6 Stabilization

 

6.1 Introduction

The response of each CAM-LW pixel strongly depends previous observations. A long-term transient response after changes in photon flux levels is a well-known characteristics of extrinsic IR photoconductors working under low background conditions (see for instance Fouks & Schubert 1995; Schubert et al. 1995; Haegel et al. 1996 and references therein). The detector used in the LW channel of ISOCAM is a gallium doped silicon photoconductor hybridized by indium bumps. The pixel pitch is 100 $\mu$m and the detectors are 500 $\mu$m thick. A physical model has been developed for the Si:Ga detector arrays used in the PHT-S instrument of the ISOPHOT experiment on board ISO (Fouks & Schubert 1995; Schubert et al. 1995). However, a physical approach to correct ISOCAM data is extremely difficult because of (1) the strong assumptions used to resolve the general equations and (2) the limited number of frames per sky position or CVF spectral position delivered by ISOCAM which generally does not allow any physical model adjustment. As detailed in Abergel et al. (1998), the response of CAM-LW pixels is known to depend on the amplitude of the step, the flux history, the direction of the flux step (upward or downward) and the local spatial gradient of illumination. We present in this section an empirical method, based on a simplified description of the pixel response after a change of the incident flux level. The spatial charge coupling between pixels is neglected.

6.2 Description of the CAM-LW response

Ground-based and in-flight measurements have shown that the pixel response after a change of the incident flux level can be separated at least in the following two phases:

• an instantaneous step to 60% of the flux step.
• a long variation for the remaining 40%, in a first order exponential similar to a time constant inversely proportional to the incident flux level.

The exponential description of the long variation for the remaining 40% after the instantaneous step is an approximation. Going from a dark level to a strong incident flux level (typically of the order of or higher than the zodiacal emission observed with the LW10 broad-band filtered centered at 12 $\mu$m, with an integration time of 2.1 s), the first readouts after the instantaneous step strongly depart from an exponential curve. This is likely due to charge coupling between pixels, which can also be responsible for the oscillations that can affect the response curve. These effects are dramatic for strong steps of flux, especially at low background. Therefore, all methods based on an exponential-like description of the pixel response fail for all steps going from the dark level, and for strong steps going from a low background. There is also a very long-term transient which affects typically $5-10\%$ the flux above the dark level which will not be discussed in this paper. This transient can introduce a memory effect with an amplitude of a few % of the input flux level, thus affecting the data over several hours.

If the number of readouts per sky position or spectral CVF position is large enough to show a significant fraction of the transient curve, it is possible to fit pixel per pixel the temporal response using a specific function to derive as well as possible a "stabilized'' value. Several methods have been developed based on the fit, and we found them less reliable when considering the flux, than the inversion method described in the following (see Sect. 6.6). Furthermore, for most of the observations, only the first part of the transient curve is observed because of the low IR emission of the sources observed with ISOCAM (especially using the CVF). The LW detector will thus never reach any stabilized value during the observation. The principle of the method presented here and developed by Abergel et al. (1996) consists in inverting a simple model of the CAM-LW response in order to recover frame by frame the successive stabilized values.

6.3 Simplified model of the CAM-LW response

The idea is to describe the response s(t) of one given pixel as a function of the input flux i(t) (in units of ADUg^-1s^-1, i.e. proportional to the incoming flux density, dark subtracted but not flat-field or distortion corrected):  

$$s(t) = r i(t) + (1-r)\int_{-\infty}^{t}i(t')\frac{{\rm e}^{-\frac{t-t'}{\tau}}}{\tau}{\rm d}t'$$ (19)

with the time constant $\tau$ (in seconds) inversely proportional to the input flux:  

$$\tau = \frac{\alpha}{i(t')}\cdot$$ (20)

It is easy to verify that, for a step of flux going from i₁ to i₂, this formula gives an instantaneous response equal to to $r \times (i_{2}-i_{1})$, followed by an exponential variation for the remaining $(1-r) \times (i_{2}-i_{1})$, with two time constants inversely proportional to i₁ and i₂.

From preliminary fitting of the model with the transient responses recorded during a dedicated orbit (revolution 16, during the Performance Verification phase, where cycles of filter sequences were performed), the values of r and $\alpha$ have been adjusted to r = 0.6 and $\alpha$ = 1200 s(ADUg^-1s^-1)^-1 (uncertainties are around 10%). The low signal to noise ratio has not allowed the detection of any significant variations of these values between pixels, though these are likely to occur.

The camera is not read continuously but rather we obtain a discrete series of readouts for all t_j. Thus we can rewrite Eq. (19) in this way:  

$$S(t_{i})= r I(t_{i}) + (1-r)\sum_{j=-\infty}^{i-1}\int_{t_{... ...ac{{\rm e}^{-(\frac{t_{i}-t' }{\tau_{j}})}} {\tau_{j}}{\rm d}t'$$ (21)

with $\tau_{j} = \alpha / I(t_{j})$.

