Phase acceleration: a new important parameter in GPS occultation technology

Based on 40 years of radio-occultation (RO) experiments, it is now recognized that the phase acceleration of radio waves (equal to the time derivative of the Doppler shift), derived from analysis of high-stability Global Positioning System (GPS) RO signals, is as important as the Doppler frequency. The phase acceleration technique allows one to convert the phase and Doppler frequency changes into refractive attenuation variations. From such derived refractive attenuation and amplitude data, one can estimate the integral absorption of radio waves. This is important for future RO missions when measuring water vapor and minor atmospheric gas constituents, because the difficulty of removing the refractive attenuation effect from the amplitude data can be avoided. The phase acceleration technique can be applied also for determining the location and inclination of sharp layered plasma structures (including sporadic E _s layers) in the ionosphere. The advantages of the phase acceleration technique are validated by analyzing RO data from the Challenging Minisatellite Payload (CHAMP) and the FORMOSA Satellite Constellation Observing Systems for Meteorology, Ionosphere, and Climate missions (FORMOSAT-3/COSMIC).     

Global morphology of ionospheric sporadic E layer from the FormoSat-3/COSMIC GPS radio occultation experiment

Ionospheric sporadic-E (Es) activity and global morphology were studied using the 50 Hz signal-to-noise ratio amplitude and excess phase measurements from the FormoSat-3/Constellation Observing System for Meteorology, Ionosphere and Climate (FS3/COSMIC) GPS radio occultation (RO) observations. The results are presented for data collected during the last sunspot cycle from mid-2006 to the end of 2017. The FS3/COSMIC generally performed more than 1000 complete E-region GPS RO observations per day, which were used to retrieve normalized L1-band amplitude standard deviation (SDL1) and relative electron density (N_e) profiles successfully. More or less 31% of those observations were identified as Es events based on SDL1 and peak SDL1 altitude criteria. We found that the peak Es-event i values are approximately proportional to the logarithms of the corresponding peak N_e differences. Five major geographical zones were identified, in which the seasonal and diurnal Es occurrence patterns are markedly different. These five zones include the geomagnetic equatorial zone (??5°?<?magnetic latitude (ML)?<?5°), two extended geomagnetic mid-latitude zones (15°?<?ML?<?55°, and ??55°?<?ML < ??15°), and two auroral zones (70°?<?ML, and ML < ??70°). The Es climatology, namely its variations with each identified zone, altitude, season, and local time has been documented.     

Artificial ionospheric wave number 4 structure below the F2 region due to the Abel retrieval of radio occultation measurements

We analyzed the effect of the Abel inversion on the wave number 4 (WN4) structure from the GPS radio occultation (RO)–measured electron densities by using the FORMOSAT-3/COSMIC (F-3/C) observations under the equinox condition. The Abel-retrieved electron density from both the F-3/C observations and the simulated results by an empirical model with an imposed WN4 structure in the F layer are investigated. It is found that the Abel inversion can reproduce the real WN4 structure well in the F2 layer. However, it will result in pseudo and reversed-phase WN4 structure in the lower altitude (F1 and E layers). Quantitatively, relative ±15% WN4 signature in the F2 layer can produce ±40% artificial WN4 in the E and F1 layers. Analysis on the F-3/C data shows about ±15% WN4 signature in the F2 layer and ±50% WN4 with reversed-phase in the E and F1 layers. The F-3/C-observed WN4 structure in the E and F1 layers might be the combinations of the real WN4 signature and the artificial effects of Abel retrieval.     

Evaluating the effect of the global ionospheric map on aiding retrieval of radio occultation electron density profiles

Radio occultation (RO) has been proven to be a powerful technique for ionospheric electron density profile (EDP) retrieval. The Abel inversion currently used in RO EDP retrieval has degraded performance in regions with large horizontal gradients because of an assumption of spherical symmetry as indicated by many studies. Some alternative methods have been proposed in the past; the global ionospheric map (GIM)-aided Abel inversion is most frequently studied. Since the number of RO observations will likely increase rapidly in the near future, it is worthwhile to continue to improve retrieval method. In this study, both the simulations and the real data test have been done to evaluate the GIM-aided Abel inversion method. It is found that the GIM-aided Abel inversion can significantly improve upon the standard Abel inversion in either the F or the E region if an accurate GIM is available. However, the current IGS GIM does not appear accurate enough to improve retrieval results significantly, because of the spherical symmetry assumption and sparse global navigation satellite system (GNSS) stations used in its creation. Generating accurate GIM based on dense GNSS network to aid the Abel inversion might be an alternative method.     

