Optimum estimation in a growth curve model with a priori unknown variance components in geodetic networks

Summary Time varying coordinates of points of a geodetic network are indirectly measured by a group of measurement devices with different characteristics of accuracy in several epochs. The design of the measurement is the same in all epochs. The ratio of the characteristics of accuracy is a priori unknown. The aim is to determine an estimator of the parameters of functions modelling changes of the coordinates, confidence regions of these functions and to construct a procedure for testing linear hypotheses on time varying coordinates of a geodetic network. As the characteristics of accuracy are a priori unknown, the problem of their estimation has to be solved simultaneously. The research of rules of recent crustal movements leads to studying the mentioned model.     

Robust estimation of geodetic datum transformation

The robust estimation of geodetic datum transformation is discussed. The basic principle of robust estimation is introduced. The error influence functions of the robust estimators, together with those of least-squares estimators, are given. Particular attention is given to the robust initial estimates of the transformation parameters, which should have a high breakdown point in order to provide reliable residuals for the following estimation. The median method is applied to solve for robust initial estimates of transformation parameters since it has the highest breakdown point. A smooth weight function is then used to improve the efficiency of the parameter estimates in successive iterative computations. A numerical example is given on a datum transformation between a global positioning system network and the corresponding geodetic network in China. The results show that when the coordinates are contaminated by outliers, the proposed method can still give reasonable results. Received: 25 September 1997 / Accepted: 1 March 1999     

Strength analysis of geodetic control networks

In strength analysis of horizontal geodetic networks it is appropriate to use pairs of functions which involve the relative position of two points and relative position of three points. Using properly chosen pairs of functions, formulae are given which allow the computation of precision criteria for the orientation and scale of the network as well as its shape. To illustrate the presentation of results, new types of errors ellipses are introduced. Analogies and correlations existing among the adopted functions are introduced by using the concept of orthogonal networks which are defined in the paper.     

Second-order design of combined linear-angular geodetic networks

The paper deals with the solution of the weight problem for linear-angular networks on the base of criterion matrices. The observation plan of combined linear-angular networks consists of distances as well as of directions in any form. In particular the role of criterion matrices of completely isotropic structure for this type of two-dimensional network is discussed. Starting from an extreme network design, the results of a least-squares approximation are described analytically. In this case, the observation plan contains all geometrical point connections. Finally, the considerations are completed by giving as an example the second-order design of a control network for a dam of a water reservoir.     

A modified Wiener-type filter for geodetic estimation problems with non-stationary noise

 One of the most basic and important tools in optimal spectral gravity field modelling is the method of Wiener filtering. Originally developed for applications in analogue signal analysis and communication engineering, Wiener filtering has become a standard linear estimation technique of modern operational geodesy, either as an independent practical tool for data de-noising in the frequency domain or as an integral component of a more general signal estimation methodology (input–output systems theory). Its theoretical framework is based on the Wiener–Kolmogorov linear prediction theory for stationary random fields in the presence of additive external noise, and thus it is closely related to the (more familiar to geodesists) method of least-squares collocation with random observation errors. The main drawback of Wiener filtering that makes its use in many geodetic applications problematic stems from the stationarity assumption for both the signal and the noise involved in the approximation problem. A modified Wiener-type linear estimation filter is introduced that can be used with noisy data obtained from an arbitrary deterministic field under the masking of non-stationary random observation errors. In addition, the sampling resolution of the input data is explicitly taken into account within the estimation algorithm, resulting in a resolution-dependent optimal noise filter. This provides a more insightful approach to spectral filtering techniques for noise reduction, since the data resolution parameter has not been directly incorporated in previous formulations of frequency-domain estimation problems for gravity field signals with discrete noisy data. Received: 1 November 2000 / Accepted: 19 June 2001     

First-order design of geodetic networks using the simulated annealing method

The general problem of the optimal design for a geodetic network subject to any extrinsic factors, namely the first-order design problem, can be dealt with as a numeric optimization problem. The classic theory of this problem and the optimization methods are revised. Then the innovative use of the simulated annealing method, which has been successfully applied in other fields, is presented for this classical geodetic problem. This method, belonging to iterative heuristic techniques in operational research, uses a thermodynamical analogy to crystalline networks to offer a solution that converges probabilistically to the global optimum. Basic formulation and some examples are studied.     

