"Thomas Smid" <[email protected]> writes:
> 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
> > time-dependent 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, T1-T2, 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 T1-T2 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
No. You are confusing the *observable*, which is the time series that
measures the intensity difference between two WMAP receiver feeds, and
the *derived products* (the angular power spectra and cross
The WMAP observable quite obviously is a signed value, since it
involves differencing two positive quantities. It is this difference
that is inherent to the WMAP hardware. The differences are also the
values used by Page et al (2003; see description of analysis therein).
Thus your original claims are erroneous. The "standard deviation" is
*not* the observable, and they contain no hidden bias.
> > 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. ...
Not true. See above.
> ... 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.
Non sequitur. The calibration observations of Jupiter indicate the
response of the instrument to a point source in one of the two
receiver feeds. For measuring the CMB, this is *exactly* what is
needed. I.e. since one desires to know the responses of each receiver
feed to a signal when the other feed is measuring noise (on average).
It's not clear what you think a "beam profile for the difference" is
meant to be. The Page et al (2003) calibrations *are* the measured
profiles of a point source as seen by a differencing instrument.
As for the derived products (cross correlations and/or angular power
spectra), the beam profiles of all the feeds and their cross
correlations are of course handled in the analysis. I already
referred to Hinshaw, et al., (2003, ApJS, 148, 135) which describes
Thus it seems that you are still providing unsubstantiated and