The linearization of IRAC was examined by making ramped exposure time observations of stars. The results were processed with known photometric corrections, and aperture photometry derived in a manner akin to normal science analysis. Results are:

1. Channels 1 and 2, the InSb arrays, are properly linearized to 0.2% over their entire usable dynamic range.
2. Channel 3 is not linearized correcly, and has errors of up to 5% introduced for pixels near the full well depth.
3. Channel 4 is also correctly linearized to 1%.
4. Both Channels 3 and 4 (the Si:As channels) have a feature similar to muxbleed. It appears above half-well, can have a very large flux, is highly non-linear, and increases roughly quadratically with exposure time. This is probably why these arrays have been so difficult to linearize. Since this effect has probably impacted the linearization of the arrays, more investigation is needed. The muxbleed appears very strong in this data, possibly stronger than in data taken normally.
5. Analysis from SWIRE indicates that all 4 arrays, including channel 3, are linearized properly. Both this result and earlier IOC results contradict point (2) above. At this time I have no real understanding of this.

1) Background

IRAC is linearized based on a pixel-by-pixel solution derived from a series of ground tests. A non-variable extended emission source (usually a test lamp) was observed using a ramped series of exposure times. From this, a model was constructed of the non-linear response of each pixel. The model assumes linear behaviour at small DN, a result determined previously during the selection and testing of the arrays during manufacture. Neither the ground-testing of the integrated instrument nor the test described here can verify this aspect of the linearization.

This model was verified in-flight based on HDR observations of extended emission clouds. This test was not able to verify the model for all pixels, due to a lack of a suitable extended emission source. However, it was able to verify it for many hundreds of pixels, which statistically indicated that the ground-derived model appeared to still be valid. The one notable exception was channel 4, which actually appeared to be more linear in flight than in ground testing. However, this array also shows other anamalous behaviour in-flight, and at least two sets of linearization ground-testing provided conflicting results. As a result, the channel 4 array is linearized with a new model derived in-flight, and which is the same for every pixel.

However, it remained unproven that the linearization solution produced good results for blue point sources, other than the anectodal evidence that so far no observers have complained.

2) The Stellar Test

An obvious test is to take a ramped exposure time series of a star, linearize the data, and verify that the measured flux is the same for all tested exposure times.

a) Identification of Targets

It is critical to the test to sample a wide range of well depths from below half full-well to well above saturation. Many previous attempts at measuring linearity produced poor results due to inadequate sampling and a failure to estimate the target brightness adequately. Several critical features for the targets were thus:

• The stars had to be previously observed by IRAC so that their brightness as seen by IRAC was well-known.
• In the interest of making the test fit into a total allocated time of order one hour, the targets needed to fill the well in 10-20 seconds.
• The targets needed to be visible.

b) Test Procedure

Because a wide range of exposure times were required, the AOT could not be used. Instead, SPOT was used to create a template AOR, the command expansion of which was then edited in SPOTIER to provide the requires frametimes. Due to concerns with possible image latents, the data were taken with each frametime on a different location on a regular grid on the array, with 5 dithered exposures at each grid point.

All of the data were taken with FOWLER=1, to simplify the analysis, which is otherwise complicated by the difference between frametime and exposure time. The IOC data were measured to derive DN/sec in each filter for the two stars. Exposure times were selected such that different well depths from 1/3 to 3x full-well were sampled. Driving the pixels well into saturation was motivated by a desire to see the effects of saturation on measured total fluxes of actual stars, which extend over many pixels.

c) Data Processing

The data were processed with pipe-0. This performed the basic steps of TRANHEAD, CVTI2R4, INSBPOSDOM, and the last vital step of pointing transfer. The data were then processed with FOWLINEARIZE, using the online pipeline versions of the calibration files. The data were additionally flat-fielded using the superflats, and multiplied by the position-dependent gain correction images (flipped for channels 1 and 2, since the corrections are designed for BCD images). The resulting images are still in DN units, but have had all the gain-related tweaks applied to them.

Photometry was then extracted with an IDL script that for each image located the position of the calibration star (using an RA/DEC coordinate transform, which has the additional wrinkle that pipe0 pointing transfer has the sign of the Y-axis wrong!). It then recentered on the star using a marginal centroiding routine. The highest value in DN (i.e. the maximum pixel value) within a 10x10 box is returned. Finally, the exact flux is measured in a circular aperture 5 pixels in radius. Note that this is not a flux in Jy, it is just the sum inside the aperture divided by the exposure time.