Accuracy of GPS-derived relative positions as a function of interstation distance and observing-session duration

 Ten days of GPS data from 1998 were processed to determine how the accuracy of a derived three-dimensional relative position vector between GPS antennas depends on the chord distance (denoted L) between these antennas and on the duration of the GPS observing session (denoted T). It was found that the dependence of accuracy on L is negligibly small when (a) using the `final' GPS satellite orbits disseminated by the International GPS Service, (b) fixing integer ambiguities, (c) estimating appropriate neutral-atmosphere-delay parameters, (d) 26 km ≤ L ≤ 300 km, and (e) 4 h ≤T ≤ 24 h. Under these same conditions, the standard error for the relative position in the north–south dimension (denoted S _n and expressed in mm) is adequately approximated by the equation S _n =k _n /T ^0.5 with k _n =9.5 ± 2.1 mm · h^0.5 and T expressed in hours. Similarly, the standard errors for the relative position in the east–west and in the up-down dimensions are adequately approximated by the equations S _e =k _e /T ^0.5 and S _u =k _u /T ^0.5, respectively, with k _e =9.9 ± 3.1 mm · h^0.5 and k _u =36.5 ± 9.1 mm · h^0.5. Received: 5 February 2001 / Accepted: 14 May 2001     

Eye on the Ionosphere: The Correlation between Solar 10.7 cm Radio Flux and Ionospheric Range Delay

Patricia Doherty joins the regular contributors of this column to discuss the correlation between measurements of solar 10.7 cm radio flux and ionospheric range delay effects on GPS. Mrs. Doherty has extensive experience in the analysis of ionospheric range delays from worldwide systems and in the utilization and development of analytical and theoretical models of the Earth's ionosphere. Ionospheric range delay effects on GPS and other satellite ranging systems are directly proportional to the Total Electron Content (TEC) encountered along slant paths from a satellite to a ground location. TEC is a highly variable and complex parameer that is a function of geographic location, local time, season, geomagnetic activity, and solar activity. When insufficiently accounted for, ionospheric TEC can seriously limit the performance of satellite ranging applications. Since the ionosphere is a dispersive medium, dual-frequency Global Positoning System (GPS) users can make automatic corrections for ionospheric range delay by computing the apparent difference in the time delays between the two signals. Single-frequency GPS users must depend on alternate methods to account for the ionospheric range delay. Various models of the ionosphere have been used to provide estimates of ionospheric range delay. These models range from the GPS system's simple eight-coefficient algorithm designed to correct for approximately 50% rms of the TEC, to state-of-the-art models derived from physical first principles, which can correct for up to 70 to 80% rms of the TEC but at a much greater computational cost. In an effort to improve corrections for the day-to-day variability of the ionosphere, some attempts have been made to predict the TEC by using the daily values of solar 10.7 cm radio flux (F_10,7). The purpose of this article is to show that this type of prediction is not useful due to irregular, and sometimes very poor, correlation between daily values of TEC and F_10.7. Long-term measurements of solar radio flux, however, have been shown to be well correlated with monthly mean TEC, as well as with the critical frequency of the inonospheric F2 region (foF2), which is proportional to the electron density at the peak of the ionospheric F2 region. ? 2000 John Wiley & Sons, Inc.     

Improving the Abel transform inversion using bending angles from FORMOSAT-3/COSMIC

The FORMOSAT-3/COSMIC satellite constellation has become an important tool toward providing global remote sensing data for sounding of the atmosphere of the earth and the ionosphere in particular. In this study, the electron density profiles are derived using the Abel transform inversion. Some drawbacks of this transform in LEO GPS sounding can be overcome by considering the separability concept: horizontal gradients of vertical total electron content (VTEC) information are incorporated by the inversion method, providing more accurate electron density determinations. The novelty presented in this paper with respect to previous works is the use of the phase change between the GPS transmitter and the LEO receiver as the main observable instead of the ionospheric combination of carrier phase observables for the implementation of separability in the inversion process. Some of the characteristics of the method when applied to the excess phase are discussed. The results obtained show the equivalence of both approaches but the method exposed in this work has the potentiality to be applied to the neutral atmosphere. Recent FORMOSAT-3/COSMIC data have been processed with both the classical Abel inversion and the separability approach and evaluated versus colocated ionosonde data.     

