Observations for CAL 7630
Dataset
Target
Filter
Exptime
U4CT0101R
NGC2419
F814W
10.
U4CT0102R
NGC2419
F814W
10.
U4CT0103R
NGC2419
F814W
U4CT0104R
NGC2419
U4CT0105R
Preflash
(e/pix)
Date
Time
PA_V3
RA_V1
Dec_V1
0.
11/18/97
07:34:13
79.283
07 38 09.381
+38 52 03.194
5.
11/18/97
07:42:13
79.283
07 38 09.381
+38 52 03.194
10.
10.
11/18/97
07:50:13
79.283
07 38 09.381
+38 52 03.194
F814W
10.
100.
11/18/97
08:43:13
79.283
07 38 09.381
+38 52 03.194
NGC2419
F814W
10.
1000.
11/18/97
09:07:13
79.283
07 38 09.381
+38 52 03.194
U4CT0106R
NGC2419
F814W
40.
0.
11/18/97
09:09:13
79.283
07 38 09.381
+38 52 03.194
U4CT0107R
NGC2419
F814W
40.
5.
11/18/97
09:17:13
79.283
07 38 09.381
+38 52 03.194
U4CT0108R
NGC2419
F814W
40.
10.
11/18/97
09:25:13
79.283
07 38 09.381
+38 52 03.194
U4CT0109R
NGC2419
F814W
40.
100.
11/18/97
10:20:13
79.283
07 38 09.381
+38 52 03.194
U4CT010AR
NGC2419
F814W
40.
1000.
11/18/97
10:32:13
79.283
07 38 09.381
+38 52 03.194
U4CT010BR
NGC2419
F814W
40.
100.
11/18/97
10:40:13
79.283
07 38 09.381
+38 52 03.194
U4CT010CR
NGC2419
F814W
40.
10.
11/18/97
10:48:13
79.283
07 38 09.381
+38 52 03.194
U4CT010DR
NGC2419
F814W
40.
5.
11/18/97
10:56:13
79.283
07 38 09.381
+38 52 03.194
U4CT010EM
NGC2419
F814W
40.
0.
11/18/97
10:58:13
79.283
07 38 09.381
+38 52 03.194
U4CT010FR
NGC2419
F814W
100.
0.
11/18/97
11:00:13
79.283
07 38 09.381
+38 52 03.194
U4CT010GR
NGC2419
F814W
100.
10.
11/18/97
11:57:13
79.283
07 38 09.381
+38 52 03.194
U4CT010HR
NGC2419
F814W
100.
100.
11/18/97
12:06:13
79.283
07 38 09.381
+38 52 03.194
U4CT010IR
NGC2419
F814W
100.
1000.
11/18/97
12:19:13
79.283
07 38 09.381
+38 52 03.194
U4CT010JR
NGC2419
F814W
300.
0.
11/18/97
12:22:13
79.283
07 38 09.381
+38 52 03.194
U4CT010KR
NGC2419
F814W
300.
100.
11/18/97
12:36:13
79.283
07 38 09.381
+38 52 03.194
U4CT010LR
NGC2419
F814W
300.
1000.
11/18/97
13:37:13
79.283
07 38 09.381
+38 52 03.194
U4CT010MR
NGC2419
F814W
300.
100.
11/18/97
13:51:13
79.283
07 38 09.381
+38 52 03.194
U4CT010NR
NGC2419
F814W
300.
0.
11/18/97
13:59:13
79.283
07 38 09.381
+38 52 03.194
U4CT010OR
NGC2419
F814W
1000.
0.
11/18/97
15:07:13
79.283
07 38 09.381
+38 52 03.194
U4CT010PR
NGC2419
F814W
1000.
100.
11/18/97
15:32:13
79.283
07 38 09.381
+38 52 03.194
U4CT0201M
NGC2419
F814W
1000.
1000.
11/18/97
16:55:13
79.283
07 38 09.381
+38 52 03.194
U4CT0202R
NGC2419
F555W
40.
0.
11/18/97
17:15:13
79.283
07 38 09.381
+38 52 03.194
U4CT0203R
NGC2419
F555W
40.
10.
11/18/97
17:23:13
79.283
07 38 09.381
+38 52 03.194
U4CT0204R
NGC2419
F555W
40.
