The effects of source structure on radio interferometry measurements were investigated. The brightness distribution measurements for ten extragalactic sources were analyzed. Significant results are reported

Thomas, J. B.

English

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SOURCE STRUCTURE ERRORS IN
RADIO-INTERFEROMETRIC CLOCK SYNCHRONIZATION
FOR TEN MEASURED DISTRIBUTIONS
J. B. Thomas
Jet Propulsion Laboratory
Pasadena, California
ABSTRACT
Source structure can in t roduce errors in radio
interferometry measurements whenever natural radio
sources are used. To increase our understanding of
this problem, an analysis previously applied to
analytical cases has been extended to the brightness
distributions measured for ten extragalactic sources.
The results of this analysis are presented along with
an approach for avoiding the largest structure errors.
INTRODUCTION
As the accuracy of the radio interferometry technique improves, the
extended structure of natural radio sources will become an increasingly
important source of error. To fur ther increase our understanding of
structure effects, this paper extends an analysis that was presented
at last year's PTTI conference (1). That earlier presentation
included a brief introduction to the theory of structure corrections
and applied the theory to the analytical cases of a double-point
source and a triple-point source. Although analytical examples are
instructive, only a thorough study of many actual source distributions
can give a complete picture of source structure effects. To begin
such an investigation, ten brightness distributions measured by the
Caltech VLBI group (2, 3) have been analyzed. This paper includes a
summary of the results of this more recent analysis and presents a
visibility-dependent limit formula that, if verified, would be
valuable in both estimating and reducing the overall structure effect
in bandwidth-synthesis (BWS) delay. Since BWS delay is currently the
p r i m a r y observable in geophysical and c lock-synchroniza t ion
applications, the analysis will focus on that observable.
635
https://ntrs.nasa.gov/search.jsp?R=19810018958 2020-03-21T12:59:37+00:00Z
STRUCTURE THEORY
In order to tie in with the earlier presentation, the theory of
structure corrections will be briefly discussed before proceeding to
the results. The effect of structure on the cross-correlation signal,
the interferometer fringes, can be obtained by evaluating the Fourier
transform of the brightness distribution:
(u, v) = f f
[u(0-e )4v(y-Y.)
R(u,  = / / D($, y)e dgdy
in which
(u, v)
where ($ , y) are plane-of-the-sky coordinates, ( & 0 > Y 0 ) is an assigned
reference position, D(B , y) is the brightness distribution, (xg , yg) is
the sky-projected baseline vector and A is the observing wavelength.
The modulus of this complex quantity gives the effect of structure on
the amplitude of the fringes while its phase gives the shift in fringe
phase. As one can see, the transform is generally a complicated
function of the instantaneous baseline vector between the two antennas.
The dependence on baseline enters as the "sky-projected" vector (xg,
yg), the two-dimensional vector obtained by projecting the baseline
vector onto the plane perpendicular to the source direction. Thus, to
obtain a general pic ture of the e f f ec t of a given br ightness
distribution, one must compute and plot structure effects as a function
of (xg, y ). As suggested by the transform, it is convenient to
express the two baseline components in units of A , the observing
wavelength at RF. When expressed in this form, the two components are
usually designated (u, v). To obtain the effect of structure on
amplitude and BWS delay, rewrite the brightness transform in the form
i2TT<j>B(u,v)
R(u, v) - |R(u, v)|e
where |R| is fringe amplitude, and 4>g is structure phase. The effect
of structure on BWS delay, which will be referred to as structure
delay, is approximately computed by taking the partial of structure
phase with respect to frequency:
9
*B
AT
 * -6T
636
(The computation of structure delay is actually carried out through an
intermediate calculation of effective position (4). For brevity, that
procedure will not be discussed here.) For the amplitude effect, it is
convenient to define a quantity called fringe visibility that gives
fringe amplitude relative to maximum amplitude:
v = (u> v)
s R(o, o)
Thus, once the brightness distribution for a source has been supplied,
one can compute visibility, structure phase and structure delay as a
function of the sky-projected baseline vector (u, v).
RESULTS FOR MEASURED BRIGHTNESS DISTRIBUTIONS
Table 1 summarizes the 10 brightness distributions measured by the
Caltech group (2, 3) and lists the source name, date of measurement,
observing wavelength, interferometer stations, and the maximum values
of (u,v) present in each measurement. In all, there are seven
separate sources observed on continental baselines at wavelengths of
2.8 to 18 cm. Figure 1 displays the brightness distribution for one of
the sources (Distribution #9, 3C3^5) and shows contours of constant
brightness on the plane of the sky, with east to the left and north
along the vertical.
The measured distributions can be passed through the structure
equations to obtain for each distribution plots of fringe visibility,
structure phase, and structure delay. Figure 2 gives the results for
visibility for the particular distribution in Figure 1. Contours of
constant visibility are plotted as a function of (u, v), the components
of the sky-projected baseline vector. A similar contour plot for
structure delay is given in Figure 3- In the plots, u is defined to
be positive to the east and v positive to the north. Since the u-v
