The MultiDrizzle Handbook

4.2 Detector Plate scales and Geometric Distortion

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4.2.1 Introduction

Apart from the uncertainties introduced by the telescope pointing accuracy when performing dither offsets, another effect which needs to be considered is the varying plate scale across each of the detectors as a result of the geometric distortion produced by the telescope optics. For example, the distortion across the WF chips in WFPC2 amounts to ~2% at the chip corners, thus a commanded telescope offset of exactly 25 pixels at the chip center would create an additional 0.5 pixel shift at the edge of the detector. In other words, objects near the edge of the chip would be subsampled by half a pixel while those near the center would not be subsampled at all, and this results in continuously varying degrees of subsampling across the image. This means that detector distortion leads to non-uniform pixel subsampling when dithering.

This additional offset introduced by geometric distortion toward the edges of the chip scales linearly with the size of the commanded telescope offset. Thus, a dither of only 5 pixels at the center (instead of 25 pixels) would produce a much less severe error of only 0.1 pixels additional shift at the edge of the chip, and the sampling would remain approximately constant across the detector. This is another reason why large dithers are discouraged - although the drizzle software is capable of dealing with changing pixel shifts and different degrees of subsampling across the image, the scientific interpretation of the data could still be impacted by the fact that not all the objects are equally subsampled. Instead, the resulting analysis is greatly simplified if small dithers are used, thereby keeping such sampling differences relatively small across the chip (e.g., < 0.1 pixel). However, the two chips that compose the WFC in ACS are separated by a gap of order 2.5 arcseconds (~50 WFC pixels). As a result, many ACS dither strategies involve the use of offsets sufficiently large to allow the detectors to cover this gap and varying degrees of subsampling across the detector will be a necessary consequence.

4.2.2 Summary of Detector Plate Scales

Multiple folds in the light path are used in the HST instruments to direct the light to selected optical elements and the detector(s) in a way that achieves the desired focal ratios within the confines of the optical bench envelope, while minimizing throughput loss and image degradation. In some instruments, the 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, depending on the tilt of the detector plane and the curvature of the folding optics. To first order, the tilt causes a square pixel on the detector to be projected as a rectangle (for rotation of the detector about one of its axes) or as a rhombus (for rotation of a square detector about one of its diagonals) or more generally as a parallelogram. The curvature of the folding optics introduces higher order terms in the distortion, causing the projected dimensions of a pixel to vary somewhat with location on the detector. The current best-determined values of the projected pixel dimensions are stored in the Science Instrument Aperture File (SIAF), specifically defined for a single pixel at specified reference locations on the detector. The geometric distortion model appropriate to a given detector then allows the calculation of projected pixel dimensions at other locations on the detector.

The Instrument Apertures tables contain parameters relating the instrument coordinates, pixels or deflection offsets, to the telescope coordinate frame. This frame, designated V2,V3, represents the projected position on the sky. If the telescope roll is zero, then V3 points North, and V2 points East. (The reason for the unconventional axis order is that V2 and V3 are part of a three-dimensional coordinate system with V1 being approximately along the optical axis.) The (X,Y) coordinate system relates to each instrument aperture and the angles and are measured anti-clockwise from the V3 axis to the x and y axes. A reference point is chosen, near the center of the aperture which has the coordinates in the instrument frame, and in the telescope frame. The reference point is the fiducial in the focal plane that HST will endeavor to place the target specified in the proposal. The scales and in x and y directions are not necessarily equal and are each tabulated. For most HST imaging instruments, the x and y axes are flipped with respect to the V2 and V3 axes (as shown in Figure 4.1) (), and their projection onto the V2,V3 frame may not be orthogonal, but this is expressed by the and angles and the transformation formula remain valid. The transformation from any point (x,y) to V2,V3 is given to first order by:

Figure 4.1: Reference Angles for the SIAF Instrument File


 
Visual representation of the frame of reference for angles in the SIAF instrument file. for ACS, NICMOS, STIS, WFC3, WF/PC1, WFPC2.
 

