Kinematic Precise Point Positioning During MarginalSatellite Availability

N.S. Kjørsvik, O. Øvstedal and J.G.O. Gjevestad

Abstract Precise Point Positioning (PPP) is a tech-nique where observations from a single GNSS receiverare used to estimate coordinates with precision rang-ing from centimeters to decimeters. PPP does not relyon data from dedicated reference receivers and is lo-gistically a very competitive alternative to differentialGNSS methods. In areas with reduced satellite avail-ability the geometrical strength of the solution will bereduced and observations from such periods shouldbe treated with special attention. In marine applica-tions the use of additional information in the formof height constraints will enhance PPP during periodswith marginal satellite availability. Additional observa-tions from GLONASS satellites or Inertial NavigationSystem (INS) will also to a certain degree compensatefor reduced availability of GPS satellites. This paperaddresses the contribution from height aiding and fromcurrent GLONASS satellites. Based on formal preci-sion and reliability as well as accuracy both height aid-ing and GLONASS improve the quality of estimatedcoordinates. The contribution to reliability is more sig-nificant than the contribution to precision and accuracy.In periods with very weak GPS constellation, heightaiding gives the most valuable contribution. The pa-per verifies that even under marginal satellite availabil-ity, PPP processing is capable of producing coordinateswith precision and accuracy at the subdecimeter leveland with reliability at the one meter level.

N.S. KjørsvikTerraTec AS, P.O. BOX 513, N-1327 Lysaker, Norway

O. ØvstedalDepartment of Mathematical Sciences and Technology,Norwegian University of Life Sciences, N-1432 As, Norway

J.G.O. GjevestadDepartment of Mathematical Sciences and Technology,Norwegian University of Life Sciences, N-1432 As, Norway

Keywords GNSS · PPP · Marine positioning

1 Introduction

Precise Point Positioning (PPP) using GPS satel-lites was first developed for static applications (e.g.Zumberge et al., 1997), but has in recent yearsalso been used in kinematic mode (e.g. Kouba andHeroux, 2001). In PPP undifferenced carrier phaseand code phase observations are used to estimatecoordinates and nuisance parameters like receiverclock errors, carrier phase ambiguities and residualtropospheric delay. The initial carrier phase offsetsin satellites and receiver make the identification ofinteger carrier phase ambiguities a very challengingtask, and in the standard PPP model the ambiguitiesare estimated as real numbers in a float solution.Compared to differential processing with fixed ambi-guities, the geometrical strength of the adjustment in afloat solution is weaker, requiring longer convergencetimes. When processing a sufficient time span of dualfrequency code- and carrier phase observations from asingle GPS receiver, sub-decimeter accuracy is never-theless readily available. Prerequisites are more than20–60 min of continuous observations of good qualityas well as a sufficient number of well-distributedsatellites. This exposes the need to treat periods withmarginal satellite availability with special attention.

In applications such as marine surveys in narrowand steep fjords, a significant number of satellitesmight be masked by the terrain. Satellites are typi-cally only available at high elevation angles, and thegeometrical strength of the solution will be severelyweakened. A typical survey pattern will consist of anumber of parallel fjord crossings. Hence, a sufficient

M.G. Sideris (ed.), Observing our Changing Earth, International Association of Geodesy Symposia 133, 691c© Springer-Verlag Berlin Heidelberg 2009

692 N.S. Kjørsvik et al.

number and distribution of satellites will only beavailable near the middle of the fjord. The situationis further complicated by the fact that different satel-lites are visible at each side of the fjord, leading toshort time spans of continuous observations to somesatellites. For other investigations of PPP in marineenvironments see e.g. Kjørsvik et al. (2006) andØvstedal et al. (2002).

In the proposed test design, almost 40 days of 1 HzGNSS observations were collected at a shuttle ferry un-der good conditions. A reference trajectory was com-puted using a local (<2 km) reference station withfixed double-differenced ambiguities. Using a DigitalTerrain Model (DTM) the topography of a nearby fjordwas simulated, and the artificial horizon was appliedto the actual data to limit the satellite availability. Wethus have real observations in a scenario with marginalsatellite availability but with a precise and reliable ref-erence trajectory.

