WFPC2 Instrument Science Report 95-02
The Geometric Distortion Correction for the WFPC Cameras
27 February 1995
Roberto Gilmozzi, Shawn Ewald, and Ellyne Kinney
Introduction
The WF/PC-1 and the WFPC2 are both subject to a geometric distortion eect which
stems from optical and mechanical design constraints (wide eld of view and small optical bench volume) and o-axis optics. In both cameras, the distortion is exaggerated
by the telescope's spherical aberration. The distortion ranges from a few tenths of a
pixel in the center of each chip, up to 2-3 pixels at the edges of each chip. The plate
scales for the WFCs are approximately 0.1 arcseconds/pixel, so the distortion eect
can cause positional errors anywhere from 20 mas to 300 mas. For the PCs, which have
plate scales of 0.043 arcseconds/pixel, these errors range from 8 mas to over 100 mas.
For both scientic and operational purposes it is desirable to have more accurate measurements. An example of a scientic use of the distortion correction would be to derive
relative positions of stars in a star cluster. On the operational side, a major use of
the distortion correction is for WFPC{ assisted Target Acquisition for one of the small
aperture SIs such as FOS or HRS. This stategy is especially useful for acquiring faint
objects in crowded or high background elds. A WFPC image is taken beforehand,
and is used to select a bright and/or isolated object near the fainter one. An accurate
oset between the two objects is derived. The small aperture SI then peaks up on the
brighter target, and then the slew to the intended target is performed. The WFC is
usually used since the PC, while having a higher pixel resolution, has a FOV that is
insucient for most oset pointing operations.
It should be stressed that the astrometry derived from WF/PC-1 and WFPC2 images
have a very high relative accuracy (4-10 mas). In an absolute sense, however, the coordinates derived from it are only as accurate as the coordinates of the guide stars used,
and the knowledge of the location of the camera in the focal plane. For WFPC-assisted
Target Acquisition, this means it is necessary to peak up on a nearby object, which in
essence zeros out these uncertainties, rather than trying to acquire the object directly
from the derived coordinates. If one wanted to derive accurate absolute astrometry, it
would be necessary to do one of two things. Either use guide stars with astrometric
coordinates, or include one or more stars with well known coordinates within the FOV.
The Data
The distortion correction was derived from a series of overlapping images of the globular
cluster NGC1850, taken in the V bandpass, as well as a handful of images in the B ,
I 1, and UV 2 bandpasses. (See Table 1) Each set of V images consisted of one central
pointing and four outer pointings. The four outer pointings were oset from the from
the central one by one quarter the FOV in both x and y (20 arcseconds for the WFC
chips and 8.5 arcseconds for the PC). (See Figure 1) The standard pipeline calibration
system at STScI was used to remove instrumental eects from the images.
The IRAF package DAOphot was used for the analysis of the data. The daond utility
was used to nd all of the stars in the central pointing. The initial coordinate list was
then edited, by examining the image at each of the coordinate values. Spurious entries
such as diraction spikes, cosmic rays, or noise in the background were eliminated.
This edited coordinate list was then used to create the coordinate lists for each of the
overlapping oset pointings, by applying the appropriate oset values to the coordinates in the initial list. These coordinate lists were run through the phot program, with
the centroid centering algorithm turned on, in order to get accurately centered coordinates and an estimate of the magnitudes. The photometry les were further processed
to eliminate stars with poor/no magnitude estimate, which indicated that the star was
too faint to get an accurate magnitude, and therefore was not accurately centered, or
that the coordinates were outside the region that overlapped with the central pointing.
The observed positions of the each star which appeared in overlapping central and
oset exposures were tabulated. The dierence between the expected and observed
positions were computed and used as the basis of the distortion mapping.
Distortion Fitting
We assume throughout that the positions of the stars are invariant across all exposures
and that any changes from one to another are attributable to: a) the distortion being
mapped, b) the oset and possible rotation of the detector from one exposure to another
due to the telescope manoeuvre, c) dierences in the position determination of the stars
in the images due to digitization of the image in space (pixels) and intensity (counts).