Measuring the robustness potential of the least-squares estimation: geodetic illustration

For a linear least-squares parametric model analysis is carried out of the structure of the projection operator transforming the vector of standardised observations into the vector of standardised residuals. On this basis the properties of the model responses to observational disturbances (i.e. gross errors or blunders) are derived. A final outcome of the research can be summarised as: (1) proposing the robustness characteristics of a model and linking them with the local measures of internal reliability, being the diagonal elements in the projection operator; (2) determining the internal reliability levels satisfying specified robustness requirements, i.e. the possibility of detecting at least one of the k observational disturbances (k=1,2,…) having most disadvantageous locations in the system. The theory and a numerical example show that for the systems which have been designed to a proper level of internal reliability, the least-squares estimation can demonstrate an accordingly high level of robustness. Received: 11 June 1996 / Accepted: 28 April 1997     

On the gravimetric inverse problem in geodetic gravity field estimation

Gravity field estimation in geodesy, through linear(ized) least squares algorithms, operates under the assumption of Gaussian statistics for the estimable part of preselected models. The causal nature of the gravity field is implicitly involved in its geodetic estimation and introduces the need to include prior model information, as in geophysical inverse problems. Within the geodetic concept of stochastic estimation, the prior information can be in linear form only, meaning that only data linearly depending on the estimates can be used effectively. The consequences of the inverse gravimetric problem in geodetic gravity field estimation are discussed in the context of the various approaches (in model data spaces) which have the common goal to bring into agreement the statistics between these two spaces. With a simple numerical example of FAA prediction, it is shown that prior information affects the accuracy of estimates at least equally as the number of input data. Received: 25 April 1994; Accepted: 15 October 1996     

Strength of long lines in terrestrial geodetic control networks

a knowledge of the à posteriori accuracy of long lines in classical triangulation networks is important not only for establishing the quality of existing primary national control networks, but also for assessing the likely contribution of satellite derived observations. Extensive tests carried out with various networks, ranging from a basic triangulation chain to Block VI of the European Triangulation Network, show definite trends. The à posteriori standard errors of both the scale and the orientation of an adjusted line diminish considerably as the length of the line increases. In the case of medium sized countries, the results compare favourably with those predicted for satellite derived data. The conclusions emphasize the need for great care in the reduction of terrestrial observations and the elimination of all possible systematic errors before proceeding with a least squares adjustment.     

Deformations of geodetic networks in gravity-invariant and potential-invariant representations

The analysis of deformations and reductions of the geodetic networks in general gravity-invariant and potential-invariant representations of the actual gravity field of the Earth by normal (theoretical) gravity field has been presented. It has been shown that the linearized formulae of the scale factor and linearized formulae of the transformed azimuth and zenith distance induce the conformal character of transformation of three-dimensional networks in isozenithal-type of gravity-invariant representation. In isozenithal-type of gravity-invariant and potential-invariant representations the two-dimensional (horizontal) networks are transformed conformally. In isoparametric cases of gravity-invariant and potential-invariant representations, the two-dimensional networks are transformed equivalently with respect to the scale and angles, but non-conformally.     

Rigorous mathematical models for the densification and integration of geodetic networks

Existing position information in a network can be integrated with the densification solution in two ways: One way is to obtain a solution of the densification network followed by a merger of this and all other solutions or vice versa. Alternatively, the existing solutions (not used as weighted constraints) can be taken to be pseudo-observations in a simultaneous adjustment with the “new” observations. In both cases, all existing solutions must first be transformed to the coordinate system of the densified network and be statistically compatible with it. Simultaneous densification and integration is discussed through mathematical adjustment models in which the geometrical strength of networks is underscored. The rationale behind densifying and integrating networks either in two different steps or simultaneously is analyzed. It is concluded that the simultaneous approach should be avoided unless the various solutions turn out to be statistically compatible.     

Consistent estimation of geodetic parameters from SLR satellite constellation measurements

In this paper, we consistently estimate geodetic parameters such as weekly 3-D station coordinates, Earth orientation parameters (EOP) including daily x/y-pole coordinates and the excess length of day \(\Delta \hbox {LOD}\), and selected weekly Earth’s gravitational field (Stokes) coefficients up to degree and order 6 from Satellite Laser Ranging measurements to up to 11 geodetic satellites. The SLR constellation consists of LAGEOS-1/2, Etalon-1/2, Stella, Starlette, Ajisai, Larets, LARES, BLITS and WESTPAC, and its observations cover a time span of 38 years ranging from February 16, 1979, to April 30, 2017. If multiple satellites with various altitudes and orbit inclinations are combined, correlations between estimated parameters are significantly reduced. This allows us (i) to investigate the ability of satellite constellations to reduce existing correlations and (ii) to estimate reliable parameters with higher precision compared to the standard 4-satellite constellation (LAGEOS-1/2, Etalon-1/2) which is currently used by the International Laser Ranging Service for the determination of the Terrestrial Reference Frame (TRF) and EOP products. In particular, the Stokes coefficients, EOP and TRF datum parameters (three translations, three rotations, one scale factor), which are highly correlated with satellite-specific orbit parameters, are improved. From our investigations, we found for an 11-satellite solution compared to the above-mentioned 4-satellite solution a decrease in the scatter of the TRF datum parameters of up to 37%, the transformation residuals are decreased by up to 22%, the scatter of the EOP is decreased by up to 22%, and their mean values are decreased by up to 84% w.r.t. the reference solutions. The largest improvement is obtained for the Stokes coefficients which significantly benefit from a combination of multiple satellites (inclinations and orbit altitudes). In total, single coefficients are improved by up to 93% and the overall improvement is up to 74%. Moreover, it could be clearly identified that Ajisai significantly disturbs the TRF solution due to an erroneous center-of-mass correction. We further quantify the impact of specific satellites on the determination of different geodetic parameters and finally evaluate the potential of the existing SLR-tracked spherical satellite constellation to support the goals of GGOS.     