coverage associated with a given distribution should not extend beyond
the maximum values allowed by baseline length considerations, the
contour plots are marked with an approximate boundary outside of which
the output has been discarded. One important point concerning the
current analysis is that all of the structure delays have been
computed on the basis of an "artificial" frequency of 8.3 GHz ( ^ = 3.6
cm). In effect, this assignment of f pretends that, for structure
637
delay computation, all distributions were measured for the specified u-
v values at 8.3 GHz even though they were not. The reason for this
assignment is that most of our work will be carried out at X-band and
it is therefore important to compare all structure delay results at
one frequency. To obtain the structure delay at the actual frequency,
one can easily scale the results by f~ (or by X ). To use the delay
plot for a given observation, one would compute the sky-projected
baseline vector for that observation and obtain the structure delay for
that point on the plot. For this particular source, the structure
delay reaches about 150 psec at its worst point but usually falls in
the range 0 to 50 psec.
Even though the results for the other nine distributions were just as
complex, some general descriptive statements can be made. The
magnitudes of the structure delays (computed for X-band relative to
the centroid) were as large as a nanosecond but typically fell between
0 and 150 psec. The largest delays (^ 1 nsec) occurred in very
local ized regions in the u-v plane whe re very smal l f r i n g e
visibilities ( ^0.03) occurred. On average, structure delay increased
as fringe visibility decreased, as one would expect (4).
A LIMIT APPROACH TO STRUCTURE DELAYS
These results indicate that, if subnanosecond clock synchronization
becomes a goal, some method must be devised for reducing or calibrating
structure effects. In geophysical applications, there is already a
need for delay accuracies better than 0.1 nsec. A possible method
for removing structure delays is the aforementioned calibration scheme
based on measured brightness distributions. The primary difficulty with
this approach is that an individual VLBI structure measurement is
currently a fairly expensive and time-consuming process. The prospect
of working with very large catalogues, possibly containing many time-
varying members, makes the calibration approach even less inviting. It
is therefore worthwhile to investigate alternate methods for overcoming
structure problems.
Another approach is suggested by the general tendency of structure
delay to increase with decreasing fringe visibility. Suppose a general
formula could be established that, purely on the basis of the value of
fringe visibility, places an upper limit on structure delay. Then, if
the limit turned out to be sufficiently small for some upper range of
visibility values, the larger, unacceptable structure delays could be
eliminated by merely deleting observations with the smaller visibility
values. Such an approach is attractive since fringe visibility can
638
often be obtained along with each VLBI observation. If the visibility
determinations were accurate enough, the experiment would not depend on
supplementary measurements of brightness distributions. One important
point that should be made is that the proposed upper limit would not
have to be an absolute limit, valid for every source. For example, it
wou ld be u s e f u l to es tabl ish , if possible, an a p p r o x i m a t e 3a
statistical limit so that structure errors could be treated like other
errors. Another example would be a limit that was valid for all
sources except for infrequent pathological cases. In fits with
redundant observations, such pathological cases could be discovered and
deleted through residual analysis.
To begin an assessment of the limit approach, a particular formula
associated with a double-point source has been tested. The
candidate limit equation is given by
1
~ 4f v
8
which can be rewritten in units of length in the form
where f and X are the observing f r e q u e n c y and w a v e l e n g t h
respectively; where vg ±3 the f r inge vis ib i l i ty for the
observation at hand; and where AT is the structure delay relative to
the ordinary centroid. A plot of the function is shown in Figure 4 for
X-band. Several arguments can be advanced to support this formula as a
general u p p e r l imi t . First , for smal l (u , v) , one can prove
analytically that the formula sets a valid (but loose) upper limit for
any source. Second, for a number of special cases, one can prove that
no greater structure delay can be obtained for a given visibility (4).
Finally, with regard to form, one would expect structure delay to
increase, on average, in inverse proportion to visibility for small
visibilities (4). Although there are these supporting arguments, one
cannot prove analytically that the equation sets a valid upper limit
for any arbitrary distribution. Thus, the next step would be
comparisons with a large number of measured distributions. When the
limit was tested with the ten measured distributions, it was found
that, for those cases, the limit was approximately valid for all vg.
639
If the same result is obtained for nearly all real sources, then the
limit equation would be of great value in both reducing and estimating
structure effects in BWS delays. For example, if one set of a lower
limit of 0.2 on fringe visibility, then the maximum structure delay
predicted by the limit formula would be about 150 psec at X-band (8.3
GHz). Many observations would have visibilities larger than 0.2 and
