Fitting 2d gaussians to crosses
The 2d gauss fit
Examples of the fit using 3C405
We use crosses on sources for the system
performance and pointing model. For the 12 meter they are:
- Each leg of the cross can be 6 or 8 beams long.
- We take 2 minutes for each leg.
- The lengths in great circle arc minutes are:
- sband beam:40' .. in the computations i used .6 degrees
- 6bms*40'= 240' or 4.0 degrees .. used 3.6
- 8bms*40'= 320' or 5.3 degrees ..
used 4.8 deg
- xband beam: 10' .. i used .15 deg or 9'
- 6bms*10' = 1.0 degrees .. used .9 degrees
- 8bms*10' = 1.3 degrees .. used 1.2 degrees
For model 3A i just fit 1d gaussians to each
az and el strip.
- This gives the pointing offsets and beam widths correctly.
- But the peaks are less than the actual values (unless the
pointing error is 0).
Using a 2d gaussian fit will:
- Give the peak with 0 pointing error.
- It also allows us to discriminate against bad crosses
(rfi, rain, etc..) since it requires both strips to fit the
same 2d gaussian.
- If the source is extended asymmetrically , it will fit for
each width, and the rotation angle as the telescope receiver
probes undergo parallatic angle rotation.
The 2d gauss fit:
The 2d gauss fit is:
- it uses the great circle offsets from the center of the
strip for the fit.
- let x=az and y= el.
- The 8 parameter fit coefficients are:
- C0 = Tsys
- C1 = Gaussian amplitude
- C2 = azimuth pointing offset
- C3 = elevation pointing offset
- C4 = Azimuth gaussian width (in sigmas)
- C5 = elevation gaussian width (in sigmas)
- C6 = is the angle (th) that rotates x,y axis into the
major,minor axis of the 2d gaussian
- C7 = change in Tsys with elevation offset (or with
- The equation is:
- xp,yp are the az,el rotated to the major,minor axis of
the gaussian, with the error offsets removed.
- xp= (x - C2)*costh + (y-C3)*sinth
- yp=-(x - C2)*sinth + (y-C3)*costh
- zfit= C0 + C1*exp(- ( xp^2/C4^2 +
yp^2/C5^2) ) + C7*elOffset (or el)
- Coefficient C7 (change in tsys with eloffset or el)
- This coef is needed because the system temperature
can change with elevation (do to ground radiation,
- For the 305meter,
- we've always used the elevation (or za) offset for
- so only the elevation strip would contribute (the
elevation offset for the az strip is 0).
- But.. the elevation does change during the azimuth
strip because we are tracking the source.
- For 1 minute strips, and elevations that never get
below 70deg, the change in tsys for the delEl in the az
strip was small.
- For the 12 meter telescope:
- the strips are 2 minutes long (to get better signal to
- The elevation goes all the way down to about 6 deg.
- For 3C405 (dec=60) the change in el during an azimuth
- rising : .3 deg
- setting: -.3 deg
of 2d fits to 3C405 (Cygnus A):
Sband crosses (@ 2270 Mhz) on 3C405 were used to
check the 2d gauss fitting.
- the cross arms were 4.8 degrees.
- the beamwidth is about .66 degrees.
- the SEFD at sband is about 4100Jy... So the source was about
25% of Tsys.
- I tried doing the linear elevation fit using the elevation
offset, and using the just the elevation.
- Bad fits (rfi, etc..) were excluded using:
- fit sigma > .005 when el > 20 degrees
- fit width outside of ( .66 +/- .04)degrees
The plots show the results of
the 2d gauss fits (.ps) (.pdf)
- Page 1: examples of strips and fits
- green is the measured strip data, black is the fit using the
elevation offset for the linear term, red is the fit using the
elevation for the linear term.
- The cross was done at 14.9 deg elevation (rising)
- frame 1,2. azimuth strip.
- frame 1 full strip
- frame 2 showing blowup of baseline fit. The black elOffset
fit is flat (since it has no el offset)
- frame 3,4 elevation strip
- frame 3 full strips
- frame 4 blowup showing baseline.
- Page 2: Tsys, src amplitude, sefd fit results:
- black shows fit using C*ElevationOffset, red shows
- + is when the source is rising, * is source setting.
- Frame 1: Tsys fit
- black (elOffset): shows little difference between rising
- red (el): shows a 10 or 20% difference in rise and set
- Frame 2: Src amplitude fit
- both black and red show the same rise, set variation.
- Frame 3: change in elevation for the azimuth strips (vs el)
- rising the el changes by .35 degrees during each az strip
- setting: the el changes by -.35 degrees for each az strip
- The elOffset fits will assume that this change is 0.
- Frame 4: change in elevation for the elevation strips (vs
- the elevation strips move +/- 2.4 degrees from the center
of the source.
- rising the el changes by 5.2 degrees, setting it changes
by 4.4 degrees.
- Frame 5: SEFD vs elevation for the two types of fits
- The black (elOffset*C ) fit shows a 15 -20%difference rise
- the red (el*C) fits track rising, setting pretty well.
- Page 3: fit sigma, tsys/degEl, az error, el error.
- frame 1: the fit sigmas are the same for the two fits
- frame 2: the Tsys/degEl (C*el) coef.. the red (C*elevation)
has a little less difference than the black C*elOffset.
- frame 3,4: Az,El pointing error fit vs el
- these are with the model removed.
- both the red and black fits track each other.
- Page 4: FWHM fits for az and el
- both fits give similar results.
- the average is around .67 degrees.
- Using the elevation offsets for the linear term (C*el)
in the fit gives SEFD values that show a rise,set variation.
- This is probably caused by forcing the elevation change in
the azimuth strip to be 0 (since it used elevation offsets).
- Using the actual elevation for the linear term gives an SEFD
that does not depend on rise set.
- The elevation liner fit does show:
- both Tsys and the srcStrength show a rise,set difference
this cancels with the SEFD
- My guess is that this is a fitting problem.. not a real
problem with the telescope.
- The rise set difference is probably related to the C*el