The rst two changes are determined by this procedure, while the third is one the
limits to the precision which can be obtained. This limitation is greater in the WFPC2
than in WF/PC-1 due to the undersampled nature of the unaberrated PSF.
The nal product of the tting procedure is a pair of functions dx(x; y) and dy(x; y),
which return the geometric distortion at any given pixel position (x; y). The corrected
star position, (xc; yc), would be derived from its measured position (xm; ym) by:
xc = xm dx(xm; ym)
(1)
yc = ym dy(xm; ym)
WF/PC-1 I images courtesy of Jim Westphal and the WF/PC-1 IDT. (Prop. 3008)
images were available for WFPC2 only.
1
2U V
2
If j represents the pointing (0 for the central pointing and 1-4 for the outer pointings) and i represents the index of the set of stars in an image, then the stars in two
overlapping exposures can be related by the expression:
xc 0 = xc + xj j > 0
(2)
where xj is the x component of the telescope pointing dierence between the central
pointing and the j th oset image. Substituting equation (1) into (2), and showing only
the equations for the x terms, since y are identical,
xm 0 dx(xm 0 ; ym 0 ) = xm dx(xm ; ym ) + xj j > 0
(3)
xm 0 xm = dx(xm 0 ; ym 0 ) dx(xm ; ym ) + xj j > 0
(4)
i;
i;
i;
i;
i:j
i;
i;j
i;j
i;
i;
i;j
i;j
i;j
i;j
The distortion, dx, is ' 1 pixel and the oset size is 200 pixels. Since dx xm
and dx xj we may omit the dx terms in (4) above and average xj over all the
measured stars in the image.
X xm
N
xj = N 1
stars
stars i=0
xm 0 xm
i;
i;j
0
i;
xm
(5)
i:j
xj = dx(xm 0 ; ym 0 ) dx(xm ; ym )
i;
i:
i;j
i;j
j >0
(6)
Which gives the dierence between the distortions in both pointings. A least-squares
best t is obtained for the measured distortions, using Legendre Polynomials. For PC-1
and WFPC2 a third order t was obtained, and a sixth order t was used for all the
detectors of WFC-1.
3
3
anPn (x)bmPm(y)
(7)
dx(x; y) =
XX
n=0 m=0
The residuals between computed and measured distortion is added to the measured x
position (central pointing only) and the manoeuvre and distortion is recomputed. This
iterative process continues until the convergence criterion is met. The convergence
criterion is expressed as xi:
xi =
X (xi; + dx(xi; ; yi; ))
N
stars
i=0
0
0
0
(xi;j + dx(xi;j ; yi;j )) ! 0
(8)
The scatter of xi about zero is an estimate of the errors of the t. (See Figure 2)
The solutions derived from each iteration were applied to a set of independent control
elds comprised of stars from the overlapping regions of adjacent oset pointings. (See
Figure 3) The iterative process was terminated when both the r.m.s. of the t and of
the control elds reached a similar value, typically 0.07-0.10 pixels. This was necessary
to check that systematisms introduced in the data by the manouver calculation in each
iteration were properly removed.
3
Camera Head Osets and Rotations
Once the Distortion Correction was determined it was possible to derive the relative
camera head osets and rotations for each of the chips. This was done by matching
stars in chip 2 of the central pointing with stars in the four chips of one of the outer
pointings. (See Figure 4) The coordinates of the stars in these data sets were corrected
for geometric distortions and the outer pointings were adjusted for the telescope oset
from the central pointing. The resulting residuals were t to yield Camera Head Osets
and Rotations, and plate scale changes across chips.