The x and y dimensions of the reference pixels in the V3 coordinate system for each of the commonly used imaging detectors are tabulated in the table below. Also listed in the table are the angles of the x and y axes of each detector, corresponding to the CCD rows and columns, measured counter-clockwise from the +V3 axis of the spacecraft (beta-x and beta-y). These dimensions are measured for the reference pixel. As a result of geometric distortion the orientation and scale of the pixel axes may change somewhat across the detector. The last column in the table lists the approximate maximum change in scale from the center pixel to elsewhere on the detector (the largest changes are generally seen between the centers of the chips and the corners). While the values in this table are intended to be current, the latest positional information can always be obtained directly through the Observatory Support Group's pages on apertures:

http://www.stsci.edu/observatory/apertures

and pointing

http://www.stsci.edu/observatory/pointing

Please note that the values for WFC3 and COS in this table are preliminary, being derived purely from the pre-flight measurements, and are subject to change pending on-orbit calibration programs. All of the numbers detailed in the table below are a snapshot of the current SIAF information as of the printing of this handbook. The SIAF file itself is an internal table which is used in the operational science database at the institute, information in which the users would be the most interested is available from the Web page listed below. If this information is insufficient to meet your needs, please contact the Space Telescope Science Institute help desk.

http://www.stsci.edu/hst/observatory/apertures/siaf.html

Table 4.2: Pixel Scales of the Primary HST Instruments

Instrument	Aperture (pixels/side)	FOV (arcsec/side)	x-scale ("/pixel)	y-scale ("/pixel)	beta-x (deg)	beta-y (deg)	Detector Tilt (deg)	Distortion Center- Corner
WFPC2	PC1 (8002)	36	0.04552	0.04550	134.908	224.908		1%
	WF2 (8002)	80	0.09950	0.09959	224.388	314.388		2%
	WF3 (8002)	80	0.09957	0.09948	314.698	44.698		2%
	WF4 (8002)	80	0.09953	0.09963	45.258	135.258		2%
ACS	WFC (40962)	200	0.0496	0.0495	92.06	177.53	22	8%
	HRC (10242)	29	0.0284	0.0248	-84.12	0.08	25	11%
	SBC (10242)	30	0.0337	0.0300	-84.56	-0.11	25	11%
NICMOS	NIC1 (2562)	11	0.04320	0.04302	225.312	315.312		0.7%
	NIC2 (2562)	19	0.07600	0.07532	224.507	314.507		0.2%
	NIC3 (2562)	52	0.20375	0.20301	224.851	314.851		0.6%
STIS	CCD (10242)	52	0.05076	0.05077	45.056	135.056		0.3%
	NUV (10242)	25	0.02475	0.02475	45	135		1%
	FUV (10242)	25	0.02474	0.02474	45	135		0.7%
WFC3	UVIS (40962)	162	0.03970	0.03952	-41.189	44.877	21	1.7%
	IR (10142)	136x123	0.13549	0.12108	-44.996	45.004	24	3.0%
COS	LFPSA	2.5 (diam)	0.022	0.094	135	145
	LNPSA	2.5 (diam)	0.0258	0.0258	135	145
	LFBOA	2.5 (diam)	0.022	0.094	135	145
	LNBOA	2.5 (diam)	0.0258	0.0258	135	145

4.2.3 Detector Distortion Models

To date, the most detailed empirical HST distortion geometry models are those that have been created for WFPC2, where distortion across the cameras is introduced not only by the HST OTA but also by the re-imaging optics within the instrument. However, the general form of the functional description used for WFPC2 has also been applied to the other instruments. The detector distortion models can be generally 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-terms:

Explicitly, these distortion polynomials expand out into:

The distortion coefficients characterizing each instrument are stored in the form of tables that are maintained by STScI, and are incorporated into the IRAF/STSDAS dither software used to analyze the data. See the description of the IDC table (Section 4.2.4) for more details. The linear component of the distortion generally reduces to:

and are zero since the two coordinate systems have the same origin. 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 ( is non-zero), a displacement in x on the detector has an X component and a Y component, while a displacement in y on the detector has only a Y component, as shown in the figure:

Figure 4.2: Projection of Detector Coordinates


 
Projection of x,y detector coordinates onto the X,Y frame using first order distortion terms.
 

WFPC2

The distortion models for WFPC2 are generally defined in terms of polynomials expressed separately for each detector, thereby providing four transformations between the detector coordinates and a common rectilinear system.

