THE CELESTIAL REFERENCE FRAME AT 24 AND 43 GHz. I. ASTROMETRY

The observing strategy evolved over the 10 sessions. All of the sessions included observations at K band. Four of the sessions alternated observing between K and Q bands, and two sessions alternated between K-band scans and simultaneous S- and X-band (hereafter S/X-band) scans. Observations at S/X-band were made in an attempt to investigate the need for corrections to the measured group delay due to the Earth's ionosphere at K band.

The initial source list was generated by extrapolating flux density data for selected sources from the current ICRF catalog to Ka band using spectral index information estimated primarily from Very Large Array (VLA) data at several frequencies. Initially, only sources with expected Ka-band flux density greater than 1.0 Jy were chosen. For the first three sessions, a total of 108 bright and well-observed ICRF sources were included.

After the first three sessions, adjustments were made in order to optimize the observing strategy for the most accurate astrometric results. To expand the pool of available sources, a list of compact sources brighter than 0.3 Jy at X band was compiled from the ICRF-Ext.2 list (Fey et al. 2004), VLBA Calibrator Survey (VCS; Beasley et al. 2002), and the Microarcsecond Scintillation-induced Variability Survey (MASIV; Lovell et al. 2003). These sources were observed during the fourth session on 2003 May 22 but only at K band. Seventy sources overlapping from the first three sessions were also observed in this survey session as astrometric calibrators in order to improve the positional determinations of the survey sources. Additional suitably strong and compact sources were thus obtained and observed in the subsequent sessions. In the last two sessions, candidate sources near the ecliptic plane were added for future use in deep space navigation.

Attempts were made to combine the K- and Q-band group delays into ionosphere-free delays in the four K- and Q-band sessions but these attempts did not improve the results. As a consequence, no more Q-band experiments were scheduled after the initial four sessions. In two sessions, simultaneous S/X-band scans were scheduled just before and just after K-band scans in an attempt to estimate the ionosphere contribution at K band via the S/X- and K-band delay differences. These sessions were quite valuable in demonstrating the magnitude of the ionosphere delay at K band and thus the need for ionosphere corrections as discussed in Section 2.4.

Initial astrometric processing was made using the NRAO Astronomical Image Processing System (AIPS; Greisen 1988), in a manner similar to that done for the VCS (Beasley et al. 2002). Briefly, the correlated data were first corrected for electronic phase variations using phase calibration measurements. The data were then processed in AIPS to determine the residual group delay (used in bandwidth synthesis) and phase delay rate and their associated uncertainties for each scan. The correlator models were then restored to yield the total delay and phase delay-rate observables. These results were written out in a form suitable for import into the Goddard Space Flight Center (GSFC) astrometric analysis system where initial single session solutions were generated. The GSFC analysis system (Ryan et al. 1980, 1993; Ma et al. 1986; Caprette et al. 1990) consists of the astrometric and geodetic VLBI reduction software CALC/SOLVE.

Accurate astrometric positions were estimated using CALC/SOLVE. The data analysis methods using the GSFC system are covered in detail in Ma et al. (1986). A "typical" global analysis combines data from many different observing sessions, allowing some parameters (e.g., source positions) to be estimated from a combination of all sessions. To obtain a solution, the individual observing sessions are combined sequentially using the "arc"-parameter elimination method (Ma et al. 1990). All solutions give weighted least-squares estimates for parameters. Time-invariant or "global" parameters, i.e., those parameters dependent on data from all sessions, are carried from step to step resulting in a single estimate derived from the combined data of all sessions in the solution. Depending on the problem at hand, these global parameters may include station positions, station velocities, source positions, source velocities (proper motions), the precession constant, and the relativistic gamma factor. Session dependent or "local" parameters depend only on the data from an individual session and are estimated separately for each epoch of observation. Local parameters can include those for the station clocks and atmospheric delay, the Earth's orientation, and nutation offsets in obliquity and longitude. Station positions and source positions can also be local parameters if the goal is to follow changes in these parameters over time.