Response of GPS occultation signals to atmospheric gravity waves and retrieval of gravity wave parameters

We show that the amplitude of the Global Positioning System (GPS) signals in the radio occultation (RO) experiments is an indicator of the activity of the gravity waves (GW) in the atmosphere. The amplitude of the GPS RO signals is more sensitive to the atmospheric wave structures than is the phase. Early investigations used only the phase of the GPS occultation signals for statistical investigation of the GW activity in the height interval 10–40 km on a global scale. In this study, we use the polarization equations and Hilbert transform to find the 1-D GW radio image in the atmosphere by analyzing the amplitude of the RO signal. The radio image, also called the GW portrait, consists of the phase and amplitude of the GW as functions of height. We demonstrate the potential of this method using the amplitude data from GPS/Meteorology (GPS/MET) and satellite mission Challenge Mini-satellite Payload (CHAMP) RO events. The GW activity is nonuniformly distributed with the main contribution associated with the tropopause and the secondary maximums related to the GW breaking regions. Using our method we find the vertical profiles of the horizontal wind perturbations and its vertical gradient associated with the GW influence. The estimated values of the horizontal wind perturbations are in fairly good agreement with radiosonde data. The horizontal wind perturbations v(h) are ±1 to ±5 m s with vertical gradients dv/dh ±0.5 to ±15 m s km at height 10–40 km. The height dependence of the GW vertical wavelength was inferred through the differentiation of the GW phase. Analysis of this dependence using the dispersion relationship for the GW gives the estimation of the projection of the horizontal background wind velocity on the direction of the GW propagation. For the event considered, the magnitude of this projection changes between 1.5 and 10 m s at heights of 10–40 km. We conclude that the amplitude of the GPS occultation signals contain important information about the wave processes in the atmosphere on a global scale.     

利用GPS掩星资料反演地球中性大气参数折射角方法研究

对利用GPS掩星资料反演地球中性大气参数的原理作了简要介绍，在此基础上对消除电离层影响的改正方法、Abel反演积分上限的确定、上边界测量折射角的优化方法进行了探讨。针对UCAR的Level2数据，在数据预处理工作的基础上，利用特定掩星事件的数据进行了反演计算，并对计算结果进行了分析。     

GPS-LEO掩星探测地球大气的方法

近年来伴随导航定位技术与无线电掩星技术的结合，一种崭新的遥感地球大气的方法-Global Positioning System/Meteoroloby(GPS/NET)应运而生，该方法可以有效地探测大气温、压、湿等气象参数。本文在介绍利用GPS-LEO掩星技术探测地球大气的原理，以及反演大气参数的方法、步骤的基础上，给出了模拟掩星数据反演出的温度廓线、密度廓线及部分GPS/NET实验数据的初步反演结果。通过分析可以看出，利用掩星技术反演出的温度廓线与大气实际情况相符，并且与ECMWF、NCEP的结果吻合较好。在干空气假设条件下，近地面大气的反演结果与实际情况还存在一定误差，有待深入研究解决。     

On the approximation of external gravitational potential with closed systems of (trial) functions

Summary Let S be the (regular) boundary-surface of an exterior regionE _e in Euclidean space ℜ³ (for instance: sphere, ellipsoid, geoid, earth's surface). Denote by {φ_n} a countable, linearly independent system of trial functions (e.g., solid spherical harmonics or certain singularity functions) which are harmonic in some domain containingE _e ∪ S. It is the purpose of this paper to show that the restrictions {ϕ_n} of the functions {φ_n} onS form a closed system in the spaceC (S), i.e. any functionf, defined and continuous onS, can be approximated uniformly by a linear combination of the functions ϕ_n. Consequences of this result are versions of Runge and Keldysh-Lavrentiev theorems adapted to the chosen system {φ_n} and the mathematical justification of the use of trial functions in numerical (especially: collocational) procedures.     

