GPS Observables Research Articles

GPS observables in Newtonian spacetime or why we do not need ‘physical’ coordinate systems

Some authors have defended the claim that one needs to be able to define ‘physical coordinate systems’ and ‘observables’ in order to make sense of general relativity. Moreover, in Rovelli (Physical Review D,65(4), 044017 2002), Rovelli proposes a way of implementing these ideas by making use of a system of satellites that allows defining a set of ‘physical coordinates’, the GPS coordinates. In this article I oppose these views in four ways. First, I defend an alternative way of understanding general relativity which implies that we have a perfectly fine interpretation of the models of the theory even in the absence of ‘physical coordinate systems’. Second, I analyze and challenge the motivations behind the ‘observable’ view. Third, I analyze Rovelli’s proposal and I conclude that it does not allow extracting any physical information from our models that wasn’t available before. Fourth, I draw an analogy between general relativistic spacetimes and Newtonian spacetimes, which allows me to argue that as ‘physical observables’ are not needed in Newtonian spacetime, then neither are they in general relativity. In this sense, I conclude that the ‘observable’ view of general relativity is unmotivated.

Smoothened Absolute Kinematic Position Estimation Of Leo Satellite Using Gps Observables

In this paper, our objective is to use the navigation signals (pseudorange and carrier phase measurements) from GPS to kinematically estimate the absolute position of a low Earth orbit satellite and its receiver clock biases. GPS observables can estimate the precise position of a satellite in near real time. The GPS satellite navigation file is downloaded from Scripps Orbit and Permanent Array Centre (SOPAC) website for starting epoch of 01-01-2016 00:00 UT. The observables are then generated for a Low-Earth-Orbit (LEO) satellite at an altitude of 901.4 km. The absolute position is estimated using pseudorange measurements. The accuracies obtained are (5.68 8.04 6.17) m in ECEF frame. This estimated absolute position is then smoothened with the time-differenced carrier phase measurements. The Kalman filter gains are computed in near real time. The smoothened accuracies obtained are (2.24 2.60 2.6982) m in ECEF frame, thus showing improved estimated absolute position accuracy.

GPS Spoofing Detection by Neural Network Machine Learning

GPS today is ubiquitous. It provides real-time positioning, navigation, and timing (PNT) data for countless military and civilian users worldwide. Yet, a proliferation of GPS degrading and denying devices threatens GPS PNT capabilities. The biggest threat is spoofing, which deceives a GPS receiver into accepting false signals as genuine. Current antispoofing techniques are highly dependent on specific spoofing scenarios, choice of mathematical models, values of thresholds, and often require significant hardware and/or software modifications to existing GPS receivers. Efforts are also underway to replace GPS with alternate sources. However, no such sensor can match the availability, accuracy, and global utility of GPS. This article examines an alternate antispoofing method using neural networks. The supervised machine learning technique uses well-understood GPS observables such as pseudorange, carrier phase, Doppler shift, and carrier-to-noise density ratio to distinguish between authentic and spoofed signals. Network architecture, training sample size, number of hidden layers, distribution of neurons between hidden layers, and number of hidden neurons are examined. Results show that the proposed method is capable of detecting spoofing signals with a high probability of correct classification and a low probability of misclassification.

A study of vTEC above Nepal exploring different calibration techniques, including a comparison with the NeQuick-2 model

In this paper, we investigate the performance of the NeQuick 2 (NeQ-2) model with respect to Ciraolo’s and Gopi’s derived ionospheric vertical Total Electron Content (vTEC) during the years 2014 and 2015. GPS observables derived from dual-frequency receivers over western Nepal (Simikot, Bhimchula, and Nepalganj) are processed to obtain the experimental vTEC utilizing Gopi’s and Ciraolo’s calibration procedures. The monthly and seasonal behavior of vTEC obtained from each calibration technique is compared with the vTEC obtained from the NeQ-2 model during a quiet period. It is observed that the vTEC value obtained from all studied approaches started to increase from 00:00 Universal Time (UT = Local Time(LT) +5:45), reached a maximum around 08:00 UT (13:45 LT), followed by a decrease, attaining a minimum value around 23:00 UT (4:45 LT). Moreover, a comparative study showed that vTEC computed using the Ciraolo calibration technique overestimates GPS vTEC, calculated in all hours and months by Gopi’s approach. In the Spring and Summer, vTEC derived using Ciraolo’s TEC calibration overestimates NeQ-2 and underestimates it in the Autumn and Winter. It is found that NeQ-2 model vTEC is favorably associated with GPS vTEC obtained using the Gopi procedure in Spring and correlates with the Ciraolo technique in Autumn. Two GPS vTEC estimations demonstrate superior consistency in the Summer and Winter seasons over the region of Nepal. It is found that the mean absolute difference between NeQ-2 prediction and GPS vTEC procured through the Gopi approach is less on the storm event day. By contrast, it is discovered less by the Ciraolo technique when the storm is recovering (except for a few cases).