100.
11/18/97
17:31:13
79.283
07 38 09.381
+38 52 03.194
U4CT0205R
NGC2419
F555W
40.
1000.
11/18/97
18:28:13
79.283
07 38 09.381
+38 52 03.194
U4CT0206R
NGC2419
F555W
300.
0.
11/18/97
18:30:13
79.283
07 38 09.381
+38 52 03.194
U4CT0207R
NGC2419
F555W
300.
100.
11/18/97
18:44:13
79.283
07 38 09.381
+38 52 03.194
U4CT0208R
NGC2419
F555W
300.
1000.
11/18/97
19:02:13
79.283
07 38 09.381
+38 52 03.194
U4CT0209R
NGC2419
F300W
100.
0.
11/18/97
19:58:13
79.283
07 38 09.381
+38 52 03.194
U4CT020AR
NGC2419
F300W
100.
100.
11/18/97
20:07:13
79.283
07 38 09.381
+38 52 03.194
U4CT020BR
NGC2419
F300W
100.
1000.
11/18/97
20:20:13
79.283
07 38 09.381
+38 52 03.194
U4CT020CM
NGC2419
F300W
1000.
0.
11/18/97
20:23:13
79.283
07 38 09.381
+38 52 03.194
U4CT020DR
NGC2419
F300W
1000.
100.
11/18/97
21:41:13
79.283
07 38 09.381
+38 52 03.194
U4CT020EM
NGC2419
F300W
1000.
1000.
11/18/97
23:27:13
79.283
07 38 09.381
+38 52 03.194
U4CT0301R
NGC2419
F814W
40.
0.
11/19/97
01:08:13
79.279
07 38 07.674
+38 53 47.371
U4CT0302R
NGC2419
F814W
40.
100.
11/19/97
01:16:13
79.279
07 38 07.674
+38 53 47.371
3
Table 1: Observations for CAL 7630
Dataset
Target
Filter
Exptime
U4CT0303R
NGC2419
F814W
40.
U4CT0304R
NGC2419
F814W
300.
-----------
--------
------
U4CT0305R
NGC2419
F814W
-----------
--------
------
---------300.
----------
Preflash
(e/pix)
Date
Time
PA_V3
RA_V1
Dec_V1
1000.
11/19/97
01:28:13
79.279
07 38 07.674
0.
11/19/97
01:30:13
79.279
07 38 07.674
+38 53 47.371
----------
---------
--------
--------
---------------
----------------
1000.
11/19/97
02:48:13
79.279
07 38 11.896
+38 53 31.096
----------
---------
--------
--------
---------------
----------------
+38 53 47.371
U4CT0401R
INTFLAT
-----
600.
3000.
11/19/97
03:38:13
------
-- -- ------
--- -- ------
U4CT0402R
NGC2419
F814W
40.
0.
11/19/97
04:23:13
79.283
07 38 09.381
+38 52 03.194
U4CT0403R
NGC2419
F814W
40.
0.
11/19/97
04:25:13
79.283
07 38 09.381
+38 52 03.194
U4CT0404R
NGC2419
F814W
40.
0.
11/19/97
04:27:13
79.283
07 38 09.381
+38 52 03.194
U4CT0405R
NGC2419
F814W
40.
0.
11/19/97
04:29:13
79.283
07 38 09.381
+38 52 03.194
U4CT0406R
NGC2419
F814W
40.
0.
11/19/97
04:31:13
79.283
07 38 09.381
+38 52 03.194
U4CT0407R
NGC2419
F814W
40.
0.
11/19/97
04:33:13
79.283
07 38 09.381
+38 52 03.194
U4CT0408R
NGC2419
F814W
40.
0.
11/19/97
04:35:13
79.283
07 38 09.381
+38 52 03.194
U4CT0409R
NGC2419
F814W
40.
0.
11/19/97
04:37:13
79.283
07 38 09.381
+38 52 03.194
U4CT040AR
NGC2419
F814W
40.
0.
11/19/97
04:39:13
79.283
07 38 09.381
+38 52 03.194
U4CT040BR
NGC2419
F814W
40.
0.
