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1.
Success probability of integer GPS ambiguity rounding and bootstrapping 总被引：26，自引：7，他引：19
P. J. G. Teunissen 《Journal of Geodesy》1998,72(10):606-612
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 相似文献
2.
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. 相似文献
3.
The success rate and precision of GPS ambiguities 总被引：8，自引：1，他引：7
P. J. G. Teunissen 《Journal of Geodesy》2000,74(3-4):321-326
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 相似文献
4.
P. J. G. Teunissen 《Journal of Geodesy》2007,81(12):759-780
In this contribution, we extend the existing theory of minimum mean squared error prediction (best prediction). This extention
is motivated by the desire to be able to deal with models in which the parameter vectors have real-valued and/or integer-valued
entries. New classes of predictors are introduced, based on the principle of equivariance. Equivariant prediction is developed
for the real-parameter case, the integer-parameter case, and for the mixed integer/real case. The best predictors within these
classes are identified, and they are shown to have a better performance than best linear (unbiased) prediction. This holds
true for the mean squared error performance, as well as for the error variance performance. We show that, in the context of
linear model prediction, best predictors and best estimators come in pairs. We take advantage of this property by also identifying
the corresponding best estimators. All of the best equivariant estimators are shown to have a better precision than the best
linear unbiased estimator. Although no restrictions are placed on the probability distributions of the random vectors, the
Gaussian case is derived separately. The best predictors are also compared with least-squares predictors, in particular with
the integer-based least-squares predictor introduced in Teunissen (J Geodesy, in press, 2006). 相似文献
5.
BRDF模型参数分阶段鲁棒性反演方法 总被引：2，自引：0，他引：2
遥感BRDF物理模型均建立于一定的假设或基于某些理想状况，其模拟的数据与观测数据之间多少会存在一些差异（误差）。利用BRDF模型反演地表参数时，如果不加选择地使用所有观测数据，势必会影响模型参数反演的准确度。遥感反演时一般都采用代价函数进行参数拟合。经典的最小二乘（LS）拟合代价函数对正态分布误差具有一定的抗干扰性，但是当观测数据含有异常值时却会导致反演结果的不稳定。最小中值平方（LMS）方法具有鲁棒性特点，反演时若将其作为代价函数，则可以有效地检测出观测数据中含有的异常值，从而可以使模型反演准确度提高。本文以遥感BRDF物理模型——SAIL模型为例，使用模拟数据与真实地面观测数据，构建LMS与LS两种代价函数，分阶段地进行地表参数的反演方法研究。结果显示，针对具有一定误差或模型不能完全表示的观测数据，本文采用的分阶段方法可以对模型参数鲁棒地反演。 相似文献
6.
The affine constrained GNSS attitude model and its multivariate integer least-squares solution 总被引：1，自引：1，他引：0
P. J. G. Teunissen 《Journal of Geodesy》2012,86(7):547-563
A new global navigation satellite system (GNSS) carrier-phase attitude model and its solution are introduced in this contribution. This affine-constrained GNSS attitude model has the advantage that it avoids the computational complexity of the orthonormality-constrained GNSS attitude model, while it still has a significantly improved ambiguity resolution performance over its unconstrained counterpart. The functional and stochastic model is formulated in multivariate form, for one-, two- and three-dimensional antenna arrays, tracking GNSS signals on an arbitrary number of frequencies with two or more antennas. The stochastic model includes the capability to capture variations in the antenna-quality within the array. The multivariate integer least-squares solution of the model parameters is given and the model’s ambiguity success-rate is analysed by means of the ambiguity dilution of precision (ADOP). A general closed-form expression for the affine-constrained ADOP is derived, thus providing an easy-to-use and insightful rule-of-thumb for the ambiguity resolution capabilities of the affine constrained GNSS attitude model. 相似文献
7.
