# Fitting 2d gaussians to crosses

28mar16
Intro
The 2d gauss fit
Examples of the fit using 3C405
Summary

# Intro:

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 or 36'
• 6bms*40'= 240' or  4.0 degrees ..  used 3.6 degrees
• 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 elevation)
• 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, atmosphere, etc..).
• For the 305meter,
• we've always used the elevation (or za) offset for this fit.
• 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 noise).
• The elevation goes all the way down to about 6 deg.
• For 3C405 (dec=60) the change in el during an azimuth strip is:
• rising  :  .3 deg
• setting: -.3 deg

# Examples 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 C*Elevation fits
• + is when the source is rising, * is source setting.
• Frame 1: Tsys fit
• black (elOffset): shows little difference between rising and setting
• 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 el)
• 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 to set
• 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.

# Summary:

• 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 linear fit

processing: aosoft/p12m/model/model3A/chk2dfit.pro

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