The accuracy inherent in the fringe phase in the determination of source positions was calculated. It is shown that accuracy can be utilized by applying differencing techniques in order to circumvent such systematic effects as clock drift and delay due to the propagation media

Wittels, J. J.

English

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PHASE DELAY ASTROMETRY AND ITS GEODETIC APPLICATIONS 
J. .I. Wittels 
Massachusetts Institute of Technology 
The goal of our experiments was to exploit the accuracy inherent in the fringe phase in the deter- 
mination of source positions.* That accuracy can be utilized by applying differencing techniques in 
order to circumvent such systematic effects as clock drift and delay due to the propagation media. 
The best cancellation of systematic effects could be attained by simultaneous observation of two or 
more sources with the same interferometer. However, having only one antenna at each end of the 
baseline, we chose to observe alternately two closely spaced sources, allowing a short cycle time 
and hence good cancellation of systematic effects by the differencing of the measured fringe phases. 
The nonsimultaneous nature of our measurements created two difficulties. First, for the best accur- 
acy, the arbitrary number of phase rotations between sequential observations of the same source 
had to be removed; or at a minimum, no rotations were to remain between sequential measurements 
of the phase difference between the two sources. Second, we had to assess the effects of systematic 
errors from the less than ideal cancellation of nonsimultaneous observations of the two sources. 
We conducted four experiments between October 1971 and May 1974, each time, cycling between 
observations of the quasars 3C 345 and NRA0 5 12, which are only one-half degree apart. Typ- 
ically, the integration time on each source was about 100 seconds, the cycle time about 4 or 5 min- 
utes, and the full span of the observations 3 to 5 hours. All four experiments were conducted at 
3.8 cm using the Haystack “MA” 37-meter antenna and the Goldstone CA 64-meter NASA antenna; 
the May 1974 experiment also used the NRA0 “WV” 43-meter antenna. All data were recorded 
and correlated with the Mark I system. 
The removal of the 2rn phase ambiguities - a process termed phase connection - was acheived for 
each source separately by manipulating a residual fringe phase, (6, calculated for each observation 
of a source from the observed total phase, pt, by the formula: 
(d = pf -@ - ,’ ~rpoly (t)dt 
to 
where (pp is the phase from a simple geometric model, and $ P”ly is the residual fringe rate calculated 
from a four or five term polynomial fit to the entire ensemble of residual fringe rates of that source 
during that experiment. The residual fringe rates are those found by subtracting, for each observa- 
tion of the source, the fringe rate calculated from the same simple geometric model from the total 
measured fringe rate. 
*See Wittels, J. J. (1975), Ph.D. Thesis, Massachusetts Institute of Technology; and Shapiro, I. I., et al. (1979),&., E, 1459. 
399 
https://ntrs.nasa.gov/search.jsp?R=19800020336 2020-03-21T17:49:35+00:00Z
400 RADIO INTERFEROMETRY 
The ambiguities were then removed by a simple computer algorithm that considered pairs of adja- 
cent observations, and, finally, were checked visually. The definitive test of proper phase connec- 
tion emerges only after parameterized modeling of the differenced phases (This will be discussed 
later.). Figure 1 shows schematically the desired results of proper phase connection. Clearly, the 
noisier the individual measurements of phase, the less reliable the phase connection. 
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After removing the ambiguities, the total phase for each observation of each source is reconstructed 
from the residual phases and the model phases. However, before the phases from the two sources 
are differenced, one other noncancelling contribution to the total phase should be removed: that 
due to the structure of the observed source. For noncylindrically symmetric sources, any structure 
resolvable by the interferometer contributes to the total measured fringe phase. Hence, the best 
accuracy can only be achieved by observing the source structure and removing its contribution. 
For our experiments, NRA0 5 12 could be considered unresolved, However, 3C 345, as we are now 
all aware, had extended and time varying structure, which was only inadequately sampled, especially 
during the first three single-interferometer experiments. Using a simple two-component model for 
the structure of 3C 345, we corrected the measured phases by removing the structure phase with re- 
spect to the model’s center of brightness. Hence, the separation determined by subsequent analyses 
is that between NRA0 5 12 and the center of brightness of the two components in the model of 3C 
345. The source 3C 345 was sufficiently resolved that the structure phase calculated from the 
model reached a value of about a third of a phase rotation. 
It must be emphasized here that the realization of centimeter accuracy in baseline determinations or 
of tenth milliarcsecond or better accuracy in relative source positions using fringe phase observa- 
tions is crucially dependent on knowledge of the structure of the sources. Also, since a very large 