During an integration, it is assumed (1) that one given pixel sees the same point in the sky and (2) that the configuration of ISOCAM is constant. The effects due to the jitter during the integration are neglected. Thus I(t) is constant between t_j and t_j+1 and the integral of Eq. (21) can be computed for all the terms in the summation. We have:  

$$S(t_{i})\!=\! r I(t_{i}) + (1-r)\! \sum_{j=-\infty}^{i-1}\!I... ... \!\left({\rm e}^{\frac{t_{j+1}-t_{j}}{\tau_{j}}}\! - 1\right).$$ (22)

This simplified model allows us to reproduce the response after a step of flux with a precision of typically $5-10\%$. It fails especially for large flux steps (typically 80ADUg^-1s^-1): upward transients strongly depart from a simple exponential behavior due to unpredicted oscillations, while downward transients seem to be described more precisely by an hyperbolic curve rather than with an exponential curve. The method is in fact not adapted to measure the brightness of strong point sources, since it neglects the spatial charge coupling, and no methods actually exists allowing this kind of transient to be correctly treated.

6.4 Inversion of the simplified model of the CAM-LW response

The transient correction consists of computing the successive values of I(t_j) from the successive CAM readouts S(t_j). It is important to note that r and $\alpha$ are fixed parameters of the model: this method does not use any fitting and can be applied whatever the state of the camera and the flux step are. The idea is to invert the Eq. (22), which can easily be done with the two following assumptions:

1.

We do not know the values of CAM readouts S(t_j) from $t=-\infty$, since the time series is in the general case cut at the beginning of each observation (at t=t₀). It is thus necessary to assume a realistic history of the input flux before the first readout of the observation to proceed further. For all the illustrating tests presented in this paper (Figs. 6 and 7), we have assumed that a constant input flux has been observed from $t=-\infty$ up to the first readout (at t=t₀), which means that the camera was assumed to be stabilized before the first readout. This translates in starting our summation at j=0 and adding to Eq. (22) a constant (derived from Eq. (19)).

2.

In Eq. (22), for the different values of j, we have $\tau_{j} = \alpha / I(t_{j})$. The successive values of $\tau_{j}$ are not known since they depend on the successive I(t_j). Thus, to allow a simple inversion of the equation, we have assumed that $\tau_{j} = \alpha / S(t_{j})$. This assumption obviously leads to inappropriate corrections since S and I can be much different due to the transient effect. A detailed assessment of the impact of this approximation on the photometric accuracy is yet to be made. An alternative approach would be to first consider that $\tau_{j} = \alpha / S(t_{j})$ and compute an estimate of the successive values of I(t_j), and then to iterate.

If our observation is made of N readouts obtained at time t₀, t₁, ..., t_N-1, we can rewrite Eq. (22) for each pixel in a matrix of the form: $S = [M] \times I + C$. S is the vector of N readouts, i.e. the data, I is the vector of N intensities, i.e the data corrected for transients, [M] is an $N \times N$ transfer matrix whose elements M_i,j are such that:

• if j > i, then M_i,j = 0,
• if j = i, then M_i,j = r,
• if j < i, then $M_{i,j} = (1-r) {\rm e}^{\frac{t_{j}-t_{i}}{\tau_{j}}} \left({\rm e}^{\frac{t_{j+1}-t_{j}}{\tau_{j}}}-1\right)$,

and C is is a vector of constants due to assumption (1) such that for all i, $C_{i} = (1-r) I(t_{0}) {\rm e}^{\frac{t_{0}-t_{i}}{\tau_{0}}}$.

Thus for all instants t_i one has $S(t_{i}) = \sum_{j=0}^{N-1} M_{i,j} \times I(t_{j}) + C_{i}$.

  

Figure 6: Example of transient correction on diffuse emission, taken from the observations of $\rho$ Oph (Abergel et al. 1996) made with the LW2 filter and with a 3 arcsec pixel field of view lens. Dots: non corrected observation. Solid line: transient corrected observation. Upper panel: a typical time series when CAM points from zodiacal background to a diffuse emission region. Lower panel: a typical time series when CAM points from a diffuse emission region to the zodiacal background, leading to a decreasing transient

The transient correction consists in computing $I= [M]^{-1} \times (S-C)$, which is always possible since the determinant of the matrix [M] is never equal to zero (all elements above the main diagonal are equal to 0). If the correction is perfect, the vector I represents the successive values measured by a detector with no memory effect.

