The IRS reads data in ramps with 4, 8, or 16 reads, which are used to calculate the slope image in electrons/s (among other uses, including cosmic ray rejection, linearity correction, etc.). Here we examine several different methods by which those reads can be used to estimate a slope, in the context of a Monte Carlo model which approximates the noise properties of real IRS ramps. The noise model includes only two components: the Poisson noise of the accumulating photo-electrons, and the read noise per sample. Note that the nature of the accumulating Poisson noise ensures that each read is correlated, in a random-walk type process, to all prior reads on the ramp (a property of time-series data points like stock prices which the statisticians call "serial correllation").

Given the number of samples in the ramp, the slope in e/s, and the read noise in electrons, the model creates sequentially a large number of sample ramps with the same input slope, by accumulating photo-electrons according to a Poisson distribution, and imposing a Gaussian read-noise on each read afterwards. Each ramp is then used to estimate the slope in four ways:

- The sample deltas (read2-read1, read3-read2, etc.) are formed, and the mean is taken.
- The sample delta mean is computed, and all deltas which deviate by two times their stddev are rejected, and a new mean computed.
- A full least-squares line fit is performed, with measurement errors formed by assuming the measured flux is a Poisson realization, along with read noise (error=sqrt(RN^2+total_photoelectrons)).
- A full least-squares line fit is performed, with no measurement errors.

The distribution of recovered slope values yields information on the fidelity of the particular slope estimator. Here are the results for a particular exposure time of LL, 30s, with 16 reads, as a function of source brightness (slope). The read noise per sample was taken to be 150e. Each image shows the distribution of measured slopes for each of the 4 methods.

All four methods provide an unbiased estimate of the slope, in that they peak at the known input value. The width of the distribution varies considerably. At this low slope level of 200e/s (roughly the zodiacal value, which sets the lower threshold), the linear fit is ~50% better at recovering the input slope than the mean, with a width of 4.9e/s vs. 7.5e/s. It is over 2x better than the sigma-trimmed mean.

As the source flux increases (here to the level of a multi-Jy source), the mean becomes a less noisy estimator, but is still somewhat inferior to the linear fit.

At the very high source flux of 5000e/s, which saturates near the end of ramp, the mean and linear fit are nearly identical. The relative error for all methods is quite small; the differences between these methods is most apparent at low source flux. Also available are model runs for other exposure time combinations:

The counter-intuitive result that sigma trimming actually worsens the
estimate holds true because all points were drawn from a pure gaussian
distribution of read noise. Trimming is effective at improving an
estimator when it is expected that some measured points deviate wildly,
far more than the Guassian probability would permit. This model does
not address that concern, but the strongest source of deviation, cosmic
rays, are treated before the slope is estimated. Even then, the mean of
all points (0 deviates rejected) is a less precise estimator of the
slope than the linear fit. At the very brightest end of the longest
ramps, the mean can actually perform *better* than the line fits, since
it ignores all of the lower fidelity early samples. This case is not
encountered in real IRS data.

Given a determination of the slope using the ramp data, estimating the
error in that slope in the context of a linear fit cannot make use of
the standard parameter errors of linear least squares fitting theory,
since these rely on the independence of each sample measurement. Since
the ramp samples are serially correlated, the calculation of the error
in the slope must be modified; see Gordon
et al., 2005 Appendix A for more information. An important point is
that only the **error** in the slope is affected by the correllation.
The slope itself is unchanged. In fact, the precision with which the
slope is determined is independent of whether measurement weights are
used in the least squares fit; accurately estimating the uncertainty in
the fit does require applying predicted measurement errors.

Last modified: Tue Jul 12 13:56:04 MST 2005