Erik W. Grafarend

World Geodetic Datum 2000

Abstract. Based on the current best estimates of fundamental geodetic parameters {W0,GM,J2,Ω} the form parameters of a Somigliana-Pizzetti level ellipsoid, namely the semi-major axis a and semi-minor axis b (or equivalently the linear eccentricity ) are computed and proposed as a new World Geodetic Datum 2000. There are six parameters namely the four fundamental geodetic parameters {W0,GM,J2,Ω} and the two form parameters {a,b} or {a,ɛ}, which determine the ellipsoidal reference gravity field of Somigliana-Pizzetti type constraint to two nonlinear condition equations. Their iterative solution leads to best estimates a=(6 378 136.572±0.053)m, b=(6 356 751.920 ± 0.052)m, ɛ=(521 853.580±0.013)m for the tide-free geoide of reference and a=(6 378 136.602±0.053)m, b=(6 356 751.860±0.052)m, ɛ=(521 854.674 ± 0.015)m for the zero-frequency tide geoid of reference. The best estimates of the form parameters of a Somigliana-Pizzetti level ellipsoid, {a,b}, differ significantly by −0.39 m, −0.454 m, respectively, from the data of the Geodetic Reference System 1980.

Three-dimensional deformation analysis: Global vector spherical harmonic and local finite element representation

Abstract Deformation elements which have global support, such as tidal induced displacements, are represented in terms of vector spherical harmonics. In contrast, local strain and local rotation are represented in terms of irregularly shaped three-dimensional finite elements. These geodetic finite elements reflect the structure of a geodetic network which is arbitrarily shaped. Various continuity classes are discussed. Numerical examples are presented.

Algebraic Geodesy and Geoinformatics

While preparing and teaching Introduction to Geodesy I and II to undergraduate students at Stuttgart University, we noticed a gap which motivated the writing of the present book: Almost every topic that we taught required some skills in algebra, and in particular, computer algebra! From positioning to transformation problems inherent in geodesy and geoinformatics, knowledge of algebra and application of computer algebra software were required. In preparing this book therefore, we have attempted to put together basic concepts of abstract algebra which underpin the techniques for solving algebraic problems. Algebraic computational algorithms useful for solving problems which require exact solutions to nonlinear systems of equations are presented and tested on various problems. Though the present book focuses mainly on the two ?elds, the concepts and techniques presented herein are nonetheless applicable to other ?elds where algebraic computational problems might be encountered. In Engineering for example, network densi?cation and robotics apply resection and intersection techniques which require algebraic solutions. Solution of nonlinear systems of equations is an indispensable task in almost all geosciences such as geodesy, geoinformatics, geophysics (just to mention but a few) as well as robotics. These equations which require exact solutions underpin the operations of ranging, resection, intersection and other techniques that are normally used. Examples of problems that require exact solutions include; three-dimensional resection problem for determining positions and orientation of sensors, e. g. , camera, theodolites, robots, scanners etc.

GPS Solutions: Closed Forms, Critical and Special Configurations of P4P

P4P is the pseudo-ranging 4-point problem as it appears as the basic configuration of satellite positioning with pseudo-ranges as observables. In order to determine the ground receiver/satellite receiver (LEO networks) position from four positions of satellite transmitters given, a system of four nonlinear (algebraic) equations has to be solved. The solution point is the intersection of four spherical cones if the ground receiver/satellite receiver clock bias is implemented as an unknown. Here we determine the critical configuration manifold (Determinantal Loci, Inverse Function Theorem, Jacobi map) where no solution of P4P exists. Four examples demonstrate the critical linear manifold. The algorithm GS solves in a closed form P4P in a manner similar to Groebner bases: The algebraic nonlinear observational equations are reduced in the forward step to one quadratic equation in the clock bias unknown. In the backward step two solutions of the position unknowns are generated in closed form. Prior information in P4P has to be implemented in order to decide which solution is acceptable. Finally, the main target of our contribution is formulated: Can we identify a special configuration of satellite transmitters and ground receiver/satellite receiver where the two solutions are reduced to one. A special geometric analysis of the discriminant solves this problem.

The Generalized Mollweide Projection of the Biaxial Ellipsoid

SummaryThe standard Mollweide projection of the sphere SR2which is of type pseudocylindrical — equiareal is generalized to the biaxial ellipsoidEA,B2.Within the class of pseudocylindrical mapping equations (1.8) ofEA,B2(semimajor axis A, semiminor axis B) it is shown by solving the general eigenvalue problem (Tissot analysis) that only equiareal mappings, no conformal mappings exist. The mapping equations (2.1) which generalize those from SR2toEA,B2lead under the equiareal postulate to a generalized Kepler equation (2.21) which is solved by Newton iteration, for instance (Table 1). Two variants of the ellipsoidal Mollweide projection in particular (2.16), (2.17) versus (2.19), (2.20) are presented which guarantee that parallel circles (coordinate lines of constant ellipsoidal latitude) are mapped onto straight lines in the plane while meridians (coordinate lines of constant ellipsoidal longitude) are mapped onto ellipses of variable axes. The theorem collects the basic results. Six computer graphical examples illustrate the first pseudocylindrical map projection ofEA,B2of generalized Mollweide type.

