Generalized maximum-likelihood estimation of variance–covariance components with non-informative prior

 A new estimator of variance–covariance components is presented. The proposed estimator is derived by applying the principle of maximum-likelihood estimation to the posterior probability density function for the case when no prior information is available. Received: 9 August 1999 / Accepted: 10 November 2000     

Estimability analysis of variance and covariance components

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.     

Atmospheric turbulence theory applied to GPS carrier-phase data

Turbulent irregularities in the lower atmosphere cause physical correlations between Global Positioning System (GPS) carrier-phase measurements. Based on turbulence theory, a variance–covariance model is developed in this paper that reflects these correlations. The main result shows that the obtained fully-populated variance–covariance matrices depend not only on the satellite-station geometry, but also on the prevailing atmospheric conditions, which are parameterised by, e.g., the von Karman spectrum of refractivity fluctuations and the wind velocity vector. It is shown that the amount of the correlation between two GPS carrier-phase observations is inversely related to the separation distance of the corresponding ray paths through the turbulent atmosphere. Furthermore, the wind velocity and direction play a key role in the correlation.     

Efficient propagation of error covariance matrices of gravitational models: application to GRACE and GOCE

We have applied efficient methods for computing variances and covariances of functions of a global gravity field model expanded in spherical harmonics, using the full variance–covariance matrix of the coefficients. Examples are given with recent models derived from GRACE (up to degree and order 150), and with simulated GOCE derived solutions (up to degree and order 200).     

Spectral accuracy of Jgm3 from satellite crossover altimetry

A detailed accuracy assessment of the geopotential model Jgm3 is made based on independent single- and dual-satellite sea-height differences at crossovers from altimetry with Jgm3-based orbits. These differences, averaged over long time spans and in latitude bands, are converted to spectra (latitude-lumped coefficients) by least-squares estimation. The observed error spectra so obtained are then compared directly to error projections for them from the Jgm3 variance–covariance matrix. It is found from these comparisons that Jgm3 is generally well calibrated with respect to the crossover altimetry of and between Geosat, TOPEX/Poseidon (T/P), and Ers 1. Some significant discrepancies at a few lower orders (namely m=1 and 3) indicate a need for further improvement of Jgm3. A companion calibration (by order) of the geopotential model Jgm2 shows its variance–covariance matrix also to be generally well calibrated for the same single- and dual-satellite altimeter data sets (but based on Jgm2 orbits), except that the error projections for Geosat are too pessimistic. The analysis of the dual-satellite crossovers reveals possible relative coordinate system offsets (particularly for Geosat with respect to T/P) which have been discussed previously. The long-term detailed seasonally averaged Geosat sea level with respect to T/P (covering 1985–1996) should be useful in gauging the relative change in sea level between different parts of the ocean over the single 4-year gap between these missions (1988–1992). Received: 16 February 1998 / Accepted: 25 November 1998     

On the sensitivity of the results of least-squares adjustments concerning the stochastic model

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     

Nonlinear analysis of the three-dimensional datum transformation [conformal group ℂ7(3)]

 The weighted Procrustes algorithm is presented as a very effective tool for solving the three-dimensional datum transformation problem. In particular, the weighted Procrustes algorithm does not require any initial datum parameters for linearization or any iteration procedure. As a closed-form algorithm it only requires the values of Cartesian coordinates in both systems of reference. Where there is some prior information about the variance–covariance matrix of the two sets of Cartesian coordinates, also called pseudo-observations, the weighted Procrustes algorithm is able to incorporate such a quality property of the input data by means of a proper choice of weight matrix. Such a choice is based on a properly designed criterion matrix which is discussed in detail. Thanks to the weighted Procrustes algorithm, the problem of incorporating the stochasticity measures of both systems of coordinates involved in the seven parameter datum transformation problem [conformal group ℂ₇(3)] which is free of linearization and any iterative procedure can be considered to be solved. Illustrative examples are given. Received: 7 January 2002 / Accepted: 9 September 2002 Correspondence to: E. W. Grafarend     