Several approaches to improve PPP accuracy duringmarginal and changing satellite availability are possi-ble. In this work we first investigate the contribution ofcurrent GLONASS satellites. The International GNSSService (IGS) (Dow et al., 2005) provides precise orbitand clock corrections for both GPS and GLONASS,however the GLONASS products are currently not onpar with the GPS products. The GLONASS constella-tion has also suffered from poor maintenance. Hence,under normal conditions GLONASS gives a very smallcontribution. The orbital inclination is however favor-able at high latitudes, and even a few extra satellitesmay be highly beneficial during marginal conditions.In a second step we investigate the effect of height con-straints during periods of marginal satellite availability.

During regular surveys a reference (“truth”) trajec-tory will not be available. Instead, the quality of thepositioning must be inferred solely from the data. Theformal precision of the estimates provides some infor-mation, but high precision does unfortunately not guar-antee high accuracy. In addition to precision, reliabilitytheory (e.g. Teunissen and Salzmann, 1989) has provento be a useful tool for quality assessment. The reliabil-ity theory uses rigorous statistics to quantify a system’sability to detect and remove erroneous observations.This statistic is often denoted Minimal Detectable Bias(MDB). In addition, the effect of an undetected erro-neous observation on the parameter estimates can alsobe computed. The latter is often denoted Minimal De-tectable Error (MDE).

2 Height Support

A marine survey vessel will usually have a moderatevertical motion (heave). Heave is due to e.g. ocean tidesand waves. A reference point (e.g. GNSS antenna) dis-placed from the vessel’s center of mass will also expe-rience some vertical motion due to roll and pitch.

The astronomical ocean tide can be modeled fairlyaccurate, provided that phases and amplitudes ofthe various constituents are known. The phases arehowever subject to large variations due to the coastlineand local seabed topography. The actual tide will inaddition be affected by local and regional weatherphenomena.

2.1 Theoretical Development

Assuming that the actual tide and wave motion can notbe accurately predicted, the preferred solution shouldconstitute a simple and robust model with well-definederror characteristics.

As a first step it can be assumed that height coor-dinates can be accurately determined during periodswith adequate satellite availability. A simple approachmay then be to extrapolate this height to the periodswith marginal satellite availability. Obviously heightsrelated to the geoid should be considered.

The error caused by the height extrapolation is ex-pected to grow with time, and be a function of e.g. tidalamplitudes and local weather conditions.

In the following development it is assumed that thetide consists of one dominating constituent with knownamplitude and period, but unknown phase. It is as-sumed that a constant height (above the mean sea sur-face) can be estimated and removed. A predicted heightusing constant projection will then have a mean squareerror given by

σ 2t (θ) = 1

2π

∫ 2π

0[A cos(α + θ) − A cos(α)]2dα

= A2[1 − cos(θ)], (1)

where A denotes the amplitude and θ = 2π t/T de-notes the prediction step (in radians) and T is the pe-riod of the tidal wave.

The period of the wave is assumed to be the semidi-urnal period of the moon (M2), i.e. approximately 12 h25 m. It can be assumed that the short period waves

Kinematic Precise Point Positioning 693

are uncorrelated with the tidal wave. As a first approx-imation we propose to model this contribution as anuncorrelated noise process with variance σ 2

w. The finalexpression for mean square error of the constant heightassumption may then be formed as a sum of the inde-pendent contributions, and reads

σ 2(t) = A2[1 − cos(2π t/T )] + σ 2w. (2)

2.2 Implementation

Representative amplitudes were derived from theNAO99b model (Matsumoto et al., 2000) using anempirical approach. The ocean tide was predicted in aglobal grid with with 0.5◦ resolution using data from arandom period of 30 days. Software from the NAO99website1 was used to predict the tides. The amplitudewas estimated for each node as 1/2 of the mean dailypeak to peak value.

In our PPP software the reference heights used forheight support are estimated in a separate Kalman fil-ter, working in parallel to the master filter.

To compensate for geoid undulations, heights areconverted to orthometric heights using the EGM96model (NIMA, 2000). The variance of the waves (σ 2

w)are estimated as the variance of the post-fit residualsfrom the height Kalman filter.

The height support is implemented by adding a con-straint (pseudo-observation) directly on the estimatedheight using an appropriate variance, as determined bythe model in Eq. (2).

Activation of the height support is triggered by test-ing a “modified” vertical dilution of precision (VDOP)against a predefined threshold of 3.0. The modifiedVDOP is computed using only satellites with a pre-cisely determined phase bias parameter. A precisionthreshold of 2 cm is currently used. To avoid rapid andrepeated switches, the height support is always kept en-abled for at least 5 min.