Comparison of reliability and geometrical strength criteria in geodetic networks

 The proper and optimal design and subsequent assessment of geodetic networks is an integral part of most surveying engineering projects. Optimization and design are carried out before the measurements are actually made. A geodetic network is designed and optimized in terms of high reliability and the results are compared with those obtained by the robustness analysis technique. The purpose of an optimal design is to solve for both the network configuration (first-order design) and observations accuracy (second-order design) in order to meet the desired criteria. For this purpose, an analytical method is presented for performing the first-order design, second-order design, and/or the combined design. In order to evaluate the geometrical strength of a geodetic network, the results of robustness analysis are displayed in terms of robustness in rotation, robustness in shear, and robustness in scale. Results showed that the robustness parameters were affected by redundancy numbers. The largest robustness parameters were due to the observations with minimum redundancy numbers. Received: 14 August 2000 / Accepted: 2 January 2001     

Effect of heteroscedasticity and heterogeneousness on outlier detection for geodetic networks

The robustness of an outlier detection method strongly depends on the weights of observations, i.e., the type of the stochastic model applied (homoscedasticity, heteroscedasticity and heterogeneousness). In this paper, we have investigated how the reliability of the robust methods and tests for outliers changes depending on the weights of the observations in geodetic networks. Furthermore, the contribution of the directions and distances to horizontal control network with regard to reliability are investigated separately. The concept of a breakdown point is used as a global measure of robustness against outliers. The mean success rate (MSR) is found to be a practical tool for confirming the breakdown point. Many different “good” data samples are generated for each network and then deliberately contaminated using a Monte-Carlo simulation. Six robust methods and Baarda’s test are applied to the corrupted samples and the degree of corruption is varied. The performance of each method is measured using both local and global MSRs. Our research shows: (1) The MSRs of Baarda’s test change depending on the strength of the heteroscedasticity, but do not change for trilateration and leveling networks, (2) the global MSRs of robust methods do not differ considerably from the local ones     

Stochastic estimation of tropospheric path delays in global positioning system geodetic measurements

Water vapor radiometric (WVR) and surface meteorological (SM) measurements taken during three Global Positioning System (GPS) geodetic experiments are used to calculate process noise levels for random walk and first-order Gauss-Markov temporal models of tropospheric path delays. Entire wet and combined wet and dry zenith delays at each network site then are estimated simultaneously with the geodetic parameters without prior calibration. The path delays and corresponding baseline estimates are compared to those obtained with calibrated data and stochastic residual delays. In this manner, the marginal utility of a priori tropospheric calibration is assessed given the ability to estimate the path delays directly using only theGPS data. Estimation of total zenith path delays with appropriate random walk or Gauss-Markov models yields baseline repeatabilities of a few parts in 10⁸. This level of geodetic precision, and accuracy as suggested by analyses on collocated baselines estimated independently by very long baseline interferometry, is comparable to or better than that obtained after path delay calibration usingWVR and/orSM measurements. Results suggest thatGPS data alone have sufficient strength to resolve centimeter-level zenith path delay fluctuations over periods of a few minutes.     

The Meissl scheme for the geodetic ellipsoid

We present a variant of the Meissl scheme to relate surface spherical harmonic coefficients of the disturbing potential of the Earth’s gravity field on the surface of the geodetic ellipsoid to surface spherical harmonic coefficients of its first- and second-order normal derivatives on the same or any other ellipsoid. It extends the original (spherical) Meissl scheme, which only holds for harmonic coefficients computed from geodetic data on a sphere. In our scheme, a vector of solid spherical harmonic coefficients of one quantity is transformed into spherical harmonic coefficients of another quantity by pre-multiplication with a transformation matrix. This matrix is diagonal for transformations between spheres, but block-diagonal for transformations involving the ellipsoid. The computation of the transformation matrix involves an inversion if the original coefficients are defined on the ellipsoid. This inversion can be performed accurately and efficiently (i.e., without regularisation) for transformation among different gravity field quantities on the same ellipsoid, due to diagonal dominance of the matrices. However, transformations from the ellipsoid to another surface can only be performed accurately and efficiently for coefficients up to degree and order 520 due to numerical instabilities in the inversion.