Singular value decomposition and cluster analysis as regression diagnostics tools for geodetic applications

It is well known that high-leverage observations significantly affect the estimation of parameters. In geodetic literature, mainly redundancy numbers are used for the detection of single high-leverage observations or of single redundant observations. In this paper a further objective method for the detection of groups of important and less important (and thus redundant) observations is developed. In addition, the parameters which are predominantly affected by these groups of observations are identified. This method thus complements other diagnostics tools, such as, e.g., multiple row diagnostics methods as described in statistical literature (see, e.g., Belsley et al. in Regression diagnostics: identifying influential data and sources of collinearity. Wiley, New York, 1980). The method proposed in this paper is based on geometric aspects of adjustment theory and uses the singular value decomposition of the design matrix of an adjustment problem together with cluster analysis methods for regression diagnostics. It can be applied to any geodetic adjustment problem and can be used for the detection of (groups of) observations that significantly affect the estimated parameters or that are of negligible impact. One of the advantages of the proposed method is the improvement of the reliability of observation plans and thus the reduction of the impact of individual observations (and outliers) on the estimated parameters. This is of particular importance for the very long baseline interferometry technique which serves as an application example of the regression diagnostics tool.     

Vertical displacement rates in the Upper Rhine Graben area derived from precise leveling

The recent vertical displacement field of the Upper Rhine Graben (URG) located in the tri-national region between Germany, France and Switzerland is investigated using repeatedly measured leveling data. We estimate vertical displacement rates at leveling benchmarks by applying a kinematic network adjustment on more than 40,000 height differences measured by German, French and Swiss surveying agencies. Focusing on an optimal solution for the adjusted rates in the URG area also historical data (measured before 1900) are used, significantly increasing the time span of available measurements and the number of transnational connections between the three countries. To account for inhomogeneities apparent in the database, we apply an iterative variance component estimation within the adjustment procedure, particularly revealing more realistic information on the accuracy of the estimated rates. A special focus within our analysis is put on the statistical testing of gross errors in the observations and model-related errors at benchmarks with non-linear movement. As some of the estimated vertical rates behave significantly different compared to the vertical rates of adjacent benchmarks, a filtering of outliers is applied after the adjustment procedure. The resulting map of linear height changes in an area of 280 km in N–S and 230 km in E–W direction provides detailed insight into the recent vertical displacements of the URG and neighboring regions. In the German part of the study area, it was possible for the first time to consistently constrain an average subsidence rate of 0.5 mm/a ( \(\pm \) 0.2 mm/a) of the Graben interior w.r.t. the Black Forest. In addition to the tectonic displacements, some man-induced surface movements, e.g., caused by oil and groundwater extraction, are observed and discussed.     

Integrated adjustment of CHAMP, GRACE, and GPS data

Various types of observations, such as space-borne Global positioning system (GPS) code and phase data, accelerometer data, K-band range and range-rate data, and ground-based satellite laser ranging data of the CHAllenging Minisatellite Payload (CHAMP) and GRAvity Climate Experiment (GRACE) satellite missions, are used together with ground-based GPS code and phase data in a rigorous adjustment to eventually solve for the ephemerides of the CHAMP, GRACE, and GPS satellites, geocenter variations, and low-degree gravity field parameters. It turns out that this integrated adjustment considerably improves the accuracy of the ephemerides for the high and low satellites, geocenter variations, and gravity field parameters, compared to the case when the adjustment is carried out stepwise or in individual satellite solutions.Acknowledgments. This study has been supported by the German Ministry of Education and Research through the Geotechnologies Programme grants 03F0333A/CHAMP and 03F0326A/GRACE.     

A test of significance for the Helmert-Kubik-Problem of Weight-determination

As it has been shown by Kubik it is possible to get an estimate, , of the reciprocal of the weight-matrix in an adjustment problem. If we want to see whether this new estimate differssignificantly from our a priori valueQ ₀ it is necessary to know the distribution function of the elements , the ’s being the elements of . This distribution is found in the present article and it is shown that it is not identical with any of the distributions well known from statistical textbooks. Furthermore a way of computing this new distribution is presented. Finally the connection with the chi-square distribution is explored and it is proved that the chi-square-distribution may be used as an approximation for a large number of over-determinations.     