3 GLONASS Modeling

A system specific bias is estimated to account forreceiver hardware biases and time system biases

1 http://www.miz.nao.ac.jp/staffs/nao99/index En.html

between GPS and GLONASS. The parameter isestimated as a random walk process, with variance3 (ns)2/h. GLONASS precise orbits and clocks areoften estimated in a separate process from GPS, andboth GPS and GLONASS clock products are alignedto their respective broadcast clocks on a daily basis.Hence, the estimated inter-system bias must be reseteach midnight to account for discontinuities betweenthe GPS and GLONASS timescales. Inter-frequencybiases in the carrier phase observations will be fullyabsorbed in the phase bias estimates, provided theyare stable over timescales of hours. Similar biases inthe code phases are currently ignored ignored in thefunctional model. The mean bias is absorbed in theinter-system bias estimate, and remaining biases areassumed to be small compared to the observation noiseand code multipath. Problems might arise during rapidconstellation changes, but should be alleviated by thestochastic model.

GLONASS satellites have a Cesium-based (Cs) fre-quency standard and will consequently have a slightlyworse short-time stability than Rubidium-based (Rb)satellites (e.g. all GPS block IIR and approx. 50%of GPS block II and IIA satellites). Short-time fre-quency instability leads to increased errors of inter-polated satellite clock corrections, yielding increasednoise in the corrected range observations.

In the analyzed data set two satellites (R17 and R21)were manually excluded throughout the processing.The receiver experienced severe tracking anomalies,with several thousand cycle slips and a vast amount ofoutliers even when observed at high elevation angleswithout signal obstruction. Both satellites were flaggedas healthy during the test period (cf. Table 1).

A satellite with unhealthy observations has the po-tential of causing divergence in the estimate of theinter-system bias parameter, in particular shortly fol-lowing a parameter reset (at midnight) in combinationwith few observed GLONASS satellites.

4 Test Data

From March to May 2006 the Norwegian Hydro-graphic Service made a full scale test of their newlyacquired PPP software TerraPos from TerraTec AS,Norway. A shuttle ferry traveling between Lauvvikand Oanes outside Stavanger, Norway was equipped

694 N.S. Kjørsvik et al.

Table 1 Status of GLONASS satellites during the test period

Slot Statusa Excluded Notes

R01 OKR02 OKR03 OKR04 OKb Unhealthy 2 h 30 mR05 UNHb Healthy for 6 daysR06 OKb Incomplete ephemeris, unhealthy

6 daysR07 OKR08 OKR17 OKb Yes Tracking anomaliesR18 UNH No ephemeris / obs.R19 OKb Unhealthy 10 h 30 mR20 OKR21 OK Yes Tracking anomaliesR22 OKb Unhealthy 30 mR23 UNH Incomplete ephemerisR24 UNHaAccording to the Russian Space Agency ftp://ftp.glonass-ianc.rsa.ru/MCC/STATUS/2006.bUnhealthy in parts of the dataset.

with one Topcon Legacy GNSS receiver. An identicalreceiver serving as reference station was deployedclose to Lauvvik. Both stations were equipped withidentical geodetic antennas. The ferry route is approx-imately 1.5 km long and the ferry repeats the routeevery half an hour. The terrain in the area is relativelyflat, giving very good satellite availability. During thisfull scale test more than 40 days of continuous obser-vations at 1 Hz were collected, with a one day breakin the middle due to repairs in Stavanger. Real-TimeKinematic (RTK) solutions were recorded along withraw data. Figure 1 shows the ferry used to collect thedata, with a map of the area shown in Fig. 2.

Fig. 1 Shuttle ferry carrying the GNSS receiver

0 0.5 1

km

Oanes

Lauvvik

4 E 8 E 12 E

58 N

60 N

62 N

Fig. 2 Map showing where the original data were collected

4.1 Reference Trajectory

A reference trajectory was computed post mission us-ing Geogenious version 2.11 from Spectra PrecisionTerrasat in 24 h batches. The post-processed solutionswere then compared to the RTK solutions. For eachepoch, the average of the two solutions were acceptedif the 3D discrepancy was less than 2 cm. The averagesolutions constitute the reference trajectory.

4.2 Terrain Simulation

In this study a subset consisting of 16 days (DoY103–119) is analyzed. In order to simulate a marginalsatellite availability with this real dataset, the morechallenging topography of a nearby fjord called Lyse-fjorden has been utilized. Lysefjorden consists of veryhigh and steep mountain sides and would generate se-vere obstructions to a GNSS receiver in the area. Inthis study a digital terrain model (DTM) of Lysefjor-den is relocated to the ferry route between Lauvvik andOanes, thus providing an artificial horizon to the ferrytrajectory.

