Improved second order design and the datum choice

It is shown that also in a rank deficient Gauss-Markov model higher weights of the observations automatically improve the precision of the estimated parameters as long as they are computed in thesame datum. However, the amount of improvement in terms of the trace of the dispersion matrix isminimum for the so-called “free datum” which corresponds to the pseudo-inverse normal equations matrix. This behaviour together with its consequences is discussed by an example with special emphasis on geodetic networks for deformation analysis.     

An algorithm for the inverse of a multivariate homogeneous polynomial of degreen

Summary Various geodetic problems (the free nonlinear geodetic boundary value problem, the computation of Gauß-Krüger coordinates or UTM coordinates, the problem of nonlinear regression) demand theinversion of an univariate, bivariate, trivariate, in generalmultivariate homogeneous polynomial of degree n. The new algorithm which is oriented towardsSymbolic Computer Manipulation is based upon the algebraic power base computation with respect toKronecker-Zehfu product structure leading to the solution of a system oftriangular matrix equations: Only the first row of the inverse triangular matrix has to be computed. TheSymbolic Computer Manipulation program of the GKS algorithm is available from the authors.     

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.     

Performance analysis of the spaceborne laser ranging system

The Spaceborne Laser Ranging System is a proposed short pulse laser on board an orbiting spacecraft.^1,2,3,4 It measures the distance between the spacecraft and many laser retroreflectors (targets) deployed on the Earth’s surface. The precision of these range measurements is assumed to be about ±2 cm (M. W. Fitzmaurice, private communication). These measurements are then used together with the orbital dynamics of the spacecraft, to derive the relative position of the laser ground targets. Assuming a six day observing period with 50% cloud cover, uncertainties in the baseline for target separations of 50 km to 1200 km were estimated to be on the order of 1 to 3 cm and the corresponding values in the vertical direction, ranged from 1 cm to 12 cm. By redetermining the measurements of the relative target positions, the estimated precision in the baseline for a target separation of 50 km is less than 0.3 cm and for a separation of 1200 km is less than 1 cm. In the vertical direction, the estimated precision ranged from 0.4 cm to 4.0 cm respectively. As a result of the repeated estimation of the relative laser target positions, most of the non-temporal effects of error sources as exemplified by the errors in geopotential are reduced. The Spaceborne Laser Ranging System’s capability of determining baselines to a high degree of precision provides a measure of strain and strain rate as shown byCohen, 1979. These quantities are essential for crustal dynamic studies which include determination and monitoring of strain near seismic zones, land subsidence, and edifice building preceding volcanic eruptions. It is evident that such a system can also be used for geodetic surveys where such precisions are more than adquate.     

A comparative study of algorithms for reducing the fill-in during Cholesky factorization

In this paper several ordering algorithms for the unknowns in geodetic least squares systems are compared. The comparison is restricted to the case of the well known Cholesky factorization of the normal matrixA into a lower triangular factorL. p ]The algorithms which were investigated are: minimum degree, minimum deficiency, nested dissection, reverse Cuthill-McKee, King's-, Snay's-, and Levy's-banker's and Gibbs-King. p ]Also some strategies are presented to reduce the time needed to compute the ordering using a priori information about the way the unknowns are connected to each other. p ]The algorithms are applied to normal matrices of the least squares adjustment of 2D geodetic terrestrial networks, photogrammetric bundle-block adjustments, and a photogrammetric adjustment using independent models. p ]The results show that ordering the unknowns yields a considerable decrease of the cpu time for computing the Cholesky factor, and that in general the minimum degree and Snay's banker's ordering perform best. Furtheron they show that a priori information about the connection structure of the unknowns speeds up the computation of the ordering substantially.Supported by the Netherlands Organization for Scientific Research (NWO)     

