Instrument Distortion Calibration File Transformation

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Instrument Distortion Calibration File Transformation David Borncamp, Vera Kozhurina-Platais, Colin Cox, Warren Hack Space Telescope Science Institute July 1, 2014 ABSTRACT The current generation of

1 Instrument Distortion Calibration File Transformation David Borncamp, Vera Kozhurina-Platais, Colin Cox, Warren Hack Space Telescope Science Institute July 1, 2014 ABSTRACT The current generation of detectors on board the Hubble Space Telescope (HST) suffer from extreme optical distortion - some of them have deviations as high as 11% across their field of view. This large effect must be accurately calibrated to obtain high precision astrometry and accurate alignment of any HST images. Corrections for the geometric distortion in the form of high-order polynomial coefficients can be found in the Instrument Distortion Coefficients Table reference file. Here we describe the transformation of polynomial coefficients of geometric distortion from the raw detector coordinate system into the HST coordinate system in the context of the IDC table. 1. Introduction High-precision astrometry and image registration with any imaging detector on board HST could not be achieved without accurate distortion calibrations. To remove distortion, the STSDAS software within DrizzlePac uses reference files called Instrument Distortion Calibration Tables Operated by the Association of Universities for Research in Astronomy, Inc., for the National Aeronautics and Space Administration Copyright 2014 The Association of Universities for Research in Astronomy, Inc. All Rights Reserved. 1

2 (IDCTABS, Hack & Cox, 2000). The relevant IDCTAB file is specified in the primary header of the HST science image. The application of this reference file coupled with STSDAS software DrizzlePac/AstroDrizzle (Gonzaga et al. 2012) allows for accurate calibration of the HST images. The IDC table contains the geometric distortion coefficients derived from a high-order polynomial solution. These coefficients are solved for each individual CCD chip and each filter. The IDCTAB reference file is unique for each HST instrument and in a uniform specific format. The required format of the reference file containing the distortion coefficients has already been described Hack & Cox (2000). The derivation of geometric distortion coefficients in the detector coordinate system has been described in detail by Kozhurina-Platais, et al. (2009). However, the transformation of polynomial coefficients into the HST system has never been fully documented. Here, we describe the transformation of distortion coefficients into the IDCTAB system, which is compatible with the HST coordinate system, called the V2V3 system, assuming that the distortions have already been modeled in the detector coordinate system. This report explains the transformations of the geometric distortion coefficients for ACS/WFC on board HST. As distortion is present in all kinds of detectors, the principles and methodology of IDCTAB formalism could be used for detectors on the James Webb Space Telescope and possibly other observatories. 2. The HST Coordinate System The HST coordinate system, usually referred to as the V2V3 system, is a Euler Angle System attached to HST and can be visualized as a tangent plane as seen in Figure 1 (from the ACS Instrument Handbook, 2013). This figure shows where each HST detector falls within the telescope s field of view in the V2V3 system. The information on the HST focal plane is maintained by the HST Telescopes group, and it contains the location and orientation of all instruments with respect to the location of the Fine Guidance Sensors on the HST coordinate system. 2

3 The orientation of the telescope and coordinate systems as seen in Figures 1 & 2 define the orientation of the instruments on the sky. When the HST roll-angle (PA_V3) is commanded to be 0, the V3 axis is parallel to North as seen in Figure 2 (Lallo & Cox, 2009). From PA_V3 we can estimate the angle of the telescope orientation on the sky. The estimated angle of the orientation is close enough for most cases since the actual initial roll-angle achieved for a given HST observation is within 0.03 degrees of the commanded value (Lallo et al., 2007). From the HST FOV, we know the location of the instruments in the system and can therefore find the location of the instrument on the sky. Thus, this information allows us to do a basic transformation between the detector coordinate system and the V2V3 coordinate system. Figure 1. The V2V3 system on the HST field of view with locations of all active instruments and the offset of each HST instrument from the optical axis in arcseconds. If the HST roll angle is 0, then V3 axis is parallel to North (modified from ACS Instrument Handbook, 2013). 3

