Tsukuba GPS Dense Net Campaign Observations: Comparison of the Stacking Maps of Post-fit Phase Residuals Estimated from Three Software Packages

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1 Journal of the Meteorological Society of Japan, Vol. 82, No. 1B, pp , Tsukuba GPS Dense Net Campaign Observations: Comparison of the Stacking Maps of Post-fit Phase Residuals Estimated from Three Software Packages T. IWABUCHI Japan Society for the Promotion of Science, University Corporation for Atmospheric Research, Boulder, CO, USA Y. SHOJI Meteorological Research Institute, Tsukuba, Japan S. SHIMADA National Research Institute for Earth Science and Disaster Prevention, Tsukuba, Japan and H. NAKAMURA Japan Meteorological Agency, Tokyo, Japan (Manuscript received 30 April 2003, in revised form 5 November 2003) Abstract The characteristics of post-fit residuals computed using three types of software have been investigated, and the behavior of multipath errors in the post-fit phase residuals evaluated. GPS data observed during the Tsukuba GPS dense net campaign were analyzed, and the post-fit phase residuals of the linear combination of the L1 and L2 GPS phase measurements carrier wave (LC), computed using the three software packages were studied. The post-fit phase residuals were stacked, and their mean value for each site was obtained, with a resolution of 1 1 for the azimuthal and elevation angles, respectively. The skymaps of the stacked post-fit phase residuals showed similar patterns among the three types of software. A random error index was introduced to evaluate random errors in the post-fit phase residuals, which showed twice as large errors using the point positioning strategy, as those obtained using the double difference strategy. The oscillation patterns shown in the stacking map were similar to those simulated using the Elosegui s multipath model (1995). This suggests that the pattern arose from multipath errors. When the stacking maps were introduced into the post-fit phase residuals, the impact of the resulting post-fit phase residuals was significant at sites that showed large multipath errors. The point positioning strategy still contained random errors that were twice as large as the double difference Corresponding author: Tetsuya Iwabuchi, Japan Society for the Promotion of Science, University Corporation for Atmospheric Research, P.O. Box 3000, Boulder, CO. USA. iwabuchi@ucar.edu ( 2004, Meteorological Society of Japan

2 316 Journal of the Meteorological Society of Japan Vol. 82, No. 1B strategy errors. Multipath errors induced biases in the zenith tropospheric delay, which resulted in larger biases of the absolute slant tropospheric delay reconstruction at lower elevation angles. This suggests the importance of introducing stacking maps in GPS analysis. 1. Introduction GPS-retrieved slant tropospheric delays have the potential to provide essential information on water vapor fields using water vapor tomography techniques (see Seko et al. 2000, for example) and the assimilation of the data into numerical weather models (see Guo et al and MacDonald et al. 2002, for example). Zenith tropospheric delay (ZTD) and post-fit phase residuals, arising from the model-fitted values are usually required to reconstruct slant tropospheric delays. However, these can be contaminated by various error sources. Many reference books concerning GPS (Leick 1995, Teunissen and Kleusberg 1998, for example) summarizes the major sources of error in GPS estimates, such as ZTD and site coordinates. There are two dominant error sources for tropospheric delay estimates; (i) the antenna characteristics showing phase center variation (PCV), which depends on the elevation angle and also the azimuthal angle of the antenna; (ii) the multipath, which depends on the properties and size of the reflectors, and the geometric conditions between the GPS satellite and the receiver. Since these errors degrade the slant tropospheric delay significantly, we have estimated the ZTD which is averaged slant tropospheric delay, mapped onto the zenith of several satellites, and have averaged them with intervals ranging from several minutes to several hours. The absolute slant tropospheric delay is generally computed from the ZTD estimates, the mapping function(s) used in the analysis, and the post-fit phase residuals. The ZTD estimates and mapping function determine the average symmetric distribution of water vapor, which depends on the elevation angle, and on the azimuthal angle, if the tropospheric delay gradient is to be estimated. Residuals lead to an inhomogeneity in the water vapor distribution by adding it to the symmetrical distribution of water vapor. Thus, it is important to compute accurate post-fit phase residuals, and to estimate unbiased ZTD values for retrieving accurate slant tropospheric delays, especially at lower elevation angles. The ZTD estimates are generally performed from tens of minutes to several hours in the least square strategy, and they are done at intervals of several tens of minutes in the stochastic strategy. The temporal resolutions of the post-fit phase residuals, are usually obtained from the sampling interval of the GPS observation, which is of the order of several ten seconds. In these time periods, multipath and antenna PCV are the most dominant sources of error. In addition, it is possible that the slant tropospheric delays have errors that depend on the GPS analysis strategy, and the models and parameters used in the analysis. For example, artificial assumptions introduced to compute the zero difference residuals from the double difference residuals may cause systematic errors in the reconstructed absolute slant tropospheric delay in some cases. This was suggested by Alber et al. (2000). In addition, Elosegui and Davis (2003) have shown that the method for computing the zero difference from the double difference residuals spreads inhomogeneous signals over all parameters estimated in the least-squares step, and therefore, over all the reconstructed GPS slant wet delays, which causes significant systematic errors. Elosegui and Davis (2003) also mentioned a common potential problem for all the methods used to reconstruct absolute slant delays determined from estimated parameters and the post-fit phase residuals, which is, generally speaking, a statistically non-robust procedure. In this work, we compare post-fit phase residuals estimated using three types of software, employing the Tsukuba GPS dense net campaign observation from the autumn of We studied the characteristics of the post-fit phase residuals, where we mainly characterized the behavior of multipath errors that appeared in the stacking maps of the post-fit phase residuals. Section 2 introduces the analysis strategies used in the study, and statistical indices showing the quantity of systematic

