The Reliability of GPS Ambiguity Resolution

GPS ambiguity resolution is the process of resolving the unknown cycle ambiguities of double-difference (DD) carrier-phase data as integers. It is the key to fast and high-precision relative GPS positioning. Critical in the application of ambiguity resolution is its reliability. Unsuccessful ambiguity resolution, when passed unnoticed, will too often lead to unacceptable errors in the positioning results. High success rates are required for ambiguity resolution to be reliable. In this contribution we will introduce and evaluate such diagnostic measures. They complement existing methods of ambiguity resolution and allow the user and/or analyst to infer their reliability. ? 1999 John Wiley & Sons, Inc.     

Integer aperture bootstrapping: a new GNSS ambiguity estimator with controllable fail-rate

In this contribution, we introduce a new bootstrap-based method for Global Navigation Satellite System (GNSS) carrier-phase ambiguity resolution. Integer bootstrapping is known to be one of the simplest methods for integer ambiguity estimation with close-to-optimal performance. Its outcome is easy to compute due to the absence of an integer search, and its performance is close to optimal if the decorrelating Z-transformation of the LAMBDA method is used. Moreover, the bootstrapped estimator is presently the only integer estimator for which an exact and easy-to-compute expression of its fail-rate can be given. A possible disadvantage is, however, that the user has only a limited control over the fail-rate. Once the underlying mathematical model is given, the user has no freedom left in changing the value of the fail-rate. Here, we present an ambiguity estimator for which the user is given additional freedom. For this purpose, use is made of the class of integer aperture estimators as introduced in Teunissen (2003). This class is larger than the class of integer estimators. Integer aperture estimators are of a hybrid nature and can have integer outcomes as well as non-integer outcomes. The new estimator is referred to as integer aperture bootstrapping. This new estimator has all the advantages known from integer bootstrapping with the additional advantage that its fail-rate can be controlled by the user. This is made possible by giving the user the freedom over the aperture of the pull-in region. We also give an exact and easy-to-compute expression for its controllable fail-rate.     

A systematic investigation of optimal carrier-phase combinations for modernized triple-frequency GPS

The upcoming modernization of the GPS signals will allow for measurements on an additional third frequency L5 located at 1176.45 MHz. To take advantage of carrier-phase measurements on this new signal, the strategies for integer ambiguity resolution, required for centimeter-level accuracy, may need to be revised. The Least-squares Ambiguity Decorrelation Adjustment method remains perhaps the most powerful tool for finding the best combinations based on a complete decorrelation of the variance–covariance matrix related to the ambiguities. However, the computational load of that method plus the opportunity to comprehensively study the interaction of multiple frequencies suggest a reconsideration of approaches using predefined combinations between frequencies is not out of place. In this paper a systematic investigation is made of all possible triple-frequency geometry-free carrier-phase combinations which retain the integer nature of the ambiguities. The concept of the lane-number is presented to unambiguously describe the wavelength of a particular combination. The propagation of the observation noise and of the ionospheric bias on these combinations is presented. These noise and ionospheric amplification factors are analysed with respect to the resulting wavelength, in an effort to highlight optimal combinations characterized by a long wavelength, low noise and limited ionospheric impact.     

Carrier-phase inter-frequency biases of GLONASS receivers

The frequency division multiplexing of the GLONASS signals causes inter-frequency biases in the receiving equipment. These biases vary considerably for receivers from different manufacturers and thus complicate or prevent carrier-phase ambiguity fixing. Complete and reliable ambiguity fixing requires a priori information of the carrier-phase inter-frequency bias differences of the receivers involved. GLONASS carrier-phase inter-frequency biases were estimated for 133 individual receivers from 9 manufacturers. In general, receivers of the same type and even receivers from the same manufacturer show similar biases, whereas the differences among manufacturers can reach up to 0.2 ns (more than 5 cm) for adjacent frequencies and thus up to 2.4 ns (73 cm) for the complete L1 or L2 frequency bands. A few individual receivers were identified whose inter-frequency biases behave differently as compared to other receivers of the same type or whose biases vary with time.     

