2 Observations and reductions

   
2.1 Observations

B, V, I observations were taken on the Canada-France-Hawaii telescope with the UH8K mosaic camera (Metzger, Luppino, & Miyazaki Metzger et al. 1995) over a series of runs from December 96-June 97; details are given in Table 1. Typically, for the VI-band exposures we used exposures of 1800 s; for the B-images we adopted exposure times of 2400 s. Individual exposures which had a FWHM >1.2'' were discarded. Observing conditions were generally quite stable: for example, for the 03 hr field observations In V and I bands, the median seeing is $\sim$1.1'' and 1.2'' respectively. Additionally, as our point-spread function (PSF) is almost always oversampled, it is not necessary to carry out PSF homogenisation before image stacking. At each pointing there is $\sim$10 exposures which allows us to carry out adequate cosmic ray removal and to fill the gaps between each CCD in the mosaic.

The UH8K camera consists of eight frontside-illuminated Loral-3 $2048\times4096$ CCDs, arranged in two banks of four devices each. Each bank is read out sequentially. The upper-right CCD (number 8) has very poor charge-transfer properties and data from this detector was discarded. The pixel scale is $0.205''\$pixel^-1. Additionally, all of the CCDs have separate amplifier and controller electronics. This arrangement, in addition to the necessity of removing the CFHT-prime focus optical distortion before stacking our images, resulted in a lengthy data reduction procedure which is outlined in the following sections.

Because of the poor blue response of the Loral-3 devices, U-band observations for the CFDF survey were taken with the Kitt Peak 4 m Mayall telescope and the Cerro Tololo Inter-American Observatory's (CTIO) 4.0 m Blanco Telescope during a series of runs in 1997. Because of the smaller field of view of these cameras, four separate pointings were needed to cover each UH8K field. While the seeing on the BVIframes is $\sim$ 0.7''-1.1'' some of the U-stacks are significantly worse ( 1.2''-1.4'') which had to be accounted for during catalogue preparation (two catalogues were prepared: one for those science objectives which required U-data and one for those which did not; this is explained in more detail in Foucaud et al.).

 

 

Table 1: Details of the images used in the CFDF fields. For each field we list the total integration time, in addition to the $3\sigma$ detection limit inside an aperture of 3''. For the 22 hr and 11 hr fields, the fields are composed of two separate stacks with bonette rotations, as detailed in the text. For the 22 hr field the U-data covers only half the field.

Field	RA (2000)	Dec. (2000)	Band	Exposure time	Seeing	$3\sigma$ measurement	Area	Date
				(hours)	(arcsec)	(AB mags)	(deg2)
0300+00	03:02:40	+00:10:21	U	10.8	1.0	26.98	0.25	06/97
			B	5.5	1.1	26.38	0.25	09/97
			V	4.2	1.3	26.40	0.25	12/96
			I	5.5	1.0	25.62	0.25	12/96
2215+00	22:17:48	+00:17:13	U	12.0	1.4	27.56	0.12	06/97
			B	5.5	0.8	25.90	0.25	09/97
			V	2.7	1.0	26.31	0.25	06/97
				1.5				09/97
			I	1.7	0.7	25.50	0.25	06/97
				2.7				09/97
1415+52	14:17:54	+52:30:31	U	10.0	1.4	27.71	0.25	03/97
			B	5.3	0.8	26.23	0.25	06/97
			V	2.3	1.0	25.98	0.25	06/97
			I	2.6	0.7	25.16	0.25	06/97
1130+00	11:30:02	-00:00:05	V	3.3	0.8	26.42	0.25	05/97
			I	4.4	1.0	25.80	0.25	12/96
				3.0				05/97

   
2.2 Preprocessing

Pre-processing followed the normal steps of overscan correction, bias subtraction and dark subtraction. We fit a fourth-order Legendre polynomial to the overscan region to allow us to remove structure in the overscan pattern. The high dark current ($\sim$0.1e^- s^-1) of the UH8K makes it essential to take dark frames. For I- and V-bandpasses, where the sky background is high, one unique dark frame (composed of the average of 5-10 individual exposures) can be used; however, with the B-band special steps have to be taken due to the "dark-current jumping'' effect which manifests itself as a either "high'' or "low'' dark current level, which can affect either right or left banks of CCDs independently. Because of the low quantum efficiency of the UH8K CCDs in B the dark current is a significant fraction of the sky level and consequently accurate dark subtraction must be performed to produce acceptable results. We achieve this by generating two sets of darks: "high'' darks and "low'' darks which we apply by a trial-and-error method to each B-exposure to determine which dark is appropriate for a given dataset. The high and low darks are identified by their statistics (mean, median). Two full reductions are done. Each kind of dark is subtracted and then a dark-independent flat (dome or twilight) is applied. The flatter final image (with smallest amplitude of residual flatness variations) indicates the kind of dark that was actually present in the data.

