ACTA ASTRONOMICA
Vol. 42 (1992) pp. 371–375

Accurate coordinates of antennas participating in Very Long Baseline Interferometry (VLBI) observations are required for a smooth processing of recorded data. The effect of positional errors on the measured observables depends primarily on the observing frequency and the baseline length. The position errors should not exceed a few meters in either of the three coordinates for typical VLBI observations. Previous experiments have indicated that the Toruń station did not satisfy this requirement. The decision to make such measurements was also based on a necessity to use of this antenna as a reference for an initial positional determination of the larger (32-meter) antenna, designed for centimeter and millimeter wavelength VLBI, that is currently being completed at Toruń radio observatory.

In this note we report the observations and data reduction results, which were undertaken for the purpose to improve the presently available coordinates of the Toruń antenna. The 26.5-meter antenna at Onsala (conventionally catalogued as ONSALA85) used in these observations, has been chosen because it has very accuarately determined geodetic coordinates, based on 25 years of geodetic VLBI observations. Also, since the staffs of these two stations cooperate in VLBI for many years now, it was relatively simple matter to organize the observations outside normal European VLBI Network activities.

The observations were performed on 27 and 28 March, 1990 at 4990.99 MHz. Over a total of 12 hours, each of four selected sources has been scanned continuously for up to 2 hours. Since the sensitivity of the 15-meter antenna is low, we were forced to observe only the strongest sources mutually visible by the two antennas: 3C84, 3C273, 3C345 and 0528+134. We aimed at as most uniform as is possible under these circumstances, coverage of the hour angles.

The recorded data were correlated at Medicina (Istituto di Radioastronomia, Bologna, Italy) in June 1990 using the 5-station Mark II VLBI correlator, formerly the Block 0 JPL/Caltech correlator. A year later, also at Bologna, the correlator output was fringe-fitted using the Caltech VLBI package installed on a VAX computer to produce residual group delay and phase delay rate observables at 1 minute intervals. These steps of data reduction included corrections for UT and UT1 difference, atmospheric propagation and for so called retarded baseline effect.

The processing showed that a large fraction (about 75 %) of the recorded data was corrupted by unknown effects (we suspect that the Toruń receiving equipment was unstable). Nevertheless, the remaining good data points (totalling to about 3 hours and including all the mentioned sources) were suitable for further analysis.

The analysis of reduced data (residual observables) have been completed at Toruń on a personal computer (IBM PC/AT compatible) using a relatively simple FORTRAN code. In this analysis correlator residual delays and delay rates were first corrected for the effects of polar motion, axis offsets of the telescopes involved (2.15 m at Onsala and 3.25 m at Toruń), and of Earth tides, all neglected during the postcorrelation processing. No attempt was made to improve correlation and postcorrelation models or include finer effects (eg. relativistic light bending) due to low expected accuracy of our solution. The corrected observables were then simultaneously least-squares-fitted to a geometric model of the Onsala-Toruń baseline error vector supplemented with a linear clock model, in order to yield best estimates of corrections to the baseline assumed in all the processing. Any systematic signals present in the observables were assumed to be purely due to clocks and geometric effects generated by the inaccurate a priori baseline vector. The observables (delay and rate) were modeled in the following manner:

τ = (  Δx c cos t –  Δy c sin t ) cosδ +  Δz c sinδ + a + bT

. τ   = –Ω (  Δx c sin t +  Δy c cos t ) cosδ + b

where Δx, Δy and Δz are the three unknown Cartesian components of the baseline error vector (y-direction being defined positive toward the East); Δ and t are the declination and Greenwich hour angle of the observed source (at its apparent position); c is the speed of light; Ω is the speed of diurnal rotation of the Earth; T is the time (UTC or TAI); and a and b represent the unknown epoch constant and rate of drift of the two clocks relative to each other. The normal matrix is constructed based on the two above equations and then inverted and multiplied by the observations vector to obtain a solution for the five unknowns. Parameter error estimates are then derived from the variances found on the diagonal of the inverted matrix.

In the fitting process, the observed data were weighted according to the inverse of the variances obtained in the standard fringe-fitting. We found that replacing these measured variances by the unweighted means of the squared postfit residuals (a constant for each scan), gave a solution that did not significantly differ from the final one. There, however, remained certain freedom in chosing an additional scaling factor for the rates relative to the weights of delays. Not to let the rates dominate the global solution we balanced the sums of weighted deviations of delay rates with that of the delays by varying the mentioned factor. Again, this ad hoc procedure did not affect in any appreciable way the final estimates (the differences were at the level of a few centimeters) but effectively allowed us to reduce unwanted influence of the rates on the estimates of z-component, which does not directly depend on the rate observable. The smallest residuals and z-coordinate estimation error have been obtained (at the acceptable cost of only slightly higher errors of the other two coordinates) with the factor discussed above (used for weighting) equal to 0.04, implying a multiplicative factor for the rate errors of 1/√{0.04} = 5. The posfit residuals of this final solution resulted in a weighted least squares deviations of 10.1 ns for the delays and 1.75 mHz (or 0.35 ps/s) for the fringe rates.