Two least-squares solutions were performed, one for the K-band (24 GHz) data and one for the Q-band (43 GHz) data. The primary geodetic parameters, the station positions, were estimated separately for each session in the solutions. In this way, any nonlinear motion of the stations (e.g., unmodeled tectonic motion, long-term antenna motion, or earthquake displacements) does not affect the integrity of the estimated source positions. Station motions within a day, from solid Earth tides and ocean loading, were obtained from a priori models (McCarthy & Petit 2003). The troposphere was modeled using the Niell Mapping Function (NMF; Niell 1996). The estimated local parameters for each session included celestial pole offsets in ecliptic longitude and obliquity to account for errors in the IAU2000 precession/nutation models (see McCarthy & Petit 2003), positions of the stations, the rate of UT1 relative to a good a priori time series, 20 minute piecewise linear continuous troposphere zenith parameters, tropospheric gradients in the east–west and north–south directions estimated once per session, quadratic clock polynomials for the gross clock behavior, 60 minute piecewise linear continuous clock parameters, and, if necessary, nuisance parameters such as clock jumps and baseline clock offsets (i.e., separate bias parameters for each VLBI baseline to accommodate small, constant, baseline-dependent instrumental and correlator errors). Source positions were the only global parameters. Only sources with three or more pairs of group delay and phase delay-rate measurements were included in the solution at either frequency band.

Radio positions at K-band were estimated from ten VLBA sessions. This data set consisted of 82,334 group delay and phase delay-rate measurements. The weighting of the data followed standard practice, i.e.,  the errors for each session were adjusted to make them consistent with the internal scatter of the data, giving a reduced χ²_ν near unity. The post-fit weighted root-mean-square (rms) residuals of the solution were 16.53 ps (46.69 fs s⁻¹) for delay (rate) with a combined reduced χ²_ν of 0.97. There were 536 global parameters, 10,391 local parameters, 2276 constraints, and 156,017 degrees of freedom. Radio positions at Q band were estimated from four VLBA sessions. This data set consisted of 19,426 group delay and phase delay-rate measurements. The post-fit weighted rms residuals of the solution were 16.17 ps (35.70 fs s⁻¹) for delay (rate) with a combined reduced χ²_ν of 0.92. There were 262 global parameters, 3986 local parameters, 1177 constraints, and 35,781 degrees of freedom.

The resulting set of source positions at each frequency band defines a coordinate frame that requires only a rotation into the International Celestial Reference System (ICRS; Arias et al. 1995). This frame alignment was achieved separately at each frequency band through an unweighted no-net-rotation constraint imposed on the sub-set of sources designated ICRF defining using their published positions from Ma et al. (1998). Only sources designated ICRF defining having 50 or more pairs of group delay and phase delay-rate measurements at K band and only sources having 30 or more pairs of group delay and phase delay-rate measurements at Q band were included in the constraint at each frequency. The number of sources used in the constraint was 68 at K band and 32 at Q band.

The Earth's ionosphere is a dispersive and refractive medium. The group delay of electromagnetic radiation traveling through this medium will experience a retardation in arrival time as compared to propagation in free space that decreases as the inverse square of the frequency of observation. The vertical (zenith) delay is given in units of seconds as 1.34 × 10⁻⁷ × TEC/ν², where TEC is the vertical total electron content in units of el  m⁻² and ν is the observation frequency in Hz. The ionospheric delay can be quite large at X band, where it can easily exceed a nanosecond of delay, or 30 cm of path length at low observation (elevation) angles. The effects are smaller but still significant at K and Q bands.

