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Keywords:

  • GPS;
  • ionospheric TEC;
  • TEC map;
  • VLBI

Abstract

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[1] Precise ionospheric total electron content (TEC) map of the Earth is useful, if it is available for calibration of ionospheric dispersive delay for space measurement techniques using microwave such as GPS, VLBI, and spacecraft navigation. Recent rapid development of GPS techniques is making it more realistic that Earth's ionosphere TEC map measured by GPS observation is practically applicable in those space measurements. For the purpose of evaluating the accuracy of the ionospheric TEC map produced from GPS measurements, two cases of TEC maps were compared with dual band VLBI TEC measurements. In one case, local TEC maps produced from observation data using TECMETERs, which are a kind of GPS receiver for TEC measurements, were compared with VLBI data. As the second case, Global Ionosphere Maps (GIMs) generated by the Center for Orbit determination in Europe (CODE) were compared with VLBI data from short baseline to intercontinental baseline. The ionospheric group delay derived from the local TEC map of the first case had about 80 % correlation with VLBI data on 109 km short baseline. Also the group delay computed by using the GIM data of the CODE (GIM/CODE) had about 90 % correlation with VLBI data on that baseline. In comparisons on intercontinental baselines, correlations between GIM/CODE data and TEC measured by VLBI indicated almost unity. Then it was found that more than 90 % of ionospheric TEC could be predictable with that TEC map. Through further statistical analysis of TEC comparison data, the error spectrum of GIM/CODE data was computed. Comparing the obtained error spectrum with error of the spherical harmonics component of the GIM/CODE data, the latter was always smaller than the former, and the error of GIM/CODE data is suspected to be underestimated, especially at low spatial frequency. It was inferred from the spectrum that more than 0.8 TECU of ionosphere perturbations remain in the higher spatial frequency region, which is not covered by the GIM/CODE model. Total accuracy of GIM/CODE data was evaluated around 3.7 – 3.9 TECU. Also phase delay rates derived from the GIM/CODE were compared with VLBI data. It indicated correlation around 0.6 – 0.8 on intercontinental baseline, but it is not enough accuracy for practical use in phase delay rate correction in VLBI observation. The reason for low coincidence is understood by the lack of small scale and short timescale TEC variation information in that TEC map model.

1. Introduction

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[2] Ionospheric plasma of the Earth is a cause of disturbance in space measurement techniques using microwave signal such as GPS, VLBI, and spacecraft navigation. Geodetic VLBI, which measures several thousands km baseline length in precision of a few centimeters or less, uses ionospheric delay correction by total electron content (TEC) measurement with simultaneous 2 and 8 GHz (S/X band) observation [Herring, 1983; Clark et al., 1985]. However, this technique cannot be applied to all of the VLBI observations. For example, astrometric VLBI observation of pulsar is often performed with single band observation at relatively lower frequency 1 – 2 GHz, because pulsar's flux density decreases rapidly at higher frequency. Then ionospheric delay correction with dual band observation in the same way with geodetic VLBI is difficult due to low signal to noise ratio at high frequency. Consequently the ionospheric delay uncertainty is one of the most significant error sources in that sort of observation. The Earth's ionosphere is also an error source for high precision spacecraft navigation by using range and range rate. High accuracy ionosphere TEC data will contribute to improvement of space position determination.

[3] Global Positioning System (GPS) technology has been developed drastically in this decade, and GPS observation sites are distributed all over the world. Now they can be a tool not only for geodesy but also for monitoring the Earth's environment such as the Earth's troposphere and ionosphere [Beutler, 1998; Feltens and Schaer, 1998]. When a microwave travels in the ionosphere, the signal is delayed by ionospheric plasma proportional to the total number of free electrons content (TEC) in the ray path. Then dual-frequency GPS receivers are good sensors for ionosphere monitoring [Lanyi and Roth, 1988; Wilson et al., 1995; Ho et al., 1997; Schaer, 1999]. GPS-based TEC measurement has potential to contribute microwave measurement system such as single frequency GPS receiver, VLBI astrometry , and space craft navigation.

[4] GPS-based TEC measurements were tried to be used for ionospheric delay correction in VLBI with precision of 9 – 14 TECU (1 TECU = 1016electrons/m2) by Kondo and Imae [1993]. Ros et al. [2000] have achieved sub milli-arc-second accuracy for astrometry in 8.4 GHz by using GPS-based TEC correction in 5.2 – 7.9 TECU precision. The 5.2 – 7.9 TECU correspond to delay of 0.1 – 0.15 nanoseconds at their 8.4 GHz radio frequency, although it corresponds 3.6 – 5.4 nanoseconds delay error at 1.4 GHz, with which pulsar observation is often performed. Thus more accurate TEC data are desirable for ionospheric delay correction in pulsar VLBI observation.

[5] We have evaluated the accuracy of two types of GPS-based ionosphere total electron content maps by comparison with S/X dual band VLBI observation. One is regional ionosphere map estimated from TECMETER, which is special GPS receiver for TEC measurement. The other is Global Ionosphere Map (GIM) generated by the Center for Orbit Determination in Europe (CODE) in Bern University, which is one of the International GPS Service (IGS) Analysis Centers.

[6] A background and direction to evaluate GPS-based TEC accuracy is briefly described in section 2. A procedure to estimate local TEC map and global ionosphere maps provided from Bern University were described in section 3.1 and section 3.2, respectively. There two types of TEC maps are compared with VLBI data in section 4. TEC and TEC-changing-rate (hereafter referred as TEC-rate) derived from GPS data are compared in section 5. The accuracy of the GIM/CODE is statistically evaluated in section 6 then conclusions are summarized in section 7.

