Carrier Phase Ambiguity Research Articles

Ambiguity Recovery Using the Triple-Differenced Carrier Phase Type Approach for Long-Range GPS Kinematic Positioning

Precise, long-range GPS kinematic positioning to centimeter accuracy requires that carrier phase ambiguities be resolved correctly during an initialization period, and subsequently to recover the “lost" ambiguities in the event of a cycle slip. Furthermore, to maximize navigational efficiency, ambiguity resolution and carrier phase-based positioning need to be carried out in real-time. Due to the presence of the ionospheric signal delay, satellite orbit errors, and the tropospheric delay, so-called absolute ambiguity resolution “on-the-fly” for long-range applications becomes very difficult, and largely impossible. However, all of these errors exhibit a high degree of spatial and temporal correlation. In the case of short-range ambiguity resolution, because of the high spatial correlation, their effect can be neglected, but their influence will dramatically increase as the baseline length increases. On the other hand, between discrete trajectory epochs, they will still exhibit a large degree of similarity for short time spans. In this article, a method is described in which similar triple-differenced observables formed between one epoch with unknown ambiguities and another epoch with fixed ambiguities can be used to derive relative ambiguity values, which are ordinarily equal to zero (or to the number of cycles that have slipped when loss-of-lock occurred). Because of the temporal correlation characteristics of the error sources, the cycle slips can be recovered using the proposed methodology. In order to test the performance of this algorithm an experiment involving the precise positioning of an aircraft, over distances ranging from a few hundred meters up to 700 kilometres, was carried out. The results indicate that the proposed technique can successfully resolve relative ambiguities (or cycle slips) over long distances in an efficient manner that can be implemented in real-time.

Precise GRACE baseline determination using GPS

Precision relative navigation is an essential aspect of spacecraft formation flying missions, both from an operational and a scientific point of view. When using GPS as a relative distance sensor, dual-frequency receivers are required for high accuracy at large inter-satellite separations. This allows for a correction of the relative ionospheric path delay and enables double difference integer ambiguity resolution. Although kinematic relative positioning techniques demonstrate promising results for hardware-in-the-loop simulations, they were found to lack an adequate robustness in real-world applications. To overcome this limitation, an extended Kalman Filter modeling the relative spacecraft dynamics has been developed. The filter processes single difference GPS pseudorange and carrier phase observations to estimate the relative position and velocity along with empirical accelerations and carrier phase ambiguities. In parallel, double difference carrier phase ambiguities are resolved on both frequencies using the least square ambiguity decorrelation adjustment (LAMBDA) method in order to fully exploit the inherent measurement accuracy. The combination of reduced dynamic filtering with the LAMBDA method results in smooth relative position estimates as well as fast and reliable ambiguity resolution. The proposed method has been validated with data from the gravity recovery and climate experiment (GRACE) mission. For an 11-day data arc, the resulting solution matches the GRACE K-Band Ranging System measurements with an accuracy of 1 mm, whereby 83% of the double difference ambiguities are resolved.

Convergence of Block Decorrelation Method for the Integer Ambiguity Fix

Because of the integer-valued nature of carrier phase ambiguities, it is essential to fix the float estimates into integer values in order for high precision DGPS positioning. A decorrelation process is necessary to solve the problem since double-differenced ambiguities are highly correlated in general. In this paper, Block Decorrelation Method (BDM) is presented and tested for its convergence. BDM divides the variance-covariance matrix into four blocks and decorrelates them simultaneously. A number of randomly selected examples show that BDM is comparable to the existing decorrelation algorithm, however its speed of convergence is relatively faster due to the computations performed on small blocks.

Penalized GNSS Ambiguity Resolution

Global Navigation Satellite System (GNSS) carrier phase ambiguity resolution is the process of resolving the carrier phase ambiguities as integers. It is the key to fast and high precision GNSS positioning and it applies to a great variety of GNSS models which are currently in use in navigation, surveying, geodesy and geophysics. A new principle of carrier phase ambiguity resolution is introduced. The idea is to give the user the possibility to assign penalties to the possible outcomes of the ambiguity resolution process: a high penalty for an incorrect integer outcome, a low penalty for a correct integer outcome and a medium penalty for the real valued float solution. As a result of the penalty assignment, each ambiguity resolution process has its own overall penalty. Using this penalty as the objective function which needs to be minimized, it is shown which ambiguity mapping has the smallest possible penalty. The theory presented is formulated using the class of integer aperture estimators as a framework. This class of estimators was introduced elsewhere as a larger class than the class of integer estimators. Integer aperture estimators, being of a hybrid nature, can have integer outcomes as well as non-integer outcomes. The minimal penalty ambiguity estimator is an example of an integer aperture estimator. The computational steps involved for determining the outcome of the minimal penalty estimator are given. The additional complexity in comparison with current practice is minor, since the optimal integer estimator still plays a major role in the solution of the minimal penalty ambiguity estimator.