The VLBA and most astrometric/geodetic VLBI networks are capable of simultaneous dual-frequency observations at S and X bands and an effective ionosphere-free delay can be obtained by combining the two group delay measurements. The S/X-band systems are optimized for ionospheric calibration, with the more sensitive X-band delay measurement contributing ≈93% of the effective ionosphere-free delay combination and the less sensitive and noisier S-band delay measurement contributing only ≈7%. The VLBA currently is not equipped with a system that is capable of simultaneous K- and Q-band observations so we looked at alternate methods of correcting for the effects due to the ionosphere at these frequencies.

One method tried was to combine the (non-simultaneous) measured K- and Q-band delays in the four sessions that were observed at both bands. These sessions alternated between K- and Q-band observations of the same source, with a short gap in between. Special data sets were made in which the K- and Q-band group delay measurements were extrapolated to the same time tag, approximately the midpoint of the gap between the K- and Q-band observations. This extrapolation added a small amount of noise to the individual group delay measurements, but allowed us to compute an effective ionosphere-free delay. Unfortunately, this Q/K-band combination is not well optimized for ionospheric calibration because the thermal fluctuations of the ionosphere-free delay are dominated by the higher noise inherent in the VLBA Q-band system. The less sensitive and noisier Q-band delay measurement contributes ≈76% of the ionosphere-free delay, while the K-band group delay measurement contributes only ≈24%. Thus, the error (noise) associated with the ionosphere correction was significant compared with the correction itself. Finally, since we were more interested in obtaining corrections for the measured K-band group delays, we did not attempt to compute corrections for the Q-band observations using this method.

Another method of ionospheric calibration tried was to schedule and use two observing sessions in which each K-band observation was bracketed by S/X-band observations just before and after the two-minute duration K-band scan. The ionospheric delays from each of the bracketing S/X-band scans were scaled to K band and interpolated to the same time tag as the K-band observation. This analysis was quite valuable in demonstrating the magnitude of the ionosphere delay at K band, and thus the need for ionosphere calibration at this frequency. However, because this method was tried for only two observing sessions, we instead concentrated on finding a method for ionosphere calibration that could be consistently applied to all 10 K-band epochs and the four Q-band epochs.

Finally, we investigated the use of ionosphere delay values derived from Global Positioning System (GPS) data to correct the K- and Q-band sessions. GPS TEC maps were obtained from the data base maintained by the International GNSS Service (IGS; Dow et al. 2005). Maps were obtained from four IGS analysis centers: the Jet Propulsion Laboratory (JPL; Mannucci et al. 1998), the Center for Orbit Determination in Europe (CODE; Schaer et al. 1995; Schaer 1999), the European Space Operations Center of ESA (ESOC; Feltens 1998), and the Polytechnical University of Catalina, Barcelona, Spain (UPC; Hernandez-Pajares et al. 1999).

The vertical TEC values were obtained from IONEX-formatted (Schaer et al. 1998) tables, which are given at 2 hr intervals with a resolution of 5° × 25 in longitude and latitude. These TEC values were interpolated to the time of observation and line of sight to a source at a given VLBI station. This was accomplished by use of software obtained from the Astronomical Institute, University of Berne, Switzerland.¹¹ The vertical TEC values were referenced to a thin ionospheric shell at h = 400 km altitude and the line-of-sight ionospheric delays were mapped from the zenith by the geometric slant-ratio mapping function M(E) = {1 − [cos E/(1 + h/R)]²}^−1/2, where E is the observation (elevation) angle of a source and R is the mean radius of the Earth. The basic principles of the thin-ionospheric-shell model based TEC determination from GPS data is described in Lanyi & Roth (1988). The final step is to calculate the ionospheric induced delays at the VLBI stations (see Thompson et al. 1986) which are then differenced, in pairs, to estimate the baseline delays. These delays are then added to the theoretical total group delays used in the least-squares analysis described in Section 2.3. After careful examination, differences between the results obtained from the TEC maps obtained from the four different analysis centers were found to be insignificant so we used only the maps obtained from JPL using the GIM 3.0 ionospheric model (Mannucci et al. 1998).