We propose a methodology for the combination of a gravimetric (quasi-) geoid with GNSS-levelling data in the presence of noise
with correlations and/or spatially varying noise variances. It comprises two steps: first, a gravimetric (quasi-) geoid is
computed using the available gravity data, which, in a second step, is improved using ellipsoidal heights at benchmarks provided
by GNSS once they have become available. The methodology is an alternative to the integrated processing of all available data
using least-squares techniques or least-squares collocation. Unlike the corrector-surface approach, the pursued approach guarantees
that the corrections applied to the gravimetric (quasi-) geoid are consistent with the gravity anomaly data set. The methodology
is applied to a data set comprising 109 gravimetric quasi-geoid heights, ellipsoidal heights and normal heights at benchmarks
in Switzerland. Each data set is complemented by a full noise covariance matrix. We show that when neglecting noise correlations
and/or spatially varying noise variances, errors up to 10% of the differences between geometric and gravimetric quasi-geoid
heights are introduced. This suggests that if high-quality ellipsoidal heights at benchmarks are available and are used to
compute an improved (quasi-) geoid, noise covariance matrices referring to the same datum should be used in the data processing
whenever they are available. We compare the methodology with the corrector-surface approach using various corrector surface
models. We show that the commonly used corrector surfaces fail to model the more complicated spatial patterns of differences
between geometric and gravimetric quasi-geoid heights present in the data set. More flexible parametric models such as radial
basis function approximations or minimum-curvature harmonic splines perform better. We also compare the proposed method with
generalized least-squares collocation, which comprises a deterministic trend model, a random signal component and a random
correlated noise component. Trend model parameters and signal covariance function parameters are estimated iteratively from
the data using non-linear least-squares techniques. We show that the performance of generalized least-squares collocation
is better than the performance of corrector surfaces, but the differences with respect to the proposed method are still significant. 相似文献
8.
建立回归模型常采用最小二乘方法并忽略自变量观测误差。尽管同时顾及自变量和因变量观测误差的总体最小二乘方法近年来得到了广泛研究,但在模型预测时,依然忽略了待预测自变量的观测误差。对此,本文提出了一种严格考虑所有变量观测误差的无缝线性回归和预测模型,该模型将回归模型的建立和因变量预测联合处理,在建立回归模型过程中对待预测自变量的观测误差进行估计并修正,从而提高了模型预测效果。理论证明,现有的几种线性回归模型都是无缝线性回归和预测模型的特例。试验结果表明,无缝线性回归和预测模型的预测效果优于现有的几种模型,尤其在变量观测误差相关性较大时,无缝模型对预测效果的改善更为显著。 相似文献
9.
On symmetrical three-dimensional datum conversion 总被引：2，自引：0，他引：2
A 3-D similarity transformation is frequently used to convert GPS-WGS84-based coordinates to those in a local datum using
a set of control points with coordinate values in both systems. In this application, the Gauss-Markov (GM) model is often
employed to represent the problem, and a least-squares approach is used to compute the parameters within the mathematical
model. However, the Gauss–Markov model considers the source coordinates in the data matrix (A) as fixed or error-free; this is an imprecise assumption since these coordinates are also measured quantities and include
random errors. The errors-in-variables (EIV) model assumes that all the variables in the mathematical model are contaminated
by random errors. This model may be solved using the relatively new total least-squares (TLS) estimation technique, introduced
in 1980 by Golub and Van Loan. In this paper, the similarity transformation problem is analyzed with respect to the EIV model,
and a novel algorithm is described to obtain the transformation parameters. It is proved that even with the EIV model, a closed
form Procrustes approach can be employed to obtain the rotation matrix and translation parameters. The transformation scale
may be calculated by solving the proper quadratic equation. A numerical example and a practical case study are presented to
test this new algorithm and compare the EIV and the GM models. 相似文献
10.