The Effects of GPS Carrier Phase Ambiguity Resolution on Jason-1

We have used GPS carrier phase integer ambiguity resolution to investigate improvements in the orbit determination for the Jason-1 satellite altimeter mission. The technique has been implemented in the GIPSY orbit determination software developed by JPL. The radial accuracy of the Jason-1 orbits is already near 1 cm, and thus it is difficult to detect the improvements gained when the carrier phase ambiguities are resolved. Nevertheless, each of the metrics we use to evaluate the orbit accuracy (orbit overlaps, orbit comparisons, satellite laser ranging residuals, altimeter crossover residuals, orbit centering) show modest improvement when the ambiguities are resolved. We conservatively estimate the improvement in the radial orbit accuracy is at the 10–20% level.

Integral GPS and QZSS Ambiguity Resolution

The US GPS and the planed Quasi Zenith Satellite System (QZSS) will enhance the capability of quickly resolving the integer cycle carrier phase ambiguities in precise differential positioning. The paper describes numerical analysis of the positioning performance of combined GPS-QZSS system. The current ambiguity resolution method is employed to resolve all cycle ambiguities of combined GPS-QZSS system. The performance of ambiguity resolution on a short base-line of 1 km in Tokyo is analyzed for different scenarios of the present GPS and combined future GPS-QZSS system. It is also analyzed in the Asian cities of Seoul, Beijing and Shanghai. The ambiguity fix percentage is adopted here for evaluating the performance of ambiguity resolution. It indicates that increasing the number of satellites has a benefit on the capability of getting quick ambiguity resolution. It also indicates that the ambiguity fix percentage with both GPS and QZSS is much higher than that with only GPS on the short-baseline operation.

On the approximation of the integer least-sqaures success rate: which lower or upper bound to use?

The probability of correct integer estimation, the success rate, is an important measure when the goal is fast and high precision positioning with a Global Navigation Satellite System. Integer ambiguity estimation is the process of mapping the least-squares ambiguity estimates, referred to as the float ambiguities, to an integer value. It is namely known that the carrier phase ambiguities are integer-valued, and it is only after resolution of these parameters that the carrier phase observations start to behave as very precise pseudorange measurements. The success rate equals the integral of the probability density function of the float ambiguities over the pull-in region centered at the true integer, which is the region in which all real values are mapped to this integer. The success rate can thus be computed without actual data and is very valuable as an a priori decision parameter whether successful ambiguity resolution is feasible or not. The pull-in region is determined by the integer estimator that is used and therefore the success rate also depends on the choice of the integer estimator. It is known that the integer least-squares estimator results in the maximum success rate. Unfortunately, it is very complex to evaluate the integral when integer least-squares is applied. Therefore, approximations have to be used. In practice, for example, the success rate of integer bootstrapping is often used as a lower bound. But more approximations have been proposed which are known to be either a lower or upper bound of the actual integer least-squares success rate. In this contribution an overview of the most important lower and upper bounds will be given. These bounds are compared theoretically as well as based on their performance. The performance is evaluated using simulations, since it is then possible to compute the ’actual’ success rate. Simulations are carried out for the two-dimensional case, since its simplicity makes evaluation easy, but also for the higher-dimensional geometry-based case, since this gives an insight to the performance that can be expected in practice.