We generate "superflats'' from the science images themselves as dome flats or twilight flats by themselves produce residual sky variations >$1\%$. These superflats are constructed by an iterative process, which begins by the division of our images by a twilight flat. On these twilight-flattened images we run the sextractor (Bertin & Arnouts 1996) package to produce mask files which identify the bright objects on each frame. For large saturated stellar objects we grow these masks well into the wings of the point-spread-function by placing down circles on these objects. In addition, we mask out non-circular transients (typically scattered light and saturated columns near bright stars) on some images. Using these mask files we combine each image to produce a superflat each pixel of which contains only contributions from the sky and not object pixels. After the division of the twilight-flattened images by the superflats, the residual variation in the sky level is <$1\%$.

Because the gain and response of each CCD in the mosaic is not the same, we scale our flat fields for each filter so that the sky background in each chip after division by the flat field is the same (normally these exposures are scaled to chip 7). In Sect. 2.5 we will quantify how successful this procedure is in restoring a uniform zero-point over the entire field of view of the image.

   
2.3 Astrometric image mapping

For each set of observations in each filter of our field, we have typically $\sim$10 pointings (we use the term "pointing'' to refer to the eight separate images which comprise each read-out of the UH8K camera). Each of these pointings are offset by $\sim$5 '' from the previous one in a random manner; these offsets allows us to remove transient events and cosmetic defects from the final stacks, and also to ensure that the gaps between the CCDs (which are $\sim$3'') are fully sampled. Additionally, on some of our fields, pointings were taken over several runs with the camera bonette in different orientations. Given the non-negligible optical distortion at the CFHT prime focus (amounting to a displacement of several pixels at the edge of the field relative to an uniform pixel scale) this means it is essential that these distortions are removed before the pointings can be coadded to produce a final stack. A further requirement is that each of the stacks for each of the filters can also be accurately co-aligned, for the purposes of measuring aperture colours reliably.

In the mapping process, the images from each CCD are projected onto an undistorted, uniform pixel plane. The tangent point in this plane is defined as the optical centre of the camera, and is the same for each of the eight CCDs. Overall, our goal is to produce an root-mean-square registration error between pointings in each dither sequence and between stacks constructed in different filters which does not exceed one pixel (0.205 $\hbox{$^{\prime\prime}$ }$) over the entire field of view.

Our astrometric mapping process is essentially a two-step process. In outline, this involves first using the United States Naval Observatory (USNO) catalogue (Monet 1998) to derive an absolute transformation between (x, y) (pixels) to  $(\alpha,\delta)$ (celestial co-ordinates). Following this, a second solution is computed using sources within each field. This method ensures that our pointings are tied to an external reference frame, but also provides sufficient accuracy to ensure that pointings can be registered with the precision we require; the surface density and positional accuracy of the USNO stars is too low to ensure this. To fully characterise the distortions which are present in the camera optics we adopt a higher order-solution, which consists of a combination of a standard tangent plane projection and higher-order polynomial terms. To prevent solution instabilities at the detector edges we use a third-order polynomial solution. To compute the astrometric solutions and carry out the image mapping we use the mscred package provided within the IRAF data reduction environment.

For each field we begin our procedure with the I-band, as these exposures normally have the highest numbers of objects. Using the external catalogue we compute an astrometric solution containing a common tangent point for each of the eight CCDs. Typically, we find 50-100 sources per CCD, with a fit rms of $\sim$0.3''. Next, using the task mscimage we project each of the eight images onto the undistorted tangent plane, using a third-order polynomial interpolation. Following this, we extract the positions in celestial co-ordinates of a large number ($\sim$1000) of sources distributed over all eight CCDs. This list forms our co-ordinate reference, and we use this list in conjunction with the task mscimatch to correct for the adjustments in the WCS (world co-ordinate system) due to slight rotation and scale change effects for each successive pointing in the dither. Before stacking we also remove residual gradients by fitting a linear surface and scale the images to photometric observations if necessary. Because each image now has a uniform pixel scale we need only apply linear offsets before constructing the final stack. Setting the gaps between each CCD to large negative values which are rejected in the stacking process allows the production of a final, contiguous image. To stack our images we use a clipped median, which although not optimal in signal-to-noise terms, provides the best rejection of outlying pixels for small numbers of pointings.