3 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 317 Fig. 1. GPS sites operated during the Tsukuba dense net GPS campaign in The topography of the network is shown with contour lines located every 10 m, using gray images. multipath errors and random errors in the slant tropospheric delays. Comparisons of the stacking maps and the indices are shown in Section 3. Such comparisons suggest the validity of the artificial assumption used in double difference strategy. In Section 4, we explain the multipath errors that commonly appear in the stacking maps of the post-fit phase residuals of the three types of analysis. The impact of applying the stacking maps of the post-fit phase residuals is shown in Section 5. The simulated ZTD errors from the multipath errors are shown in Section 6, and the results are summarized in Section Data and analysis Figure 1 shows the distribution map of the GPS sites operated during the Tsukuba dense net campaign in the autumn of The GPS sites were deployed in the field with small height differences of less than 60 m. The network was divided into three networks, according to the receiver-antenna type. Of these three, we used the two major networks: the Ashtech, and Trimble networks. We did not use the AOA network, because it consisted of only two sites. A detailed explanation of the observations is summarized in Shoji et al. (submitted manuscript to JMSJ, the current issue). The GPS data were analyzed, and the post-fit phase residuals were computed using the following three types of software, (i) the Bernese GPS Software (BERNESE, Bern University, Beutler et al. 2000) employing the zero difference method (Alber et al. 2000), (ii) the GAMIT GPS Analysis software (GAMIT, Massachusetts Institute of Technology, King & Bock 1997), and (iii) the Precise Point Positioning (PPP) of GIPSY-OASIS II software (GIPSY, Jet Propulsion Laboratory/NASA, Zumberge et al. 1997). Tables 1 and 2 show the analysis strategies, models, and parameters used in the analyses, and the tropospheric delay estimation, respectively. The LC wave, formed from Table 1. Analysis strategies, models, and parameters used in the three types of software package: BERNESE, GAMIT, and GIPSY (GIPSY1, GIPSY2, and GIPSY3). Software package Basic Satellite strategy PCV model OTL model Orbit Clock errors BERNESE DD* 1 NOAA* 3 GOTIC2* 5 IGS DD cancellation GAMIT DD IGS_01* 4 Onsala Hans-Georg IGS DD cancellation Scherneck* 6 GIPSY1 PPP* 2 NOAA GOTIC2 JPL JPL (5 min) CIPSY2 PPP NOAA GOTIC2 JPL JPL (5 min) GIPSY3 PPP NOAA GOTIC2 JPL JPL (30 sec) * 1 Double difference * 2 Precise point positioning * 3 Mader, G.L., and J.R. MacKay, Calibration of GPS antennas, * 4 * 5 Matsumoto et al. (2001) * 6 Created by Hans-Georg Scherneck at Onsala Space Observatory