Boundary-value problems in the complex world of geodetic measurements

A review of recent progress and current activities towards an improved formulation and solution of geodetic boundary value problems is given. Improvements stimulated and required by the dramatic changes of the real world of geodetic measurements are focused upon. Altimetry–gravimetry problems taking into account various scenarios of non-homogeneous data coverage are discussed in detail. Other problems are related to free geodetic datum parameters, most of all the vertical datum, overdetermination or additional constraints imposed by satellite geodetic observations or models. Some brief remarks are made on pseudo-boundary value problems for geoid determination and on purely gravitational boundary-value problems. Received: 17 March 1999 / Accepted: 19 April 1999     

The gravitational attraction of any polygonally shaped vertical prism with inclined top and bottom faces

 A closed formula for computing the gravitational attraction of a general vertical prism with N+2 faces (N faces are vertical planes, the other two are the inclined top and bottom planes) in Cartesian coordinates is presented. In addition, the special case of a triangular prism is discussed. Algebraic differences and overlooked singularity conditions of a previously published formula of this computation (which was only for the triangular special case) were identified and are also presented. Received: 22 March 1999 / Accepted: 2 February 2000     

Accuracy analysis of vertical deflection data observed with the Hannover Digital Zenith Camera System TZK2-D

This paper analyses the accuracy of vertical deflection measurements carried out with the Digital Zenith Camera System TZK2-D, an astrogeodetic state-of-the-art instrumentation developed at the University of Hannover. During 107 nights over a period of 3.5 years, the system was used for repeated vertical deflection observations at a selected station in Hannover. The acquired data set consists of about 27,300 single measurements and covers 276 h of observation time, respectively. For the data collected at an earlier stage of development (2003 to 2004), the accuracy of the nightly mean values has been found to be about 0′′.10−0′′.12. Due to applying a refined observation strategy since 2005, the accuracy of the vertical deflection measurements was enhanced into the unprecedented range of 0′′.05 − 0′′.08. Accessing the accuracy level of 0′′.05 requires usually 1 h of observational data, while the 0′′.08 accuracy level is attained after 20 min measurement time. In comparison to the analogue era of geodetic astronomy, the accuracy of vertical deflection observations is significantly improved by about one order of magnitude.     

An alternative algebraic algorithm to transform Cartesian to geodetic coordinates

The algorithm to transform from 3D Cartesian to geodetic coordinates is obtained by solving the equation of the Lagrange parameter. Numerical experiments show that geodetic height can be recovered to 0.5 mm precision over the range from −6×10⁶ to 10¹⁰ m. Electronic Supplementary Material: Supplementary material is available in the online version of this article at     

Models for controlling national and continental networks

Doppler derived geocentric and relative geodetic positions are now widely used for detecting and controlling systematic scale and orientation errors in large classical terrestrial triangulation networks. However, the combined adjustment of terrestrial and space data raises several theoretical problems, including the choice of appropriate reference systems, the a priori weighting of the various types of observations, the modelling of systematic errors and the conditioning of the network in terms of internal and external rank deficiencies. Tests with large national networks show conclusively that, without correct modelling, systematic errors will largely be unaffected by “higher order” observations.     

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     

The parallel sum of matrices and its application in geodetic adjustments

Summary . It is well known that for the comparison and combination of geodetic networks their heterogeneous datum definitions are well to be considered. Various algorithms have been developed for this purpose. As an alternative concept to deal with hybrid datum problems, the operator parallel sum of matrices is introduced in this paper. To begin with, a definition is given and some basic properties are explained. To demonstrate the usefulness of the operator, two practical applications are given. The first deals with the estimation of parameters describing the deformation of two networks which are to be compared to each other. The second one treats the estimation of parameters representing the heterogeneous datum definitions of two networks which are to be merged into a hybrid network. It will be shown that – regardless of the datum definitions of the preadjusted individual networks – the parallel sum of matrices can be used to simplify the algorithms applied for the estimation of those parameters. Received 15 December 1995; Accepted 24 September 1996     