Theory of integer equivariant estimation with application to GNSS

Carrier phase ambiguity resolution is the key to high-precision global navigation satellite system (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. The so-called ‘fixed’ baseline estimator is known to be superior to its ‘float’ counterpart in the sense that its probability of being close to the unknown but true baseline is larger than that of the ‘float’ baseline, provided that the ambiguity success rate is sufficiently close to its maximum value of one. Although this is a strong result, the necessary condition on the success rate does not make it hold for all measurement scenarios. It is discussed whether or not it is possible to take advantage of the integer nature of the ambiguities so as to come up with a baseline estimator that is always superior to both its ‘float’ and its ‘fixed’ counterparts. It is shown that this is indeed possible, be it that the result comes at the price of having to use a weaker performance criterion. The main result of this work is a Gauss–Markov-like theorem which introduces a new minimum variance unbiased estimator that is always superior to the well-known best linear unbiased (BLU) estimator of the Gauss–Markov theorem. This result is made possible by introducing a new class of estimators. This class of integer equivariant estimators obeys the integer remove–restore principle and is shown to be larger than the class of integer estimators as well as larger than the class of linear unbiased estimators. The minimum variance unbiased estimator within this larger class is referred to as the best integer equivariant (BIE) estimator. The theory presented applies to any model of observation equations having both integer and real-valued parameters, as well as for any probability density function the data might have.

Impact of Real-Time Ionospheric Determination on Improving Precise Navigation with GALILEO and Next-Generation GPS

: Ionospheric differential refraction at distances of tens of kilometers or more is one of the main problems affecting the capability of instantaneous carrier-phase ambiguity resolution. As a consequence, the ionosphere affects the feasibility of centimeter-accuracy navigation with dual-frequency global navigation satellite systems (GNSSs), such as present-day GPS, as well as future systems with three frequencies, such as GALILEO and modernized GPS. We have shown in previous work that a real-time tomographic model of the ionosphere, computed from GPS reference network data, is accurate enough to allow GPS real-time carrier-phase ambiguity resolution for a rover at hundreds of kilometers from the nearest reference site. In this work, we study the extension of such techniques to the improvement of instantaneous three-carrier-phase ambiguity resolution algorithms, such as three-carrier ambiguity resolution (TCAR), at medium and long distances (from tens to hundreds of kilometers) and with a minimum of geodetic computation. The datasets used were generated with a modified GNSS satellite signal generator. In particular, the success rate of instantaneous ambiguity resolution with three frequencies can be improved from about 60 percent of all attempts in previous work to about 100 percent with the proposed technique at distances of about 60 km and with low ionospheric values, and to more than 90 percent at distances of more than 100 km under solar maximum conditions.

Feasibility of wide-area subdecimeter navigation with GALILEO and modernized GPS

Precise corrections with a three-dimensional voxel model of the ionosphere based on Global Navigation Satellite System (GNSS) data from a wide-area network of ground receivers can help resolve differential carrier-phase ambiguities over very long baselines of hundreds of kilometers in present two-frequency systems [Global Positioning System (GPS) and Global Orbiting Navigation Satellite System (GLONASS)] or in planned three-frequency systems (GALILEO, Modernized GPS). A study based on simulated three-frequency data from a modified GNSS signal generator indicates that all the phase ambiguities could be resolved successfully more than 90% of the time. This should be useful in surveying large areas with instruments that require very precise geolocation (e.g. radar or lidar altimetry, interferometric synthetic aperture radar, interferometric sonar, etc.).

Optimizing a network-based RTK method for OTF positioning

Differential GPS is able to provide cm-level positioning accuracies, as long as the carrier phase ambiguities are resolved to integer values. Classical methods are based on the use of a single reference station located in the vicinity of the rover. Due to the spatial decorrelation of the errors, the distance between the reference station and the user is generally limited to within 20–30 km or even less, mainly due to the ionosphere. The MultiRef method, developed at the University of Calgary, uses a network of reference stations to generate regional code and carrier phase corrections, which can be transmitted to users in order to increase the distance over which integer ambiguity resolution is possible. In the original method, the correlated errors, due to the satellite orbits, troposphere, and ionosphere are modeled together using the L1 and wide-lane observables.

Relative positioning of multiple moving platforms using GPS

To obtain subdecimeter level accuracy in relative kinematic positioning, the use of double differenced GPS carrier phase measurement with carrier phase ambiguities fixed to their correct integer values must be adopted. If multiple platforms are available in the configuration, the redundancy provided by the multiplicity of platforms can speed up the time to integer ambiguity fixing while at the same time improve the reliability of the solution. An approach to effectively construct ambiguity constraints through the multiplicity of platforms is presented herein. The use of these ambiguity constraints to position multiple moving platforms with respect to each other is then discussed. A series of simulations and field tests are designed and conducted to investigate the effects of different system parameters on this approach, with a configuration of up to 10 moving platforms. The test results show that the use of ambiguity constraints can improve the time to integer ambiguity fixing by up to 67%, relative to the case when no constraints are used. In addition, the use of ambiguity constraints is found to enhance the ability of the multiple platform system to detect wrong ambiguity fixes.