4 318 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Table 2. Analysis models and methods used in the estimation of tropospheric delay, and one-way post-fit phase residual computation for the three software packages, BERNESE, GAMIT, and GIPSY (GIPSY1, GIPSY2, and GIPSY3). ZTD estimation Mapping function Software package Initial Estimation Time Intervals Computation of post-fit phase residuals Method Stacking maps (degree) Time Intervals Pre-app. Post-appl. 30 sec N/A sec N/A 1 1 BERNESE Niell dry* 1 Niell wet* 2 60 min Zero mean assumption* 3 GAMIT Niell dry Niell wet 60 min Sum of clock errors ¼ 0* 4 GIPSY1 Niell dry Niell wet 5 min Post-fit (O-C) 5 min N/A 1 1 residuals GIPSY2 Niell dry Niell wet 5 min Post-fit (O-C) 5 min residuals GIPSY3 Niell dry Niell wet 30 sec Post-fit (O-C) residuals 30 sec * 1,2 Niells mapping function (Niell 1996). * 3 Sum of residuals of double differences and single differences are assumed to be zero every sampling epoch. See Alber et al. (2000). * 4 Sum of satellite clock errors are assumed to be zero every sampling epoch. linear combinations of the L1 and L2 waves, is used by all three software packages to remove the contribution from the ionospheric delay by utilizing the dispersion characteristics of the ionospheric delay. The minimum elevation angle was set to approx. 7 to avoid large fluctuations in the slant tropospheric delays at lower elevation angles, and some sites were not recorded at lower elevation angles. Although the estimation of the tropospheric delay gradient decreases the bias in the ZTD values (Iwabuchi et al. 2003), this was not estimated in this case, to avoid an increase in the number of estimated parameters in the least square strategy. It should be noted that two different PCV models (NOAA and IGS_01), and two different ocean tide loading models (GOTIC2 and Onsala Hans-Georg Schemeck) were used in the analyses. The effect of any differences is expected to be negligible in the post-fit phase residuals, because most of the effects should be absorbed in the estimated parameters. The BERNESE and GAMIT methods employ double differences as the basic observables (for example, see Leick 1995). By using the double difference values, the receiver and satellite clock errors can be canceled out. The BERNESE and GAMIT analysis, first divides the networks into the Ashtech and Trimble networks, with the Trimble network being further divided into several sub-networks. Analysis of each network was performed independently. On the other hand, the GIPSY method can directly solve the receiver clock errors, along with the other parameters, by using a sophisticated stochastic filtering technique with precise satellite orbit and satellite clock error information. The estimated parameters in the PPP method, however, strongly depend on the clock error information and the quality of the global parameters that are mostly canceled out using double difference strategies. The details of the differences in the analysis strategies, have also been summarized by Iwabuchi et al. (2003). In the analysis of the GIPSY method, three types of analysis were performed: (i) the standard PPP analysis method, denoted as GIPSY1 in this study; (ii) a similar method to GIPSY1, but one that introduces stacking maps in the analysis, denoted as GIPSY2; and, (iii) a special analysis using a high rate satellite clock information of 30 s, which also introduces stacking maps in the analysis, denoted as GIPSY3. This last method has a resolution of the introduced stacking maps of 5 by 2 for the azimuthal and

