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Q: What is the optimal number of dither positions to use?
A: There is no single answer to this question. The best choice for the number of dithers depends on the amount of time available and the goals of the project. However, in general, the user can be sure that a single (two position) dither -- from the original pixel position of (0,0) to one offset by half a pixel in both x and y, (0.5,0.5), -- will produce a substantial gain in spatial information. On the other hand, little extra information is gained from obtaining more than four positions, if the standard four point dither is used, and if the telescope has successfully executed the dither. Therefore the recommended number of dither positions is between 2 and 4.
Q: What is the chance that the telescope will accurately perform the dither operation?
A: We do not yet have good statistics on this subject. However, during the observations of the Hubble Deep Field, the telescope was commanded to nine different pointings, with the true number of moves closer to twenty. The dithers commanded were generally substantially larger than those that will be requested by the average user. Seven of the nine pointings were within 10 mas of the requested position, while two pointings were off by more than 25 mas. In general the telescope reacquired these positions without any noticable offset; however, on two reacquisitions the telescope moved to pointings a substantial fraction of an arcsecond from the requested position. On these two occasions the telescope also showed rolls of about 5 arcminutes from the orientation seen in the other pointings. The telescope did not return to the correct pointing until the next full acquisition. There has been a GO proposal performing a three position dither and two images per position and all frames were within 10 mas of the requested position.
Q: What happens if the telescope rolls between my dithered observations or does not point exactly at the requested position -- will I still be able to use the data?
A: The pointing and orientation of the telescope can be determined directly using a sensitive cross-correlation technique that we have developed. Usually, however, one only needs to determine the shift between images. The chance of a roll appears small -- many long programs have been performed without experiencing the roll seen in the HDF. Software developed by Andy Fruchter and Richard Hook for the HDF is capable of combining rotated, as well as dithered, data. This software program, known as "Drizzle", can be installed under IRAF/STSDAS. You can obtain more information on this package and dithering in general by going Andy Fruchter's Dither Page. The Drizzle code will be included in the next general release of STSDAS, expected in late July 1997.
Q: The WFPC2 suffers from geometric distortion, and as a result pixels near the edges are smaller than those in the center of the field. Doesn't this mean that the dither varies across the field? How does this affect the standard recommended dithers?
A: The pixels near the edge of the field do indeed differ in size on the sky from those near the center. Thus a shift of (10,10) pixels at (400,400) corresponds to a shift of about (10.2,10.2) pixels at (700,700). The default dither-line spacing produces a shift of (2.5,2.5) WF pixels and (5.5,5.5) PC pixels. Therefore, over nearly the entire field of view the difference in offset -- even on the PC -- is less than 0.1 pixels, and the shift will be essentially optimal across the whole field. However, the standard dither-box spacing offsets the telescope by as much as 0."75 or 15.5 PC pixels. This means that at (700,700) the shift differs from that at the center by ~0.3 pixels in x and y. While the "drizzling" software developed for the HDF removes this geometric offset, it cannot change the fact that the sampling will not be optimal across the entire field of view. The dither-box defaults were chosen to avoid repeating the placement of objects on the same columns (to reduce the effects of bad columns). However, if one is willing to live with the possibility that a given position of interest may fall twice on one of the several bad columns per chip, then one can use smaller scans to produce a box that is more nearly perfect across the entire chip. For instance, to observe the field of Omega Cen using a square 2x2 box with side of 2.5 WF pixels (equivalently 5.5 PC pixels), with two 40s exposures at each corner of the square (ie a total of 8 CCD frames) one could use the following RPS2 Spatial Scan information:
Scan_Data Scan_Number: 1 FGS_Scan: Cont_or_Dwell: D Dwell_Points: 2 Dwell_Secs: 40 Scan_Width: 0.250 Scan_Length: 0.250 Sides_Angle: 90.0 Number_Lines: 2 Scan_Rate: First_Line_PA: 270 Scan_Frame: S/C Length_Offset: 0.0 Width_Offset: 0.0 Visit_Number: 01 Visit_Requirements: On_Hold_Comments: Visit_Comments: . . . . Exposure_Number: 100 Target_Name: OMEGA-CEN-1 Config: WFPC2 Opmode: IMAGE Aperture: WF4 Sp_Element: F555W Wavelength: Optional_Parameters: CR-SPLIT=NO,ATD-GAIN=7,SCAN-READ=ALL Number_of_Iterations: 2 Time_Per_Exposure: 40S Special_Requirements: SPATIAL SCAN 1 Comments:
Q: Dithering naturally provides many images of the same field. Do I still need to take several images at a single pointing to remove cosmic rays?
A: In principle, it should be possible to do a fairly good job in removing cosmic rays using only dithered data. And we are making good progress at developing code, using the "Drizzle" program which is capable of doing this. However, this code is still under development. Furthermore, it is not clear if this code will ever be able to obtain better than ~ 2% photometry on point sources. And in all cases, use of this code will require significant more work than standard cosmic ray rejection using aligned images. Therefore, at present, we recommend that the user consider taking AT LEAST two, and preferably more images at each pointing. A rule of thumb is that about 4% of pixels are hit by cosmic rays in a 2000s exposure. The number of pixels affected is linear with dark time (which is nearly equal to the exposure time). You can therefore use the binomial theorem to estimate the number of pixels per chip that are likely to be affected in a given number of images.
Q: The sub-pixel positions for two and four point dithers are clear, but what should I do if the time available naturally divides itself into a three-point dither?