综合利用双频载波相位观测值进行周跳的探测与修复

针对参考站单站静态双频的特点,提出了一种综合利用双频载波相位观测值求差法和载波相位变化率来探测周跳并修复的新方法;介绍了两种方法的数学模型,给出了具体实施步骤;最后用算例验证了方法的可靠性和有效性。     

GNSS satellite-based augmentation systems for Australia

We provided an overview of various satellite-based augmentation systems (SBAS) options for augmented GNSS services in Australia, and potentially New Zealand, with the aim to tease out key similarities and differences in their augmentation capabilities. SBAS can technically be classified into two user categories, namely SBAS for aviation and “non-aviation” SBAS. Aviation SBAS is an International Civil Aviation Organization (ICAO) certified civil aviation safety-critical system providing wide-area GNSS augmentation by broadcasting augmentation information using geostationary satellites. The primary aim was to improve integrity, availability and accuracy of basic GNSS signals for aircraft navigation. On the other hand, “non-aviation” SBAS support numerous GNSS applications using positioning techniques such as wide-area differential-GNSS (DGNSS) and precise point positioning (PPP). These services mainly focus on delivering high-accuracy positioning solutions and guaranteed levels of availability, and integrity remains secondary considerations. Next-generation GNSS satellites capable of transmitting augmentation signals in the L1, L5 and L6 frequency bands will also be explored. These augmentation signals have the data capacity to deliver a range of augmentation services such as SBAS, wide-area DGNSS and PPP, to meet the demands of various industry sectors. In addition, there are well-developed plans to put in place next-generation dual-frequency multi-constellation SBAS for aviation. Multi-constellation GNSS increases robustness against potential degradation of core satellite constellations and extends the service coverage area. It is expected that next-generation SBAS and GNSS will improve accuracy, integrity, availability and continuity of GNSS performance.     

卫星双向载波相位时间频率传递及其误差分析

深入解析了基于码伪距测量的卫星双向时间频率传递(TWSTFT)原理,对其误差进行了分析,其精度主要受码速率的限制,难于满足更高精度要求;文章提出了基于载波相位的卫星双向时间频率传递方法,载波频率高,波长短,从理论上分析了以其作为TWSTFT伪距测量的参数可大幅度提高时间传递的精度;并对TWSTFT载波相位测量存在的误差源进行了初步分析,得出影响最大的因素是转发器误差、电离层误差、多径效应误差及星站相对运动引起的误差,参考GPS载波相位测量中误差消除的方法,对误差的消除方法提出建议。     

Short-term analysis of GNSS clocks

A characterization of the short-term stability of the atomic frequency standards onboard GNSS satellites is presented. Clock performance is evaluated using two different methods. The first method derives the temporal variation of the satellite’s clock from a polynomial fit through 1-way carrier-phase measurements from a receiver directly connected to a high-precision atomic frequency standard. Alternatively, three-way measurements using inter-station single differences of a second satellite from a neighboring station are used if the receiver’s clock stability at the station tracking the satellite of interest is not sufficient. The second method is a Kalman-filter-based clock estimation based on dual-frequency pseudorange and carrier-phase measurements from a small global or regional tracking network. Both methods are introduced and their respective advantages and disadvantages are discussed. The analysis section presents a characterization of GPS, GLONASS, GIOVE, Galileo IOV, QZSS, and COMPASS clocks based on these two methods. Special focus has been set on the frequency standards of new generation satellites like GPS Block IIF, QZSS, and IOV as well as the Chinese COMPASS/BeiDou-2 system. The analysis shows results for the Allan deviation covering averaging intervals from 1 to 1,000 s, which is of special interest for real-time PPP and other high-rate applications like processing of radio-occultation measurements. The clock interpolation errors for different sampling rates are evaluated for different types of clocks and their effect on PPP is discussed.     

基于加权的伪距和历元间相位差分模型GPS单点定位方法

在星载GPS精密定轨或单点定位中,尤其单频接收机的情况下,仅利用相位观测值,由于需要解算模糊度方程通常奇异,仅利用伪距观测资料,由于伪距观测值的噪声影响使得难以实现高精度定位要求。鉴于此,本文讨论了基于加权的伪距和历元间相位差分模型的GPS单点定位方法,该方法既能改善方程奇异性,又无需考虑模糊度因素,能确保观测值的精度。本文分别探讨了伪距观测方程和历元间差分相位观测方程,并给出了联合误差模型,推导了权值的计算公式和参数解算公式,最后基于动态单点定位考虑,探讨了基于加权的伪距和历元间相位差分模型的序贯最小二乘参数解算一般表达式。     

