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\begin{opening}
\title{DETECTION LIMITS IN THE FAR INFRARED}

\author{G. H. Rieke}
\author{E. T. Young}
\institute{Steward Observatory\\ University of Arizona\\ Tucson,
AZ. 85721 }
\author{T. N. Gautier}
\institute{Jet Propulsion Laboratory\\ Pasadena, CA. 91109}
\date{}
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\begin{abstract}

The advent of far infrared arrays will change fundamentally the
means of analyzing observations in this spectral region.  Sources
much fainter than traditional "confusion limits" will be extracted
from images by using computer algorithms similar to CLEAN or
DAOPHOT.  We have conducted numerical experiments to evaluate these
techniques and show that they will permit long integrations ($\sim$
10,000 sec at 60\um, $\sim$ 200 sec at 100\um) to achieve nearly
photon-background-limited performance and hence very deep detection
limits.  The dominant noise sources -- photon noise, confusion by
distant galaxies, and confusion by IR cirrus -- scale with nearly
the same power of the telescope aperture.  As a result, the
integration times required to reach confusion limits are nearly
aperture-independent.    

\keywords confusion limit, infrared arrays 
\end{abstract}

\section{Introduction}

     In planning far infrared astronomy missions, it is important
to understand the types of noise they will encounter.  Confusion by
faint galaxies and by infrared cirrus emission can limit the signal
to noise achievable in the far infrared.  We calculate the
magnitude of these effects where the telescope beam is limited
purely by diffraction and the confusing galaxies appear as
unresolved point sources.  These assumptions are appropriate for
the majority of envisioned far infrared telescopes, which are
designed with image quality and pointing stability required for
near infrared operation and which are therefore virtually perfect
optical systems in the far infrared.

     A number of previous works have argued that 
natural-background-limited telescopes will be severely limited by
confusion noise in a few seconds of integration.  These authors
have neglected the improvements in performance that will be
possible with imaging arrays in the far infrared.  When used with
arrays in an optimum manner, a telescope cold enough to be photon
noise limited from celestial backgrounds has far superior
performance to warmer telescopes throughout the far infrared. 
Telescopes sufficiently cold to be natural background limited can
be essentially photon noise limited at wavelengths of 100\um~and
shorter for long integrations (e.g., as long as 10,000 sec at
60\um).

\section{Point Source Confusion Noise}

     Condon (1974) demonstrates that estimating point source
confusion noise requires only knowledge of the integral source
density and of the field of view of the instrument, so long as the
point source response is taken to be the "effective beam solid angle",
$\Omega_e$, which accounts for the wings of the beam.  In the
regime of interest to us, the integral source counts go
approximately as a power law with index $\beta = -1$.  In this
case, the limiting noise goes as the square of the beam diameter. 
This steep dependence can lead to substantial variations in
estimates of the confusion noise limit if differing assumptions are
made about the achievable beam diameter. 

     IRAS observations provide a direct measure of the far infrared
source density.  At 60\um, Hacking and Houck (1987) and Hacking
(1994, private communication) find $\sim 20$ objects deg$^{-2}$
brighter than 50 mJy. Above this flux limit, the integral counts
follow a power law as $S^{-3/2}$, as expected for Euclidean space. 
However, below this flux level the most prevalent galaxies become
increasingly more distant, and cosmological corrections will become
important so that the counts will deviate from the Euclidean
dependence.  Therefore, a model of the behavior of galaxies must be
constructed from the IRAS data and used with an appropriate
cosmology to extrapolate to lower flux densities and determine the
density of faint galaxies on the sky.  The model we have used will
be described elsewhere (Rieke, Young, and Gautier, in preparation);
here, we note that the source counts from our model are in good
agreement (within a factor of $\sim 1.5$) with the predictions of
Franceschini et al. (1991) and Helou and Beichman (1991).  At the
flux densities and wavelengths of interest for limiting far
infrared measurements with cryogenic observatories, all the counts
behave roughly as power laws with $\beta \sim -1$.  The predictions
are not strongly dependent on cosmological assumptions or on the
amount of galaxy evolution when normalized to the 60\um~counts
described above.

