The conditions for the applicability of the phase observable in very long baseline interferometry are discussed and examples of adequate observation programs are given. These considerations apply in particular to the Mark 2 system, which is already in operation at many observatories

Campbell, J.

English

NASA Technical Reports Server

PHASE VERSUS DELAY IN GEODETIC VLBI 
James Campbell 
Geodetic Institute 
University of Bonn 
ABSTRACT 
For high precise baseline and source position determinations with VLBI, large bandwidth techniques 
are required to achieve the necessary delay resolution of S 0.1 ns. In local interferometry, the 
interferometric phase observable offers an equally high resolution in spite of the much smaller 
bandwidths used. In the present paper, the conditions for the applicability of the phase observable 
in very long baseline interferometry (VLBI) are discussed and examples of adequate observation 
programs are given. These considerations apply in particular to the Mark II system, which is already 
in operation at many observatories. 
385 
https://ntrs.nasa.gov/search.jsp?R=19800020335 2020-03-21T17:49:30+00:00Z
386 RADIO INTERFEROMETRY 
DELAY AND PHASE OBSERVATIONS 
In radio interferometry the travelling time r of a wavefront between the antennas at two sites is 
measured by maximizing the cross-correlation function of the two signal streams received at the two 
antennas. Simultaneously, .the phase and the rate of the interference fringes are determined in the 
processor as they vary with the Earth’s rotation. During correlation, the data stream from one sta- 
tion is delayed quasi-continuously in such a way that the changing geometric delay 7s is almost 
completely compensated for. This gives rise to a rather low residual fringe frequency the phase of 
which slowly varies on the scale of a few turns per minute. In order to be able to observe those 
fringes which have maximum amplitude, the correlation is carried out simultaneously in a number 
of delay channels separated by 
1 
AT = - 
2 Bo 
(1) 
where B, is the effective bandwidth of the processed signal streams. The output of the correlator is 
usually described by the complex cross correlation functions, which translates the response of an 
interferometer system to a point source*: 
R (t, Ti) -jGf (t, 7i) = WSNR l D (r - ri) * e , (2) 
where 
WSNR = a factor depending essentially on the signal-to-noise ratio (SNR) of the interferom- 
eter system observing a given source, 
t = observing time (i.e., UTC), 
7 = actual delay including all instrumental and atmospheric effects, 
Ti = discrete time delays as computed for the different delay channels, 
D (7 - ri) = 
sin a B, (T - Ti) 
7T Bo (7 - 71) 
D = delay resolution function, 
@f (t, 71) = residual fringe phase (including all instrumental and atmospheric effects), 
and $f = Wf l t 
(2a) 
(2b) 
where Of = residual fringe frequency. 
(For more details the reader is referred to Thomas, 1972 or Moran, 1976.) 
For the discussion that follows, it is important to note that the interferometric observables used in 
geodetic VLBI are all contained in equation (2), namely the group delay r(t), the phase $f (t), and 
the fringe frequency Wf (t). 
*The source structure is expressed by an extra amplitude and phase term which is omitted here for simplicity. 
WAVES OF THE FUTURE AND OTHER EMISSIONS 387 
The group delay is estimated from the delay resolution function (2a), which becomes a maximum 
for 71 = 7. The accuracy with which the exact fraction of ri can be determined, depends mainly on 
the halfwidth of the delay function, which is given by 
Arh =$. 
0 
(3) 
There are different possible methods of delay estimation* (i.e., Whitney et al., 1976) all of which 
yield a precision c7 of roughly 1 percent of Arh, depending on the SNR and the available integra- 
tion time per observation. The following table is meant to give an overall idea on the delay estima- 
tion performance of different VLBI systems. 
Table 1 
**high-degree of flexibility (Mk III: 0.2 - 300 MHz!) 
The instrumental and atmospheric errors are discussed in connection with the phase observable. 
The residual fringe phase and the fringe rate are obtained from the sine and cosine parts of the 
cross-correlation function (last term in equation (2)). Due to the close relationship of uf and wf 
(equation (2b)), these observables are determined simultaneously, either from an ordinary sine wave 
adjustment or using the fourier transform into the frequency domain, where the maximum of S 
(Wfi) is estimated. After convolution with edwft, the phase for a given instant t can be computed 
from the real and imaginary components at lag zero. These methods, which allow to establish the 
function @ (t) over a certain interval of time (usually the duration of an uninterrupted source scan), 
are often referred to as “phase tracking” methods (Thomas, 1972). The accuracy of phase tracking, 
