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Subsections

Appendix A: Flat fielding

   
A.1. Averaging on a sliding window

One way to estimate a flat field for each readout of a data cube consists of computing a sliding window average of the data cube:

\begin{displaymath}F(x,y,t_i) = \frac{1}{N}\sum_{t_j=t_i-N/2}^{t_i+N/2} I_{\rm obs}(x,y,t_j),
\end{displaymath} (A1)

where N is the size of the sliding window.

The variable flat field is normalized to 1 over the 11$\times$11 central part of F(x,y,t):

\begin{displaymath}F(x,y,t_i) = \frac{F(x,y,t_i)}{\left< F(10:21,10:21,t_i) \right>}\cdot
\end{displaymath} (A2)

Here we suppose that during the N readouts the camera has observed many positions on the sky and each pixel of the detector has seen the same flux on average. This is a good approximation if the sky observed is rather smooth and does not contain too much small-scale structure. If the emission is fairly uniform on the detector, the number of images required to compute the flat field may be relatively small. The choice of N depends on the number of readouts taken per sky position and on the contrast of small scale structures (in our case we use N=100). Extreme high and low flux pixels were rejected (top and bottom 15%) in order to exclude glitches and point sources. The main limitation of this method is that sky structures may be still present in the variable flat field.

A.2. Adaptive flat field

 

The adaptive single flat field method is based on the following assumption. We suppose that the variable flat field is, to a first order, a single flat field F(x,y) computed on the whole data cube (see Sect. 3). The temporal variations of the flat field are treated as perturbations of the single flat, and we have:

\begin{displaymath}I_{\rm obs}(x,y,t) = [1 + \delta(x,y,t)]F(x,y)I_{\rm sky}(x,y,t),
\end{displaymath} (A3)

where $\delta(x,y,t)$ is the perturbation term that take into account flat field deformations.

Flat deformations are mainly due to slow glitches and bad short term transient correction (especially on point sources). Therefore, for one given readout, they are considered as small scale defects, and $\delta(x,y,t)$is dominated by high frequency structures in the (x,y) space.

On a large-scale, the quantity

\begin{displaymath}\frac{I_{\rm obs}(x,y,t)}{F(x,y)} = I_{\rm sky}(1 + \delta(x,y,t))
\end{displaymath} (A4)

is dominated by real large scale structures ( $I_{\rm sky}(x,y,t)$) and not by flat defects ( $\delta(x,y,t)$).

The low frequency sky emission $I_{\rm SLF}(x,y,t)$ is estimated by smoothing (median filtering) each readout of $I_{\rm obs}(x,y,t)/{\rm Flat}(x,y)$. The size of the smoothing window has to be smaller than the smallest sky structures and larger than the largest flat defects. In most cases, a compromise must be found (typically a $7 \times 7$ window). The flat field deformations then are estimated using:

 \begin{displaymath}
(1+\delta(x,y,t)) \simeq \frac{I_{\rm obs}(x,y,t)}{F(x,y)I_{\rm SLF}(x,y,t)}\cdot
\end{displaymath} (A5)

We get rid of residual high frequency real sky structures by applying a temporal sliding average (see Appendix A.1) over the right hand side term of Eq. (A5).

The variable flat field is finally obtained with:

\begin{displaymath}{\rm Flat}(x,y,t) = [1 + \delta(x,y,t)]F(x,y).
\end{displaymath} (A6)

The averaging on a sliding window method (see Appendix A.1) follows low frequency temporal flat field deformations but, as a limited number of readouts are averaged together, sky structures are still present in the flat field. The adaptive single flat field is a better estimate since low frequencies of the incident flux cube are removed.


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