Once these were determined it was possible to transform the coordinate systems of the
four chip in each camera to one metachip coordinate system with chip 2 (or chip 6 in
WF/PC-1 in its PC mode) as the reference. The following transformation equations
were used:
x0 = xi cos i yi sin i + xoff
y0 = xi sin i + yi cos i + yoff
x = x0 scalei
y = y0 scalei
Where i is the number of the chip being transformed. Tables 2-4 give the values of
xoff , yoff , i, and scalei for each of the three cameras. (In the WF1 detector of
WF/PC-1, the camera head had a slight tilt out of the focal plane and thus needed
dierent scale factors for x and y.)
i
i
i
i
Wavelength Dependance of the Distortion
In addition to the images taken in V three other images were taken with the UV, B, and
I lters at the same pointing as the central pointing. These images were used to check if
there was any wavelength dependance of the geometric distortion. The only eect that
was seen was the expected change in plate scale and a slight oset between the dierent
lters for WFPC2. In WF/PC-1 some lters had shifts of their metachip coordinate
systems with respect to the F555W metcachip coordinates. This was probably due to
small, xed, tilts of the lters in their lter wheel mounting. Tables 5 and 6 show the
wavelength dependance.
We stress again that the solutions presented here were derived from observations at
5500 A, which is the one we recommend be used for astrometric purposes.
Images taken with UV lters have PSF that are highly aected by the position of it's
position in the FOV. The Wood's lter (f 160bw) shows a strong variation of the PSF
across the eld of view, while in the distortion is not as sever f 170w lter. This eect
is not addressed in this distortion correction, but is very prominent, in particular near
the apex of the pyramid. This distortion may be removed through the use of a variable
PSF in one's astrometric centrioding program.
4
References
Ewald, S.P., Kinney, E.K., and Gilmozzi, R., 1994, BAAS, in press.
Fosbury, R., in "The Future of Space Imaging", Brown, R., ed., 1993.
Gilmozzi, R., Ewald, S.P., 1995, in press.
Holtzman, J.A., et. al., The Performance and Calibration of the WFPC2, PASP, 1995,
in press.
Wide Field and Planetary Camera Final Orbital/Science Verifcation Report, WF/PC-1
IDT, 1991.
Wide Field and Planetary Camera Instrument Handbook, J.W. MacKenty (Ed.), STScI
publication (April 1992).
Wide Field and Planetary Camera 2 Instrument Handbook, C.J. Burrows (Ed.), STScI
publication (May 1994).
5
Figure 1a. WFPC-2 takes four images each time the shutter is opened and the CCDs
read out to the on-board tape recorder. The eld of view is divided into four sections,
one per CCD. The sections are labelled here by the CCD names, PC1 (which has a
higher magnication than the other CCDs), WF2, WF3 and WF4. The "L"-shaped
area adjacent to PC1 is not observed by WFPC2.
Figure 1b. For each CCD, the pattern of overlapping exposures taken in order to
derive the geometric distortion of its optics is shown in this diagram. The dark outline
indicates the position of the CCD's eld of view in the 'central pointing' exposure.
The lighter outlines are the areas of the sky observed in the oset pointings. The
cross-hatched area if the region where the same stars are seen in one part of the CCD
in the central pointing and another part of the CCD in an oset pointing.
6
Figure 2. The two panels show the dierence between positions of the same 600+ stars
observed in one of the control elds for chip 1, without the correction applied. The rst
is a vector plot (magnied 25 times) and the second is a residuals plot (x vs y1). The
bottom panel shows the residuals after correction.pThe r:m:s: is 0:11 pixels (indicating
accuracies on the single measurements of 0:11= 2).
7
Figure 3. The full eld of view for two oset pointings is detailed in this diagram.
The shaded region is the overlap of one CCD in one exposure with a dierent CCD in
another exposure. After the geometric distortions associated with each CCD's optics
are removed from the positions of the stars seen in the two images in the shaded region,
the pairs of observed positions of each star matched. The dierence between these pairs
of positions is the sum of the telescope manoeuvre between the two exposures and the
relative displacement, plate scale and rotation between the two cameras.
8
Figure 4. The outline of the central pointing exposure's full eld of view has been
added to the previous gure to show the relationship of the two oset pointings used
to measure the camera head position dierences. If the shaded region contains WF2
in Oset2 and WF3 in Oset3, then portions of those CCDs' images overlap with
dierent portions of the same CCDs in the central pointing. The upper-right quarter
of WF2 in Oset2 overlaps the lower-left quarter of WF2 in the Central pointing, and
the lower-right quarter of WF3 in Oset3 overlaps the upper-left quarter of WF3 in the
Central pointing. The information in the overlaps between central and oset pointings
contribute to the geometric distortion solution for each camera, but is independent of
the information contained in the overlap between two oset pointings of two dierent
CCDs.