A proposed GPS method with multi-antennae and single receiver

A method based on multi-antennae linked to a common GPS receiver is proposed. The goal of the technique is to improve height determination for baselines a few kilometres in length. The advantage of this technique resides in the elimination of relative clock parameters in the between-antenna single difference observations. Because single difference observations are free of clock errors more geometrical strength remains to determine the baseline components. This statement is valid as long as intercable biases can be carefully calibrated. For millimetre height determination, the intercable calibration must be done at the same level of accuracy. Under this assumption it is shown that in general the height standard deviation improves by a factor of about three compared to standard GPS data processing. With the proposed method, the effect of relative tropospheric zenith delay errors becomes a bit smaller (in absolute value), compared to standard data processing. To absorb this error, a relative tropospheric zenith delay parameter may be estimated. Even with this additional parameter in the solution the height standard deviation remains two times smaller than the results of standard processing techniques (without tropospheric zenith delay parameter), and at least five times smaller than in the results obtained from standard processing including one tropospheric zenith delay parameter.     

On the separation of gravitation and inertia and the determination of the relativistic gravity field in the case of free motion

Summary The authors explored the possibility of separating gravitation from inertia in the frame of general relativity. The Riemann tensor is intimately related with gravitational fields and has nothing to do with inertial effects. One can judge the existence or nonexistence of a gravitational field according as the Riemann tensor does not vanish or vanishes. In the free fall case, by using a gradiometer on a satellite, gravitational effects can be separated from inertia completely. Furthermore, the authors put forward a general method of determining the relativistic gravity field by using gradiometers mounted on satellites. At the same time the following two statements are proved: in the case of using gradiometers on a satellite, with some kind of approximation the Riemann tensorR can be found; in the case of free motion, if the measured Riemannian componentsR _(i0j0) are equal to zero, the Riemann tensorR equals zero.     

Validating ocean tide loading models using GPS

Ocean tides cause periodic deformations of the Earths surface, also referred to as ocean tide loading (OTL). Tide-induced displacements of the Earths crust relying on OTL models are usually taken into account in GPS (Global Positioning System) data analyses. On the other hand, it is also possible to validate OTL models using GPS analyses. The following simple approach is used to validate OTL models. Based on a particular model, instantaneous corrections of the site coordinates due to OTL are computed. Site-specific scale factors, f, for these corrections are estimated in a standard least-squares adjustment process of GPS observations together with other relevant parameters. A resulting value of f close to unity indicates a good agreement of the model with the actual site displacements. Such scale factors are computed for about 140 globally distributed IGS (International GPS Service) tracking sites. Three OTL models derived from the ocean tide models FES95.2.1, FES99, and GOT00.2 are analyzed. As expected, the most reliable factors are estimated for sites with a large loading effect. In general, the scaling factors have a value close to unity and no significant differences between the three ocean tide models could be observed. It is found that the validation approach is easy to apply. Without requiring much additional effort for a global and self-consistent GPS data analysis, it allows detection of general model misfits on the basis of a large number of globally distributed sites. For detailed validation studies on OTL models, the simultaneous estimation of amplitudes and phases for the main contributing partial tides within a GPS parameter adjustment process would provide more detailed answers.     

Probability distribution of eigenspectra and eigendirections of a twodimensional,symmetric rank two random tensor

Let there be given a twodimensional symmetric rank two tensor of random type (examples:strain, stress) which is either directly observed or indirectly estimated from observations by an adjustment procedure. Under the assumption of normalityof tensor components we compute the joint probability density functionas well as the marginal probability density functionsof its eigenspectra (eigenvalues) and eigendirections (orientation parameters). Due to the nonlinearity of the relation between eigenspectra-eigendirections and the random tensor components, via the inverse nonlinear error propagationbiases and aliases of their first and centralized second moments (mean value, variance-covariance) are expressed in terms of Jacobianand Hessianmatrices. The joint probability density function and the first and second moments thus form the fundamental of hypothesis testing and qualify control of eigenspectra (eigenvalues, principal components) and eigendirections (orientation parameters, eigenvectors, principial direction) of a twodimensional, symmetric rank two random tensor.     

