Why source structure affects VLBI observations and how source structure corrupts the geodetic observables are discussed. Evidence is presented which shows that corrections for source structure are possible and necessary for centimeter level accuracy VLBI measurements

Cotton, W. D.

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SOURCE STRUCTURE CORRECTIONS TO THE GEODETIC 
VERY LONG BASELINE INTERFEROMETRY OBSERVABLES 
William D. Cotton 
Department of Earth h Planetary Sciences 
Massachusetts Institute of Technology 
This paper will attempt to address two questions: (1) why does source structure affect very long 
baseline interferometry (VLBI) observations at all, and (2) how does source structure corrupt the 
geodetic observables? Then, finally, evidence will be presented which shows that corrections for 
source structure are possible and necessary for centimeter-level accuracy VLBI measurements. 
Why does structure affect VLBI observations ? How is it possible to determine this structure? Be- 
fore considering more complex sources, let us first review the response of an interferometer to a 
point source. In this case, the received signal can be considered to contain two parts, (1) noise 
from the source which will be identical at the two antennas (see figure l), and (2) the receiver noise, 
etc., which is independent at the two antennas. Thus, when the signals from the two antennas are 
correlated, the peak of the amplitude of the correlation function is proportional to the ratio of the 
system power due to the source to the total system power, independent of the observing geometry: 
P source 
@Ia p 
total 
Unfortunately, nature was not generous enough to supply point sources useful for geodetic applica- 
tions, and we are forced to consider more complex situations. 
POINT SOURCE 
Figure 1 . 
193 
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194 RADIO INTERFEROMETRY 
The simplest possible extended source is an equal brightness double point source, which will be suf- 
ficient to show the general features of extended structure. In this case, the signal from each compo- 
nent is still identical at the two antennas but with different delays. This is demonstrated in figure 2 
on a greatly exaggerated scale. The delay difference causes the two patterns to interfere, reducing 
the observed correlation amplitude. In this case, the correlation amplitude is of the form: 
P 
IPI a F cos(27r~Ar) 
total 
where tr = the observing frequency. The delay difference depends on the observing geometry so the 
observed correlation amplitude will be different on different baselines and vary with time on a given 
baseline. 
EQUAL DOUBLE 
Figure 2. 
In the most general case, the observed correlation function is proportional to the so-called visibility 
function which is the Fourier transform of the source brightness distribution: 
P a v(u,v) = IIB(x,y) exp[-2ni(ux + vy)] dxdy 
where B(x,y) is the brightness distribution. Note that the visibility is complex as this point will be 
relevant to the following discussion. It is possible to deduce the source structure if a sufficient 
number of observations of the correlation function are made and properly calibrated. (See Cotton, 
1979, Astron. J., 84, 1122.) - 
MOVEMENTS TERRESTRIAL AND CELESTIAL 195 
The second question posed at the beginning of the paper was what effect does source structure 
have on the geodetic observables, group delay, and fringe rate. The group delay is defined as the 
derivative of fringe phase with respect to angular frequency: 
Since the visibility, or structure phase, can vary arbitrarily rapidly with observing frequency, espe- 
cially near amplitude minima, the measured group delay can be affected to an arbitrary degree. Ex- 
treme cases of this phenomenon are easily detectable because of the very pronounced minimum in 
the observed correlation amplitude. The structure contribution to the observed group delay is given 
by the following relationship: 
WV) 
dim(V) dRe(V) 
dw - ImW) dw 
rs = 
WI2 
Likewise, the contribution of the source structure to the observed fringe rate is given by: 
aRe(V) du 
-WV) au 1 [ a Im(V) WV) av aRe(V) dv dt - WV) 7 1 dt t WI2 WI2 
Nature has added a further complication; that is, that many sources have time variable structure. 
This has two effects: (1) the corrections to group delay and fringe rate are time variable and (2) the 
apparent position of the source can change. To resolve these difficulties, we need to monitor the 
source structure and to pick, on physical grounds, a feature in the source which is most likely to be 
at a constant position in the sky. 
The bottom line of this paper is, of course, how much effect does source structure really have? To 
address this question, figure 4 shows the effects due to the structure of 3C 345 on observations on 
the Goldstone-Sweden baseline in May 1974. The map of 3C 345 used is shown in figure 3. The 
reference feature for this source is the component on the left which has had a relatively constant 
brightness and size for the last few years and is probably fixed in the nucleus of the parent object. 
In figure 4, the corrections to the delay and rate as well as the visibility amplitude and phase are 
plotted as a function of Greenwich Sidereal Time. Figure 4 indicates that corrections to individual 
delay observations can be many centimeters, especially near minima in the visibility amplitude. 
In order to avoid giving the impression that the problem is always this severe, figure 5 shows perhaps 
a more nearly typical case, Haystack-Goldstone, for the same source and epoch in which the maxi- 
mum corrections are only a few centimeters. Since geodetic measurements involve a number of 
196 
c 
RADIO INTERFEROMETRY 
I I I I I Y, 
0 
0 
= 0:‘OOl 
3C 345 MAY 1974 7.9 GHz 
Figure 3. 
sources, the effects of structure on the final results will be somewhat diluted. While we have not 
performed conclusive tests, we feel that structure has probably not had a serious effect on results to 
date. However, if the full potential of the Mark III system is to be realized, source structure correc- 
tions must be made. 
MOVEMENTS TERRESTRIAL AND CELESTIAL 197 
GROUP DELAY (CM) 
lo 0 lo 
GROUP DELAY (CM) ,n 
lo n 


Source structure corrections to the geodetic very long baseline interferometry observables

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