2. Background of GPS-Based Ionospheric TEC Evaluation

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[7] When electro-magnetic wave travels in a plasma, the refractive index is expressed with Appleton-Hartree's formula [e.g., Maeda and Kimura, 1984]. If radio frequency of the electro-magnetic wave is much higher than both geomagnetic cyclotron frequency and plasma frequency of the Earth's ionosphere, excess delay due to propagation in the dispersive medium is approximately expressed as follows:

  • equation image

where e: elementary charge, ε0: dielectric constant in vacuum, c: speed of light, f: observation frequency, and TEC: total electron content in the propagation path. VLBI and GPS techniques can measure the TEC along the line of sight by using 2 and 8 GHz and L1(1575.42MHz)/L2(1227.60MHz) signals, respectively. Let f1 and f2 be two observation frequency (f1 < f2) and Δτion be an excess delay difference between the two frequencies, TEC along the propagation path is derived as follows:

  • equation image

where equation image and equation image It is seen from this equation that as the observation frequency (f2) is lower and as bandwidth ratio (fr) is larger, the error of TEC become smaller, if the delay measurement error does not depend on frequency. This is the case of group delay measurements (in geodetic VLBI and code delay of GPS), because the error of observable depend on bandwidth and signal to noise ratio (SNR) but not on observation frequency itself. It is also valid for phase delay measurements, since phase error depend on signal to noise ratio, but not directly on observation frequency. The coefficient part composed from f2 and fr in right hand side of equation (2) is about 4.3 in VLBI using 2/8 GHz and 3.9 for GPS using L1/L2. If observable of geodetic VLBI observation has precision of a few hundreds picoseconds at 2 GHz, then TEC measured by VLBI have several hundreds of milli-TECU precision. Actually, TEC errors of VLBI are mostly less than 1 TECU and RMS was 0.8 TECU in the intercontinental VLBI experiment data used for comparison in section 4.2.2.

[8] The GPS technique can measure the TEC along the ray path from GPS satellite to an observation station, whereas VLBI measure the difference of TEC in the ray path to the radio source between two stations. Then TEC measured by VLBI and that by GPS were compared with the following procedures: (i) TEC map above the station is estimated from GPS observation data (section 3.1). (ii) The coordinates of ionospheric point (ISP), which is an intersection between ionospheric layer and line of sight from VLBI station to an observing radio source at VLBI observation epoch, are calculated. Then vertical TEC value is computed from TEC maps obtained by GPS measurements (section 3.1 or section 3.2). (iii) Slant TEC is computed from the vertical TEC by taking into account an ionosphere mapping function, then difference of the slant TECs between two stations is taken as VTECy · Fm(Ely, H) − VTECx · Fm(Elx, H), where VTECi and Fm(Eli, H) are ionospheric TEC in zenith direction and mapping function at station i defined in the next section. (iv) Finally the derived GPS-based TEC is compared with VLBI-based TEC. Step (i) is skipped in the case of the GIM/CODE data, since we used the global ionosphere map provided from Bern University.

[9] The group observable of VLBI is derived from signal processing with wide bandwidth, thus the frequencies (f1, f2) used for conversion between TEC and dispersive group delay have to be effective ionospheric frequency [Herring, 1983] of each band, which is given by

  • equation image

where equation imagef0 is arbitrary reference frequency, equation image And j, k are index of frequency channels. The ionospheric effective frequencies are hereafter used for VLBI data conversion between TEC and dispersive delay.

[10] Precision of VLBI-based TEC measurement is simply determined by group delay measurement precision and its error is evaluated by signal to noise ratio. Then by supposing that error of VLBI-based TEC measurements is known, error of GPS-based TEC map can be evaluated statistically. Even the comparison of TEC map with VLBI data is relative difference of slant TEC at two points of Earth's ionosphere, that has sensitivity to vertical absolute TEC through mapping function. Thus as far as the error of the TEC map follows appropriate mapping function, the error in slant TEC can be projected upon the vertical absolute TEC value. This is basic idea of evaluating TEC map accuracy by comparison with VLBI data.

3. GPS-Based Ionosphere Map

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

3.1. Local Ionosphere Map Estimated From TECMETER Data

3.1.1. TECMETER Observation

[11] TECMETER is a kind of special GPS receiver [Imae et al., 1991], to measure TEC in line of sight to GPS satellite by measuring difference of arrival time of P-code between L1 and L2 signal. Since bit rate of P-code is 10.23 MHz (clock width is about 98 ns), its time resolution is lower than geodetic GPS receiver, which uses carrier phase as observable. However, the ease in getting the TEC value without complicated processing is an advantage of TECMETER except for the hardware offset described in the next section. Two sets of TECMETER were placed at Kashima and Koganei Key Stone Project (KSP [Yoshino, 1999]) VLBI stations. (KSP VLBI observation data are available at http://ksp.crl.go.jp/index.html, 2000.) The baseline is 109 km length in east-west direction. TEC observation was performed with the same time with dual band geodetic VLBI observation with KSP 11 m diameter antenna from 7th to 17th April 2000. The data acquisition interval was 3 minutes, and elevation cutoff angle of the TECMETER observation was 30 degrees to reduce multi-path effect, whereas VLBI observation elevation cutoff angle was 7 degrees.

3.1.2. Calibration of Instrumental Differential Code Biases of Satellites and Receivers

[12] The observed delay difference of P-code between L1 and L2 GPS signal contains not only the ionospheric dispersive delay of equation (2) but also instrumental hardware biases. Differential code bias of each GPS satellite (DCBS) and receiver (DCBR) are the main error source to estimate absolute TEC by using dual frequency code observation [Coco et al., 1991; Sardón et al., 1994]. The DCBSs are different from satellites to satellites and they are order of several nanoseconds [e.g., Coco et al., 1991; Sardón et al., 1994]. Taking into account the DCBSs and DCBRs, the TECMETER output data are expressed as follows:

  • equation image
  • equation image

where equation image is observed delay difference of P-code in the ith observation along the line of sight from station k to the GPS satellite l. Fm(El, H) is mapping function of single layer spherical ionosphere model described in the next section, where El is elevation angle of the observation, R is the Earth radius, and H is height of ionosphere from the Earth's surface. The mapping function depend on H, which changes between daytime and night time. Although, the height fixed model was used to derive TEC in section 3.1 and section 3.2, thus the mapping function is hereafter referred by Fm(El). equation image is the DCBS of satellite l and τk,DCBR is the DCBR of receiver k.

