The VZERO algorithm	- JPH 960212

Summary
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	Following a query by Jayaram Chengalur, this note provides an exegesis of W.N. Brouw's algorithm for determining the phase-zero-difference ('PZD') between the XX and YY interferometer subsets as implemeneted in the Newstar program NCALIB. The 180-degree ambiguity suggested by Chengalur appears indeed to exist; I suggest that a check could be implemented to insure selection of the  proper value for the PZD.
				-------------


Brouw's algorithm
-----------------
	I reproduce here the relevant lines of code from the program module NCAPVZ where the PZD is calculated:


 1	    DO I1=0,STHJ(STH_NIFR_J)-1			!ALL IFRS
 2	      IF (WGT(I1,1).GT.0 .AND. WGT(I1,2).GT.0 .AND. !XY/YX PRESENT
 3	1		ABS(DATC(I1,2)).NE.0) THEN	!SOME POWER
 4		CF=1					!COEFFICIENT
 5		CW=-DATC(I1,2)*DATC(I1,1)		!-YX.XY
 6		R0=ABS(CW)				!WEIGHT
 7		IF (R0.NE.0)
 8	1		CALL WNMLMN(MAR,LSQ_C_COMPLEX,CF,R0,
 9	1			(DATC(I1,2)-CONJG(DATC(I1,1)))/SQRT(CW))
			.
			solve 
			.
	IF (ABS(CSOL).NE.0) THEN
10	  XYDIF=ATAN2(AIMAG(CSOL),REAL(CSOL))		!GET ANGLE
	ELSE
	  XYDIF=0
	END IF
11	CALL WNCTXT(F_TP,'A complex angle of !EC9.2\(!EC9.2) '//
	1		'or !EAR9.2 degrees',
	1		CSOL,CME,XYDIF)
C

	1: Process all interferometers, I1 = interferometer index
	2: DATC = cpomplex visibility, WGT = weight; the first index is the 
	   interferometer number, second index is 1 for XY, 2 for YX. Check 
	   validity of data: if weight WGT equals 0, point has been deleted.	
	3: This check seems to be redundant: If either XY or YX is 0, so will 
	   be R0 in line 6 and consequently lines 8 and 9 will be skipped.
	5: CW = -XY.YX
	6: R0 = weight of this interferometer in the solution.
	7: If the weight is 0, bypass lines 8 and 9 including the division by
	   sqrt(CW).
	8: Accumulate sum of (XY-YX*)/sqrt(-XY.YX) with weights |XY.YX|.
	10: The result is a complex number, the weighted average of the 
	   quantities of line 9 over all interferometers; its argument is the 
	   phase difference we were after.
	11: The test displaying this result is a bit sloppy!
 
So the quantity that is averaged (line 8) is

		P = sqrt(XY/YX) - sqrt(YX*/YX).sqrt(YX/XY)


Interpretation
--------------

	I suggest the following interpretation:

We seek to minimise |V| by multiplying X with a factor a.e^(i.phi), so

		YX' = YX.a^2.e^(2i.phi)

making

		P' = sqrt(XY/YX) - a.e^(i.phi).sqrt(YX/XY)

Setting P' = 0 gives us 

		a.e^(i.phi) = sum[sqrt(XY/YX)] / sum[sqrt(YX/XY)]

The amplitude factor a should be 1 after XX and YY Selfcal, and could be used for a consistency check. NCALIB implicity displays its value by printing out a.e^(i.phi) as a 'complex angle', but otherwise ignores it. 


The PZD ambiguity
-----------------
 
	The 180-degree ambiguity in the VZERO determination noted by Chengalur is represented in the algorithm above by the sign ambiguity of the sqrt() function. 

	It seems to me that whatever algorithm one uses, there is probably no way to avoid the appearance in one form or another of a factor 2 on the unknown phase phi. This means that one must check afterwards in some way that the algorithm selected the right value; for a calibrator there should be enough flux in a single scan to do this straightaway.