Interferometric Baseline Performance Estimations for Multistatic Synthetic Aperture Radar Configurations Derived from GRACE GPS Observations

Recent studies have demonstrated the usefulness of global positioning system (GPS) receivers for relative positioning of formation-flying satellites using dual-frequency carrier-phase observations. The accurate determination of distances or baselines between satellites flying in formation can provide significant benefits to a wide area of geodetic studies. For spaceborne radar interferometry in particular, such measurements will improve the accuracy of interferometric products such as digital elevation models (DEM) or surface deformation maps. The aim of this study is to analyze the impact of relative position errors on the interferometric baseline performance of multistatic synthetic aperture radar (SAR) satellites flying in such a formation. Based on accuracy results obtained from differential GPS (DGPS) observations between the twin gravity recovery and climate experiment (GRACE) satellites, baseline uncertainties are derived for three interferometric scenarios of a dedicated SAR mission. For cross-track interferometry in a bistatic operational mode, a mean 2D baseline error (1σ) of 1.4 mm is derived, whereas baseline estimates necessary for a monostatic acquisition mode with a 50 km along-track separation reveal a 2D uncertainty of approximately 1.7 mm. Absolute orbit solutions based on reduced dynamic orbit determination techniques using GRACE GPS code and carrier-phase data allows a repeat-pass baseline estimation with an accuracy down to 4 cm (2D 1σ). To assess the accuracy with respect to quality requirements of high-resolution DEMs, topographic height errors are derived from the estimated baseline uncertainties. Taking the monostatic pursuit flight configuration as the worst case for baseline performance, the analysis reveals that the induced low-frequency modulation (height bias) fulfills the relative vertical accuracy requirement (σ<1 m linear point-to-point error) according to the digital terrain elevation data level 3 (DTED-3) specifications for most of the baseline constellations. The use of a GPS-based reduced dynamic orbit determination technique improves the baseline performance for repeat-pass interferometry. The problem of fulfilling the DTED-3 horizontal accuracy requirements is still an issue to be investigated. DGPS can be used as an operational navigation tool for high-precision baseline estimation if a geodetic-grade dual-frequency spaceborne GPS receiver is assumed to be the primary instrument onboard the SAR satellites. The possibility of using only single-frequency receivers, however, requires further research effort.Deutsche Forschungsgemeinschaft (DFG) research fellow until Sept. 2004 at the Microwaves and Radar Institute, Deutsche Zentrum für Luft- und Raumfahrt (DLR) e.V., 82234 Weßling, Germany     

Triple-frequency GPS precise point positioning with rapid ambiguity resolution

At present, reliable ambiguity resolution in real-time GPS precise point positioning (PPP) can only be achieved after an initial observation period of a few tens of minutes. In this study, we propose a method where the incoming triple-frequency GPS signals are exploited to enable rapid convergences to ambiguity-fixed solutions in real-time PPP. Specifically, extra-wide-lane ambiguity resolution can be first achieved almost instantaneously with the Melbourne-Wübbena combination observable on L2 and L5. Then the resultant unambiguous extra-wide-lane carrier-phase is combined with the wide-lane carrier-phase on L1 and L2 to form an ionosphere-free observable with a wavelength of about 3.4 m. Although the noise of this observable is around 100 times the raw carrier-phase noise, its wide-lane ambiguity can still be resolved very efficiently, and the resultant ambiguity-fixed observable can assist much better than pseudorange in speeding up succeeding narrow-lane ambiguity resolution. To validate this method, we use an advanced hardware simulator to generate triple-frequency signals and a high-grade receiver to collect 1-Hz data. When the carrier-phase precisions on L1, L2 and L5 are as poor as 1.5, 6.3 and 1.5 mm, respectively, wide-lane ambiguity resolution can still reach a correctness rate of over 99 % within 20 s. As a result, the correctness rate of narrow-lane ambiguity resolution achieves 99 % within 65 s, in contrast to only 64 % within 150 s in dual-frequency PPP. In addition, we also simulate a multipath-contaminated data set and introduce new ambiguities for all satellites every 120 s. We find that when multipath effects are strong, ambiguity-fixed solutions are achieved at 78 % of all epochs in triple-frequency PPP whilst almost no ambiguities are resolved in dual-frequency PPP. Therefore, we demonstrate that triple-frequency PPP has the potential to achieve ambiguity-fixed solutions within a few minutes, or even shorter if raw carrier-phase precisions are around 1 mm. In either case, we conclude that the efficiency of ambiguity resolution in triple-frequency PPP is much higher than that in dual-frequency PPP.     