An Invariant Upperbound for the GNSS Bootstrapped Ambiguity Success Rate

Carrier phase ambiguity resolution is the keyto fast and high precision GPS positioning. Critical in the application of ambiguity resolution is the quality of the computed integer ambiguities. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. The success or failure of carrier phase ambiguity resolution can be predicted by means of the probability of correct integer estimation, also referred to as the ambiguity success-rate. Upperbounds of the success-rate can be used to decide that ambiguity resolution has become unreliable. In this contribution we prove an upperbound for the bootstrapped success-rate. The upperbound is easy to compute and it is invariant for the class of admissible ambiguity transformations.

A performance comparison of single and dual frequency GPS ambiguity resolution strategies

The impact of observation selection, observation combination and model parameterization on GPS carrier phase ambiguity resolution and position accuracy under operational conditions is investigated. The impact of an ionospheric bias for a generic linear combination of L1 and L2 measurements is assessed and the results are used to clearly outline the desirable characteristics for improving ambiguity resolution versus positioning accuracy performance. Ambiguity resolution performance and position accuracy are shown for widelane (WL), L1-only, and ionospheric-free (IF) combinations. Several techniques for dealing with the ionospheric bias are also presented and compared, including stochastic ionospheric modelling. Multiple carrier phase combination solutions estimated in the same filter are also compared. The concept of an optimal processing strategy—in terms of both reliable ambiguity resolution and high accuracy positions—is presented. In total, eight strategies, which vary in observables and parameters, are tested on several datasets ranging from 13 km to 43 km.

Precise GPS/GNSS positioning solution for airborne data acquisition systems

The precise positioning of aircrafts during flights belongs to the great challenges with respect to the development of airborne data acquisition systems. Satellite positioning systems like GPS offers a unique capability for precise positioning but requires in depth knowledge of GPS in airborne applications, e.g. GPS for high dynamic application, integration of GPS with other sensors, dynamic behaviour of aircrafts or antenna location. For its positioning reference system of Flight Inspection systems Aerodata AG has developed a robust GPS carrier phase ambiguity solution P-DGPS, Precise Differential GPS combined with complementary sensors like INS, barometers, radio altimeters or laser altimeters as well as laser trackers. Using recorded data during the flight the algorithm offers also the capability to calculate more accurate positions in post-processing. The presented sensor fusion algorithm using GPS without differential corrections (SGPS, standalone GPS) offers a precise height reference solution for approach calibration based only on aircraft-based sensors. SGPS data are combined in post-processing with inertial, pressure, radio and laser altimeter data. Flight trials with a Bombardier “Global Express” at Braunschweig Airport on May 2002 shows the achieved accuracies of the height reference solution calculated by SGPS in comparison to P-DGPS. The SGPS solution for precise height calculation of special mission aircrafts provides accuracies in the order of 5 m and at the runway’s threshold in the order of 30 cm.

Theory of carrier phase ambiguity resolution

Carrier phase ambiguity resolution is the key to high precision Global Navigation Satellite System (GNSS) positioning and navigation. It applies to a great variety of current and future models of GPS, modernized GPS and Galileo. A proper handling of carrier phase ambiguity resolution requires a proper understanding of the underlying theory of integer inference. In this contribution a brief review is given of the probabilistic theory of integer ambiguity estimation. We describe the concept of ambiguity pull-in regions, introduce the class of admissible integer estimators, determine their probability mass functions and show how their variability affect the uncertainty in the so-called ‘fixed’ baseline solution. The theory is worked out in more detail for integer least-squares and integer bootstrapping. It is shown that the integer least-squares principle maximizes the probability of correct integer estimation. Sharp and easy-to-compute bounds are given for both the ambiguity success rate and the baseline’s probability of concentration. Finally the probability density function of the ambiguity residuals is determined. This allows one for the first time to formulate rigorous tests for the integerness of the parameters.