Altimetry–gravimetry problems with free vertical datum

 The definition and connection of vertical datums in geodetic height networks is a fundamental problem in geodesy. Today, the standard approach to solve it is based on the joint processing of terrestrial and satellite geodetic data. It is generalized to cases where the coverage with terrestrial data may change from region to region, typically across coastlines. The principal difficulty is that such problems, so-called altimetry–gravimetry boundary-value problems (AGPs), do not admit analytical solutions such as Stokes' integral. A numerical solution strategy for the free-datum problem is presented. Analysis of AGPs in spherical and constant radius approximation shows that two of them are mathematically well-posed problems, while the classical AGP-I may be ill posed in special situations. Received: 2 December 1998 / Accepted: 30 November 1999     

Vector method to compute the Cartesian (X, Y, Z) to geodetic ({\phi}, λ, h) transformation on a triaxial ellipsoid

The vector-based algorithm to transform Cartesian (X, Y, Z ) into geodetic coordinates (, λ, h) presented by Feltens (J Geod, 2007, doi:) has been extended for triaxial ellipsoids. The extended algorithm is again based on simple formulae and has successfully been tested for the Earth and other celestial bodies and for a wide range of positive and negative ellipsoidal heights.     

On the evaluation of stationary sea surface topography using geodetic techniques

One of the principal problems in separating the non-tidal Newtonian gravitational effects from other forces acting on the ocean surface with a resolution approaching the 10 cm level arises as a consequence ofall measurements of a geodetic nature being taken eitherat orto the ocean surface. The latter could be displaced by as much as ±2 m from the equipotential surface of the Earth’s gravity field corresponding to the mean level of the oceans at the epoch of observation— i.e., the geoid. A secondary problem of no less importance is the likelihood of all datums for geodetic levelling in different parts of the world not coinciding with the geoid as defined above. It is likely that conditions will be favourable for the resolution of this problem in the next decade as part of the activities of NASA’s Earth and Ocean Physics Applications Program (EOPAP). It is planned to launch a series of spacecraft fitted with altimeters for ranging to the ocean surface as part of this program. Possible techniques for overcoming the problems mentioned above are outlined within the framework of a solution of the geodetic boundary value problem to ±5 cm in the height anomaly. The latter is referred to a “higher” reference surface obtained by incorporating the gravity field model used in the orbital analysis with that afforded by the conventional equipotential ellipsoidal model (Mather 1974 b). The input data for the solution outlined are ocean surface heights as estimated from satellite altimetry and gravity anomalies on land and continental shelf areas. The solution calls for a quadratures evaluation in the first instance. The probability of success will be enhanced if care were paid to the elimination of sources of systematic error of long wavelength in both types of data as detailed in (Mather 1973 a; Mather 1974 b) prior to its collection and assembly for quadratures evaluations.     

Modeling and estimation of C1–P1 bias in GPS receivers

 Modern dual-frequency global positioning system (GPS) receivers are capable of providing direct measurements of both L1 C/A (C1) and P code (P1) without the use of the Y-codes under Anti-Spoofing. A discrepancy or bias between the C1 and P1 measurements from these receivers has however been of concern to operators and users of GPS reference networks. For the purpose of modeling and estimation, the nature and characteristics of the discrepancy must be investigated. The research results presented indicate that the discrepancy between the C1 and P1 measurements contains two different types of components: one is of constant type while another is time variant. A method has been developed for their modeling and estimation. The residual C1–P1 time series after a satellite-dependent bias removal agree at a few-centimeter level, indicating the effectiveness of the proposed model. This allows the C1–P1 discrepancy, both constant and non-constant components, to be removed from GPS reference network solutions. Numerical results are provided to support the analysis. Received: 12 October 1999 / Accepted: 11 May 2000     