11/19/97
04:41:13
79.283
07 38 09.381
+38 52 03.194
U4CT040CR
NGC2419
F814W
40.
0.
11/19/97
04:43:13
79.283
07 38 09.381
+38 52 03.194
U4CT040DR
NGC2419
F814W
40.
0.
11/19/97
04:45:13
79.283
07 38 09.381
+38 52 03.194
U4CT040ER
NGC2419
F814W
40.
0.
11/19/97
04:47:13
79.283
07 38 09.381
+38 52 03.194
3. Data Analysis 1: Characterization
Image calibration
The images were first reduced through the standard STSDAS WFPC2 pipeline task
calwp2, using the normal bias, dark, and flat field. In order to conserve observing time, we
did not take cosmic ray splits, and therefore we did not carry out cosmic ray rejection on
the images. Since WFPC2 has no “faint” cosmic rays (WFPC2 Handbook, Biretta et al
1996, Section 4.9), and the vast majority of CR hits deposits a charge of at least 700 e, we
were able to identify the stars affected by cosmic ray hits as outliers in the subsequent
analysis. We also increased the robustness of the analysis against outliers by using median
and quartiles, instead of average and rms dispersion, wherever practical.
Photometry
After the pipeline processing, a master list of stars - containing 3470 objects - was constructed from the longest images available, and aperture photometry was performed using
the IRAF task noao.digiphot.apphot at the position of each star, using aperture radii
ranging from 1 to 5 pixels; the 2 pixel aperture was used as the primary measurement. The
sky level was determined in an annulus between 7 and 12 pixel radius, using the option
ofilter as recommended by Ferguson (1996). All photometry results were then placed in a
master database used for all subsequent analysis.
4
Reference magnitude
The next step was to establish a reference magnitude for each object. We started by
assuming that the count rates in the longest exposures would be least affected by the nonlinearity, and thus used the combination of all 1000s exposures (300s in F555W) to determine an initial reference magnitude in each filter and aperture. When using CTE-corrected
magnitudes (see Section 3), the reference magnitudes were also computed from CTE-corrected raw magnitudes.
Subsequent analysis shows that even the magnitude measured in the longest exposure
may be affected by the non-linearity for faint enough stars. The correction formula derived
in Section 4 to bring the long and short exposures into agreement has been applied to both
the short and the long exposure data; more details are given in Section 4.
Basic measurement
We then proceeded to determine a baseline measurement of the magnitude difference
as a function of the various observational variables: exposure time, reference magnitude,
background, history, filter, and position in the detector. We examine each one in turn.
Magnitude and exposure time
Figure 1 gives the most basic representation of the anomaly: the magnitude difference
(short minus long) is plotted as a function of magnitude in the long exposure for non-preflashed F814W images. Magnitudes are defined in a 2 pixel aperture; individual panels
compare short exposures of 10s, 40s, 100s, and 300s, respectively, against the reference
long exposure which is always 1000s. The crosses indicate the median difference in 0.5
magnitude bins, and the vertical bars span the interquartile range (between the 25% and
75% percentile) for each bin.
Figure 1 shows clearly that 1) the magnitude difference is a strong function of magnitude, with a larger difference found for fainter objects, and 2) for a given magnitude, the
magnitude discrepancy is larger for larger ratios of exposure times. This suggests that the
magnitude discrepancy is largest when the “short” exposure has the fewest counts, either
because the object is faint or because the exposure time is short. We will quantify this
dependence in Section 4.
5
Figure 1: Magnitude discrepancy vs. reference magnitude for various exposure times,
without preflash. Magnitude I=21 corresponds to 1.7 counts/second.
Dependence on aperture
The magnitude discrepancy occurs for all apertures, and generally seems to increase
with the size of the aperture. This is illustrated in Figure 2, where the median and interquartile range for the magnitude difference between non-preflashed 40s and 1000s
exposures is plotted as a function of the magnitude in the long exposure. The median is
shown for each aperture, and the interquartile range for the 1 pixel aperture only. For
example, the median magnitude discrepancy for I ~ 21 mag (about 1.7 counts/second in
the 2-pixel aperture) nearly doubles, from 0.18 mag in the 1 pixel aperture to 0.35 mag in
the 5 pixel aperture. Note that there is a slight reversal of this trend for the faintest stars,
where the discrepancy in the 5 pixel aperture is smaller than in the 3 pixel aperture. Other
apertures and preflash levels show very similar results, as shown in Figure 3 for the 100s
exposure with a preflash of 100e/pixel.