)

)

)

)) )

Kinematic Precise Point Positioning 695

Fig. 3 Oblique view of Lysefjorden, viewed towards east

Figures 3 and 4 are included to give an impressionof the challenging topography in the simulation.

The starting point of the terrain reduction is YUMAformat almanacs for both GPS and GLONASS. The ar-tificial horizons are based on a 3 × 6 arc sec. DTM(i.e. approximately 100 m grid size). A horizon is cal-culated for each grid position at sea level. Each horizonis stored with 2◦ azimuthal resolution.

After the satellite almanacs are loaded the positionsof the vessel trajectory are read epoch by epoch. Foreach vessel position a list of potentially visible satel-

lites is computed by assuming an open sky. Using theactual vessel position, a horizon is interpolated using anearest neighbor strategy. A new list of visible satellitesis constructed, and unavailable satellites are maskedand used as information to reduce the real observationfile.

5 Data Processing

5.1 Software

The data were processed using the TerraPos softwarefrom TerraTec AS, Norway. TerraPos is compatiblewith the latest IERS conventions (McCarthy andPetit, 2004), as well as IGS recommendations. Ter-raPos utilizes an optimal Kalman filter and smoothercombination (e.g. Gelb, 1974). The data were pro-cessed in a single filter/smoother run, ensuring optimalestimation of all parameters.

Effects that are currently handled include, e.g. solidearth tides, pole tide displacements, receiver and satel-lite antenna offsets and phase center variations andphase wind-up.

Fig. 4 Map showing the terrain used in the simulation

696 N.S. Kjørsvik et al.

Satellites in eclipses and noon-turn maneuversare included in the processing. Satellite attitude isnormally determined using the model from Bar-Sever (1996). For periods or satellites where theattitude cannot be reliably predicted, the phase bias isassigned a random walk process. This is the case for allGLONASS satellites, as well as GPS satellites wherethe shadow entry or exit times can not be determined.

No corrections are applied for ocean tide loadingdisplacements.

Tropospheric delay is predicted using the UNB3model (Collins and Langley, 1997) in combination withthe Global Mapping Function (Boehm et al., 2006).Residual tropospheric zenith delay is estimated as a1st order Gauss-Markov process with a large time con-stant, i.e. a process approaching random walk. Table 2gives an overview of the spectral amplitudes associatedwith the white noise driving functions of the variousstate parameters.

Table 2 Filter parameters

Process Spectral amplitude

Cross-track velocity (2.0)2 m2/s3

Along-track velocity (2.0)2 m2/s3

Vertical velocity (0.2)2 m2/s3

Tropospheric ZD (3.0)2 mm2/hInter-system bias (3.0)2 ns2/h

An elevation cutoff angle of 5◦ is used throughoutthis study.

Minimal Detectable Errors (MDE) are computed us-ing a Minimal Detectable Bias (MDB) approach as de-scribed in e.g. Teunissen and Salzmann (1989). TheMDBs are computed using α ≈ 0.1% (probability ofcommitting a type I error) and β = 80.0% (probabilityof committing a type II error). For a discussion of typeI and II errors see e.g. Teunissen (2000).

The reliability computations are performed duringthe forward filter run. Although somewhat suboptimal,steady-state is reached within minutes. Efficient imple-mentation of an optimal computation of reliability isconsidered to be a non-trivial task.

The absolute level of reliability is to a large ex-tent governed by the parameters characterizing the dy-namic model, the observation noise as well as the cho-sen levels of confidence. We therefore base most of theanalysis of performance relative to that of a GPS-onlysystem.

Dual-frequency code and carrier phases as well asDoppler observations are employed using their respec-tive ionosphere-free linear combinations.

5.2 Ephemeris and Satellite ClockCorrections

In this study we use final ephemerides and satelliteclock corrections from the Center of Orbit Determina-tion in Europe (CODE) for the GPS satellites, due tothe availability of high-rate (30 s) satellite clock cor-rections. Unfortunately, only broadcast clock correc-tions are included for GLONASS satellites. Hence,GLONASS ephemerides and clock corrections fromthe European Space Agency/European Space Opera-tions Center (ESA/ESOC) are used. These files dohowever only have 15 min resolution, which is not op-timal for high-rate kinematic positioning. Also, onlythe quality of the ephemeris is stated in the SP3-filesas overall values for each day. The precision of theclock corrections are assumed to be at the same levelas that of the orbits, i.e. 0.5 ns, corresponding to 0.15 min terms of range error.

