Precise satellite positioning requires that carrier phase data be used and that the integer ambiguities associated with the carrier phase measurements be resolved in some way. However, the distance from the user receiver to the nearest reference receiver may range from a few kilometres to hundreds of kilometres. As the receiver separation increases, the problems of accounting for distance-dependent biases increase, and reliable ambiguity resolution for carrier phase-based satellite positioning becomes an even greater challenge.

'Wide area modelling' for precise satellite positioning requires either long observation spans to estimate all biases in the functional model, or multiple reference stations. For the first approach, all error sources, such as orbit bias, atmospheric parameters, receiver inter-channel biases, along with the user’s trajectory, should be estimated simultaneously. This is the approach used for geodetic static positioning (e.g., IGS-based site coordinate determination, and precise GPS orbit determination). The second approach provides more opportunities to either estimate the different biases individually and then apply interpolated biases (at the user location) to the measurements, or generate a so-called 'virtual reference station', by using the data from a multiple reference station network. Some of the concepts have been studied in the past by previous IAG SSGs, both separately and in combination, and with respect to various applications. In 1999 the IAG established SSG 1.1.79 to focus on investigations of the GPS functional model, the stochastic model, and ambiguity resolution procedures. The website of the Special Study Group 1.179 is http://www.gmat.unsw.edu.au/snap/gps/iag_section1/ssg1179.htm.


Objectives of the SSG 1.179

1.        Error modelling through the improvement of functional models for medium-range, and long-range high precision satellite positioning using multiple reference stations, including:

·         multipath mitigation algorithms,

·         troposphere model refinement,

·         regional ionosphere modelling algorithms,

·         orbit bias modelling,

·         parametric modelling algorithms (for each error source), and

·         integer bias estimation and validation, e.g. cycle slip detection/repair and ambiguity resolution.

2.        Error modelling through stochastic model refinement, including:

·         correlation analysis of carrier phase measurements from satellite positioning systems,

·         stochastic modelling algorithms suitable for post-processing applications, and

·         stochastic modelling algorithms suitable for real-time applications.

3.        The continued study of ambiguity resolution techniques in order to develop:

·         more efficient means of searching integer ambiguities, and

·         validation procedures for ambiguity resolution.

4.        The application of these improvements to:

·         short-range satellite positioning applications,

·         differential correction generation from multiple reference GNSS receiver network, in support of medium-range high precision navigation,

·         precise long-range GPS kinematic positioning, and

·         sub-centimetre engineering applications, e.g. construction deformation monitoring, volcano monitoring, etc.


Members and Corresponding Members


Members: Shaowei Han (Chair, USA), Oscar Colombo (USA), Paul Cross (UK), Paul de Jonge (U.S.A), Hans-Jürgen Euler (SWITZERLAND), Yanming Feng (AUSTRALIA), Yang Gao (CANADA), Yongil Kim (KOREA), Donghyun Kim (CANADA), Dennis Odijk (THE NETHERLANDS), Günter Seeber (GERMANY), Dariusz Lapucha (USA), Jingnan Liu (CHINA), Nigel Penna (AUSTRALIA), Rock Santerre (CANADA), Julia Talaya (SPAIN), Jinling Wang (AUSTRALIA), Xinhua Qin (USA), Peiliang Xu (JAPAN).

Corresponding Members: Changdon Kee (KOREA)


Activities of the SSG1.179

 Error Modelling Through Improvement of Functional Models

Error modelling through the improvement of functional models for medium-range, and long-range high precision satellite positioning using multiple reference stations includes the study of topics such as multipath mitigation, troposphere modelling, regional ionosphere modelling, and orbit bias modelling. These biases could be estimated individually through some special approaches, or by setting different parameters in the functional model for the different error biases.

Absolute field calibration of GPS antennas is based on the controlled antenna motion of a robotic arm, and is now a mature calibration technique. The technique can be used to calibrate all antennas in a multiple reference station network. With (absolutely) calibrated antennas it is possible to separate phase centre variations and multipath. An approach for multipath calibration based on controlled antenna motion was proposed.

Investigations into the use of 'semi-parametric least squares' for the mitigation of systematic errors in GPS processing have been conducted. Current focus is the lumping together of all systematic errors as a single smoothing function, estimated over the processing session. Initial results from a 'short' 30km baseline are encouraging, and tests have commenced on more data sets.

An adaptive Finite-duration Impulse Response filter, based on a least-mean-squares algorithm, has been developed to derive a relatively noise-free time series from continuous GPS results. This algorithm is suitable for real time applications. Numerical simulation studies indicate that the adaptive filter is a powerful signal decomposer, which can significantly mitigate multipath effects.

Increased use has been made of ionospheric regional modelling to improvement on-the-fly ambiguity resolution over long distances, as part of initiatives within the GEOIDE project (website: www.scg.ulaval.ca/gps-rs/). Ionospheric tomography has also been used to help resolve GPS ambiguities on-the-fly at distances of hundreds of kilometres during increased geomagnetic activity. An approach, referred to as the "grand solution", which estimates orbit, refraction, and local bias error states, along with the uer's trajectory, was proposed. The modelling and estimation of the tropospheric zenith delay, both for more accurate real time and post-processed navigation, and for rapid and precise meteorological updates, has been implemented.


With respect to Real-Time Kinematic (RTK) positioning using multiple reference stations, the results of a survey conducted by Dr. Euler, Chair of the RTCM SC104 Working Group "Network RTK", of working group members found:

The expected RTK accuracy could be at sub-decimetre to centimetre level (one sigma).