Design of a high-stability heterogeneous clock system for small satellites in LEO

We describe our investigation into the performance of low-power heterogeneous timing systems for small satellites, using real GPS observables from the GRACE Follow-On mission. Small satellites have become capable platforms for a wide range of commercial, scientific and defense missions, but they are still unable to meet the needs of missions that require precise timing, on the order of a few nanoseconds. Improved low-power onboard clocks would make small satellites a viable option for even more missions, enabling radio aperture interferometry, improved radio occultation measurements, high altitude GPS navigation, and GPS augmentation missions, among others. One approach for providing improved small satellite timekeeping is to combine a heterogeneous group of oscillators, each of which provides the best stability over a different time frame. A hardware architecture that uses a single-crystal oscillator, one or more Chip Scale Atomic Clocks (CSACs) and the reference time from a GPS receiver is presented. The clocks each contribute stability over a subset of timeframes, resulting in excellent overall system stability for timeframes ranging from less than a second to several days. A Kalman filter is used to estimate the long-term errors of the CSACs based on the CSAC-GPS time difference, and the improved CSAC time is used to discipline the crystal oscillator, which provides the high-stability reference clock for the small satellite. Simulations using GRACE-FO observations show time error standard deviations for the system range from 2.3 ns down to 1.3 ns for the clock system, depending on how many CSACs are used. The results provide insight into the timing performance which could be achieved on small LEO spacecraft by a low power timing system.

Multi constellation GNSS precise point positioning and prediction of propagation errors using singular spectrum analysis

The present paper investigates GPS, GLONASS, and Beidou observables from the Multi-GNSS Experiment (MGEX) network of stations in the East Asia region. Precise Point Positioning (PPP) estimates, Dilution of Precision (DOP), Slant Total Electron Content (STEC), and Zenith Total Delay (ZTD), along with their prediction from the Singular Spectrum Analysis (SSA), were studied to analyze and understand their effectiveness on the overall positioning accuracy over the region. The analysis confirms that the PPP accuracy does not solely depend on the DOP values, but is also affected by seasonal changes or partial corrections of the ionospheric and tropospheric delays. The accuracy is improved by approximately 70% for the GPS (G), GPS+GLONASS (GR), and GPS+GLONASS+Beidou (GRC) constellations after applying the SSA method. The GNSS-derived STEC and ZTD values were also predicted with the SSA methods to evaluate the trustworthiness of the approach for mitigation of atmospheric errors. The results show that the STEC and ZTD seem to be in perfect agreement with the SSA model. The technique could be applicable for improved navigational measurements augmenting other available constellations such as the Galileo, IRNSS, and QZSS, as well as satellite-based augmentation systems across the globe.

RECURSIVE LEAST SQUARES WITH REAL TIME STOCHASTIC MODELING: APPLICATION TO GPS RELATIVE POSITIONING

Abstract. Geodetic data processing is usually performed by the least squares (LS) adjustment method. There are two different forms for the LS adjustment, namely the batch form and recursive form. The former is not an appropriate method for real time applications in which new observations are added to the system over time. For such cases, the recursive solution is more suitable than the batch form. The LS method is also implemented in GPS data processing via two different forms. The mathematical model including both functional and stochastic models should be properly defined for both forms of the LS method. Proper choice of the stochastic model plays an important role to achieve high-precision GPS positioning. The noise characteristics of the GPS observables have been already investigated using the least squares variance component estimation (LS-VCE) in a batch form by the authors. In this contribution, we introduce a recursive procedure that provides a proper stochastic modeling for the GPS observables using the LS-VCE. It is referred to as the recursive LS-VCE (RLS-VCE) method, which is applied to the geometry-based observation model (GBOM). In this method, the (co)variances parameters can be estimated recursively when the new group of observations is added. Therefore, it can easily be implemented in real time GPS data processing. The efficacy of the method is evaluated using a real GPS data set collected by the Trimble R7 receiver over a zero baseline. The results show that the proposed method has an appropriate performance so that the estimated (co)variance parameters of the GPS observables are consistent with the batch estimates. However, using the RLS-VCE method, one can estimate the (co)variance parameters of the GPS observables when a new observation group is added. This method can thus be introduced as a reliable method for application to the real time GPS data processing.