Since the TEC maps use GPS data averaged over 2 hr, obtaining ionosphere corrections for the VLBI observations required interpolating between two maps, effectively smoothing the ionosphere contribution over a period of 4 hr. Thus, it was not possible to account for short-term ionospheric variations. However, applying these GPS derived ionosphere corrections removes a large fraction of the more slowly varying long-term components of the ionosphere.

Figure 1 shows the effect of including the GPS obtained ionosphere calibration on declination at K band. As can be seen, the GPS-based ionosphere calibration produces a significant effect. Hence, all results given in this paper will include this GPS ionosphere calibration. The effectiveness of the GPS obtained ionosphere corrections to the K- and Q-band data is analyzed further in Section 4.4.

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To assess the quality of our K- and Q-band astrometric catalogs, we compared these catalogs to the two S/X-band catalogs described in Section 3. For the various catalog comparisons, we computed position differences in αcos δ and δ, respectively. From these position differences, the weighted mean difference and the weighted rms about the weighted mean were determined. The weights were derived from the root sum square of the positional uncertainties from the two catalogs being compared. These uncertainties were just the formal errors resulting from the astrometric solutions for all catalogs except for the ICRF defining source catalog. The coordinate uncertainties for the ICRF were adjusted by Ma et al. (1998) in an attempt to obtain a realistic error estimate, i.e.,  the formal uncertainties from the ICRF least-squares global solution were inflated by a factor of 1.5 followed by a root-sum-square increase of 0.25 mas. The results for all catalog comparisons are summarized in Table 6. The first column of Table 6 lists the two catalogs being compared and the second column lists the number of sources in common for each catalog pair. Columns 3 and 4 list the weighted mean difference, Δ, in αcos δ and δ, respectively, while Columns 5 and 6 list the associated weighted rms differences about the weighted mean. Columns 7 and 8 list the median absolute differences for comparison with the weighted rms differences. Finally, the last two columns in Table 6 list the statistical significance of the weighted mean position differences for each comparison in each coordinate. Also listed are values for K- and Q-band catalogs with no GPS ionospheric calibration applied, designated K band_NoIon and Q band_NoIon, respectively.

Examination of the values listed in Table 6 shows that the weighted rms differences between all catalogs are less than about 0.3 mas in both coordinates for all comparisons but that there are statistically significant weighted mean offsets between some catalogs. We also find that the weighted rms is consistently higher in δ than αcos δ for all comparisons listed in Table 6 as one might expect given the more limited north–south geometry of the VLBA as compared to some of the networks used in the S/X-band observations.

Listed in Table 6 are the statistics of the differences between the K- and S/X-band catalogs. There were 221 common sources. These results show that there are small but statistically significant differences in the weighted mean offsets between these two catalogs. The comparisons listed in Table 6 show that application of ionospheric corrections to the K-band data reduces the significance of the weighted mean difference between the K- and S/X-band catalogs in δ by more than a factor of 2. The weighted mean difference in αcos δ is statistically significant but does not appear to be affected by the ionosphere calibration. Residual ionospheric/tropospheric effects are typically more pronounced in δ than in αcos δ, but these results suggest that residual effects are still present in the K-band catalog in both coordinates even after application of ionospheric calibration. The ionosphere contribution to the K- and Q-band catalogs is discussed further in Section 4.4.

A similar comparison is listed in Table 6 between the Q-band catalog and the S/X-band catalog. Because there were fewer Q-band sessions and fewer Q-band sources that were observed relative to K band, there are fewer sources in common than at K band. Of the 131 sources for which Q-band positions were derived, 117 were found to be in common with the S/X-band catalog. Again, the comparisons listed in Table 6 show that, similarly to the K-band catalog, application of ionospheric corrections to the Q-band data reduces the significance of the weighted mean difference between the Q- and S/X-band catalogs, primarily in δ, but residual effects are still present in the Q-band catalog in both coordinates.

