4.2.1
κσ-clipping
Whenever averages of presumably normal distributions have to be calculated, i.e. to correct for
additive constants or to divide by normalisation factors, iterative rejection of “outliers” (κσ-
clipping) helps to reduce systematic errors. Asymmetric outliers can be rejected even more
effectively by clipping relative to the median (and not the mean) of a sample. Whenever asym-
metric outliers are a probable contaminant (e.g. particle events in the overscan region evaluated for the bias level), we apply median clipping, but, as a final iteration, calculate the average.
4.2.2
Bad-Pixel Mask
We mask saturated (and blooming affected) pixels, as well as CCD-defects (hot, cold pixels etc.).
While saturated pixels will be flagged as a dominant error the CCD-defects will be corrected later (Sect. 4.2.8).
52 4. Data Reduction
4.2.3
Bias Correction
We subtract the bias level of individual frames estimated from the overscan region and (for
CA 1.23 images) a master bias, i.e. theκσ-clipped mean image of multiple bias level corrected
bias frames. The bias pattern for MONICA/WST (Sect. 2.2.2) images varies over very short
timescales (already between two consecutive exposures). This is because of the strong immission from a nearby radio station, so no master bias is applicable there.
4.2.4
Gain
To calculate with more meaningful numbers we multiply the flux [ADU] with the “gain” (electronsI
ADU )
of the detector system. So we have numbers in units of electrons or detected photons to start with.
A propagated error (next paragraph) can now always be compared with a naive √flux error.
4.2.5
Initial Error Estimate
The initial error estimate δI(x,y) for each pixel (x,y) in every image I is calculated from the
pixel’s photon noise1, the bias noise of the image (clipped r.m.s. of the overscan), and the uncer-
tainties of bias level and bias pattern determination.
δI(x,y)= s
countsI(x,y)−biasI +σ2bias
I +
σ2 biasI
nbiasI
,where (4.1)
countsI = flux of pixel (x,y) in imageIine−,
biasI = bias of the image,
σbiasI = we use theκ(=3)σ-clipped r.m.s. of a
suitable part of the overscan as an estimation for the bias noise (i.e. readout noise),
nbiasI = number of pixels actually used for the bias
determination.
Errors are propagated throughout the complete reduction pipeline with Gaussian error prop- agation.
4.2.6
Flat field Calibration
The relative sensitivity of resolution elements (pixels) is calibrated by applying a flat field cali- bration.
Since the time to get twilight flat field calibration images is very limited (< 1h) we also used
short timed exposures where the movement time of non-photometric shutters can no longer be
1“NOISE, n. A stench in the ear. Undomesticated music. The chief product and authenticating sign of civ-
ilization.” [Bierce, 1906] With the noiseN being the denominator of the signal-to-noise (S/N) ratio civilisation greatly hampers scientific insight as can be seen in the problems of optical astronomy with light pollution or radio astronomy with radio broadcasting.
4.2 Standard Image Processing 53
neglected. Whenever the shutter movement could be proven to have a predictable time depen- dency, the flat fields were deconvolved from the two-dimensional shutter function as proposed
by Surma [1993]2.
To achieve a high signal-to-noise ratio (S/N) for a combined flat field of an epoch we first
calculate in each pixel the error weighted mean of normalised and illumination corrected twilight
flat fields. After rejecting all 5×5 pixels regions where the centre pixel exceeds this mean by
more than 5σ, the final calibration image is built by 3σclipping of the remaining pixels. The il-
lumination correction is applied twice: Per filter references-of-the-epoch are selected containing as few as possible stars but nevertheless having a comparatively long exposure time. All flat field images of a single epoch are then transformed to the reference illumination pattern by dividing
through a 25-parameter 2D polynomial fit to a rigorously clipped flat field/reference ratio. This
procedure is repeated adjusting all flat field epochs to an unique illumination pattern in order to
minimise illumination effects on the final images.
