
Craig Markwardt wrote:
> "Thomas Smid" <[email protected]> writes:
> > The point is that in this case the standard deviation *is* the observed
> > variable:
> > if telescope A measures a temperature T with standard deviation dT and
> > telescope B measures the same, then the most probable absolute value
> > for the difference signal is sqrt(2)*dT (assuming the measurements are
> > uncorrelated, which they should be as the telescopes are pointing into
> > different directions)
>
> Your claims continue to be erroneous. Let's take them one at a time.
>
> "standard deviation *is* the observed variable"  False. If you had
> done some research on WMAP, you would know that the observable is the
> timedependent flux difference between the "A" and "B" telescopes as
> the spacecraft spins. This is an inescapable consequence of the
> design of the observatory.
> [ E.g. http://map.gsfc.nasa.gov/m_mm/ob_techradiometers.html ]
>
> "the most probable absolute value for the difference signal is
> sqrt(2)*dT"  False. It's a simple computer experiment to show that
> the difference you describe is not sqrt(2)*dT. HOWEVER,
>
> Since WMAP's receivers and analog electronics do not measure the
> absolute value, but rather the *signed* difference between the A and B
> fluxes, your whole line of reasoning is irrelevant. If T1 and T2 are
> two random variables with the same expectation (mu) and same standard
> deviation (sigma), then the signed difference, T1T2, has an expected
> value of *zero* and standard deviation of sqrt(2)*sigma.
The sign of the differences between A and B is irrelevant for the
angular power spectrum (or do you see any negative values there?). The
power spectra are obtained by means of a 'quadratic estimator' i.e. the
sign of T1T2 doesn't matter (see
http://lambda.gsfc.nasa.gov/product/map/dr1/ang_power_spec.cfm ). The
power spectrum thus reflects directly the standard deviation
sqrt(2)*sigma.
> However, since you are discussing detections of Jupiter, the
> measurement is clearly not noise. Jupiter has a systematically strong
> signal. Your discussions of gaussian distributions are irrelevant in
> that case.
>
> And to be clear, one WMAP feed detects the strong signal of Jupiter
> (plus the CMB); while the other feed detects only the CMB. The
> measurements displayed by Page et al are the difference signals,
> averaged over many observations and scan patterns. (see Page et al 2003)
>
Jupiter only determines the beam profile for an individual telescope,
not for the difference signal. The whole point of the WMAP design is
subtracting equal (or almost equal) intensities and only in this case
does the statistical noise become relevant. The Jupiter signal is much
stronger than the CMB signal and thus the differential signal is
practically identical to the Jupiter signal. This is why the beam
profile obtained by Jupiter can not be used as the beam profile for the
difference of (on average) equal signal strenghts.
Thomas