4 Figure 2. Angles of the V2V3 System. North on the sky points up and the roll of the telescope relative to V3 is PA_V3. The angle between North and the Y axis of the detector (Y image ) is ORIENTAT which is calculated and put in the primary header of HST images (HST FOV Lallo & Cox, 2009). The V2V3 system is a spherical, three-dimensional space with the third dimension (the V1 axis) extended along the optical axis of HST. It allows us to simplify the visualization of the focal plane as looking down the optical axis (as seen in Figures 1 & 2). For most cases, the tangent plane approximation is sufficient for transformations within a single field of view. The differences between the tangent plane and spherical systems are small, on the order of parts per million (0.0001%) within an ACS/WFC field of view. However, these small differences become significant when transforming between instruments or within large mosaics. Therefore, we must account for the spherical properties of the V2V3 system during the transformation of polynomial coefficients into V2V3 system. In this case, to visualize the three-dimensional space (V1, V2, V3) would be a more advanced version of Figures 1 & 2, with the optical V1 axis and V2 & V3 axis on spherical coordinate system. 4

5 3. Geometric Distortion Coefficients The large-scale geometric distortion of all HST imaging instruments such as WFPC2, ACS/WFC and WFC3 are due to the optical assembly of the telescope. In the case of ACS/WFC, the detector focal plane is tilted with respect to the incoming beam by 22. As a result, the ACS/WFC images are optically distorted by about 8% across the entire detector (ACS Instrument Handbook, 2013). As described by Kozhurina-Platais et al. (2009), an easy and simple way to derive an accurate geometric distortion is to use an astrometric standard catalog (e.g. standard astrometric catalog in the vicinity of globular cluster 47Tuc, Anderson, 2007). In this way, optical distortions can then be represented by high-order polynomials relative to a reference point for the CCD chip. This point, which we will call point Q, is located in the center of the chip. In the case of ACS point Q is located: X= x 2048 and Y= y 1024, where x and y are measured positions on the detector. The detector based geometric distortion can be represented by high-order polynomials as seen in Equations 1 & 2: X c = A 1 +A 2 X +A 3 Y +A 4 X 2 +A 5 XY +A 6 Y 2 +A 7 X A 21 Y 5 (1) Y c = B 1 +B 2 X + B 3 Y +B 4 X 2 +B 5 XY +B 6 Y 2 +B 7 X B 21 Y 5 (2) Where X c and Y c are tangential-plane positions in an undistorted reference frame (the astrometric catalog frame); X,Y are raw measured pixel positions in the observed ACS/WFC frame normalized to the center of the CCD chip; A j and B j are arrays of the polynomial coefficients in the detector coordinate system. This gives us one set of arrays of coefficients (A j and B j ) per one polynomial solution. In order to properly use these arrays of coefficients in the IDCTAB reference file, they need to be transformed from the detector coordinate system X image & Y image (Figure 2) into the HST V2V3 coordinate system (Figure 2). Then derived geometric distortion coefficients will be applied in the same way for each instrument and provides a common coordinate system for cross-instrument image alignment. 5

6 4. Instrument Distortion Coefficients Table (IDCTAB) 4.1 Angles of Rotation In order to transform the polynomial coefficients from the detector coordinate system into the V2V3 system, we must rotate the polynomial coefficients (Eqs.1-2). First, we have to find an accurate angle of rotation, accounting for all angles between two coordinate systems which includes: 1) HST roll-angle, PA_V3 shown in Figure 2; 2) ORIENTAT angle between North and Y detector axis shown in Figure 2; 3) Correction angle to ORIENTAT, which is related to the reference point on the reference CCD chip. The HST roll-angle, commanded to the telescope can be found in the image header as the PA_V3 keyword. The next angle, which is calculated as the angle between North and the Y-axis of the image as seen in Fig. 1, 2 can also be found in the header. Both of these values have errors in the pointing due to the Guide Star Catalog errors, as discussed in section 2. The small correction to the ORIENTAT angle is calculated from the high-order polynomial solution in the detector system. The PA_V3 and ORIENTAT angles are related to the aperture of the instrument, the keywords of the aperture RA_APER (α 0 ) & DEC_APER (δ 0 ) could be found in the primary header of the image. However, the reference point for the polynomial solutions are in the center of the reference CCD chip, so we will need to move from point P on the sphere to point Q as seen in Figure 3 where the aperture of ACS/WFC is located in point P(α 0,δ 0 ) but the location of the reference point for geometric distortion solution is point Q. 6

