Statistical characterization of composite protection levels for GPS

A statistical characterization is presented of Global Positioning System (GPS) user range error as a normally distributed random variable with non-zero mean over the length of the aircraft precision approach operation, correlated from one GPS measurement epoch to another and from one satellite to another. This leads directly to modeling GPS error in the position domain as multivariate normal with non-zero mean. Based on this model, a vertical composite protection level VPL_c and a horizontal composite protection level HPL_c are each calculated as scalar values from a univariate normal distribution displaced from the origin by the worst-case position domain bias combination possible, given the maximum possible individual satellite bias magnitudes and the satellite geometry. A method is then presented by which exact values—that is, values accurate to a user-defined error tolerance and consistent with statistical assumptions—of VPL_c and HPL_c are obtained, and by which computationally efficient approximations may be evaluated. A statistical quadratic form under the multivariate normal distribution is used to derive a new class of protection levels based on the probability enclosed within a radius defined in two or more dimensions. A central Chi-square representation of this quadratic form is also presented and is incorporated into a six-step computational procedure for the two-dimensional composite radial protection level RPL_c. This procedure is extended to the spherical protection level (SPL_c) and the ellipsoidal protection level (EPL_c).     

MINI-MACtm—A new generation dual—Band surveyor

Summary Litton Divisions presently produce both high accuracy GPS surveyors and low—cost GPS navigation sets. Aero Service'sMACROMETER ^R Interferometric Surveyors, have become the standard against which GPS surveying equipment is measured. Litton Aero Products has developed a highly digitized, low costL ₁,C/A code GPS card set. The integration of these technologies had led to the development of a low-cost, high-precision, GPS survey system which can be configured with or without a codelessL ₂ capability. TheMINI-MAC surveying system is the first member of the new generation of GPS survey systems resulting from this joint development. The system design is described in this paper, and initial survey test results using a prototypeMINI-MAC surveying system are presented.     

GPS/Pseudolite/INS integration: concept and first tests

This paper discusses the introduction of pseudolites (ground-based GPS-like signal transmitters) into existing integrated GPS/INS systems in order to provide higher availability, integrity, and accuracy in a local area. Even though integrated GPS/INS systems can overcome inherent drawbacks of each component system (line-of-sight requirement for GPS, and INS errors that grow with time), performance is nevertheless degraded under adverse operational circumstances. Some typical examples are when the duration of satellite signal blockage exceeds an INS bridging level, resulting in large accumulated INS errors that cannot be calibrated by GPS. Such a scenario, unfortunately, is a common occurrence for certain kinematic applications. To address such shortcomings, both pseudolite/INS and GPS/pseudolite/INS integration schemes are proposed here. Typically, the former is applicable for indoor positioning where the GPS signal is unavailable for use. The latter would be appropriate for system augmentation when the number and geometry of visible satellites is not sufficient for accurate positioning or attitude determination. In this paper, some technical issues concerned with implementing these two integration schemes are described, including the measurement model, and the appropriate integration filter for INS error estimation and correction through GPS and pseudolite (PL) carrier phase measurements. In addition, the results from the processing of simulated measurements, as well as field experiments, are presented in order to characterize the system performance. As a result, it has been established that the GPS/PL/INS and PL/INS integration schemes would make it possible to ensure centimeter-level positioning accuracy even if the number of GPS signals is insufficient, or completely unavailable. Electronic Publication     

The probability distribution of the GPS baseline for a class of integer ambiguity estimators

In current global positioning system (GPS) ambiguity resolution practice there is not yet a rigorous procedure in place to diagnose its expected performance and to evaluate the probabilistic properties of the computed baseline. The necessary theory to bridge this gap is presented. Probabilistic statements about the `fixed' GPS baseline can be made once its probability distribution is known. This distribution is derived for a class of integer ambiguity estimators. Members from this class are the ambiguity estimators that follow from `integer rounding', `integer bootstrapping' and `integer least squares' respectively. It is also shown how this distribution differs from the one which is usually used in practice. The approximations involved are identified and ways of evaluating them are given. In this comparison the precise role of GPS ambiguity resolution is clarified. Received: 3 August 1998 / Accepted: 4 March 1999     

Stability analysis of GPS carrier tracking loops by phase margin approach

Stability, which is significantly related to the loop parameters, is an important factor in the traditional GPS tracking loop design. Through the analysis of phase margin values in the discrete GPS PLL tacking loop, we are able to theoretically reveal the relationship between loop stability, equivalent noise bandwidth B _n , predetection integration time T, and loop parameters. We calculate the theoretical limitations for B _n T, that is, the product of equivalent noise bandwidth multiplied by predetection integration time, for second- and third-order phase-locked loop, respectively. The results are verified by actual data from GPS receivers.     