Adaptive Kalman Filtering for INS/GPS

After reviewing the two main approaches of adaptive Kalman filtering, namely, innovation-based adaptive estimation (IAE) and multiple-model-based adaptive estimation (MMAE), the detailed development of an innovation-based adaptive Kalman filter for an integrated inertial navigation system/global positioning system (INS/GPS) is given. The developed adaptive Kalman filter is based on the maximum likelihood criterion for the proper choice of the filter weight and hence the filter gain factors. Results from two kinematic field tests in which the INS/GPS was compared to highly precise reference data are presented. Results show that the adaptive Kalman filter outperforms the conventional Kalman filter by tuning either the system noise variance–covariance (V–C) matrix `Q' or the update measurement noise V–C matrix `R' or both of them. Received: 14 September 1998 / Accepted: 21 December 1998     

Simplified formulae for the BIQUE estimation of variance components in disjunctive observation groups

 General rigorous and simplified formulae are reported for the best invariant quadratic unbiased estimates of the variance–covariance components, which can be applied to all least-squares adjustments with the general linear stochastic model. Simplified procedures are given for two cases frequently recurring in geodetic applications: uncorrelated groups of correlated or uncorrelated observations, with more than one variance component in each group. Received: 19 November 1998 / Accepted: 21 March 2000     

Rigorous transformation of variance–covariance matrices of GPS-derived coordinates and velocities

With the advances in the field of GPS positioning and the global densification of permanent GPS tracking stations, it is now possible to determine at the highest level of accuracy the transformation parameters connecting various international terrestrial reference frame (ITRF) realizations. As a by-product of these refinements, not only the seven usual parameters of the similarity transformations between frames are available, but also their rates, all given at some epoch t _k . This paper introduces rigorous matrix equations to estimate variance–covariance matrices for transformed coordinates at any epoch t based on a stochastic model that takes into consideration all a priori information of the parameters involved at epoch t _k , and the coordinates and velocities at the reference frame initial epoch t ₀. The results of this investigation suggest that in order to attain maximum accuracy, the agencies determining the 14-parameter transformations between reference frames should also publish their full variance–covariance matrix. Electronic Publication     

GPS vector configuration design for monitoring deformation networks

 The performance of geodetic monitoring networks is heavily influenced by the configuration of the measured GPS vectors. As an effective design of the GPS measurements will decrease GPS campaign costs and increase the accuracy and reliability of the entire network, the identification of the preferred GPS vectors for measurement has been highlighted as a core problem in the process of deformation monitoring. An algorithm based on a sensitivity analysis of the network, as dependent upon a postulated velocity field, is suggested for the selection of the optimal GPS vectors. Relevant mathematical and statistical concepts are presented as the basis for an improved method of vector configuration design. A sensitivity analysis of the geodetic geodynamic network in the north of Israel is presented, where the method is examined against two deformation models, the Simple Transform Fault and the Locked Fault. The proposed method is suggested as a means for the improvement of the design of monitoring networks, a common practice worldwide. Received: 30 July 2001 / Accepted: 3 June 2002 Acknowledgments. It is my pleasant duty to thank the Survey of Israel and Dr. E. Ostrovsky for providing the variance–covariance matrix of the G1 network in northern Israel. I would like to thank the reviewers of this paper for their constructive and helpful remarks.     

Efficient satellite orbit modelling using pseudo-stochastic parameters

If the force field acting on an artificial Earth satellite is not known a priori with sufficient accuracy to represent its observations on their accuracy level, one may introduce so-called pseudo-stochastic parameters into an orbit determination process, e.g. instantaneous velocity changes at user-defined epochs or piecewise constant accelerations in user-defined adjacent time subintervals or piecewise linear and continuous accelerations in adjacent time subintervals. The procedures, based on standard least-squares, associated with such parameterizations are well established, but they become inefficient (slow) if the number of pseudo-stochastic parameters becomes large. We develop two efficient methods to solve the orbit determination problem in the presence of pseudo-stochastic parameters. The results of the methods are identical to those obtained with conventional least-squares algorithms. The first efficient algorithm also provides the full variance–covariance matrix; the second, even more efficient algorithm, only parts of it.     

