6.1. TEC Map Accuracy of Cases I and II
 As discussed in Appendix A, the relation between ensemble average of the RMS difference (GPS-VLBI) and uncorrelated error of TEC map is expressed by equation (A6) and (A8) ((A6) is special case of (A8)). The uncorrelated error of TEC map can be statistically evaluated with these relations.
 The GPS-VLBI difference data set was divided into 9 subsets by 10 degrees step of elevation angle. Ensemble averages 〈Fm2(Elx) + Fm2(Ely)〉, 〈ΔTEC〉, and were computed for each data subset and estimate of uncorrelated error σGPS is given by
The uncorrelated TEC error σGPS is plotted with respect to elevation angle in Figure 10 for cases I and II. It is seen that elevation dependency due to mapping function is mostly removed and uncorrelated error level is obviously seen in that figure. As discussed in section 4.1 the error still increases to lower elevation in both cases. It corresponds to that the error increases as the ionospheric point become far from observation point. Slightly rapid increase of the error below 30 degrees should be caused from the elevation cutoff angle 30 degrees in TECMETER observation. From average in elevation angle range 40 – 90 degrees, uncorrelated errors is inferred as 2.7 TECU in case I and 0.5 TECU in case II. Since two stations TEC maps in the case I were estimated independently from independent data set, then common error cancellation effect did not work in case I. Hence the uncorrelated error 2.7 TECU (in range of 1.0 – 5.3 TECU discussed in section 4.1 ) of case I approximately represents the accuracy of our TEC map estimation method using TECMETER described in section 3.1.
Figure 10. Uncorrelated error of TEC map in cases I and II is plotted with respect to elevation angle. The open circles are case I, and the crosses are case II.
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6.2. Accuracy of the GIM/CODE Model
 The relation between RMS difference of GPS - VLBI and GIM/CODE error is expressed equation (A8) on long baseline. The data were analyzed in a similar procedure as in the previous section. The whole data set including both IVS and KSP was divided into 9 subsets of 10 degrees intervals of elevation angle, which was taken from smaller one of VLBI stations pair. Each data subset was divided again into 4 groups by the baseline length (i) 0-500 km (KSP), (ii) 500-4000 km, (iii) 4000 – 8000 km, and (iv) longer than 8000 km. Then σGPS was computed by equation (8) for each subsets. The obtained σGPS is plotted in the upper panel of Figure 11. It is seen that the σGPS increases with baseline length as was seen in Figure 8. As expected from Figure 6, the error of GIM/CODE is almost constant with respect to the elevation angle for cases (i) and (ii). This implies that mapping function is almost appropriate and TEC map error is constant regardless the distances of ISPs from the VLBI stations. The increase of σGPS at lower elevation (at below 20 degrees), which is especially significant at longer baseline, would be caused from error of mapping function. The mapping function (equation (5)) is modeling ionosphere with thin single spherical layer at fixed hight. Also ray path bending effect, which becomes significant at lower elevation angle [Brunner and Gu, 1991], is not included in the model. The reason that effect larger at longer baseline will be understood as follows: Occurrence of low elevation observation increases with baseline length longer. And both stations are observing at low elevation at almost Earth diameter baseline, whereas one station observing at low elevation and the other is at high elevation on middle range baseline. Consequently the error of mapping function affect in double as baseline increases to Earth diameter.
Figure 11. Estimated GIM/CODE TEC error is plotted in respect to elevation angle (upper panel). TEC difference (GPS - VLBI) data set was divided into data subset by 10 degrees intervals of lower elevation angle of the VLBI stations pair. The data subsets were again divided into 4 groups by the baseline length, open circles: 0 – 500 km (KSP), crosses: 500 – 4000 km, open squares: 400 – 8000 km, and open triangles: longer than 8000 km. The GIM/CODE error was computed by using equation (8). Estimated GIM/CODE errors are plotted in respect to baseline length in lower panel. The data were divided into subset by 1200 km interval of baseline length except for the KSP data (109 km). Elevation cutoff test was also performed at elevation limit 20 (open circles), 40 (crosses), and 60 (open squares) degrees. The error bars in the plots (upper and lower panels) indicate 95 % confidence interval.