The North American datum of 1983: Project methodology and execution

A new adjustment of the geodetic control networks in North America has been completed, resulting in a new continental datum—the North American Datum of 1983 (NAD 83). The establishment ofNAD 83 was the result of an international project involving the National Geodetic Survey of the United States, the Geodetic Survey of Canada, and the Danish Geodetic Institute (responsible for surveying in Greenland). The geodetic data in Mexico and Central America were collected by the Inter American Geodetic Survey and validated by the Defense Mapping Agency Hydrographic/Topographic Center. The fundamental task ofNAD 83 was a simultaneous least squares adjustment involving 266,436 stations in the United States, Canada, Mexico, and Central America. The networks in Greenland, Hawaii, and the Caribbean islands were connected to the datum through Doppler satellite and Very Long Baseline Interferometry (VLBI) observations. The computations were performed with respect to the ellipsoid of the Geodetic Reference System of 1980. The ellipsoid is positioned in such a way as to be geocentric, and its axes are oriented by the Bureau International de l'Heure Terrestrial System of 1984. The mathematical model for theNAD readjustment was the height-controlled three-dimensional system. The least squares adjustment involved 1,785,772 observations and 928,735 unknowns. The formation and solution of the normal equations were carried out according to the Helmert block method. [Authors' note:This article is a condensation of the final report of the NAD 83 project. The full report (Schwarz,1989) contains a more complete discussion of all the topics.]     

A method of error adjustment for marine gravity with application to Mean Dynamic Topography in the northern North Atlantic

International compilations of marine gravity, such as the International Gravity Bureau (BGI) contain tens of millions of point data. Lemoine et al. (The Development of the Joint NASA GSFC and the National Imagery and Mapping Agency (NIMA) Geopotential Model EGM96, NASA/TP-1998-206861) chose not to include any marine gravity in the construction of the global gravity model EGM96. Instead they used synthetic anomalies derived from altimetry, so that no independent information about Mean Dynamic Topography (MDT) can be deduced. Software has been developed not only to identify and correct those aspects of marine gravity data that are unreliable, but to do so in a way that can be applied to very large, ocean-wide data sets. First, we select only straight-line parts of ship-tracks and fit each one with a high-degree series of Chebyshev polynomials, whose misfit standard deviation is σ _line and measures the random error associated with point gravity data. Then, network adjustment determines how the gravity datum is offset for each survey. A free least squares adjustment minimises the gravity anomaly mismatch at line-crossing points, using σ _line to weight the estimate for each line. For a long, well crossed survey, the instrumental drift rate is also adjusted. For some 42,000 cross-over points in the northern Atlantic Ocean, network adjustment reduces the unweighted standard deviation of the cross-over errors from 4.03 to 1.58 mGal; when quality weighted, the statistic reduces from 1.32 to 0.39 mGal. The geodetic MDT is calculated combining the adjusted gravity anomalies and satellite altimetry, and a priori global ocean model through a new algorithm called the Iterative Combination Method. This paper reports a first demonstration that geodetic oceanography can characterise the details of basin wide ocean circulation with a resolution better than global ocean circulation models. The result matches regional models of ocean circulation from hydrography measurements (Geophys Res Lett 29:1896, 2002; J Geophys Res 108:3251, 2003).     

Defining the basic entities in a geodetic data base

The term “entity” covers, when used in the field of electronic data processing, the meaning of words like “thing”, “being”, “event”, or “concept”. Each entity is characterized by a set of properties. An information element is a triple consisting of an entity, a property and the value of a property. Geodetic information is sets of information elements with entities being related to geodesy. This information may be stored in the form ofdata and is called ageodetic data base provided (1) it contains or may contain all data necessary for the operations of a particular geodetic organization, (2) the data is stored in a form suited for many different applications and (3) that unnecessary duplications of data have been avoided. The first step to be taken when establishing a geodetic data base is described, namely the definition of the basic entities of the data base (such as trigonometric stations, astronomical stations, gravity stations, geodetic reference-system parameters, etc...). Presented at the “International Symposium on Optimization of Design and Computation of Control Networks”, Sopron, Hungary, July 1977.     

A fill-in storage technique for geodetic normal equations

A minimum fill-in scheme, published by Yale University and not previously discussed in the geodetic literature, has been tested on four typical triangulation networks. Its computer storage requirement has been compared with that of the profile method using Snay's station reordering algorithm. Tests have been carried out for terrestrial and mixed terrestrial/satellite networks. The conclusion is that the fill-in scheme is a viable method for geodetic networks and, in some cases, it needs less computer storage than the profile method.