Biases and accuracy of, and an alternative to, discrete nonlinear filters

The biases and accuracy of the extended Kalman filter (EKF) and a second-order nonlinear filter (SONF) are discussed from the point of view of a frequentist; these are often derived by applying the relevant conditional quantities to the linear Kalman algorithm under the Bayesian framework. The EKF and the SONF are biased, although the SONF has been derived in the hope of improving first-order filters. Unfortunately the biases of the SONF may be magnified further, because the second-order terms of the relevant Bayesian conditional quantities have never been properly used to derive the SONF from the frequentist point of view. The variance–covariance matrix of the SONF given in the literature is proven to be incorrect up to the second-order approximation, and the correct one is derived. Finally, also from the point of view of a frequentist, an alternative, almost unbiased SONF is proposed, if the randomness of partials is neglected. Received: 12 July 1997 / Accepted: 5 October 1998     

A systematic investigation of optimal carrier-phase combinations for modernized triple-frequency GPS

The upcoming modernization of the GPS signals will allow for measurements on an additional third frequency L5 located at 1176.45 MHz. To take advantage of carrier-phase measurements on this new signal, the strategies for integer ambiguity resolution, required for centimeter-level accuracy, may need to be revised. The Least-squares Ambiguity Decorrelation Adjustment method remains perhaps the most powerful tool for finding the best combinations based on a complete decorrelation of the variance–covariance matrix related to the ambiguities. However, the computational load of that method plus the opportunity to comprehensively study the interaction of multiple frequencies suggest a reconsideration of approaches using predefined combinations between frequencies is not out of place. In this paper a systematic investigation is made of all possible triple-frequency geometry-free carrier-phase combinations which retain the integer nature of the ambiguities. The concept of the lane-number is presented to unambiguously describe the wavelength of a particular combination. The propagation of the observation noise and of the ionospheric bias on these combinations is presented. These noise and ionospheric amplification factors are analysed with respect to the resulting wavelength, in an effort to highlight optimal combinations characterized by a long wavelength, low noise and limited ionospheric impact.     

A constrained LAMBDA method for GPS attitude determination

An improved method to obtain fixed integer ambiguity in GPS attitude determination is presented. Known conditions are utilized as constraints to acquire attitude information when the float solution and its variance–covariance matrix are not accurate enough. The searching ellipsoidal region is first expanded to compensate for errors caused by the inaccurate float solution. Then the constraints are used to shrink the region to a proper size, which maintains the true integer ambiguity. Experimental results demonstrate that this scheme gives a fast search time and a higher success rate in determining the fixed integer ambiguity than the unconstrained method. The accuracy of attitude angles is also improved.

ISO2002: an analytical stochastic model of multi-difference GPS carrier-phase data

The differencing technique is useful in global positioning system (GPS) positioning when two or more GPS receivers collect simultaneous observables from common satellites at each epoch, and all carrier-phase observables have the same normal distribution. An analytical probability distribution of the single-, double-, triple- and multi-difference GPS observables is obtained. This analytical model, called ISO2002, has a good matrix structure, in which I indicates the number of receivers, S indicates the number of observed satellites, and O indicates the number of epochs. The variance–covariance matrix can be expressed as the Kronecker product of several small matrices, so its inverse is equal to the Kronecker product of the inverses of these sub-matrices. Moreover, these small matrices are circulant or symmetric diagonal Toeplitz matrices, so their inverses have analytical solutions. The analytical model ISO2002 proposed to compute the inverse variance–covariance matrix is shown to be very effective.     

Combination of spatial networks using an estimated covariance matrix

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.     

Multi-level arc combination with stochastic parameters

 The method of square root information filtering and smoothing (SRIF/S) is reviewed and has been implemented in the combined square root information filter and smoother (CSRIFS) program. CSRIFS is a part of the GEOSAT space geodesy software developed at Forsvarets forskningsinstitutt (FFI, The Norwegian Defence Research Establishment). The state vectors and complete variance–covariance matrices from the analyses of a number of independent arcs of space geodesy data can be combined using CSRIFS. Four parameter levels are available and any parameter can, at each level, be represented as either a constant or a stochastic parameter (white noise, colored noise, or random walk). The batch length (i.e. the time interval between the addition of noise to the SRIF array) can be made time and parameter dependent. CSRIFS was applied in the combination of 623 very long baseline interferometry (VLBI) observing sessions. The purpose of this test was to validate the computer implementation of the SRIF/S method and to give an example of how this method can be used in the analysis of a large number of space geodetic observations. The results show that the implementation is very satisfactory. Received: 28 May 1999 / Accepted: 15 June 2000