The reference station distances should be of the order of 50-70 km for centimetre accuracy, or about 200 km and above for decimetre accuracy.

The size of a reference station area should be of the order of 500 km x 500 km. However, target could be nationwide to continentwide coverage.

The medium for distribution of data could be unidirectional techniques (Broadcast like UHF, VHF, TV, DARC, etc) or bi-directional techniques (GSM, UTMS, etc.).

The baud rates for transmission are from 2400 Baud upwards, including 1Hz observation data.

The tolerated latency is up to 10 seconds without SA, or up to 2 seconds with SA. However, the orbit information can be delayed by up to 120 seconds, ionosphere by up to 10 to 60 seconds, troposphere by up to 30 seconds. The real-time positioning output is expected within 100 milliseconds.

The requirement for reference station equipment is dual-frequency receivers with clear sky view.


With regards to GPS/Glonass surveying and navigation applications using multiple reference stations, a new method was proposed, in which the distance-dependent biases have been separated into the frequency-dependent errors (ionospheric bias) and frequency-independent errors (e.g. troposphere bias and orbit bias). The separate estimates of the two types of errors, which are generated from the carrier phase measurements using the multiple reference stations, can be used to model the user distance-dependent biases for L1, L2 carrier phase and pseudo-range measurements in different ways.


Error Modelling Through Stochastic Model Refinement

High quality estimation results using least squares require the correct selection of the functional and stochastic models. The stochastic model should represent the statistical characteristics of the modelling errors. It is dependent on the choice of observation functional model, hence for a different choice of functional model, a different stochastic model may be needed. For example, if the ionospheric delay is considered an unknown parameter in the functional model, the modelling errors will not include the residual (double-differenced) ionospheric bias, and hence they will more likely have random properties.


The SIGMA-D model has been developed for stochastic modelling of GPS signal diffraction errors in high precision GPS surveys. The basic information used in the SIGMA-D model is the measured carrier-to-noise power-density ratio (C/N0). Using the C/N0 data and a template technique, the proper variances are derived for all phase observations. Thus the quality of the measured phase is automatically assessed and if phase observations are suspected of being contaminated by diffraction effects they are downweighted in the least-squares adjustment. An extended weight model for GPS phase observations was also proposed.


Mathematical and statistical modelling has also been investigated. Using a multipath estimation method based on the signal-to-noise ratio and an elevation-dependent stochastic model, the height accuracy of a typical RTK session has been improved by approximately 44% and the fidelity of quality measures has been increased.


A stochastic assessment procedure has been developed to take into account the heteroscedastic, space- and time-correlated error structure of the GPS measurements. Test results indicate that by applying the stochastic assessment procedure developed , the reliability of the estimated positioning results is improved. In addition, the quality of ambiguity resolution can be more realistically evaluated.


Magellan's new product Instant-RTKTM has reportedly overcome the functional and stochastic modelling problem through empirical knowledge and a real-time learning procedure which can used to adapt the model when the environment is changing.


On the other hand, the stochastic modelling approach has been applied to the parameters in the functional model. For example, the residual ionospheric delay after applying ionospheric delay corrections could be accounted for through processing the residual ionospheric delay correction as stochastic observables. The stochastic model to be applied for the corrections could be provided by multiple reference stations. First results show indeed an enormous improvement in the success rate of ambiguity resolution.


Continued Study of Ambiguity Resolution Techniques

GPS ambiguity resolution (AR) techniques have been intensively investigated. The integer ambiguity searching techniques have been dramatically improved over the last decade, especially by the contribution of the LAMBDA method. However, it has to be recognised that all search algorithms are likely to result in identical integer ambiguity candidates under comparable setups, e.g. using like search windows/volumes and similar parameters. Continued study is now focused on AR in integrated systems: GPS, Glonass, pseudolite or other systems, and more powerful validation criteria to ensure correct ambiguity resolution.


For example, Magellan's new product Instant-RTKTM appears to have successfully addressed the functional and stochastic modelling problem through empirical knowledge and a real-time learning procedure. A series of validation criteria have been implemented, in addition to the commonly used ratio test, which can be adapted based on the reliability requirements, number of satellites, observation time and baseline length. The Instant-RTK validation criteria have successfully traded off the requirements of observation span on the one hand, and RTK solution reliability on the other. Moreover, the algorithm to detect, identify and adapt the outliers to guard against incorrect integer ambiguity determination has been implemented, and the success rate of AR has been increased significantly.


Leica Geosystems' System 500 has implemented a repeated search processing technique to shorten ambiguity initialisation time and to improve AR reliability, especially in difficult environments. This method repeats its internal determination of the integer ambiguity using significantly shorter observation times. Once the AR algorithm has verified that they are identical, the system can output its coordinates.


On the theoretical side, a method was proposed to evaluate the probabilities of correct integer estimation based on the variance matrix of the (real-valued) least-squares ambiguities. These success rates are given for the ambiguity estimator that follows from integer bootstrapping. Although less optimal than integer least-squares, integer bootstrapping provides useful and easy-to-compute approximations to the integer least-squares solution. In a similar manner, the bootstrapped success rates provide bounds for the probability of correct integer least-squares estimation.


New Development and Future Trends

In the next few years, more commercial system will be developed to generate corrections from multiple reference stations for surveying and precise navigation applications. RTCM SC104 Working Group "Network RTK" will propose a new format to transmit correction information from multiple reference station networks. This is not only beneficial to RTK systems, but also to single-frequency, low-cost GPS systems. Moreover, once the additional civilian frequency is transmitted by Block IIF satellites, the wide area error modelling for precise satellite positioning will be significantly improved.


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