6 Results

A total of 14 days of continuous observations at 1 Hzare merged into one observation file and processed inkinematic PPP-mode using the TerraPos software. Theobservations are processed using four different strate-gies:

◦ GPS only◦ GPS with height support◦ GPS and GLONASS◦ GPS and GLONASS with height support

6.1 Overall Statistics

In Tables 3, 4, 5 and 6 quality indicators from the fourprocessing are presented. Precision and MDE are basedon formal statistics from the processing software whilethe accuracy is based on differences between positions

Kinematic Precise Point Positioning 697

Table 3 95%-quantile for the GPS-only solution, using 1294820epochs

STD (m) MDE (m) Error (m)

Hor. 0.064 1.43 0.067Vert. 0.068 0.75 0.100

Table 4 95%-quantile for the GPS solution with height support,using 1299197 epochs

STD (m) MDE (m) Error (m)

Hor. 0.067 1.46 0.067Vert. 0.069 0.75 0.098

Table 5 95%-quantile for the GPS/GLONASS solution, using1299766 epochs

STD (m) MDE (m) Error (m)

Hor. 0.063 1.14 0.074Vert. 0.064 0.60 0.103

Table 6 95%-quantile for the GPS/GLONASS solution withheight support, using 1299767 epochs

STD (m) MDE (m) Error (m)

Hor. 0.062 1.14 0.073Vert. 0.064 0.59 0.100

from the PPP processings and the reference trajectory.The statistics are given as 95% quantiles.

For all strategies the precision and accuracy are atthe sub-decimeter level horizontally and at the decime-ter level in height. The MDEs are approximately atthe meter level. Compared to the GPS-only solution,the improvements in formal precision and accuracyare at the millimeter level when including heightaiding, GLONASS and GLONASS+height aiding.The improvement in MDEs are at the level of severaldecimeters.

The apriori process noise for vertical coordinates issmaller than for horizontal coordinates, leading to bet-ter MDEs for vertical coordinates.

6.2 Quality of Marginal Solutions

It should be noted that Tables 3, 4, 5 and 6 are based ona very large number of epochs where the majority of es-timated positions are of reasonably good quality. In or-der to assess the epochs with somewhat marginal satel-lite availability, quality indicators are also computedbased on epochs where a GPS-only solution does notmeet the following quality criteria:

◦ Formal precision (1σ ) below 0.10 m horizontallyand 0.15 m in height.

◦ MDE better than 0.30 m horizontally and 0.45 m inheight.

Epochs where the quality criteria were not met inthe GPS-only solution were therefore extracted andused to compute another set of quality indicators.Table 7 shows formal precision, MDE, accuracy,HDOP and VDOP as well as total number of epochsfor the marginal set of data. The contribution fromheight aiding, GLONASS and GLONASS+heightaiding is more significant for the marginal set of thedata.

The results in Table 7 are based on epochs wherea marginal GPS-only solution is possible. Due to aninsufficient number of GPS-satellites, there are 5237epochs where the GPS-only processing is not possible.The extra information from height aiding, GLONASSand GLONASS+height aiding respectively leads to es-timated positions for a substantial part of these miss-ing epochs. Table 8 presents quality indicators for theepochs where a GPS-only solution is not possible. Theformal precision as well as accuracy are at the decime-ter level for the height aided strategies while the MDEis at the 2–3 m level. The contribution of GLONASSis not as significant as the contribution from heightaiding.

Table 7 Statistics (95%) for the epochs rejected in a GPS-onlysolution

GPS GPS GPS GPSGLO GLO

HS HS

H prec. 0.086 m 0.084 m 0.081 m 0.081 mV prec. 0.094 m 0.092 m 0.088 m 0.086 mH rel. 2.11 m 2.08 m 1.70 m 1.70 mV rel. 1.03 m 0.97 m 0.73 m 0.69 mH err 0.080 m 0.079 m 0.085 m 0.084 mV err. 0.123 m 0.116 m 0.125 m 0.119 mHDOP 6.1 6.1 3.1 3.1VDOP 8.1 8.1 5.0 5.0# epochs 392854 392854 392854 392854