which is inherent to the system, only weakly depends on system bandwidth through SNR consider- 
ations. So in the case of observations on strong sources, no marked differences in accuracy can be 
found comparing the different VLBI systems. The tracked phase can be converted into a so-called 
phase delay by using 
1 
rdt) - 2 xv0 - - d (0 + f$o (4) 
*Rigorous delay estimation is done iteratively together with the fringe rate estimation: D = D(T, 9). 
388 RADIO INTERFEROMETRY 
where ve = center of observing frequency band, and 
$0 = constant phase zero term, accounting for the unknown number of turns at the start 
of each source scan. 
The phase delay accuracy, which is equivalent to the phase accuracy, is composed of the small phase 
tracking error plus the much larger instrumental and atmospheric phase errors listed below: 
- clock stability (phase variations of the local oscillators) 
- instrumental delay (electronics, cables + mechanical structure) 
- atmospheric delay (ionosphere; troposphere, dry and wet) 
- resolved sources add phase variations due to their structure. 
Typical phase tracking errors on strong sources are about one-tenth of a turn or smaller; this trans- 
lates into phase delay errors of some tens of picoseconds only, corresponding to the observing fre- 
quency used. The overall phase variations due to the above mentioned effects, however, amount to 
about 0.1 ns (if hydrogen masers are used as frequency standards). It is difficult to give a represen- 
tative number here because of the wide variability of atmospheric and instrumental conditions. It 
may be possible in the near future to reduce this number by a factor of two or three using satellite 
LO links, instrumental delay calibration, dual-frequency receiving systems and water vapor radiom- 
etry. But even in the less favourable case of 07@ = 0.1 ns, the phase delay accuracy of any narrow 
band system is equivalent to a group delay precision of a wide bandwidth system with B, = 100 
MHz. 
What can be done to take advantage of the phase observables for meaningful geodetic work, will be 
shown in the next paragraphs. 
GEODETIC BASELINE DETERMINATIONS USING PHASE OBSERVATIONS 
There are several ways in which phase observations are made following the purpose of the experi- 
ment. 
Method A: Uninterrupted Scans for Mapping the Structure of Compact Radio Sources 
In most of these experiments the telescopes are aimed at the same source for many hours, with the 
exception of a few short calibration scans. During each of these scans, the phase can be tracked 
without ambiguity problems, even at higher frequencies, as long as interference fringes are seen. 
For a unique baseline solution with this kind of data, the following conditions have to be met: 
1. The polar component (z = component) of the baseline has to be kept fixed. 
2. At the start of each source scan, a new phase zero unknown (which is equal to a clock offset 
unknown) has to be introduced. 
WAVES OF THE FUTURE AND OTHER EMISSIONS 389 
3. The duration of the sources scans should not be shorter than 3 to 4 hours. An optimum is 
reached if the hour angle range covers 180” (z 12 hours). 
These conditions can be deduced from the observation equations, where the baseline terms ex- 
pressed in cylindrical coordinates are 
dbl = +sin6 (polar component) 
db2 = $ cos 6 cos h (equatorial component) 
dhh = b2 - cos 6 sin h (longitude of baseline vector) 
C 
with 6 = declination and h = hour angle of the observed source. 
An example of a baseline solution with 13 cm data on an intercontinnetal baseline is shown in fig- 
ure 1. These data are the first that could be used with a program completed only very recently. In 
spite of the poor quality of these data (rubidium clocks were used at both stations), a comparison 
with the delay solution of the same experiment demonstrates the substantial gain in accuracy: 
delay solution : 
*r = kg.5 ns 
dbl = 47.52 f 1.62 m 
db2 = 16.00 f 2.86 m 
dhb = 4’.‘97 * 0’.‘32 
phase delay solution: dbl = - - - 
% = 20.20 ns db2 = 19.32 f 0.24m 
dXb = 4’.‘42 +- 0’.‘03 
Another example is shown in figure 2, where phase data on the 832 km baseline Effelsberg-Onsala 
at 1.3 cm (22.2 GHz) were converted into phase delays. Due to the small ambiguity spacing at this 
high frequency, it proved to be difficult to connect the phases across data gaps larger than 5 minutes 
(coherence time of the interferometer). 
The phase delay solution with scan on 3C273 and 3C84 gave the following results: 
% 
= +0.15 ns db2 = 0.15 f 0.054m 
dhb = 3’.‘3 6 +_ 0’.‘008 
The drawback of method A, however, is twofold: the baseline solution is incomplete and only a 
few sources can be observed during one experiment. Moreover, very severe bounds have to be 
placed on the systematic effects due to the instrumentation and the atmosphere because there is a 
high sensitivity of these solutions to any non-linear effects in the phase delay observables. 
3c
 