5 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 319 elevation angles, respectively, due to the limitation of available dimension from the GIPSY package (see Tables 1 and 2, and Shoji et al. 2003). Post-fit phase residuals computed using the three types of software over 10 days from 18 October to 27 October 2000 were stacked, and their mean values and standard deviations obtained with a resolution of 1 1 for the azimuthal and elevation angles, respectively. We defined three types of statistical indices, the multipath error index (MEI), the multipathcaused ZTD equivalent error index (MZI), and the random error index (REI), which are expressed as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MEI ¼ S y S f ðmapðy; fþþ 2 /N; ð1þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi MZI ¼ S y S f ðmapðy; fþ sinðyþþ 2 /N; ð2þ REI ¼ S y S f ðsd_mapðy; fþ sinðyþþ/n; ð3þ where y is the elevation angle, j is the azimuthal angle, MAPðy; jþ and SD_MAPðy; jþ show the stacked post-fit phase residuals in each grid of the skymap, and the standard deviation of MAPðy; jþ in each grid, respectively, and N is the number of valid grids that have data points greater than 10. The sin y term in Equations (2) and (3) is the inverse of the classical mapping function used to map slant tropospheric delays in the zenith direction. It should be noted that not only the multipath errors, but also imperfect modeling, and mis-modeling in the GPS analyses, contributed to MEI, MZI, and REI. The MEI values mainly show the magnitude of the contributions of the multipath errors and PCV errors from the imperfect PCV model to the post-fit residuals, while the MZI values mainly represent the magnitude of the ZTD equivalent biases caused by the multipath errors, and the PCV errors from the imperfect PCV model. The REI values show the errors of small-scale multipaths, which cannot be detected by the 1 1 resolution of the stacking maps. It should be noted that the REI values also include the contributions of all other random errors, and the true variability of water vapor. 3. Comparison of the post-fit phase residuals The three maps at the left edge of Fig. 2 show the stacking maps of the post-fit phase residuals computed using the three types of software packages for a typical Trimble site, Site #58. Figure 2 shows that there were similar patterns among the maps of the three software packages, despite the different analysis strategies used. Among these maps, an especially strong similarity can be seen between the BERNESE and the GAMIT data. Although the magnitude of the stacked post-fit phase residuals in the GIPSY1 package was larger than that of the BERNESE and GAMIT packages, the spatial patterns were similar in the three skymaps. The upper right panels in Fig. 2 show the standard deviation of the post-fit phase residuals of Site #58 computed using the software packages for each degree of elevation angle. The variation in the residuals of the BERNESE and GAMIT packages was twice as small as that in the GIPSY package at elevation angles above 30. It is interesting to note that similar patterns were observed using the BERNESE and GAMIT packages. It is assumed that the computation of the post-fit phase residuals using the double difference residuals, produces similar results when two software packages (BERNESE and GAMIT), are utilized. The discrepancy at lower elevation angles between the two software packages is mainly caused by the difference in the different limits for data rejection (GAMIT uses smaller data limits for rejection than the BERNESE package in computing the post-fit phase residuals), and this difference causes an increase in the standard deviation of the BERNESE data, at low elevation angles. We also found a difference in the post-fit phase residuals, between the ground plane antenna in the Trimble network and the choke ring antenna in the Ashtech network. The two lower right panels in Fig. 2 show the dependency on the elevation angle of the post-fit phase residuals of the BERNESE data at Site #58 (Trimble ground plane antenna) and at Site #93 (Ashtech choke ring antenna). Site #93 was located less than 10 m north of Site #58. The oscillation pattern at Site #58, with azimuth angle was considered with the PCV of the ground plane antenna, which had an azimuthal variation with a wave number of three (Schupler et al. 1994), while the Ashtech choke ring antenna did not show any significant azimuthal PCV. Some large scattering, at elevation angles