Integer least-squares theory for the GNSS compass 总被引：7，自引：2，他引：5
P. J. G. Teunissen 《Journal of Geodesy》2010,84(7):433-447
Global navigation satellite system (GNSS) carrier phase integer ambiguity resolution is the key to high-precision positioning
and attitude determination. In this contribution, we develop new integer least-squares (ILS) theory for the GNSS compass model,
together with efficient integer search strategies. It extends current unconstrained ILS theory to the nonlinearly constrained
case, an extension that is particularly suited for precise attitude determination. As opposed to current practice, our method
does proper justice to the a priori given information. The nonlinear baseline constraint is fully integrated into the ambiguity
objective function, thereby receiving a proper weighting in its minimization and providing guidance for the integer search.
Different search strategies are developed to compute exact and approximate solutions of the nonlinear constrained ILS problem.
Their applicability depends on the strength of the GNSS model and on the length of the baseline. Two of the presented search
strategies, a global and a local one, are based on the use of an ellipsoidal search space. This has the advantage that standard
methods can be applied. The global ellipsoidal search strategy is applicable to GNSS models of sufficient strength, while
the local ellipsoidal search strategy is applicable to models for which the baseline lengths are not too small. We also develop
search strategies for the most challenging case, namely when the curvature of the non-ellipsoidal ambiguity search space needs
to be taken into account. Two such strategies are presented, an approximate one and a rigorous, somewhat more complex, one.
The approximate one is applicable when the fixed baseline variance matrix is close to diagonal. Both methods make use of a
search and shrink strategy. The rigorous solution is efficiently obtained by means of a search and shrink strategy that uses
non-quadratic, but easy-to-evaluate, bounding functions of the ambiguity objective function. The theory presented is generally
valid and it is not restricted to any particular GNSS or combination of GNSSs. Its general applicability also applies to the
measurement scenarios (e.g. single-epoch vs. multi-epoch, or single-frequency vs. multi-frequency). In particular it is applicable
to the most challenging case of unaided, single frequency, single epoch GNSS attitude determination. The success rate performance
of the different methods is also illustrated. 相似文献
11.
An optimality property of the integer least-squares estimator 总被引：36，自引：15，他引：21
P. J. G. Teunissen 《Journal of Geodesy》1999,73(11):587-593
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 相似文献
12.
The 3D similarity coordinate transformation with the Gauss–Helmert error model is investigated. The first-order error analysis of an analytical least-squares solution to this problem is developed in detail. While additive errors are assumed in the translation and scale estimates, a 3 × 1 multiplicative error vector is defined to effectively parameterize the rotation matrix estimation error. The propagation of the errors in the coordinate measurements to the errors in the estimated transformation parameters is derived step-by-step, and the formulae for calculating the variance–covariance matrix of the estimated parameters are presented. 相似文献
13.
This paper examines the influence that certain omission and commission errors can have on the gravity field models estimated from the initial release of data (RL01) from the Gravity Recovery And Recovery Experiment (GRACE) satellite mission. The effects of omission errors were analyzed by limiting the degree and order to which the GPS and K-band range-rate (KBR) measurement partials were extended in the solution process. The commission error studies focused on the impact of an imperfect mean reference gravity field model on the solution. Combinations of both of these error sources were also explored. The nature of these errors makes them difficult to distinguish from the true gravity signal, so the exploration of these error sources was performed using simulations; however, comparisons to real-data solutions are provided. The results show how each of the specific error sources investigated influences the gravity field solution. The simulations also show how all of the errors examined can be sufficiently mitigated through the appropriate choice of processing parameters. 相似文献
14.