3) Results

Channel 1

It is apparent from Figure 1 that pixel phasing is extremely important in terms of the "saturation" point for unresolved sources. Because of IRAC's undersampling, the location of the exact center of the beam relative to the pixels can induce a spread in observed peak pixel DN for a given exposure time of factors of 2.5 in channel 1. Thus, there is no one "saturation" value per se, but values of exposure times which will definitely saturate under all circumstances, and those which will never saturate under any circumstance.

Figure 1. - Channel 1 peak pixel value vs. exptime. As expected, images start to saturate at around 12.5 seconds, but not until 25 seconds do all images saturate.

Shown in Figure 2 is the measured "flux" vs the flux in the peak pixel. The open circles show raw pipe-0 output, the closed red circles show fully corrected data. The non-linearity in the uncorrected data is obvious. After correction, the data are linear to 0.2%. The scatter in the observed fluxes is about 1.3%. One must conclude that the Channel 1 calibration is very good.

Figure 2. - Flux integral in a 5-pixel radius aperture divided by the exposure time. After all corrections are applied, the fluxes are linear to 0.2% with a scatter of 1.3%.

Finally, figure 3 illustrates the effect of saturation on photometry. Although saturation occurs for exposure times greater than roughly 12 seconds, even when exposed 3x longer than this, the measured flux is only 20% low. This is expected, because a large fraction of the flux in the IRAC psf is located outside the peak pixel. Hence, saturating this pixel results in a relatively small underestimation of the flux.

Figure 3. - Ratio between measured and real flux vs exptime. Note that the effect of saturation leads to an error smaller than might otherwise have been expected.

Channel 2

Channel 2 mirrors the behaviour of channel 1 very closely. The linearity solution linearizes point sources to 0.2% over the entire usable DN range. The scatter in individual measurements is about 2%.

Figure 4. - Channel 1 peak pixel value vs. exptime. As expected, images start to saturate at around 30 seconds, but not until 40 seconds do all images saturate.

Figure 5. - flux integral in a 5-pixel radius aperture divided by the exposure time. After all corrections are applied, the fluxes are linear to 0.2% with a scatter of 2%.

Figure 6. - ratio between measured and real flux vs exptime. Note that the effect of saturation leads to an error smaller than might otherwise have been expected.

Channel 3

The reader is advised to read the section on Channel 4 below first, since Channel 3's behaviour is very similar. In particular, it shows almost identical "muxbleed" behavior, as shown in Figures 8, 9, and 11. The array saturates as expected after about 15 seconds.

Prior to flight, the Si:As arrays appeared to be much more non-linear than the InSb arrays, with non-linearities as high as 10% or more near the full well depth. In flight Channel 4 was seen to be demonstrably much more linear, and in fact was about as linear as the InSb arrays, with deviations of only a few %. Channel 3, however, appeared about the same. In the current test, however, Channel 3 also appears to be very linear, and the pre-flight linearization produces very poor results. It overcompensates for the non-linearity in the array by 5-10%.

Frankly, I am completely baffled by why the Channel 3 linearization looks so bad in this test, yet looked good during IOC and also looks good for SWIRE. I am somewhat convinced that there is an additional effect, akin to the muxbleed and scattering, which is affecting the inner few pixels.

Figure 7. - Channel 3 peak pixel value vs. exptime.

Figure 8. - Flux integral in a 5-pixel radius aperture divided by the exposure time. Something is not healthy!	Figure 9. - Flux integral in a 3-pixel radius aperture divided by the exposure time. Solution is still not very good.

Figure 10. - Ratio between measured and real flux vs exptime.

Figure 11.- (left) peak DN for the star and the location 4 pixels to the right of the brightest stellar pixel. (right) ratio between the peak of the star and 4 pixels to the right.

Channel 4

First, the saturaton curve appears fairly well-behaved. Because the beamsize at 8 microns is fairly large, pixel phasing is not very important, so almost all of the individual images at a given exposure time saturate at the same time.

Figure 12. - Channel 4 peak pixel value vs. exptime. As expected, images start to saturate at around 12 seconds, and with the larger beam pixel phasing is much less important.

Figure 13, however, is very alarming. Note that even in the raw data, the flux is increasing as the array is driven closer to saturation. This is not what is supposed to happen! Previously it was noted that the array seemed to behave differently in flight than it did on the ground, most notably that it looked more linear in flight than it did on the ground. As this plot shows, if anything aperture fluxes of stars actually increase rather than decrease.

Figure 13. - Flux integral in a 5-pixel radius aperture divided by the exposure time. Something is not healthy!	Figure 14. - Image of a near-saturated point source, with a "muxbleed" pixel located at n+4 and n+8 pixels.