6 320 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Fig. 2. Stacking maps of the post-fit phase residuals of Site #58 using a ground plane antenna for the three types of software, BERNESE, GAMIT, and GIPSY1, respectively (left panels, see Tables 1 and 2), where the center of the sky maps show the zenith angle. Each circle is plotted for each 15 of elevation angle, and the radial lines from the center of the maps show the azimuthal angle. The standard deviation of the post-fit phase residuals for the three types of software (but not GIPSY3 for the GIPSY solutions due to its having the same sampling interval as the BERNESE and GAMIT packages) were computed for each degree of elevation angle, and are shown in the upper right panels with number of data used. The dependency of the elevation angle on the residuals is also shown in the lower right panels for Site #58 and for Site #93 located several meters north of Site #58. The antenna type at Sites #93 and #58 were the choke ring and ground plane type, respectively. from 50 to 85 at Site #98, seems to be caused by multipath errors having a high frequency, because the skymap of the standard deviation plot for the site (not shown here), shows a large variability in this range. Figure 3 shows a comparison of the stacking maps from Site #57 (Trimble), showing that some specific azimuth angle, dependent oscillation patterns can be seen. The RMS deviation computed from each grid is also shown in the lower skymaps. Similar oscillation patterns appeared in the southwestern part of the sky in the three maps. As there was a pole that interrupted the view of the GPS satellites in the

7 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 321 Fig. 3. Skymaps (upper panels) and their RMS distribution (middle panels) of the post-fit phase residuals for the BERNESE (left), GAMIT (center), and GIPSY (right) packages at Site #57, where significant azimuth angle dependent oscillation patterns were observed. The pictures in the lower panels show the environment near the antenna (left), and the visibility of the sky taken with an equiangular fisheye lens (right). southwestern part of the GPS antenna, the systematic oscillation pattern in the map is expected to be caused by the phase shift of synthetic wave, between the direct wave and the reflected wave from this pole. Figures 4 and 5 show the MEI and MZI values defined in Equations (1) and (2), respectively. Although the magnitude of the index is different, the relative magnitudes among the sites are similar for the five strategies used. GIPSY1 shows much larger MEI and MZI values than those computed for BERNESE and GAMIT data. The MEI and MZI values for GIPSY2 showed a similar magnitude to those of the BERNESE and GAMIT packages, and the MEI and MZI values of the GIPSY3 were sometimes higher than those of the BERNESE and GAMIT packages. The impact of introducing the stacking maps in the GIPSY2 and GIPSY3 analysis on post-fit phase residuals can be clearly seen. Figure 6 shows the REI values defined in Equation (3) for all the sites. The index shows the quality of the post-fit phase residuals,

8 322 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Fig. 4. The multipath error index (MEI, Equation 1) showing systematic errors in the post-fit phase residuals of the three software packages: BERNESE (solid squares), GAMIT (solid triangles), and GIPSY (crosses for GIPSY1, diamonds for GIPSY2, and asterisks for GIPSY3). The sites with thick underlines are reference sites for the Trimble and Ashtech networks. Fig. 5. The same as Fig. 4, but showing the multipath-caused ZTD equivalent error index (MZI, Equation 2), showing systematic errors in the zenith-mapped post-fit phase residuals, i.e., the ZTD residuals, for the three software packages. whose systematic errors are subtracted by the stacking maps, with a 1 1 resolution for the azimuthal and elevation angle, respectively. The REI values for most of the sites were similar to the pattern observed using the BERN- ESE and GAMIT packages. Multipaths, with a