Robust estimation by expectation maximization algorithm 总被引：2，自引：2，他引：0
Karl Rudolf Koch 《Journal of Geodesy》2013,87(2):107-116
A mixture of normal distributions is assumed for the observations of a linear model. The first component of the mixture represents the measurements without gross errors, while each of the remaining components gives the distribution for an outlier. Missing data are introduced to deliver the information as to which observation belongs to which component. The unknown location parameters and the unknown scale parameter of the linear model are estimated by the EM algorithm, which is iteratively applied. The E (expectation) step of the algorithm determines the expected value of the likelihood function given the observations and the current estimate of the unknown parameters, while the M (maximization) step computes new estimates by maximizing the expectation of the likelihood function. In comparison to Huber’s M-estimation, the EM algorithm does not only identify outliers by introducing small weights for large residuals but also estimates the outliers. They can be corrected by the parameters of the linear model freed from the distortions by gross errors. Monte Carlo methods with random variates from the normal distribution then give expectations, variances, covariances and confidence regions for functions of the parameters estimated by taking care of the outliers. The method is demonstrated by the analysis of measurements with gross errors of a laser scanner. 相似文献
15.
陈永奇 《武汉大学学报(信息科学版)》1997,(4)
检验解算的整周模糊度是ＧＰＳ精密快速定位的一个重要课题。过去已提出了不少方法，然而这些方法都存在一定的缺陷。本文提出一种新的方法，它是基于模型可区分度的概念。该方法利用可接受的第一、二类错误概率，计算边界值，并与固定解中最小残差二次型和次最小残差二次型比较，以确认相应于最小残差二次型的整周模糊度的有效性。 相似文献
16.
On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms 总被引：2，自引：1，他引：1
The multivariate total least-squares (MTLS) approach aims at estimating a matrix of parameters, Ξ, from a linear model (Y−E
Y
= (X−E
X
) · Ξ) that includes an observation matrix, Y, another observation matrix, X, and matrices of randomly distributed errors, E
Y
and E
X
. Two special cases of the MTLS approach include the standard multivariate least-squares approach where only the observation
matrix, Y, is perturbed by random errors and, on the other hand, the data least-squares approach where only the coefficient matrix
X is affected by random errors. In a previous contribution, the authors derived an iterative algorithm to solve the MTLS problem
by using the nonlinear Euler–Lagrange conditions. In this contribution, new lemmas are developed to analyze the iterative
algorithm, modify it, and compare it with a new ‘closed form’ solution that is based on the singular-value decomposition.
For an application, the total least-squares approach is used to estimate the affine transformation parameters that convert
cadastral data from the old to the new Israeli datum. Technical aspects of this approach, such as scaling the data and fixing
the columns in the coefficient matrix are investigated. This case study illuminates the issue of “symmetry” in the treatment
of two sets of coordinates for identical point fields, a topic that had already been emphasized by Teunissen (1989, Festschrift
to Torben Krarup, Geodetic Institute Bull no. 58, Copenhagen, Denmark, pp 335–342). The differences between the standard least-squares
and the TLS approach are analyzed in terms of the estimated variance component and a first-order approximation of the dispersion
matrix of the estimated parameters. 相似文献
17.
分析了多点灰色模型利用最小二乘估计原理进行参数解算时无法顾及起算数据误差带来的影响。将混合最小二乘与多元整体最小二乘应用到多点灰色模型的参数估计中。首先利用QR分解将起算数据中常数列和误差项相分离;采用最小二乘和多元整体最小二乘分别进行解算建模;最后通过实验证明了优化的MGM(1,n)模型具有较高的建模和预测精度,能够为精密工程变形分析提供一定的参考和借鉴。 相似文献
18.
19.
20.
The non-linear 2D symmetric helmert transformation : An exact non-linear least-squares solution 总被引：1，自引：1，他引：0
Peter J. G. Teunissen 《Journal of Geodesy》1988,62(1):1-16
In this paper a particular class of non-linear least-squares problems for which it is possible to take advantage of the special
structure of the non-linear model, is discussed. The non-linear models are of the ruled-type (Teunisson, 1985a). The proposed solution strategy is applied to the2D non-linear Symmetric Helmert transformation which is defined in the paper. An exact non-linear least-squares solution, using
a rotational invariant covariance structure is given. 相似文献