therefore would have smaller limits on structure delay. Further, for
the cases analyzed here, actual structure delays were almost always
much less than the limit, with the RMS value being about 1/3 of the
limit or less at a given value of visibility. If we hypothesize on the
basis of this information that the aforementioned maximum delay of 150
psec is an approximate estimate of the 3 a delay for all observations,
then the 1 a error in delay for all observations would be about 50
psec. This calculation suggests one might be able to reduce the
1 a structure delay to about 50 psec simply by deleting
observations with fringe visibilities less than 0.2.
SUMMARY AND CONCLUSIONS
To begin a study of structure effects for natural sources, 10
brightness distributions measured by the Caltech group have been
analyzed. The magnitude of structure delay (the BWS delay effect
computed for X-band relative to the ordinary centroid) was as large as
a nanosecond but typically fell between 0 and 150 psec. On average,
structure delay tended to increase as fringe visibility decreased.
These features suggest the possibility of a limit approach to structure
delay. To begin an investigation of this approach, a visibility-
dependent limit formula has been suggested that appears to correctly
set an approximate upper limit on structure delay for 10 particular
brightness distributions. The results for these sources suggest that
source structure effects in BWS delay might be reduced to about 50
psec (1a ) simply by deleting observations with fringe visibilities
less than 0.2. If this technique or some refinement proves successful
in reducing the 1a structure delay to 50 psec or below, then it would
not be necessary to make supplementary brightness distribution
measurements for the purpose of calibrating a source catalogue until
the goal for total delay error (10) falls below about 100-150 psec.
In the near future, these high accuracies will probably be in greater
demand in geophysical and astrometric applications than in clock-
synchronization applications.
640
A final word of caution would be that the limit approach in its present
form has not yet been verified. It may turn out that, at the large u-
v values associated with intercontinental baselines, too few natural
sources possess sufficiently large fringe visibilities in the required
u-v regions. Further, the measured distribitions considered here are
too few in number and too limited in u-v coverage and observing
frequency to provide a general verification. The u-v coverage is too
limited because (a) the absence of short baselines (< 300 km) can
allow important large scale structure to be resolved out and missed and
(b) the absence of intercontinental baselines can allow important
small scale structure to remain unresolved. One example of a source
distribution that would violate the limit formula would be a large,
strong diffuse component displaced by a considerable distance from a
weaker compact component. In this case, the effective position would
move from the centroid of both components to the center of the compact
component as baseline length increased. If the separation of
components were great enough, large changes in structure delay could
occur. More survey data is required to determine whether the
percentage of sources that fall in this category, or other disallowed
categories, is too large for the limit equation in its present form to
be valid. Even if the present form is unsuitable, it should be
possible to construct another visibility-dependent limit that is valid
for nearly all sources. How useful the the final form will be in
overcoming source structure problems remains to be determined.
In future work, we plan to investigate more sources with more detailed
analysis with the hope that the present limit approach, or some refined
version, can be generally established.
ACKNOWLEDGEMENTS
The author wishes to thank S. G. Finley for her enthusiastic and
skillful support in generating the software for the contour plots.
This paper presents results from one phase of research carried out at
the Jet Propulsion Laboratory, California Institute of Technology,
under contract number NAS7-100, sponsored by the National Aeronautics
and Space Administration.
641
REFERENCES
(1) J. B. Thomas, "Source Structure Errors in the Synchronization
of Clocks by Radio Interferometry", Proceedings of Eleventh
Annual PTTI Meeting, NASA Conference Publication 2129, p. 599,
November 1979
(2) T. J. Pearson, A. C. S. Readhead, and P. N. Wilkinson, "Maps of
the Quasars 3C119, 3C286, 3C345, 3C454.3 and CTA102 with a
Resolution of 5 milliarcseconds at 1.67 GHz", Astrophysical
Journal 236f 714, March 1980.
(3) A. C. S. Readhead, T. J. Pearson, M. H. Cohen, M. S. Ewing and
T. A. Moffet, "Hybrid Maps of the Milliarcsecond Structure
of 3C120, 3C273 and 3C345", Astrophysical Journal 231. 299,
July 1979.
J. B. Thomas, "An Analysis of Source Structure Effects in Radio
Interferometry Measurements", A JPL report to be published.
642
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643
AN EXAMPLE BRIGHTNESS DISTRIBUTION
3C 345 DISTRIBUTION 9
, 0".OQ3
DECEMBER 1977, 5011 MHZ E
FIGURE 1
FRINGE VISIBILITY
3C 345 DISTRIBUTION 9
40
00
530
520
10
U-V BOUNDARY
60 40 20 0 -20
U (MEGAWAVELENGTHS)
FIGURE 2
-40 -60
644
STRUCTURE DELAY* RELATIVE TO
BRIGHTNESS CENTROID
3C 345 DISTRIBUTION 9
60 40
FOR X-BAND IN PSEC
20 0 -20
U (MEGAWAVELENGTHS)
FIGURE 3
-40 -60
A LIMIT FORMULA FOR
STRUCTURE EFFECTS* IN BWS DELAY
RELATIVE TO ORDINARY CENTROID
OF BRIGHTNESS DISTRIBUTION
FRINGE VISIBILITY, v$
FIGURE 4
645


Source structure errors in radio-interferometric clock synchronization for ten measured distributions

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