Space Telescope Science Institute
DrizzlePac 2012 Handbook
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The DrizzlePac Handbook > Appendix A: Plate Scales and Polynomial Distortions > A.5 Detector Distortion Models

A.5 Detector Distortion Models
Detector distortion models can generally be characterized in terms of the dependence of the true location (X, Y) of an object, (e.g., in arcseconds), as a function of the measured pixel position (x, y) of the object on the detector relative to some reference pixel (x0, y0), usually as a polynomial of order m in x, in y, and including all their cross-term
Explicitly, these distortion polynomials expand out to:
X = a00 + a10y + a11x + a20y2 + a21xy + a22x2 + a30y3 + a31xy2 + a32x2y + a33x3 + ...
Y = b00 + b10y + b11x + b20y2 + b21xy + b22x2 + b30y3 + b31xy2 + b32x2y + b33x3 + ...
The distortion coefficients characterizing each instrument are stored in the form of tables called IDCTAB reference tables that are maintained by STScI, and are incorporated into the IRAF/STSDAS DrizzlePac software. The linear component of the distortion generally
reduces to:
X = a11x
Y = b10y + b11x
a00 and b00 are zero since the two coordinate systems have the same origin. a10 is zero when the Y axis is defined to lie along the y axis of the detector, which is generally the case. If the projected x and y axes are not perpendicular (b11 is non-zero), a displacement in x on the detector has a X component and a Y component, while a displacement in y on the detector has only a Y component, as shown in Figure A.4:
Figure A.4: Projection of x, y Detector Coordinates onto the X, Y Frame Using First Order Distortion Terms
The ACS/WFC is significantly more distorted than the WFPC2, STIS or NICMOS cameras. The magnitude of the distortion is predominantly a result of the large format of the detector, together with its off-axis location in the HST focal plane. The distortion of ACS/WFC is a combination of an 8% elongation along the detector diagonal, resulting from the detector inclination with respect to the optical axis, together with an increasing radial distortion away from the center of the WFC. The ACS distortion, characterized in terms of polynomials (Hack & Cox 2000), has been described in the ACS Data Handbook.
In particular, the WFC distortion is illustrated in this figure, 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 five 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 four arcseconds.
Figure A.5: Non-linear Component of the ACS Distortion for the WFC Detector using a F475W Quadratic Fit
Note that this figure is rotated 180 degrees with respect to the default orientation of the drizzled product, where WFC2 would be the lower half of the detector
This distortion model has been stored in IDCTAB reference files for use by the AstroDrizzle software to correct the distortion in ACS observations and allow for the combination of dithered images.
The strong distortion implies that the degree of subsampling will vary across the image even for small dithers. For example, an offset of five pixels at the center of the WFC already introduces an additional shift of 0.4 pixels near the edge of the detector. A larger dither, e.g., 12 pixels at the center, will correspond to an integral shift near the edge (one entire additional pixel), but will provide half-pixel subsampling midway between the center and the edge. Thus, varying degrees of subsampling across the image will be unavoidable with any single pair of dither offsets. Obtaining a larger number of offsets can help produce more uniform sub-sampling across the entire ACS field of view.
Time-dependent Distortion for ACS
A small change in the skew of the ACS images has been detected over time, as have small distortions, which are of higher frequency than can be corrected by the low-order polynomial presently incorporated into the IDCTAB reference files. This time-dependent distortion can result in residual alignment errors between two images taken in different visits of up to 0.3 pixels, and therefore needs to be corrected. Full details of the calibration of this effect were documented in the ACS ISR 2007-08 on the “Variation of the Distortion Solution of the WFC.”
This distortion correction term has been included in the latest versions of the IDCTAB reference table as additional keywords in the Primary header of the IDCTAB file itself. The polynomial distortion coefficients get read from the IDCTAB by updatewcs when running AstroDrizzle (or tweakreg or when running updatewcs by itself) and these time-dependent terms get computed and used to adjust the polynomial distortion terms before they get written out as the linear WCS (CD* keywords) and SIP keywords. Thus, the distortion model represented in the keywords in each science image reflects the distortion specific to that image and should correct the image with residual alignment errors of one ACS image to another of much less than 0.1 pixels (with RMS of 0.02 being noted with some datasets).
Wide Field Camera 3 was installed during Hubble’s fourth servicing mission. The UVIS detector is tilted about one of its diagonals with respect to the light path. This tilt produces a projected rhombus with a diagonal elongation of ~7%, reminiscent of the projected parallelogram of the ACS/WFC detector. The IR detector is tilted about its x axis, creating a projected rectangle of elongation of ~8%. Both WFC3 detectors have substantial non-linear geometric distortion, in addition to these projected elongations. The maximum displacement from the rhomboidal projection of the UVIS detector, occurring at two opposite corners, is about 62 pixels (2.5 arcseconds) along the diagonal. The maximum displacement from the rectangular projection of the IR detector, occurring at all four corners, is about 20 pixels (2.4 arcseconds) along the diagonals. These effects make it critical to calibrate the distortion model and remove it in order to be able to combine WFC3 data using AstroDrizzle.
The distortion model has been calibrated to an RMS of roughly 0.1 pixels for both the UVIS and IR channels. Full details on how these models were calibrated can be found in the WFC3 Instrument Science Reports for the WFC3 ISR 2009-33 and WFC3 ISR 2009-34. These calibrations indicated that a simple polynomial solution provided by the IDCTAB reference table would be sufficient for use with AstroDrizzle.
As previously described, the ACS/WFC geometric distortion varies with time. WFC3/UVIS and IR geometric distortion, on the other hand, appears to be stable, with no evidence of secular changes. However, during each interval of the orbital target visibility, there is a linear trend in the UVIS and IR X and Y scale at the level of 0.05 UVIS and IR pixels. These linear changes are related to focus variations over an HST orbital time scale, known as orbital breathing, and will not effect the quality of the drizzled products (Kozhurina-Platais and Petro, WFC3 ISR 2012-03).
An additional feature of the WFC3/UVIS geometric distortion is the fine-scale systematic pattern in the residuals from the best-fitting polynomial solutions. These systematic residuals are typically 0.15 pixels in amplitude and depend on positions of UVIS CCD chips and UVIS passbands. A complicated and correlated residuals pattern is due to the photo-lithographic pattern with size of 700 x 1000 pixels, imprinted onto detector itself during the manufacturing process (Kozhurina-Platais, 2010 in “HST 2010 Calibration Workshop”).
This complicated structure of residuals cannot be removed by a polynomial model. The simple way to remove the fine scale variations is to model the fine scale structure with a look-up table, which can be linearly interpolated at any point in the image. This look-up table would be analogous to ACS/WFC and ACS/HRC (Anderson & King, 2002) as a reference file NPOLFILE in AstroDrizzle.
Figure A.6: Lithographic Pattern of Residuals Seen After Applying WFC3/UVIS Polynomial Distortion Model
The top panel shows the X,Y residuals for the WFC3/UVIS1 CCD chip and the bottom panel, XY residuals for WFC3/UVIS2 CCD chip, after geometric distortion have been removed. The largest vector is ~0.15 pixels, magnified by 2500. The red lines indicate the boundaries of the lithographic pattern The units are WFC3/UVIS pixels.

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