4.1 ACS Geometric Distortions
ACS produces the largest single images ever taken with HST. However, because the optics were designed with a minimum number of components, ACS focal surfaces are far from normal to the principal rays. This results in an image of the sky which is distorted in a large, but predictable manner. The distortion consists of two effects, discussed previously in Section 1.1.3. The first is the elongation of the ACS apertures, causing the pixel scale to be smaller along the radial direction of the Optical Telescope Assembly (OTA) field of view than along the tangential direction. The second effect is the variation of pixel area across the detector.
The results presented in the discussion below are derived from multiple observations of dense star fields with the images displaced from each other by amounts comparable with the size of the field of view. The following discussion is based on reports from the HST Calibration Workshop Proceedings: Calibration of Geometric Distortion in the ACS Detectors (Meurer et al. 2002) and Astrometry with the Advanced Camera: PSFs and Distortion in the WFC and HRC (Anderson 2002) at:
http://www.stsci.edu/hst/HST_overview/documents/calworkshop/workshop2002/CW2002_TableOfContents
4.1.1 On-orbit Calibration Program
The ACS geometric distortion campaign observed the core of 47 Tucanae with the WFC and HRC (program 9028, Meurer). The exposures were designed to detect stars on the main sequence turn-off, allowing a high density of stars with relatively short exposures. The F475W filter (Sloan g') was used to minimize the number of saturated red giant branch stars in the field. For calibrating the distortion in the SBC, exposures of NGC6681 (program 9027, Hartig) were chosen for the relatively high density of UV emitters (hot horizontal branch stars).
For the WFC and HRC pointings, the dither pattern was designed so that the offsets between all pairs of images adequately sample all spatial scales from about 5 pixels to 3/4 the detector size. For the SBC pointings, a more regular pattern is used, with a series of 5 pixel offsets.
Data obtained with F775W for the WFC and HRC detectors were used to check the wavelength dependence of the distortion, but little or no increase in the fit rms was found. While it was expected that the same distortion solution would be applicable to all filters except the polarizers, recent work has shown that at least one other optical filter (F814W) induces a significant plate scale change (factor of ~4x10-5).
In the future, separate IDC tables will be available for each filter, and the CALACS pipeline will select the appropriate solution automatically. For now, the solution derived for F475W is used for all filters. Preliminary investigations reveal that filter-dependent changes in the distortion pattern exist at the level of 0.1 pixel.
4.1.2 Distortion Model
The heart of the distortion model relates detector pixel position (x, y) to sky position (see ACS ISR 2000-11) using a polynomial transformation given by:
where k is the polynomial order of the fit, (xr , yr) is the reference pixel position taken to be the center of each detector or WFC chip, and (xc , yc) is the undistorted position in arcseconds. The coefficients to the fits, ai,j and bi,j, are free parameters. A similar form is used for the inverse relation which provides the conversion from units of arcseconds to distorted input pixels.
For the WFC, an offset is applied to put the two CCD chips on the same coordinate system:
, 
.
The offsets 
are 0,0 for chip 1 and correspond to the separation between chips 1 and 2 for chip 2. The chip 2 offsets are free parameters in the fit. 
correspond to tangential plane positions in arcseconds which are tied to the HST 
coordinate system. Next the positions are corrected for velocity aberration:
, 
, where
,
u is the unit vector towards the target, and v is the velocity vector of the telescope (heliocentric plus orbital). Neglect of the velocity aberration correction can result in misalignments on the order of a pixel for WFC images taken six months apart for targets near the ecliptic. For a further discussion of the effect of velocity aberration, see The Effect of Velocity Aberration Correction on ACS Image Processing: HST Calibration Workshop Proceedings (Cox & Gilliland 2002) at:
http://www.stsci.edu/hst/HST_overview/documents/calworkshop/workshop2002/CW2002_Papers/CW02_cox
Finally, all frames are transformed to the same coordinate grid on the sky:
,
,
where the free parameters
are the position and rotation offsets of frame i.
4.1.3 Analysis
The positions of stars observed multiple times in the dithered star fields are used to iteratively solve for the free parameters in the distortion solution: fit coefficients ai,j , bi,j ; chip 2 offsets 
(WFC only); frame offsets 
; and tangential plane position 
of each star used in the fit.
Originally, only images taken with a single roll angle were used to define the distortion solutions. The solution using only these data is degenerate in the zeroth (absolute pointing) and linear terms (scale, skewness). Therefore the largest commanded offsets with a given guide star pair were used to set the linear terms. However, comparison of corrected coordinates to astrometric positions showed that residual skewness in the solution remained. Hence, as of November 2002, the IDC tables for the WFC and SBC are based on data from multiple roll angles, incorporating data obtained from the calibration outsourcing program 9443 (King & Anderson). The overall plate scale is set by the largest commanded offset. For the HRC, the linear scale is set by matching HRC and WFC coordinates, since the same field was used in the SMOV observations. The zeroth order terms (position of ACS apertures in the V2,V3 frame) were determined from observations of an astrometric field.