9
Figures 5-8. Camera Head Osets and Rotations The camera head osets were calculated by comparing the position of a bright star that appeared in chip 2 of the central
pointing and one of the chips in the outer pointing. Renements to theses values were
made by looking at the plots of dx vs. x and dy vs. y, which helped rene the osets
and relative scales. An oset of dx or dy from 0 indicated the amount of oset due to
the camera heads. A slope in dx or dy indicated a scale change. A slope in plots of dx
vs. y or dy vs. x indicated a rotation in the camera heads.
10
Figure 6.
11
Figure 7.
12
Figure 8.
13
Table 1: Camera Head Osets and Rotations for PC-1
Camera
PC5
PC6
PC7
PC8
xoff
yoff
i
scalei
(pixels) (pixels) (degrees)
i
-0.15
0.0
63.75
51.61
i
71.36
0.0
3.70
64.58
269.815 1.00260
0.0
1.0
90.753 0.99935
180.407 1.00000
Table 2: Camera Head Osets and Rotations for WFC-1
Camera
xoff
yoff
i
(pixels) (pixels) (degrees)
i
i
WF1
-14.054 55.884
WF2
WF3
WF4
0.0
45.262
36.585
0.0
4.933
61.145
scalei
269.449 x : 1.001671
y : 1.001276
0.0
1.0
90.187
1.000000
180.305 x : 1.000000
y : 1.000400
Table 3: Camera Head Osets and Rotations for WFPC2
Camera
PC1
WF2
WF3
WF4
i
yoff
xoff
(pixels) (pixels) (degrees)
i
i
scalei
58.676 101.347 270.496 x : 0.45730450
y : 0.4571768
0.0
0.0
0.0
1.0
98.627 -6.509 90.327
0.9994667
95.430 75.480 180.899
0.9999972
14
Table 4: Wavelength Dependance of the Distortion for WF/PC-1
Filter Chip
f 439w
1
2
3
4
5
6
7
8
1
2
3
4
f 785lp
scalei
xoff
0.99987902
0.99990500
0.99999432
0.99996953
0.99985987
1.00019560
0.99985199
1.00016730
1.0009088
1.0008418
1.0009332
1.0009749
1.209
1.188
1.449
1.385
4.074
4.096
4.194
4.200
3.073
3.045
3.257
3.260
i
yoff
i
1.458
1.382
1.305
1.431
1.397
1.207
1.250
1.391
2.708
2.521
2.547
2.724
Table 5: Wavelength Dependance of the Distortion for WFPC2
Filter Chip
f 791w
f 439w
f 170w
1
2
3
4
1
2
3
4
1
2
3
4
scalei
xoff
1.00451
1.00209
1.00111
1.00259
1.00046
1.00019
1.00013
1.00018
0.999718
0.999734
0.999744
0.999848
0.2119613
0.248912
0.183698
0.193212
0.1506212
0.182048
0.153802
0.129359
0.5386210
0.646853
-0.227277
-0.170283
15
i
yoff
i
0.2405904
0.321801
0.303697
0.265783
0.0948799
0.124968
0.146408
0.110601
0.2327530
0.512240
0.793233
-0.200359
Table 6: Co-ecients for the PC-1 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 5
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.276987E-01
1.80270
0.472875E-01
0.839371
-0.194826E-01
0.205235
-0.538380E-01
-0.585192E-01
0.102121
1.57496
0.996310E-03
-0.202905E-01
0.359101E-01
-0.263058E-01
0.515688E-01
0.556717E-01
0.887961E-01
-0.258017
0.187032E-01
-0.432581E-02
1.68511
0.169656
1.48879
0.166961
0.212737
-0.552163E-01
-0.114369
0.379805E-01
0.969871
0.426470E-02
-0.275033E-01
-0.228036E-01
Table 7: Co-ecients for the PC-1 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 6
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.826583E-01
1.56328
0.112179
0.931775
0.149869
0.783824E-01
0.155961
-0.474787E-01
0.195649E-01
1.39522
-0.785402E-01
0.155761
0.736697E-01