Confidence regions for GPS baselines by Bayesian statistics

 The global positioning system (GPS) model is distinctive in the way that the unknown parameters are not only real-valued, the baseline coordinates, but also integers, the phase ambiguities. The GPS model therefore leads to a mixed integer–real-valued estimation problem. Common solutions are the float solution, which ignores the ambiguities being integers, or the fixed solution, where the ambiguities are estimated as integers and then are fixed. Confidence regions, so-called HPD (highest posterior density) regions, for the GPS baselines are derived by Bayesian statistics. They take care of the integer character of the phase ambiguities but still consider them as unknown parameters. Estimating these confidence regions leads to a numerical integration problem which is solved by Monte Carlo methods. This is computationally expensive so that approximations of the confidence regions are also developed. In an example it is shown that for a high confidence level the confidence region consists of more than one region. Received: 1 February 2001 / Accepted: 18 July 2001     

大范围GPS水准的综合模型解法

函数模型逼近和统计模型逼近是GPS水准拟合中常用的方法,前者对趋势性和规律性变化参数求解较为理想,后者对随机变化的逼近较为合适。对大范围的GPS水准拟合,综合模型法兼顾二者优点,将获得更高精度,给出了具体模型和计算方法,并对一个大测区的GPS水准数据进行解算,获得了满意的结果。     

Numerical problems in the computation of ellipsoidal harmonics

The correct use of ellipsoidal coordinates and related ellipsoidal harmonic functions can provide a representation of linearized Geodetic Boundary Value Problems (GBVP) much closer to the exact ones than what is usually done in spherical approximation: this becomes important in the present age, since terms of the type e²N, possibly amounting to several dozens of centimetres, are nowadays observable.Although the theory of ellipsoidal harmonics has been introduced into geodesy by several authors to treat gravity global models, the numerical computation of ellipsoidal harmonics of high degree and order seems to be more critical than it has been recognized. In particular, exact recursive relations display a quite unstable behaviour, no matter what normalization constants are used; it is only through particular representation of hypergeometric functions that it is possible to find a sound method for numerical manipulation. Also the asymptotic approximations, exploiting the smallness of the eccentricity, e², are analysed in relation to their critical behaviour for particular values of degree and order, it is shown that a limit layer theory can provide a simpler, better, and stable approximation of the exact values of ellipsoidal harmonics.     

Ocean tide loading (OTL) displacements from global and local grids: comparisons to GPS estimates over the shelf of Brittany, France

In this paper we examine OTL displacements detected by GPS stations of a dedicated campaign and validate ocean tide models. Our area of study is the continental shelf of Brittany and Cotentin in France. Brittany is one of the few places in the world where tides provoke loading displacements of ∼10–12 cm vertically and a few cm horizontally. Ocean tide models suffer from important discrepancies in this region. Seven global and regional ocean tide models were tested: FES2004 corrected for K2, TPXO.7.0, TPXO.6.2, GOT00.2, CSR4.0, NAO.99b and the most recent regional grids of the North East Atlantic (NEA2004). These gridded amplitudes and phases of ocean tides were convolved in order to get the predicted OTL displacements using two different algorithms. Data over a period of 3.5 months of 8 GPS campaign stations located on the north coast of Brittany are used, in order to evaluate the geographical distribution of the OTL effect. We have modified and implemented new algorithms in our GPS software, GINS 7.1. GPS OTL constituents are estimated based on 1-day batch solutions. We compare the observed GPS OTL constituents of M₂, S₂, N₂ and K₁ waves with the selected ocean tide models on global and regional grids. Large phase-lag and amplitude discrepancies over 20° and 1.5 cm in the vertical direction in the semi-diurnal band of M₂ between predictions and GPS/models are detected in the Bay of Mont St-Michel. From a least squares spectral analysis of the GPS time-series, significant harmonic peaks in the integer multiples of the orbital periods of the GPS satellites are observed, indicating the existence of multipath effects in the GPS OTL constituents. The GPS OTL observations agree best with FES2004, NEA2004, GOT00.2 and CSR4.0 tide models.     