6.3 Post-Fit RMS Statistics

During the filter run an overall RMS of the post-fitresiduals is computed for each observable and systemseparately. These overall values are hence averaged

698 N.S. Kjørsvik et al.

Table 8 Statistics (95%) for the epochs not available in a GPS-only solution

GPS GPS GPSGLO GLO

HS HS

H prec. 0.211 m 1.659 m 0.174 mV prec. 0.236 m 4.141 m 0.223 mH rel. 3.54 m 3.35 m 3.43 mV rel. 2.82 m 2.67 m 1.92 mH err 0.111 m 2.159 m 0.108 mV err. 0.100 m 6.205 m 0.113 mHDOP ∞ 5.1 5.1VDOP ∞ 14.2 14.2# epochs 4377 4951 4951

Table 9 Overall post-fit RMS statistics. All observables are usedwith their respective ionosphere-free linear combinations

Observable GPS GLONASS

Carrier phase 0.01 m 0.15 mCode phase 2.5 m 3.5 mDoppler 0.01 m/s 0.01 m/s

over all elevation angles and satellites in a system. Theresults are shown in Table 9.

The carrier phase errors are dominated by satelliteclock errors. A value of 0.01 m for GPS demonstratesthe excellent quality of the IGS GPS products. A valueof 0.15 m for GLONASS carrier phases thus verifiesthe assumed quality of the GLONASS satellite clockcorrections.

The slightly higher RMS of the GLONASS codephase observations as compared to GPS is to some ex-tent explained by the lower chip rate, but also suggeststhe presence of small inter-frequency biases. The ef-fect of satellite clock errors (<0.15 m, as seen from thecarrier phase statistics) is negligible compared to thecombined level of multipath and noise.

Both GPS and GLONASS Doppler observationsseem to have comparable performance.

7 Summary and Conclusion

We have optimized the memory usage of our softwarein order to process huge kinematic datasets withoutloss of optimality. The software is now capable of pro-cessing several months of 1 Hz kinematic GNSS data inan integrated and optimal adjustment on a standard 32bit PC-based workstation, with only modest memoryrequirements. This achievement can be important in

the study of several geophysical signals observed byGNSS.

The software’s ability to process GLONASS ob-servations has been verified, and valuable experiencewith multi-system modeling and parameter estimationhas been gained. We have identified issues with time-scales and inter-system biases that arise when usingclock products spanning several day and week bound-aries. These artifacts are due to current analysis anddata combination strategies of the analysis centers thatcontribute to the IGS. In light of the current state ofGLONASS ephemeris and clock products the currenthandling of code phase inter-frequency biases seemsadequate, as justified by the post-fit RMS statistics.

A refined model for handling of inter-frequency bi-ases (in satellites and in receivers) must be used by allIGS analysis centers to ensure a high degree of con-sistency in satellite clock and calibration products. Nosuch convention is currently available.

The current contribution of GLONASS to PPP ishampered by the number of available satellites and thequality of ephemeris and satellite clock corrections.The quality of the satellite clock corrections is partlydue to the temporal resolution, but also to the short-time stability of the Cesium frequency standard. Thenet result is that the low precision of the correctedranges limits the impact on the parameter estimates andtheir precision. Redundancy is nevertheless improved,yielding improved MDE. In this study we found thatthe MDE was improved by approximately 20%. Incases where GPS alone was unable to provide a solu-tion, the combined solutions were accurate only to themeter-level.

In contrast to the addition of a GLONASS ob-servation, the height support constrains the heightcomponent directly. A GLONASS observation hasa functional relation to many parameters, includingposition, receiver clock bias, inter-system bias, tro-pospheric zenith delay etc. The activation of heightsupport was found to consistently improve the statisticsof the marginal solutions, in particular by improvingthe MDE by approximately 5%.

The high-resolution terrain simulation provides adetailed model of the topography within Lysefjorden,yielding a very realistic obstruction scenario. How-ever, one should keep in mind that the quality of theobservations, e.g. multipath and cycle slips, might betoo optimistic when applying artificial terrain obstruc-tions. In a real-world degraded environment a receiver

Kinematic Precise Point Positioning 699

would occasionally experience low signal power andcycle slips. One should also note that in situations withmarginal satellite availability observation domain er-rors may have severe impact on estimated parameters.

Acknowledgments The authors greatly acknowledge the Inter-national GNSS Service for its continuing efforts. The artificialhorizons were simulated by Lars K. Nesheim and Arne E. Ofs-tad of the Norwegian Hydrographic survey.

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