Determination of the ionospheric foF2 using a stand-alone GPS receiver

The critical frequency of ionospheric F2 layer (foF2) is a measure of the highest frequency of radio signal that may be reflected back by the F2 layer, and it is associated with ionospheric peak electron density in the F2 layer. Accurate long-term foF2 variations are usually derived from ionosonde observations. In this paper, we propose a new method to observe foF2 using a stand-alone global positioning system (GPS) receiver. The proposed method relies on the mathematical equation that relates foF2 to GPS observations. The equation is then implemented in the Kalman filter algorithm to estimate foF2 at every epoch of the observation (30-s rate). Unlike existing methods, the proposed method does not require any additional information from ionosonde observations and does not require any network of GPS receivers. It only requires as inputs the ionospheric scale height and the modeled plasmaspheric electron content, which practically can be derived from any existing ionospheric/plasmaspheric model. We applied the proposed method to estimate long-term variations of foF2 at three GPS stations located at the northern hemisphere (NICO, Cyprus), the southern hemisphere (STR1, Australia) and the south pole (SYOG, Antarctic). To assess the performance of the proposed method, we then compared the results against those derived by ionosonde observations and the International Reference Ionosphere (IRI) 2012 model. We found that, during the period of high solar activity (2011–2012), the values of absolute mean bias between foF2 derived by the proposed method and ionosonde observations are in the range of 0.2–0.5 MHz, while those during the period of low solar activity (2009–2010) are in the range of 0.05–0.15 MHz. Furthermore, the root-mean-square-error (RMSE) values during high and low solar activities are in the range of 0.8–0.9 MHz and of 0.6–0.7 MHz, respectively. We also noticed that the values of absolute mean bias and RMSE between foF2 derived by the proposed method and the IRI-2012 model are slightly larger than those between the proposed method and ionosonde observations. These results demonstrate that the proposed method can estimate foF2 with a comparable accuracy. Since the proposed method can estimate foF2 at every epoch of the observation, it therefore has promising applications for investigating various scales (from small to large) of foF2 irregularities.     

Ionospheric climatology derived from gps occultation observations made by the ionospheric occultation experiment

The ionospheric occultation experiment (IOX) is a GPS occultation sensor with an ionospheric mission focus. IOX measurements of GPS L1 and L2 carrier phase during Earth limb views of setting GPS satellites are used together with the Abel transform to determine vertical profiles of electron density from which F-region peak parameters are determined. Data from a four and a half month period beginning in November 2001 are statistically binned and compared with a climatological model. To account for potential errors in interpretation that could arise from violation of the Abel transform assertion of spherical symmetry, the data are compared to both the climatology and to statistics of simulated ionospheric inversions using the climatological model. General characteristics of the climatology are reproduced by the occultation data. However, several significant discrepancies between the model and the data are observed during this near-solar maximum time period. In particular, average mid-latitude daytime densities are shown to be higher than the climatological prediction and the height of F2 layer in the post-sunset equatorial region is underestimated by up to 150 km.

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Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree n _max = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), Δr _CF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of Δr _CF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.     

Electron density profiles in the equatorial ionosphere observed by the FORMOSAT-3/COSMIC and a digisonde at Jicamarca

We examine for the first time the ionospheric electron density profiles concurrently observed by the GPS occultation experiment (GOX) onboard the FORMOSAT-3/COSMIC (F3/C) and the ground-based digisonde portable sounder DPS-4 at Jicamarca (12°S, 283°W, 1°N geomagnetic) in 2007. Our results show that the F3/C generally underestimates the F2-peak electron density NmF2 and the F2-peak height hmF2. On the other hand, when the equatorial ionization anomaly (EIA) pronouncedly appears during daytime, the total electron content (TEC) derived from the radio occultation of the GPS signal recorded by the F3/C GOX is significantly enhanced. This results in the NmF2 at Jicamarca being overestimated by the Abel inversion on the enhanced TEC during the afternoon period.     