12
0 
I--
--
-3
h 
- 
I 
-i 
Fi
gu
re
 
1.
 
Ph
as
e 
de
la
y 
re
si
du
al
s 
at
 2
.3
 G
H
z 
(1
3 
cm
) 
on
 t
he
 b
as
el
in
e 
Ef
fe
ls
be
rg
-J
oh
an
ne
sb
ur
g 
(8
02
4 
km
). 
R
b-
st
an
d.
 
at
 E
ffb
 
. : E
 
(Y
 
II c 3 
3C
 
84
 
c 
I 
ph
w
 
dr
lo
y 
; 9 f 0
 
- 
.- 
0 
- 
3 
- 
8 
- - 
2z
k 
t3
* 
O
h 
1’
 
2’
 
3h
 
u-
l- 
Fi
gu
re
 2
. 
Ph
as
e 
de
la
y 
re
si
du
al
s 
at
 2
2.
2 
G
H
z 
(1
.3
 c
m
) 
on
 t
he
 b
as
el
in
e 
Ef
fe
ls
be
rg
-O
ns
al
a 
(8
32
 k
m
). 
H
-m
as
er
s 
392 RADIO INTERFEROMETRY 
Method 6: Interrupted Scans on Source Pairs (Source Switching Between Pairs) for Accurate Deter- 
mination of Relative Source Positions* 
In this case, conditions for a unique baseline solution are: 
1. Polar baseline component is fixed. 
2. A phase zero unknown for each observed source has to be introduced. 
3. Same duration minimum as method A, the optimum being again 12 hours. 
4. The length of the individual scans (including telescope drive time) should not exceed a limit set 
by the possibility to interpolate the phase change up to the correct number of turns between 
two consecutive scans on the same source. This condition is satisfied if the integrated phase 
error is substantially smaller than 180”. Remaining phase ambiguities within the scans pertain- 
ing to one source can be removed in an iterative process. A portion of a plot still containing 
some ambiguity errors is shown in figure 3. 
The number of sources that can be observed in this scheme is larger by a factor of two, which in- 
creases the stability of the solution. Some comparative computations based on a constant observa- 
tional error of 0 
ro 
= 0.1 ns are listed in the following table: 
Source 
Pair 
VRO 
3c 371 
CTA 102 
3c 371 
VRO 
CTA 102 
Table 2 
Baseline Effelsberg - Green Bank (6 335 km) 
Baseline Component 
Decl. Total Observation Accuracy 
1 “3 Time 
b2 lb 
42” 
70” 
-7 h f 8cm *0:‘015 
11” 
70” 
-7h +-10cm +0:‘0.17 
42” 
11” 
-7 h +13 cm + OYOO4 
(duration of individual source scans: 24 mm) 
*Shapiro, Wittels, et al., 1979. 
-5
 
f 
* 
/ 
1.
 