-0.187339E-01
-0.149583E-01
-0.644191E-01
16
-0.192901E-01
0.954678E-01
0.260649E-01
0.713268E-01
1.64822
0.186032
1.29882
-0.104328
0.500916E-01
0.139403
0.640607E-01
-0.282517E-01
0.929756
-0.161168
-0.224210E-01
0.467203E-01
Table 8: Co-ecients for the PC-1 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 7
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.123508
1.63335
0.262432
0.809902
-0.238150E-01
0.170262
-0.119149
-0.158564
-0.216911E-01
1.31328
-0.733799E-01
-0.725823E-01
-0.227318E-01
-0.993457E-01
-0.667001E-01
-0.523880E-01
0.716987E-01
-0.577290E-01
0.838524E-01
-0.418771E-01
1.69257
0.432265
1.60774
0.185271
0.942161E-01
-0.127476
-0.628792E-01
-0.131979
0.851954
0.350061E-01
0.468264E-01
0.184103
Table 9: Co-ecients for the PC-1 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 8
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.826797E-01
0.963531
-0.985975E-01
0.963147
0.258219
0.246323
0.114268
0.890538E-03
-0.705783E-01
1.52589
0.160118
0.201436E-01
-0.168445
-0.141034
-0.370968
-0.339899
17
0.618644E-01
0.205632
0.181418E-01
-0.857678E-01
0.882170
-0.183242
1.54962
0.387620E-01
0.112948
0.233455
-0.186104E-01
-0.149138
1.02215
0.445508E-01
-0.387411E-01
-0.227995
Table 10: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 1
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
-0.233644E-01
0.527532
0.160644
0.901084
-0.216011E-01
-0.140963E-01
-0.142361E-01
0.558357E-02
0.139696
-0.194561E-01
-0.111291
0.503659E-01
0.438467E-05
0.194187E-01
-0.263032E-01
1.48265
-0.760040E-01
0.325426E-01
-0.779825E-01
0.725623E-03
0.100961E-01
0.149592E-03
-0.170976
0.602980E-02
-0.123613
0.875220E-01
-0.115254
0.195115
-0.681991E-01
0.723664E-01
-0.414895E-01
-0.161967E-01
0.733362E-01
-0.982506E-01
-0.156723
-0.165164E-01
-0.220650
18
-0.229461E-02
-0.241854E-01
-0.392640E-01
-0.235227E-01
-0.642825E-02
-0.840310E-02
0.291674E-01
0.548335
0.387662
1.41894
-0.105085
0.440937E-02
0.843286E-01
-0.497410E-02
0.700789E-01
-0.528973E-01
0.701003E-01
-0.154925
-0.264293E-01
-0.846450E-01
0.152589
0.877111
-0.132327
-0.106160
0.196699
-0.121649
-0.173855E-01
0.199421E-01
-0.122222E-02
-0.661174E-02
-0.325303E-02
-0.269248
0.614570E-01
-0.758538E-01
0.163976
0.282092E-01
-0.202283
Table 10: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 1
a
b
38
39
40
41
42
43
44
45
46
47
48
49
-0.267263E-01
-0.918068E-01
-0.647043E-02
0.218341
0.490153E-01
0.448921E-01
0.328306E-01
-0.735550E-01
0.674810E-01
-0.241444
0.688317E-02
-0.681448E-01
19
0.222040E-01
0.531590E-01
-0.127389
0.298571
0.379930E-01
-0.182698E-01
0.332307E-02
-0.123486
-0.427196E-01
0.277558
-0.104449
-0.152209
Table 11: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 2
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
0.550319E-02
0.432135
0.105880
0.905788
0.949131E-02
0.274268E-01
0.265834E-01
-0.936640E-01
0.286223
-0.595855E-01
0.661712E-02
0.404042E-01
0.596573E-02
0.137155
-0.715509E-01
1.58508
0.483154E-02
-0.171728
0.548727E-01
0.129194
-0.119332
-0.311638E-01