In addition to the K- and Q-band comparisons with the S/X-band catalogs, we compared these catalogs to the K- and Q-band catalogs with no ionospheric calibration applied, designated K band_NoIon and Q band_NoIon, respectively. These comparisons, also listed in Table 6, show that there are no significant differences between the catalogs with and without ionosphere calibration in αcos δ but there are statistically significant differences in δ.

Next, we compared the K- and Q-band catalogs with each other as listed in Table 6. In this case, there is no significant bias between the two catalogs, i.e., the weighted mean of the differences are not significant in either αcos δ or δ. Also, the weighted means are smaller than those found for the comparisons with the S/X-band catalogs, especially in the case of the declination. This result suggests that application of the GPS ionosphere calibration successfully removed a large fraction of the ionospheric contribution.

For reference purposes, the last three entries in Table 6 compare the K-, Q- and S/X-band catalogs to the ICRF defining source catalog. There were 77 sources in common between the K band and ICRF defining source catalogs and 41 sources in common between the Q band and ICRF defining source catalog. All 212 ICRF defining sources are common to the S/X-band catalog. The main conclusion to draw is that there are no significant differences between these catalogs and the sources that define the ICRF. However, as previously noted, the ICRF formal uncertainties were inflated in order to account for systematic source position errors and unmodeled troposphere errors and are thus larger than the formal errors from any of the other catalogs.

As a final consistency check, we compared catalogs from preliminary solutions produced internally for testing purposes at JPL, GSFC, and USNO using the same calibrated K- and Q-band VLBA data described in this paper. The GSFC and USNO solutions used the same software and physical models as described in Section 2.2 and hence provided a quantitative cross-check on the effect of having independent analysts set up similar solutions. The JPL solution used software and current updates of physical models described in Sovers et al. (1998). The JPL solution provided a quantitative cross-check on the effect of having both an independent analyst and independent software. Parameterization of these preliminary solutions did not differ significantly from that described in Section 2.2. The weighted rms differences between the resultant catalogs were at the 1σ level in both coordinates thereby providing confidence in the integrity of the results that we present.

Overall, the position comparisons made between the various catalogs suggest that the K- and Q-band catalogs are reasonably consistent despite the limited number of observations and sources.

One of the goals of these observations is to determine if the source positions at K and Q bands are more stable with time than those at S/X-band. In a manner similar to that described in Section 2.2, several additional K-band solutions were made using the GSFC CALC/SOLVE software in order to investigate the astrometric positional stability of the sources over the course of the data span. The difference between these solutions and those used to generate the astrometric catalogs is that here a large fraction of the source positions were treated as local parameters (i.e., a position was estimated for each session in which a particular source was observed) and the remaining sources were treated in a global sense as described earlier in Section 2.2. These sources served to create a coordinate frame in which the varying positions of the local sources could be measured. In each of four K-band solutions, ∼1/4 of the sources, evenly distributed in right ascension, were treated as local parameters allowing all sources to be treated in the local parameter sense. Thus, a time series of positions in right ascension and declination was produced for all sources.

To be considered eligible for statistical estimation, we required a source to have been observed in at least 5 of the 10 K-band sessions with a minimum of four observations (scans) per session. The five session minimum was considered sufficient for a reliable estimate of the positional stability of a source. A total of 88 sources met this criteria. For these 88 sources, we computed the weighted mean positions, the weighted rms variations about the weighted mean positions, and the reduced χ²_ν in both right ascension and declination. Due to the reduced amount of data available at Q band as compared to K band, a position time series analysis was not done for the Q-band positions.

The results for all 88 sources at K band are presented in Paper II together with comparisons with source structure indicators determined from the images. Here, the goal was to compare the K-band astrometric positional stability to the stability at S/X band. To accomplish this, similar solutions were made using all available S/X-band VLBI astrometric/geodetic data between 2002 and 2007 (i.e., a similar range in time over which the K-band data were taken albeit with many more S/X-band sessions to consider). For these solutions, the same minimum number of sessions and observation selection criteria were applied. A total of 61 of the 88 K-band sources were found to have time series determined at S/X band over the 2002–2007 period. Whereas the number of sessions for the K-band data ranged from 5 to 10 with a median of 6, the 61 overlapping sources at S/X band had session counts ranging from 9 to 851 with a median number of sessions of 228.

