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.

/pdf/least-squares-variance-component-estimation-45k4p2blaz.pdf

Least-squares variance component estimation

We propose a methodology to assess the noise characteristics in time series of position estimates for permanent Global Positioning System (GPS) stations. Least squares variance component estimation (LS?VCE) is adopted to cope with any type of noise in the data. LS?VCE inherently provides the precision of (co)variance estimators. One can also apply statistical hypothesis testing in conjunction with LS?VCE. Using the w?test statistic, a combination of white noise and flicker noise turns out in general to best characterize the noise in all three position components. An interpretation for the colored noise of the series is given. Unmodelled periodic effects in the data will be captured by a set of harmonic functions for which we rely on the least squares harmonic estimation (LS?HE) method and parameter significance testing developed in the same framework as LS?VCE. Having included harmonic functions into the model, practically only white noise can be shown to remain in the data. Remaining time correlation, present only at very high frequencies (spanning a few days only), is expressed as a first?order autoregressive noise process. It can be caused by common and well?known sources of errors like atmospheric effects as well as satellite orbit errors. The autoregressive noise should be included in the stochastic model to avoid the overestimation (upward bias) of power law noise. The results confirm the presence of annual and semiannual signals in the series. We observed also significant periodic patterns with periods of 350 days and its fractions 350/n, n = 2, , 8 that resemble the repeat time of the GPS constellation. Neglecting these harmonic signals in the functional model can seriously overestimate the rate uncertainty.

/pdf/assessment-of-noise-in-gps-coordinate-time-series-cfr9bicdzk.pdf

Assessment of noise in GPS coordinate time series : Methodology and results

Global positioning system (GPS) carrier phase measurements are used in all precise static relative positioning applications. The GPS carrier phase measurements are generally processed using the least-squares method, for which both functional and stochastic models need to be carefully defined. Whilst the functional model for precise GPS positioning is well documented in the literature, realistic stochastic modelling for the GPS carrier phase measurements is still both a controversial topic and a difficult task to accomplish in practice. The common practice of assuming that the raw GPS measurements are statistically independent in space and time, and have the same accuracy, is certainly not realistic. Any mis-specification in the stochastic model will inevitably lead to unreliable positioning results. A stochastic assessment procedure has been developed to take into account the heteroscedastic, space- and time-correlated error structure of the GPS measurements. Test results indicate that the reliability of the estimated positioning results is improved by applying the developed stochastic assessment procedure. In addition, the quality of ambiguity resolution can be more realistically evaluated.

Stochastic assessment of GPS carrier phase measurements for precise static relative positioning

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.

/pdf/noise-in-multivariate-gps-position-time-series-529o89dxy5.pdf

Noise in multivariate gps position time-series

This contribution can be seen as an attempt to apply a rigorous method for variance components in a straightforward manner directly to GPS observables. Least-squares variance component estimation is adopted to assess the noise characteristics of GPS observables using the geometry-free observation model. The method can be applied to GPS observables or GNSS observables in general, even when the navigation message is not available. A realistic stochastic model of GPS observables takes into account the individual variances of different observation types, the satellite elevation dependence of GPS observables precision, the correlation between different observation types, and the time correlation of the observables. The mathematical formulation of all such issues is presented. The numerical evidence, obtained from real GPS data, consequently concludes that these are important issues in order to properly construct the covariance matrix of the GPS observables. Satellite elevation dependence of variance is found t...

Application of Least-Squares Variance Component Estimation to GPS Observables

Due to the steady progress in global positioning system (GPS) technology and methods of data evaluation, it is possible to obtain highly precise relative point positions also for extensive geodetic networks. However, some limiting influences such as temporal correlations of observational data are neglected in most of the GPS processing programs. Therefore, it is necessary to consider the impact of these neglected correlations on the coordinates and their accuracy measures. In this paper the sensitivity and reliability of GPS positioning over long baselines when temporal correlations are modelled with the aid of a correlation function are discussed. The implementation in the variance–covariance matrix and the subsequent evaluation process require a considerable amount of computing time and memory. Therefore it is necessary to use appropriate numerical methods such as approximated matrix inversion in order to reduce the numerical requirements. After the description of the methodical and numerical handling of the temporal correlations, exemplary results are given to illustrate the effects on position and accuracy of GPS stations.

Impact of temporal correlations on GPS-derived relative point positions

Due to increased demands on the quality of the results of Global Positioning System (GPS) evaluations, various authors have studied improvements of the stochastic model of GPS carrier-phase observations. These improvements are based on the reasonable assumption that the commonly used stochastic model with independent and homoscedastic (i.e. equal variance) errors is unrealistic. However, this has not been proved rigorously so far. A statistical test procedure based on uncorrelated least–squares residuals, which allows verification of the hypothesis of a heterogeneous variance, is provided. The statistical test procedure is of interest in its own right, and is independent of the practical problem considered. The presented technique is applied to GPS carrier-phase observations. Results show that the variances of the investigated observations are far from homogeneous. It is indicated that the error variances of the presented data increase with decreasing GPS satellite elevation. These results confirm the assumption that the commonly used stochastic model of GPS observations is inadequate and has to be improved.

A procedure for testing the assumption of homoscedasticity in least squares residuals: 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.

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

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Impact of temporal correlations on GPS-derived relative point positions

A procedure for testing the assumption of homoscedasticity in least squares residuals: a case study of GPS carrier-phase observations

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