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 note on adjustment of free networks

The present paper deals with the least-squares adjustment where the design matrix (A) is rank-deficient. The adjusted parameters \(\hat x\) as well as their variance-covariance matrix ( \(\sum _{\hat x} \) ) can be obtained as in the “standard” adjustment whereA has the full column rank, supplemented with constraints, \(C\hat x = w\) , whereC is the constraint matrix andw is sometimes called the “constant vector”. In this analysis only the inner adjustment constraints are considered, whereC has the full row rank equal to the rank deficiency ofA, andAC ^T =0. Perhaps the most important outcome points to the three kinds of results

1. A general least-squares solution where both \(\hat x\) and \(\sum _{\hat x} \) are indeterminate corresponds tow=arbitrary random vector.
2. The minimum trace (least-squares) solution where \(\hat x\) is indeterminate but \(\sum _{\hat x} \) is detemined (and trace \(\sum _{\hat x} \) corresponds tow=arbitrary constant vector.
3. The minimum norm (least-squares) solution where both \(\hat x\) and \(\sum _{\hat x} \) are determined (and norm \(\hat x\) , trace \(\sum _{\hat x} \) corresponds tow?0

     

Reliability checking for GNSS baseline and network processing

Reliability analysis is inseparably connected with the formulation of failure scenarios, and common test statistics are based on specific assumptions. This is easily overlooked when processing observation differences. Poor failure identification performance and misleading pre-analysis results, mainly meaningless minimum detectable biases and external reliability measures, are the consequence. A reasonable failure scenario for use with differenced GNSS observations is formulated which takes into account that individual outliers in the original data affect more than one processed observation. The proper test statistics and reliability indicators are given for use with correlated observations and both batch processing and Kalman filtering. It is also shown that standardized residuals and redundancy numbers fail completely when used with double differenced observations.

Groebner-basis solution of the three-dimensional resection problem (P4P)

The three-dimensional (3-D) resection problem is usually solved by first obtaining the distances connecting the unknown point P{X,Y,Z} to the known points P_i{X_i,Y_i,Z_i}i=1,2,3 through the solution of the three nonlinear Grunert equations and then using the obtained distances to determine the position {X,Y,Z} and the 3-D orientation parameters {,, }. Starting from the work of the German J. A. Grunert (1841), the Grunert equations have been solved in several substitutional steps and the desire as evidenced by several publications has been to reduce these number of steps. Similarly, the 3-D ranging step for position determination which follows the distance determination step involves the solution of three nonlinear ranging (`Bogenschnitt') equations solved in several substitution steps. It is illustrated how the algebraic technique of Groebner basis solves explicitly the nonlinear Grunert distance equations and the nonlinear 3-D ranging (`Bogenschnitt') equations in a single step once the equations have been converted into algebraic (polynomial) form. In particular, the algebraic tool of the Groebner basis provides symbolic solutions to the problem of 3-D resection. The various forward and backward substitution steps inherent in the classical closed-form solutions of the problem are avoided. Similar to the Gauss elimination technique in linear systems of equations, the Groebner basis eliminates several variables in a multivariate system of nonlinear equations in such a manner that the end product normally consists of a univariate polynomial whose roots can be determined by existing programs e.g. by using the roots command in Matlab.Acknowledgments.The first author wishes to acknowledge the support of JSPS (Japan Society of Promotion of Science) for the financial support that enabled the completion of the write-up of the paper at Kyoto University, Japan. The author is further grateful for the warm welcome and the good working atmosphere provided by his hosts Professors S. Takemoto and Y. Fukuda of the Department of Geophysics, Graduate School of Science, Kyoto University, Japan.     

