共查询到20条相似文献，搜索用时 890 毫秒
1.
Klaus-Peter Schwarz 《Journal of Geodesy》1974,48(2):171-186
When combining satellite and terrestrial networks, covariance matrices are used which have been estimated from previous data.
It can be shown that the least-squares estimator of the unknown parameters using such an estimated covariance matrix is not
necessarily the best. There are a number of cases where a more efficient estimator can be obtained in a different way. The
problem occurs frequently in geodesy, since in least-squares adjustment of correlated observations estimated covariance matrices
are often used.
If the general structure of the covariance matrix is known, results can often be improved by a method called covariance adjustment.
The statistical model used in least-squares collocation leads to a type of covariance matrix which fits into this framework.
It is shown in which way improvements can be made using a modified approach of principal component analysis.
As a numerical example the combination of a satellite and a terrestrial network has been computed with varying assumptions
on the covariance matrix. It is shown which types of matrices are critical and where the usual least-squares approach can
be applied without hesitation. Finally, a simplified representation of covariances for spatial networks by means of a suitable
covariance function is suggested.
Paper presented at the International Symposium on Computational Methods in Geometrical Geodesy-Oxford, 2–8 September, 1973. 相似文献
2.
GPS Solutions - The true covariance matrix of the GPS phase observations is unknown and has to be assumed or estimated. The variance of the least-squares residuals was empirically shown to have an... 相似文献
3.
G. Fotopoulos 《Journal of Geodesy》2005,79(1-3):111-123
The well-known statistical tool of variance component estimation (VCE) is implemented in the combined least-squares (LS) adjustment of heterogeneous height data (ellipsoidal, orthometric and geoid), for the purpose of calibrating geoid error models. This general treatment of the stochastic model offers the flexibility of estimating more than one variance and/or covariance component to improve the covariance information. Specifically, the iterative minimum norm quadratic unbiased estimation (I-MINQUE) and the iterative almost unbiased estimation (I-AUE) schemes are implemented in case studies with observed height data from Switzerland and parts of Canada. The effect of correlation among measurements of the same height type and the role of the systematic effects and datum inconsistencies in the combined adjustment of ellipsoidal, geoid and orthometric heights on the estimated variance components are investigated in detail. Results give valuable insight into the usefulness of the VCE approach for calibrating geoid error models and the challenges encountered when implementing such a scheme in practice. In all cases, the estimated variance component corresponding to the geoid height data was less than or equal to 1, indicating an overall downscaling of the initial covariance (CV) matrix was necessary. It was also shown that overly optimistic CV matrices are obtained when diagonal-only cofactor matrices are implemented in the stochastic model for the observations. Finally, the divergence of the VCE solution and/or the computation of negative variance components provide insight into the selected parametric model effectiveness. 相似文献
4.
Parameter estimation in 3D affine and similarity transformation: implementation of variance component estimation 总被引：1，自引：0，他引：1
A. R. Amiri-Simkooei 《Journal of Geodesy》2018,92(11):1285-1297
Three-dimensional (3D) coordinate transformations, generally consisting of origin shifts, axes rotations, scale changes, and skew parameters, are widely used in many geomatics applications. Although in some geodetic applications simplified transformation models are used based on the assumption of small transformation parameters, in other fields of applications such parameters are indeed large. The algorithms of two recent papers on the weighted total least-squares (WTLS) problem are used for the 3D coordinate transformation. The methodology can be applied to the case when the transformation parameters are generally large of which no approximate values of the parameters are required. Direct linearization of the rotation and scale parameters is thus not required. The WTLS formulation is employed to take into consideration errors in both the start and target systems on the estimation of the transformation parameters. Two of the well-known 3D transformation methods, namely affine (12, 9, and 8 parameters) and similarity (7 and 6 parameters) transformations, can be handled using the WTLS theory subject to hard constraints. Because the method can be formulated by the standard least-squares theory with constraints, the covariance matrix of the transformation parameters can directly be provided. The above characteristics of the 3D coordinate transformation are implemented in the presence of different variance components, which are estimated using the least squares variance component estimation. In particular, the estimability of the variance components is investigated. The efficacy of the proposed formulation is verified on two real data sets. 相似文献
5.
Assessing receiver noise using GPS short baseline time series 总被引：14，自引：2，他引：14
6.
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
7.