• The Trauger (1995) solution is derived from a model of ray tracing through the camera optics, and includes wavelength dependence parameterized by the refractive index of the MgF₂ field-flattener windows. The geometric distortion in this model is defined in terms of third-order polynomials of the above form, derived separately for each detector and expressed relative to the central pixel of each chip. The r.m.s. residuals of this solution are of the order ~5 m.a.s. across each of the cameras, with larger systematics extending up to ~20 m.a.s. toward some of the corners of the detectors.
• The solutions of Gilmozzi et al. (1995) and Holtzman et al. (1995) are based on multiple overlapping observations of globular clusters (NGC 1850 and w Centauri respectively), and are derived for a single wavelength (555nm). An important difference between these two solutions is that the Gilmozzi solution contains combined Legendre polynomials of third order, thus it includes individual terms up to 6 orders, while the Holtzman solution uses a complete third-order polynomial in x and y. The Holtzman solution is also defined in terms of a single meta-frame coordinate system, relative to the center of the PC camera. The r.m.s. residuals of this solution are of the order 2 m.a.s. across the WF chips and 5 m.a.s. across the PC, although some systematics up to 10 - 15 m.a.s. are present toward the edges of the chips.
• The solution of Casertano and Wiggs (2001) is based on more recent observations of Centauri, obtained in June 1997. In addition to providing an updated solution that takes into account detector motions since 1995, this work also made use of a much larger number of stars to substantially improve the sampling of the distortion across the entire area of each detector, and reduce the large systematic errors toward the corners. The resulting solution (derived at 555nm) has r.m.s. residuals of ~1.3 m.a.s. across the entire extent of each of the WF chips, and ~3 m.a.s. across the entire PC chip, thereby effectively eliminating the systematics that had been present in the earlier solutions.

These solutions have been converted into reference files (Section 4.2.4) for use in the calibration pipeline. Work to refine these solutions for a broader wavelength range has been published in a report by the WFPC2 team (Kozhurina-Platais 2003). In addition, time-dependent effects of the chip-to-chip positions (Section 4.2.4) have been calibrated and included as a separate reference file, the OFFTAB.

ACS

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 which can be downloaded from:

http://www.stsci.edu/hst/acs/documents/handbooks/

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 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.

Figure 4.3: Non-linear Component of the ACS WFC Detector


 
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 the IDCTAB reference file (Section 4.2.4) for use by the MultiDrizzle 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 5 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 subsampling across the entire ACS field of view.

A small change in the skew of the ACS images has been detected over time, as have small distortions which are higher frequency than can be corrected by the low-order polynomial presently incorporated into the IDCTAB reference files. These effects are discussed in more detail in (Section 4.2.4).

NICMOS

For NICMOS the degree of distortion is somewhat less than for WFPC2, partly owing to the smaller areas covered by the NICMOS detectors (Cox et al. 1997). For all three cameras (NIC1 and NIC2, as well as NIC3 when used at the optimal focus position), the distortion is less than 1 pixel at the chip corners relative to the center and appears to be best modeled by a quadratic polynomial. However, the NIC3 camera at its nominal out-of-focus position has substantially higher geometric distortion, and the initial NIC3 geometric distortion solution reported by Cox et al. (1997) was compromised by the effects of vignetting which forced those authors to discard stellar position measurements over a significant portion of the field of view.

The geometric distortion for NIC3 was remeasured during the January 1998 NIC3 refocus campaign, at which time the Field Offset Mirror was repositioned to substantially reduce the amount of vignetting across the field. A new set of distortion coefficients was measured using these data. These are reported in Chapter 5 of the NICMOS Data Handbook (Dickinson et al. 2002), and have been used to derive the in-focus NIC3 geometric distortion coefficient files.

The NICMOS plate scale has been more variable than that of the other instruments, particularly during the early part of its lifetime, due to path length changes introduced by the NICMOS dewar anomaly. More detailed information on these and other geometric effects are all available from the NICMOS Data Handbook which can be downloaded from this Web page:

http://www.stsci.edu/hst/nicmos/documents/

The latest distortion models for NICMOS have also been converted into reference files(Section 4.2.4) for use by PyDrizzle and MultiDrizzle to correct the distortion of NICMOS data.

STIS

The geometric distortions for the three detectors of STIS in imaging-mode have been measured on-orbit (Malumuth and Bowers 1997; Walsh, Goudfrooij and Malumuth 2001). The CCD, NUV-MAMA and FUV-MAMA display maximum distortions less than about 0.3 - 1% at the edges relative to the detector center. The distortions for these cameras have been modeled in terms of cubic polynomials, similar in their functional form to those used for NICMOS and WFPC2, and stored as IDCTAB reference files (Section 4.2.4) for use with the MultiDrizzle software.