-0.189211E-01
0.512064E-01
0.166421
0.146718
0.380757E-01
-0.349375E-02
0.226666E-01
-0.690692E-01
0.104447
0.128583
0.861257E-01
0.843517E-01
-0.143859
0.342341E-01
0.182830
20
0.176582E-01
-0.128684
-0.183566E-01
-0.233728E-01
0.174834E-01
0.102550
0.484545E-01
0.598599
0.277103
1.63174
0.148166E-01
-0.634429E-01
0.833309E-01
0.578246E-01
0.142916
0.189755E-01
0.332725E-01
0.883261E-01
0.421103E-01
0.181612
-0.106995E-01
0.937254
-0.183634E-01
-0.130179
0.809492E-01
0.879568E-01
-0.433506E-01
0.145550
0.665690E-01
0.897547E-01
0.208942E-01
0.245311
-0.754056E-01
-0.939822E-01
-0.124732
-0.508283E-01
0.454969E-01
Table 11: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 2
a
b
38
39
40
41
42
43
44
45
46
47
48
49
0.981716E-01
0.736910E-01
0.592615E-01
-0.351944
-0.141686
0.478025E-02
0.185782E-01
-0.363186E-01
0.122674E-02
-0.221890
0.344632
0.229562
21
0.145580E-01
0.217669E-02
-0.999051E-02
0.154702E-01
0.250810
0.102983E-01
0.778874E-01
0.285723E-01
0.516451E-01
-0.226409E-01
0.253033
-0.249701
Table 12: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 3
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
-0.264266E-01
0.336880
0.104388
0.812891
0.599160E-01
0.688058E-01
-0.970845E-01
-0.261449E-01
0.292877
0.307018E-01
-0.353839E-01
0.833566E-02
-0.523576E-01
0.755655E-02
-0.423350E-01
1.43097
0.741312E-02
0.298880E-01
-0.129647
-0.159938E-01
0.143802E-01
-0.161040E-01
-0.350987E-01
-0.132448E-01
0.136141E-01
0.380449E-01
0.122190E-01
-0.198862E-01
-0.462074E-01
0.392152E-01
-0.794818E-01
-0.888443E-01
0.348593E-01
0.177125E-02
0.156821
0.402841E-02
-0.232923E-01
22
-0.295294E-03
-0.347180E-01
-0.369894E-01
-0.249304E-01
-0.593540E-02
0.993049E-02
0.281742E-01
0.345864
0.169266
1.44205
0.498738E-01
0.568238E-01
-0.180031
-0.246800E-01
0.203811
-0.133171E-01
-0.634889E-01
-0.593117E-02
0.420348E-02
-0.402736E-01
0.434928E-01
0.764973
0.460570E-01
0.586922E-01
-0.851932E-01
-0.396292E-01
0.164158
-0.569705E-01
0.502268E-01
0.588378E-01
-0.172513E-01
-0.336568E-02
0.102801
-0.814054E-01
0.192949E-01
0.664845E-02
-0.258540E-01
Table 12: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 3
a
b
38
39
40
41
42
43
44
45
46
47
48
49
-0.414563E-01
-0.777913E-01
-0.874515E-01
0.129185
0.157718
0.152659E-01
0.259774E-01
0.808500E-01
0.335988E-01
0.207165
-0.996049E-01
-0.264415
23
-0.323719E-01
0.348109E-02
-0.556088E-01
0.177595
-0.277053E-01
-0.637379E-01
0.496433E-01
-0.275070E-01
-0.904321E-02
0.288531E-01
-0.133526
-0.241295E-01
Table 13: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 4
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
-0.201227E-01
0.422125
-0.787926E-02
0.712238
0.654487E-01
-0.105002
-0.136475E-01
0.523019E-01
0.459140
0.148413E-01
-0.725054E-01
-0.104992
0.140289
-0.191440
-0.927930E-01
1.38338
-0.921004E-02
0.669445E-01
0.128159E-01
0.530097E-01
0.869855E-01
-0.126787E-01
0.180397E-01
-0.166184
0.252978E-01
0.146418E-01
0.411792E-01
-0.760605E-01
-0.256522E-02
0.795980E-02
0.453194E-01
-0.742488E-01
0.609984E-01
-0.403499E-01