ENHANCING STAND-ALONE GPS CODE POSITIONING USING STAND-ALONE DOUBLE DIFFERENCING CARRIER PHASE RELATIVE POSITIONING

Pseudo-range GPS code observables can provide absolute stand-alone positioning with accuracy of a few meters, which may not suitable for a wide range of engineering applications. Differencing GPS observations (DGPS) can be used for reducing or removing some of GPS errors based on the high correlation between these errors over short baselines providing accurate relative positioning. Stand-Alone Double Differences Carrier Phase (SADDCP) is an accurate velocity estimation method based on single frequency stand-alone GPS observables. Precise GPS relative positioning can then be achieved by integrating the velocity over epoch. SADDCP is a double differences technique including two epochs, two satellites and one receiver. In SADDCP, the ambiguity and receiver clock errors are removed, whereas satellite clock error, orbit errors, and ionospheric and tropospheric delays are reduced significantly. Multipath remains and can be reduced based on the multipath correlation over time, and receiver noise is increased. In this paper, SADDCP will be used to enhance the performance of stand-alone GPS code positioning, where the two positioning techniques are integrated using Kalman filter. The precise relative positioning provided by SADDCP will be utilized to smooth the absolute low accurate stand-alone GPS code positioning, providing enhanced absolute single frequency stand-alone GPS positioning. Tests in different GPS environments will be carried out for reliable investigations and the results will be discussed in details showing the advantages and limitations of this integration.

A Forward GPS Multipath Simulator Based on the Vegetation Radiative Transfer Equation Model.

Global Navigation Satellite Systems (GNSS) have been widely used in navigation, positioning and timing. Nowadays, the multipath errors may be re-utilized for the remote sensing of geophysical parameters (soil moisture, vegetation and snow depth), i.e., GPS-Multipath Reflectometry (GPS-MR). However, bistatic scattering properties and the relation between GPS observables and geophysical parameters are not clear, e.g., vegetation. In this paper, a new element on bistatic scattering properties of vegetation is incorporated into the traditional GPS-MR model. This new element is the first-order radiative transfer equation model. The new forward GPS multipath simulator is able to explicitly link the vegetation parameters with GPS multipath observables (signal-to-noise-ratio (SNR), code pseudorange and carrier phase observables). The trunk layer and its corresponding scattering mechanisms are ignored since GPS-MR is not suitable for high forest monitoring due to the coherence of direct and reflected signals. Based on this new model, the developed simulator can present how the GPS signals (L1 and L2 carrier frequencies, C/A, P(Y) and L2C modulations) are transmitted (scattered and absorbed) through vegetation medium and received by GPS receivers. Simulation results show that the wheat will decrease the amplitudes of GPS multipath observables (SNR, phase and code), if we increase the vegetation moisture contents or the scatters sizes (stem or leaf). Although the Specular-Ground component dominates the total specular scattering, vegetation covered ground soil moisture has almost no effects on the final multipath signatures. Our simulated results are consistent with previous results for environmental parameter detections by GPS-MR.

A dynamic model for GPS based attitude determination and testing using a serial robotic manipulator

A computational algorithm is developed for estimating accurately the attitude of a robotic arm which moves along a predetermined path. This algorithm requires preliminary input data obtained in the static mode to yield phase observables for the precise, 3-axis attitude determination of a swinging manipulator in the dynamic mode. Measurements are recorded simultaneously by three GPS L1 receivers and then processed in several steps to accomplish this task. First, artkconv batch executable converts GPS receiver readings into RINEX format to generate GPS observables and ephemeris for multiple satellites. Then baseline vectors determination is carried out by baseline constrained Least-Squares Ambiguity Decorrelation (LAMBDA) method that uses double difference carrier phase estimates as input to calculate integer solution for each baseline. Finally, attitude determination is made by employing alternatively Least-squares attitude determination (LSAD) in the static mode and extended Kalman filter in the dynamic mode. The algorithm presented in this paper is applied to recorded data on Mitsubishi RV-M1 robotic arm in order to produce attitude estimates. These results are confirmed by another set of Euler angles independently evaluated from robotic arm postures obtained along the predefined trajectory.

