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.     

Grid point search algorithm for fast integer ambiguity resolution

A Grid Point Search Algorithm (GRIPSA) for fast integer ambiguity resolution is presented. In the proposed algorithm, after the orthogonal transformation of the original ambiguity parameters, the confidence ellipsoid of the new parameters is represented by a rectangular polyhedron with its edges parallel to the corresponding axes. A cubic grid covering the whole polyhedron is then identified and transformed back to the original coordinate system. The integer values of the corresponding transformed grid points are obtained by rounding off the transformed values to their nearest integer values. These values are then tested as to whether they are located inside the polyhedron. Since the identification of the grid points in the transformed coordinate system greatly reduces the search region of the integer ambiguities, marked improvements are obtained in the computational effort. Received: 13 October 1997 / Accepted: 9 June 1998     

Orthogonality defect and reduced search-space size for solving integer least-squares problems

In the context of ambiguity resolution (AR) of global navigation satellite systems (GNSS), decorrelation among entries of an ambiguity vector, integer ambiguity search, and ambiguity validations are three standard procedures for solving integer least-squares problems. This paper contributes to AR issues from three aspects. Firstly, the orthogonality defect is introduced as a new measure of the performance of ambiguity decorrelation methods and compared with the decorrelation number and with the condition number, which are currently used as the judging criterion to measure the correlation of ambiguity variance–covariance matrix. Numerically, the orthogonality defect demonstrates slightly better performance as a measure of the correlation between decorrelation impact and computational efficiency than the condition number measure. Secondly, the paper examines the relationship of the decorrelation number, the condition number, the orthogonality defect, and the size of the ambiguity search space with the ambiguity search candidates and search nodes. The size of the ambiguity search space can be properly estimated if the ambiguity matrix is decorrelated well, which is shown to be a significant parameter in the ambiguity search progress. Thirdly, a new ambiguity resolution scheme is proposed to improve ambiguity search efficiency through the control of the size of the ambiguity search space. The new AR scheme combines the LAMBDA search and validation procedures together, which results in a much smaller size of the search space and higher computational efficiency while retaining the same AR validation outcomes. In fact, the new scheme can deal with the case there are only one candidate, while the existing search methods require at least two candidates. If there are more than one candidate, the new scheme turns to the usual ratio-test procedure. Experimental results indicate that this combined method can indeed improve ambiguity search efficiency for both the single constellation and dual constellations, respectively, showing the potential for processing high-dimension integer parameters in multi-GNSS environment.     

整周模糊度去相关的两种实现方法

整周模糊度搜索一直是GPS快速精确定位的关键问题。短时间的观测会导致观测方程和整周模糊度方差、协方差矩阵的高相关性，因而急剧增大整周模糊度的搜索空间，对整周模糊度未知数方差、协方差矩阵进行去相关性处理，可以有效地压缩搜索空间。本文对整周模糊度去相关的迭代法和联合变换法从原理上进行了阐述，并结合实际算倒进行了分析和比较。     

Integer estimation in the presence of biases

 Carrier phase ambiguity resolution is the key to fast and high-precision GNSS (Global Navigation Satellite System) kinematic 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. Very high success rates are therefore required for ambiguity resolution to be reliable. Biases which are unaccounted for will lower the success rate and thus increase the chance of unsuccessful ambiguity resolution. The performance of integer ambiguity estimation in the presence of such biases is studied. Particular attention is given to integer rounding, integer bootstrapping and integer least squares. Lower and upper bounds, as well as an exact and easy-to-compute formula for the bias-affected success rate, are presented. These results will enable the evaluation of the bias robustness of ambiguity resolution. Received: 28 September 2000 / Accepted: 29 March 2001     

Fast integer least-squares estimation for GNSS high-dimensional ambiguity resolution using lattice theory

GNSS ambiguity resolution is the key issue in the high-precision relative geodetic positioning and navigation applications. It is a problem of integer programming plus integer quality evaluation. Different integer search estimation methods have been proposed for the integer solution of ambiguity resolution. Slow rate of convergence is the main obstacle to the existing methods where tens of ambiguities are involved. Herein, integer search estimation for the GNSS ambiguity resolution based on the lattice theory is proposed. It is mathematically shown that the closest lattice point problem is the same as the integer least-squares (ILS) estimation problem and that the lattice reduction speeds up searching process. We have implemented three integer search strategies: Agrell, Eriksson, Vardy, Zeger (AEVZ), modification of Schnorr–Euchner enumeration (M-SE) and modification of Viterbo-Boutros enumeration (M-VB). The methods have been numerically implemented in several simulated examples under different scenarios and over 100 independent runs. The decorrelation process (or unimodular transformations) has been first used to transform the original ILS problem to a new one in all simulations. We have then applied different search algorithms to the transformed ILS problem. The numerical simulations have shown that AEVZ, M-SE, and M-VB are about 320, 120 and 50 times faster than LAMBDA, respectively, for a search space of dimension 40. This number could change to about 350, 160 and 60 for dimension 45. The AEVZ is shown to be faster than MLAMBDA by a factor of 5. Similar conclusions could be made using the application of the proposed algorithms to the real GPS data.     