fraction of observed compact radio sources are time-dependent, the structure must necessarily be 
observed at the same epoch as the geodetic and astrometric results. 
WAVES OF THE FUTURE AND OTHER EMISSIONS 401 
Having removed the ambiguities and source structure, for each experiment, we differenced each 
measured fringe phase of one source with the following measured fringe phase of the other source. 
The ensemble of differenced phases for each experiment was then subjected to a least squares 
analysis using a comprehensive parameterized model. * The only parameters varied were the right 
ascension and declination separations of the sources and a constant delay offset, referred to as 
a constant clock; the parameters for the antenna locations, atmospheric models, earth tides, UTl, 
etc. were kept fixed. Examination of the post-fit residuals was the final determining test of proper 
phase connection: When all ambiguities had been properly removed, the post-fit residuals would 
scatter approximately randomly between about +20 psec, where one full rotation comprised 127 
psec. If a phase rotation was inserted between observations, such as in a gap in the October 1971 
experiment, the post-fit residuals became highly systematic and spanned most of a full rotation, 
clearly pointing to an error in phase connection. 
In testing the sensitivity of the results from the least squares analyses to changes in the values of the 
fixed parameters and other effects, we found that the greatest change in the solution values resulted 
from reversing the order of the two sources in the differencing of the phases. Clearly, this sensi- 
tivity resulted from the imperfect cancellation of clock drifts and/or propagation media effects due 
primarily to nonsimultaneous observations. The problem could be best treated by using both the 
sum and difference of the phases in the solution. However, due to a computer program limitation, 
we combined instead the differenced variable with the measured phases from only the stronger 
source 3C 345. The set of 3C 345 phases alone primarily determined a four or five term polyno- 
mial “clock”, while the differenced phases determined the right ascension and declination separa- 
tions of the two sources. The constant clock offset was now fixed at the integral number of phase 
rotations closest to the value determined for this clock by the earlier solution, in which it had been 
varied. 
The values determined for the right ascension and declination separation, Aar and A6, respectively, 
from each of the four experiments combine to give a weighted mean of: 
AaM = zrn 29:436680 f 0~000008 
A6M = 1’ 40’.‘72630 f 0:‘00020 
for the difference in the position of NRA0 5 12 from that of 3C 345. The distribution of the four 
values about this mean is shown in table 1, along with the root-mean-square of the post-fit resi- 
duals for both the differenced and the undifferenced observations. Also shown are the comparable 
values obtained from the group delay data for the two experiments in which wide band synthesis** 
was used. 
*See Robertson, D. S. (1975) Ph.D. Thesis, Massachusetts Institute of Technology. 
**Whitney, A. R., et al. (1976), Radio Science, 11,421. 
402 RADIO INTERFEROMETRY 
Table 1 
Experiment 
Date 
10/71 
05172 
07172 
05174 
Ao - AoM AS - A6M 
(x lo6 set) (x lo5 arcsec) 
Group 
-40 + 80 
150 f 430 
Phase 
5+4 
4?8 
28 f 4 
-45 f 5 
Group 
50+ 40 
180 f 260 
Phase 
5+ 11 
-72 9 
-14+ 19 
12* 15 
Post-fit RMS 
Difference 
WC) T 
12 
19 
13 
14 
Single 
(psec) 
15 
36 
18 
18 
1 
The values in table 1 show that the distribution in As is consistent with the formal errors for the 
individual experiments and for the mean. However, the formal errors for A.cu are too small com- 
pared with the actual scatter. The likely explanation for this discrepancy underlines the importance 
of simultaneous measurements of source structure: 3C 345 is extended primarily in the right ascen- 
sion direction. The structure of this source is only poorly determined for the epochs of the first 
three experiments yet structural scale changes during the 2% years spanned by the experiments are 
of the same order as the scatter in Ao. Taking the conservative view of errors, we estimate our ac- 
curacy as 0’.‘0003 in both coordinates. A conservative limit on the relative proper motion of the 
two sources is 0’.‘00025 + 0!‘00020/year along a position angle of 220” f 60”. 
In conclusion, the accuracy of the differenced phase technique could be maximized by the following: 
1. Full sky coverage for each experiment (11 hours in the case of these two sources with these 
interferometers) would significantly decouple the determination of Aa, and A6. 
2. Simultaneous source mapping would keep the measured separations consistently defined. Of 
course, the reverse application to understanding of the motions within the variable radio sources 
is also enticing. 
3. Good models and values for earth tides, UT1 , polar motion, the atmosphere, etc. 
4. Simultaneous observations and, or at least, more stability in clocks and receivers, short cycle 
times, and cooperative propagation media. Fortunately, indications are that we are presently 
more limited by the equipment than the uncontrollable whims of the propagation media. 


Phase delay astrometry and its geodetic applications

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