Tensor structure and the least squares

It is shown that for linear parametric adjustment models all the least-squares equations can be obtained from a commutative diagram, where the observation and parameter spaces are regarded as covariant. Their contravariant counterparts are defined through the metric property of the covariance matrix of the observations.     

Nonlinear least-squares method via an isomorphic geometrical setup

The resolution of a nonlinear parametric adjustment model is addressed through an isomorphic geometrical setup with tensor structure and notation, represented by a u-dimensional “model surface” embedded in a flat n-dimensional “observational space”. Then observations correspond to the observational-space coordinates of the pointQ, theu initial parameters correspond to the model-surface coordinates of the “initial” pointP, and theu adjusted parameters correspond to the model-surface coordinates of the “least-squares” point . The least-squares criterion results in a minimum-distance property implying that the vector Q must be orthogonal to the model surface. The geometrical setup leads to the solution of modified normal equations, characterized by a positive-definite matrix. The latter contains second-order and, optionally, thirdorder partial derivatives of the observables with respect to the parameters. This approach significantly shortens the convergence process as compared to the standard (linearized) method.     

A generalisation of the least-squares method for the adjustment of free networks

Summary The system of normal equations for the adjustment of a free network is a singular one. Therefore, a number of coordinates has to be fixed according to the matrix. The mean square errors and the error ellipses of such an adjustment are dependent on this choice. This paper gives a simple, direct method for the adjustment of free networks, where no coordinates need to be fixed. This is done by minimizing not only the sum of the squares of the weighted errorsV ^T PV=minimun but also the Euclidean norm of the vectorX and of the covariance matrixQ X ^T X=minimum trace (Q)=minimum This last condition is crucial for geodetic problems of this type.     

MLAMBDA: a modified LAMBDA method for integer least-squares estimation

The least-squares ambiguity Decorrelation (LAMBDA) method has been widely used in GNSS for fixing integer ambiguities. It can also solve any integer least squares (ILS) problem arising from other applications. For real time applications with high dimensions, the computational speed is crucial. A modified LAMBDA (MLAMBDA) method is presented. Several strategies are proposed to reduce the computational complexity of the LAMBDA method. Numerical simulations show that MLAMBDA is (much) faster than LAMBDA. The relations between the LAMBDA method and some relevant methods in the information theory literature are pointed out when we introduce its main procedures.     

The Null method applied to GNSS three-carrier phase ambiguity resolution

The Null method is a technique to fix the ambiguity in L1 phase measurements of the global positioning system (GPS). The method is adapted to new global navigation satellite systems (GNSS) which offer phase measurements at three frequencies. In order to validate the efficiency of the adapted method, results obtained using a software simulator and an emulator are presented. The results are then compared to those obtained with the least-squares ambiguity decorrelation adjustment (LAMBDA) method. Good performance of the Null method in new GNSS systems is shown.Acknowledgments. The measurements used were provided by the European Space Agency, and were generated by Spectra Precision Terrasat under contract No. 12.406/77/NL/DS. The authors thank Maria Belmontes Rivas for her comments and the reviewers for their suggestions.     

附不等式约束最小二乘平差的一种新算法

在大地测量数据处理中,很多情况下可根据先验知识建立合理的不等式约束,能够改善平差结果,提高精度。首先简要总结了附不等式约束平差的各种方法及存在的问题。根据有效约束和库恩塔克条件,提出了解决不等式约束平差的新算法,把不等式约束平差转化为等式约束平差问题,从而得到解的显示表达。最后用一数值算例证明了该算法的可行性。     

基于选权迭代法的抗差整体最小二乘及其应用

在测量数据处理中,观测向量与系数矩阵同时存在偶然误差时,整体最小二乘法能够得到更高精度的参数解,但整体最小二乘法无抗差能力,观测向量中的粗差将对参数求解产生较大影响。为解决上述问题,采用拉格朗日极值法推导了基于选权迭代法的抗差整体最小二乘计算公式,通过三维坐标转换参数求解实例对3种选权迭代法进行分析。结果表明,IGG法在抗差整体最小二乘解法中抗差效果最好。     