6
Figure 2: Dependence of magnitude discrepancy on aperture: 40s exposure, no preflash.
The general pattern thus is that the discrepancy increases with the size of the aperture
used, with the exception of very faint stars in apertures > 3 pixel. In the following, we will
use almost exclusively a 2 pixel aperture, which is a good compromise for most projects
requiring accurate photometry: larger apertures generally increase the statistical noise
because of the additional background and read noise, and smaller apertures may be excessively sensitive to the focus position (Suchkov and Casertano 1997). The 2 pixel aperture
has been often used in the literature for WFPC2 data, and a high-quality CTE correction is
available (Whitmore and Heyer 1997, Whitmore 1998).
Dependence on background level
The other well-known WFPC2 non-linearity, the CTE effect, lessens its impact with
increasing background, presumably because the increase in background helps fill the traps
that are likely responsible for the loss of charge. This is an important element to establish
7
Figure 3: Aperture dependence for 100s exposure with preflash of 100e/pixel.
the impact of this anomaly on real-world observations, since observations designed to
detect faint sources usually include long exposures, which often have significant background (up to about 150 e/pixel for normal broad-band V and I exposures). Since we
observed relatively bright stars, the natural background in our short exposures - designed
to test the magnitude discrepancy for real-world faint objects - is typically less than 5 e/
pixel. In order to test the influence of the background, we added an artificially elevated
background to some of the images, in the form of an image preflash. The preflash was
obtained by turning on the INTFLAT lamp, which illuminates the chip via light reflected
off the shutter blades, for a predetermined amount of time. A narrow-band filter (F502N)
was used to obtain a convenient count rate. We used preflash levels of 5, 10, 100 and 1000
e/pixel, bracketing the values that are likely to occur in real observations. Since the illumination produced by the INTFLAT lamp is not uniform, we measure the actual background
individually for each star.
8
Indeed, the non-linearity appears to decrease somewhat with increasing background, at
least at very high background levels. In Figure 4 we plot the magnitude discrepancy
between the 40s and 1000s exposures within a 2-pixel aperture for stars with 20.5 < I <
21.0, as a function of background around the star.
Figure 4: Magnitude discrepancy vs. background.
However, a substantial decrease of the discrepancy is only seen in the high background
case: 150 DN/pixel (about 1000 e/pixel) are required for the median discrepancy to drop
from 0.25 mag to 0.15 mag. At such high background, the ability to measure individual
faint sources is impaired by the additional shot noise produced by the preflash, as shown
by the increased size of the error bars in Figure 4. For example, an I=20 source produces
about 50 ± 5 DN in 40s within a 2 pixel aperture, where the rms error includes read noise,
shot noise, and a background of about 5 DN/pixel; the source magnitude can thus be measured with a statistical uncertainty of about 0.1 mag. With the large preflash used here
9
(about 150 DN/pixel), the signal is increased by about 1800 DN, bringing the statistical
error to about 17 DN - or 0.3 mag. Therefore, preflashing is not a viable method to reduce
the impact of the long vs. short anomaly on magnitude measurements for faint sources.
Similarly, an extremely high background - unlikely to occur, except in extremely crowded
fields or on within an extended source, such as a galaxy - is required to suppress slightly
the magnitude discrepancy. Therefore, it appears that the background does not play a significant role in the long vs. short magnitude discrepancy.
Figure 5: Effect of noiseless preflash (thin lines) compared with non-preflashed exposures
(vertical bars).
Dependence on history and noiseless preflash
Comparison of images taken with the same parameters at different points in the orbit,
before or after long exposures, and before or after preflash shows no measurable change.
10
The magnitude discrepancy appears to have no significant history term: it depends on the
properties of each exposure and not on the status of the camera at the time.