Least-squares variance component estimation 总被引：19，自引：15，他引：4
Least-squares variance component estimation (LS-VCE) is a simple, flexible and attractive method for the estimation of unknown
variance and covariance components. LS-VCE is simple because it is based on the well-known principle of LS; it is flexible
because it works with a user-defined weight matrix; and it is attractive because it allows one to directly apply the existing
body of knowledge of LS theory. In this contribution, we present the LS-VCE method for different scenarios and explore its
various properties. The method is described for three classes of weight matrices: a general weight matrix, a weight matrix
from the unit weight matrix class; and a weight matrix derived from the class of elliptically contoured distributions. We
also compare the LS-VCE method with some of the existing VCE methods. Some of them are shown to be special cases of LS-VCE.
We also show how the existing body of knowledge of LS theory can be used to one’s advantage for studying various aspects of
VCE, such as the precision and estimability of VCE, the use of a-priori variance component information, and the problem of
nonlinear VCE. Finally, we show how the mean and the variance of the fixed effect estimator of the linear model are affected
by the results of LS-VCE. Various examples are given to illustrate the theory. 相似文献
8.
Estimability analysis of variance and covariance components 总被引：1，自引：1，他引：1
Although variance and covariance components have been extensively investigated and a number of elegant formulae to compute
them have been derived, nothing is known, without any ambiguity, about their estimability in the case of a fully unknown variance–covariance
matrix. We prove that variance and covariance components in this case are not estimable, thus clarifying the ambiguity of
the literature on the topic and correcting some erroneous statements in the literature. We also give a new theorem on the
estimability of a linear function of variance and covariance components. Then we propose a new method to estimate the variance–covariance
matrix with special structure, which can presumably be represented by, at most, r(r + 1)/2 independent parameters to guarantee its estimability in such a subspace, by directly implementing the positive definiteness
of the matrix as constraint to the restricted maximum likelihood method, where r is the number of redundant measurements. Therefore, our estimates of the variance and covariance components always reconstruct
a positive definite matrix and are always physically meaningful. 相似文献
9.
An iterative solution of weighted total least-squares adjustment 总被引：9，自引：0，他引：9
Total least-squares (TLS) adjustment is used to estimate the parameters in the errors-in-variables (EIV) model. However, its
exact solution is rather complicated, and the accuracies of estimated parameters are too difficult to analytically compute.
Since the EIV model is essentially a non-linear model, it can be solved according to the theory of non-linear least-squares
adjustment. In this contribution, we will propose an iterative method of weighted TLS (WTLS) adjustment to solve EIV model
based on Newton–Gauss approach of non-linear weighted least-squares (WLS) adjustment. Then the WLS solution to linearly approximated
EIV model is derived and its discrepancy is investigated by comparing with WTLS solution. In addition, a numerical method
is developed to compute the unbiased variance component estimate and the covariance matrix of the WTLS estimates. Finally,
the real and simulation experiments are implemented to demonstrate the performance and efficiency of the presented iterative
method and its linearly approximated version as well as the numerical method. The results show that the proposed iterative
method can obtain such good solution as WTLS solution of Schaffrin and Wieser (J Geod 82:415–421, 2008) and the presented numerical method can be reasonably applied to evaluate the accuracy of WTLS solution. 相似文献
10.
Spectral analysis by least squares as developed by Vaníček is applied to a series of transit times measurements obtained with
a suspended gyrocompass (Wild) electronically equipped with three photocells and a printing chronograph. Instead of being
the Fourier transform of the autocovariance function as in the usual spectral analysis of time series (Wiener theory), the
spectral function used here is a function of an estimator of the variance factor obtained after a least squares fitting of
a sinusoid to the data. That function is normalized to values between zero and one. For step-by-step spectral analysis by
least squares each time a significant frequency appears in the spectrum it is removed by least squares fitting of the corresponding
sinusoid including a damping coefficient, the residuals being again examined by spectral analysis by least squares. We find
four significant frequencies: the well known principal period of about 7min in the spinning case; a very strong component with a period nearly exactly half the principal period and an amplitude of
about 70″, explained by taking into account the second-order term in the theory developed by Jeudy, and two remaining periods
with much smaller amplitudes (2″.9 and 0″.9). It is shown that the shortest period (0s.021), predicted by theory, exists in the measurements and cannot be neglected. The smallest component is considered to correspond
to the wobble which can easily be observed in the perturbed motion. 相似文献
11.