[13] It is not impossible to estimate the DCBS of each GPS satellite from our observation data, but the stability of the estimated value was not so good in our case. Thus we used the DCBS values provided by the Deutsches Zentrum für Luft- unt Raumfahrt (DLR) [Sardón et al., 1994]. The DLR is one of the institutes analyzing GPS data for geodesy and producing regional ionospheric map above the Europe region. A set of DCBS is obtained as a by-product of the procedure generating the regional TEC map. When one uses the DCBS by the DLR, it need to be aware that their DCBS data are not absolute values but relative values to average of them. Because DCBSs and DCBRs are easily coupled and it is difficult to estimate them separately.

[14] After the DCBSs were subtracted from the observation data, τk,DCBR was estimated by least squares method with observed TECMETER data of two weeks. The data during nighttime were used for the estimation, because ionosphere TEC is more stable in nighttime than in daytime. Consequently the DCBRs were derived as −10.8 ± 0.6 ns for Kashima and −7.9 ± 0.8 ns for Koganei.

3.1.3. Estimation of Ionosphere Map

[15] The Earth's ionosphere consists of D, E, and F layers of plasma. The maximum electron density occurs in the F layer at a height between 200 and 500 km. The electron density decreases rapidly with distance from the F layer. Thus most of electrons in ionosphere plasma is assumed to be concentrated in a thin layer at the first approximation. Based on this approximation, we used single-layer spherical shell model for ionosphere structure. The height of the ionosphere changes in a day, although here we assumed fixed ionosphere height (300 km) for simplicity.

[16] The vertical TEC (VTECi) obtained from equation 2 after calibration of the DCBSs and DCBRs is TEC in zenith direction at ionospheric point (ISP), where line of sight across the ionosphere layer. When ISP coordinates are expressed by (λ, φ), it is useful to use local time λ in longitude direction. Because a pattern of TEC distribution in the ionosphere moves with longitude of the Sun, thus TEC distribution can be expressed by slowly changing TEC model in the sun-fixed coordinates. The local time is λ = UT + λg, where UT and λg stand for universal time and geographical longitude of the ISP. Then λ has two meaning as space axis in longitude direction and time axis at the observation point. The φ indicates coordinate of geographical latitude.

[17] Lanyi and Roth [1988] introduced a thin-shell ionosphere model expressed with two dimensional polynomials. Wilson et al. [1995] proposed the model with spherical harmonics (SH) expansion instead of polynomials to express global TEC distribution. In our case, the GPS data received at only two stations cover very narrow belt shaped region of about 10 degrees width in latitude. Kondo and Imae [1993] have used mixed model of Fourier expansion plus linear trend in one dimension in longitude (time) direction. Two-dimensional VTEC distribution model, which is expressed by product of independent longitudinal and latitudinal functions on the thin ionospheric layer was used in our case. The model is expressed as follows:

  • equation image

Constant offset, linear trend, and 4th order of harmonic variation were assumed in longitude direction by using function f(λ). Constant offset and linear trend were assumed in latitude direction by g(φ). Since proved region of ionospheric layer from single GPS station is in narrow belt shape region about 10 degrees width in latitude (it is seen in TEC map in Figure 1), linear trend or polynomial modeling in latitude direction will be appropriate in this case. Then, equation (6) was fitted to observed data VTECi by using Gauss-Newton non linear least squares method to solve 11 coefficients, ak, bk(k = 1..4), and ck(k = 0..2).

image

Figure 1. These panels are results of least squares model fitting to combined data set Kashima and Koganei TECMETER data from 7th to 9th of April 2000 (case II; see text). The left panel shows the result of model fitting to vertical TEC (VTEC) data on the (local time)-latitude plane. The black pluses, red line, and green pluses, respectively, indicate the VTEC data, fitted model, and residual. The upper right panel shows the common VTEC map above Kashima and Koganei stations. Origin of the horizontal coordinate is local time 0 h of 8th in April. The lower right panel shows the error map of the VTEC model calculated from the correlation matrix of the analysis.

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[18] We have examined two cases in estimating ionospheric TEC with the data sets. In case I, ionospheric maps were obtained from the Kashima and Koganei data sets, separately. In case II, the data sets of both stations were combined and a common TEC map was produced, because the distance between two VLBI sites is only 109 km and the ionospheric conditions were almost the same for both stations. An example TEC map obtained in the case II is shown in Figure 1.

3.2. Global Ionosphere Map Generated by the CODE

3.2.1. Global Ionosphere Model of the CODE

[19] The CODE at Bern University is one of the IGS analysis centers. This institute determines precise GPS orbit by using IGS network observation data and provides the orbit information for world wide GPS users. The CODE has been routinely generating global ionosphere maps (GIM) on a daily basis since 1 January 1996 [Schaer et al., 1998a; Schaer, 1999] by using more than 130 IGS station's data. Their GIM has an advantage that instrumental delay biases are not included in the observable, since double difference of pseudo range measured by carrier phase is used in the analysis. The CODE uses single-layer spherical shell model for the ionosphere structure and the shell height is assumed to be constantly 450 km. The GIM of the CODE (GIM/CODE) is modeled with 149 coefficients of SH expansion up to 12 degrees and 8 orders. Solar-geomagnetic reference frame or Solar-geographic reference frame is used to express the GIM/CODE. Twelve GIMs and their errors in 2 hour intervals (12 maps per day) are included in a GIM/CODE data. And the data are provided in two kinds of formats, in the IONEX format [Schaer et al., 1998b] and in the Bernese ION file format. Daily GIM data from 1 January 1995 and related subroutines are provided on the Internet (http://www.aiub.unibe.ch/ionosphere.html. Figure 2 displays an example of TEC map and its error of GIM/CODE data on 5th July 2000. It is expressed in geographical coordinates, hence it is seen that the TEC structure is meandering along geomagnetic equator.

image

Figure 2. Global ionosphere map data on 5th July 2000 generated by the CODE (upper panel) and the TEC estimation error (lower panel). Horizontal axis is geographical longitude and vertical axis is geographical latitude.