We tested separately a special case of dependence on history: the so-called noiseless
preflash, which consists of preflashing the detectors with charge that is read out before the
actual exposure is taken. This test was performed in the hope that the charge generated by
a heavy preflash would be persistent enough in time to reduce the magnitude discrepancy
even if read out. In the surface trap picture (Biretta and Mutchler 1998), the interpretation
would be that the traps responsible for the charge loss would be filled, and perhaps remain
filled for long enough that subsequent exposures would lose less charge, just as preflashed
exposures lose less charge.
We carried out a noiseless preflash test by illuminating the detectors with a total charge
of 3000 e/pixel, three times more than the highest normal preflash we used, before the start
of a series of fourteen 40s exposures. The 40s exposures were then reduced similarly to all
other exposures, and the magnitude discrepancy compared with that of the “standard” 40s
exposures. However, there was a 35 minute interval between the preflash and the start of
the 40s exposures; based on the recent results of Biretta and Mutchler (1998), this is more
than twice the half-life of the trapped charge, so that any benefit of the noiseless preflash
would have been greatly reduced.
Figure 5 shows the result of the noiseless preflash test. Each of the thin lines plots the
median magnitude discrepancy between one of the fourteen 40s exposures and the standard 1000s long exposure. The solid line with error bars shows the median and
interquartile range for the normal, non-preflashed 40s exposure. There is no measurable
difference between the exposures with or without noiseless preflash, or within the set of
consecutive exposures after the noiseless preflash. While this indicates no effect of the
noiseless preflash on the current data, the long delay between the preflash and the exposures, combined with the new information on the decay time of trapped charge, indicate
that a new test of the noiseless preflash, with a short interval between preflash and test
exposures, is required.
Dependence on position on the chip
Another test of interest is the dependence of the magnitude discrepancy on the x and y
position within the chip. The CTE effect is well known to depend primarily on row number (or the y position), as it consists of a charge loss proportional to the number of rows
traversed during readout; this is the parallel-register CTE, or y-CTE. Whitmore and Heyer
(1997) have recently published an extensive quantitative characterization of the CTE
effect using a set of images in which stars were shifted from one CCD to another, and thus
moved from the top to the bottom or from the right to the left of the CCD (since each CCD
has a different orientation on the sky). They also find a CTE effect in the x direction (serial
register CTE, or x-CTE).
11
Figure 6: x and y dependence of the short vs. long magnitude discrepancy.
If the data used here are affected by the CTE loss, we expect to see a dependence of
the long vs. short magnitude discrepancy on the x and y coordinate as a second-order
effect, even for images taken at the same pointing. The reason is that the CTE charge loss
is larger for fainter images (fewer total counts), and therefore sources at high row numbers
will be affected more in the short exposures than in the long exposures, thus contributing a
row-dependent term to the ‘‘long vs. short’’ anomaly.
For example, consider two stars of the same magnitude, one at small y and the other at
large y. A long and a short exposure are taken. The star at small y will see very little y-CTE
effect. The star at large y will see a smaller fractional charge loss in the long exposure
(CTE is small for bright objects), and a larger fractional charge loss in the short exposure.
Thus the CTE loss will appear as a “long vs. short” effect at high y, but not at low y. Similar considerations apply to the x-CTE if present.
12
Figure 7: x and y dependence of the magnitude discrepancy after CTE correction
This effect can be seen in Figure 6 (top), where the median and interquartile ranges of
the magnitude discrepancy are plotted against the y coordinate for 40s and 100s exposures
without preflash. On the other hand, the magnitude discrepancy appears to vary only very
weakly with the x coordinate (Figure 6, bottom), possibly due to the smaller amplitude of
the x-CTE.
In order to determine the ‘‘true’’ dependence of the long vs. short anomaly on position
in the chip, we correct the measured magnitudes for the CTE loss as recommended by
Whitmore and Heyer (1997), with the modifications suggested in Whitmore (1998) to
account for the time variation in the CTE loss. Specifically, we apply the following
corrections:
∆ f y = 1.47 ⋅ ( y ⁄ 800 ) ⋅ 10
1.0474 – ( 0.2564 ⋅ log BKG ) – ( 0.0987 ⋅ log CTS )
∆f x = 1.26 ⋅ ( x ⁄ 800 ) ⋅ ( 7.373 – 1.57 ⋅ log CTS )
13
(1)
where ∆f is the percent change in flux, BKG are the total counts in DN (gain 7) per
pixel in the background, and CTS are the total counts in DN in the source (2 pixel radius).