Quality-control issues relating to instantaneous ambiguity resolution for real-time GPS kinematic positioning

An integrated method for the instantaneous ambiguity resolution using dual-frequency precise pseudo-range and carrier-phase observations is suggested in this paper. The algorithm combines the search procedures in the coordinate domain, the observation domain and the estimated ambiguity domain (and therefore benefits from the integration of their most positive elements). A three-step procedure is then proposed to enhance the reliability of the ambiguity resolution by: (1) improving the stochastic model for the double-differenced functional model in real time; (2) refining the criteria which distinguish the integer ambiguity set that generates the minimum quadratic form of residuals from that corresponding to the second minimum one; and (3) developing a fault detection and adaptation procedure. Three test scenarios were considered, one static baseline (11.3 km) and two kinematic experiments (baseline lengths from 5.2 to 13.7 km). These showed that the mean computation time for one epoch is less than 0.1 s, and that the success rate reaches 98.4% (compared to just 68.4% using standard ratio tests). Received: 5 June 1996; Accepted: 16 January 1997     

On the sensitivity of the location, size and shape of the GPS ambiguity search space to certain changes in the stochastic model

In this contribution we analyse in a qualitative sense for the geometry-free model the dependency of the location, the size and the shape of the ambiguity search space on different factors of the stochastic model. For this purpose a rather general stochastic model is used. It includes time-correlation, cross-correlation, satellite elevation dependency and the use of an a priori weighted ionospheric model, having the ionosphere-fixed model and the ionosphere-float model as special cases. It is shown that the location is invariant for changes in the cofactor matrix of the phase observables. This also holds true for the cofactor matrix of the code observables in the ionosphere-float case. As for time-correlation and satellite elevation dependency, it is shown that they only affect the size of the search space, but not its shape and orientation. It is also shown that the least-squares ambiguities, their variance matrix and its determinant, for, respectively, the ionosphere-fixed model, the ionosphere-float model and the ionosphere-weighted model, are all related through the same scalar weighted mean, the weight of which is governed by the variance ratio of the ionospheric delays and the code observables. A closed-form expression is given for the area of the search space in which all contributing factors are easily recognized. From it one can infer by how much the area gets blown up when the ionospheric spatial decorrelation increases. This multiplication factor is largest when one switches from the ionosphere-fixed model to the ionosphere-float model, in which case it is approximately equal to the ratio of the standard deviation of phase with that of code. The area gives an indication of the number of grid points inside the search space. Received: 11 November 1996 / Accepted: 21 March 1997     

一种用于GPS整周模糊度OTF求解的整数白化滤波改进算法

提出一种用于整周模糊度OTF求解的整数白化滤波改进算法。该算法首先对整周模糊度的协方差矩阵进行整数白化滤波处理 ,以降低整周模糊度间的相关性 ,然后构造搜索空间来判定是否需要进行搜索。如果需要 ,则通过搜索来确定变换后的整周模糊度 ;如果不需要 ,则通过直接取整来确定整周模糊度 ,进而得到原始的整周模糊度和基线分量的固定解。初步试验结果显示 ,采用改进方法解算整周模糊度可以提高成功率和解算效率     

基于格论的GNSS模糊度解算

快速、准确地解算整周模糊度是实现GNSS载波相位实时高精度定位的关键,由于模糊度之间的强相关,基于整数最小二乘估计准则时,需要较长的时间才能搜索出最优的整周模糊度向量。为了提高模糊度的搜索效率,本文在扼要介绍格论的理论框架基础上,引入基于格论的模糊度解算方法,通过格基规约来降低模糊度之间的相关性,从而快速搜索出最优的整数模糊度向量。与此同时,将GNSS领域的主要降相关方法统一到格论框架下,探讨了并建议采用Bootstrapping成功率作为格基规约的性能指标之一。最后实验分析了三频多系统长基线相对定位情况下,不同格基规约可获得的性能。     

Precision, volume and eigenspectra for GPS ambiguity estimation based on the time-averaged satellite geometry

In this contribution we consider the time-averaged GPS single-baseline model and study in a qualitative sense its relation with the geometry-free model and the geometry-based model. The least-squares estimators of the model are derived and their properties discussed. Special attention is given to the ambiguity search space, since it plays such a crucial role in the problem of integer ambiguity estimation and validation. Easy-to-evaluate, closed-form expressions are presented for the volumes of the ambiguity search spaces that belong to the geometry-free model, the single-epoch geometry-based model and the time-averaged model. By means of an eigenvalue analysis, the geometry of the ambiguity search spaces is revealed and its impact on the search for the integer least-squares ambiguities discussed. Received: 3 April 1996; Accepted: 6 January 1997     

回代解算的LAMBDA方法及其搜索空间

基于回代解算的序贯条件最小二乘技术和上三角的Cholesky分解算法,提出求解载波相位模糊度的一种新算法--回代解算的LAMBDA方法.该方法同Teunissen提出的LAMBDA方法相比,有着不同的目标函数和不同的搜索空间.实例试算表明这两种方法有着不同的执行过程,但有着相同的模糊度整数解和相当的搜索效率.     