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3.2.2. Computation of TEC in VLBI Observation From the GIM/CODE

[20] We have computed the ionospheric delays of VLBI observations from the GIM model (hereafter referred as case III) and compared them with the dual-frequency VLBI observation results. The comparison results are discussed in the next section.

[21] One set of GIM/CODE data contains 12 GIMs with 2 hours intervals in a day. And recent GIM/CODE are expressed in Sun-fixed geomagnetic reference frame, which takes z-axis in direction of geomagnetic pole and x-axis is fixed in direction to the Sun. Vertical TEC value at arbitrary epoch t at geographical coordinates (λ,φ) is calculated by linear interpolation of nearest two GIM data by following formula.

  • equation image

where tk, and tk+1 are earlier and latter nearest epoch to t in the GIM data, respectively. A coordinates (λ, φ) in geographical reference frame at at epoch t corresponds to coordinates (l(t), φm) in sun-fixed geomagnetic reference frame.

4. Comparison of TEC Maps and VLBI Data

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

4.1. Comparison Between Local Ionosphere Map Generated by TECMETER Observation and VLBI

[22] Two cases of TEC estimation I, and II by using TECMETER data were compared with TEC data measured by dual band KSP-VLBI 109 km baseline observation. The number of scans used for the comparison were 2442 from total 2872 scans of VLBI observation data. Root-mean-square and maximum of the TEC measurement error by the KSP-VLBI observation were 0.21 TECU and 0.88 TECU, respectively. Data of outliers and of low signal to noise ratio were excluded. The results of statistical comparisons are listed in Table 1. When common ionosphere map were estimated with joint data set (case II), coincidence with VLBI-based TEC was increased to about 80 % of correlation and the root-mean-square (RMS) of the difference from VLBI TEC was about 1.2 TECU. This RMS value correspond to excess delay of 25 ps at 8 GHz. Figure 3 shows comparison between VLBI and GPS-based measurement of the case II on 7th April 2000 as an example. The reason of the drastically smaller RMS difference of case II than case I is because common error of the TEC map from two stations are canceled in case II, whereas errors of TEC maps generated for each stations do not correlate each other in case I. In the process to compute RMS TEC difference, offsets between VLBI and GPS-based TEC map data (third lines of each case in Table 1) were subtracted in advance. These offsets are come from instrumental delays accompanied with VLBI receivers as discussed in section 6.3.

image

Figure 3. Comparison of ionospheric TEC measured by VLBI and by TECMETER (case II) on 7th April 2000. The lower left panel shows the ionospheric TEC difference measured by dual band VLBI observation and the upper left panel is the corresponding difference computed from the TEC map generated from combined TECMETER data of two stations (case II; see text). The upper right panel shows the correlation between the TEC measured by VLBI and the TEC measured by the TECMETERs. The correlation coefficient, proportional coefficient, and offset were respectively 0.79, 0.99, and −3.2 TECU. The lower right panel shows difference of the VLBI and TECMETER observation. The RMS of the residual was 1.3 TECU.

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Table 1. TEC Comparison Results Between VLBI TEC Measurements and GPS-Based TEC Measurements on KSP (109 km) Baselinea
  7–8 April9–10 April11–12 April13–14 April15–16 April17–18 AprilAverage
  • a

    In case I, two TEC maps were generated from TECMETER observations at each VLBI station. In case II, the TECMETER data from two stations were combined, and a single TEC map was generated. In case III, the global ionosphere map model produced by the CODE was used. The difference of the TEC along the line of sight from each VLBI station was compared with the corresponding difference in the three cases of GPS-based TEC measurements.

Case IRMS (TECU)5.65.77.55.08.29.56.92
(TECMETER)Correlation0.400.00.100.350.15-0.10.15
 offset (TECU)-3.9-3.9-2.7-3.4-3.2-1.9-3.2
 proportional coefficient0.1-0.10.00.10.00.00.0
Case IIRMS (TECU)1.31.21.31.31.31.01.2
(TECMETER)Correlation0.790.780.780.800.760.840.79
 offset (TECU)-3.2-3.5-3.2-2.9-2.7-3.1-3.1
 proportional coefficient0.990.920.870.960.750.950.91
Case IIIRMS (TECU)0.770.630.770.800.880.730.76
(GIM/CODE)Correlation0.930.940.910.920.870.920.92
 offset (TECU)-3.4-3.3-2.8-2.9-3.0-3.0-3.1
 proportional coefficient1.010.821.030.860.730.740.87

[23] To test elevation dependency of TEC difference (GPS-VLBI), the data set of each experiment was divided into subsets by 10 degree intervals of elevation angles, and mean-square of the difference is computed for each subset. VLBI receiver offset (see section 6.3) was also subtracted prior to taking the difference for each baseline and for each experiment. (This procedure was also taken for the all the other TEC difference data (GPS-VLBI) in this paper.) Mean-square of error of VLBI data was subtracted from it, and then square-root was taken as suggested in equation (A6). Therefore plots in Figure 4 correspond to the first term of the right-hand side of equation (A6). Since differences of elevation angles between the two VLBI observations are less than 1 degree on this KSP 109 km baseline, Figure 4 approximately indicates by factor equation image larger elevation dependency of uncorrelated error of the TEC map. Broken lines in the figure indicate constant error of 1.5, 4.5, and 7.5 TECU in case I (upper panel) and 0.25, 0.55, and 0.85 TECU in case II (lower panel) magnified by mapping function (5). The distribution of RMS TEC difference in the upper panel almost follows the mapping function's elevation dependency. This is interpreted as follows: The two ionospheric TEC models were produced independently, hence, those errors are uncorrelated each other. Consequently, error derived above the observation station represents the error of the TEC map. Thus an accuracy of TEC map by our estimation method is inferred from the figure to be in range of 1.0 – 5.3 TECU (divided by factor equation image Faster increase than mapping function at low elevation angle implies that the accuracy of TEC map decreases with the ISP goes far away from the observation point.