W0: an estimate in the Finnish Height Datum N60, epoch 1993.4, from twenty-five GPS points of the Baltic Sea Level Project

Based upon a data set of 25 points of the Baltic Sea Level Project, second campaign 1993.4, which are close to mareographic stations, described by (1) GPS derived Cartesian coordinates in the World Geodetic Reference System 1984 and (2) orthometric heights in the Finnish Height Datum N60, epoch 1993.4, we have computed the primary geodetic parameter W ₀(1993.4) for the epoch 1993.4 according to the following model. The Cartesian coordinates of the GPS stations have been converted into spheroidal coordinates. The gravity potential as the additive decomposition of the gravitational potential and the centrifugal potential has been computed for any GPS station in spheroidal coordinates, namely for a global spheroidal model of the gravitational potential field. For a global set of spheroidal harmonic coefficients a transformation of spherical harmonic coefficients into spheroidal harmonic coefficients has been implemented and applied to the global spherical model OSU 91A up to degree/order 360/360. The gravity potential with respect to a global spheroidal model of degree/order 360/360 has been finally transformed by means of the orthometric heights of the GPS stations with respect to the Finnish Height Datum N60, epoch 1993.4, in terms of the spheroidal “free-air” potential reduction in order to produce the spheroidal W ₀(1993.4) value. As a mean of those 25 W ₀(1993.4) data as well as a root mean square error estimation we computed W ₀(1993.4)=(6 263 685.58 ± 0.36) kgal × m. Finally a comparison of different W ₀ data with respect to a spherical harmonic global model and spheroidal harmonic global model of Somigliana-Pizetti type (level ellipsoid as a reference, degree/order 2/0) according to The Geodesist's Handbook 1992 has been made. Received: 7 November 1996 / Accepted: 27 March 1997     

Monitoring selective availability dither frequencies and their effect on GPS data

Summary The signals transmitted by Block II satellites of the Global Positioning System (GPS) can be degraded to limit the highest accuracy of the system (10 m or better point positioning) to authorized users. This mode of degraded operation is called Selective Availability (S/A). S/A involves the degradation in the quality of broadcast orbits and satellite clock dithering. We monitored the dithered satellite oscillator and investigated the effect of this clock dithering on high accuracy relative positioning. The effect was studied over short 3-meter and zero-baselines with two GPS receivers. The equivalent S/A effects for baselines ranging from 0 to >10,000 km can be examined with short test baselines if the receiver clocks are deliberately mis-synchronized by a known and varying amount. Our results show that the maximum effect of satellite clock dithering on GPS double difference phase residuals grows as a function of the clock synchronization error according to: S/A _effect =0.04 cm/msec, and it increases as a function of baseline length like: S/A _effect =0.014 cm/100 km. These are equations for maximum observed values of post-fit residuals due to S/A. The effect on GPS baselines is likely to be smaller than the 0.14 mm for a baseline separation of 100 km. We therefore conclude, for our limited data set, and for the level of S/A during our tests, that S/A clock dithering has negligible effect on all terrestrial GPS baselines if double difference processing techniques are employed and if the GPS receivers remain synchronized to better than 10 msec. S/A may constitute a problem, however, if accurate point processing is required, or if GPS receivers are not synchronized. We suggest and test two different methods to monitor satellite frequency offsets due to S/A. S/A modulates GPS carrier frequencies in the range of-2 Hz to +2 Hz over time periods of several minutes. The methods used in this paper to measure the satellite clock dither could be applied by the civilian GPS community to continuously monitor S/A clock dithering. The monitored frequencies may aid high accuracy point positioning applications in a postprocessing mode (Malys and Ortiz 1989), and differential GPS with poorly synchronized receivers (Feigl et al. 1991).     

