Jeffrey R. Pier,
This document presents a brief summary of how SDSS does astrometry, discusses the scientific requirements for astrometry and how well those requirements are currently being met, and points out work yet to be done and prospects for future improvement.
The key task of SDSS astrometry is to ensure that the positions of the holes drilled for spectroscopic fibers are sufficiently accurate that spectroscopic targets are well-centered on the fibers. For a variety of reasons (see the Astrometry Chapter of the SDSS Project Book for details), this means that SDSS must determine the absolute positions of objects detected by the imaging camera. (While the key science drivers for SDSS are clearly driven by the spectroscopy, the imaging part of the Survey will result in a large database which will be a significant resource for decades to come. Obviously, we would like the positions of all detected objects to be determined as accurately as feasible.)
Spectroscopic targets are chosen from objects detected by the main (photometric) array of CCDs in the imaging camera, which saturate for point-source objects brighter than about 14th magnitude. Sufficiently accurate all-sky astrometric catalogs are not available for stars fainter than about 12th magnitude. Consequently, the imaging camera was designed with a second (astrometric) array of CCDs. These astrometric CCDs precede and trail the photometric CCDs in camera columns. The bandpass of the astrometric CCDs is similar to the SDSS photometric r' bandpass, but the saturation level is about 5.5 magnitude brighter. Objects on the sky pass down a camera column as the telescope is driven along a great circle and the CCDs are read-out synchronously in drift-scan mode. Hence an object will pass through, in sequence, the imaging area of the leading astrometric CCD, then the five photometric CCDs (in order r', i', u', z', g'), and finally, the trailing astrometric CCD.
Stars detected by the astrometric CCDs span a range of magnitudes from about 8.5 to nearly 17 while those on the photometrics reach from about 14th magnitude to beyond 20th (depending on the bandpass). Thus the astrometric CCDs can detect and measure not only bright stars from an astrometric catalog (currently Tycho-2), but also a significant number of stars between 14th and 17th magnitude which also appear on the photometric CCDs.
The astrometric pipeline (astrom) takes, as input, a set of measured parameters (centroids, counts, flags, shape information) for bright objects, matches these up against an astrometric reference catalog, and derives, as outputs, transformations from detected coordinates (CCD row/column) to celestial coordinates.
The original design and implementation astrom works in two stages called Primary Match and Secondary Match. In primary match, stars detected on the astrometric CCDs are matched against the astrometric reference catalog (Tycho-2) stars. Corrections to the detected (CCD row/column) positions are made to account for known optical distortions in the focal plane and for differential chromatic refraction (DCR). A least-squares polynomial fit is derived which follows, as a function of time, the mapping from CCD (row/column) to celestial coordinates. When we began reducing early commissioning data, we discovered that the residuals from the fits showed systematic trends (coherent image motion in the focal plane over various time-scales) due mostly, we believe, to atmospheric effects (seeing and anomalous refraction). To partially correct these effects, we apply cubic splines to smooth the residuals. From these smoothed fits, the CCD row/column positions of non-catalog stars are converted to celestial coordinates, and these star positions are used to construct a boot-strapped secondary catalog.
The secondary match works much the same way, but it matches the secondary catalog stars with detections of point-source objects on the photometric CCDs. Again, a least-squares fit is derived and smoothed and from those fits an affine transformation (row/column to great circle coordinates) is generated for each photometric CCD for each frame (a frame consists of 2048 columns and 1361 rows, approximately 36 seconds of scanning).
While the original pipeline design was fundamentally sound, experience with commissioning data has led us to recently implement a couple of changes in the way the astrom pipeline works which have helped reduced the residuals by 10 to 20 milli-arcseconds: * the least squares polynomial fits have been supplanted by a running means technique which is better behaved, particularly in regions were the catalog star density is low, or when the telescope tracking drifts off of the desired great circle;
* we now map the astrometric (as well as photometric u', g', i', z') CCD star detections onto the photometric r' star detections in CCD row/column (making appropriate corrections for optical distortion terms and DCR). Thus, the bright catalog reference stars are mapped directly onto the photometric CCDs. The running means fit is then applied directly to the re-mapped row/column coordinates on the photo CCDs. This step was taken when we noticed that the atmospheric fluctuations were smoothed differently for astrometric and photometric CCDs due to the different chip widths (the astrometric CCDs are 2048 columns by 400 rows and have integration times of about 10.5 seconds, whereas the the photometrics CCDs are 2048 by 2048 and have an integration time of 54 seconds).
The output of the astrometric pipeline is simply the set of affine transformations, one per photometric CCD per frame. These transformations are applied to the raw device coordinates (CCD row/column). An early database design choice was to store observed quantities in observed coordinates so that if/when transformations were changed it would not be necessary to re-write the object entries (a huge undertaking). Instead it is only necessary to change the transformations used when exporting object information from the database.
There are two categories of requirements for SDSS astrometric performance. One stipulates how well the survey must do in the determination of absolute positions on the sky (as measured in the r' band) of point-source objects brighter than r' = 19, and the other how well the relative positions of objects measured in the four other bandpasses (u', g', i',z') must match up with the r' position.