image

Figure 4. The RMS difference of TEC between VLBI and TECMETER measurements on short baseline (109 km) are plotted as a function of elevation angle. The elevation angles are almost the same for both VLBI stations. The each RMS were computed from data subsets in each 10 degrees step of elevation angle. Upper panel is for case I and lower panel is for case II. Broken lines indicate constant error magnified by mapping function factor (equation (5)). Each line corresponds to constant errors 1.5, 4.5, and 7.5 TECU from the bottom in the upper figure. Each line corresponds to constant errors 0.25, 0.55 and 0.85 TECU from the bottom in the lower figure. Each mark indicates the data of observation on 7th, 9th, 11th, 13th, 15th, and 17th of April 2000.

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[24] The magnitude of RMS TEC difference between case II is almost one order smaller than that of case I, and it increases rapidly at low elevation angle. These are understood as follows: Large part of error of the map in case II is canceled out as common error. According to decrease of elevation angle, ISPs of the two VLBI observation stations apart each other and this cancelation less works. And the TEC map accuracy decreases as the ISP becomes far away from the observation point.

4.2. Comparison Between the GIM/CODE and VLBI

4.2.1. Comparison on 109 km Baseline

[25] Results of comparison between GIM/CODE data and TEC derived by KSP VLBI experiments for 6 days of 24 hours (2683 scans are used) are also summarized in Table 1. An example of scattered plot and difference residual plot is shown in Figure 5. The GIM/CODE data show better coincidence with VLBI-based TEC than TECMETER measurement in case I and II. The correlation is mostly larger than 0.9, RMS difference is about 0.8 TECU. The offset between the GIM/CODE data and VLBI is −3.1 TECU in average, and it is the same with that of case II. The offset 3.1 TECU corresponds to 0.73 ns difference in delay between S- and X-band and to ten and a few centimeters in electric length. Since GIM/CODE data do not contain instrumental bias in principle, this range of delay difference can be explained by signal transmission delay difference between X-band and S-band VLBI receiver system. More detail on VLBI receiver offset is discussed in section 6.3.

image

Figure 5. Comparison of the ionospheric TEC measured by VLBI and the global ionosphere map model by the CODE (case III) on 7 April 2000. The lower left panel shows TEC difference measured by dual band VLBI observation. The upper left panel is the corresponding difference computed from the global ionosphere map by the CODE. The upper right panel shows the correlation between the TEC measured by VLBI and the TEC derived from the GIM/CODE model. The correlation coefficient, proportional coefficient, and offset were respectively 0.93, 1.01, and −3.4 TECU. The lower right panel is difference of the VLBI-based TEC and the TEC derived from GIM/CODE model. The RMS of the difference was 0.77 TECU.

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[26] As is the same with the previous section, the data set of each experiment is divided into 9 subsets by elevation angle, and corresponding quantities of square-root of the first term of equation (A6) are computed. The upper panel of Figure 6 shows a plot of them with respect to elevation angle. As is obvious from equation (A6), it shows elevation dependency of the uncorrelated error of GIM on 109 km baseline. Broken lines in that panel indicate constant TEC error of 0.3, 0.4, 0.5, and 0.6 TECU magnified by mapping function (formula (5)). Elevation dependency of the errors coincide well with these lines. This implies that the uncorrelated error of GIM is constant regardless of distance from the observation point to ISP. This is reasonable because: (i) The GIM/CODE data have no relation with the VLBI observation station position. (ii) Distance from observation station to ISP will be within minimum spatial resolution of GIM/CODE, which is about 2500 km × 1700 km scale. As described in section 4.1, the small RMS difference between VLBI and the GIM/CODE is caused from cancellation effect of common error, since the baseline is short and most part of the ray path in the Earth's ionosphere is the shared by both VLBI stations. The uncorrelated error of the GIM/CODE data on 109 km baseline is estimated to be no larger than equation image TECU from this figure.

image

Figure 6. The RMS difference of TEC between VLBI and GIM/CODE data are plotted as a function of smaller one of elevation angles of the two observation stations. Upper panel is RMS plot of 109 km baseline and lower panel is that of intercontinental baseline. Broken lines in the upper panel indicate constant error magnified by factor of mapping function (equation (5)). Each line corresponds constant error 0.3, 0.4, 0.5, and 0.6 TECU. The lines in lower panel indicate constant error magnified by 1 + Fm(El) (see text), and they correspond to 1, 4, and 7 TECU error from the bottom. Each mark indicates different baselines.

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4.2.2. Comparison on Intercontinental (Thousands of Kilometers) Baseline

[27] Intercontinental geodetic VLBI experiment data were downloaded from the International VLBI Service (IVS) data center (IVS products are available electronically at http://ivscc.gsfc.nasa.gov/service/products2.html, 2000), and totally 6855 scans of 37 baselines were used for TEC comparison with the GIM/CODE data. The name of the VLBI experiments used for the comparison is listed in Table 2. VLBI TEC measurement error were mostly less than 1 TECU, and RMS and maximum of the error were 0.85 TECU and 4.3 TECU respectively. An example of scattered plot and difference residual plot is shown in Figure 7. Since two radio signals shared almost no common ray path in the Earth's ionosphere on intercontinental baseline, contribution of dispersive excess delay became larger on longer baseline. The comparison result indicated that the correlation coefficient approached to unity as the baseline became longer.

image

Figure 7. Comparison of the ionospheric TEC measured by VLBI and the GIM/CODE model on 18th July 2000 for Algonquin - Wettzell about 6000 km baseline. Upper panel shows scattered plot of TEC measured by VLBI and GPS. The correlation coefficient, proportional coefficient, and offset were respectively 0.99, 1.13, and 57.6 TECU. Lower panel is difference residual plot of the VLBI-based TEC and the GIM/CODE model. The RMS of the difference was 5.4 TECU.