Research Article. Improved Dual Frequency PPP Model Using GPS and BeiDou Observations

Abstract This paper introduces a new dual-frequency precise point positioning (PPP) model, which combines GPS and BeiDou observations. Combining GPS and BeiDou observations in a PPP model offers more visible satellites to the user, which is expected to enhance the satellite geometry and the overall PPP solution in comparison with GPSonly PPP solution. However, combining different GNSS constellations introduces additional biases, which require rigorous modelling, including GNSS time offset and hardware delays. In this research, ionosphere-free linear combination PPP model is developed. The additional biases, which result from combining the GPS and BeiDou observables, are lumped into a new unknown parameter identified as the inter-system bias. Natural Resources Canada’s GPSPace PPP software is modified to enable a combined GPS/BeiDou PPP solution and to handle the newly introduced biases. A total of four data sets at four IGS stations are processed to verify the developed PPP model. Precise satellite orbit and clock products from the IGS-MGEX network are used to correct both of the GPS and BeiDou measurements. It is shown that a sub-decimeter positioning accuracy level and 25% reduction in the solution convergence time can be achieved with combining GPS and Bei-Dou observables in a PPP model, in comparison with the GPS-only PPP solution.

Multimodal data fusion for SB-JPALS status prediction under antenna motion fault mode

High positioning accuracy and system integrity are crucial for the Sea-based Joint Precision Approach and Landing System (SB-JPALS) to ensure life safety in carrier landing. The antenna motion of shipboard receivers may introduce large positioning error, which is one of the primary failure modes affecting the SB-JPALS integrity. Multimodal data fusion based on dual-frequency GPS observables and antenna motion information is applied to monitor the failure and enables the system to alarm message once position domain error caused by antenna movement exceeds a threshold. Monitoring algorithm uses multiple reference station architecture, selects integrated translating error of all valid reference antennas as the monitor measurement, and establishes a monitor decision-making model based on fault vertical protection level to evaluate the integrity performance. Multiple antenna motion errors are fused to be the integrated translating error and relevant methodology is described. Firstly the associated equation is built between individual antenna motion error and inter-antenna baseline motion error. The latter is regarded as known variables due to feasibility to capture them on the mobile ship platform. Then each antenna motion error is calculated and projected onto satellite line-of-sight direction. Finally the integrated error is obtained by averaging all antenna motion errors. The experiment data analysis result shows that individual antenna motion error derived from inter-antenna baseline errors is consistent with its real value, and the vertical protection level of the integrated translating error computed by multimodal data fusion processing based on multiple reference antennas is smaller than that by individual antenna information, namely, the system achieves better integrity.

Denoising GPS-Based Structure Monitoring Data Using Hybrid EMD and Wavelet Packet

High-frequency components are often discarded for data denoising when applying pure wavelet multiscale or empirical mode decomposition (EMD) based approaches. Instead, they may raise the problem of energy leakage in vibration signals. Hybrid EMD and wavelet packet (EMD-WP) is proposed to denoise Global Positioning System- (GPS-) based structure monitoring data. First, field observables are decomposed into a collection of intrinsic mode functions (IMFs) with different characteristics. Second, high-frequency IMFs are denoised using the wavelet packet; then the monitoring data are reconstructed using the denoised IMFs together with the remaining low-frequency IMFs. Our algorithm is demonstrated on a synthetic displacement response of a 3-story frame excited by El Centro earthquake along with a set of Gaussian random white noises on different levels added. We find that the hybrid method can effectively weaken the multipath effect with low frequency and can potentially extract vibration feature. However, false modals may still exist by the rest of the noise contained in the high-frequency IMFs and when the frequency of the noise is located in the same band as that of effective vibration. Finally, real GPS observables are implemented to evaluate the efficiency of EMD-WP method in mitigating low-frequency multipath.

Impact of Tropospheric Delay Gradients on Total Tropospheric Delay and Precise Point Positioning