A: The best placement of a three point dither is somewhat controversial. This is because there is no natural way to tile the plane using three placements of a rectangular grid (the CCD). We therefore generally recommend a two or four point dither. A calculation performed by A. Fruchter suggests, however, that if the user wants to do a three point dither, the best sub-pixel placements are along the diagonal at (0,0), (1/3,1/3) and (2/3,2/3) pixel offsets. (The symmetric diagonal works just as well, of course.) The dither-line option in RPS2 does not properly handle CR-SPLIT when more than two dither positions are used. One can surmount this problem by setting CR-SPLIT=NO, and setting Number_of_Iterations to 2.
Q: What software is available for analyzing dithered data?
A: If the dithers are small enough that the effects of geometric distortion can be ignored, then simply shifting and adding the images (on a sub-sampled grid) will gain much of the information that can be obtained through a linear reconstruction technique (reconstruction refers to recreating the image after it has been convolved by the instrumental PSF, including the pixel response function, deconvolution refers to attempting to remove the effects of the PSF on the ideal image). The "drizzling" technique can handle large dithers where geometric distortion is important, and also does a more sophisticated linear reconstruction, which can allow one to gain a bit in resolution over shift-and-add (the typical gain is about 15% in the final FWHM of the PSF). Furthermore, the "correlated noise" in the Drizzled image, will typically be much smaller than that in an image produced by "shift-and-add".
Small regions of dithered images can be deconvolved using the Lucy-Hook version of the Lucy-Richardson algorithm (see acoadd in stsdas.contrib). Memory limitations, and changes in the shape of the PSF over the WFPC2 field, presently prevent this routine from handling full WFPC2 dithered images.
Q: What are reasonable dither strategies for cosmic rays, warm pixels, undersampling?
A: There is no single observing strategy that is entirely satisfactory in all circumstances for WFPC2. One must consider cosmic rays, hot pixels (i.e. pixels with high and time variable dark count), spatial undersampling of the image, and trading signal-to-noise for ability to recognize and deal with these features. The optimal strategy chosen depends crucially on the scientific question: is the underlying structure totally unknown, is spatial resolution of paramount importance, is photometric accuracy the most crucial aspect, etc.?
1) Cosmic Rays: The best way to deal with cosmic rays is to CR-SPLIT the exposures (take multiple exposures at a FIXED image location). Note that even with two exposures taken at a fixed position there will be some cosmic rays that overlap. As an example, for an observer that has two 2000s exposures, about 1000 pixels per chip will be unrecoverable because they have been hit in both images. Furthermore, because CR events typically affect ~7 pixels per event, these pixels will not be independently placed, but rather will frequently be adjacent to other unrecoverable pixels.
2) Hot Pixels: There are three ways to deal with hot pixels: correct using "dark frames" that bracket the observation (presently obtained weekly), obtain a second image (or pair of images to best reject cosmic rays) shifted by a small amount spatially (e.g. about 5 pixels), or use a program such as 'cosmicrays' in IRAF to filter out the obvious hot pixels.
3) Undersampling: In order to maximize spatial resolution, an observing strategy that is being used by a number of observers is to shift images by sub-pixel amounts. In principle, the information provided by this method can be used to minimize the problems of undersampling and obtain a higher spatial resolution than from a single location image.
4) Sensitivity Variation: There is a variation in the sensitivity across each individual pixel. Since the PSF is undersampled, this can limit the photometric accuracy (and also helps explain why optimal cosmic ray reject is not consistent with sub-pixel shifting).
For related articles on dither strategies, see the January, 1995 issue of the WFPC2 Space Telescope Analysis Newsletter and the February, 1995 issue of the ST-ECF Newsletter. Also, the following two FAQ items address the issue of dithering.
Q: I want to dither my exposures by exactly integral pixel amounts. What is the exact relationship between the POS TARG's we specify in the proposal and the CCD rows and columns?
A: The POS TARG axes run exactly along the CCD rows and columns on the specified aperture. For example, if you specify aperture WF3, the POS TARG axes will run *exactly* along the rows and columns on WF3. For the other CCD's the POS-TARG's will run only approximately along rows and columns, since there are small rotations (<0.5 degree) of the CCD's from their nominal alignments. Note that if WFALL is specified, then the rotation for WF3 is used. For small dithers (<0.3 arcsec.) the rotations between CCD's are unimportant, as they imply pixel registration errors less than 3 milliarcseconds, which is roughly the nominal pointing and guiding stability. But such small dithers do not allow integral pixel stepping simultaneously on both the WFC and PC. A dither of 0.5 arcseconds (5 WFC pixels or 11 PC pixels) gives near-integral stepping on both the WFC and PC, though the CCD rotations will then introduce registration errors up to 5 milliarcseconds. For more detailed information please see the report Dithering: Relationship Between POS TARG's and CCD Rows/Columns.
Q: How accurate are dithers between observations?
A: For observations within a single visit of less than 8 orbits, the dither accuracy is about 3 milliarcseconds. For programs exceeding 8 orbits, or for different visits to the same target, position errors up to 500 milliarcseconds and field rotations up to ~0.1 degree can occur, although experience indicates the errors are typically tens of milliarcseconds.
Note that large dithers will incur other errors. The camera distortion increases with strength toward the CCD corners, and alters the image scale by about 2% at the corners. Hence a 1.993 arcsecond dither will be 20.0 WFC pixels at the field center, but suffer a 0.4 pixel error at the CCD corners. The individual CCD's are misaligned by up to ~0.5 degrees from their nominal orientations, and again, this implies errors when attempting to dither by certain pixel amounts. A POS TARG = 1.993, 0.000 arcsecond dither in X on WF3 would cause spurious motion in Y of 0.17 pixel on WF4, due to the rotation. Large dithers may also require a different set of guide stars, and then the pointing accuracy is only that of the guide star catalog (~1 arcsecond).