4.2.7
Cosmic-Ray Rejection
We fit five-parameter (and three-parameter3) Gaussians to preselected4 local maxima of an im-
age. Sources with a full-width-half-maximum (FWHM) along the minor axis of the fit function smaller than a threshold (chosen according to the best, i.e. sharpest, encountered point-spread- function, PSF) and, in addition, an amplitude of the fit function exceeding the expected noise by a certain factor (chosen according to the additional noise i.e. due to crowding) correspond to
cosmics5. We replace the pixels, where the fit function exceeds the fitted surface constant by
more than two times the expected photon noise with the fitted surface constant. Additionally a standard spatial median filter is applied, i.e. we replace pixels which exceed the median of the directly adjacent pixels by more than a suitable factor times the expected noise. In order to mark the “erased” cosmics as a “guess-of-flux” they get the geometric sum of their former error and
the proposed naive √e−noise assigned as new error.
4.2.8
Approximation of bad pixel areas
Bad pixel areas would spread to larger areas after aligning dithered frames to a common po- sitional reference and especially deep stacks, combining many images, would hence look like Swiss cheese. Therefore previously masked bad pixel areas have to be interpolated before align- ment and stacking somehow. If all images had the same depth, background and PSF they could
2See also Riffeser [2006], Sect. 5.5.5.
3Most “cosmics” leave a “trail” on the detector. However, there are also many “circular hits”, and those yield a singular matrix if centred on a pixel when solving a Levenberg-Marquardt algorithm [Levenberg, 1944, Marquardt, 1963] with too many degrees of freedom. Those cases are re-fit with only surface, amplitude, and “overall” width, neglecting the second width and angular orientation parameter of the five-parameter fit.
4Maxima have to exceed the minimum adjacent pixel by a factor depending on their propagated error. They will still be ignored if they have more than four masked neighbours.
5Actually, those “cosmics” are mostly due to particle events in the detector itself and its surroundings, and only partly due to real high energy cosmic radiation.
54 4. Data Reduction
easily be combined. Unfortunately, this is not at all applicable for ground based observations at moderate sites. We therefore replace all bad pixel areas, but saturated ones, with a distance and error weighted linear approximation of the closest neighbours. The fit box is selected as small
as possible with the restriction that more than 2/3 of the fit box pixels minus the centre pixel
must be valid pixels. Interpolated pixels get comparably large errors assigned derived from the supporting pixels input errors and the quality of the fit. As with the case of cosmics this helps to keep track of “guessed” fluxes and minimises its impact further on.
4.2.9
Astrometric Alignment
To register frames for astrometric alignment suitable sources have to be identified in the DG fields: In a first step the brightest not saturated sources besides those in regions of high crowding
have been selected. After going through all the next steps up to convolved difference images
and even variability masks (see next Sections) sources with parallaxes and/or proper motion are
identified and disregarded. Unfortunately many of the brightest sources are “nearby” and have shown considerable proper motions over the more than five years of observations, especially
in the fields at higher galactic latitudes6, leaving only few suitable sources. The actual cross
identification had to be done manually for every frame by pinpointing at least two reference stars.
All frames are then shifted to a common positional reference frame for each observed field:
A linear projection proves to be sufficient7. To achieve sub pixel alignment without degrading
the images we redistribute the flux by fitting a suitable polynomial to the actual flux distribution
with integral flux conservation as a side condition as described in Riffeser [2006, Sect. 5.8.6].
We link the references grid to celestial coordinates by identifying USNO stars [Monet, 1996, 1998] and calculating a WCS solution according to the FITS standard [Greisen and Calabretta, 2002, Calabretta and Greisen, 2002].
4.2.10
Signal-to-Noise Optimised Stacks
When building our per-epoch stacks, as outlined in Sect. 2.5, we maximise the S/N ratio by
applying individual weightswto the input frames according to
w= f
(FWHM·δsky)2 ,with (4.2)
f = the flux of a bright and stable “standard” source in the field8,
FWHM of the position reference PSF, and
δsky = the clipped median of the “error” image.
While f scales with the overall atmospheric and instrumental throughput, i.e. the “signal”, the
6For obvious reasons: While the galactic disk acts almost like a rigid rotator and therefore the “disk” stars more or less corotate with the sun, halo stars follow “random” motions.
7Because the image planes of both, Wendelstein 0.8 m and Calar Alto 1.23 m, RC-configuration telescopes can be considered flat and undistorted for the rather small field-of-view of the cameras used.