     Given the densities of far infrared galaxies on the sky, to
determine the confusion limit of an instrument we need to determine
the smallest effective beam area that it can use.  For future missions, the
effective beam area should be determined using source extraction techniques
appropriate to observations with imaging arrays such as will soon
be available for the far infrared (Young et al. 1993).  In the following, we
investigate whether optimal source extraction with oversampled array images
will allow useful data to be obtained in crowded fields that might be
considered confusion limited with a single detector.  We have taken a
numerical approach that allows us to simulate the operation of
various source extraction procedures using Monte Carlo methods.  We
have verified that our results agree with the analytic treatment of
Condon (1974) where the results can be compared.  In our Monte
Carlo simulations, we have combined the effects of confusion and
photon noise so that the results realistically model the effects of
amplification of the photon noise as the extraction technique operates in
increasingly crowded fields. In comparing results with varying pixel sizes, we
have assumed that the pixels operate at the photon noise background limit.  We
have also simulated problems where the required signal to noise is modest, so
that small errors in determining the beam profile do not affect our results. 
These latter two assumptions allow us to address a "fundamental" confusion
limit that will not depend critically on the assumed parameters for the
instrument.  

     The definition of confusion limit depends on the nature of the
investigation.  For example, an astronomer making an unbiased
survey for infrared-bright galaxies would be less annoyed at being
"confusion limited" in such objects than would one wishing to
determine the flux density of a quasar in the same set of data.  In
the following, we have considered only the second type of
observation since it yields the most stringent measure of the
confusion limit. 

     In our experiments, we generated artificial data by drawing
sources randomly from a power law distribution N($> S$) = C $S^{-
\beta}$, where $S$ is the flux density and C is a constant. 
Sources were placed at random positions in a field of 57x57 pixels at an
average
density of 25 per $\lambda/D$ diameter beam, where $D$ is the
telescope aperture.  The faintest sources were therefore roughly
only 1$\%$ of the flux densities at the achievable detection
limits.  We left out of this field any sources at more than 1000
times the rms noise level, under the assumption that any deep
survey field would be selected to avoid extremely bright sources. Each source
was convolved with an Airy pattern of full width at
half maximum of 10 pixels and gaussian-distributed random noise was
added to each pixel.  This noise was scaled with pixel size under the
assumption of background limited operation.  A test source of known amplitude
and with an
Airy pattern profile was placed in the center of this "data" array. 

     Initial tests set $\beta = -1$.  We used a source extraction
method closely related to the CLEAN algorithm.  The data array was
deconvolved by identifying the brightest pixel and subtracting an
Airy pattern of amplitude 1/3 the rms gaussian noise.  The position
and amplitude of this subtracted flux contribution were stored in
another "deconvolved" array and the procedure was repeated,
incrementing the amplitudes in the deconvolved array as small
amplitude sources were subtracted from the data array.  The
subtraction was stopped when the variance in the data array
increased with subtraction of an additional small source.  

     The program then computed the estimated flux density of the
test source in the deconvolved array.  The sky level was determined
as the average of the surface brightness between radii of 1 and 2.5
$\lambda/D$, after rejecting all peaks at greater than 3 times the
rms noise in this region.  This rejection was based on the
hypothesis that one would avoid obvious neighboring sources in
computing a sky level.  The source flux density was determined by
integrating the signal within apertures of various sizes and
subtracting the sky contribution.  400 Monte Carlo integrations
were run for each value of assumed noise.  It was found that the
optimum extraction procedure in the deconvolved image was to integrate the
signal within a sharp-sided angular aperture of diameter 0.5
to 0.7 $\lambda/D$.  Extractions in this size (and neighboring
size) apertures reproduced the input source strengths accurately,
with no significant biases toward under- or over-estimation.

