Astrometric and Photometric Distortion for WFCAM and VISTA

Document number: VDF-TRE-IOA-00009-0002 Version 1.2


This report analyses the expected astrometric distortions for WFCAM and VISTA and discusses some of the consequences. One of the many conclusions from the analysis is that there is approximately a 10 arcsec diameter limit to dithering patterns for WFCAM before non-linear resampling at the detector level is required. For VISTA, no practically useful dither is possible without non-linear resampling. The radial scale change implicit in the distortions also has an impact on photometric measurements. With conventional processing WFCAM is within photometric error budget without further correction, but for VISTA the induced systematics vary from 0\% to 3.5\% and without correction would overflow the entire systematic photometry error budget.

Field Distortion

Optical systems, particularly those involving refractive elements, do not have a uniform plate scale over the field and generally have a radial distortion term which takes the form

r_true = k1 * r + k3 * r**3 + k5 * r **5

where r_true is an idealised angular distance from the optical axis and r is the measured distance. k1 is the scale at the centre of the field, usually quoted in arcsec/mm.

WFCAM and VISTA are no exception and prior to on-sky measurements being available, ray-tracing enables an estimate of the expected field distortion to be made.

Expected radial distortion for WFCAM

The above figure shows the non-linear distortion (ie. % scale change) expected for WFCAM as a function of wavelength. For VISTA there is negligible variation with wavelength and the following figure shows the tabulated K-band distortion, (red splodges), with an "r-cubed" fit superimposed (green circles).

Expected radial distortion for VISTA

In the K-band WFCAM and VISTA will have scales, ie. k1 values, of 22.30 and 17.09 arcsec/mm respectively. The term due to k5 is usually negligible and until real sky data is available is not worth pursuing, since other similarly sized distortions may be present. Rearranging the preceding equation to a more convenient form gives

r_true = r' * (1 + k3/k1**3 * r'**2) = r' + k * r'**3

where r' is the measured distance from the optical axis in arcsec using the k1 scale. If we convert all units to radians the "r-cubed" coefficient is conveniently scaled (in units of radians/radian**3 - honest!) and has values of 76.1 and 42.5 in the K-band for WFCAM and VISTA respectively.

[Measurements on real WFCAM data indicate that this coefficient is more like -50.0, ie. apart from the sign and magnitude it was about right. Note that subsequent numbers for WFCAM have been updated to correspond to the real on-sky values.]

Although this type of distortion generally presents no problem for accurate calibration of individual pointings, it can lead to various complications when stacking data taken at various locations, eg. dither sequences. This is caused by the differential non-linear distortions across individual detectors being comparable to, or larger than, the pixel size of the detector. In these cases stacking involves resampling and interpolation of some form. While these are inevitable in combining pointings to form contiguous tiled regions, they may be avoided at earlier stages, such as stacking individual detector dither sequences, by suitably limiting dither offsets and thereby both simplify and speed up the data processing.

The following figures demonstrate the level of this optical distortion over the field of view.

VISTA radial distortion

For WFCAM the non-linear term introduces a (pincushion/barrel) distortion amounting to roughly 10 arcsec from the centre to the edge of the 0.8 degree diameter field, denoted by the blue circle.

VISTA radial distortion

For VISTA the corresponding numbers are a distortion of around 25 arcsec at the edge of the 1.67 degree diameter field, denoted by the blue circle.

Differential Distortion

An example of the differential non-linear distortion on sky arising from a shift along a diagonal of 10,10 arcsec is shown below. This would be the extreme ends of, say, dithering within a +/-5 arcsec box-like region centred on a target pointing. As a reference the pixel scales for WFCAM and VISTA are 0.40 arcsec/pixel and 0.34 arcsec/pixel respectively.

WFCAM differential field distortion

VISTA differential field distortion

We can quantify this effect in a more useful manner by forming

dr_true/dr' = 1 + 3k * r'**2

which describes the local change in relative pixel scale as a function of radial distance. For example, for WFCAM at around 0.45 deg radius, the second term is about 1.0% in size (ie. 3x the radial distortion value shown in the earlier plots). This means that a 10 arcsec shift in the centre corrsponds to a 10.1 arcsec shift at the outer corners of the arrays. The majority (~90%) of this distortion occurs across individual detectors. Therefore, even after re-registering the detector centre, there would still be a distortion at the +/-0.05 arcsec level across it.

