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. Received: 31 October 1995 / Accepted: 21 March 1997     

On the probability density function of the GNSS ambiguity residuals

Integer GNSS ambiguity resolution involves estimation and validation of the unknown integer carrier phase ambiguities. A problem then is that the classical theory of linear estimation does not apply to the integer GPS model, and hence rigorous validation is not possible when use is made of the classical results. As with the classical theory, a first step for being able to validate the integer GPS model is to make use of the residuals and their probabilistic properties. The residuals quantify the inconsistency between data and model, while their probabilistic properties can be used to measure the significance of the inconsistency. Existing validation methods are often based on incorrect assumptions with respect to the probabilistic properties of the parameters involved. In this contribution we will present and evaluate the joint probability density function (PDF) of the multivariate integer GPS carrier phase ambiguity residuals. The residuals and their properties depend on the integer estimation principle used. Since it is known that the integer least-squares estimator is the optimal choice from the class of admissible integer estimators, we will only focus on the PDF of the ambiguity residuals for this estimator. Unfortunately the PDF cannot be evaluated exactly. It will therefore be shown how to obtain a good approximation. The evaluation will be completed by some examples.     

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     

Generalized integer aperture estimation for partial GNSS ambiguity fixing

In satellite navigation, the key to high precision is to make use of the carrier-phase measurements. The periodicity of the carrier-phase, however, leads to integer ambiguities. Often, resolving the full set of ambiguities cannot be accomplished for a given reliability constraint. In that case, it can be useful to resolve a subset of ambiguities. The selection of the subset should be based not only on the stochastic system model but also on the actual measurements from the tracking loops. This paper presents a solution to the problem of joint subset selection and ambiguity resolution. The proposed method can be interpreted as a generalized version of the class of integer aperture estimators. Two specific realizations of this new class of estimators are presented, based on different acceptance tests. Their computation requires only a single tree search, and can be efficiently implemented, e.g., in the framework of the well-known LAMBDA method. Numerical simulations with double difference measurements based on Galileo E1 signals are used to evaluate the performance of the introduced estimation schemes under a given reliability constraint. The results show a clear gain of partial fixing in terms of the probability of correct ambiguity resolution, leading to improved baseline estimates.     

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.     

长距离GPS/BDS参考站网多频载波相位整周模糊度解算方法

长距离网络RTK是实现GPS/BDS高精度实时定位的主要手段之一,其核心是长距离参考站网GPS/BDS整周模糊度的快速准确确定。本文提出了一种长距离GPS/BDS参考站网载波相位整周模糊度解算方法,首先利用GPS双频观测数据计算和确定宽巷整周模糊度,同时利用BDS的B2、B3频率观测值确定超宽巷整周模糊度。然后建立GPS载波相位整周模糊度和大气延迟误差的参数估计模型,附加双差宽巷整周模糊度的约束,解算双差载波相位整周模糊度,并建立参考站网大气延迟误差的空间相关模型。根据B2、B3频率的超宽巷整周模糊度建立包含大气误差参数的载波相位整周模糊度解算模型,利用大气延迟误差空间相关模型约束BDS双差载波相位整周模糊度的解算。克服了传统的使用无电离层组合值解算整周模糊度的不利影响。采用实测长距离CORS网GPS、BDS多频观测数据进行算法验证,试验结果证明该方法可实现长距离参考站网GPS/BDS载波相位整周模糊度的准确固定。     

An integer ambiguity resolution procedure for GPS/pseudolite/INS integration

In order to achieve a precise positioning solution from GPS, the carrier-phase measurements with correctly resolved integer ambiguities must be used. Based on the integration of GPS with pseudolites and Inertial Navigation Systems (INS), this paper proposes an effective procedure for single-frequency carrier-phase integer ambiguity resolution. With the inclusion of pseudolites and INS measurements, the proposed procedure can speed up the ambiguity resolution process and increase the reliability of the resolved ambiguities. In addition, a recently developed ambiguity validation test, and a stochastic modelling scheme (based on-line covariance matrix estimation) are adapted to enhance the quality of ambiguity resolution. The results of simulation studies and field experiments indicate that the proposed procedure indeed improves the performance of single-frequency ambiguity resolution in terms of both reliability and time-to-fix-ambiguity.     