The parameters estimated with the described downweighting of rates were as follows

Δx = –5.72 ± 0.19   m, Δy = –5.12 ± 0.54   m, Δz = –123.45 ± 1.66   m, a = –0.232 ± 0.003 μs        and b = –1.469 ± 0.007 μs/day.

(for comparison, in the solution without downweighting we found Δz = –123.42 ± 4.04 m for the coordinate exhibiting the greatest difference).

By adding the baseline correction (Δx, Δy, Δz) to the a priori baseline vector itself we have obtained a new estimate of the Onsala-Toruń baseline given in Table 1. The baseline vector is added in turn to the position vector of the ONSALA85 antenna to get position of the Toruń antenna. During correlation and later processing of our data the coordinates of ONSALA85 were fixed at (3370960.13 m, 711467.60 m, 5349664.45 m), values somewhat offset from the best determinations. The position is better approximated from systematic geodetic VLBI experimments performed with the nearby 20-meter antenna (ONSALA60). The ONSALA85 coordinates given in Table 1 were taken from a current geodetic catalogue kept at Haystack and included in the CDP project (e.g. Caprette et al. 1990). They were derived from special VLBI experiments involving the two Onsala telescopes.

Table 1 Baseline vector and coordinates of the antennas Vector  x [m]   y [m]   z [m] Onsala85 (best estimate)    3370968.18 711464.92   5349664.11 Baseline 267641.43 510308.31   –272639.61 Toruń (corrected) 3638609.62   1221773.23   5077024.50      One sigma error 0.19 0.54 1.66

So obtained coordinates of the Toruń antenna are expressed in the IERS Terrestrial Reference Frame and refer specifically to the point on the radio telescope polar axis where it intersects the (perpendicular) plane containing the declination axis. The rectangular coordinates of Table 1 can be readily converted to polar geocentric and geodetic coordinates. Assuming the IAU reference ellipsoid (with 6378140 m for the semimajor axis and 1/298.257 for the flattening) we obtain for the reference point of the Toruń antenna:

Radial distance    6364619.98    ± 1.42 m East longitude    18°33′39.72′′  ± 0.03′′ Geocentric latitude 52°54′37.93′′  ± 0.03′′ Geodetic latitude    53°05′43.79′′  ± 0.03′′ Height above ellipsoid     112.35    ± 1.43 m

The uncertainties given above are one sigma weighted errors derived from the errors of the (Δx, Δy, Δz) vector and the correlations between its components.

Results reported here differ by only a few meters from earlier, much rougher, estimates (Borkowski 1987). To asses the validity of our determination we have also used an independent and more elaborate set of programs called OCCAM, worked out by VLBI geodesists in Spain and Germany (Sardón et al. 1989, Zarraoa 1991). Though OCCAM is designed for somewhat different input data (in principle it accepts only restored observed delays and rates rather then residual ones) we were able to fit our residual observables with it at the expense of introducing certain typical corrections twice (those corrections that were introduced in preprocessing of our data and that OCCAM has automatically performed). In this situation the differences, between the OCCAM output and our results, of a few decimeters in x and y coordinates and a few meters in z might still be considered satisfactory.

Acknowledgements. We would like to thank Prof. A. Kus who made the observations used in this report at Onsala. Thanks also go to Dr. N. Zarraoa for making the OCCAM package available to us and for his advice on running it.

REFERENCES

Borkowski, K.M. and Graham, D.A. 1987, in: Proc. V National Symp. Radio Sci. — URSI ed. by B. Krygier, Uniw. M. Kopernika, Toruń, p. 345 (in Polish).

Caprette, D.S., Ma, C. and Ryan, J.W. 1990, Crustal Dynamics Project Data Analysis-1990, NASA Techn. Mem. 100765, GSFC, Greenbelt.

Sardón, E., Zarraoa, N. and Rius, A. 1989, in: Proc. 7th Working Meeting European VLBI Geod. Astron. ed. by A. Rius, Instituto de Astronomia y Geodesia, C.S.I.C – U.C.M., Madrid p. 103.

Zarraoa, N. 1991, OCCAM v 2.0 – User's Guide, Instituto de Astronomia y Geodesia, C.S.I.C – U.C.M., Madrid.