4.1.4 Results
The most obvious component of the ACS distortion is a marked skew which results in the X/Y detector axes lying at ~85 degrees to each other on the sky. In addition, the distortion in all ACS detectors is highly non-linear (see Figure 4.1-4.4). A quartic fit (k=4) is adequate for characterizing the distortion to an accuracy much better than 0.2 pixels over the entire field of view.
WFC
The WFC detector is tilted at 22 degrees giving an elongation of 8% along the diagonal. The non-square projected aperture shape is evident from Figure 1.1 in Chapter 1, where the x axis is approximately in the V2 direction and the y axis is in the -V3 direction.
The WFC distortion is also illustrated in Figure 4.1, a vector displacement diagram which shows the contribution of the non-linear part of a quartic fit to the data. The vectors represent the degree of distortion to be expected in the WFC beyond the directional dependence of the plate scale. For display, the vectors are magnified by a factor of 5 compared to the scale of the x and y axes. The largest displacement indicated at the top left corner of the figure is ~82 pixels or about 4 arcseconds. While a quartic solution is adequate for most purposes, binned residual maps (Figure 4.4) show that there are significant coherent residuals in the WFC solutions which have amplitudes up to ~0.1 pixels. These residuals are not currently accounted for in pipeline calibration, although efforts are underway to do so in the future.
The pixel scales and array sizes are summarized in Table 4.1 for each ACS detector. The resulting variation of the projected pixel area on the sky requires corrections to the photometry of point sources in images which are still distorted. A contour plot of relative pixel area across the WFC, normalized to the central pixel, is shown in Figure 4.5. The range of area is from 0.89 to 1.08 times the central value.
When deriving the ACS flat fields, an implicit correction is made for the distortion which is equivalent to dividing each pixel by the effective pixel area in Figure 4.5. Thus, point source photometry extracted from a flat fielded image must be multiplied by the effective pixel area. This correction is accounted for in pipeline processing by PyDrizzle which uses the distortion solution to correct all pixels to equal areas.
Table 4.1: Pixel scale (arcsec/pixel) and array size (pixels) for each ACS detector.
| Detector |
Pixel Scale at Center Xscale, Yscale |
Average Pixel Scale Xscale, Yscale |
Array size (pixels) |
Array size (arcsec) |
|
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Figure 4.1: Non-linear component of the ACS distortion for the WFC detector. Note that this figure is rotated 180 degrees with respect to the pipeline calibration products, where WFC2 is the lower half of the detector. 
HRC
The High Resolution Channel has its edges aligned approximately along the V2 and V3 axes (see Figure 1.1). In this case, the center of the aperture lies on a line passing through the V2/V3 origin and making an angle of 22 degrees with the V3 axis. Because the aperture diagonal does not correspond to a radius of the HST field of view, the distortion has no particular symmetry with respect to the detector axes. Because the focal plane, and therefore the detector plane is 25 degrees away from the plane normal to the light path, the scales along the axes differ by ~14%. However, since the HRC is less than 30 arcsec across, the total variation in scale across the detector is much less than for the WFC, being only about 1%.
A vector plot of the deviation from linearity is given in Figure 4.2 in which the deviations have been magnified by a factor of 10 for illustrative purposes. The largest displacement is 4.9 pixels in the top left corner and corresponds to about 0.1 arcsec. While a quartic solution is adequate for most purposes, binned residual maps (Figure 4.4) show that there are significant coherent residuals in the HRC solutions which have amplitudes up to ~0.1 pixels. These residuals are not currently accounted for in pipeline calibration, although efforts are underway to do so in the future.
The pixel scales and array sizes are summarized in Table 4.1 for the HRC. The variation of pixel area across the detector is shown in Figure 4.5 and is required for photometric correction of point sources in distorted, uncorrected images. The maximum deviation from the central value is just over 2%.
Figure 4.2: Non-linear component of the ACS distortion for the HRC detector. 
SBC
The Solar Blind Channel contains the MAMA detector. It is centered at approximately the same place as the HRC in the V2/V3 plane and is slightly larger, about 35 x 30 arcsec, versus 29 x 25 arcsec for the HRC. The average scales in the x and y directions are given in Table 4.1 and are calculated by matching observations of the same sky area by HRC and SBC.
The optical distortions are close to those displayed by the HRC. The only difference in their light paths is the presence of a plane M3 fold mirror which reflects light away from the SBC onto the HRC. Because there is one less mirror in the SBC light path, the distortion corrected image (in sky coordinates) in Figure 4.3 appears flipped with respect to the HRC image in Figure 4.2. The MAMA has a small amount of extra distortion in the detector itself, arising from irregularities in the multi-channel plate. The largest difference from a square pattern is 2 pixels, or 0.06 arcsec.