After these corrections, the residual magnitude discrepancy (Figure 7) appears essentially
independent of position; the y dependence is almost entirely corrected by the Whitmore
and Heyer formulae, while the x dependence appears slightly overcorrected. We will use
the CTE-corrected measurements in the remainder of the analysis.
Figure 8: The magnitude discrepancy in F555W (solid), compared with F814W (dashed)
Dependence on filter
Finally, we wish to check whether the magnitude discrepancy is the same in different
filters. The bulk of the data for program 7630 were taken in F814W, but we also collected
a limited amount of data in F300W and F555W. Figure 8 shows the magnitude discrepancy for F555W, plotted as a function of instrumental magnitude - thus points at the same
14
abscissa have the same number of counts in equal-length exposures in the two filters. The
magnitude discrepancy is essentially the same in both filters for a given instrumental magnitude. Unfortunately, the data in F300W proved insufficient to characterize the magnitude
discrepancy, because of the low flux and small number of usable stars.
Figure 9: Median and interquartile range for the magnitude discrepancy in F814W for different exposure times, plotted vs. the total counts in the short exposure.
4. Data Analysis 2: A Unified Description
As illustrated in Section 3, the basic features of the so-called long vs. short anomaly
are: strong dependence on counts, weak dependence on background, and no measurable
dependence on history, filter, and on position in the chip (the latter after correcting for the
well-known CTE anomaly). Here we present a simple description that explains the major
features of the long vs. short anomaly and offers an avenue towards correcting its effects in
real-world data.
15
In this view, the long vs. short anomaly is due to a non-linearity of the WFPC2 detectors and/or signal chain. The non-linearity causes a loss of signal which is proportionately
larger for fainter signal. The result is a fractional signal loss (thus magnitude discrepancy)
which is primarily a function of the total signal itself. In Figure 9 we illustrate this functional dependence by plotting the magnitude discrepancy in non-preflashed images vs. the
total signal in the short exposure. The curves for different exposure times overlap very
well, indicating that the primary determinant of the magnitude discrepancy is indeed the
total observed signal. There is a slight deviation for faint stars in the 300s exposure, which
can be explained as residual non-linearity in the reference image (a faint star in the 300s
image will be rather faint in the 1000s image as well). Figure 9 is for non-CTE-corrected
magnitudes; the plot for CTE-corrected magnitudes is very similar.
With this interpretation of the behavior of the WFPC2 detectors, there will be a functional relation between the total flux impinging on the detector and the measured flux,
defined by the shape of the curves in Figure 9. Therefore it is possible to define a correction function that can be applied to the observed flux to recover the true flux.
The correction function can be expressed in terms of the magnitude discrepancy dm
expected at given observed counts and background. After considering various functional
forms, we find that a convenient expression is the following:
A
dm = ---------------------------------------------------------------------21 + B ⋅ counts + C ⋅ counts
(2)
where counts are the background-subtracted source counts in a 2-pixel aperture, measured
in DN at gain 7. There does appear to be some background effect for very high background and at low count levels (see Figure 4), but it is not statistically significant in the
overall fit. The functional form of Equation (2) is not unique, but offers a very simple correction and fits well the measurable characteristics of the long vs. short anomaly. The
quantities A, B, and C are adjustable parameters whose best values, found by a non-linear
least squares optimization, are:
A = 0.629
B = 0.0315
C = 0.000333
(3)
With this choice of parameters, the magnitude discrepancy is fitted quite well, except
for very faint objects (< 30 counts). The residuals for all points are shown in Figure 10; the
median and quartiles, as a function of the total source counts, are given in Figure 11. Note
that the correction in Equations (2) and (3) has been applied self-consistently to both long
and short exposures. The residual error has no obvious systematic component above 30
16
counts, but the discrepancy is somewhat undercorrected below 30 counts; we will discuss
this point again at the end of the Section.
Figure 10: Residual magnitude discrepancy after correction for dm in Equation 2.