WFC3

Wide Field Camera 3 is due to be installed during Hubble's fourth servicing mission. An initial distortion model has already been created for each channel, UVIS and IR. These models are described in the WFC3 Data Handbook, which can be downloaded from:

http://www.stsci.edu/hst/wfc3/documents/

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 ~10%. 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 30 pixels (1.3 arcseconds) along the diagonal. The maximum displacement from the rectangular projection of the IR detector, occurring at all four corners, is about 10 pixels (1.4 arcseconds) along the diagonals.

The full distortion solutions will be updated during Servicing Mission Orbital Verification (SMOV4) in the Winter of 2008/9. For the most current information, please see the WFC3 Web page at:

http://www.stsci.edu/hst/wfc3

4.2.4 The IDC Table

The Instrument Distortion Correction table file (represented by the FITS header keyword IDCTAB) was created to support the description of the geometric distortion models for the instruments. The IDCTAB is stored as a FITS file with a table in the first extension. It contains the minimum information necessary to correct each pixel in the given science image to its true location on the sky. This takes the form of the coefficients for the Science Instrument Aperture File (SIAF) polynomial fit. The SIAF file contains descriptions of the distorted and undistorted aperture positions. Any instrument whose distortion has been modeled using a polynomial fit can use this table as a reference file.A complete description of the IDC reference file can be found in Hack & Cox (2001).

This file can support any order polynomial, with the order of the polynomial being specified in the primary header using the NORDER keyword. The table itself consists of separate rows for each configuration which has been calibrated. The first few columns of each row indicate the chip identification (of a multi-chip detector) and filter selection for which the coefficients of this row apply. The next 6 columns specify the physical size of that chip and the position of the reference pixel on that chip both in pixels on the chip and in arcseconds on the sky relative to the center of the telescope's field of view; in other words, its position in the V2 and V3 coordinate system. The column labeled THETA describes the orientation of the detector's Y axis relative to the telescope's V3 axis. This allows each chip of a multi-chip detector, such as WFPC2 or ACS, to be properly oriented in the final output mosaic. The remaining columns specify undistorted plate-scale and the coefficients of the polynomial which need to be applied to the chip to correct for the distortion. These reference files were originally developed for use with ACS and STIS imaging data, with IDCTAB reference files being subsequently generated for WFPC2 and (if not already available) NICMOS.

The coefficients from the IDCTAB reference file form the basis for the description of the distortion model, however, the MultiDrizzle software and the SIP convention require the model in different forms. The reference file allows for updates to the distortion model as new calibrations produce higher-quality models, updates which can be easily provided to the HST operational calibration pipeline and made available for use in archive retrievals at any time. The reference files themselves, though, only serve as the primary source of the model, while the MultiDrizzle code and "makewcs" in particular interprets those models and updates the headers with the distortion coefficients and generates ASCII files for use with IRAF's `drizzle' task to actually perform the re-sampling. The distortion model will be written to the header using the SIP convention (Section 4.2.5) and will eventually serve as the sole source for the model without having to rely on the actual reference file itself once the SIP keywords are updated.

Time Dependent Motions of WFPC2

The IDCTAB reference files for WFPC2, however, does not contain the complete description of the model as the position of each chip varies over time relative to the other chips. Calibrations have determined the long-term movements of the chips as described in WFPC2 ISR 2001-10. The WFPC2 Data Handbook section on astrometry, reports that these motions have been tracked using K-spot imaging and started out with shifts of 2-4 pixels in the first few months of operation. Those motions settled down to a rate of about 0.1 pixels/year, which has added up to more than a pixel over the course of the operational life of WFPC2.

These motions are not accounted for in the WCS information in the raw (d0h) or calibrated (c0h) WFPC2 images generated in the archive by CALWP2. However, the generation of drizzled WFPC2 products in the archive updates the WCS information in the headers to account for these time-dependent motions. Those time dependent motions have been calibrated and recorded in an ancillary reference file, specified by the header keyword OFFTAB, which gets used along with the IDCTAB to completely define the distortion for the observation. This secondary reference file contains as its first column the date for which that particular row can first be applied to WFPC2 data. There are sets of rows corresponding to various dates when the chip positions have been determined. The remaining columns specify the chip and the V2 and V3 coordinates of the chip's reference position. The positions are then determined for any given observation date by linearly interpolating between V2,V3 positions corresponding to dates that bracket the observation date. The latest calibrations, available as of Fall 2008, will support chip-to-chip alignment to within 0.2 pixels relative to the WF3 chip for all observations up until the removal of WFPC2 during SM4.