To first order the size of this effect clearly scales as the size of the offset (though of course the pattern induced depends on the direction). Confining dither offsets (and microsteps) to within +/-5 arcsec of a target centre (a reasonable practical compromise of various constraints) would enable nonlinear resampling for "dither" stacks of each detector to be avoided for WFCAM but possibly not for VISTA. Of course tiling contiguous regions by filling in the gaps between detectors with additional pointings will always involve non-linear resampling as will large offset pattern dither sequences.

Effect of Scale Change on Photometry

In addition to astrometric effects the change in scale as a function of radius also creates photometric complications. The aim of conventional flatfielding is to create a flat background by normalising out perceived variations from uniformly illuminated frames. If the sky area per pixel changes then this is reflected in a systematic error in the derived photometry. However, since it much simpler to deal with "flat" backgrounds, this problem is either usually ignored or corrected during later processing stages, together with other systematic photometry effects. The effect is simplest to envisage by considering what happens to the area of an annulus on sky when projected onto the detector focal plane. The sky annulus of 2.pi.s.ds becomes 2.pi.r'.dr' on the detector, which using k to denote k3/k1 leads to an relative area of

(1 + k r'**2).(1 + 3k r'**2) =~ (1 + 4k r'**2)

or in other words roughly 4x the linear scale distortion.

At the corners of the field WFCAM has a worst case (K-band) linear scale distortion of about 0.3% leading to photometry systematics due to this effect of at most ~1.2% relative to the centre, and can probably be safely ignored for most applications. If required it is also feasible for this correction to be factored in with the "mesostep" correction for scattered light photometric systematics. To illustrate the effect the following colourful image shows the systematics in photometry that would arise from a standard WFCAM 4 pointing tile.

WFCAM photometry systematics

The systematics vary from 0% in the centre up to 0.5% lighter shades of green, through purple and red at 1.0% to yellow, 1.5% in the corners. The field of view shown is roughly 0.8 deg x 0.8 deg. The exposure map below shows how this relates to the final field.

WFCAM tile exposure map

Grey-green equates to 1; puprle 2; and yellow 4.

For VISTA however, the scale distortion at the edge amounts to some 0.8% - 0.9% implying worst case photometry systematics of order 3.5% from centre to edge of field.

Given that tiling of 6 pointings to create a "uniform" contiguous coverage of the 1.67 degree diameter field is likely to be the baseline mode of operation for VISTA, further modelling is required to see if the resultant systematics are large enough to warrant special attention during pipeline processing and at the mosaicing stage. Stacking individual detector dithers is not a problem (other than needing to cope with non-linear astrometric distortions). The next colourful image shows the systematics in photometry arising from a standard VISTA 6 pointing tile.

VISTA photometry systematics

The systematics vary from 0% in the centre up to 1% lighter shades of green, through purple and red at 2-2.5% to yellow, 3% toward the corners. The field of view shown is roughly 1.2 deg x 1.5 deg. The exposure map below shows how this relates to the final field.

VISTA tile exposure map

Green equates to 1; grey-green 2; purple 3; red; 4 and yellow 6.

World Coordinate Systems and FITS Headers

After a mere 20 years the FITS community finally agreed upon a standard way of defining a World Coordinate System (WCS) for telescopes with focal stations requiring a general radial distortion model. An example of this is given below, but for more details consult: Calabretta & Greisen 2002 A&A 395 1077 and Greisen & Calabretta 2002 A&A 395 1061

CTYPE1  = 'RA---ZPN'           / Zenithal polynomial projection 
CTYPE2  = 'DEC--ZPN'           / Zenithal polynomial projection

CRPIX1  =     3914.68291287068 / Reference pixel on axis 1                     
CRPIX2  =     2958.23149413302 / Reference pixel on axis 2                     
CRVAL1  =             325.5563 / Value at ref. pixel on axis 1                 
CRVAL2  =            0.2084674 / Value at ref. pixel on axis 2                 
CD1_1   =  -1.4011089128826E-6 / Transformation matrix                         
CD1_2   =  -9.2625825594913E-5 / Transformation matrix                         
CD2_1   =  -9.2750922381494E-5 / Transformation matrix                         
CD2_2   =  1.31362241659925E-6 / Transformation matrix                         

PV2_1   =      1.0             / coefficient for r term
PV2_3   =    220.0             / coefficient for r**3 term

and is the style we are proposing to adopt for both WFCAM and VISTA.

Mike Irwin (
Last modified: Thu Sep 28 13:05:44 2006