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In this invited contribution a brief review will be presented of the integer estimation theory as developed by the author over the last decade and which started with the introduction of the LAMBDA method in 1993. The review discusses three different, but closely related classes of ambiguity estimators. They are the integer estimators, the integer aperture estimators and the integer equivariant estimators. Integer estimators are integer aperture estimators and integer aperture estimators are integer equivariant estimators. The reverse is not necessarily true however. Thus of the three types of estimators the integer estimators are the most restrictive. Their pull-in regions are translational invariant, disjunct and they cover the ambiguity space completely. Well-known examples are integer rounding, integer bootstrapping and integer least-squares. A less restrictive class of estimators is the class of integer aperture estimators. Their pull-in regions only obey two of the three conditions. They are still translational invariant and disjunct, but they do not need to cover the ambiguity space completely. As a consequence the integer aperture estimators are of a hybrid nature having either integer or non-integer outcomes. Examples of integer aperture estimators are the ratio-testimator and the difference testimator. The class of integer equivariant estimators is the less restrictive of the three classes. These estimators only obey one of the three conditions, namely the condition of being translational invariant. As a consequence the outcomes of integer equivariant estimators are always real valued. For each of the three classes of estimators we also present the optimal estimator. Although the Gaussian case is usually assumed, the results are presented for an arbitrary probability density function of the float solution. The optimal integer estimator in the Gaussian case is the integer least-squares estimator. The optimality criterion used is that of maximizing the probability of correct integer estimation, the so-called success rate. The optimal integer aperture estimator in the Gaussian case is the one which only returns the integer least-squares solution when the integer least-squares residual resides in the optimal aperture pull-in region. This region is governed by the probability density function of the float solution and by the probability density function of the integer least-squares residual. The aperture of the pull-in region is governed by a user defined aperture parameter. The optimality criterion used is that of maximizing the probability of correct integer estimation given a fixed, user-defined, probability of incorrect integer estimation. The optimal integer aperture estimator becomes identical to the optimal integer estimator in case the success rate and the fail rate sum up to one.The best integer equivariant estimator is an infinite weighted sum of all integers. The weights are determined as ratios of the probability density function of the float solution with its train of integer shifted copies. The optimality criterion used is that of minimizing the mean squared error. The best integer equivariant estimator.

Carrier phase ambiguity resolution is the key to 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.

In real time kinematic (RTK) GPS positioning the reference station(s) is (are) static, and the moving receivers must not be far from the reference station(s). But in some cases, such as formation flying, satellite-to-satellite orbit determination, etc, it is difficult to establish a static reference station. GPS kinematic-to-kinematic positioning (KINRTK) will meet such requirements. The key work of ambiguity resolution for KINRTK is to obtain an ambiguity float solution rapidly. The float solution can be estimated using either the ‘Geometry-based’(GB) or ‘Geometry-free’(GF) approach, requiring the construction of a ‘GB’ or ‘GF’ ambiguity search space. These two spaces are different but have the same true integer ambiguity result. Searching in two spaces at the same time, referred to here as Dual-space Ambiguity Resolution Approach (DARA), will be faster than in the individual spaces because only a few ambiguity candidates meet the conditions of both spaces simultaneously. It can be shown that DARA can dramatically reduce ambiguity candidates even if the C/A-code pseudo-range observables are used. The results of a vehicle test confirm that our approach is promising, resulting in millimeter-level misclosure of the KINRTK run.

Multipath, MAI (Multiple Access Interference) and near-far effects are the three main influences on the performance of CDMA-based communication and navigation systems. A great deal of research has been conducted to develop advanced signal processing algorithms and novel receiver structures useful for mitigation of these effects in mobile land wireless communication systems, such as UMTS. Although the multipath effects on PRN code ranging in GNSS have been investigated for about two decades, the MAI and near-far effects have only been discussed in pseudolite applications.In this paper, the impairments of the satellite-mobile receiver channel with multipath-selective fading, and shadowing/attenuation effects by objects such as trees/forests and buildings, are theoretically analysed, under a more general and practical definition of the ‘near-far’ effect. The MAI-mitigation and near-far resistant receiver structures for Galileo/GNSS applications are presented. The principles of such receiver structures and their applications in GNSS are discussed. Both theoretical analyses and computer simulations are presented and show the applicability of the proposed receiver structures.

A diagonalisation algorithm of the least squares normal equation is proposed in this paper. The equivalent observation equations related to the diagonalised normal equations are also derived in detail. For the equivalent observation equations and their normal equations, the related equivalent ambiguity search criteria are outlined. Theoretical application of the proposed algorithm in ambiguity search is briefly summarised. Using this algorithm, the ambiguity search turns out to be a search in a diagonal space so that the search can be done very quickly. Numerical examples to illustrate the diagonalisation process of the normal equation and observation equation are also given.

Kalman filters have been widely used for navigation and system integration. One of the key problems associated with Kalman filters is how to assign suitable statistical properties to both the dynamic and the observational models. For GPS navigation, the manoeuvre of the vehicle and the level of measurement noise are environmental dependent, and hardly to be predicted. Therefore to assign constant noise levels for such applications is not realistic.In this paper, real-time adaptive algorithms are applied to GPS data processing. Two different adaptive algorithms are discussed in the paper. A number of tests have been carried out to compare the performance of the adaptive algorithms with a conventional Kalman filter for vehicle navigation. The test results demonstrate that the new adaptive algorithms are much robust to the sudden changes of vehicle motion and measurement errors.

This paper demonstrates the concept and practical examples of instantaneous mapping of regional ionosphere, based on GPS observations from the State of Ohio continuously operating reference stations (CORS) network. Interpolation/prediction techniques, such as kriging (KR) and the Multiquadric Model (MQ), which are suitable for handling multi-scale phenomena and unevenly distributed data, were used to create total electron content (TEC) maps. Their computational efficiency (especially the MQ technique) and the ability to handle undersampled data (especially kriging) are particularly attractive. Presented here are the preliminary results based on GPS observations collected at five Ohio CORS stations (~100 km station separation and 1-second sampling rate). Dual frequency carrier phase and code GPS observations were used. A zero-difference approach was used for absolute TEC recovery. The quality of the ionosphere representation was tested by comparison to the International GPS Service (IGS) Global Ionosphere Maps (GIMs), which were used as a reference.