Time transfer using GPS carrier phase: error propagation and results

A joint time-transfer project between the Astronomical Institute of the University of Berne (AIUB) and the Swiss Federal Office of Metrology and Accreditation (METAS) was initiated to investigate the power of the time transfer using GPS carrier phase observations. Studies carried out in the context of this project are presented. The error propagation for the time-transfer solution using GPS carrier phase observations was investigated. To this purpose a simulation study was performed. Special interest was focussed on errors in the vertical component of the station position, antenna phase-center variations and orbit errors. A constant error in the vertical component introduces a drift in the time-transfer results for long baselines in east–west directions. The simulation study was completed by investigating the profit for time transfer when introducing the integer carrier phase ambiguities from a double-difference solution. This may reduce the drift in the time-transfer results caused by constant vertical error sources. The results from the present time-transfer solution are shown in comparison to results obtained with independent time-transfer techniques. The interpretation of the comparison benefits from the investigations of the error propagation study. Two types of solutions are produced on a regular basis at AIUB: one based on the rapid orbits from CODE, the other on the CODE final orbits. The rapid solution is available the day after the observations and has nearly the same quality as the final solution, which has a latency of about one week. The differences between these two solutions are below the nanosecond level. The differences from independent time-transfer techniques such as TWSTFT (two-way satellite time and frequency transfer) are a few nanoseconds for both products.

Spurious periodic horizontal signals in sub-daily GPS position estimates

Periodic horizontal signals have been identified in sub-daily estimates of position obtained from GPS data using batch least-squares solutions. The estimation of sub-daily position is commonly used for ocean loading and ice movement studies, for example. These spurious motions are artefacts of the processing methodology due to the presence of unmodelled, tidally-induced vertical signals in the GPS data. The periodic horizontal signals have magnitudes of about 40–50% of the amplitude of the vertical periodic signal. The shape of the periodic horizontal signals is related to the first derivative and the square of the vertical signal. The artefacts are mostly evident in the east–west component and can be removed by fixing the carrier phase ambiguities to integer values. We introduce a sensitivity analysis to study in detail the effects of ambiguity resolution and latitudinal variation of the artefacts. The overall conclusions are that for high precision batch processing of time series of geodetic positions, using sub-daily data spans, care needs to be taken to avoid aliasing of horizontal coordinates due to the vertical motion of the site.

Assessment of pseudolite layout under urban environments using a simulation system for seamless positioning

Since the launch of the first Global Positioning System satellite, the demand for higher accuracy and wider applicability has been continuously increasing. In particular, personal navigation stresses requirements on “seamless” or continuous positioning with higher reliability are needed in urban areas. Unfortunately, urban canyons causing shadowing and multipath represent an adverse condition for Global Navigation Satellite System (GNSS) signal reception, thus leading to poor satellite visibility and low positioning accuracy. In order to overcome such problems, pseudolites can be set up to provide additional ranging signals and aid the carrier phase ambiguity resolution which is crucial for utilizing the full accuracy and wide range capability of GNSS. Pseudolites, which are ground-based instruments that transmit GPS-like signals, can improve the satellite-receiver layout and be used as additional range observations to improve the performance of a GPS-based deformation monitoring system. However, due to economic and environment constraints, the number of pseudolites that can be installed will be limited. More importantly, in order to reduce the areas affected by multi-path of pseudolite signals, locations and antenna pattern of pseudolites have to be carefully examined. This paper shows the development of a simulation system for evaluating the appropriateness and economics of installing pseudolites using precise orbital information of the satellite and a three-dimensional digital map.

Predicting atmospheric biases for real-time ambiguity resolution in GPS/GLONASS reference station networks

When using multiple reference station networks to support real-time kinematic positioning, the global positioning system (GPS) and/or GLONASS carrier-phase ambiguities associated with between-reference-receiver processing have to be resolved. At the initialisation stage it is not difficult to reliably resolve these ambiguities, in the post-processing mode, using data sets of several hours or days in length. However, due to significant residual atmospheric delays in the double-differenced measurements, it is a big challenge to resolve the ambiguities within the reference station network in real time, particularly in the case of the ambiguities of the newly risen satellites. Both temporal and spatial correlation characteristics for the atmospheric delays are discussed, and appropriate atmospheric delay prediction models are proposed. The experimental results show that the proposed bias prediction models can be used to reliably and efficiently resolve the integer ambiguities within reference station ne tworks in real time, on a single-epoch basis.