This method of correction allows us to derive in a systematic way, a first order estimate of the sky brightness observed with ISOCAM-LW, without any parameter adjustment. However, one should say right away that the resulting correction is not perfect, since it is limited in precision by the precision of the model of the CAM-LW response we use which is obviously oversimplified. In particular, for large flux steps, the memory effect is predicted to be damped too fast. The model of the response we use is approximate. We have already pointed out that upward and downward transients resulting from large flux steps (typically 80ADUg^-1s^-1) are not symmetrical and depart from a simple exponential behavior, while the model is symmetrical and exponential. The photometric accuracy of the LW channel of ISOCAM is actually of $5-10\%$ after applying the transient correction algorithm we have presented in this paper. It is not better for strong sources than for faint sources, because of the over-simplification of the model of the response we use. We are currently working on algorithms to correct ISOCAM-LW data from transient effects taking into account the charge coupling, which introduces most of the features in the response curves not taken into account at the present time.

6.5 Examples

Figure 6 shows an example of transient correction in the two general cases of observation of diffuse emission: when CAM first points at a region of zodiacal background, then to a diffuse emission region, then back. The correction appears to be good. Various levels of flux that were blurred due to transients in the original data, because CAM did not spend enough time on each position to become stabilized, become obvious on the corrected data, even for small flux steps. As mentioned earlier, the representation that has been adopted to describe the response of ISOCAM-LW is not fully adequate. In particular it is unable to correctly predict the behavior of transients due to strong flux steps. This is illustrated in Fig. 7 where we show data coming from two pixels that see the brightest region of an observation of Centaurus A in the LW3 filter and with the 3'' pixel field-of-view lens. In the upper panel a strong step occurs and as a result, the corrected data (thick line) show an overshoot for the upward transient and an undershoot for the downward transient. This effect is much reduced in the lower panel where the maximum flux steps are 1/2 those in the upper panel.

  

Figure 7: Pixel response in ADUGain-1s-1 versus time during an observation on Centaurus A using the LW3 filter and the 3 arcsec PFOV lens. In both cases the thin line is the original data dark-corrected and deglitched, while the thick line is the transient corrected data. As mentioned in the text, data previous to the first frame of the cube are assumed to have a constant value, identical to that found in the first frame. Hence there is little difference between the original and corrected data at the start of the observation. The upper panel shows how the formula used to described the transient behavior of the detector fails on strong steps: it creates an overshoot, or undershoot, for respectively upward steps, or downward steps. In the lower panel, where the flux steps are smaller (the y axis scale is half that of the upper panel) the overshoots and undershoots are gone and the correction is much cleaner

6.6 A short word on fitting methods

  A number of fitting algorithms have been created to tackle the problem of transient correction (see, e.g., van Buren 1996; Lari 1997). It is not our point to elaborate on the merit of these methods, yet we can use the modeling of ISOCAM's response developed above to investigate the possibility of fitting the observed signal rather than inverting it. A fit offers the further possibility of letting some of the model parameters vary.

We therefore selected a number of observations that satisfy the following constraint: the source is extended but does not fill the array, and it is observed using a long raster. Thus it is easy to find pixels that will spend enough time on the background so that they are nearly stabilized, see part of the source for one raster position, and then observe the background again. Therefore the incoming flux history is very simple and can be written as: $l_{\rm back}\,\rightarrow\,l_{\rm src}\,\rightarrow\,l_{\rm back}$, where $l_{\rm back}$ is the background flux, and $l_{\rm src}$ the source flux. Given the selection criteria we have applied, $l_{\rm back}$ can safely be assumed to be the flux measured on the pixel prior to the raster position that sees the source. The instant at which the fluxes change are also easily derived from the data given the existence of the fast component in the pixel's response.

We have then tried to fit the response model (19) to the observed signal, letting the model parameters r and $\alpha$ vary. We thus have three parameters to derive from the fit: $l_{\rm src}$, r and $\alpha$. The results of this study have been presented elsewhere (Sauvage 1997) but we summarize them here.

1.

Very satisfactory fits can be obtained for a range of $l_{\rm src}$going from a few to $\simeq$60 ADUg^-1s^-1. Above that level we observe that the transient starts to significantly depart from the predicted behavior.

2.

No single couple of values for r and $\alpha$ can fit all the transients we selected. That is still true when we restrict the study to one pixel, in order to eliminate pixel to pixel variations.

3.

$\alpha$ was found to depend on $l_{\rm src}$ in a way that is contrary to the expectations, i.e. the smaller the flux the faster the stabilization.

Point (3) is in total contradiction with the transient behavior deduced from ground-based controlled experiment; in these we observed that the lower the flux, the longer the detector took to stabilize. It therefore casts serious doubts on the values of $l_{\rm src}$ found in the fit.

The main conclusion of this study is thus that the quality of the fit cannot guarantee that the result is photometrically correct. We thus caution observers against fitting methods for transient correction as they offer little control on the validity of the resulting photometry.

In the present case, we interpret the discrepancy between point (3) and our experimental knowledge of the detector as beeing due to an oversimplification of the response function (Eq. 19): it does not account for a slow part in the transient behavior and fails to reproduce transients observed on strong sources (see Fig. 7). Nevertheless, the values of r and $\alpha$are such that (1) memory effects are quite significantly reduced, and (2) residual photometric errors are limited and can be quantified to $\simeq$10%.