L 
._
- 
I : 
. 
P J 
42
’3
 ? 
? i 
- - I ” ; 
- - z?
! 
- ! ;r -r
 
10
 r
&n
 
/ 
D
ir.
 
Fr
in
,e
 ro
te
 
. 
? 
I L 18’ 58 I phase
 d
rlo
y 
Fi
gu
re
 3
. 
Ph
as
e 
de
la
y 
re
si
du
al
s 
at
 2
.3
 G
H
z 
(E
ffb
.-J
ob
.).
 
So
ur
ce
 S
w
itc
hi
ng
 
M
od
e 
(3
C
27
3 
an
d 
3C
27
9)
 
394 RADIO INTERFEROMETRY 
Method C: Interrupted Scans on Many Different Sources 
The sampling of sources all over the commonly visible portion of the sky would, of course, provide 
the most stable least squares solution, a fact which can best be demonstrated by comparing the cor- 
relation matrices of methods A, B and C: unless the observation period is extended over 12 hours 
or more, the solutions of methods A and B tend to show strong correlations (0.9) between some of 
the estimated parameters. It is interesting, however, that the equatorial baseline projection b2 in all 
cases correlates only very weakly with the other parameters. 
The realization of method C becomes increasingly difficult with an increasing number of sources 
involved in the switching scheme. This is easily understood because the gaps between scans on the 
same source widen considerably, preventing correct phase interpolation. In this case, more sophis- 
ticated methods of ambiguity elimination are required. 
AMBIGUITY ELIMINATION 
In order to obtain a complete baseline solution including the z-component, the remaining phase 
ambiguities between scans to different sources must be resolved. In Rogers et al., 1978 an interest- 
ing method of phase ambiguity elimination is described, which relies on a preliminary wide band 
delay solution. At the relatively high observing frequency of 7.85 GHz, one phase turn corresponds 
to an ambiguity spacing of 3.8 cm or 0.13 ns. This was roughly equivalent to the uncertainty of the 
adjusted group delay observations from the preliminary solution. Now these adjusted delay obser- 
vations were converted to reference phases which were then subtracted from the observed phases. 
After suppression of clock drift effects, the phase differences were treated in a special function 
f(g) = 2 COSA@i. 
i 
cl)* 
If f ($) is maximized by varying the baseline solution parameters, all the different A@i tend to crop 
around zero phase. The values obtained for 3 in this way are used to remove the ambiguities from 
the original phase observations, which are then converted to phase delays and processed in the final 
solution. The accuracy gain proved to be considerable: the postfit rms of the phase delay residuals 
was down to about 10 ps compared to some 200 ps** with the group delay solution. 
From the above experiences, which were obtained with the Mark I system on a very short baseline 
(1.24 km), we may draw one important conclusion (which is independent of baseline length): it is 
possible to resolve phase ambiguities if a preliminary solution with a delay phase accuracy of about 
one turn (or even somewhat more) can be achieved. It is obvious that the requirements for the pre- 
liminary delay solution will become less stringent with decreasing frequency. However, a lower 
*This sum was weighted if the observations are not equally spaced. 
**The synthesized bandwidth was -100 MHz. 
WAVES OF THE FUTURE AND OTHER EMISSIONS 395 
limit of the observing frequency is set by the ionospheric path length variations, which increase with 
l/vc2. Experiments are planned by our group to test the available frequencies at the lower end of 
the GHz-scale. 
With reference to the limited Mark II system’s performance, the following configuration appears to 
hold promise without showing much complexity: A basic observing frequency could be chosen at 
1.66 GHz (18 cm) with an ambiguity spacing of 0.6 ns. In order to resolve this ambiguity, a band- 
width of about 10 MHz would have to be synthesized. This could be achieved with four channels 
spread over the band (see figure 4). In order to reduce SNR losses, two recording sets are used in 
parallel, each switching rapidly between two channels. The resulting delay ambiguity of about 300 
