Double-differenced Ambiguities Research Articles

Spatially convergent GPS kinematic positioning

It is a known fact that obtaining accurate GPS carrier-phase measurements involves fixed, unknown whole-cycle ambiguity parameters. As the use of cosine functions to eliminate any double-difference integer ambiguities causes spatial ambiguity problems, both reasonably approximated positions and wavelength-dependent convergence ranges are of the utmost importance. Differential GPS-based position solutions are first smoothed to create a polynomial trajectory, leading to less variable position approximations. Long-wavelength wide-lane phase combinations will then be utilized to facilitate convergent GPS positioning, on a stage-by-stage basis. Although double-difference ionospheric path delays are often interpreted as nuisance parameters, they can be obtained when the respective cosines of the original L1 and L2 carrier phases undergo a simultaneous least-squares estimation. In particular, quadratic forms of the estimated phase residuals will be linked with hypothesis testing to allow for a meaningful statistical inference. Some low-dynamics experiments are then performed to prove the feasibility of the proposed hierarchical positioning concept.

Improving satellite orbit solution using double-differenced GPS carrier phase in kinematic mode

The limit of orbit accuracy associated with processing the ionospheric-free double-differenced (DD) GPS carrier phase observable in pure kinematic mode is investigated. The kinematic orbit determination problem becomes unobservable when the satellite positions at all epochs and all of the DD ambiguities are estimated simultaneously without any constraints. Although searching for some of the DD ambiguities can make the problem observable, this is not practical when the DD ambiguities are not integers. However, if a reasonably accurate a priori satellite orbit solution is available, epoch-by-epoch satellite positions and the DD ambiguities can be estimated by constraining a geometrically strong DD pass. The technique was demonstrated with TOPEX GPS data, which gave a final radial RMS orbit accuracy of 4 cm with respect to a fully dynamic solution.

A Modular Approach to Precise GPS Positioning

A modular approach consisting of three steps is presented for precise positioning with GPS: integrity, monitoring (data validation), ambiguity resolution, and positioning. Results of one step are used as input for the next. After a brief description of reliability and testing and techniques for GPS ambiguity resolution, the observation equations for the geometry-free GPS model are given, used for integrity monitoring, followed by those for single-difference relative positioning with the double-difference ambiguities removed. © 1999 John Wiley & Sons, Inc.

SATELLITE GEOMETRY CONSIDERATION FOR EFFICIENT GPS AMBIGUITY RESOLUTION

In this paper, a formula to analyse the effect of satellite geometry to the change in position derived from double-difference phase ambiguities is derived analytically, and geometrical interpretation is given. Based on the analysed results, a conclusion is drawn that the number of trial points within a defined ambiguity search space is satellite geometry dependent, therefore research on satellite selection for trial position generation for efficient ambiguity resolution is necessary.

The Ionosphere-weighted GPS baseline precision in canonical form

In this contribution we consider the precision of the floated and the fixed GPS baseline for the case of weighted ionosphere. Canonical forms of the baseline variance matrices are developed for different measurement scenarios. These forms make the relation between the various variance matrices transparent and thus present a simple way of studying their relative merits. It is also shown how these canonical forms give an intrinsic description of the gain in baseline precision which is experienced when the carrier-phase double-differenced ambiguities are treated as integers instead of as reals. The dependence of this gain on the various contributing factors, such as the decorrelation of the ionosphere, is also included.

The least-squares ambiguity decorrelation adjustment: its performance on short GPS baselines and short observation spans

The least-squares ambiguity decorrelation adjustment is a method for fast GPS double-difference (DD) integer ambiguity estimation. The performance of the method will be discussed, and although it is stressed that the method is generally applicable, attention is restricted to short-baseline applications in the present contribution. With reference to the size and shape of the ambiguity search space, the volume of the search space will be introduced as a measure for the number of candidate grid points, and the signature of the spectrum of conditional variances will be used to identify the difficulty one has in computing the integer DD ambiguities. It is shown that the search for the integer least-squares ambiguities performs poorly when it takes place in the space of original DD ambiguities. This poor performance is explained by means of the discontinuity in the spectrum of conditional variances. It is shown that through a decorrelation of the ambiguities, transformed ambiguities are obtained which generally have a flat and lower spectrum, thereby enabling a fast and efficient search. It is also shown how the high precision and low correlation of the transformed ambiguities can be used to scale the search space so as to avoid an abundance of unnecessary candidate grid points. Numerical results are presented on the spectra of conditional variances and on the statistics of both the original and transformed ambiguities. Apart from presenting numerical results which can typically be achieved, the contribution also emphasizes and explains the impact on the method's performance of different measurement scenarios, such as satellite redundancy, single vs dual-frequency data, the inclusion of code data and the length of the observation time span.

A canonical theory for short GPS baselines.

The present contribution is the second of four parts. It considers the precision and correlation of the least-squares estimators of the carrier phase ambiguities. It is shown how the precision and correlation of the double-differenced ambiguities as well as of the widelane ambiguities are effected by the observation weights, by the number of satellites tracked, by the number of observation epochs used, and by the change over time of the relative receiver-satellite geometry. Also the ability of the widelane transformation to decorrelate and to improve the precision is investigated.

A canonical theory for short GPS baselines.

The present contribution is the first of four parts. It considers the precision of the floated and the fixed baseline. A measure is introduced for the gain in baseline precision which is experienced when the carrier phase double-differenced ambiguities are treated as integers instead of as reals. The properties of this measure are analyzed, and it is shown by means of principal angles how it relates to the change over time of the relative receiver-satellite geometry. We also present canonical forms of the baseline variance matrices for different measurement scenarios. These canonical forms make the relation between the various variance matrices transparent and thus present a simple way of studying their relative merits.

The least-squares ambiguity decorrelation adjustment: a method for fast GPS integer ambiguity estimation

The GPS double difference carrier phase measurements are ambiguous by an unknown integer number of cycles. High precision relative GPS positioning based on short observational timespan data, is possible, when reliable estimates of the integer double difference ambiguities can be determined in an efficient manner. In this contribution a new method is introduced that enables very fast integer least-squares estimation of the ambiguities. The method makes use of an ambiguity transformation that allows one to reformulate the original ambiguity estimation problem as a new problem that is much easier to solve. The transformation aims at decorrelating the least-squares ambiguities and is based on an integer approximation of the conditional least-squares transformation. This least-squares ambiguity decorrelation approach, flattens the typical discontinuity in the GPS-spectrum of ambiguity conditional variances and returns new ambiguities that show a dramatic improvement in correlation and precision. As a result, the search for the transformed integer least-squares ambiguities can be performed in a highly efficient manner.