单频GPS动态定位中整周模糊度的一种快速解算方法

针对单频GPS动态定位中常用模糊度求解方法存在的问题,提出一种整周模糊度快速解算方法。首先通过对双差观测方程中坐标参数的系数阵进行QR分解变换以消除坐标参数,从而仅对模糊度参数建立Kalman滤波方程进行估计,然后利用排序和双Cholesky分解对滤波得到的模糊度进行降相关处理,并结合收缩模糊度搜索空间的思想来搜索固定整周模糊度。以实测的动态数据为例对该方法进行测试。分析结果表明,该方法不但可以改善模糊度浮点解精度,而且具有良好的模糊度降相关效果,可正确有效地实现整周模糊度的快速解算。     

The success rate and precision of GPS ambiguities

 An application of a theorem on the optimality of integer least-squares (LS) is described. This theorem states that the integer LS estimator maximizes the ambiguity success rate within the class of admissible integer estimators. This theorem is used to show how the probability of correct integer estimation depends on changes in the second moment of the ambiguity `float' solution. The distribution of the `float' solution is considered to be a member of the broad family of elliptically contoured distributions. Eigenvalue-based bounds for the ambiguity success rate are obtained. Received: 11 January 1999 / Accepted: 2 November 1999     

GNSS模糊度降相关性能的条件方差平稳度评价法

GNSS模糊度降相关通过整数变换优化条件方差的排列顺序,提高搜索效率。降相关和条件方差的关系及其评价是关键问题之一。针对这一问题,本文从理论上分析了排序后模糊度降相关与条件方差之间的数值关系,发现降相关性能与条件方差数值序列的平稳性有关,降相关性能越强,条件方差数值序列越平稳。基于这一理论关系,给出了"条件方差平稳度"定义,并将其作为评价降相关性能的指标。通过模拟和实测数据验证,并采用条件方差变化趋势图和搜索时间来定性和定量评价降相关性能,用以判定条件方差平稳度的合理性。试验结果表明,条件方差平稳度可以较精确直观地衡量模糊度的降相关性能。本文定义的指标揭示了模糊度降相关的本质。     

The probability distribution of the ambiguity bootstrapped GNSS baseline

 The purpose of carrier phase ambiguity resolution is to improve upon the quality of the estimated global navigation satellite system baseline by means of the integer ambiguity constraints. However, in order to evaluate the quality of the ambiguity resolved baseline rigorously, its probability distribution is required. This baseline distribution depends on the random characteristics of the estimated integer ambiguities, which in turn depend on the chosen integer estimator. In this contribution is presented an exact and closed-form expression for the baseline distribution in the case that use is made of integer bootstrapping. Also presented are the bootstrapped probability mass function and easy-to-compute measures for the bootstrapped baseline's probability of concentration. Received: 28 September 2000 / Accepted: 11 January 2001     

On the GPS widelane and its decorrelating property

In this contribution we consider the popular widelaning technique from the viewpoint of ambiguity decorrelation. It enables us to cast the technique into the framework of the least-squares ambiguity decorrelation adjustment (LAMBDA) and to analyse its relative merits. In doing so, we will provide answers to the following three questions. Does the widelane decorrelate? Does it explicitly appear in the automated transformation step of the LAMBDA method? Can one do better than the widelane? It is shown that all three questions can be answered in the affirmative. This holds true for the ionosphere-fixed case, the ionosphere-float case, as well as for the ionosphere-weighted case. Received: 11 November 1996 / Accepted: 23 April 1997     

Success probability of integer GPS ambiguity rounding and bootstrapping

Global Positioning System ambiguity resolution is usually based on the integer least-squares principle (Teunissen 1993). Solution of the integer least-squares problem requires both the execution of a search process and an ambiguity decorrelation step to enhance the efficiency of this search. Instead of opting for the integer least-squares principle, one might also want to consider less optimal integer solutions, such as those obtained through rounding or sequential rounding. Although these solutions are less optimal, they do have one advantage over the integer least-squares solution: they do not require a search and can therefore be computed directly. However, in order to be confident that these less optimal solutions are still good enough for the application at hand, one requires diagnostic measures to predict their rate of success. These measures of confidence are presented and it is shown how they can be computed and evaluated. Received: 24 March 1998 / Accepted: 8 June 1998     

An optimality property of the integer least-squares estimator

A probabilistic justification is given for using the integer least-squares (LS) estimator. The class of admissible integer estimators is introduced and classical adjustment theory is extended by proving that the integer LS estimator is best in the sense of maximizing the probability of correct integer estimation. For global positioning system ambiguity resolution, this implies that the success rate of any other integer estimator of the carrier phase ambiguities will be smaller than or at the most equal to the ambiguity success rate of the integer LS estimator. The success rates of any one of these estimators may therefore be used to provide lower bounds for the LS success rate. This is particularly useful in case of the bootstrapped estimator. Received: 11 January 1999 / Accepted: 9 July 1999