From this final, combined stack, we extract a second catalogue of ($\sim$1000) sources (with $\alpha$, $\delta$ computed from the astrometric solution) which we use as an input "astrometric catalogue'' for the dither sequences observed in other filters (as opposed to the USNO catalogues which we use for the first step). Typically, the rms of the fit these cases is <0.1''. We then proceed as before, mapping each of CCD images from each pointing in the dither set onto the undistorted tangent plane and constructing a final stack.

For the final mapping between the stacks taken in different filters, we find a residual of $\sim$0.06'', or $\sim$0.3 pixels over the whole field of view, which is with our aim of a root-mean-square of one pixel or less; this is illustrated in Fig. 1. Our large grid of reference stars extracted from the I-band image ensures that the derived WCS for the other filters is very well matched to the I-band exposure. We also find that this method allows us to successfully register and combine observations distributed over separate runs containing bonette rotations.

Figure 1: The difference, in pixels, of the positions of non-saturated stellar sources with 18 < I < 22 in the 03 hr I-band stack compared with the 03 hr B-stack. The rms in both co-ordinate directions is $\sim$0.3 pixels, or $\sim$0.06''.

For the U-band exposures, each of the four corners were stacked separately and scaled to have the same photometric zero-point. Then, using the I-band reference list described previously, a mapping was computed between each of the corners and the undistorted I-stack. During this process the image was also resampled (using the same third-order polynomial kernel employed above) to have the same pixel scale as the UH8K data. These four U-images were then stacked to produce the final U-mosaic. Overall, we find that the rms of the mapping between I- and U- is not as good as between the UH8K data, with an rms $\sim 1$ pixel in the region of CCD 8 (the CCD suffering cosmetic defects) but still within our stated goal.

Each of the final images have a scale of 0.204 $\hbox{$^{\prime\prime}$ }$/pixel and cover $\sim$ $8000 \times 8000$ pixels (the scale of the final stack is determined from the linear part of the astrometric solution of the image which is closest to the tangent point of the camera). In all the analyses that follow we exclude the region covered by CCD 8 as this chip has very bad charge transfer properties and is highly photometrically non-linear. However, for the 11 hr-I and 22 hr-I stacks we are able to use the full area of UH8K because these final stacks consist of two separate stacks with bonette rotation, allowing us to cover the region lost by the bad CCD.

   
2.4 Calibration

 

 

Table 2: The (l,b) for each CFDF field, together with the galactic dust extinction corrections from Schlegel, Finkbeiner, & Davis Schlegel et al. (1998)(^a) compared to the values of Burstein & Heiles (1982) (^b, BH).

Field	b	l	E(B-V)a	E(B-V)b
			Schlegel et al.	BH
0300+00	-48	179	0.071	0.040
2215+00	-44	63	0.061	0.040
1415+52	+60	97	0.011	0.000
1130+00	+57	264	0.026	0.013

In this section we will describe how we derive the relationship between magnitudes measured in our detector/filter combination (which we denote by u_cfdf, b_cfdf, v_cfdf, i_cfdf) and the standard Johnson UBVI system. Our zero-points are computed from observations of the standard star fields of Landolt (1992). Of the four observing runs with UH8K which are discussed here, only the data from May were not photometric and for the runs of June and October sufficiently large numbers of observations of standards were taken ($n\sim 30$) it was possible to determine the colour equation.

We apply the same data reduction procedure to the standard star fields as we do to the science frames. This involves bias and dark subtraction, followed by flat-fielding which is necessary to account for the sensitivity and gain variations from CCD to CCD and the application of the astrometric solution derived previously to produce a single, undistorted image. This procedure assures that a uniform pixel scale is restored when computing the photometric zero-point, and has added advantage of that we may use a catalogue of standards in  $\alpha,\delta$ to derive the zero-point in a semi-automated fashion. All standards used were visually inspected and faint or saturated objects were rejected. Our zero-points are corrected for galactic extinction using the E(B-V) values provided by Schlegel, Finkbeiner, & Davis Schlegel et al. (1998).

In Fig. 2 we show sample plots of standard star observations taken during the October observations. For the I and V band for all three runs we derive zero-point rms of $\sim$0.05magnitudes, with no evidence for position-dependent residuals (as would happen if an error had occurred in the flat-fielding process).