Improved amplitude- and phase-scintillation indices derived from wavelet detrended high-latitude GPS data

Accuracy and validity of scintillation indices estimated using the power and phase of the GPS signal depend heavily on the detrending method used and the selection of the cutoff frequency of the associated filter. A Butterworth filter with a constant cutoff frequency of 0.1?Hz is commonly used in detrending GPS data. In this study, the performance of this commonly used filter is evaluated and compared with a new wavelet-based detrending method using GPS data from high latitudes. It was observed that in detrending high-latitude GPS data, a wavelet filter performed better than Butterworth filters as the correlation between amplitude- and phase-scintillation indices in S ₄ and ?? _? improved significantly from 0.53, when using a Butterworth filter, to 0.79, when using the wavelet filtering method. We also introduced an improved phase-scintillation index, ?? _CHAIN, which we think is comparatively a better parameter to represent phase scintillations at high latitudes as the correlation between S ₄ and ?? _CHAIN was as high as 0.90. During the analysis, we also noted that the occurrence of the ??phase scintillation without amplitude scintillation?? phenomenon was significantly reduced when scintillation indices were derived using the wavelet-based detrending method. These results seem to indicate that wavelet-based detrending is better suited for GPS scintillation signals and also that ?? _CHAIN is a better parameter for representing GPS phase scintillations at high latitudes.     

Approximating the Bayesian estimate of the standard deviation in a linear model

The Bayesian estimates _b of the standard deviation σ in a linear model—as needed for the evaluation of reliability—is well known to be proportional to the square root of the Bayesian estimate (s ²) _b of the variance component σ² by a proportionality factor involving the ratio of Gamma functions. However, in analogy to the case of the respective unbiased estimates, the troublesome exact computation ofa _b may be avoided by a simple approximation which turns out to be good enough for most applications even if the degree of freedom ν is rather small. Paper presented to the Int. Conf. on “Practical Bayesian Statistics”, Cambridge (U.K.), 8.–11. July 1986.     

重力异常的一种逼近方法

函数模型逼近和统计模型逼近在大地测量均有广泛的应用。前者对趋势和性和规律性变化参数的求解较为理想。后者对随机变化物逼近较为合适，本文首先简单介绍了各种函数模型逼近和统计模型逼近的特性，提出了一种在函数模型逼近的基础上，辅以统计逼近的综合逼近法，作为一个例子，文中利用移动曲面模型和多面函数模型分别与拟合推估结合进行了重力异常的综合逼近计算。理论与计算表明，新的综合逼近法优于单独的函数模型逼近和统计模型逼近。     

Empirical comparison of noise reduction techniques for NDVI time-series based on a new measure

This study empirically compared noise reduction techniques for the normalized difference vegetation index (NDVI) time-series based on a new absolute measure using a time-series of 16-day composite NDVI images extracted from the Terra Moderate Resolution Imaging Spectroradiometer (MODIS) products covering the Poyang Lake area in China. We proposed an approach to accurately extract representative NDVI temporal profiles for the 12 land cover cluster types by clustering profiles, selecting optimal number of clusters, merging and labeling clusters, and selecting the representative NDVI profiles. The geometric average of the mean average distance between the reconstructed profile and the raw profiles, and the mean average distance between the reconstructed profile and the upper envelope (D_g(nr, c)) was selected as the most appropriate measure substitutive to RMSE for the evaluation of the noise reduction effects, when the ‘true’ profiles were not available. The running median, mean value, maximum operation, end point processing, and Hanning smoothing (RMMEH) filter and iterative Savitzky–Golay filter were the two most appropriate noise reduction techniques for the NDVI temporal profiles of the study area in the evaluation of noise reduction effects by the seven techniques. The robust framework using the proposed approach for the accurate extraction of representative NDVI temporal profiles and (D_g(nr, c)) in this study, is applicable in the evaluation of noise reduction effects using different techniques and in other study areas.     

