Predicting signal-to-noise

We'd like to know the expected signal-to-noise level for the sources
we are searching for.  So we want to be able to predict the expected
source signal in "mac-units", i.e. as stored in the data-files, and
then we can combine this with the expected noise level (again, in
mac-units), to get a signal-to-noise estimate.  We can determine the
expected noise level in mac-units by looking at a file taken under
optimum noise conditions.  The resulting predicted signal-to-noise
will help us decide what level of integration time we need to find our
pointing sources, how wide to search, how out-of-focus we can afford
to be, what kind of sky-noise (if any) we can tolerate, etc.

It is often possible to estimate the flux, in Jy per SPARO beam, for a
source.  We need to figure out how to convert from this flux (Jy) to
the signals stored in the data file (Mac units). To do this we will
need to make an observation of a source of known flux (calibrator,
e.g. Moon), and also we need to understand how to correct for
atmospheric absorption.  We'd also like to have a handle on the
uncertainties involved in the signal-to-noise estimated obtained in
this way.

We start with the atmospheric absorption, then begin to address the
calibration of Moon vs. rcw 57 as a *first step* in developing a
method to predict the signal we will get from any source.


Atmospheric absorption:

The signal is proportional to the flux times exp(-tau sec[Z]), where Z
is the zenith angle.

In order to see if the tipper accurately measures the tau that applies
to SPARO observations, we consider two detections of the Moon, on
separate days with very different taus, and, for completeness, two
"probable" detections of RCW 57, on separate days with quite
different taus.  Observations are from Nov 1999.  We compute the
*predicted* signal ratio, assuming that the tau is as given by the
tipper, and the *measured* signal ratio.  Here is how they compare:


SI/SII= .0181	Actual signal ratio: 0.05

SI: Data were taken on November 11, 1999. Tau was 2.56. el=18:34:47.9
    peak signal: 3,000,000 integ units (see "units document")
SII: Data were taken on November 13, 1999. Tau was 1.38. el=19:42:48.9
    peak signal: 60,000,000 integ-units (see "units document")

Note that the Moon was at very high airmass (3-4) so the above discrepancy
can be accounted for by a small fractional error in tau, e.g.
something like 25%.  Such an error is plausible given the different
passbands for the 350 micron tipper and the 450 micron SPARO.  Also,
note that the gain of SPARO depends on time since start of cycle, and
the Moon signal depends on phase of the Moon.  All things considered,
the agreement is reasonable, and thus tau measaured by the tipper can
be used as a reasonable estimate of the actual tau that applies to
SPARO observations.

RCW 57:

SI/SII= .42  Actual signal ratio: .77

SI: Data were taken on November 12, 1999. Tau was 2.14
SII: Data were taken on November 13, 1999. Tau was 1.38
(All signals in Integ Units)

Calibration of Moon vs. rcw 57

We also calculated 

-> the *expected* ratio of Moon flux to RCW 57 flux, based on known
far-ir/submillimeter fluxes for these sources.

-> the *inferred* ratio of moon flux to rcw 57 flux, based on data from
Nov. 1999.  I.e., we corrected the raw Moon signals and raw rcw 57
signals to correct for atmospheric absorption, before taking the ratio.


The theoretical prediction is that the Moon should have 10,000 times
more flux into a SPARO beam.

W.D. Eve et al(1977) measured the brightness temperature of the moon in the 
350 and 450 micron atmospheric windows.  The brightness temperature depends
on the zenith angle of the sun relative to the moon's surface, so as long
as the moon is between half and full, one should see a peak of Tb~300K. 
This, converted to Intensity units is about 4.09 x 10^6 MJy/sr.  

RCW57's flux is known at 60 and 100 microns from iras maps. The color temperature
can be derived knowing this and an emissivity law(~lambda^-3/2) assuming a
thermal profile. The best guess flux for the peak of RCW57 is about 450 MJy/sr.

The ratio of the two intrinsically suggests that the moon is 11,000 times brighter,
but when taking into account the difference in elevation at a tau of 1.38(good weather)
the moon should only be about 600 to 700 times as bright as RCW57.

Note that this means that the measured *signal* from rcw 57 should be
typically about 1000 times smaller than the Moon signal, as it is at a
much higher elevation.


Here are the data for the ratio of the moon flux to rcw57 flux, as we
measured it in Nov. 1999.  All four combinations of [moon]/[rcw 57]
are used.  (We mapped each source twice.)

1.81 e4
6.54 e4
3.34 e4
1.21 e5

The average of all possible combinations is 2.95 e 4, i.e. Moon is ~
30,000 x brighter.  This is a factor of 3 lower than above prediction.
One of many possible possible explanations is that we were out of focus.

The actual measurements taken on the moon (in integ-units) are
60,000,000 for a tau of 1.38(11/13/99) and 3,000,000 for a tau of 2.56
(11/11/99). (See "units document" for discussion of units used for SPARO
signal.)  Our recent detection of rcw57 was around 110,000 on 3/13/00.
Tau is still unknown for this observation.

Using the predicted 450 micron flux of rcw57, the signal as a function of tau is

S(rcw57)=4.57x10^5*exp(-1.14*tau) integ-units (see "units document")

If our prediction is correct, then tau on 3/13/00 would have been 1.25. 


In the March 13, 2000 measurements of rcw57, histogram measurements have
indicated that sigma is around 40,000 integ-units(10,000 mac-units).  Further
tests will examine whether this is consistent with errors determined by
examining frame rate data.  For each position, one nod pair was taken
and there were 3 frames per int. 



If rcw 57 is 1000 times fainter than the Moon, then it will be barely
detectable as a "strip-chart object".  I.e. you should (barely) see it in
real time on stripchart when you do a right and left.  So it should be
easily detected with a few sec of integration.  This is approx. consistent
with our expectations (see Tom's info on Web where detection times of ~10
sec are claimed).

Note if we are a few inches out of focus, or if there is a factor of
three-ish excess noise, or if tau is high, then RCW 57 may not be detectable
in a few seconds and thus may not show up in wide searches.  Finding this
guy is do-able but everything needs to be reasonably close to optimum.


Units document