     The artificial central source was selected so that the final
signal to noise (including confusion noise) would be near five. 
This case is appropriate to a deep survey where source detections
are achieved down to the limiting noise.  For each Monte Carlo run of 400
cases, we determined the rms fluctuations in the estimated brightness of the
central source, so the suite of runs yielded the relation between density of
confusing sources on the sky and noise in the measurement of the central
source. We fitted this relation to the expected theoretical behavior, with the
effective beam diameter as a free parameter; an excellent fit was achieved
with an assumed beam of angular diameter 0.8 $\lambda/D$.  That
is, in this application and with the CLEAN method of source
extraction, we would successfully predict the total noise of the
system including amplification of the photon shot noise if we assumed we were
observing with a sharp sided aperture
of diameter 0.8 $\lambda/D$.  The results were found to be largely
independent of the selection of reference field, so long as the
inner radius was $\geq$ 0.6 $\lambda/D$. Our results represent a modest amount
of superresolution over the full width at half maximum of 1.03 $\lambda/D$ for
an Airy function. 

     A second set of simulations was run with a power law source
index of $\beta = -1.5$, with results very similar to those just
described.  Again, the behavior is adequately described by the
simple theory if a beam diameter of 0.8 $\lambda/D$ is assumed. 
The experiments described above assumed an extreme degree of
oversampling, i.e., pixels of 0.1 the diffraction-limited beam
diameter.  An additional set of experiments addressed the use of
larger pixels.  Here, the data frames and CLEAN beam were both
measured with pixels respectively of 0.3, 0.5, and 0.7 $\lambda/D$. 
It was assumed that the telescope was substepped to maintain the
same sampling interval as before, that is 0.1 $\lambda/D$.  For
example, with the 0.5 $\lambda/D$ pixels, it would be required that
data be taken at 25 telescope pointings on a 5x5 grid with 0.1
$\lambda/D$ between points.  200 iterations were made for each
case, all at the same confusing source density where the previous
experiments indicated confusion noise would degrade the photon
noise limit by a factor of 3.  These experiments produced similar
results to those with the 0.1 $\lambda/D$ pixels, showing that
imagers can make efficient use of the far infrared arrays by using
pixels that are a reasonably large portion of the Airy pattern. 
>From other experiments, we conclude that this favorable result
arises only if the telescope is substepped on a finer grid than the
pixel-to-pixel spacing. In addition, the requirements on the accuracy of
calibration and the knowledge of the point spread function will be increased
as the pixel size is increased.  These factors must all be considered in the
design of realistic systems for high sensitivity operation in the far
infrared. 

     Although we have treated our simulations as if there is a hard
confusion limit, at higher ratios of signal to noise it is likely
that the source could be localized more precisely than in our
experiments, leading to a smaller effective beam diameter and an
improvement in limiting flux density.  Again, achieving these goals will place
greater demands on the calibration and knowledge of the point spread function
and therefore will depend on the details of system design. 

     As examples, we compute for an 85 cm telescope at 60, 100, and
150\um~the rms confusion noise limits due to distant galaxies given
in Table I.  The confusion noise from point sources with
uncorrelated positions on the sky will scale in this case inversely
as the effective aperture area, or as $D^{-2}$.  

\begin{table}[ht]
\begin{center}
\caption{Noise Components for 1000 Sec Integrations}
\begin{tabular}{llll}
Noise Source & 60\um & 100\um & 150\um \\ \hline
Galaxies & $\sim 25\mu$Jy & $350\mu$Jy & $1500\mu$Jy \\
IR Cirrus & $6\mu$Jy & $89\mu$Jy & $300\mu$Jy \\
Photon Noise & $90\mu$Jy & $140\mu$Jy & $400\mu$Jy \\ \hline
Net Noise & $94\mu$Jy & $387\mu$Jy & $1580\mu$Jy \\ \hline
\end{tabular}
\end{center}
\end{table}

\section{Confusion by Infrared Cirrus}

     We base our discussion of confusion by infrared cirrus on the
formalism of Gautier et al. (1992).  To apply these results, we
need to set three parameters: 1.) the power law index, $\alpha$, of
the cirrus emission power; 2.) the power at 0.01 arcmin$^{-1}$,
$P_o$(0.01/arcmin); and 3.) the observing geometry.