GPS signals are electromagnetic waves that are affected by the Earth’s atmosphere. The Earth’s atmosphere can be categorized, according to its effect on GPS signals, into the ionosphere (ionospheric delay) and neutral atmosphere (tropospheric delay). The first-order ionospheric delay can be eliminated by linear combination of GPS observables on different frequencies. However, tropospheric delay cannot be eliminated because it is frequency-independent. The total tropospheric delay can be divided into three components. The first is the dry component, the second part is the wet component, and the third part is the horizontal gradients which account for the azimuthal dependence of tropospheric delay. In this paper, the effect of modeling tropospheric gradients on the estimation of the total tropospheric delay and station position is investigated. Long session, one month during January 2015, of GPS data is collected from ten randomly selected globally distributed IGS stations. Two cases are studied: the first case, the coordinates of stations are kept fixed to their actual values and the tropospheric delay is estimated twice, with and without tropospheric gradients. In the second case, the station position is estimated along with the total tropospheric delay with and without tropospheric gradients. It is shown that the average bias of the estimated total tropospheric delay when neglecting tropospheric gradients ranges from ?1.72 mm to 2.14 mm while the average bias when estimating gradients are ?0.898 mm to 1.92 mm which means that the bias is reduced by about 30%. In addition, the average standard deviation of the bias is 4.26 mm compared with 4.52 mm which means that the standard deviation is improved by about 6%.

Stochastic Analysis of Low-Cost Single-Frequency GPS Receivers

Typically, dual-frequency geodetic grade GNSS receivers are utilized for positioning applications that require high accuracy. Single-frequency high grade receivers can be used to minimize the expenses of such dual-frequency receivers. However, user has to consider the resultant positioning accuracy. Since the evolution of low-cost single-frequency (LCSF) receivers is typically cheaper than single-frequency high grade receivers, it is possible to obtain comparable positioning accuracy if the corresponding observables are accurately modelled. In this paper, two LCSF GPS receivers are used to form short baseline. Raw GPS measurements are recorded for several consecutive days. The collected data are used to develop the stochastic model of GPS observables from such receivers. Different functions are tested to determine the best fitting model which is found to be 3 parameters exponential decay function. The new developed model is used to process different data sets and the results are compared against the traditional model. Both results from the newly developed and the traditional models are compared with the reference solution obtained from dual-frequency receiver. It is shown that the newly developed model improves the root-mean-square of the estimated horizontal coordinates by about 10% and improves the root-mean-square of the up component by about 39%.

ON THE REALISTIC STOCHASTIC MODEL OF GPS OBSERVABLES: IMPLEMENTATION AND PERFORMANCE

Abstract. High-precision GPS positioning requires a realistic stochastic model of observables. A realistic GPS stochastic model of observables should take into account different variances for different observation types, correlations among different observables, the satellite elevation dependence of observables precision, and the temporal correlation of observables. Least-squares variance component estimation (LS-VCE) is applied to GPS observables using the geometry-based observation model (GBOM). To model the satellite elevation dependent of GPS observables precision, an exponential model depending on the elevation angles of the satellites are also employed. Temporal correlation of the GPS observables is modelled by using a first-order autoregressive noise model. An important step in the high-precision GPS positioning is double difference integer ambiguity resolution (IAR). The fraction or percentage of success among a number of integer ambiguity fixing is called the success rate. A realistic estimation of the GNSS observables covariance matrix plays an important role in the IAR. We consider the ambiguity resolution success rate for two cases, namely a nominal and a realistic stochastic model of the GPS observables using two GPS data sets collected by the Trimble R8 receiver. The results confirm that applying a more realistic stochastic model can significantly improve the IAR success rate on individual frequencies, either on L1 or on L2. An improvement of 20% was achieved to the empirical success rate results. The results also indicate that introducing the realistic stochastic model leads to a larger standard deviation for the baseline components by a factor of about 2.6 on the data sets considered.

Performance of the L2C civil GPS signal under various ionospheric scintillation effects

As GPS is modernizing, there are currently fourteen satellites transmitting L2C civil code and seven satellites transmitting L5 signal. While the GPS observables are subject to several sources of errors, the ionosphere is one of the largest error sources affecting GPS signals. Small irregularities in the electrons density along the GPS radio signal propagation path cause ionospheric scintillation that is characterized by rapid fluctuations in the signal amplitude and phase. The ionospheric scintillation effects are stronger in equatorial and high-latitude geomagnetic latitude regions and occur mainly due to equatorial anomaly and solar storms. Several researchers have analyzed the L2C signal quality since becoming available in December, 2005. We analyze the performance of L2C using GPS data from stations in the equatorial region of Brazil, which is subject of weak, moderate and strong ionospheric scintillation conditions. The GPS data were collected by Septentrio PolaRxS---PRO receivers as part of the CIGALA/CALIBRA network. The analysis was performed as a function of scintillations indexes S4 and Phi60, lock time (time interval in seconds that the carrier phase is tracked continuously without cycle slips), multipath RMS and position variation of precise point positioning solutions. The analysis shows that L2C code solutions are less affected by multipath effects than that of P2 when data are collected under weak ionospheric scintillation effects. In terms of analysis of positions, the kinematic PPP results using L2C instead P2 codes show accuracy improvements up to 33 % in periods of weak or strong ionospheric scintillation. When combining phase and code collected under weak scintillation effects, the results by applying L2C against P2 provide improvement in accuracy up to 59 %. However, for data under strong scintillation effects, the use of L2C for PPP with code and phase does not provide improvements in the positioning accuracy.