GPS sidereal filtering: coordinate- and carrier-phase-level strategies

Multipath error is considered one of the major errors affecting GPS observations. One can benefit from the repetition of satellite geometry approximately every sidereal day, and apply filtering to help minimize this error. For GPS data at 1 s interval processed using a double-difference strategy, using the day-to-day coordinate or carrier-phase residual autocorrelation determined with a 10-h window leads to the steadiest estimates of the error-repeat lag, although a window as short as 2 h can produce an acceptable value with > 97% of the optimal lag’s correlation. We conclude that although the lag may vary with time, such variation is marginal and there is little advantage in using a satellite-specific or other time-varying lag in double-difference processing. We filter the GPS data either by stacking a number of days of processed coordinate residuals using the optimum “sidereal” lag (23 h 55 m 54 s), and removing these stacked residuals from the day in question (coordinate space), or by a similar method using double-difference carrier-phase residuals (observational space). Either method results in more consistent and homogeneous set of coordinates throughout the dataset compared with unfiltered processing. Coordinate stacking reduces geometry-related repeating errors (mainly multipath) better than carrier-phase residual stacking, although the latter takes less processing time to achieve final filtered coordinates. Thus, the optimal stacking method will depend on whether coordinate precision or computational time is the over-riding criterion.     

A Procedure for Estimating the Variance Function of Linear Models and for Checking the Appropriateness of Estimated Variances: A Case Study of GPS Carrier-phase Observations

In many applications of linear model theory, homogeneous variances are assumed. In practice, however, the variances are frequently heterogeneous. Therefore, to improve the results, the unknown variances have to be estimated. The appropriateness of the estimated variances has then to be checked by a suitable statistical test procedure. Such a procedure is also useful to study models of global positioning system (GPS) carrier-phase observations. While the functional model of GPS carrier-phase observations is widely accepted, the stochastic model is still under development. As well as the neglected correlations of GPS observations, a homogenous variance function is frequently assumed. In Bischoff et al. (J Geod 78:397–404, 2005), we showed by statistical testing that the assumption of constant variances is not appropriate. In this paper, we give a procedure to estimate an individual variance function for a pair of satellites and a procedure to check the appropriateness of the estimated variances. As an example, the approach is applied to double-differenced carrier-phase GPS observations.     

Higher order ionospheric effects in precise GNSS positioning

With the increasing number of precise navigation and positioning applications using Global Navigation Satellite Systems (GNSS) such as the Global Positioning System (GPS), higher order ionospheric effects and their correction become more and more important. Whereas the first-order error can be completely eliminated by a linear combination of dual- frequency measurements, the second- and third-order residual effects remain uncorrected in this approach. To quantify the second-order residual effect, a simple formula has been derived for GNSS users in Germany. Our proposed correction algorithm reduces the second-order effects to a residual error of fractions of 1 mm up to 2 mm at a vertical total electron content level of 10¹⁸ electrons/m² (100 TECU), depending on satellite azimuth and elevation angles. The correction formula can be implemented in real-time applications as it does not require the knowledge of the geomagnetic field or the electron density distribution in the ionosphere along the signal path. It is expected that the correction will enable more accurate positioning using the line-of-sight carrier-phase measurements.     

Investigation of ionosphere TEC distribution in China region by GIM model

In this paper, the TEC in China region is calculated by final GIM model and plotted as contour maps, the TEC value calculated from dual frequency GPS measurements（carrier-phase and pseudorange） is used as the reference truth and the TEC values from GIM models are then compared with the reference truth, and the accuracy and performance of GIM model in China region are analyzed and concluded.     