9 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 323 Fig. 6. The same as Fig. 4, but showing the random error index (REI, Equation 3), showing the variability and random errors that cannot be explained by the 1 1 resolution of the stacking maps in the post-fit phase residuals for the three software packages. high frequency, were dominant at the sites showing larger indices in the BERNESE and GAMIT data. We found that REI values of the BERNESE and GAMIT data were twice as large at the reference sites as those obtained at sites in the Tsukuba area. The reference sites are located far away from the Tsukuba area (more than 400 km). The contributions of the post-fit residuals at the reference sites, to those of the Tsukuba area network, were much smaller due to the small number of reference sites used in the zero mean assumption. It is supposed that the more marked difference in the water vapor distribution at the reference sites from that of the large number of sites located in the Tsukuba area network, accounts for the larger REI values at the reference sites. On the other hand, the REI values of the GIPSY (1, 2, and 3) data, were twice as large as those of the BERNESE and GAMIT data, and the magnitude of the REI values from the GIPSY data, depended on the type of receiver and antenna, while the REI values of the BERNESE and GAMIT data also showed site dependent differences. The observed differences between the point positioning strategy (GIPSY), and the double difference strategies (BERNESE and GAMIT), seemed to agree with the results presented by Braun et al. (2001). One of the major contributors to the large random errors in GIPSY is considered to arise from satellite clock errors and global parameter errors. The difference in the random errors among the receivers and antenna types in the GIPSY data, is expected to reflect different receiver clock modeling for each type of receiver. The observations also suggest that the general noise level of the post-fit phase residuals in the GIPSY data, is larger than that of the multipath, with a higher frequency than the 1 1 azimuthal and elevation angles. 4. Reproduction of the multipath A similar oscillation pattern to that seen in the skymaps shown in Fig. 3, was observed at about one third of the sites over the entire network. Figure 7 shows the stacking maps of the BERNESE analysis and photographs of the visibility of the sky for Sites #33 and #61. Azimuthal angle dependent oscillation patterns in the skymaps were observed on both sides of the poles in the pictures. A similar result was also obtained for Site #57 in Figure 3. It is interesting to note that those sites showed relatively higher MEI and MZI values than the

10 324 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Fig. 7. Stacking maps of the post-fit phase residuals from the BERNESE package (left), and pictures showing the visibility of the sky (right) at Sites #33 and #61. The azimuthal angle dependent oscillation patterns can be seen in the stacking maps. There are several poles, such as lightning rods, near the sites. other sites, but the same magnitude of REI values, as shown in Figs. 4, 5 and 6. The facts suggest that the oscillation patterns were not random variations, but systematic variations. These systematic oscillation patterns are therefore strongly anticipated to be caused by the multipath. Figure 8 shows the skymaps of the BERN- ESE data and photographs of the visibility of the sky for Sites #68 and #60. Different oscillation patterns from those shown in Fig. 7 can be seen in the maps. The MEI values at those sites showed a much larger value than those at Sites #33, #61 (Fig. 7) and #57 (Fig. 3), and showed elevation angle dependent oscillation patterns. The MZI values were of the same magnitude at these two groups of sites. This was caused by a mapping effect of the slant tropospheric delay into the zenith delay, i.e., biases at lower elevation angles were smaller than those at higher elevation angles. The REI values at those sites were of similar magnitude to those at the other sites, showing that the oscillation pattern was also systematic. Figure 9(a) shows the raw post-fit phase residuals for the satellite Pseudo Random Noise (PRN) 21 at Site #57 (see also Fig. 3). Azimuth angle dependent oscillation patterns, which are symmetrical with respect to the pole direction, can be seen in the figure. We computed the multipath phase shift using a simple model, which was a modification of the multipath model of Elosegui et al. (1995), which is expressed as dfðe; Dc; a; R; lþ R a sin 2p sin eð1 þ cos DfÞ ¼ tan 1 l 1 þ a cos 2p R ð4þ sin eð1 þ cos DfÞ l where e is the elevation angle of the satellite, Dj is the azimuth angle of the satellite from the pole, a is a damping factor, and R and l are the projective distance between a pole and an antenna on the surface, and its wavelength, respectively. Figure 9(b) shows a simulated mul-

11 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 325 Fig. 8. Stacking maps of the post-fit phase residuals from the BERNESE package (left), pictures showing the visibility of the sky (middle), and pictures showing the environment near the antennas (right) at Sites #68 and #60. These sites show elevation angle dependent oscillation patterns in the stacking maps. The solid and dashed lines in the photographs of Site #68 are of direct and reflected signals from the rooftop, respectively, which cause conceptual multipath phase shift errors in the post-fit phase residuals. Fig. 9. (a) The azimuthal angle dependence of the raw post-fit phase residuals for satellite PRN 21 (pseudo random noise used in the identification number of the GPS satellite) at Site #57 (Fig. 3); and (b) the azimuthal angle dependence of the post-fit phase residuals simulated from the multipath model defined in Equation (4) for elevation angles of 15 (thin solid curve) and 30 (thick solid curve), respectively. tipath for elevation angles of 15 and 30 for R ¼ 5:0 m. A damping factor of 0.2 was determined by comparing the amplitude of the observed, and the simulated waves. A similar trend in oscillation patterns to that observed was reproduced by the simulation, although the azimuthal phase does not wholly agree. Therefore, the oscillation patterns shown in the