0.342597E-01
0.811365E-02
-0.175822E-01
24
0.691057E-01
0.156087
0.195515E-01
-0.526593E-02
0.839561E-02
-0.132709
0.101332
0.521700
0.617597E-02
1.44366
0.531254E-01
-0.155309
0.130997
0.507035E-02
0.309520
-0.106961
0.154650
-0.205095
0.226045
-0.280692
0.225150
0.811252
0.364762E-01
-0.814805E-01
0.135707
-0.729369E-01
0.138829
-0.222966
0.545485E-01
-0.174782
0.133649
-0.140565
0.381033
0.141552
-0.651064E-01
0.278417E-01
0.130193
Table 13: Co-ecients for the WFC-1 Geometric Distortion Correction
Sixth Order Polynomial Fit
Chip 4
a
b
38
39
40
41
42
43
44
45
46
47
48
49
-0.659592E-01
-0.769669E-01
-0.295574
-0.238558E-01
0.498107
0.915286E-01
0.239522E-01
-0.984721E-02
0.560242E-01
-0.193011
-0.393026E-01
0.107539
25
0.547447E-01
0.255356
0.123642E-01
-0.556008
-0.399270
-0.164447E-01
-0.635381E-01
0.335267E-01
-0.471332E-01
-0.825712E-01
-0.834433E-01
0.284922
Table 14: Co-ecients for the WFPC2 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 1
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.668149E-01
2.16326
0.380947E-01
0.940948
0.516275E-01
0.616919E-01
-0.468250E-02
-0.799010E-02
0.106746
1.48615
0.224224E-01
0.569497E-01
-0.871714E-02
0.343500E-01
0.151337E-01
-0.315461E-01
0.574160E-01
0.176289E-01
0.130385
0.169268E-01
2.36346
0.806029E-01
1.51212
0.224606E-01
-0.139148E-01
0.253915E-01
-0.169642E-01
0.759251E-01
0.965949
-0.443844E-01
0.131287
-0.532698E-01
Table 15: Co-ecients for the WFPC2 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 2
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.834638E-01
1.93344
0.171628
0.846194
0.499745E-01
0.156076
0.267013E-02
-0.976506E-02
-0.122987E-01
1.46568
-0.151957E-01
0.423389E-01
-0.336837E-02
-0.113084E-01
0.325430E-01
-0.816739E-01
26
-0.961072E-02
-0.432376E-01
-0.572591E-01
0.234889E-02
1.91277
0.335549
1.45848
-0.272907E-01
0.414067E-01
0.239586E-01
0.673926E-02
0.207205E-01
0.834137
-0.613613E-01
0.326873E-01
-0.337791E-01
Table 16: Co-ecients for the WFPC2 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 3
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
-0.638637E-01
1.92847
-0.403292E-01
0.880849
-0.672139E-01
0.153023
0.588188E-02
-0.102037E-01
-0.861200E-01
1.46324
-0.330019E-02
0.756068E-01
0.863599E-03
-0.493526E-01
-0.144707E-02
-0.338435E-01
-0.376068E-01
-0.441374E-01
-0.768074E-01
0.420585E-02
1.95842
0.163623
1.48087
-0.272244E-01
0.197844E-01
0.251328E-01
0.241439E-02
0.335336E-01
0.868325
-0.235801E-01
0.584494E-01
-0.320786E-01
Table 17: Co-ecients for the WFPC2 Geometric Distortion Correction
Third Order Polynomial Fit
Chip 4
a
b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
0.537544E-02
1.95195
0.754186E-01
0.877816
-0.533245E-01
0.171163
0.295295E-01
0.180753E-01
-0.578378E-01
1.46786
0.136570E-01
0.367823E-02
0.131553E-01
-0.137205E-01
0.340074E-01
-0.213247E-01
27
-0.209039E-01
0.458124E-01
-0.552349E-01
-0.150328E-01
2.01643
0.269249
1.46365
0.174260E-01
0.260297E-01
0.371601E-01
0.177419E-01
-0.374130E-01
0.886505
0.229406E-02
0.108477E-01
0.483584E-01