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 To test how σGPS will change with baseline length, the total data set was again divided into subsets of 109 km (KSP) and of 1200 km intervals of baseline length. Also for the purpose of elevation cutoff test, σGPS were computed for three data set with elevation cutoff angle 20, 40, and 60 degrees. Baseline dependency of σGPS for each elevation cutoff subset is shown in the lower panel of Figure 11. Three elevation cutoff data sets coincide within their 95 % error bars. The error of mapping function becomes significant below 20 degrees of elevation angle (upper panel of Figure 11), so these data will be basically free from the error of inappropriate mapping function including the ray path bending effect. This panel displays vertical TEC error expected on a certain baseline, when the GIM/CODE model is used for ionospheric TEC correction in interferometric measurements. For instance, GIM/CODE model gives ionospheric calibration data about 5 TECU RMS accuracy on Earth diameter baseline. In taking into account the magnification factor by mapping function, GIM/CODE data will provide the same order of ionospheric correction with the results of Ros et al.  on several thousands km baseline.
Figure 12. Square-root of structure function of TEC error computed from TEC difference between VLBI measurements and GIM/CODE model. Three kinds of marks correspond to data of elevation cutoff test (20, 40, and 60). Solid and broken lines are two kinds of models of structure function composed of three truncated functions. Small screen at right corner shows auto-correlation functions derived from two model of structure functions.
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 A Fourier component n of TEC error power spectrum S(n) is computed from Auto-correlation function with Wiener-Khintchine's relation: where To comparison with the TEC error SH coefficient of GIM/CODE data, power spectrum S(n) computed from autocorrelation model is converted to corresponding error amplitude with equation (B4). Figure 13 displays TEC error amplitude spectrum of GIM/CODE model derived from comparison with VLBI measured TEC. Root-sum-square (RSS) of SH components and RSS of errors are taken for each TEC maps, and each mean of them over 12 GIMs were computed for each GIM/CODE files. The mean RSSs of TEC map coefficients and error coefficients are plotted with crosses and pluses on the same figure. Wave number of GIM/CODE data was chosen from larger one between degree and order of SH index set. The obtained TEC error spectrum between two structure function models are not significantly different each other except for the point n = 2. The sink of the error spectrum at n = 2 in model-2 (dashed line) was caused from lack of that component in the model, although it might be related with the similar sink of SH components of GIM/CODE at n = 2, 3. It is seen from the figure that the error attached with GIM/CODE model is always smaller than the error spectra computed from the comparison data. Thus the GIM/CODE data seem to underestimate the error of SH components, especially at low spatial frequency. The TEC measured by VLBI and GPS has inherent difference contributed by interplanetary and interstellar plasma from outside of GPS orbit to the radio source at far distance. It is not easy to evaluate accurately the contribution to VLBI delay observable from them, however it is supposed to be less than one TECU. Because the spatial structure of turburance of cold plasma in the those spaces will be mostly larger than Earth diameter. Hence they affect commonly to the signals of both stations and canceled out. Addtionally electron density is much less than the Earth's ionosphere, so those contributions are expected to be small. Even taking into account that our evaluation of TEC error is slightly over estimate due to above reasons, the GIM/CODE data seem to have larger error than the error at least at DC (plotted at 0.1) and first order SH components. If it is true, the reason for the larger error at lower frequency component might be in the procedure of double differences of GPS data, since it was originally developed for eliminating the error due to intermediate propagation medium.
Figure 13. TEC error spectrum of GIM/CODE model computed from comparison with VLBI measured TEC data. Solid line and broken line correspond to the structure function models in Figure 12. Crosses indicate root-sum-square of SH components of GIM/CODE, and pluses indicate root-sum-square of errors attached with GIM/CODE data. Larger index between degree and order of SH index set was used as wave number and the summation was taken for the same wave number. Horizontal coordinate 0 is indicated at coordinate of 0.1 to express DC component of the error spectrum.