In this contribution, using the example of the Mátern covariance matrices, we study systematically the effect of apriori fully populated variance covariance matrices (VCM) in the Gauss–Markov model, by varying both the smoothness and the correlation length of the covariance function. Based on simulations where we consider a GPS relative positioning scenario with double differences, the true VCM is exactly known. Thus, an accurate study of parameters deviations with respect to the correlation structure is possible. By means of the mean-square error difference of the estimates obtained with the correct and the assumed VCM, the loss of efficiency when the correlation structure is missspecified is considered. The bias of the variance of unit weight is moreover analysed. By acting independently on the correlation length, the smoothness, the batch length, the noise level, or the design matrix, simulations allow to draw conclusions on the influence of these different factors on the least-squares results. Thanks to an adapted version of the Kermarrec–Schön model, fully populated VCM for GPS phase observations are computed where different correlation factors are resumed in a global covariance model with an elevation dependent weighting. Based on the data of the EPN network, two studies for different baseline lengths validate the conclusions of the simulations on the influence of the Mátern covariance parameters. A precise insight into the impact of apriori correlation structures when the VCM is entirely unknown highlights that both the correlation length and the smoothness defined in the Mátern model are important to get a lower loss of efficiency as well as a better estimation of the variance of unit weight. Consecutively, correlations, if present, should not be neglected for accurate test statistics. Therefore, a proposal is made to determine a mean value of the correlation structure based on a rough estimation of the Mátern parameters via maximum likelihood estimation for some chosen time series of observations. Variations around these mean values show to have little impact on the least-squares results. At the estimates level, the effect of varying the parameters of the fully populated VCM around these approximated values was confirmed to be nearly negligible (i.e. a mm level for strong correlations and a submm level otherwise). 相似文献
12.
H. Kutterer 《Journal of Geodesy》1999,73(7):350-361
A proper perturbation theory of a mathematical model and the quantities derived by means of least-squares adjustments is
indispensable if the results have to be interpreted in a wider context. The sensitivity of some characteristic results of
least-squares adjustments such as the estimated values of the parameters and their variance–covariance matrix due to imminent
uncertainties of the stochastic model is discussed in detail. Linearizations are used with rigorous error measures and interval
mathematics. Numerical examples conclude the investigations.
Received: 27 December 1997 / Accepted: 19 April 1999 相似文献
13.
The second-order derivatives of the Earth’s potential in the local north-oriented reference frame are expanded in series of
modified spherical harmonics. Linear relations are derived between the spectral coefficients of these series and the spectrum
of the geopotential. On the basis of these relations, recurrence procedures are developed for evaluating the geopotential
coefficients from the spectrum of each derivative and, inversely, for simulating the latter from a known geopotential model.
Very simple structure of the derived expressions for the derivatives is convenient for estimating the geopotential coefficients
by the least-squares procedure, at a certain step of processing satellite gradiometry data. Due to the orthogonality of the
new series, the quadrature formula approach can be also applied, which allows avoidance of aliasing errors caused by the series
truncation. The spectral coefficients of the derivatives are evaluated on the basis of the derived relations from the geopotential
models EGM96 and EIGEN-CG01C at a mean orbital sphere of the GOCE satellite. Various characteristics of the spectra are studied
corresponding to the EGM96 model.
Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users. 相似文献
14.
Application of least squares variance component estimation to errors-in-variables models 总被引：3，自引：2，他引：1
A. R. Amiri-Simkooei 《Journal of Geodesy》2013,87(10-12):935-944
15.
Noise in multivariate GPS position time-series 总被引：4，自引：2，他引：2
A. R. Amiri-Simkooei 《Journal of Geodesy》2009,83(2):175-187
A methodology is developed to analyze a multivariate linear model, which occurs in many geodetic and geophysical applications.
Proper analysis of multivariate GPS coordinate time-series is considered to be an application. General, special, and more
practical stochastic models are adopted to assess the noise characteristics of multivariate time-series. The least-squares
variance component estimation (LS-VCE) is applied to estimate full covariance matrices among different series. For the special
model, it is shown that the multivariate time-series can be estimated separately, and that the (cross) correlation between
series propagates directly into the correlation between the corresponding parameters in the functional model. The time-series
of five permanent GPS stations are used to show how the correlation between series propagates into the site velocities. The
results subsequently conclude that the general model is close to the more practical model, for which an iterative algorithm
is presented. The results also indicate that the correlation between series of different coordinate components per station
is not significant. However, the spatial correlation between different stations for individual components is significant (a
correlation of 0.9 over short baselines) both for white and for colored noise components. 相似文献
16.
