4.7  Error estimates.

The various noise sources associated with the NI spectrometers are described in detail in the CDS Software Note #49: Deriving Statistics from NIS data.

As well as Poisson noise associated with the photons which interact with the detector, there is another source of noise due to the fluctuation in the amount of amplification in the detector system, known as the pulse height distribution (PHD). This latter effect can be assumed to be comparable to the Poisson noise. Other sources of noise are only detector-dependent. The readout noise R has been found to vary from 0.62 to 0.83 photon-events/pixel, and therefore has an effect only at low light levels. A value of 0.7 is assumed in this work.

Another major effect is that the detector resolution makes each photon affect more than one pixel on the CCD. The line profiles therefore appear smoother than would be expected from the photon statistics, and it is more appropriate to consider the statistics of the whole group of pixels where each line is spread.

This makes error estimates based on curve fitting appear smaller than the true values, in many cases.

If N is the total number of photon-events in a line, the Poisson noise is ÖN, and assuming that the PHD component is comparable, and adds in quadrature, the total photon noise is s_N = Ö{2*N + R²*n_pix}, where n_pix is the number of pixels being summed over and R is the readout noise per pixel. In reality, the main source of error in determining intensities in NIS spectra is the scattered light in the detector, and what we want to measure is the intensity of each background-subtracted line I=N-B , where B is the total background intensity over the line. The noise associated with the background is again s_B=Ö{2*B + R²*n_pix} and the error on the intensity I is therefore

sI = Ö   sN2 + sB2   = Ö   2*(I+B) + R2*npix + 2*B + R2*npix

(1)

In the case of n_avg spectra averaged, we have to divide s_I by Ö{n_avg}.

Another way to determine the error on the intensity is directly from the errors on the parameters of the fit. If the line profile is gaussian, the total intensity in the line is derived from the peak amplitude I₀ and the line full-width-half-maximum FWHM (in pixels):

I =  1 2   æ Ö  p ln2     I0   FWHM

(2)

Since the peak amplitude I₀ and the FWHM are not statistically independent, in the sense that too high FWHM favors too low I₀ and vice versa, the error on the intensity s_I can be calculated from:

(  sI I )2 =  1 2 é ë (  sI0 I0 )2 + (  sFWHM FWHM )2 ù û

(3)

For unblended lines the errors calculated in this way are similar to those calculated from the total line intensities.