基于整周模糊度概率特性的有效性检验

准确确定载波相位整周模糊度是快速高精度GPS定位的关键,已有的检验GPS整周模糊度有效性的方法几乎均是基于其为非随机常量建立的,因而都存在一定的缺陷。本文在研究整周模糊度概率特性的基础上,提出一种基于LABMBAD算法的整周模糊度概率分布函数的检验方法。实际演算表明该方法简单有效,统计概念明确。     

GPS/GLONASS组合静态相位相对定位算法

论文介绍了GPS/GLONASS组合静态相位相对定位模型,将GLONASS双差观测方程的模糊度参数表示成参考卫星的单差模糊度和双差模糊度参数;用误差分析法证明了单差模糊度按实参数估计不影响基线解算精度,而GLONASS双差模糊度必须按整参数进行解算;用Helmert方差分量估计确定GPS和GLONASS观测值的合理权比。实际观测数据处理结果表明：GPS/GLONASS组合定位较单一系统解算的基线精度均有提高,尤其比GLONASS单系统的解算精度有显著提高,比GPS单系统的精度也有适当提高,其中单历元基线解算精度约提高了10%,当单一系统的可用卫星数少于4颗时,GPS/GLONASS组合定位更具有应用价值。     

Improving carrier-phase ambiguity resolution in global GPS network solutions

Integer carrier-phase ambiguity resolution is one of the critical issues for precise GPS applications in geodesy and geodynamics. To resolve as many integer ambiguities as possible, the ‘most-easy-to-fix’ double-difference ambiguities have to be defined. For this purpose, several strategies are implemented in existing GPS software packages, such as choosing the ambiguities according to the baseline length or the variances of the estimated real-valued ambiguities. Although their efficiencies are demonstrated in practice, it is proven in this paper that they do not reflect all effects of varying data quality, because they are based on theoretical considerations of GPS data processing. Therefore, a new approach is presented, which selects the double-difference ambiguities according to their probability of being fixed to the nearest integer. The probability is computed from estimates and variances of wide-lane and narrow-lane ambiguities. Together with an optimized ambiguity fixing procedure, the new approach is implemented in the routine data processing for the International GPS Service (IGS) at GeoForschungsZentrum (GFZ) Potsdam. Within a sub-network of about 90 IGS stations, it is demonstrated that more than 97% of the independent ambiguities are fixed correctly compared to 75% by a commonly used method, and that the additionally fixed ambiguities improve the repeatability of the station coordinates by 10–26% in regions with sparse site distribution.     

Modified ambiguity function approach for GPS carrier phase positioning

This paper presents a new strategy for GPS carrier phase data processing. The classic approach generally consists of three steps: a float solution, a search for integer ambiguities, and a fixed solution. The new approach is based on certain properties of ambiguity function method and ensures the condition of integer ambiguities without the necessity of the additional step of the integer search. The ambiguities are not computed explicitly, although the condition of “integerness” of the ambiguities is ensured in the results through the least squares adjustment with condition equations in the functional model. An appropriate function for the condition equations is proposed and presented. The presented methodology, modified ambiguity function approach, currently uses a cascade adjustment with successive linear combinations of L1 and L2 carrier phase observations to ensure a correct solution. This paper presents the new methodology and compares it to the three-stage classic approach which includes ambiguity search. A numerical example is provided for 25 km baseline surveyed with dual-frequency receivers. All tests were performed using an in-house developed GINPOS software and it has been shown that the positioning results from both approaches are equivalent. It has also been proved that the new approach is robust to adverse effects of cycle slips. In our opinion, the proposed approach may be successfully used for carrier phase GPS data processing in geodetic applications.     