Statistical modeling for the mitigation of GPS multipath delays from day-to-day range measurements 总被引：1，自引：0，他引：1
Based on a least-squares model for double-difference GPS pseudoranges and carrier-phases, measurement residuals expressed
in time series during an observation session are positively correlated between one sidereal day and the preceding days. As
a result of the satellite’s period, the phenomenon, which takes place at a user receiving site, is attributed to multipath
interference. Examples from a weekly measurement dataset of control baselines are shown, where the known end-point coordinates
also serve as a benchmark for assessing positioning accuracy. The system of error equations for mixed-model adjustment is
divided into two subsystems. One set of the error equations is related to the real range measurements, while the other involves
the pseudo-observation with an empirical sample variance. According to the existing correlation between day-to-day residual
estimates, a multipath-mitigating algorithm is proven to improve the accuracy of the GPS height determination by at least
40%. It is also found that the algorithm depends on a variance-component estimator that adaptively scales an error covariance
matrix for both the real range and empirical delay measurements. 相似文献
17.
18.
基于等价条件闭合差的VCE通用解析法 总被引：1，自引：1，他引：0
分析指出了现有方差-协方差分量估计(VCE)方法在计算效率与?2统计量统计性质两方面的固有缺陷。利用零空间算子消去概括平差模型中的参数向量,建立了等价条件平差模型。由此定义了等价条件闭合差(ECC),并导出了以ECC表示的?2统计量计算式。进而,基于等价条件闭合差与新构造的可逆方差分量模型提出了方差-协方差分量估计的通用解析法,简称为VCE-ECC法。同时,给出了对应四种基本平差模型的VCE-ECC法简化计算式。实例与仿真结果表明：VCE-ECC法与现有VCE方法的方差-协方差分量估计值在统计意义上无明显差异,并有效地克服了现有VCE方法的固有缺陷。 相似文献
19.
Although total least squares (TLS) is more rigorous than the weighted least squares (LS) method to estimate the parameters in an errors-in-variables (EIV) model, it is computationally much more complicated than the weighted LS method. For some EIV problems, the TLS and weighted LS methods have been shown to produce practically negligible differences in the estimated parameters. To understand under what conditions we can safely use the usual weighted LS method, we systematically investigate the effects of the random errors of the design matrix on weighted LS adjustment. We derive the effects of EIV on the estimated quantities of geodetic interest, in particular, the model parameters, the variance–covariance matrix of the estimated parameters and the variance of unit weight. By simplifying our bias formulae, we can readily show that the corresponding statistical results obtained by Hodges and Moore (Appl Stat 21:185–195, 1972) and Davies and Hutton (Biometrika 62:383–391, 1975) are actually the special cases of our study. The theoretical analysis of bias has shown that the effect of random matrix on adjustment depends on the design matrix itself, the variance–covariance matrix of its elements and the model parameters. Using the derived formulae of bias, we can remove the effect of the random matrix from the weighted LS estimate and accordingly obtain the bias-corrected weighted LS estimate for the EIV model. We derive the bias of the weighted LS estimate of the variance of unit weight. The random errors of the design matrix can significantly affect the weighted LS estimate of the variance of unit weight. The theoretical analysis successfully explains all the anomalously large estimates of the variance of unit weight reported in the geodetic literature. We propose bias-corrected estimates for the variance of unit weight. Finally, we analyze two examples of coordinate transformation and climate change, which have shown that the bias-corrected weighted LS method can perform numerically as well as the weighted TLS method. 相似文献
20.
Graph theory is useful for analyzing time-dependent model parameters estimated from interferometric synthetic aperture radar (InSAR) data in the temporal domain. Plotting acquisition dates (epochs) as vertices and pair-wise interferometric combinations as edges defines an incidence graph. The edge-vertex incidence matrix and the normalized edge Laplacian matrix are factors in the covariance matrix for the pair-wise data. Using empirical measures of residual scatter in the pair-wise observations, we estimate the relative variance at each epoch by inverting the covariance of the pair-wise data. We evaluate the rank deficiency of the corresponding least-squares problem via the edge-vertex incidence matrix. We implement our method in a MATLAB software package called GraphTreeTA available on GitHub (https://github.com/feigl/gipht). We apply temporal adjustment to the data set described in Lu et al. (Geophys Res Solid Earth 110, 2005) at Okmok volcano, Alaska, which erupted most recently in 1997 and 2008. The data set contains 44 differential volumetric changes and uncertainties estimated from interferograms between 1997 and 2004. Estimates show that approximately half of the magma volume lost during the 1997 eruption was recovered by the summer of 2003. Between June 2002 and September 2003, the estimated rate of volumetric increase is \((6.2 \, \pm \, 0.6) \times 10^6~\mathrm{m}^3/\mathrm{year} \). Our preferred model provides a reasonable fit that is compatible with viscoelastic relaxation in the five years following the 1997 eruption. Although we demonstrate the approach using volumetric rates of change, our formulation in terms of incidence graphs applies to any quantity derived from pair-wise differences, such as range change, range gradient, or atmospheric delay. 相似文献