Each astrometric requirement is expressed as an rms figure per coordinate in units of milli-arcseconds (hereinafter mas) along a great circle. The requirements apply to objects with ``normal'' spectral distributions (e.g. stars of spectral type O5 through M0). Extremely blue or red objects and/or emission-line objects will occasionally fail to meet the requirements due to chromatic differential refraction (DCR). It is expected that the requirements herein will be satisfied for each ``acceptable imaging run.'' Here an acceptable imaging run is a scan of length no less than 20 minutes (exclusive of ``ramp-up'' frames), airmass less than or equal to 2.0, and seeing less than or equal to 1.5 arcsec (FWHM), with the absolute telescope pointing error no worse than 3 arcsec rms (2-D on the sky) and relative telescope tracking error no worse than 50 mas (1-D on the sky) on time scales of 1-10 minutes.
There are three different levels of requirement for absolute astrometry:
1. The ``Drop-Dead'' requirement - if the astrometry does not meet this, the run will be deemed unacceptable since core survey science would be jeopardized:
The absolute astrometry on the sky should be no worse than 180 mas.
2. The ``Science Goals'' requirement - the astrometry should routinely meet this requirement to enable non-core science:
The absolute astrometry on the sky should be no worse than 100 mas.
3. The ``Enhanced Goal'' requirement - if all goes well (i.e., the telescope tracks well, the atmosphere (seeing, anomalous refraction) are favorable, the catalog density is at or above 10 stars/deg2) the astrometry should be limited by the atmosphere:
Reasonable effort should be expended to attain absolute astrometry on the sky of 60 mas. (N.B. - the number that appears here is really no more than an educated guess about the properties of the atmosphere.)
There are also three different levels of requirement for relative astrometry:
4. The ``Drop-Dead'' requirement - if the relative astrometry does not meet this, the run will be deemed unacceptable since core survey science would be jeopardized:
The relative astrometry between adjacent colors should be no worse than 180 mas on the time-scale over which the relative astrometry is determined.
5. The ``Science Goals'' requirement - the relative astrometry should routinely meet this requirement to enable non-core science:
The relative astrometry between adjacent colors should be no worse than 100 mas.
6. The ``Enhanced Goal'' requirement.
Reasonable effort should be expended to attain relative astrometry between adjacent colors of 40 mas.
The ideal, direct way to determine astrometric performance is simply to compare star positions determined with SDSS with those determined externally. Since positions of objects in the r' photometric bandpass are required, it is necessary to compare positions of stars which are fainter than 14th magnitude (and indeed down to 19th to fully satisfy the requirement). There are no wide-area astrometric catalogs available for stars fainter than 14th which are accurate enough to verify SDSS astrometry.
However, the Naval Observatory is presently constructing an all-sky CCD catalog ( The USNO CCD Astrograph Catalog or UCAC) which reaches fainter than 16th magnitude. UCAC appears to have systematic errors of no worse than about +/- 15 mas, and the precision ranges from 20 to 70 mas for stars between 14 and 16th magnitude.
At present, the survey is underway in the Southern hemisphere and ome of the SDSS equatorial scans covered regions where UCAC observations have been made. For example Run 745 was an equatorial parked scan conducted UT March 20, 1999. SDSS positions were matched against the UCAC preliminary catalog and the results are shown in the following figure:
These results are extremely encouraging, but we should realize that such a run represents a best-case scenario for the following reasons:
* The telescope was parked and clamped on the equator at the meridian during the observations so that no telescope motion was involved: telescope pointing or tracking errors were not a consideration.
* A number of astrometric calibration fields lie along the equator (so-called FASTT fields, dense astrometric grids of stars produced by the USNO for the purpose of calibrating the SDSS imaging camera geometry), so that a well-calibrated focal-plane model, generated from the very data being reduced, was available to be used for the astrom pipeline reductions.
For a worst-case scenario, we turn to Run 1449, which was observed recently (UT May 3, 2000) on SDSS' southern-most stripe (Stripe 1). A comparison of astrometry from that run with 5555 UCAC star positions show a mean offset in RA of -11 mas, rms of 119 mas, and in DEC 7 +/- 139 mas. Frankly, this run was not particularly good from an astrometric viewpoint. The atmosphere was particularly troublesome. This is probably due to how far south the stripe is (DEC of -22.5), which means that the observations must be conducted at an airmass of 2.0. The following figure shows data from one of six Camera Columns for Run 1449. Note the significant excursions traced by the astrometric residuals. We believe that these are due to atmospheric fluctuations. The grossest such excursions are fitted and removed by the astrom pipeline, and the points here show the residuals after such fitting.
A possible further contributing factor with run 1449 is that the telescope was required to track a great circle which changed very little in azimuth or elevation during the run. The altitude axis rate ranged from only -0.1 arcsec/sec to 1.5 arcsec/sec, while the azimuth axis rate ranged from 1.3 to 1.4 arcsec/sec. Such slow rates may be difficult to sustain uniformly.
