Joseph L. Awange

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.

Compatibility of NMEA GGA with GPS Receivers Implementation

The NMEA GGA standard for global positioning system (GPS) and GLONASS receiver interfaces provides smooth data transfer from receiver to computer for postprocessing. Specifically, the NMEA GGA specifies that the orthometric height and the undulation be listed in addition to other quantities. The Ashtech manual (Ashtech, 1997; p. 104), however, specifies that the respective field in the NMEA GGA message contains the ellipsoidal height when outputting from Ashtechs GG24 receiver. Because such an inconsistency between the official NMEA GGA specifications and the implementation by a manufacturer can potentially confuse the user, we carried out a numerical test to confirm Ashtechs implentation. The result indicates that Ashtech indeed gives the ellipsoidal height in the fielt that should actually contain the orthometric height according to the NMEA GGA specifications. Firmware version after and including GF02 have corrected this situation.

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.

Nonlinear Adjustment of GPS Observations of Type Pseudo-Ranges

The nonlinear adjustment of GPS observations of type pseudo-ranges is performed in two steps. In step one a combinatorial minimal subset of observations is constructed which is rigorously converted into station coordinates by means of Groebner basis algorithm or the multipolynomial resultant algorithm. The combinatorial solution points in a polyhedron are reduced to their barycentric in step two by means of their weighted mean. Such a weighted mean of the polyhedron points in ℝ3 is generated via the Error Propagation law/variance-covariance propagation. The Fast Nonlinear Adjustment Algorithm (FNon Ad Al) has been already proposed by Gauss whose work was published posthumously and Jacobi (1841). The algorithm, here referred to as the Gauss-Jacobi Combinatorial algorithm, solves the over-determined GPS pseudo-ranging problem without reverting to iterative or linearization procedure except for the second moment (Variance-Covariance propagation). The results compared well with the solutions obtained using the linearized least squares approach giving legitimacy to the Gauss-Jacobi combinatorial procedure.

Analytic solution of GPS atmospheric sounding refraction angles

The nonlinear system of equations for solving GPS atmospheric sounding’s bending angles are normally solved using Newton’s method. Because of the nonlinear nature of the equations, Newton’s method applies linearization and iterations. The method assumes the refraction angle to be small enough such that the dependency of the doppler shift on these angles are linear. The bending angles are then solved iteratively. Since the approach assumes the dependency of doppler shift on bending angles to be linear, which in actual sense is not, some small nonlinearity error is incurred. The Newton’s iterative method is often used owing to the bottleneck of solving in exact form the nonlinear system of equations for bending angles. By converting this system of trigonometric nonlinear equations into algebraic, the present contribution proposes an analytic (algebraic) algorithm for solving the bending angles and presents the geometry of the solution space. The algorithm is tested by computing bending angles of three CHAMP occultation data and the results compared to those of iterative Newton’s approach. Occultation 133 of 3rd May 2002 is selected as it occurred during diurnal solar radiation maximum past afternoon. During this time, the effect of ionospheric noise is high. Occultations number 3 of 14th May 2001 and number 6 of 2nd February 2002 were selected since they occurred past mid-night, a time of low solar activity and thus less effect of ionospheric noise. The results for occultation 133 of 3rd May 2002 indicate that the nonlinearity errors in bending angles increase with decrease in height to a maximum absolute value of 0.00069° (0.1%) for the region 5–40 km during period of high solar activity. Such nonlinearity errors are shown to impact significantly on the computed impact parameters to which the bending angles are referred. During low solar activity period, the nonlinearity error was relatively small for occultation number 3 of 14th May 2001 with maximum absolute value of 0.00001°. The analytical algorithm thus provide an independent method for controlling classical iterative procedures and could be used where very accurate results are desired.

Ranging algebraically with more observations than unknowns

In the recently developed Spatial Reference System that is designed to check and control the accuracy of the three-dimensional coordinate measuring machines and tooling equipment (Metronom US., Inc., Ann Arbor: http://www.metronomus.com), the coordinates of the edges of the instrument are computed from distances of the bars. The use of distances in industrial application is fast gaining momentum just as in Geodesy and in Geophysical applications and thus necessitating efficient algorithms to solve the nonlinear distance equations. Whereas the ranging problem with minimum known stations was considered in our previous contribution in the same Journal, the present contribution extends to the case where one is faced with many distance observations than unknowns (overdetermined case) as is usually the case in practise. Using the Gauss-Jacobi Combinatorial approach, we demonstrate how one can proceed to position without reverting to iterative and linearizing procedures such as Newton’s or Least Squares approach.

Role of algebra in modern day Geodesy

Algebra, in particular the Abelian Group and the Semi Group (also known as Monoid) axioms, which form a “ring with identity”, are employed to define the polynomial ring. Polynomial ring theory enables the solution of geodetic observations that can be converted into (algebraic) polynomials. The advantages of algebraic approaches are that they provide exact solutions to problems requiring closed form approaches (e.g. solving for geocentric coordinates from Helmerts projection through minimum distance mapping) and also act as tools to control iterative procedures. As a motivation, we present several examples of geodetic problems solved algebraically. These examples include; nonlinear analysis of bending angles in GPS-Meteorology, transformation of geocentric Cartesian coordinates into ellipsoidal, densification problems etc. The overriding advantage of the algebraic approach is the removal of the requirement of approximate starting values; they are non-iterative and enable detection of outliers.

The Sixth Problem of Probabilistic Regression the random effect model – “errors-in-variable”

“In difference to classical regression error-in-variables models here measurements occurs in the regressors. The naive use of regression estimators leads to severe bias in this situation. There are consistent estimators like the total least squares estimator (TLS) and the moment estimator (MME).

The First Problem of Probabilistic Regression: The Bias Problem

Minimum Bias solution of problems with datum defects. LUMBE of fixed effects. The bias problem in probabilistic regression has been the subject of Sect. 437 for simultaneous determination of first moments as well as second central moments by inhomogeneous multilinear, namely bilinear, estimation. Based on the review of the first author “Variance-covariance component estimation: theoretical results and geodetic application” (Statistical and Decision, Supplement Issue No. 2, pages 401–441, 105 references, Oldenbourg Verlag, München 1989), we collected 5 postulates for simultaneous determination of first and second central moments. A first reference is J. Kleffe (1978). It forms the basis of Sect. 4-37:

Overdetermined System of Nonlinear Equations on Curved Manifolds

Here we review a special issue of distributions on manifolds, in particular the spherical problem of algebraic regression or analyse the inconsistent of directional observational equations. The first section introduces loss functions on longitudinal data ((Phi= 1)) and (p = 2) on the circle or on the sphere as a differential manifold of dimension Φ = 1 and p = 2. Section 6.2 introduces the minimal distance mapping on S 1 and S 2 and constructs the related normal equations. Section 6.3 reviews the transformation from the circular normal distribution to an oblique normal distribution including a historical note to von Mises analyzing data on a circle, namely atomic weights. We conclude with note on the “angular metric.” As a case study in section four we analysze 3D angular observations with two different theodolites, namely Theodolite I and Theodolite II. The main practical result is the set of data from ((tan {wedge }^{mathrm{n}},tan {Phi }^{mathrm{n}})), its solution (({wedge }^{mathrm{n}},{Phi }^{mathrm{n}})) is very different from the Least Squares Solution.