     Analysis of the IRAS far infrared data shows that the cirrus
power spectrum has a spectral index of $\alpha$ = $-2.6$ to $-3.2$
(Gautier et al. 1992, T. N. Gautier unpublished work).  The IRAS
data do not extend to the high spatial frequencies that are
important for future missions that will have both telescopes of
larger aperture and imaging far infrared arrays.  However, the
power spectrum can be estimated at these high frequencies by
comparison with deep CCD exposures in the visible, which can reach
1 arcsec$^{-1}$.  Cutri (private communication) finds that the
power spectrum steepens slightly from the low spatial frequencies
(i.e., � becomes more negative).  In the following, we will assume
$\alpha = -3.0$.

     The level of cirrus contamination is characterized by $P_o$. 
As a medium level of cirrus contamination, we adopt the average
$P_o$ for $\mid$b$\mid > 50^o$, namely $3 \times 10^6$ Jy$^2$/sr
(Gautier et al. 1992).  Other levels of cirrus emission are listed
in Table II.  

\begin{table}[ht]
\begin{center}
\caption{Cirrus Confusion Noise at 100\um}
\begin{tabular}{lll}
Type of Area & $P_o$(0.01 arcmin$^{-1})$ & RMS Confusion \\
{} & (Jy$^2$/sr) & ($\mu$Jy/beam) \\ \hline
Best 100 deg$^2$ in Baade's Hole & $\leq 1.2 \times 10^4$ & $\leq
6$ \\
500 additional deg$^2$ in Baade's Hole & $\leq 5 \times 10^5$ &
$\leq 37$ \\
Average, $\mid$b$\mid$ $\geq 50^o$ & $3 \times 10^6$ & 89 \\
Moderately Bright Cloud & $10^8$ & $510$ \\
Bright Cloud & $10^{10}$ & 5100 \\ \hline
\end{tabular}
\end{center}
\end{table}

     Because of the steep power spectrum of the cirrus emission,
there is a premium in keeping the reference field as tightly
coupled to the source as possible.  We have therefore computed the
cirrus confusion noise assuming an annular reference field that
lies between 0.6 and 1.6 $\lambda/D$.  For comparison with the
galaxy confusion limit, we have interpolated to an effective beam
diameter of 0.8 $\lambda/D$.  Because of the low power at high
spatial frequencies, the noise from cirrus confusion is only weakly
dependent on the beam diameter but very strongly on the reference
area size.

     For the case of an 85 cm telescope, the resulting cirrus
confusion noise limit at the average level of high latitude cirrus
emission is 89$\mu$Jy per beam, with these assumptions.  Values for
the rms confusion noise for the 85 cm telescope at other
wavelengths and in other parts of the sky are entered in Table II. 
The confusion noise will scale as the square root of $P_o$ and as
$D^{(\alpha /2 - 1)}$, i.e., as $D^{-2.5}$ for $\alpha = -3$.

\section{Photon Noise}

     A far-infrared astronomical system will also be limited by
photon noise.  The cosmic background at infrared wavelengths can be
broken down into a number of important components.  At shorter
wavelengths scattered solar radiation and thermal emission from
zodiacal dust dominate, while beyond $\sim$ 40\um~emission from
cold galactic dust must be included.  Each of these components has
a distinctive positional dependence on the sky, and large
variations in the relative contributions are present.  To
investigate a case appropriate for extragalactic astronomy, we have
taken the COBE Diffuse Infrared Background Experiment (DIRBE)
measurements of the South Galactic Pole (Hauser et al. 1991).  The
noise from this background is computed in a conventional manner
(e.g., Rieke 1994); because the astronomical backgrounds are due to
very dilute graybodies, the Boson correction can be ignored (van
Vliet 1967).  

