[daip] forwarded message from Mark Reid

[Mark Reid]

Leonia,
 
Let me try to answer your questions:

On Mon, 23 Sep 2002, Leonia Kogan wrote:

> Mark,
> 
> Right now I have found a time to infer more precise formulae for delay 
> at the given elevation (ZA) having had the vertical delay.
> 
> These formulae consider a model of the atmosphere: the homogenios belt 
> around the Earth with the belt thick equaled to the vertical delay.
> 
> The formulae are full precise in comparison with the aproximate 'cosec(el)'
> for the plane model of the atmosphere.
> 
> My question is:
> The actual model of the atmosphere can be more complicate than the homogenious 
> sphere belt around the Earth and can be variable in time. 

    True.  However, we are looking at small corrections, typically  
3 centimeters of vertical path length.  These corrections cannot 
be measured to much better than about 0.1 cm.  So the accuracy does 
not need to be any better than about 3% for the correction.  Also,
since the measurements are based on fitting delay residuals to a 
simple model, it does not make sense to think that it would be 
better use a complicated model to apply them to the data.  So, a 
simple model is just fine.

    As for time variations, I proposed that the user supply a time
sequence of vertical delay errors (for each station).  Since these
are coming from observations of quasars, one can assume that there 
will be a small number of measurement times.  The best that one can
do with data like this is to do a simple linear 
interpolation/extrapolation in time (as I proposed).

    I'll attach a simple fortran subroutine that we use to
calculate the effect of a vertical delay change on the data.

Mark



-------------------------------------------------------------
Mark J. Reid                 Phone: 617-495-7470 
Harvard-Smithsonian CfA      Fax  : 617-495-7345
60 Garden Street             Email: reid at cfa.harvard.edu
Cambridge, MA 02138, USA     Web  : cfa-www.harvard.edu/~reid                  
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	subroutine atmosphere_shift(stm,ra,dec,	
     +				    xm,ym,zm,del_tau_z,
     +				    tau_atm)

c	Returns excess atmospheric delay (tau_atm) in radians of phase shift
c       for given station owing to a vertical path error (del_tau_z) in cm's
c       at a GST (stm)

c       This version assumes the excesss path is from a dry
c	atmospheric term.  (Commented out equations shows what
c       form it would be for a wet component instead.)

c       See Thompson, Moran, Swenson Eq 13.41 

	implicit real*8 (a-h, o-z)

	pi      = 4.d0*atan(1.d0)
	rad_hr	= 12.d0/pi 
	deg_rad = pi/180.d0

	stat_eq = sqrt( xm**2 + ym**2 )
	stat_lat = atan2( zm, stat_eq )		! station latitude (rad)
	stat_w_long = atan2( ym, xm )		! station west longitude (rad)

c		local sidereal time of observation...
	sha  = stm - ra - stat_w_long		! source hour angle (rad)

c		calculate source zenith angle...
	cos_za = sin(stat_lat)*sin(dec) + 	
     +			cos(stat_lat)*cos(dec)*cos(sha)

	sec_za = 1.d0 / cos_za
	zenith_angle	= acos( cos_za )/deg_rad
        tan_za          = tan( zenith_angle*deg_rad ) 

c       For consistency with TMS equations, convert to mBars (and back to cm)...
	Po      = del_tau_z  / 0.228d0		                          ! mBars
	tau_dry = 0.228d0* Po*sec_za*(1.d0-0.0013d0*tan_za*tan_za)	  ! cm

c       Were we to think the excess delay was from a wet atmosphere, then...
c	pVo     = ??			! partial pressure H2O at surface in mBar
c					! 5 mBar ~ 0.72 cm prec H20 => approx 5 cm path delay
c	T       = 280.d0		! Surface temperature (K)         
c	tau_wet = 7.5d4*pVo/(T*T)       ! wet delay at Zenith          
c       tau_wet = 7.5d4*pVo*sec_za*(1.d0-0.0003d0*tan_za*tan_za)/(T*T)	  ! cm
c       NB: the only difference is the smaller coefficient of tan^2(za)

	tau_cm	  = tau_dry	                        ! cm
	tau_waves = tau_cm / wavelength			! wavelengths 
	tau_atm	  = tau_waves * twopi			! radians of phase shift


	return
	end