Unscented Kalman filter with nonlinear dynamic process modeling for GPS navigation

This paper preliminarily investigates the application of unscented Kalman filter (UKF) approach with nonlinear dynamic process modeling for Global positioning system (GPS) navigation processing. Many estimation problems, including the GPS navigation, are actually nonlinear. Although it has been common that additional fictitious process noise can be added to the system model, however, the more suitable cure for non convergence caused by unmodeled states is to correct the model. For the nonlinear estimation problem, alternatives for the classical model-based extended Kalman filter (EKF) can be employed. The UKF is a nonlinear distribution approximation method, which uses a finite number of sigma points to propagate the probability of state distribution through the nonlinear dynamics of system. The UKF exhibits superior performance when compared with EKF since the series approximations in the EKF algorithm can lead to poor representations of the nonlinear functions and probability distributions of interest. GPS navigation processing using the proposed approach will be conducted to validate the effectiveness of the proposed strategy. The performance of the UKF with nonlinear dynamic process model will be assessed and compared to those of conventional EKF.     

Ultra-tight GPS/INS/PL integration: a system concept and performance analysis

The architecture of the ultra-tight GPS/INS/PL integration is the key to its successful performance; the main feature of this architecture is the Doppler feedback to the GPS receiver tracking loops. This Doppler derived from INS, when integrated with the carrier tracking loops, removes the Doppler due to vehicle dynamics from the GPS/PL signal thereby achieving a significant reduction in the carrier tracking loop bandwidth. The bandwidth reduction provides several advantages such as: improvement in anti-jamming performance, and increase in post correlated signal strength which in turn increases the dynamic range and accuracy of measurements. Therefore, any degradation in the derived Doppler estimates will directly affect the tracking loop bandwidth and hence its performance. The quadrature signals from the receiver correlator, I (in-phase) and Q (quadrature), form the measurements, whereas the inertial sensor errors, position, velocity and attitude errors form the states of the complementary Kalman filter. To specify a reliable measurement model of the filter for this type of integrated system, a good understanding of GPS/PL signal characteristics is essential. It is shown in this paper that phase and frequency errors are the variables that relate the measurements and the states in the Kalman filter. The main focus of this paper is to establish the fundamental mathematical relationships that form the measurement model, and to show explicitly how the system error states are related to the GPS/PL signals. The derived mathematical relationships encapsulated in a Kalman filter, are tested by simulation and shown to be valid.

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Improved relativistic transformations in GPS

For GPS satellite clocks, a nominal (hardware) frequency offset and a conventional periodic relativistic correction derived as a dot product of the satellite position and velocity vectors, are used to compensate the relativistic effects. The conventional hardware clock rate offset of 38,575.008 ns/day corresponds to a nominal orbit semi-major axis of about 26,561,400 m. For some of the GPS satellites, the departures from the nominal semi-major axis can cause an apparent clock rate up to 10 ns/day. GPS orbit perturbations, together with the earth gravity field oblateness, which is largely responsible for the orbit perturbations, cause the standard GPS relativistic transformations to depart from the rigorous relativity transformation by up to 0.2 ns/day. In addition, the conventional periodic relativistic correction exhibits periodic errors with amplitudes of about 0.1 and 0.2 ns, with periods of about 6 h and 14 days, respectively. Using an analytical integration of the gravity oblateness term (J₂), a simple analytical approximation was derived for the apparent clock rate and the 6-h periodic errors of the standard GPS gravity correction. For daily linear representations of GPS satellite clocks, the improved relativistic formula was found to agree with the precise numerical integration of the GPS relativistic effects within about 0.015 ns. For most of the Block IIR satellites, the 6-h periodical errors of the GPS conventional relativistic correction are already detectable in the recent IGS final clock combinations.     

GPS navigation processing using the quaternion-based divided difference filter

Divided difference filter (DDF) with quaternion-based dynamic process modeling is applied to global positioning system (GPS) navigation. Using techniques similar to those of the unscented Kalman filter (UKF), the DDF uses divided difference approximations of derivatives based on Stirling’s interpolation formula which results in a similar mean but different posterior covariance compared to the extended Kalman filter (EKF) solutions. The second-order divided difference is obtained from the mean and covariance in second-order polynomial approximation. The quaternion-based dynamic model is adopted for avoiding the singularity problems encountered in the Euler angle method and enhancing the computational efficiency. The proposed method is applied to GPS navigation to increase the navigation estimation accuracy at high-dynamic regions while preserving (without sacrificing) the precision at low-dynamic regions. For the illustrated example, the second-order DDF can deliver about 41–82% accuracy improvement as compared to the EKF. Some properties and performance are assessed and compared to those of the EKF and UKF approaches.     