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 Sum-squares of error spectrum components taken at larger wave number than 12 give TEC variation with high spatial frequency, which is not modeled by GIM/CODE (GIM/CODE data used in this paper are expressed by spherical harmonics up to 12 degrees). These values are 0.76 TECU and 0.75 TECU for solid line and broken line models, respectively. Since these two models of structure function were obtained by smoothed interpolation of data, thus they are eliminating fine structure of real one, hence, the obtained power spectrum at higher spatial frequency might be underestimate than real one. Therefore sum of high spatial frequency component of ionospheric TEC, which is not modeled by GIM/CODE, is inferred to be more than 0.8 TECU. Saito et al. have visually demonstrated that smaller (about 100 km) scale TEC wave is propagating over the Japanese Islands [Saito et al., 1998] by using the very dense Japanese GPS observation network (GEONET) [Miyazaki et al., 1997]. The amplitude was about 1 TECU from peak to peak. This scale of perturbation is understood as one example of appearance of such high spatial frequency components. Precise ionospheric information is useful for accurate astrometry VLBI with microwave. If accuracy of ionospheric delay correction at L-band is required to be the same level with current X-band observation, no more than 20 picoseconds accuracy of ionospheric delay correction is required. This corresponds to 0.03 TECU in 1.4 GHz, This means ionospheric map need to be expressed with spherical harmonics up to several hundreds of wave number. Even this has not reality for global ionospheric map, though we expect regional ionosphere map will approach to that level in future.
6.3. S/X VLBI Receiver Offset
 From comparison between GPS-based TEC measurements and VLBI data, difference of signal transmission delay between X-band and S-band of VLBI receiver system is derived as by-product (section 4.2). Since most of the offsets derived from these comparison were almost constants for each baselines regardless of difference of experiments, it is sure that these offsets are originated from each VLBI stations. This hardware delay offset between S and X band VLBI receivers has not been made aware in normal geodetic VLBI observation, because it is absorbed in station clock offset in the analysis. These have already been pointed out by Herring . But these offsets were not actually measured due to lack of independent measurements of ionospheric TEC to distinguish the VLBI receiver offsets from dual frequency delays. Attention may have to be paid to these offsets when the VLBI experiment is used for such a precise time transfer.
 The difference of the offsets was inferred as 0.73 ns (-3.1 TECU in Table 1) on the Koganei-Kashima KSP baseline. Regarding the stations that participated in the intercontinental VLBI experiments used in this analysis, available data are differences of the offsets between VLBI station pairs, thus the VLBI receiver offsets for each station can be determined by fixing one station's offset to known value or giving an additional independent condition. We do not have any prior knowledge on VLBI receiver offset for any stations, then we put a condition that sum of all station's offset equals to 0. VLBI receiver offset is derived with where A is constant used in equation (2). And radio frequencies f1 and f2 are effective ionospheric frequency computed by equation (3). Accordingly, the obtained offsets in TECU were converted to delay then VLBI receiver offsets were solved by weighted least squares method. The offset was almost constant for each baseline except for the data of Kokee-Wettzell baseline on 12th July, which indicated 15 nanoseconds jump from other experiments on the same baseline. This jump of offset cannot be explained by group delay ambiguity, which is order of 50 nanoseconds. Two explanations are possible as the cause of the offset jump. One is that several meter of signal transmission line might be changed only for this experiment on Kokee or Wettzell station, because it corresponded to several meters of cable length. The other possible reason is manual pcal phase operation. Phase delay calibration signal (pcal) is used to calibrate dispersive instrumental delay in each band [Clark, 1976]. However, constant phase delay set is artificially used in case pcal data are not available for that station. That was the case for Kokee station, and it is known that manual pcal phase was used for Kokee station. Manual pcal phase operation introduces arbitrary group delay in observables. The latter reason is likely to be the cause of jump of offset on Kokee-Wettzell baseline in CORE-3001 session. VLBI data of NEOS sessions were processed at Washington correlator and CORE sessions were at Haystack observatory. If manual pcal is applied differently between Washington correlator and Haystack observatory, different delay offset is introduced between NEOS and CORE sessions. And it will cause inconsistency in least-squares network solution of the VLBI receiver offset. Thus Kokee related baseline data of the CORE-3001 session were excluded from least squares analysis of VLBI receiver offset. To make reduced-χ2 equal to unity, square of extra 1.2 nanoseconds was added to square error of each data. The solution set is listed in Table 3. Since the manual pcal operation was taken for Kokee station's data, the estimated VLBI receiver offset of Kokee station in Table 3 may not be related with instumental delay.
Table 3. VLBI Receiver Offset Derived From Comparison Between VLBI-Based TEC and GPS-Based TECa
|Station||Offset, ns||Error, ns|