We expect that the absolute astrometry may also suffer when the telescope is required to track at very high axis rates (i.e. near the zenith). In such circumstances, it is difficult to keep the instrument rotator positioned so that stars track straight down the camera CCD columns. Unfortunately there are no calibrated astrometric catalogs above the equator which reach faint enough magnitudes yet, and so we cannot do a direct comparison.
We can, however, compare SDSS-determined positions of stars which appear within the overlap regions of adjacent or interleaved strips. Run 1345 is notorious (at least among the astrometrists) for its high tracking rates (the azimuth rate exceeded 50 arcsec/sec at the beginning of the run). Comparing the positions of 953 stars from this run with matching positions from Run 1331 (an interleaved strip of the same Great Circle) yields a mean error in RA of -22 +/- 103 mas, and in DEC of 60 +/- 120 mas.
In summary, the absolute astrometry consistently meets the Drop-Dead requirement, surpasses the Science Goals Requirement when extreme conditions are not being encountered, and approaches the Enhanced Goals requirement under ideal conditions.
Relative Astrometry Performance
The relative astrometry between bands currently being delivered consistently betters both the Drop-Dead and Science Goals requirements and is flirting with the Enhanced Goals requirement in all but the u' band, even for the aforementioned run 1449, as shown in the following table. (The numbers here are taken directly from the calibrated object files for stars brighter than 19th magnitude.) Relative (CCD-to-CCD) Astrometry
Run 1449: Relative (CCD-to-CCD) Astrometry (mas)
It is worth noting, however, that while these formal results are encouraging, they hide much that is interesting if not troubling. The next figure shows the distribution of offsets in Right Ascension and Declination between the r' and g' CCDs in column 3 for run 1449. The distribution of errors appears to be nicely behaved.
However, the next figure shows the same offsets as a function of RA for about one degree (in RA) of the scan. While Run 1449 was observed on a great circle with a modest amount of inclination to the equator (22.5 degrees), changes in RA correspond quite closely to changes in clock time at nearly a sidereal rate (1 degree of RA is nearly 4 minutes of scanning). Note the swings on the order of +/- 100 mas over short time scales. We believe, though are not 100% convinced, that this is the atmosphere at work. We continue to investigate these problems.
The astrometric pipeline has changed rather significantly as a result of the commissioning phase of the Survey. While we hope that the pipeline is now quite stable, we are not naive enough to think that our job is done. As we encounter different observing conditions we may well need to make additional changes.
In addition to unanticipated modifications, there is one area where we need and intend to make changes in the next few months. As mentioned above, a fundamental problem in attempting to evaluate astrometric performance is that there is no "truth table." There is no wide-area astrometric catalog which goes faint enough with which we can directly compare SDSS positions. What we can do, however, is to compare the positions of objects determined on adjacent scans to get some measure of the internal consistency, if not the accuracy, of SDSS astrometry (there is a fair amount of overlap throughout the Survey regions, something like 40% of the Survey area will be imaged twice, not counting any duplicate scans). We can also compare SDSS astrometry with that of other wide-area surveys (e.g. digitized scans of Palomar Observatory Sky Surveys, 2MASS, etc.), though none of these surveys can claim sufficient astrometric precision to do more than assure us that our errors are not gross. At present, we are performing such tests in an off-line fashion. We need to integrate such QA tools into the pipeline for routine execution.
A possible means of improving the astrometry might be to change the transformations. The astrometric solution is modeled as a linear transformation from pixel coordinates to celestial coordinates, one transformation per CCD per frame. High-order terms model optical distortions (as a function of column) and DCR as a function of color. Various systematic effects cannot be completely removed by the adopted model. Typical seeing introduces systematics of order 20-50 mas. The rms error measure is, in fact, dominated by systematic errors due to seeing. DCR in the g' filter can introduce systematic errors of up to 20 mas with color. Optical distortions can also introduce systematics of the same order. CCD charge transfer inefficiencies are also problematic and are currently under active investigation. We are currently considering whether it would be worthwhile to adopt a non-linear transformation to cope with these systematics.
When faint, all-sky astrometric catalogs, such as UCAC, become available, it will be possible to determine the transformation from photometric CCD row/column to celestial coordinates directly, by-passing the mapping step from astrometric CCDs to photometric. Atmosphere aside, it is the astrometric CCD mapping which is the largest contributor to our error budget. This change should decrease our astrometric residuals for at least two reasons: * the density of stars increases dramatically with magnitude so that the new catalog contains many times the number of reference stars thus allowing us to better track atmospheric excursions and,
* the differences in integration time for the astrometric and photometric CCDs will cease to be an issue.
When UCAC is completed in the Northern hemisphere (this should happen in 2003), it will be a straight-forward exercise to re-derive the astrometric transformations from the new catalog (software is already in the pipeline to do this). Since the positions of objects in the databases are stored in device (CCD row/column) coordinates, it is a simple matter to generate new celestial coordinates by merely changing the transformation coefficients which are applied to CCD row/column to generate celestial coordinates.