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Table 2. Intercontinental VLBI Experiments Used for TEC Comparison
DateExperiment NameStation Name
5–6 July 2000NEOS-A375Algonquin, Fortleza, Kokee, Wettzell, Gilcreek
10–11 July 2000CORE-1001Algonquin, Gilcreek, Hartrao, Hobart, Matera, Tsukuba
11–12 July 2000NEOS-A376Algonquin, Fortleza, Kokee, Nyales, Wettzell,
12–13 July 2000CORE-3001Gilcreek, Kokee, Onsala, Westford, Wettzell
18–19 July 2000NEOS-A377Algonquin, Fortleza, Kokee, Wettzell, Gilcreek,
25–26 July 2000NEOS-A378Algonquin, Fortleza, Kokee, Nyales, Wettzell

[28] Lower panel of Figure 6 shows elevation dependency of RMS TEC difference on intercontinental baseline made in the same way with the upper panel. Total data set is divided into subsets by 10 degrees intervals with smaller elevation angle of two VLBI stations. Then mean-square of TEC difference (GPS-VLBI) were computed for each data subset regardless of difference of experiments and mean-square of VLBI TEC measurement errors were subtracted before taking square-root. Based on the statistical assumption of equation (A8), the data consist of elevation dependency of mapping function and error of GIM/CODE model. And elevation angle difference between two stations becomes larger on longer baseline. That is to say, one station is observing at low elevation when the counter part station is observing at high elevation angle. Vertical TEC error is magnified by mapping function, thus the error budget is mostly dominated by TEC error of lower elevation station. Hence the elevation dependency may approximately follow a function (1 + Fm(El)). Broken lines in the figure indicate 1, 4 and 7 TECU error magnified by this factor. Error distribution almost follows the elevation dependency of that function. This implies again that vertical TEC error is constant regardless the distance of ISP from observation point. The larger error in longer baseline than in shorter baseline (upper panel) is because common error does not canceled in longer baseline as it does in shorter baseline.

[29] Based on the discussion of Appendix A, baseline dependence of TEC errors, which are actually computed by equation (8) of the section 6.2, are plotted for each baseline in upper panel of Figure 8. It is seen that the TEC error increases with baseline length. Lower panel of the figures shows ratio between the TEC error after normalized by mapping function and mean of vertical TEC between two stations. The error bars indicate error ratios to smaller vertical TEC of the station pair and error ratios to larger one, respectively. The reason of quite small error ratio at KSP (109 km) baseline is understood by cancelation effect due to almost common ray path for both VLBI stations. The ray paths from two stations will no more share common ionosphere on a few thousands km baselines. This is caused from error in low spatial frequency component of the Earth's ionosphere. Because even at longer than thousand km baseline, the ray path shares some common SH components of that TEC model. The ionospheric TEC contribution is in order of 100 TECUs on intercontinental baseline, thus several percent of error ratio corresponds to from a few to several TECUs of error. The baseline dependency of TEC error ratio suggest that GIM model has a few TECUs of error in low spatial frequency component. The same conclusion is inferred from error spectrum discussed in section 6.2. Thus plots in Figure 8 are interpreted to represent a kind of error spectrum of the GIM/CODE model. In other words, the error at short baseline represents error of high spatial component, and that at longer baseline stands for error summed up to the corresponding spatial frequency components. The lower panel of the figure indicates that the error ratio is mostly less than 10 % for any baseline. This implies more than 90 % of the ionosphere can be modeled by the GIM/CODE, and still 10 % or less of error remains in it.

image

Figure 8. Baseline dependency of the uncorrelated TEC error σGPS(equation 8) is plotted as a function of baseline length in upper panel. Mean error ratio of TEC, which is a ratio of uncorrelated error of GIM/CODE to mean vertical TEC of VLBI stations pair, is plotted with respect to baseline length in lower panel. Upper and lower ends of error bars indicate ratios of mean errors to smaller vertical TEC of the VLBI station pair and error ratios to larger one, respectively.

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5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[30] If phase delay rate caused from TEC of the Earth's ionosphere is predicted by ionosphere model with high precision, it will be also useful information for correction of that effect in several kinds of space measurements with microwave. The SELENE project [Matsumoto et al., 1999; Heki et al., 1999] is one of those projects, in which phase delay rate need to be determined at precision of 0.1 mm/s or more at S-band. The phase changing rate due to the Earth's ionosphere needs to be corrected as accurately as possible.

[31] VLBI observation can measure phase delay rate caused by ionosphere, which correspond to TEC changing rate (hereafter referred as TEC rate) in line of sight. It is computed by phase delay rate difference between X-band and S-band. We compared the TEC rates between VLBI measurements and counterparts computed from GIM/CODE data. VLBI-based TEC rate consists of the time variation of TEC and spatial change of the line of sight due to tracking of radio source. We computed TECs in line of sight to radio source at desired epoch and at other 4 epochs with 10 minutes intervals before and after the desired epoch. Then TEC rate was derived by numerical derivation by using the 5 points TEC data.