北斗广域差分分区综合改正数定位性能分析

目前北斗广域分米级星基增强系统在钟差改正数、轨道改正数的基础上,提出了基于相位观测值的分区综合改正数,介绍了分区综合改正数的概念及单频、双频用户的使用方法与定位模型。利用中国范围不同地区的北斗观测数据和对应的分区综合改正信息,统计了单频和双频用户分区综合改正精密单点定位的精度,并对其收敛性进行了分析。通过与使用GFZ提供的北斗超快速精密星历的定位效果比较,验证了分区综合改正定位在实时定位中的优势。在此基础上进一步对中国范围内分区综合改正定位效果与分区中心距离的关系进行了分析,并对不同观测时间长度的定位效果进行比较。结果表明,经分区综合改正后的双频用户平均25 min内动态定位三维误差能收敛至0.5 m以内,收敛后的定位精度为水平0.15 m,高程0.2 m;单频用户平均20 min内动态定位三维误差能收敛至0.8 m以内,收敛后的定位精度为水平0.3 m,高程0.5 m。随着用户站距离分区中心越远,定位效果总体呈现变差的趋势。总体上,当用户在分区中心1 000 km范围内时,北斗广域分区综合改正数将能提供实时分米级定位服务。     

Calibration errors on experimental slant total electron content (TEC) determined with GPS

The Global Positioning System (GPS) has become a powerful tool for ionospheric studies. In addition, ionospheric corrections are necessary for the augmentation systems required for Global Navigation Satellite Systems (GNSS) use. Dual-frequency carrier-phase and code-delay GPS observations are combined to obtain ionospheric observables related to the slant total electron content (sTEC) along the satellite-receiver line-of-sight (LoS). This observable is affected by inter-frequency biases [IFB; often called differential code biases (DCB)] due to the transmitting and the receiving hardware. These biases must be estimated and eliminated from the data in order to calibrate the experimental sTEC obtained from GPS observations. Based on the analysis of single differences of the ionospheric observations obtained from pairs of co-located dual-frequency GPS receivers, this research addresses two major issues: (1) assessing the errors translated from the code-delay to the carrier-phase ionospheric observable by the so-called levelling process, applied to reduce carrier-phase ambiguities from the data; and (2) assessing the short-term stability of receiver IFB. The conclusions achieved are: (1) the levelled carrier-phase ionospheric observable is affected by a systematic error, produced by code-delay multi-path through the levelling procedure; and (2) receiver IFB may experience significant changes during 1 day. The magnitude of both effects depends on the receiver/antenna configuration. Levelling errors found in this research vary from 1.4 total electron content units (TECU) to 5.3 TECU. In addition, intra-day vaiations of code-delay receiver IFB ranging from 1.4 to 8.8 TECU were detected.     

Efficiency of carrier-phase integer ambiguity resolution for precise GPS positioning in noisy environments

Precise GPS positioning relies on tracking the carrier-phase. The fractional part of carrier-phase can be measured directly using a standard phase-locked loop, but the integer part is ambiguous and the ambiguity must be resolved based on sequential carrier-phase measurements to ensure the required positioning precision. In the presence of large phase-measurement noise, as can be expected in a jamming environment for example, the amount of data required to resolve the integer ambiguity can be large, which requires a long time for any generic integer parameter estimation algorithm to converge. A key question of interest in significant applications of GPS where fast and accurate positioning is desired is then how the convergence time depends on the noise amplitude. Here we address this question by investigating integer least-sqaures estimation algorithms. Our theoretical derivation and numerical experiments indicate that the convergence time increases linearly with the noise variance, suggesting a less stringent requirement for the convergence time than intuitively expected, even in a jamming environment where the phase noise amplitude is large. This finding can be useful for practical design of GPS-based systems in a jamming environment, for which the ambiguity resolution time for precise positioning may be critical.     

Investigation of ionosphere TEC distribution in China region by GIM model

In this paper, the TEC in China region is calculated by final GIM model and plotted as contour maps, the TEC value calculated from dual frequency GPS measurements (carrier-phase and pseudorange) is used as the reference truth and the TEC values from GIM models are then compared with the reference truth, and the accuracy and performance of GIM model in China region are analyzed and concluded.     

Part 3 Specialized aspects of GIS and spatial analysis

GIS and spatial analysis is suited mainly for static pictures of the landscape, but many of the processes that need exploring are dynamic in nature. Dynamic processes can be complex when put in a spatial context; our ability to study such processes will probably come with advances in understanding complex systems in general. Cellular automata and agent-based models are two prime candidates for exploring complex spatial systems, but are difficult to implement. Innovative tools that help build complex simulations will create larger user communities, who will probably find novel solutions for understanding complexity. A significant source for such innovations is likely to be from the collective efforts of hobbyists and part-time programmers, who have been dubbed “garage-band scientists” in the popular press.