After correction, the median residual systematic discrepancy is less than 3% for
sources with more than 30 counts; note that the shot noise alone is about 7% for a source
with 30 counts. The residual dispersion, as measured by the chi-squared value, is about 1.7
times the nominal photometric error; however, part of the excess dispersion is due to a
broad tail to the distribution of magnitude errors, possibly related to the non-standard data
reduction procedures needed for cosmic ray rejection in the present dataset. The interquartile range compares well with the expected nominal error.
Since the CTE-related charge loss also depends on the measured flux and background,
it would have been preferable to fit simultaneously for both CTE and ‘‘long vs. short’’
non-linearity. In practice, the dataset used here is not well suited for an independent deter-
17
mination of the CTE correction, and therefore we have adopted the CTE prescription of
Whitmore and Heyer (1997) and Whitmore (1998). The long vs. short magnitude correction given here applies after the appropriate CTE corrections (Equation 1) have been
carried out, but please note that the measured flux counts in the equation above is the
detected flux without any form of correction.
Figure 11: Median and quartiles of the residual magnitude discrepancy, before and after
correction.
The correction thus derived works reasonably well for F555W data as well. Figure 12
shows the median of the residual magnitude discrepancy for the F555W data after correction with Equations (2) and (3). The short exposure in F555W was 40s and the long
exposure was 300s. The magnitude discrepancy is well corrected above 30 counts, and - as
in the case of F814W - undercorrected at lower source counts.
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Figure 12: The discrepancy in F555W data after correction with the F814W equation.
The undercorrection at low count values cannot be easily avoided with the chosen
functional form. Forcing a better correction below 30 counts causes an overcorrection in
the range 100-400 counts, where the magnitude is measured with much smaller statistical
uncertainty; thus we prefer to adopt the correction that works best where the magnitude
measurements are better.
5. Recommendations
The procedure to correct the photometry of point sources for the “long vs. short” nonlinearity is the following:
1. Carry out the standard aperture photometry using an aperture of radius 2 pixel; do
not apply any aperture corrections. Retain the value of measured flux and sky
background;
19
2. Determine (do not apply yet) the CTE correction as detailed in Whitmore and
Heyer (1997) and Whitmore (1998); see also Equation (1) in this document;
3. Determine the magnitude correction dm according to the following formula:
0.629
dm = --------------------------------------------------------------------------------------------------------------2
1 + 0.0315 ⋅ ( counts ) + 0.000333 ⋅ ( counts )
(4)
where counts are the total source counts (after background subtraction) within the 2pixel radius aperture, expressed in DN at gain 7; subtract dm from the magnitude;
4. Now apply the CTE and aperture corrections, as well as any other relevant corrections (focus, contamination).
Caveats
The correction expressed in Equation (4) has been derived for a specific dataset, and it
must be applied with care under more general circumstances. For example, scientific
WFPC2 data often consist of numerous images with similar or different exposure times, or
have been taken at gain 15; neither of these circumstances is directly addressed by our test
dataset.
Multiple exposures
If multiple exposures of the same length are available, we expect that Equation (4) can
still be applied, as long as the average value of counts is used. If the exposures are of dissimilar length, most likely the total counts will be dominated by the longest exposures,
which therefore determine the value of dm to be used.
An untested refinement is the following procedure:
•
Group together exposures of similar length;
•
Record the count rates obtained in each exposure or average of similar-length exposures;
•
Apply to each a multiplicative correction equal to 100.4 ⋅ dm , using the value of counts
appropriate to each image group;
•
Average the corrected count rates (weighted according to their S/N, if desired) and
convert to magnitudes.
This procedure assumes that the magnitude correction of Equation (4) brings the count
rates measured in different exposure times into agreement, and thus should in principle
work; however, we have not been able to test this procedure in practice, and cannot quantify its accuracy.
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Gain 15 data
All observations for CAL 7630 were taken at Gain 7. If the anomaly originates in the
detector, then Equation (4) should apply for Gain 15 data as well, as long as the source
counts are expressed at Gain 7 equivalent by multiplying them by the gain ratio (approximately 2; Holtzman et al 1995). However, if the anomaly originates elsewhere in the
signal chain, we cannot quantify its effect on Gain 15 data, which use different electronics.
Which zero point should I use?