Ground- and space-based GPS data ingestion into the NeQuick model

This paper presents a technique for ingesting ground- and space-based dual-frequency GPS observations into a semi-empirical global electron density model. The NeQuick-2 model is used as the basis for describing the global electron density distribution. This model is mainly driven by the F2 ionosphere layer parameters (i.e. the electron density, N _m F2, and the height, h _m F2 of the F2 peak), which, in the absence of directly measured values, are computed from the ITU-R database (ITU-R 1997). This database was established using observations collected from 1954 to 1958 by a network of around 150 ionospheric sounders with uneven global coverage. It allows computing monthly median values of N _m F2 and h _m F2 (intra-month variations are averaged), for low and high solar activity. For intermediate solar activity a linear interpolation must be performed. Ground-based GNSS observations from a global network of ~350 receivers are pre-processed in order to retrieve slant total electron content (sTEC) information, and space-based GPS observations (radio occultation data from the FORMOSAT-3/COSMIC constellation) are pre-processed to retrieve electron density (ED) information. Both, sTEC and ED are ingested into the NeQuick-2 model in order to adapt N _m F2 and h _m F2, and reduce simultaneously both, the observed minus computed sTEC and ED differences. The first experimental results presented in this paper suggest that the data ingestion technique is self consistent and able to reduce the observed minus computed sTEC and ED differences to ~25–30% of the values computed from the ITU-R database. Although sTEC and ED are both derived from GPS observations, independent algorithm and models are used to compute their values from ground-based GPS observations and space-based FORMOSAT-3/COSMIC radio occultations. This fact encourages us to pursue this research with the aim to improve the results presented here and assess their accuracy in a reliable way.     

Low-degree earth deformation from reprocessed GPS observations

Surface mass variations of low spherical harmonic degree are derived from residual displacements of continuously tracking global positioning system (GPS) sites. Reprocessed GPS observations of 14 years are adjusted to obtain surface load coefficients up to degree n _max = 6 together with station positions and velocities from a rigorous parameter combination. Amplitude and phase estimates of the degree-1 annual variations are partly in good agreement with previously published results, but also show interannual differences of up to 2 mm and about 30 days, respectively. The results of this paper reveal significant impacts from different GPS observation modeling approaches on estimated degree-1 coefficients. We obtain displacements of the center of figure (CF) relative to the center of mass (CM), Δr _CF–CM, that differ by about 10 mm in maximum when compared to those of the commonly used coordinate residual approach. Neglected higher-order ionospheric terms are found to induce artificial seasonal and long-term variations especially for the z-component of Δr _CF–CM. Daily degree-1 estimates are examined in the frequency domain to assess alias contributions from model deficiencies with regard to satellite orbits. Finally, we directly compare our estimated low-degree surface load coefficients with recent results that involve data from the Gravity Recovery and Climate Experiment (GRACE) satellite mission.     

Anomaly analysis of 18 years of newly merged GPS ephemeris from four IGS data centers

The merged GPS navigation files from the International GNSS Service (IGS) data centers, i.e., the Crustal Dynamics Data Information System (CDDIS), the Bundesamt für Kartographie und Geodäsie (BKG), the Scripps Institution of Oceanography (SIO), and the Institut Geographique National (IGN) are occasionally contaminated by anomalies and inconsistent user range accuracy (URA). This contamination impairs the performance assessment of GPS service, especially the system integrity. We remerged these files starting Day of Year (DOY) 1, 2000 using all available navigation data files from IGS stations. To effectively get the upper bound URA, a frequency-dependent pattern recognition method was developed. In addition, a comprehensive comparison between the navigation data remerged by us and those provided by the four IGS data centers was performed. The compared results revealed that T_GD and Issue of Data Clock (IODC) were the dominating anomalies in the merged navigation data from CDDIS and SIO for the first several years after 2000, and M₀, Ω₀, ω, and a_f0 were the dominant anomalies in the merged data from IGN. In addition to a number of missing records, many records with incorrect PRN (pseudo-random noise number), identifying a GPS satellite, were found in files from the IGS data centers. Although the number of anomalies in the merged files from CDDIS has continued to decrease in recent years, they have not disappeared and would affect system-level assessment and scientific applications to a certain extent. The results also revealed that our remerged files were more complete, clean, compact and consistent, making them more suitable for GPS system performance assessment and related research studies. Moreover, those data are now openly available.