Influence of ambiguity precision on the success rate of GNSS integer ambiguity bootstrapping

In this contribution, we study the dependence of the bootstrapped success rate on the precision of the GNSS carrier phase ambiguities. Integer bootstrapping is, because of its ease of computation, a popular method for resolving the integer ambiguities. The method is however known to be suboptimal, because it only takes part of the information from the ambiguity variance matrix into account. This raises the question in what way the bootstrapped success rate is sensitive to changes in precision of the ambiguities. We consider two different cases. (1) The effect of improving the ambiguity precision, and (2) the effect of using an approximate ambiguity variance matrix. As a by-product, we also prove that integer bootstrapping is optimal within the restricted class of sequential integer estimators.     

改进的ARCE方法及其在单频 GPS快速定位中的应用

基于TIKHONOV正则化原理，设计了一种正则化矩阵的构造方法，将ARCE(ambiguity resolution using constraint equation)方法进行了改进。通过新的正则化矩阵的作用，减弱了GPS快速定位中少数历元情形下法矩阵的病态性，得到了比较准确的模糊度浮动解，大大减小了模糊度的搜索范围，利用ARCE方法固定模糊度的成功率高。并结合一个算例，验证了本文改进方法的效果。     

Estimation of satellite position,clock and phase bias corrections

Precise point positioning with integer ambiguity resolution requires precise knowledge of satellite position, clock and phase bias corrections. In this paper, a method for the estimation of these parameters with a global network of reference stations is presented. The method processes uncombined and undifferenced measurements of an arbitrary number of frequencies such that the obtained satellite position, clock and bias corrections can be used for any type of differenced and/or combined measurements. We perform a clustering of reference stations. The clustering enables a common satellite visibility within each cluster and an efficient fixing of the double difference ambiguities within each cluster. Additionally, the double difference ambiguities between the reference stations of different clusters are fixed. We use an integer decorrelation for ambiguity fixing in dense global networks. The performance of the proposed method is analysed with both simulated Galileo measurements on E1 and E5a and real GPS measurements of the IGS network. We defined 16 clusters and obtained satellite position, clock and phase bias corrections with a precision of better than 2 cm.     

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。     

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     

一种整周模糊度快速求解的改进LAMBDA方法

本文提出了一种改进LAMBDA方法:在确定Z变换后的模糊度时,改变以往对所有历元的模糊度全部进行搜索的做法,而是通过设置合理的条件,将搜索与直接归整有效地结合起来,从而减少了模糊度的解算时间,提高了解的效率。文章最后利用实测GPS数据验证了改进效果。     

单频GPS快速定位中病态问题的解法研究

研究只利用少数历元GPS载波相位观测值进行快速定位时的新解法.在分析病态法矩阵结构特性的基础上,基于TIKHONOV正则化原理,提出一种选择正则化矩阵R的新方法,减弱法方程的病态性.与其他方法相比,新方法得到与模糊度准确值更接近的浮动解及其相应的均方误差矩阵.结合LAMBDA方法,用均方误差矩阵代替协方差阵确定模糊度的搜索范围,可准确快速地确定模糊度,最后得到基线向量的解.结合算例,将新解法与最小二乘估计、岭估计和截断奇异值法分别结合LAMBDA方法解算模糊度的结果进行比较分析,展示新解法的效果.     

短基线GPS定向系统及搜索模糊度优化算法的研究

GPS载波相位测量相对定位可以达到毫米级精度,利用GPS载波相位测量方向可以达到2密位的精度。研究了载波相位双差测量方向的原理和应用最小二乘法解算基线矢量的算法,详细讨论快速解算整周模糊度的优化算法。实验结果表明,应用双GPS测量方向的原理和搜索模糊度优化算法正确,其定向精度达2密位,解算时间小于0.3秒,并运用于产品中。