Marcos Diaz - Oct/97

The advantages of using Integral Field Units in stellar spectrophotometry and high dispersion spectroscopy are discussed in this study. This document is intended to give an idea of the S/N and resolution trades that are relevant when comparing stellar slit spectroscopy with IFU spectroscopy. In order to adequately address this point, the issue of optimal extraction was also explored, since it is a common procedure applied to improve S/N in standard stellar spectroscopy. Many approximations made here may be removed in a future version. Of course, the definitive reduction procedures and performance values will be established on the basis of real data analysis.

When operating a slit spectrograph with a single optical configuration, the only way of dealing with the wide range in seeing conditions of a particular site is to limit the image angular size by narrowing or widening the slit. Of course, a significant fraction of the target light is lost, increasing noise and imposing the need of extra exposures for photometric calibration. The IFU image slicing allow the use of all the light in the stellar profile over a wide range of seeing values while exploiting the full instrumental resolution.

Lets suppose that the stellar profile at the focal plane is sampled by N contiguous hexagonal microlenses. With respect to the profile flux center, each lens may be identified by a radial index i and an azimuthal index j. In addition, each image sample is positioned at an arbitrary position in a "virtual slit" and each one produces a dispersed image in the detector that cover M rows in the "spatial" direction (see figure).

For a slit width (w) matched to the FWHM of a gaussian seeing profile the gain in stellar signal is of about 32%. In addition, the sky background can be determined closer to the star with the same S/N or within the same spatial scale with a higher S/N ratio. For example, a circular IFU with angular diameter (l) should define the background with S/N improved by a factor of ~(w/l)^½ when compared with a slit of the same size.

The optimal extraction of stellar spectra obtained through narrow slits has been applied as an efficient technique to suppress background noise while preserving the photometric properties of the data (Horne 1988). The statistical benefits of the method rely on the knowledge of the stellar profile (as a function of wavelength, if there is a significant dependence). However, when the data set is obtained under time-varying seeing, most of the work is dedicated to estimate the spatial (x) light distribution in the low S/N stellar science exposures. A practical limitation of the method is evident in the cases where it is not possible to derive a reliable, "noise-free", estimation of the stellar profile. The direct imaging (or zero order) IFU remapped stellar image would be the counterpart of the stellar profile in the extraction of stellar IFU spectra. Possibly, the estimation of the actual IFU PSF is less S/N constrained than the stellar profile evaluation in standard slit spectroscopy data.

II.1 What is the best weighting function?

In order to extract the intensity values at a given wavelength one may simply sum the counts in the detector rows (l) corresponding to all (N) IFU elements (i,j) at a given wavelength (lambda):

(1)

then the associated variance would be:

(2)

where the variance in the sum is (sigma)² on each detector pixel.

Alternatively, we may weight the sum. Due to the PSF shape each image element is biased in the sense that it receives more or less light depending on its position (alpha,delta) in the focal plane. An unbiased estimator of the incoming element flux as a function of the position (r,theta) may be given by:

(3)

where

i.e. the sum of all pixels corresponding to one IFU element.

The PSF is normalized by the enclosed energy: ,

These f estimators are statistically independent because they assemble information from different pixels (assuming that there is no element overlapping) and they depend linearly on the accumulated counts. For the sake of simplicity lets assume that M is <=2, or in other words, lets assume the profile of each image slice at the detector is narrow so readout noise is minimized and the profile shape dependence on wavelength is negligible. In addition, lets consider in this preliminary analysis that the counts have been flatfielded and the typical flatfield correction does not affect the statistical weight of each image element. The variance of the estimator is:

(4)

The estimated intensity per image element may be weighted by a function w(r_i, theta_j, lambda) that minimize the variance of the sum "along the whole" PSF. Therefore, at a given lambda, the measured intensity is described by the simple weighted mean:

(5)

Here we remove one dimension from the weighting function by assuming an axially symmetric PSF. Dropping indices we find the following intensity estimator and its variance.

(6)

(7)

now minimizing the intensity variance with respect to the weights:

(8)

(9)

and (10)

this is the classical inverse variance weighting factor for each IFU annulus.

Then using 10, 4 and 2 the variance ratio between the sum and optimal extraction is the following:

(11)

II.2 Case 1 - background limited observations

Now, lets us assume that the background and, eventually, the readout noise are constant along the PSF and they follow a Poisson distribution. When the variance per pixel is dominated by the sky background we have (sigma)² = n_counts ~ constant (the number of counts per pixel). Summing over the N IFU elements (i, j), and their detector columns l at lambda.

with a gaussian PSF approximation we get:

(12)

Example: Suppose we integrate over twice the stellar FWHM and the stellar profile is sampled by 2.5 IFU elements per FWHM. In addition, each IFU element is recorded in 2 CCD columns. The corresponding ratio would be 2.7 (in variance) or 1.6 (in sigma).

The above formulae shows that optimal extraction is much more important for IFU stellar spectra than for slit spectra where the same exercise yield a variance ratio of about 1.7. This result is not surprising since the number of profile elements increase is quadratic with the PSF radius.

II.3 Case 2 - Negligible background and readout noise

This is the case of bright stars (flux standards for example) where source photon statistics is the major contribution to the measurement noise. By assuming Poisson statistics, the variance of each IFU element would be proportional to the local PSF value:

applying this in our variance ratio expression (11) and using the normalization condition for the PSF model one find:

(13)

Therefore, where there is no background to suppress nothing can be gained from original data statistics.