As documented previously, the in-flight behaviour of channel 4 appears to differ from it's ground performance. An examination of frames linearized with a solution derived from ground data shows that the array behaves as if it is more linear than it was. As a result, our linearization noticeably overcorrects data that exceeds about 3/4 full-well. In Figure 1 I confirm this again using science data from N 2024. The ratio between long and short frametimes in an HDR pair is shown. Most of the ratio image is uniform, showing correct linearization. However, near some of the brighter areas of the image we see the ratio increase (overcorrect; white), and then fall (turn black) as expected once we reach saturation.

Figure 1. (left) An image of N2024. (right) Image of the ratio of long and short HDR frames. Black is low, white is high.

I describe here a method for deriving a new linearity solution from data such as this. There are several points to make:

1. It is not practical in flight to derive a linearity curve for every pixel separately. Instead, we derive one for the detector as a whole. From ground testing we know that the pixels are in fact mostly all the same, so this is probably ok.

2. It is desirable to derive a solution that works with existing software. The existing software module FOWLINEARIZE accepts quadratic and cubic models.

3. Without the use of the shutter, we must use science data to derive the correction.

Since the linearity seems to be better than expected, we fall back to the simpler quadratic model. This has a big benefit in that the functional form that needs to be fit to the data is much easier to derive.

Two datasets were used in order to ensure that the observed linearity curve was repeatable. These data were from observations of SL140 and N2024. These have prodigous nebular emission, and provide a gradient across the array such that many possible well depths are sampled. Furthermore, they were taken in HDR mode, so that two different depths were sampled with a single flux rate per pixel. It turns out that we have very little data so far that is suitable, since the ideal data set needs to be extended and have a gradient that extends up to full-well, ideally saturating at least a few thousand pixels. Very few observations actually drive the detector close to saturation. I suppose this actually indicates that this treatment of linearization is not that important, since almost no one will ever notice!!

The solution is based on the idea that for linear data the ratio of the observed DN in the long and short frames of an HDR pair should just be the ratio of exposure times, and should thus be the same for every pixel. The systematic variation of this ratio as a function of the flux (DN) is a result of non-linearity. The trick is in getting from this observed variation to the needed form for correction.

FOWLINEARIZE expects a model, for a quadratic function, of

S=At+Bt^2

where S is the observed DN, A is the flux rate (effectively in linear DN/sec), and B is the non-linearity. When we had a ramp of exposure times, this was the functional form fit to the observables (S and t), and was very easy to use. Using At = S', where S' is the linear DN, we have

S = S' + (B/A^2) S'^2

let B/A^2=C, then

S = S' + C S'^2

which is a more intuitive form that just says the observed DN are the linear DN, plus a non-linearity C multiplied by the square of the linear DN. Since S'=rate t, we can solve the for the flux rate based on the observed DN and the exposure time, using the roots of the quadratic equation (I leave it to the reader to think what happens when we use the cubic form). This then allows us to compute a predicted observed DN in the short frame time, and hence a predicted flux ratio between long and short frames. We can then perform a fit to the data allowing S(in the long exptime frame) to be the independent variable, and the ratio between long and short frames as the dependent variable. In reality we solve for C as well as a bias level offset in both the long and short frames. The functional form used, assuming 0.4 and 10.4 second exposure times, is:

function ratio3(w,x)
wave w
variable x

variable rate,shortdn,out

rate=(-10.4 + sqrt (108.16 + 432.64 * (x-w[2]) * w[0]))/ ( 2* w[0] * 108.16) 
shortdn=(rate*0.4)+(rate*rate*0.4*0.4*w[0])
out=x/(shortdn+w[1])


return (out)
end

where w[0] is the non-linearity parameter, w[1] is the bias offset in the short frame, and w[2] is the bias offset in the long frame. The input "x" is the observed DN in the long frametime, and the the output "out" is the observed ratio. The result of such a fit is shown in Figure X below. The non-linearity derived is on the order of 1e-6, which is similar to that of the InSb arrays. As expected, the bias offset in the long frame in this particular example is poorly constrained (24+/-26 DN), as it affects this measurement very little. The bias offset in the short frame is tightly constrained at 15+/-1 DN as it affects the observed ratio throughout much of the dynamic range, particularly at low DN. In fact, most of the observed variation in the flux ratio is actually due to small errors in the bias level, not the non-linearity (note that we are assuming the arrays are linear at small DN, if not, it would be degenerate with the above).

Figure 2. Ratio between observed DN in individual pixels in long and short HDR frames vs. DN in the long frame. The red line is a fit of the funtional form described above.

Figure 3 shows the result of applying the new linearity solution. The result linearizes the data over the usable DN range. The right hand side shows DN that could not have been linearized because they exceed the saturation point (46k DN) in the long frame. Most of the low-level striping still evident is probably due to variations in the bias level.

Figure 3. (left) Flux ratio after linearization with the new curve. (right) A mask showing saturated pixels that could not have been linearized.