7 Figure 3. Reference points for ACS/WFC. Point P is the center of the aperture and point Q is the reference point on the reference chip. To calculate the difference from the center of the aperture P(α 0,δ 0 ) to the reference point Q, we must recognize that the V2V3 system is spherical and use proper spherical trigonometry. It is useful to consider the triangles on a spherical surface NPQ and PQR where R is an arbitrary point on the sphere, as illustrated in Figure 4. N is at the North Pole, so the arc NP is equal to 90 δ 0 degrees. The angle r 0 is the angle between the North direction and the V3 axis. The sides PR and RQ represent roll displacements of V2 and V3 as seen in Figure 2. And PRQ is a right angle so the arc length ρ between P and Q is given by the relationship: cos(ρ)=cos(v2) cos(v3) Then the angles β and γ are given by the Sine Rule in which sin(prq) = 1 and will be expressed as follows : sin(β) =sin(v3)/sin(r 0 ) sin(γ)=sin(v2)/sin(r) Knowing β we can calculate A within NPQ because r = A+β. We now have two sides and the included angle of triangle NPQ, so we can find B. We can now calculate B by working out the length of NQ using the Cosine Rule, and then using the Sine rule as in Equation 3: tan(b) = sin(a)cos(δ 0 ) sin(δ 0 )sin(ρ) cos(δ 0 )cos(ρ)cos(a) (3) 7

8 Since the arc from R to Q points in the V3 direction at Q, γ + B + r =180, then the angle r = 180 γ B. It is important to place angles in the proper quadrant on the sphere that is attached to HST. For instance if V2 is calculated as a negative then γ is correctly calculated as a negative angle but β must be replaced by 180 β. If this is the case, in the formula for tan(b), the arctan2 formula should be used with the numerator and denominator calculated exactly as indicated in Eq. 3. Thus it will place the result in the proper quadrant on the sphere. This consideration is especially important in detectors like ACS/WFC whose orientation is almost aligned with the V3 axis but whose Y-axis orientation is anti-parallel to V3. Figure 4. Schematic illustration of the angles on the V2V3 system. Point N is located at the North Pole of the sphere, point P is at the center of the aperture on the system, point Q is our reference point, point R is an arbitrary point elsewhere on the system. 8

9 Where: All of these angles culminate in the final angle (θ) as seen in Eq. 4: θ = (ORIENTAT ξ) + arctan( tan Δα sin(δ! )) B (4) ξ is the correction to ORIENTAT, calculated from the polynomial solution (Eq. 1 & 2), and accounts for the inaccuracies in HST pointing described in Lallo et al., (2007); Δα is the difference between Right ascension (α) of the target and Right ascension (α c ) of the center of the standard astrometric catalog; δ c is the Declination of the standard astrometric catalog; B is the angle from Eq. 3 as PA_V3 angle with respect to reference point Q on the chosen reference chip. Therefore, the final angle θ will account for all angles of rotation from detector coordinate system into the spherical V2V3 system. 4.2 Transformation of Coefficients Now that we have the computed angle of rotation, the transformation from the detector coordinate system into the V2V3 system can be done. The rotation and transformation for each set of array of coefficients from the detector coordinate system into the V2V3 system can be performed as follows: A j B j = cos θ + sin θ sin θ + cos θ A j B j (5) Where A j and B j are the j th element in the arrays of coefficients (Eqs. 1-2) per each polynomial solution. The angle θ is found from Eq. 4. After the coefficients have been rotated into the V2V3 system, they are averaged from all sets of polynomial solutions. 9

10 4.3 Detector Scale Next, the averaged coefficients in the V2V3 system must be scaled by the adopted instrument s plate-scale, to put them into the sky tangent plane. The scale of the detector is a first order determination of the optical distortion inherent to the instrument. As discussed earlier, the ACS/WFC is tilted with respect to the optical axis and rotated on the focal plane, so even if the detector was created with exactly square pixels, the X and Y scales will not be exactly the same on the sky. It also means that the X and Y scale terms must be solved independently using the linear part of the polynomial terms that have been transformed into the V2V3 system. Since the scale can change across the detector on the sky, we will be finding the scale at the reference point Q. This means we are solving for scale at the center of each CCD chip. It is important to mention here, that the higher order-polynomial terms will have no effect. Then X and Y scale can be calculated as follow: X!"#$% = A!! + B!! Y!"#$% = A!! + B!! (6) (7) Where A!, A!, B! and B! are the coefficients defined in Eq. 1 & 2 after they have been transformed into the V2V3 system (Eq. 5). The calculated scales should be very close to the plate scale of the detector. However, it is likely that the scales will not perfectly match. At this point we can also calculate the angle of the X and Y-axes relative to V3, or the so-called β x and β Y as follows: β! = arctan (!!!!!! ) (8) β! = arctan (!!!!!! ) (9) Where A!, A!, B! and B! are the linear terms from Eq. 1&2 after they have been transformed into the V2V3 system (Eq. 5) 10