12 326 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Fig. 10. Elevation angle dependence of the raw post-fit phase residuals for three satellites: PRNs 22, 30, and postfit phase residuals simulated using the multipath model defined in Equation (5). stacking maps in Figs. 3 and 7, are expected to be caused by multipath effects. On the other hand, we also computed the multipath error shown in Fig. 10, which can explain the elevation angle dependent oscillation patterns shown in Fig. 8, by using the multipath model developed by Elosegui et al. (1995), which is expressed as H a sin 4p dfðe; a; H; lþ ¼tan 1 l sin e 1 þ a cos 4p H ð5þ l sin e where H is the height of the antenna phase center from the reflecting surface. The observed raw post-fit phase residuals are shown for two satellites in Fig. 10. They are approximately reproduced by the multipath model, as shown in Fig. 10, where the damping factor was 0.12 and H ¼ 80 cm, which is the true height (see the picture shown in Figure 8). The smaller damping factor used compared with the azimuth angle oscillation patterns is considered to be due to a difference in the strength of the reflected signals. In addition, the antenna gain was smaller at low elevation angles, which also contributed to a smaller damping factor. 5. Application of the stacking maps Figures 11 and 12 show the impact of postapplication of the stacking maps for two sites: one is a site with good sky visibility (#16), and the other is a site showing a dominant multipath (#33). The raw post-fit phase residuals shown in the upper panels in Fig. 11 show significant differences in variability between the double difference strategy (BERNESE and GAMIT) and the point positioning strategy (GIPSY3). The residuals in the middle panels, whose stacking maps have been subtracted, show a decrease in systematic oscillation patterns, as shown in the upper panels. However, the impact of applying the stacking maps to the site with a good sky visibility is small, as shown in the lower panels. On the other hand, the site where the multipath errors were dominant showed a much larger impact when the stacking maps were subtracted, as shown in Fig. 12. The variability of the raw post-fit phase residuals in the double difference strategy (BERNESE and GA- MIT), for this site was considerably larger than those represented in Fig. 11. The results obtained by subtracting the stacking map, showed a significant decrease in the variability, as shown in the middle panels of Fig. 12. In the lower panel, it can be seen that the solid lines showing the standard deviation of the residuals, from subtracting the sky maps, is close to the same level as that at the good visibility site in Fig. 11. The GIPSY3 analysis, which had a resolution of 5 2 pre-applied to the stacking map also showed an impact from subtracting the stacking map with a resolution of 1 1, although the impact was not as large as that achieved using the double difference strategies. This suggests that the resolution of the stacking map should be 1 1, or finer, to remove such oscillation patterns as shown in Figures 3, 7 and ZTD errors caused by the multipath Some ZTD errors are expected to be caused by the multipath and/or the PCV. ZTD estimates can be contaminated by these errors and cause biases in reconstructing the absolute slant tropospheric delays mapped from the ZTD, as previously discussed, especially at the lower elevation angles. The mapping function would also be one of the systematic sources of error for elevation angles less than 10. Figure 13 shows the ZTD biases simulated by using satellite geometry information, and