Role of stochastic model on GPS integer ambiguity resolution success rate

An important step in the high-precision GPS positioning is double-difference integer ambiguity resolution (IAR). The fraction or percentage of success among a number of integer ambiguity fixing is called the success rate. We investigate the ambiguity resolution success rate for the GPS observations for two cases, namely a nominal and a realistic stochastic model of the GPS observables. In principle, one would expect to have higher reliability on IAR success rates if a realistic GPS observables stochastic model is employed. The GPS geometry-based observation model is employed in which a more realistic stochastic model of GPS observables is determined using the least-squares variance component estimation. Two short and one GPS long baseline datasets and one simulated dataset are employed to evaluate the efficacy of the proposed algorithm. The results confirm that a more realistic stochastic model can significantly improve the IAR success rate on individual frequencies, either on L1 or on L2. An improvement of 25 % was achieved to the empirical success rate results. The results are of interest for many applications in which single-frequency observations can be used. This includes applications like attitude determination using single frequency single epoch of GPS observations.

An Improved Between-Satellite Single-Difference Precise Point Positioning Model for Combined GPS/Galileo Observations

AbstractThis article introduces a new model for precise point positioning (PPP), which combines dual-frequency GPS and Galileo observations. Our model is based on the between-satellite single-difference (BSSD) linear combination, which cancels out some receiver-related biases, including receiver clock error and non-zero initial phase bias of the receiver’s oscillator. Two different scenarios are considered when forming BSSD linear combinations. In the first scenario, either a GPS or a Galileo satellite is selected as a reference for both GPS and Galileo observables. The second scenario, on the other hand, selects two reference satellites: a GPS reference satellite for the GPS observables and a Galileo satellite for the Galileo observables. Natural Resources Canada’s GPSPace PPP software is modified to enable a combined GPS/Galileo PPP solution and to handle the newly introduced biases. A total of 12 data sets representing two-day GPS/Galileo measurements at six IGS stations are processed to verify the developed PPP model. Precise satellite orbit and clock products from the IGS-MGEX network are used to correct both of the GPS and Galileo measurements. It is shown that using one reference satellite to form the BSSD linear combinations improves the precision of the estimated parameters by about 25 % compared with the GPS-only PPP solution. When two reference satellites are used, however, the precision of the estimated parameters improves by about 50 % compared with the GPS-only PPP solution. Additionally, the solution convergence time is reduced to 10 min for both BSSD scenarios, which represents about 50 % improvement in comparison with the GPS-only PPP solution.

An Improved Model for Single-Frequency GPS/GALILEO Precise Point Positioning

This paper introduces a new precise point positioning (PPP) model, which combines single-fre- quency GPS/Galileo observations in between-satellite single-difference (BSSD) mode. In the absence of multipath, all receiver-related errors and biases are cancelled out when forming BSSD for a specific constellation. This leaves the satellite originating errors and atmospheric delays un- modelled. Combining GPS and Galileo observables introduces additional biases that have to be modelled, including the GPS to Galileo time offset (GGTO) and the inter-system bias. This paper models all PPP errors rigorously to improve the single-frequency GPS/Galileo PPP solution. GPSPace PPP software of Natural Resources Canada (NRCan) is modified to enable a GPS/Galileo PPP solution and to handle the newly introduced biases. A total of 12 data sets representing the GPS/Galileo measurements of six IGS-MEGX stations are processed to verify the newly developed PPP model. Precise satellite orbit and clock corrections from IGS-MEGX networks are used for both GPS and Galileo measurements. It is shown that sub-decimeter level accuracy is possible with single-frequency GPS/Galileo PPP. In addition, the PPP solution convergence time is improved from approximately 100 minutes for the un-differenced single-frequency GPS/Galileo solution to approximately 65 minutes for the BSSD counterpart when a single reference satellite is used. Moreover, an improvement in the PPP solution convergence time of 35% and 15% is obtained when one and two reference satellites are used, respectively.