11 4.4 Time Dependency ACS/WFC geometric distortion have shown a time dependency of the linear terms of distortion as found by Anderson (2007), van der Marel, et al., (2007) and re-characterized by U beda, et al., (2013). In the above papers, the time dependence of the linear terms of distortion is characterized in the detector coordinate system, then the derived linear time correction is applied to the X and Y positions which have been already corrected for geometric distortion. It is numerically awkward to apply those corrections in IDCTAB format, since the IDCTAB is already in the V2V3 system. To characterize the time dependent distortion in the V2V3 system and correctly apply it in the telescope system, each set of coefficients (Eqs. 1 & 2) must first be transformed into the V2V3 system individually. This means that the transformed coefficients are not averaged for each set of coefficients as they are in section 4.2 but instead used exactly as they appear in Eq. 5, i.e. transformed for each individual set into the V2V3 system. It is then easy to look for time dependency in each coefficient. It is important to note that there will be some noise in the time series of coefficients due to orbital breathing of the telescope. This creates slight changes in the focus, making the linear coefficients, particularly plate-scale change with it. In the case of ACS/WFC, the linear term in the Y solution (B 2 coefficient in Eq.2) is changing linearly with time. Once it is fit as a function of time, the linear parameter of the fit for the time dependency can be added as keywords in the primary header of the IDCTAB reference file and then to be correctly accounted for by the DrizzlePac software. 5. Summary After transforming the coefficients into the IDC system, it is a simple matter of correctly arranging them, putting them into a FITS table using the format described in Hack & Cox (2000), and putting the time dependency keywords into the header. The V2V3 system is a relatively simple system that uses a combination of spherical trigonometric identities to transform detector 11

12 coordinates into it. Once the coefficients are transformed into this system, optical distortion within an instrument is represented as a table of coefficients and corrected with the application of the STSDAS software. The geometric distortion for any imaging instrument can be obtained using the same basic method. And using the method described in Hack & Cox (2000), it can be correctly formatted into a reference file that can be interpreted by the DrizzlePac software package to be applied to the image. Acknowledgements We would like to thank Norman Grogin for his keen interest in ACS calibrations. We also would like to thank Jay Anderson for his geometric distortion comments and the ACS team for their valuable comments, which significantly improved the clarity of the text. References Anderson, J., 2007, ACS Instrument Science Report, ACS-ISR (Baltimore: STScI). Gonzaga, S., et al., 2012, DrizzlePac Handbook (Baltimore: STScI). Hack, W., Cox, C., 2000, ACS Instrument Science Report, ACS-ISR (Baltimore: STScI). Lallo, M., Cox, C., 2009, HST Field of View (Baltimore: STScI). M. Lallo, E. Nelan, E. Kimmer, C. Cox, S. Casertano, Improving Hubble Space Telescope s Pointing & Absolute Astrometry, Bulletin of the American Astronomical Society 38, 194 (2007). Kozhurina-Platais, V., Cox, C., McLean, B., Petro, L., Dressel L., Bushouse, H., E. Sabbi, 2009, WFC3 Instrument Science Report, WFC3 ISR (Baltimore: STScI). Úbeda, L., Kozhurina-Platais, V. & Bedin, L., 2013, ACS Instrument Science Report, ACS-ISR (Baltimore: STScI). Ubeda, L., et al., 2014, ACS Instrument Handbook, Version 13.0 (Baltimore: STScI). 12

13 Van der Marel, R., Anderson, J., Cox, C., Kozhurina-Platais, V., Lallo, M., Nelan, E., 2007, ACS Instrument Science Report, ACS-ISR (Baltimore: STScI). 13

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