As shown in Figure 14, each bright star is followed by a phenomenon we have seen many times but remains fairly poorly characterized. I will discuss this in detail in the next section. In Figure 15 we see the same curves as Figure 13, only now the photometry is performed in an aperture 3 pixels in radius. In fact, the non-linearized data are now nearly completely linear, and if anything continue to increase slightly. The linearization exacerbates this slightly. The fully-corrected data are linear to about 1%, so long as they are extracted within this small window. It is also worth noting that no linearization is applied above 46k DN, as this is defined as saturated.

Figure 15. - flux integral in a 3-pixel radius aperture divided by the exposure time. Now the curves are better behaved.

Figure 16. - Ratio between measured and real flux vs exptime.

Behaviour of the Ch. 4 "muxbleed" Si:As pixels

After we got in-flight results, we discovered that what looked like the "bandwidth effect" was orders of magnitude worse than we expected. In order to examine this more carefully from this dataset, I extracted the DN of the brightest pixel associated with the calibration star in the pipe-0 data, and also the value 4 pixels to it's right (later in the fast readout direction). In order to measure the "extra" flux, I subtract the value 4 pixels to the left of the stellar peak, assuming roughly circular symmetry of the PSF. The results are shown in Figure 17.

1. The number of DN in the x+4 pixel continues to increase even after the pixel that it echoes has saturated. Although in the figure it appears to saturate at a lesser value, in fact this is because the PSF from the star is eating up some of the well-depth and what is shown on the figure is pixel minus background. I believe this argues that this is not a cable-related bandwidth effect.

2. The ratio between the x+4 pixel and the stellar peak is not constant. It is very nearly zero below 20,000 or so DN, and then increases until it is fully 40% of the stellar peak value by the time the peak saturates. Eventually they will both saturate and they will be equal.

3. The "echo" of the star contains a great deal of flux, easily reaching 10 or 20% of the total stellar flux, depending on the brightness of the star.

Figure 17. (left) peak DN for the star and the location 4 pixels to the right of the brightest stellar pixel. (right) ratio between the peak of the star and 4 pixels to the right.

This probably explains why it has been so difficult to linearize this channel. We have used extended sources for calibration, but extended sources have this additional non-linear, increasing source of flux in them. Additionally, note that the magnitude of the muxbleed in the InSb channels varies greatly as a function of fowler number. It is not clear if the results shown here carry over to all of our operational values of fowler number and wait periods. Also, the IOC checks were performed using clouds that were extended, but also had small-scale structure. It is unclear what effect that may have had on the results.

Given the above curves, one might consider code to correct for the above effect. I have written a piece of IDL code (you can ask me for it) which works like the bandwidth corrector. It

1. Steps through the image in readout order, and if it finds a pixel above 20k DN it applies a correction to the x+4 pixel.
2. For pixels between 20k and saturation it uses the right-hand curve of Figure 17.
3. For saturated pixels, it computes the total integrated flux of the star and uses a similar relation between integrated flux and the x+4 pixel brightness.

Unfortunately, in reality it doesn't work very well. The noise in the relation used for step (2) results in a very noisy subtraction of a big number from what is usually a very small background number. While statistically this may produce a superior result, it doesn't look very good. Also, in step (3) the integrated stellar flux is only useful for correcting the central pixel. If pixels adjacent to the peak have also saturated, they will be over-corrected. Solving this requires wrapping in additional spatial information. In principle it should be possible to include a look-up table with the shape of the PSF, and then something that computes radial distance from the center of the saturated stellar core.

4) SWIRE Results

For completeness, I include the following analysis performed as part of the SWIRE survey. From our first data release of the ELAIS-N1 field, I cross-ID'd all of the 2MASS sources in the field vs. the SWIRE sources. Photometry is derived in small apertures approximately 2 arcsec in radius, and a mesh background is used. Figure 18 shows the difference in observed IRAC flux vs. what would have been predicted from 2MASS based on the Vega spectrum. The locus of points around a color of zero is made up of stars. Note that for clarity I have subtracted a small constant (usually about 0.05) from this color to force the locus to zero. In reality, the stars are observed to not be strictly colorless.

Assuming 2MASS to be properly linearized, then any deviation from a straight line is either due to a variation in stellar colors as a function of observed flux or due to non-linearity in the IRAC data. The former is certainly possible - galactic number counts and makeup near the SWIRE depth limit remain contentious. However, the sharp downturn on these color diagrams is due to saturation of the IRAC bands. Overall, the curves are well-behaved and are consistent within measurement error of having a slope of zero. The Si:As channels have some evidence of an upturn at high fluxes, and may be over-corrected at the level of a couple %. Puzzlingly, Channel 3 does not show the large expected upturn shown in Figure 8. Overall, the Si:As arrays are difficult to understand, as different tests seems to indicate different linearity curves.

Figure 18. - (2MASS-IRAC) colors of stars seen in the SWIRE survey.