方差分量估计分析北斗伪距信号精度

针对北斗伪距精度和时间相关性研究较少的问题,该文引入最小二乘方差分量估计方法,对伪距与载波相位形成的电离层残差组合序列求高阶差分,构造伪距观测量的关系方程,并分析了北斗卫星伪距观测量的精度和时间相关性。结果表明,北斗伪距观测量的测量噪声在0.1~0.5m范围内,随高度角的增大而减小;不同频率的伪距噪声各有差异,高度角增大时差异减小,所有测站中B1的伪距噪声最大,B2、B3的大小关系随测站而异;地球同步轨道卫星的伪距精度优于倾斜地球同步轨道和中地球轨道卫星,且后两者差距较小;市场上两款北斗接收机的伪距测量精度相当。相关分析表明,当接收机采样频率等于或低于1Hz时,伪距观测量不存在明显的时间相关性;当采样频率高于1Hz时,伪距观测量表现出较强的时间相关性。     

Estimation of asymmetry and kurtosis coefficients in the process of geodetic network adjustment by the least-squares method

 This paper presents an extension of the geodetic network adjustment model. The proposed extension makes possible the estimation of the 3-rd and 4-th central moments for the vector of measurement errors in the process of network adjustment by the least-squares method with application of orthogonal matrices. It allows to estimate the asymmetry and kurtosis of the measurement errors distribution. Received 13 April 1993; Accepted 8 July 1996     

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.     

Integer least-squares theory for the GNSS compass

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.     

On the multivariate total least-squares approach to empirical coordinate transformations. Three algorithms

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.     

Tuning a gravimetric quasigeoid to GPS-levelling by non-stationary least-squares collocation

This paper addresses implementation issues in order to apply non-stationary least-squares collocation (LSC) to a practical geodetic problem: fitting a gravimetric quasigeoid to discrete geometric quasigeoid heights at a local scale. This yields a surface that is useful for direct GPS heighting. Non-stationary covariance functions and a non-stationary model of the mean were applied to residual gravimetric quasigeoid determination by planar LSC in the Perth region of Western Australia. The non-stationary model of the mean did not change the LSC results significantly. However, elliptical kernels in non-stationary covariance functions were used successfully to create an iterative optimisation loop to decrease the difference between the gravimetric quasigeoid and geometric quasigeoid at 99 GPS-levelling points to a user-prescribed tolerance.     

Impact of network geometry, observation schemes and telescope structure deformations on local ties: simulations applied to Sardinia Radio Telescope

The 64-m Sardinia Radio Telescope (SRT) is currently under construction in Sardinia (Italy). To ensure future surveying and monitoring operations at an utmost level of accuracy, we aim at selecting the optimal design and the most cost-effective solution for the establishment of the local ground control network (LGCN). We simulate and test 45 data sets corresponding to 5 different network configurations. We investigate the influence of 2 LGCN geometries (14 or 8 ground markers) and 3 terrestrial observation schemes (based on redundant forward intersections or side shots) on the precision and accuracy of the conventional reference point (CRP) of SRT and the simulated tie vector with a global navigation satellite system (GNSS) station. In addition, thermal and gravitational deformations of the radio telescope structure are simulated as systematic errors introduced into the observations and their effects on the CRP estimates are quantified. The state-of-the-art of CRP surveying and computation, based on terrestrial indirect methods, is applied. We show how terrestrial indirect methods can estimate the position of the radio telescope CRP to the millimeter precision level. With our simulations, we prove that limiting the LGCN to a 8-point configuration ensures the same precision on the CRP obtained with a 14-point network. Furthermore, we demonstrate that in the absence of telescope deformations, side shots, despite the lower redundancy, preserve a precision similar to that of redundant forward intersections. We show that the deformations due to gravitational flexure and thermal expansion of the radio telescope cannot be neglected in the tie vector computation, since they may bias the CRP estimate by several millimeters degrading its accuracy but not impacting on its formal precision. We highlight the dependency of the correlation matrices of the solutions on the geometry of the network and the observation schemes. Similarly, varying the extent of telescope deformations, we show that the CRP estimate again depends on the combination of the network geometry and the observation schemes.     

Public participation applied to the environmental planning

This paper focuses on the public participation in environmental planning. After the decade for inaccessible information related to the decision taken, actually, the program of public parti,cipation is the reference of all the decision making process. However, there are some factors that limit this process, such as poverty, illiteracy, ignorance and often the social inequality. Therefore, this study focuses first on the benefits of public participation in environmental planning, then the involvement of the local population, and finally the decision making access using a case study of Madagascar.     

Public participation applied to the environmental planning

This paper focuses on the public participation in environmental planning. After the decade for inaccessible information related to the decision taken, actually, the program of public participation is the reference of all the decision making process. However, there are some factors that limit this process, such as poverty, illiteracy, ignorance and often the social inequality. Therefore, this study focuses first on the benefits of public participation in environmental planning, then the involvement of the local population, and finally the decision making access using a case study of Madagascar.     

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