     Since the focal plane fully samples the point spread function,
the signal is spread over a number of pixels.  Therefore, we
combine the measurements from the number of pixels needed to
synthesize a point source, quadratically adding the noise.  An
additional noise penalty of a factor of 1.1 was added to allow for
noise in the flat field.  We have assumed a diffraction limited
beam (for a telescope with 15$\%$ areal obscuration) and have
computed the net signal in apertures centered on the source after
subtraction of the average surface brightness in a reference area
between 1 and 2.5 $\lambda/D$.  The result is that the signal to
noise for a point source is 8.2 times worse than that computed for
a single 0.4 $\lambda/D$ pixel (assuming all the signal fell on the
single pixel) if the signal is measured in a 0.8 $\lambda/D$ beam
and 7.1 times worse if measured in a 1.2 $\lambda/D$ beam.  The
latter beam size has the maximum ratio of signal to noise if photon
noise alone is considered; however, to maximize the ratio of signal
to noise that can be achieved in a confusion-limited situation, we
will use the former beam size in the following. 

     Some degradation of the photon noise will occur in space
because cosmic rays striking the detector will limit the
integration times.  A discussion of the extent of this effect is
given by Herter (1990).  The results depend on detector geometry,
read noise, and the method of operation of the readout.  We assume
appropriate parameters for the far infrared arrays under
development for SIRTF (Young et al. 1993), i.e., that a cosmic ray
hit destroys all information in the integration after the hit, a
hit affects the pixels on either side of the hit one so that the
resulting net pixel area is 0.13 cm$^2$, and the read noise is 50
electrons; the result is that the pure background-limited photon
noise will be degraded by a factor of $\sim$ 1.5. 

     The photon noise limits in Table I are computed according to
the description above and include the degradation by cosmic
radiation.  Of course, for a diffraction limited telescope, these
detection limits scale as $D^{-2}$. 

\section{Combined Noise}

     The various sources of noise are combined in Table I, where
the cirrus emission has been taken at the average for high latitude
sky.  The behavior across the far infrared band progresses rapidly
toward increasing levels of confusion with increasing wavelength,
both because of the increase in the density of galaxies of a given
flux density and because of the increase in the diffraction limited
beam size.  At 60\um, integrations greater than 10,000 seconds are
required to approach serious levels of confusion, while 200 seconds
and 100 seconds are required respectively at 100 and 150\um.  It is
possible with an 85 cm telescope to reach limits about two orders
of magnitude deeper than the data in IRAS bands 3 and 4.  

     We note that the scaling of the noise sources, for a given and
roughly optimized method of source extraction, is $D^{-2}$ for
point source confusion noise and photon noise and $D^{-2.5}$ for
cirrus noise.  Thus, the relative importance of these differing
noise sources will change only slowly with telescope aperture.  Any
telescope designed to operate at fundamental detection limits in
the far infrared should be cold enough so that the photon noise
does not increase significantly compared with the natural
background.  

\section{Conclusion}

     The advent of far infrared detector arrays will allow
qualitatively new approaches to sensitive astronomical detection at
these wavelengths.  By applying source extraction techniques
similar to CLEAN or DAOPHOT, the effective beam size can be reduced
substantially with a resulting reduction in confusion noise
compared with predictions for instruments with a small number of
detectors or for aperture photometers.  The same benefits can be
achieved for operating in regions where there is strong infrared
cirrus.  To operate in this mode, it is essential that far infrared
telescopes have highly reproducible, preferably diffraction
limited, beam profiles. 

     Table I shows as an example the sensitivity that can be
achieved with a cryogenic 85 cm telescope using a CLEAN algorithm
and operating at the average level of IR cirrus at high Galactic
latitude.  At 60\um, the instrument will not become confusion
limited unless integrations longer than 10,000 sec are attempted. 

     Because the three types of limiting noise scale with almost
the same power of the telescope aperture, their relative
contributions will not change significantly for telescopes of other
sizes.  As a result, the well-recognized benefits of operating
meter-class telescopes at temperatures cold enough that they are
natural background limited also apply to larger instruments.  For
example, although a telescope operating at $<$ 5K at 160 -
200\um~will become confusion limited in about 60 seconds, the same
telescope at 8K would require 1000 seconds to reach the same
sensitivity limit; at 15K it would require 100,000 seconds. 
Therefore, designs for future far infrared telescopes should
emphasize operating them at low temperatures. 

\acknowledgements

We greatly appreciate the hospitality of the conference organizers.
The work reported here was supported by NASA as part of the SIRTF
project. 

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\end{thebibliography}
\end{document}