The free versus fixed geodetic boundary value problem for different combinations of geodetic observables

Various formulations of the geodetic fixed and free boundary value problem are presented, depending upon the type of boundary data. For the free problem, boundary data of type astronomical latitude, astronomical longitude and a pair of the triplet potential, zero and first-order vertical gradient of gravity are presupposed. For the fixed problem, either the potential or gravity or the vertical gradient of gravity is assumed to be given on the boundary.The potential and its derivatives on the boundary surface are linearized with respect to a reference potential and a reference surface by Taylor expansion. The Eulerian and Lagrangean concepts of a perturbation theory of the nonlinear geodetic boundary value problem are reviewed. Finally the boundary value problems are solved by Hilbert space techniques leading to new generalized Stokes and Hotine functions. Reduced Stokes and Hotine functions are recommended for numerical reasons. For the case of a boundary surface representing the topography a base representation of the solution is achieved by solving an infinite dimensional system of equations. This system of equations is obtained by means of the product-sum-formula for scalar surface spherical harmonics with Wigner 3j-coefficients.

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

SummaryVarious 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-Zehfuβ 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.

The oblique azimuthal projection of geodesic type for the biaxial ellipsoid : Riemann polar and normal coordinates

SummaryRiemann polar/normal coordinates are the constituents to generate the oblique azimuthal projection of geodesic type, here applied to the “reference” ellipsoid of revolution (biaxial ellipsoid).Firstly we constitute a minimal atlas of the biaxial ellipsoid built on {ellipsoidal longitude, ellipsoidal latitude} and {metalongitude, metalatitude}. TheDarboux equations of a 1-dimensional submanifold (curve) in a 2-dimensional manifold (biaxial ellipsoid) are reviewed, in particular to represent geodetic curvature, geodetic torsion and normal curvature in terms of elements of the first and second fundamental form as well as theChristoffel symbols. The notion of ageodesic anda geodesic circle is given and illustrated by two examples. The system of twosecond order ordinary differential equations of ageodesic (“Lagrange portrait”) is presented in contrast to the system of twothird order ordinary differential equations of ageodesic circle (Proofs are collected inAppendix A andB). A precise definition of theRiemann mapping/mapping of geodesics into the local tangent space/tangent plane has been found.Secondly we computeRiemann polar/normal coordinates for the biaxial ellipsoid, both in theLagrange portrait (“Legendre series”) and in theHamilton portrait (“Lie series”).Thirdly we have succeeded in a detailed deformation analysis/Tissot distortion analysis of theRiemann mapping. The eigenvalues — the eigenvectors of the Cauchy-Green deformation tensor by means of ageneral eigenvalue-eigenvector problem have been computed inTable 3.1 andTable 3.2 (Λ1, Λ2 = 1) illustrated inFigures 3.1, 3.2 and3.3. Table 3.3 contains the representation ofmaximum angular distortion of theRiemann mapping. Fourthly an elaborate global distortion analysis with respect toconformal Gauβ-Krüger, parallel Soldner andgeodesic Riemann coordinates based upon theAiry total deformation (energy) measure is presented in a corollary and numerically tested inTable 4.1. In a local strip [-lE,lE] = [-2°, +2°], [bS,bN] = [-2°, +2°]Riemann normal coordinates generate the smallest distortion, next are theparallel Soldner coordinates; the largest distortion by far is met by theconformal Gauβ-Krüger coordinates. Thus it can be concluded that for mapping of local areas of the biaxial ellipsoid surface the oblique azimuthal projection of geodesic type/Riemann polar/normal coordinates has to be favored with respect to others.

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 propagation”biases 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.

Direct polynomial approach to nonlinear distance (ranging) problems

In GPS atmospheric sounding, geodetic positioning, robotics and photogrammetric (perspective center and intersection) problems, distances (ranges) as observables play a key role in determining the unknown parameters. The measured distances (ranges) are however normally related to the desired parameters via nonlinear equations or nonlinear system of equations that require explicit or exact solutions. Procedures for solving such equations are either normally iterative, and thus require linearization or the existing analytical procedures require laborious forward and backward substitutions. We present in the present contribution direct procedures for solving distance nonlinear system of equations without linearization, iteration, forward and backward substitution. In particular, we exploit the advantage of faster computers with large storage capacities and the computer algebraic softwares of Mathematica, Maple and Matlab to test polynomial based approaches. These polynomial (algebraic based) approaches turn out to be the key to solving distance nonlinear system of equations. The algebraic techniques discussed here does not however solve all general types of nonlinear equations but only those nonlinear system of equations that can be converted into algebraic (polynomial) form.