基于等价分析法的稳健估计权函数设计

在观测值中加入粗差，粗差的影响可以通过调整观测值的权加以消除，对含有粗差的观测值利用稳健估计处理后的平差结果应与加粗差前的利用最小二乘原理处理的平差结果一致，依据这样的思想，本文利用间接平差函数模型，借用经典最小二乘原理，推导出了基于等价分析方法的稳健估计的等价权函数。     

Validation of ETOPO5U in the Hellenic area

Mean 5 × 5 heights and depths from ETOPO5U (Earth Topography at 5 spacing Updated) Digital Terrain Model (DTM) were compared with corresponding quantities of a local DTM in the test area [38° 40°, 21° 24°]. From this comparison a shift of ETOPO5U with respect to the local DTM in the longitudinal direction equal to 5 min was found after applying an efficient fast Fourier transform (FFT) technique. Furthermore, sparse mean height differences larger than 1,000 m were observed between ETOPO5U and the local DTM due rather to errors of ETOPO5U. The effect of these errors on gravity and height anomalies was computed in a subregion of the area under consideration.     

On the use of iterative re-weighting least-squares and outlier detection for empirically modelling rates of vertical displacement

The proper identification and removal of outliers in the combination of rates of vertical displacements derived from GPS, tide gauges/satellite altimetry, and GRACE observations is presented. Outlier detection is a necessary pre-screening procedure in order to ensure reliable estimates of stochastic properties of the observations in the combined least-squares adjustment (via rescaling of covariance matrices) and to ensure that the final vertical motion model is not corrupted and/or distorted by erroneous data. Results from this study indicate that typical data snooping methods are inadequate in dealing with these heterogeneous data sets and their stochastic properties. Using simulated vertical displacement rates, it is demonstrated that a large variety of outliers (random scattered and adjacent, as well as jointly influential) can be dealt with if an iterative re-weighting least-squares adjustment is combined with a robust median estimator. Moreover, robust estimators are efficient in areas weakly constrained by the data, where even high quality observations may appear to be erroneous if their estimates are largely influenced by outliers. Four combined models for the vertical motion in the region of the Great Lakes are presented. The computed vertical displacements vary between  − 2 mm/year (subsidence) along the southern shores and 3 mm/year (uplift) along the northern shores. The derived models provide reliable empirical constraints and error bounds for postglacial rebound models in the region.     

Atmospheric tides and rotation of the Earth

The six-hourly values of the atmospheric angular momentum (AAM) functions computed by the U.S. National Meteorological Center (NMC) were used to estimate the effects of the atmospheric tides on the Earth's rotation. Variations of the equatorial components ₁ and ₂ of the AAM have periods close to gravitational tidesP ₁ andK ₁.The amplitudes of the detected variations in ₁ and ₂ functions have been found to be much larger than the theoretical ones, the reason of this amplification remains unexplained. According to theoretical formulations, these waves can be expressed only as retrograde motions. Because of frame effects, there is a correspondance between diurnal retrograde polar motion and precession-nutations and the atmospheric effect on polar motion cannot be detected from observations.The second part of this paper deals the effects of atmospheric tides in Earth rotation. High-frequency UT1 variations have been derived from VLBI and GPS techniques during the SEARCH'92 campaign (Study ofEarth-AtmosphereRapidCHanges) (Dickey et al. 1994). They have been compared to values derived by Ray et al. (1994) from global ocean tide model. The results obtained in the present paper show the existence of variations of thermal origin with an amplitude of about 1µs in Universal Time UT1. The agreement between observed and theoretical values is better when the determined thermal atmospheric tides are taken into account.Oceanic tidal signal explains a large part (60% of the signal variance) of the diurnal and sub-diurnal variations. Our results show that only a small part of the residuals (5%) accounts for the atmospheric tidal effects. The residual signal remains unexplained; it might be due to mismodelization of oceanic or atmospheric tides or effect of other geophysical phenomena.