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## 9.6.1 The Uncertainty on the Flat Fields

We would like to calculate how well we are able to flat field our science images, i.e. what the uncertainty (or noise) on the flat fields are.

Given the read out noise in ADU, RON , the conversion factor in e-/ADU, CF, and the number of counts, N , the uncertainty on N is given by

 (9.7)

If we make an unweighted average of n images with individual uncertainties , the uncertainty on the combined image is

 (9.8)

The Gunn r night 2+ flats (of which there are 11), where combined in one round. The combination was made using combine with the following parameters different from the default: lsigma = 2, hsigma = 2, scale = mean, weight = mean, nkeep = 1, grow = 2. The intention with scaling was, that the different flats, which have different levels, should be scaled to a common level, so that sigma rejection can be used, after which they should be scaled back, and then made into a weighted average. Unfortunately, it seems that combine does not scale back, leading to the output being just an unweighted mean. The uncertainty should not be much higher due to this, since the individual flat field images have similar levels (within a factor of less than 4 at worst). It also means, that the uncertainty on the combined image is very easy to calculate.

Table  lists the levels in ADU, Ni, and the uncertainties on the individual flats, , with is calculated from Eq. () (using N = Ni). They also lists the scale factors, si, and the weights, ai, but these quantities do not enter the uncertainty equations.

Table: The levels and uncertainties on the individual Gunn r night 2+ flat fields.
 i exptime Ni si ai 1 30 2896.5 1.289 0.071 38.6 2 25 4046.9 0.922 0.099 45.6 3 20 5307.4 0.703 0.129 52.2 4 10 4127.3 0.905 0.101 46.1 5 60 3029.7 1.232 0.074 39.5 6 45 4004.2 0.932 0.098 45.4 7 25 3737.3 0.999 0.091 43.8 8 15 3447.3 1.083 0.084 42.1 9 10 3388.5 1.102 0.083 41.7 10 7 3500.6 1.066 0.085 42.4 11 5 3579.7 1.043 0.087 42.9

Notes: exptime is in seconds. Ni, si, and ai, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. was calculated from Eq. () using CF = 1.95 e-/ADU and RON = 2.25 ADU.

To calculate the uncertainty on the combined GR flat, we apply Eq. () to the individual uncertainties, (calculated from Eq. ), which yields

 (9.9)

What we really want is the relative uncertainty, , and since the Gunn r combined flat had a level before normalization of 3723 ADU, we get

 (9.10)

The Johnson B and U night 2+ flats (of which there are 12 and 13, respectively), were combined in two rounds: first in 3 groups of 4 or 5 images, and then these 3 images were combined. The combination on both levels were made with combine using the same parameters as for Gunn r (the so-called scaled weighted mean). The two-level combination was used to make sure that no objects (stars) would make it to the final flat field, since there was not applied any offset of the telescope between the individual flats from night 6 (where only JB and JU flats were taken). Table  and Table  lists the levels in ADU, Nij, and the uncertainties on the individual flats, , with calculated from Eq. () (using N = Nij). They also lists the scale factors, tij, and the weights, bij, but these quantities do not enter the uncertainty equations.

Table: The levels and uncertainties on the individual Johnson B night 2+ flat fields.
 i j exptime Nij tij bij 1 1 15 5665.3 0.844 0.296 53.9 1 2 30 4348.0 1.099 0.227 47.3 1 3 5 4840.9 0.987 0.253 49.9 1 4 17 4264.5 1.121 0.223 46.8 2 1 7 5064.0 0.871 0.287 51.0 2 2 30 2305.3 1.914 0.131 34.5 2 3 20 5632.8 0.783 0.319 53.8 2 4 10 4649.1 0.949 0.263 48.9 3 1 20 2174.1 2.271 0.110 33.5 3 2 20 3982.4 1.240 0.202 45.2 3 3 15 7823.4 0.631 0.396 63.4 3 4 7 5767.3 0.856 0.292 54.4

Notes: exptime is in seconds. Nij, tij, and bij, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. was calculated from Eq. () using CF = 1.95 e-/ADU and RON = 2.25 ADU.

To calculate the uncertainty the combined JB flat, we apply Eq. () to the individual uncertainties, , in each of the 3 groups, which yields

 = (9.11) = (9.12) = (9.13)

Using Eq. () one more time, we get

 (9.14)

The relative uncertainty is:

 (9.15)

Table: The levels and uncertainties on the individual Johnson U night 2+ flat fields.
 i j exptime Nij tij bij 1 1 25 4645.9 1.047 0.239 48.9 1 2 15 5130.9 0.948 0.264 51.3 1 3 18 4618.8 1.053 0.237 48.7 1 4 7 5065.0 0.961 0.260 51.0 2 1 8 4833.0 1.018 0.246 49.8 2 2 5 4935.4 0.997 0.251 50.4 2 3 9 4014.5 1.226 0.204 45.4 2 4 14 5898.6 0.834 0.300 55.0 3 1 3 4685.0 0.851 0.235 49.1 3 2 5 3731.4 1.068 0.187 43.8 3 3 3 3544.5 1.125 0.178 42.7 3 4 20 4546.8 0.877 0.228 48.3 3 5 3 3426.9 1.163 0.172 42.0

Notes: exptime is in seconds. Nij, tij, and bij, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. was calculated from Eq. () using CF = 1.95 e-/ADU and RON = 2.25 ADU.

To calculate the uncertainty the combined JU flat, we follow the same procedure as for JB:

 = (9.16) = (9.17) = (9.18)

and

 (9.19)

The relative uncertainty is:

 (9.20)

The levels used to calculate the relative uncertainties (Eq. , , and ), are the levels in the section [25:800,25:925]. This section is in some sense normal'', as opposed to the right and upper edge, which is more sensitive in the blue than the rest of the chip. The levels in the entire frames are 3733 ADU (GR), 4710 ADU (JB), and 4591 ADU (JU). If these values are used for calculating the relative uncertainty, we get 0.35% (GR), 0.30% (JB), and 0.30% (JU), so there is only a small difference.

Even though the chip is more blue-sensitive in the right and upper side, it is still linear in that area. This was checked using 10 JU dome flats with levels of ADU (dfsc1649-1658, exptime 10 sec), and 10 with levels of ADU (dfsc1659-1663, exptime 5 sec; dfsc1664-1668, exptime 7 sec). The high and the low level images were separately combined, and the quotient of the 2 resulting images was made. This quotient images had a gradient in the x-direction of < 0.05%, and in the y-direction of < 0.1%, which is very small. The quoted numbers are (maximum - minimum)/2.

The relative uncertainties are summarized in Table  (p. ).

Next: 9.6.2 Illumination Correction Up: 9.6 Flat Fields Previous: 9.6 Flat Fields

Properties of E and S0 Galaxies in the Clusters HydraI and Coma
Master's Thesis, University of Copenhagen, July 1997

Bo Milvang-Jensen (milvang@astro.ku.dk)