ns can be solved with the coarse delay of each of the channels, while the fine delay itself will have a 
halfwidth of + 100 ns. The accuracy of the preliminary delay solution of this configuration should 
be about 0.8 ns with good SNR, which is in agreement with the above mentioned requirements. 
The phase delay accuracy of the proposed system will be limited by the same effects as considered 
in paragraph 1, this amounts to a few cm or about 0.1 ns on intercontinental baselines. 
The proposed Mark II phase delay system could be equivalent to an economy version of the Mark III 
system with the advantage of being at a lower cost level. On the other hand, if used in conjunction 
with the highly flexible Mark III system, the proposed scheme should result in a substantial reduc- 
tion of tape consumption because only two channels would have to be recorded simultaneously 
(factor of 14). 
The 38 MHz Mark II system (bandwidth synthesis), as it is used at present by the JPL-Caltech group, 
should be particularly suitable for phase delay analysis with ambiguity elimination. 
CONCLUSIONS 
There is no doubt that with wide band group delay techniques the optimal geodetic baseline results 
can be achieved, but it is also clear that these techniques, due to their high costs and complexity, 
will not be in reach of many observatories. 
The test computations with real and simulated data presented here are meant to demonstrate that 
meaningful geodetic work can be done with the existing Mark II technology if the phase observable 
is used. In spite of the incomplete baseline solution (i.e., the z-component cannot be determined), 
it is possible, for example, to measure Earth tides and continental drift on east-west baselines if the 
changes in the equatorial baseline component are considered. Furthermore, the longitude of the 
baseline vector can be used for the determination of UT 1 variations. 
An additional advantage of the proposed phase observation scheme resides in the fact that source 
structure analysis is possible using the closure phase method on the same data. Thus, the structure 
is monitored simultaneously and yet independently of the baseline solution. However, due to the 
396 
RADIO INTERFEROMETRY 
0 
a) 10 MHz synthesis 
b) fine delay function 
Figure 4. System characteristics for ambiguity elimination at 18 cm. 
WAVES OF THE FUTURE AND OTHER EMISSIONS 397 
sensitivity of the pure phase solutions to instrumental and atmospheric effects, the full benefit of 
the high resolution of the phase observable can only be obtained if these effects are monitored with 
great care. 
As an alternative, a system based on Mark II technology is outlined, which is capable of (a posteriori) 
resolving the phase ambiguities in order to achieve a complete baseline solution. With this system, 
it is possible to optimize observing strategies to obtain well conditioned baseline solutions that are 
less sensitive to long term instrumental and atmospheric effects. 
REFERENCES 
Moran, J. M.: Very Long Baseline Interferometric Observations and Data Reduction. Methods of 
Experimental Physics, Vol. 12, Part C, Radio Observations, Academic Press, New York, 1976. 
Rogers, A. E. E. et al.: Geodesy by Radio Interferometry : Determination of a 1.24-km Baseline 
Vector with -5 mm Repeatability. J. Geophys. Res., Vol. 83, p. 325, 1978. 
Shapiro, I. I., Wittels, J. J., et al.: Submilliarcsecond Astrometry via VLBI : I Relative Position of 
the Radio Sources 3C345 and NRA0 512. Preprint submitted to The Astronomical Journal 
April 1979. 
Thomas, J. B.: An Analysis of Long Baseline Radio Interferometry. The Deep Space Network Pro- 
gress Report, Technical Report 32-1526, Vol. VII, VIII, XVI, Jet Propulsion Laboratory, 
Pasadena, Calif., 1972. 
Whitney, A. R. et al.: A Very-Long-Baseline Interferometer System for Geodetic Applications. 
Radio Science, Vol. 11, p. 421-432, 1976. 


Phase versus delay in geodetic VLBI

Abstract

Similar works

Full text

Available Versions

NASA Technical Reports Server