[32] The TEC rates comparison was performed from short baseline (KSP) to intercontinental baseline (IVS data listed in Table 2). Unfortunately correlation between VLBI-based TEC rate and TEC rate computed from GIM/CODE was almost close to zero for short (109 km) baselines. Some correlation around 0.6 – 0.8 were found in longer baseline, although it was not enough accuracy for phase delay rate correction. Figure 9 shows an example of TEC rate comparison result of Algonquin - Wettzell baseline on 18th July 2000. RMS difference of TEC rate between the GIM/CODE and VLBI is several milli-TECU per second, whereas error of TEC rate measured by VLBI is less than 1 milli-TECU/sec. The reason for the poor coincidence of TEC rate is understood by the following: (i) The GIM/CODE data consist of 12 global ionosphere maps with 2 hours interval. Thus shorter timescale TEC variation is not included in the data. (ii) The GIM/CODE data are expressed with up to 12 degrees 8 orders of SH components. Then ionospheric TEC structure smaller than a minimum spatial scale (2500 km × 1700 km) is not contained in the data. The derivative operation to derive the TEC rate is equivalent to high pass filtering, which suppress lower frequency components and enhances higher frequency components. Since GIM/CODE data do not contain higher frequency component in both time and space domain, then TEC rate derived from the GIM/CODE shows poor coincidence with actual VLBI-based TEC rate.

image

Figure 9. Comparison of TEC rate between GIM/CODE and VLBI data on 18 July 2000 on Algonquin - Wettzell baseline. The TEC rate of GIM/CODE were computed by numerical derivation. Upper panel shows scattered plot of TEC rate measured by VLBI and counterpart computed from GIM/CODE. Lower panel is difference residual plot of the TEC rate between GIM/CODE model and VLBI data. Correlation was 0.7 and the RMS residual was 2.9 × 10−3 TECU/sec.

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6. Discussion

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

6.1. TEC Map Accuracy of Cases I and II

[33] 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.

[34] 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 equation image were computed for each data subset and estimate of uncorrelated error σGPS is given by

  • equation image

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.

image

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

[35] 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.

image

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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[36] 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. [2000] on several thousands km baseline.

[37] Here, we suppose a structure function of TEC error as equation image where er(x) is vertical TEC error of GIM/CODE model at coordinate x (error of TEC map is supposed to be statistically isotropic on the spherical surface for simplicity). Then, the plot in the lower panel of Figure 11 can be interpreted as measurement data of equation image A structure function R(l) and auto-correlation function defined by equation image has relation equation image To make comparison of each error of spherical harmonics components of GIM/CODE model, we change the parameter of structure function from baseline length l to geocentric angle θ by a relation l = 2Rsin (θ/2) where R is the Earth's radius. Figure 12 shows plot of σGPS as function of geocentric angle θ. We suppose two structure function models for the data. The error increases rapidly below 20 degrees, then it gradually increases up to 60 degrees if we ignore the data around 50 degrees. After 60 degrees the slope becomes slightly steep to further baseline length. The first model (solid line ) is based on this interpretation. The second structure function model (dashed line) is simply increases linearly from 10 to 140 degrees and then saturates. The small screen at the right bottom of the figure indicates auto-correlation function (C(θ)) derived from the two structure function models. Total accuracy of GIM/CODE model is inferred from equation image as 3.7 (model-1) – 3.9 (model-2) TECU.

image

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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[38] A Fourier component n of TEC error power spectrum S(n) is computed from Auto-correlation function with Wiener-Khintchine's relation: equation image where equation image 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.

image

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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[39] 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

[40] 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 [1983]. 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.

[41] 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 equation image 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
StationOffset, nsError, ns
  • a

    Each station's offsets were derived with assuming that the sum of all stations offsets is equal to 0. Error is formal error of least squares solution. To make reduced-χ2 equal to unity, the square of extra 1.2 nanoseconds of error was added to the square of each error.

Algonquin-0.430.4
Fortleza-5.30.4
Gilcreek0.130.4
Hobart1.20.6
Hartrao-16.50.6
Kokee-18.40.4
Wettzell13.60.4
Westford0.51
Tsukuba5.60.6
Nyales-0.40.6
Onsala14.40.8
Matera5.50.5

7. Conclusions

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[42] Accuracies of two cases of GPS-based ionosphere TEC map were evaluated by comparison with VLBI TEC measurements on 109 km baseline (2683 scans) and on intercontinental baselines (37 baselines, 6855 scans). The analysis was based on comparison of TEC difference between two VLBI stations and GPS-based TEC map, although, these observables had sensitivity to vertical absolute TEC through mapping function. As far as the error distribution follows appropriate mapping function, the error is projected upon vertical error of the TEC map, and accuracy of absolute vertical TEC was discussed in this paper.

[43] In one case, local TEC maps were estimated by using TECMETER observation data. The accuracy of TEC map generated by single TECMETER observation was evaluated 2.7 TECU as mean and in a range of 1 – 5.3 TECU. When common local TEC map was estimated with joint data set of two TECMETER observation data (case II in section 3.1), correlation of the TEC data with VLBI-based TEC measurements on 109 km baseline reached to 80 %. Most of common error of TEC map were canceled on such short baseline, then RMS difference between VLBI and TEC map was less than 0.75 TECU near zenith direction but it degraded up to 3 TECU as elevation angle decreases (lower panel of Figure 4).

[44] As the second case, global ionosphere map generated by the CODE (GIM/CODE) was compare with VLBI data. The GIM/CODE had better accuracy than the case of TECMETER observation compared here. The RMS difference with VLBI data was less then 0.6 TECU near zenith direction and increased up to 1.4 TECU at low elevation on 109 km short baseline (upper panel of Figure 6). On intercontinental baseline, the RMS difference was 3 – 20 TECU at elevation angle above 20 degrees (lower panel of Figure 6).

[45] The vertical TEC error of GIM/CODE was computed statistically from slant TEC error by projecting with mapping function. The ratio of vertical TEC error to the vertical absolute TEC was less than 10 %, i.e. more than 90 % of vertical TEC distribution on the Earth could be predictable with GIM/CODE model. Since mean-square of the difference between TEC measured by VLBI and ionospheric map was interpreted as structure function of the TEC map, auto-correlation function and power spectrum of TEC map error could be computed from the data by appropriate interpolation with models. Precision of GIM/CODE data was evaluated from auto-correlation as 3.7 – 3.9 TECU. Estimated error spectrum of GIM/CODE showed that high spatial frequency component of the TEC, which was not included in the model, remained more than 0.8 TECU. From a comparison between obtained error spectrum and error attached with GIM/CODE data, it was suspected that GIM/CODE model underestimated its error, especially at low spatial frequency.