Time Dependent Distortion for ACS

The IDC table refers to the state of the geometric distortion at a given date. Observations taken at different times and/or different orientations, however, exhibit some residual distortions. The residual distortions for ACS WFC observations have been determined to have a time-dependent component to them, as reported by ACS ISR 2007-08. The time-dependent correction to the polynomial coefficients gets derived from 2 coefficients computed using the equations reported by Anderson (2007) in ACS ISR 2007-08:

where the date used in the calculation refers to the observation date in decimal years.

The values of and get recorded in the header keywords TDDALPHA and TDDBETA when they are derived by the "makewcs" task (described in the section on Implementation) when updating the header to be consistent with the latest IDCTAB model. The SIP coefficients written out by "makewcs" can be updated to include this time-dependence so that any SIP-aware software (like DS9) should correctly account for this effect. This is not turned on in pipeline use prior to SM4. The keyword TDDCORR will be added to the ACS/WFC primary image header with a value of "PERFORM" to turn on this correction for pipeline use. However, "makewcs" can be run to apply these corrections to the SIP coefficients when needed by manually adding the TDDCORR keyword to the ACS/WFC image primary header.

ACS ISR 2007-08 also identified errors in the overall distortion solution at a level of a fraction of a pixel that were higher frequency than can be represented in the low order polynomial presently used by the IDCTAB or SIP header (see below). These are represented by two tables which are linearly interpolated for the position at any given pixel in a ACS chip. At present these corrections are represented as full reference images in the archive, with a suffix of "_dxy". Under the present transition to the use of the fits header to represent the geometric distortion of HST images, the linearly interpolated look-up table will be placed in the fits image header and will replace the "_dxy" images.

Addition of the look-up table and time dependent distortion corrections substantially improves relative ACS astrometry; however, even with the full Anderson (2007) solution, the user can expect up to pixel or mas systematic errors in the astrometric distortion correction near the edges of the field. These errors are due to time variable changes in the distortion which have yet to be characterized. Comparison of two ACS images taken at different times or orientations may show somewhat larger offsets near the edges, since the small remaining systematic errors may add rather than cancel.

4.2.5 SIP Coefficients in the Image Headers

The distortion model used to calibrate HST images can be represented primarily by a polynomial, and in the case of ACS, a 4th-order polynomial. This polynomial has been calibrated by the instrument teams at STScI and recorded in reference files for use with HST data. This information can now be included in the headers of each image using the Simple Imaging Polynomial (SIP) convention as implemented for Spitzer data (Shupe et al., 2005), and understood by many packages, such as DS9, to report distortion-corrected positions from the image. These SIP coefficients allow MultiDrizzle to operate without requiring external reference files for describing the polynomial distortion model for each image.

SIP Convention

As described in the paper by Shupe et al. (2005), the use of the SIP coefficients can be recognized by 2 changes to the image header. The format of the coordinates described in the FITS header is provided by the CTYPE1 and CTYPE2 keywords. The extra characters -SIP are simply appended to the existing keyword values to report that the SIP transformation should be used for interpreting the coordinates of objects in the image. For HST images, this results in:

  CTYPE1 = 'RA---TAN-SIP' / the coordinate type for the first axis
 CTYPE2 = 'DEC--TAN-SIP' / the coordinate type for the second axis

The pixel coordinates are transformed to sky coordinates using the CD matrix specified in the header of the image. The CD matrix already includes the linear terms of the distortion model as well as any skew, rotation and scaling. The non-linear terms, defined as "f(u,v)" and "g(u,v)", get applied using the following transformation:

where "u" and "v" are the initial position from the image.

The non-linear coefficients are recorded in the image header using keywords "A_p_q" and "B_p_q" for the polynomial terms . The polynomial functions for use in the transformation then come out to be:

where p+q <= A_ORDER, and

where p+q <= B_ORDER

where A_ORDER and B_ORDER are keywords reporting the order of the polynomial used for the model.

Implementation

The SIP coefficients have been implemented for use with HST data through the use of the Python task "makewcs". This task has been released with STScI_Python and MultiDrizzle runs this task when the parameter "updatewcs=yes". However, the CTYPE keywords were not being appended with the -SIP string until more software was available to interpret the SIP coefficients. This default behavior changed during the Summer of 2008 when MultiDrizzle was modified to work exclusively with the SIP convention. Any HST data processed with versions of MultiDrizzle (and "makewcs") installed with STScI_Python 2.6 (released in Feb. 2008) or earlier can append -SIP to the CTYPE keywords in all science extensions to allow DS9 and other SIP-aware software to use the SIP coefficients recorded in the science headers.