4 Transient correction: Inversion method

4.1 Introduction

We have seen in the previous section that the FS model allows an adjustment of the response curves with an accuracy better than 1% for more than 90% of the pixels for uniform illuminations on the whole useful dynamic range. The two parameters of the model ($\beta$ and $\lambda$) have been adjusted pixel by pixel. No significant temporal variations have been found during the ISO mission. We now present correction methods based on this model.

Basically two methods are possible to correct the data from the transient effects: readout by readout and block by block. One readout corresponds to the output value measured at the end of one integration. One block (typically a few to several hundreds of readouts) corresponds to a constant configuration of the whole system (parameters of the camera, pointing of the satellite). To take into account this information it may help to improve the stability of the correction and the signal to noise ratio.

At the present time and in this paper, we only indicate methods which allow us to correct the data readout by readout. Obviously, we need to assume that during each integration, the input flux is constant. It is wrong during the slew of the satellite during two successive sky positions, or when a change of one parameter of the instrument (lens and filter wheels position) is commanded. Generally, only one or two successive readouts are affected.

4.2 Readout by readout correction method

The correction method is based on the FS model described in Sect. 3. We use the same notations as in Eq. (6). We consider a set of N successive readouts $J_{n}^{{\rm out}}$, in units of ADU/G/s. Each term $J_{n}^{{\rm out}}$ can be identified with J _n(t)in Eq. (6), if we take t equal to the time of the end of the integration number n which starts at time t_n (see Fig. 5). The transient correction consists of computing, for n going from 0 to N-1, $J _{n} ^\infty$ from the measured responses J _n(t). Practically we need to invert Eq. (6) in order to compute $J _{n} ^\infty$from J _n(t), $J _{n-1} ^\infty$ and $J _{n-1} ^{{\rm end}}$. Equation (6) is non-linear because of the exponential term and there is no analytical solution.

We have tested three different methods to find the solution:

• A second order development of the exponential around 0 gives a third order polynomial in $J _{n} ^\infty$. The closest root to the measured value J _n(t) gives the solution. This is the quickest method, but only applicable for low values of $(t-t_{n})J _{n} ^\infty /\lambda$, so of $J _{n} ^\infty$;
• The monotonic property of Eq. (4) allows us to use an iterating algorithm. First we start by replacing in the exponential $J _{n} ^\infty$ by J _n(t). We obtain a second order equation, so a first estimate of $J _{n} ^\infty$: $J _{n,1} ^\infty$. Then we replace in the exponential $J _{n} ^\infty$ by $J _{n,1} ^\infty$, and we iterate. This method converges after a few iterations for any steps of flux with no or very limited noise, but practically fails for any steps of flux on experimental data because of the noise level;
• The Müller method (in Numerical Recipes, Press et al. 1992) finds roots of non-linear equations when three initial points are provided. This numerical method is more time consuming.

In order to study the inversion quality and its noise robustness, we have tested these three correction methods on simulated data for different fluxes, different steps of fluxes and different signal to noise ratio. Practically we choose an input flux history: $J _{n,{\rm input}} ^\infty$, for n going from 0 to N-1. The detector response ( J _n (t), for n going from 0 to N-1) is computed, readouts by readouts, using Eq. (6). Then we add a Gaussian noise with a standard deviation up to $\sigma =1$ ADU/G/s. All the values J _n (t) were positive before the noise is added but not necessarily after. For n going from 0 to N-1, we compute readout by readout the corrected response $J _{n} ^\infty$using the three methods described above. Finally we compare $J _{n,{\rm input}} ^\infty$ and $J _{n} ^\infty$. Practically we have used the same kind of signal as in Fig. 4, with typically 600 readouts, upward steps at readout 200, downward step at readout 400, an integration time $t_{{\rm int}}$ of 2.1 s for each readout, $\beta$ = 0.55 and $\lambda$ = 600 ADU/G. The low levels are equal to 0.1, 0.5, 1, 2, 5 and 10 ADU/G/s, while high levels are 5, 25, 50, 100, 250, 500 or 1000 ADU/G/s.

The Müller method gives always the more accurate result even for high steps (typically for $10^{-3}< J _{n} ^\infty / J _{n-1} ^\infty <10^3$) and strong noise ( $\sigma =1$ ADU/G/s). But this method is also the slowest one. Sometimes divergences occur for levels below 1 ADU/G/s.

The iterating approach is very sensitive to the amplitude of the noise. On the simulations, divergence could occur for any step ratio when $\sigma$ is greater than $\sim 0.1$ ADU/G/s. On real data, divergences occur quite often. Thus we conclude that this method has to be discarded.

The method based on a second order development of the exponential around 0 must be used only for limited fluxes $J _{n} ^\infty$, since the factor $(t-t_{n})J _{n} ^\infty /\lambda$must be significantly smaller than 1. This method presents the advantage that at low flux it never diverges, even with noise of high amplitude ( $\sigma \sim 1$ ADU/G/s), and is extremely fast.

Practically, we use this last method at small flux, and the Müller one at high fluxes. In any case, the computing time is typically ten times lower than with the old IAS method (Abergel et al. 1999) for an accuracy typically ten times better. On simulations, the accuracy is better than $1 \%$ for each readout; on real data, the accuracy is better than $\sim 3 \%$for each readout when we have not problem due to glitches, spatial gradients or inaccurate dark levels...

Figures 9 and 10 present two results to illustrate the quality of the correction of real data for uniform illumination. For observations with the Circular Variable Filter (CVF), after correction we see that the data can be used on the whole spectral range. It was generally not the case with previous methods.