Finally, note that the published WFPC2 zero points and its estimated system throughput are based on exposures with a total of several thousand counts, for which dm in
Equation (4) is very small (< 0.0001). Therefore the published zero points are directly
applicable after the correction in Equation (4) applied to the observed data.
6. Open issues
The observations taken for program 7630 attempted to cover as broad a range of exposure times and preflash values in the F814W filter as practical. However, due to limitations
of both observations and analysis, we could not cover all conditions that could conceivably occur in real observations. A number of open issues remain, some of which could be
addressed with more data or with the use of archival data, while others are amenable to a
more sophisticated analysis. Some of the questions still open are:
1. The magnitude discrepancy in filters other than F814W. We have limited data in
F555W, which are fitted reasonably well by Equation (4). However, we have insufficient data for F300W, and no data for other filters, such as the extreme UV or narrow-band filters. There is no reason to expect that the non-linearity of the detectors
and signal chain depend strongly on the wavelength - note that the photons
detected in UV observations are in reality 500 nm photons emitted by the lumogen
coating of the CCD - but it is possible that the optimal correction parameters might
depend on the wavelength.
2. The time dependence of the magnitude discrepancy. Data taken in 1994 and in
1997 appear to show comparable amounts of magnitude discrepancy, but we do not
have enough information to put quanitative constraints on the change in the magnitude discrepancy over the life of WFPC2.
3. The interrelation between CTE and the magnitude discrepancy. As stated in Section 4, ideally it would be best to fit simultaneously for both magnitude discrepancy and CTE-related charge loss. We plan to carry out such a fit in the future.
4. Quantifying the magnitude discrepancy as a function of aperture radius. The discrepancy generally increases with the radius of the aperture used for the photometry, but a quantitative formula requires a general formula for the CTE-related
charge loss, which is not yet available.
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5. The physical interpretation of the non-linearity. The strong correlation between
magnitude discrepancy and total source counts suggests that understanding the
microphysics of the detectors and/or non-linearities in the signal chain might shed
light on the discrepancy - and possibly yield a better correction formula - but work
in this direction has not been done yet.
6. The effect of noiseless preflash. A new test is required with a short time interval
between preflash and observations.
7. The behavior of the magnitude discrepancy in extreme cases. For example, a large
number of repeated exposures can be used to achieve good signal-to-noise ratio for
very faint sources; because of the prohibitive cost of such an experiment, we were
unable to reproduce these conditions in our calibration program.
8. Extended sources. The data presented here refer exclusively to point sources; a test
for the long vs. short nonlinearity in an extended source was carried out by Stiavelli and Mutchler (1997), who found some marginal evidence for a difference of
about 10% between long and short exposures at count levels of about 5 DN/pixel.
However, they were unable to provide a systematic description of the discrepancy
they found. Furthermore, they used images of the bright galaxy NGC 4472, which
nearly fills the WFPC2 field of view; it is unclear how their results apply to the
more common case of a faint source extending over a few tens of pixels.
7. References
Biretta, J. A., et al, 1996, The WFPC2 Instrument Handbook, version 4.0
Biretta, J., and Mutchler, M., 1998, WFPC2 Instrument Science Report 97-05
Casertano, S., 1995, internal STScI report, available at URL http://www.stsci.edu/
ftp/instrument_news/WFPC2/Wfpc2_cte/shortnlong.html
Casertano, S., 1997, in The 1997 HST Calibration Workshop, eds. Casertano, Jedrzejewski, Keyes and Stevens (Baltimore: STScI), p. 327
Holtzman, J., et al., 1995, PASP 107, 1065
Kelson, D. D., el al., 1996, ApJ 463, 26
Ratnatunga, K.U., 1995, private communication
Saha, A., et al., 1996, ApJ 463, 26
Suchkov, A., and Casertano, S., WFPC2 Instrument Science Report 97-01
Stetson, P., 1995, unpublished (reported in Kelson et al. 1996 and Saha et al. 1996)
Stiavelli, M., 1995, unpublished
Stiavelli, M., and Mutchler, M., 1997, WFPC2 Instrument Science Report 97-07
Whitmore, B., and Heyer, I., 1997, WFPC2 Instrument Science Report 97-08
Whitmore, B., 1998, WFPC2 Technical Instrument Report 98-01
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