13 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 327 Fig. 11. The impact of post-application of the stacking maps on the post-fit phase residuals at the site with good visibility (Site #16). The left, center, and right panels are results from the BERN- ESE, GAMIT, and GIPSY3 packages. The upper and middle panels show the elevation angle dependent residual distributions for the raw and post-application of the stacking map data, respectively. The lower panels show the standard deviation over 1 intervals of the elevation angle for the upper panel (solid line) and the middle panel (dashed line). stacking maps of the post-fit phase residuals for a site with good conditions (#16, Fig. 11), and for a site where the dominant multipath was observed (#33, Fig. 12). The strategy of the simulation used here was the point positioning method, due to its simplicity in explaining errors at one site, and the stacked post-fit residuals were used as observations in the simulation. The variability of the ZTD biases was larger when the time-interval of the ZTD estimation was shorter, especially at the site with poor conditions. At this site, the ZTD bias reached 7 mm when the ZTD was estimated using an interval of 5 min, which is equal to about 1 mm of PWV. These results suggest that if we compute an absolute slant tropospheric delay, based on a ZTD value that has not been corrected for multipath and PCV errors in the analysis, then the computed absolute slant tropospheric delays will have larger biases when mapping the ZTD into slant tropospheric delays.

14 328 Journal of the Meteorological Society of Japan Vol. 82, No. 1B Fig. 12. The panel descriptions are the same as those for Fig. 11, but at the site where the azimuthally dependent oscillation patterns were dominant (Site #33, see Fig. 7). In addition, an estimation of the zenith tropospheric delay, with a piecewise-step function is considered to cause biases in the absolute slant tropospheric delay. This is one of the major problems when we reconstruct absolute slant tropospheric delays, using analysis strategies in which temporal resolutions of the ZTD estimation (from several minutes to several hours), and residuals (sampling intervals) are not equal. 7. Summary and conclusion GPS data observed during the Tsukuba GPS dense net campaign in 2000, were analyzed using three types of software for evaluation of the sources of error, such as the multipath in slant tropospheric delay. Stacking maps of the post-fit phase residuals and statistical indices based on the stacking maps, were used to evaluate the post-fit phase residuals. We compared the residuals of linear combinations of the L1 and L2 GPS phase measurement carrier waves, by computing them using three types of analysis: BERNESE, GAMIT, GIPSY-OASIS II. We observed the following results: (1) A comparison of the stacking maps (resolution ¼ 1 1 ) showed similar multipaths and PCV patterns among the three analyses. This suggests that the zero mean as-

15 March 2004 T. IWABUCHI, Y. SHOJI, S. SHIMADA and H. NAKAMURA 329 Fig. 13. Biases of the ZTD estimates for Sites #16 (left, see Fig. 11) and #33 (right, see Fig. 12) expected from the point positioning simulation using satellite orbit information and stacking maps of the post-fit phase residuals from the BERNESE package. The ZTD errors were estimated using intervals of 5, 10, 30, and 60 min. sumption works for such site-specific and systematic signals; (2) The random error index shows that network analysis using a strategy employing double difference in relative positioning (BERNESE and GAMIT), shows an RMS deviation that is twice as small as the point positioning strategy (GIPSY). This is considered to be caused by clock error cancellation in the double difference strategy; (3) Some sites show azimuth angle and/or elevation angle dependent oscillation patterns in the stacking maps. Since similar oscillation patterns can be simulated by using Elosegui s multipath model (1995), these are expected not to be tropospheric signals, but systematic effects caused by the multipath; (4) Carrier phase residuals contaminated with the multipath and/or PCV of the antenna, where the amplitude of the systematic variations reaches about 20 mm in the tropospheric delay, can be cleared by applying the stacking map to the raw residuals; and, (5) Systematic multipaths cause errors in the PWV of more than 1 mm when the ZTD is estimated using a time interval of 5 min. Stacking maps and their statistical indices shown in this work, are useful for evaluating systematic errors, such as multipaths, PCV, and random errors in the GPS estimates. The stacking maps should also be introduced in the relative positioning strategy to utilize their precise post-fit residuals for meteorological studies. Acknowledgements We thank Dr. Matsumoto of NAO for providing the GOTIC 2 ocean tidal loading coefficients. We also express our gratitude to UCAR/GST for providing the BERNESE Zero Difference program, MIT for providing the GAMIT program, and JPL for providing the

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