Our observations do not indicate that the presence of a colour term for either the V or I filters and in what follows we assume V=v_cfdf and I=i_cfdf. However, we find that b_cfdf is different from the standard Johnson B- and for this reason we derive colour terms.

$$b_{cfdf}\sim B-0.07(B-V)-23.2 .$$ (1)

We convert our magnitudes to the AB system using U_AB=u_cfdf+0.73, B_AB=b_cfdf-0.1, V_AB=v_cfdf and I_AB=i_cfdf+0.43 (computed based on our filter response functions).

Figure 2: Standard star calibration plots for the I and V filters for the October 97 data (upper and lower panels respectively). For each panel, we plot the difference between the standard star magnitude and the instrumental magnitude (corrected to one airmass) as a function of the true I,V magnitudes. The solid line shows the adopted zero-point; this is $24.24\pm 0.04$ (V) and $24.26\pm 0.06$ (I), per second and at one airmass. Atmospheric extinction coefficients are taken from the CFHT handbook, although most of our standards are at or near the zenith.

Figure 3: B-bcfdf vs. (B-V), based on observations of 51 standard stars during the October 1997 observations. The slope and offset of the fitted line is 0.07 and 23.26, respectively. The slope is not well determined due to the small range in (B-V) colours spanned by our observations and we expect errors of $\sim$0.02.

   
2.5 Systematic photometric errors

To measure accurately the galaxy clustering signal it is essential that the photometric zero-point is uniform across the stacked images. Zero-point variation across the mosaic will introduce excess power on large scales and contribute to a flattening of the $\omega (\theta )$ on large scales. For single-CCD images, improperly flattened data can produce this effect; in our case we have the additional complication that we must correctly account for the different responses and amplifier gains for the eight CCDs in our mosaic. As outlined above, this is accomplished by scaling each CCD image before co-addition to have the same sky background. Our standard star reductions detailed in Sect. 2.4 have already indicated that this procedure produces zero-point variations on order 0.05 mag rms. However, further observations allow a more rigorous test of this effect. On two separate occasions we have observed the same field (11 hr, 22 hr) after the camera bonnette had been rotated $180\deg$. These data provide an excellent opportunity to verify that there are no residual systematic magnitude zeropoint variations in our final stacks after co-addition and stacking have been carried out.

To carry out these tests we prepare two separate stacked mosaics. For the field at 11 hrs, we have 4.4 hrs of integration in I from December 1996 and 3 hrs total integration from May 1997. By using sources extracted from the December run to compute our astrometric solution following methods outlined above, we can produce final stacked mosaics which are aligned with a standard deviation of <1 pixel over the entire field of view. By carrying out the detection process on the sum of these two images, and photometry on the two separate mosaics, we are able to measure the difference in magnitude between sources located in the same part of the sky but falling on different elements of the detector-telescope system. Note that because we place our photometric apertures on the same positions on each of the two stacks, this test also allows us to investigate magnitude errors introduced by mapping inaccuracies between the two stacks, which are expected to be present for the measurement of aperture colours.

The results of this test are illustrated in Fig. 4 where we plot the difference in magnitude for non-saturated stellar sources with 18.5 < I_AB < 22.5 between the two 11 hr stacks as a function of position in both x and y directions. We find that the systematic magnitude errors, measured as the dispersion of these residuals is $\sim$0.04 magnitudes, which corresponds to the limit of our CCD-to-CCD calibrations, as explained in Sect. 2.4.

Towards the $50\%$ completeness limit of our catalogues, $I_{AB}\sim25.5$, differential incompleteness becomes a significant bias in the measurement of  $\omega (\theta )$. This effect arises from the differing read-out electronics and detector gains used in each of the eight individual CCDs in UH8K. Neuschaefer & Windhorst (1995), using the Palomar four-shooter camera, discuss this effect in more detail. However, we emphasise that all our scientific analysis is only carried out where our completeness is >80%, as determined from the simulations and source counts detailed in the following sections. Furthermore have verified that this effect is only significant at the faintest magnitudes by adding 40000 objects with the same clustering amplitude as galaxies at $I{_{AB}\sim25}$ to one of our images. This test is described in detail in Sect. 5.2.

Figure 4: The magnitude difference as a function of x and y position between non-saturated stellar sources with 18.5 < IAB<22.5 in the December 1996 11 hr stack and the May 1997 data covering the same field. We find $\sigma (\delta \rm _m)\sim 0.04$ magnitudes.