It should be noted that historically the VLBI arrays used to obtain the S/X-band data usually consisted of heterogeneous networks of dissimilar radio telescopes and that the vast majority of sessions were designed specifically for the purpose of geodesy. Although the K- and Q-band sessions reported here were not designed solely for astrometry (they were designed to optimize the competing goals of mutual visibility for imaging and sky coverage for astrometry), the VLBA is a uniform array of identical radio telescopes. Consequently, the temporal consistency of K- and Q-band observations might be expected to be better than at S/X band but systematic errors might be larger.

Shown in the top half of Figure 6 are the distributions of the weighted rms variations in αcos δ and δ for the 61 sources at K band. The distributions at S/X band are shown in the lower half of the figure. Indicated on each plot are the mean and median values of each weighted rms variation distribution. These values are also listed in Table 7 along with the mean and median values of the reduced χ²_ν distributions for both S/X and K band. Comparing the S/X- and K-band results, we find that in αcos δ the distributions are very similar with nearly identical values for the mean and median weighted rms variations. In δ, the K-band variations are roughly a factor of 1.6 and 1.4 greater than at S/X band in mean and median weighted rms variations, respectively.

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In Section 4.2, we evaluated K-band position times series to investigate the variation of source positions over the entire data set. The resulting reduced χ²_ν values for these time varying source positions, listed Table 7, are greater than unity by a statistically significant margin. These reduced χ²_ν values provide evidence that our statistical process significantly underestimates the source position errors.

The least-squares estimation process implemented in CALC/SOLVE, described in Section 2.3, adds white noise in quadrature to the thermal delay measurement errors for each observing session until the reduced χ²_ν is equal to unity for each session. These delay errors are then statistically transformed into parameter errors using the partial derivatives of delay with respect to a given parameter (in this case source position). To compensate for the deficiencies in this process, we recommend that the position formal errors listed in Table 2 be further increased by a scale factor equal to the square root of the applicable mean reduced χ²_ν values listed in Table 7. Since these values are similar for right ascension and declination at K band, the source formal error scale factor at K band is estimated to be about 1.6 for both coordinates. An equivalent scale factor can also be estimated for the S/X-band positions by averaging the mean reduced χ²_ν values from Table 7 with a resultant S/X-band scale factor of about 1.3 for both coordinates. This additional noise is significantly larger than the position scatter expected due to variations in the intrinsic structure of the sources, e.g., Sovers et al. (2002), hence it is more likely to be caused by small unmodeled systematic effects such as residual ionosphere and troposphere. Because a position time series analysis was not done for the Q-band positions a scale factor for the Q-band position formal errors was not estimated.

As discussed in Section 2.4, we applied ionosphere corrections taken from GPS TEC maps to correct the 10 K-band and the 4 Q-band sessions. Figures 7 and 8 show the declination differences, Δδ, as a function of αcos δ and δ, respectively, between K-band positions with/without GPS ionosphere calibration applied and their corresponding S/X-band positions (the S/X-band catalog is described and a reference given in Section 3). No systematic declination dependent effect is expected or seen for Δδ as a function of αcos δ. However, Figure 8 clearly shows a systematic effect in Δδ as a function of δ which increases from north to south and that application of the GPS ionosphere corrections greatly mitigates the magnitude of this effect but does not totally remove it. North of −20° declination, the systematic declination offset between the K-band and S/X-band positions decreases from about 0.4 mas to about 0.1 mas between the uncorrected and the corrected K-band results, respectively. Near the celestial equator, the systematic declination differences decrease from about 1.0 mas to about 0.4 mas. This effect is also seen in the weighted mean declination differences at K band listed in Table 6, which show that the average declination bias decreases from 0.35 to 0.10 mas (∼70%) through the application of GPS ionospheric corrections. As expected, this is consistent with the estimated contribution of −0.24 mas between the K-band differences with and without GPS ionospheric corrections.