Figure 4.3: Non-linear component of the ACS distortion for the SBC detector. 
Figure 4.4: Binned residuals to quartic distortion model fits for the ACS WFC and HRC detectors. Note that the WFC figure is rotated 180 degrees with respect to the pipeline calibration products, where WFC2 is the lower half of the detector. The large residuals close to the upper edge of the HRC map, left of center, are due to the presence of the Fastie occulting finger. There is insufficient data to make the equivalent plot for the SBC. 
Figure 4.5: Variation of the WFC and HRC effective pixel area with position, in detector coordinates. 
4.1.5 Distortion Table (IDCTAB)
The Science Instrument Aperture File (SIAF) contains position and scale information for every aperture of each of HST's science instruments. Thus, it supports accurate target acquisitions, image processing, and photometry. The conversion between distorted and undistorted positions in the SIAF is controlled by a polynomial expansion which is derived by the individual instrument teams. A simple, common reference file can therefore be created for each instrument which describes the geometric distortion of each detector. In this section, we describe the format of this common reference file which has been developed to be as general as possible so that it can be used for many different instruments or detectors.
For ACS, the coefficients to the polynomial fit are found in the IDCTAB reference file. The distortion correction is not a part of CALACS. A separate task, PyDrizzle (see Section 4.3), converts the distortion model in the IDCTAB into a form which is used by drizzle to produce geometrically corrected images.
Application of the Model
The steps performed in the application of the IDCTAB can be summarized in the following steps:
- First, the offset in position of each pixel relative to the reference position is computed.
- These deltas are used to compute the undistorted positions in arcseconds with the coefficients of the polynomial fit given in the IDCTAB reference file.
- The undistorted positions are divided by the desired output plate scale, where the default value is taken from the SCALE column in the IDCTAB unless specified by the user.
- Finally, both chips (WFC only) are combined into a single image by adding the appropriate offset to each pixel to get the final position.
The fit for each chip ensures that the distorted and corrected Y axes for a chip remain at the same angle with respect to the telescope V3 axis. This allows each chip to have a different rotation relative to the other, though this effect has been empirically shown to be less than 0.001 degree.
Format of the IDCTAB Reference file
The IDCTAB is stored as a FITS binary table (calibration file with '_idc.fits' extension), where the primary header contains keywords INSTRUME and DETECTOR which define the model appropriate to a specific instrument and detector. The NORDER keyword gives the order of the polynomial stored in the table. While the original laboratory data provided a cubic solution to the distortion, more accurate in-flight data now supports a fourth-order solution (NORDER=4).
The first extension of the IDCTAB contains the geometric distortion solution, where the table columns are listed in Table 4.2. The table contains distinct rows for each detector chip and filter combination specified by the DETCHIP and FILTER columns. If the DIRECTION column is 'INVERSE' rather than 'FORWARD', the coefficients specify the conversion from undistorted to distorted pixel positions. The SCALE keyword gives the default pixel scale of the drizzled image, where the default output scale for drizzled WFC images is 0.05 arcsec/pixel and for HRC/SBC images is 0.025 arcsec/pixel.
Table 4.2: Columns in the IDCTAB
| Column |
Description |
|
ID of chip/detector used for observation |
|
Application direction of coefficients (FORWARD,INVERSE) |
|
Name of filter in filter wheel 1 |
|
Name of filter in filter wheel 2 |
|
Raw image size in X direction (pixels) |
|
Raw image size in Y direction (pixels) |
|
X position of reference pixel |
|
Y position of reference pixel |
|
V2 position of reference point (arcsec) |
|
V3 position of reference point (arcsec) |
|
Scale of square corrected pixel (arcsec/pix) |
|
Distortion Coefficients for X position |
|
Distortion Coefficients for Y position |
The distortion coefficients provide a correction for each pixel relative to the detector chip reference point specified in the XREF and YREF columns. The relationship between each chip must be factored in separately. HST relies on a V2/V3 coordinate system to define each detector's position (in arcseconds) relative to the center of HST's field of view. The V2REF and V3REF columns contain the reference position of each detector in the V2/V3 system. The absolute position is secondary to the relative position of this reference point, because only the offset between V2REF/V3REF positions is used for combining multiple chips into a single output image. The XREF/YREF and V2REF/V3REF positions represent the same point in different coordinate systems.
The columns CXij and CYij contain the coefficients of the polynomial fit for each chip and convert an input pixel position to an undistorted position. The column names contain the indices for the coefficients from the polynomial fit, where CX11 and CY11 correspond to the a11 and b11 coefficients. For subarray data, the input pixel position must be corrected for the subarray offset in order to put it in the coordinate frame of the full chip.
