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 $_{\mbox{\scriptsize ADU}}$, the conversion factor in e^-/ADU, CF, and the number of counts, N $_{\mbox{\scriptsize ADU}}$, the uncertainty on N $_{\mbox{\scriptsize ADU}}$ is given by

$$\sigma_{N_{\mbox{\tiny ADU}}} = \left( \frac{N_{\mbox{\scrip... ...{CF}} + \mbox{RON}^2_{\mbox{\scriptsize ADU}} \right)^{1/2} .$$ (9.7)

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

$$\sigma = \frac{1}{n} \left( \sigma^2_{x_1} + \ldots + \sigma^2_{x_n} \right)^{1/2} .$$ (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, N_i, and the uncertainties on the individual flats, $\sigma_i$, with $\sigma_i$ is calculated from Eq. () (using N = N_i). They also lists the scale factors, s_i, and the weights, a_i, 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	$\sigma_i$
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. N_i, s_i, and a_i, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. $\sigma_i$ 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, $\sigma_i$ (calculated from Eq. ), which yields

$$\sigma_{\rm GR} = 13.21 \, {\rm ADU} \enspace .$$ (9.9)

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

$$\left( \frac{\sigma}{N} \right)_{\mbox{\scriptsize GR}} = \frac{13.21\, {\rm ADU}}{3723\, {\rm ADU}} = 0.35\% \enspace .$$ (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, N_ij, and the uncertainties on the individual flats, $\sigma_{ij}$, with $\sigma_{ij}$ calculated from Eq. () (using N = N_ij). They also lists the scale factors, t_ij, and the weights, b_ij, 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	$\sigma_{ij}$
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. N_ij, t_ij, and b_ij, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. $\sigma_{ij}$ 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, $\sigma_{ij}$, in each of the 3 groups, which yields

$$\sigma_{\mbox{\scriptsize JB},1} 24.8 \, {\rm ADU}$$ = (9.11)

$$\sigma_{\mbox{\scriptsize JB},2} 23.8 \, {\rm ADU}$$ = (9.12)

$$\sigma_{\mbox{\scriptsize JB},3} 25.2 \, {\rm ADU}$$ = (9.13)

Using Eq. () one more time, we get

$$\sigma_{\mbox{\scriptsize JB}} = \frac{1}{3} \left( \sigma^2... ...criptsize JB},3} \right)^{1/2} = 14.20 \, {\rm ADU} \enspace .$$ (9.14)

The relative uncertainty is:

$$\left( \frac{\sigma}{N} \right)_{\mbox{\scriptsize JB}} = \frac{14.20 \, {\rm ADU}}{4645 \, {\rm ADU}} = 0.31\% \enspace .$$ (9.15)

 

Table: The levels and uncertainties on the individual Johnson U night 2+ flat fields.

i	j	exptime	Nij	tij	bij	$\sigma_{ij}$
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. N_ij, t_ij, and b_ij, are the level (mean) in ADU, the scale factor, and weight, respectively, as reported from combine. $\sigma_{ij}$ 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:

$$\sigma_{\mbox{\scriptsize JU},1} 25.0 \, {\rm ADU}$$ = (9.16)

$$\sigma_{\mbox{\scriptsize JU},2} 25.1 \, {\rm ADU}$$ = (9.17)

$$\sigma_{\mbox{\scriptsize JU},3} 20.2 \, {\rm ADU}$$ = (9.18)

and

$$\sigma_{\mbox{\scriptsize JU}} = \frac{1}{3} \left( \sigma^2... ...criptsize JU},3} \right)^{1/2} = 13.61 \, {\rm ADU} \enspace .$$ (9.19)

The relative uncertainty is:

$$\left( \frac{\sigma}{N} \right)_{\mbox{\scriptsize JU}} = \frac{13.61 \, {\rm ADU}}{4384 \, {\rm ADU}} = 0.31\% \enspace .$$ (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 $\sim 13000$ ADU (dfsc1649-1658, exptime 10 sec), and 10 with levels of $\sim 6000$ 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. ).