   
2.6 Random photometric errors

We may also use these repeated observations of the same field to investigate what random photometric errors are present in our data. At fainter magnitudes these errors dominate. We have used three separate methods to estimate the magnitude errors computed in our data; firstly, we may use the errors computed directly by sextractor; secondly, we can use the errors computed from the simulations detailed in Sect. 2.7 in which stellar objects are added to our fields and recovered; and lastly, we may use the our two independent stacks of the same field to estimate our errors.

Figure 5 shows magnitude errors for these three different estimators: sextractor (circles), the simulation (stars) and from the direct measurement (squares). We have carried out these tests on both the 11 hr stacks and the 22 hr stacks. In many magnitude ranges, the sextractor errors are lower than the other two measurements. We believe the origin of this discrepancy is due to the image resampling and interpolation process which produces images with correlated background noise. By contrast, the sextractor magnitude errors are computed assuming white background noise.

Figure 5: Root-mean-square (rms) magnitude errors as a function of IAB magnitude. The upper and lower panels shows measurements made on the 11 hr and 22 hr fields respectively. Magnitude errors were computed using a variety of techniques: from sextractor (open circles); from a simulation in which stars were added to random patches of the field (stars) and the dispersion in the recovered total magnitudes is calculated; and finally as measured between two independent stacks covering the same field (open squares). The direct magnitude error measurements have been multiplied by $\sqrt 2$ to account for the shorter exposure time for the individual stacks. At all magnitudes the sextractor errors appear to underestimate the both the direct and simulation error by at least a factor of two.

   
2.7 Incompleteness simulations and limiting magnitudes

In Table 1 we list the $3\sigma$ values for detection in a 3 $\hbox{$^{\prime\prime}$ }$ aperture. These should be regarded as lower limits on the detectability of the galaxies in our catalogues. To better characterise the photometric properties of our images we have carried out an extensive set of simulations. These simulations involved adding artificial stars and galaxies to our single-band images and measuring the fraction recovered as a function of magnitude. In Fig. 6 we show the results of one set of such simulations for the 03 hr field.

We note that this result should be regarded as lower limit to the completeness in our data as our actual catalogues are constructed using the chisquared technique described in Sect. 3.1 and can be expected to be slightly deeper (but note also that in constructing the chisquared catalogues all images must first be convolved to the worst seeing). From a rough comparison of the I-band galaxy counts presented in Fig. 9, we see that the we see that the simulations provide a good estimate of the magnitude at which the the observed counts begin to fall off.

Figure 6: Derived completeness in four bands for the 03 hr field as a function of AB magnitudes. These simulations were derived by adding 1000 artificial stars in each half-magnitude interval to the original images and measuring how many objects were recovered as a function of magnitude. The detection threshold used for this test (number of sigmas above the noise background and minimum number of connected pixels) were the same as used for the actual detections on the $\chi ^2$ image.

   
2.8 Comparison with CFRS photometry

Three of our fields (22 hr, 03 hr, 14 hr) cover the original survey fields of the Canada-France Redshift Survey (CFRS; Lilly et al. 1995). For the 22 hr and 14 hr field we have BVIphotometry from the CFRS; for the 03 hr field, data exists in the VIbandpasses.

For all these fields we have carried out a detailed comparison of our photometry with CFRS photometry. In Fig. 7 we compare V and I photometry from our stacked images with the CFRS for V and I filters in the 03 hr and 14 hr fields. For the 14 hr fields, the agreement with the CFRS photometry is <0.1 magnitudes or better. In the 03 hr field, however, we find that our magnitudes are $\sim$0.2 and 0.1 magnitudes brighter than CFRS magnitudes in the V and I filters respectively. We suspect the origin of this discrepancy is that the CFRS fields were selected to have low galactic extinction as measured in the maps of Burstein & Heiles (1982) (BH). In Table 2 we show that the difference between the BH extinction and the more recent E(B-V) values given in Schlegel, Finkbeiner, & Davis Schlegel et al. (1998) is non-negligible (amounting to $\sim$0.15 in I_AB magnitudes). In all our fields we apply extinction corrections based on E(B-V) values from Schlegel, Finkbeiner, & Davis Schlegel et al. (1998).

Figure 7: Comparison between CFDF and CFRS magnitudes measured in 3 $^{\prime \prime }$ aperture diameter for the V-band (right panels) and I-band (left panels). The $\sim$0.2 magnitude offset between CFRS and CFDF magnitudes for the 03 hr field is a consequence of the significant galactic extinction in this field, which the CFRS magnitudes have not been corrected for.