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Similarly, comparison of the weighted mean declination differences listed in Table 6 between the Q-band positions with/without GPS ionospheric calibration and the S/X-band positions shows an increase from 0.12 mas to 0.25 mas, suggesting that the ionosphere, on average, increases source declinations at Q band by approximately 0.13 mas. Again, this is consistent with the weighted mean declination difference of −0.13 mas obtained directly from the comparisons of the Q-band positions with/without GPS ionospheric calibration. However, the Q-band value is about a factor of 1.7 times greater than would be expected by simply scaling the K-band contribution by the inverse square of the observing frequencies. Note that increased solar activity when the sunspot cycle was closer to maximum during the 2002–2003 Q-band observations contributed more to ionospheric effects at Q band as compared to K band.

The gross effect of the Earth's ionosphere on astrometric position estimation by VLBI will, on average, cause southern hemisphere sources observed from a northern hemisphere array at low elevation angles to appear systematically higher in the sky relative to their actual positions. The resultant effect for astrometric observations made at low elevation angles using the VLBA is that source declinations will be systematically biased in a positive sense. The effect of the ionosphere on right ascension should be of a more random nature, provided sources are observed symmetrically around transit. As a further complication, atmospheric gradients (azimuthally dependent tropospheric propagation delays) can have similar effects on source positions as the ionosphere. Most VLBI sites, including all of the VLBA sites, are believed to have north–south gradients which do not average out (MacMillan & Ma 1997). On average, tropospheric delays increase toward the equator. If uncorrected, this will cause southern hemisphere sources to appear systematically higher in the sky when observed with a primarily northern hemisphere array, biasing declinations in a positive sense, as was seen for the ICRF (Ma et al. 1998). At K band at least, the ionosphere and gradient effects on source declinations are similar in magnitude. Thus, it is possible for some of the ionosphere contribution to be absorbed in the troposphere and/or gradient models that are used in the CALC/SOLVE analysis but separating these effects is problematic at best and was not attempted.

One of the motivations for constructing a catalog at high frequencies is to diminish the effects of the ionosphere. At X band, the magnitude of ionospheric delays on a VLBI baseline can easily exceed a nanosecond (30 cm of path length). However, as discussed in Section 2.4, this can be corrected to first order by making simultaneous S-band observations, and combining the two measurements into an effective ionosphere-free delay. The VLBA and most astrometric/geodetic VLBI networks are capable of simultaneous dual-frequency S- and X-band observations. Simultaneous dual frequency observing at K and/or Q bands is not currently possible. Evaluation of GPS TEC maps at the epochs of the 10 K-band sessions showed that the absolute value of ionospheric delays at K band reached a peak of around 200 ps on long baselines and averaged around 30 ps (1 cm of path length) for all baselines. Further, the ionosphere corrections decreased by a factor of about 2–3 from the first to the last of the 10 sessions, likely due to decreased solar activity as a consequence of the sunspot cycle.

Additional data obtained using simultaneous dual-frequency observing systems in order to properly remove the ionosphere component from the observations will help determine whether the biases in the declination coordinates of the observed sources can be definitively attributed to ionospheric refraction or other causes (e.g., unmodeled tropospheric effects, tropospheric gradients, etc.). However, it is clear from Table 6 and Figure 8 that the use of GPS ionospheric calibration systematically improves the K-band and the Q-band positions (not shown), primarily in the declination coordinate, with respect to the reference S/X-band catalog.

The detailed effects of the ionosphere on the accuracy of astrometric positions are difficult to determine but there really is no need since almost all VLBI astrometric observations are made using simultaneous dual-frequency systems for which the ionosphere contribution can be easily removed.