[46] TEC rates derived from GIM/CODE were tested by comparison with VLBI-based TEC rates. Unfortunately the correlation between them was not so high and TEC rate derived from GIM/CODE did not have enough accuracy for TEC rate correction in other space measurement technique. The reason of that could be understood by lack of high frequency TEC variation component in the GIM/CODE model in both time and space domain.

[47] If GIM/CODE data are used for ionospheric delay correction on about 8000 km baseline, the expected error will be around 4*(1 + Fm(el)) ∼ 12 TECUs. That corresponds to about 0.2 ns excess delay at 8.4 GHz. This is the same order of error as with Ros et al. [2000] on similar baseline length. Twelve TECU correspond to about 8 ns at 1.4 GHz. When these calibration data are applied to astrometric VLBI observation data for pulsar at 1.4 GHz observation, unfortunately the precision of the measurement will not be improved, but precision of astrometry is expected to be improved by removing 90 % of the ionospheric delay contribution, which can potentially cause systematic error. Additionally GIM/CODE data have the advantage that TEC map is available at any time and any place on the Earth without additional own investment to measure the ionosphere TEC. Improvement of ionospheric TEC model will also benefit to space navigation and geodetic measurement with single frequency GPS receiver. It is often employed for dense array geodetic observation after the earthquakes or volcanic eruptions, since it is better than two frequency receiver in terms of power consumption and price.

[48] After submission of this paper, the GIM/CODE data were upgraded to use SH components with both degree and order up to 15. Thus the precision of the map might be more improved than that evaluated in this paper.

Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[49] The relation between TEC map error and RMS difference is discussed for accuracy evaluation of the TEC maps. TEC observed by VLBI is expressed by

  • equation image

where dTEC is true difference of TEC along ray paths between two VLBI stations, and ϵVLBI is error of VLBI measurement. Vertical TEC value of GPS-based TEC map is expressed by

  • equation image

where VTEC is true vertical TEC and ϵGPS is error of GPS-based TEC map. TEC difference corresponding to VLBI observation (A1) is expressed by using mapping function Fm(El) as

  • equation image

When effect of ray path bending and electron density height profile are eliminated here, true TEC difference observed by VLBI is expressed by equation image Then error related terms are remain in the difference between VLBI and GPS as

  • equation image
A1. In the Case of Short Baseline

[50] Error of GPS-based TEC map ϵGPS consists of two components ϵGPS = ϵGPS,u + ϵGPS,c, where ϵGPS,c and ϵGPS,u represent correlated and uncorrelated error of TEC on that baseline. When baseline is short and the elevation angles at both stations are almost the same, equation image and equation image thus common error component are canceled. Then ensemble average of ΔTEC2 at an elevation angle El equation image is

  • equation image
  • equation image

Here 〈〉 means ensemble average and equation image, equation image, and the equation image are variances of uncorrelated error of TEC map for x, y stations, and error of VLBI observation, respectively. equation image were used for equation (A5) and (A6).

A2. In the Case of Long Baseline

[51] When the baseline is long and the elevation angles at two stations are different, common error is not canceled out. Ensemble average of square of equation (A4) at an elevation angle pair Elx, Ely is

  • equation image
  • equation image

where σx is error variance of GPS-based TEC map. An approximation of σx ≈ σy and an assumption of no correlation between ϵx and ϵy (i.e. 〈ϵxϵy〉 = 0), are used in approximation from (A7) to (A8).

Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[52] For comparison between our estimation of GIM/CODE error spectrum and the spherical harmonics (SH) coefficients of TEC map error included in GIM/CODE data, here we will make clear the relation between Fourier error amplitude components and discrete power spectrum computed from auto-correlation function.

[53] When error is expressed with respect to function of angle θ(−π ≤ θ < π) as

  • equation image

auto correlation function is expressed by

  • equation image

where 〈〉 indicates ensemble average. Power spectrum is given by Wiener-Khintchine's relation as

  • equation image

Therefore TEC error amplitudes, which are appropriate to be compared with SH coefficient of GIM/CODE error data, are expressed by components of power spectrum as

  • equation image

Acknowledgments

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information

[54] The authors thank Stefan Schaer at Bern University for providing GIM data and related subroutines, and thank Stefan Schlueter at the DLR for kindly providing the satellite DCB data. We thank Yoshiki Igarashi and Hisamitsu Minakoshi of the CRL for providing TECMETER data. We appreciate Yuri P. Ilyasov, Vasilli V. Oreshko, and Alexander E. Rodin, of the Lebedev Physical Institute of Russia, and Boris A. Poperechenko for supporting this study. We also greatly appreciate members of the Key Stone Project Team, the Radio Astronomy Applications Section, and the Time and Space Measurement Section for supporting and encouraging this research. In addition, we thank Wayne Cannon for supporting this study in Canada. This research has made use of international geodetic VLBI data provided by the international VLBI Service for Geodesy and Astrometry. Finally, we thank the referees of Radio Science for constructive criticism, which has greatly contributed to improving the quality of this paper.

References

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information
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Supporting Information

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Background of GPS-Based Ionospheric TEC Evaluation
  5. 3. GPS-Based Ionosphere Map
  6. 4. Comparison of TEC Maps and VLBI Data
  7. 5. Comparison of Phase Delay Rate Derived From GPS-Based TEC and VLBI
  8. 6. Discussion
  9. 7. Conclusions
  10. Appendix A: TEC Map Error and RMS Difference Between VLBI-Based TEC and GPS-Based TEC Map
  11. Appendix B: Wiener-Khintchine's Relation and Discrete Error Spectrum
  12. Acknowledgments
  13. References
  14. Supporting Information
FilenameFormatSizeDescription
rds4707-sup-0001tab01.txtplain text document1KTab-delimited Table 1.
rds4707-sup-0002tab02.txtplain text document1KTab-delimited Table 2.
rds4707-sup-0003tab03.txtplain text document1KTab-delimited Table 3.

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