Ground- and space-based GPS data ingestion into the NeQuick model

This paper presents a technique for ingesting ground- and space-based dual-frequency GPS observations into a semi-empirical global electron density model. The NeQuick-2 model is used as the basis for describing the global electron density distribution. This model is mainly driven by the F2 ionosphere layer parameters (i.e. the electron density, N _m F2, and the height, h _m F2 of the F2 peak), which, in the absence of directly measured values, are computed from the ITU-R database (ITU-R 1997). This database was established using observations collected from 1954 to 1958 by a network of around 150 ionospheric sounders with uneven global coverage. It allows computing monthly median values of N _m F2 and h _m F2 (intra-month variations are averaged), for low and high solar activity. For intermediate solar activity a linear interpolation must be performed. Ground-based GNSS observations from a global network of ~350 receivers are pre-processed in order to retrieve slant total electron content (sTEC) information, and space-based GPS observations (radio occultation data from the FORMOSAT-3/COSMIC constellation) are pre-processed to retrieve electron density (ED) information. Both, sTEC and ED are ingested into the NeQuick-2 model in order to adapt N _m F2 and h _m F2, and reduce simultaneously both, the observed minus computed sTEC and ED differences. The first experimental results presented in this paper suggest that the data ingestion technique is self consistent and able to reduce the observed minus computed sTEC and ED differences to ~25–30% of the values computed from the ITU-R database. Although sTEC and ED are both derived from GPS observations, independent algorithm and models are used to compute their values from ground-based GPS observations and space-based FORMOSAT-3/COSMIC radio occultations. This fact encourages us to pursue this research with the aim to improve the results presented here and assess their accuracy in a reliable way.     

Performance of the improved Abel transform to estimate electron density profiles from GPS occultation data

The Abel inversion is a straightforward tool to retrieve high-resolution vertical profiles of electron density from GPS radio occultations gathered by low earth orbiters (LEO). Nevertheless, the classical approach of this technique is limited by the assumption that the electron density in the vicinity of the occultation depends only on height (i.e., spherical symmetry), which is not realistic particularly in low-latitude regions or during ionospheric storms. Moreover, with the advent of recent satellite missions with orbits placed around 400 km (such as CHAMP satellite), an additional issue has to be dealt with: the treatment of the electron content above the satellite orbits. This paper extends the performance study of a method, proposed by the authors in previous works, which tackles both problems using an assumption of electron-density separability between the vertical total electron content and a shape function. This allows introducing horizontal information into the classic Abel inversion. Moreover, using both positive and negative elevation data makes it feasible to take into account the electron content above the LEO as well. Different data sets involving different periods of the solar cycle, periods of the day and satellites are studied in this work, confirming the benefits of this improved Abel transform approach.     

The M 2 ocean tide loading wave in Alaska: vertical and horizontal displacements, modelled and observed

Crustal deformations caused by surface load due to ocean tides are strongly dependent on the surface load closest to the observation site. In order to correctly model this ocean loading effect near irregular coastal areas, a high-resolution coastline is required. A test is carried out using two GPS sites located in Alaska, where the ocean tide loading effect is large and consequently observed easily by relative positioning with GPS. The selected sites are Fair (Fairbanks) and Chi3 (located on an island that separates Prince William Sound from the Gulf of Alaska). Processing of hourly baseline solutions between Fair and Chi3 over a period of 49 days yields a significant ocean tide loading effect. The data are processed using different strategies for the tropospheric delay correction. However, the best results are obtained when 1-h ZTD (Zenith Tropospheric Delay) parameters for hourly solutions are used. In this case ocean tide loading is not absorbed into the ZTD parameters. Hence, ocean tide loading can be well resolved in the GPS data analysis. In addition, the M ₂ ocean tide wave in the Gulf of Alaska has a very large amplitude. Although the horizontal M ₂ ocean tide loading amplitude in general is only about 1/4 of the vertical M ₂ ocean tide loading amplitude, the differential horizontal M ₂ ocean tide loading displacements are nevertheless measurable using differential GPS (DGPS). When using the GOT99.2 ocean tide model and taking the coastal structure into account, the predicted differential vertical M ₂ amplitude and Greenwich phase lag due to ocean tide loading are 19.3 mm and 110.2 degrees respectively, while GPS measurements yield 21.3 ± 1.0 mm and 99.7±2.8 degrees. Similarly, the predicted differential horizontal M ₂ amplitude and Greenwich phase lag (in the north–south direction) are 4.5 mm and –77.0 degrees, while GPS yields 5.4 ± 0.3 mm and –106.3±3.3 degrees. Only the north-south component of the differential horizontal M ₂ ocean tide loading wave is considered, because the east–west component is too small for the processed baseline and not detectable using DGPS.