From kinematical geodesy to inertial positioning

WhenH. Moritz (1967, 1971) studied “kinematical geodesy” for the purpose of separation of gravitation and inertia, especially within combined accelerometer-gradiometer systems, it was hard to believe that within five years time inertial survey systems would be available, exactly operating according to his theoretical design. Here, we attempt to give a geodetic introduction into the fundamental equation of inertial positioning materialized by inertial survey systems with emphasis on a careful error model, including 36 parameters of type time interval, initial positions, initial gravity, varying acceleration, varying gravity gradients, accelerometer bias, accelerometer random uncertainty, accelerometer non-orthogonality, initial misalignment angles, accelerometer scale factor uncertainty. The notion of “multipoint” boundary value problem and initial value problem of inertial positioning is reviwed. So-called “post-mission” adjustment techniques for inertial surveys are discussed.

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Sommaire QuandH. Moritz (1967, 1971) a étudié la géodésie cinématique dans le but de séparer la gravitation et l’inertie, spécialement en combinant accéléromètres et gradiomètres, il était difficile de croire qu’en cinq ans les systèmes d’arpentage inertiels seraient disponibles et fonctionneraient exactement selon ses prévisions théoriques. Ici, nous allons tenter de donner une introduction géodésique à l’équation fondamentale du positionnement inertiel, matérialisée par un système d’arpentage inertiel en soulignant l’importance d’un modèle d’erreur incluant 36 paramètres du genre intervalle de temps, positions initiales, gravité initiale, accélération variable, gradient de la gravité variable, déviation des accéléromètres, incertitude aléatoire des accéléromètres, non-orthogonalité des accéléromètres, angles initiaux des défauts d’alignement, incertitude du facteur d’échelle des accéléromètres. La notion de problème “multipoint” aux limites et du problème de la valeur initiale du positionnement inertiel y est revue. Les techniques de compensation “après la mission” y sont discutées.

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Presented at the 2nd International Symposium on Inertial Technology for Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

Resultats de la determination de mouvements des continents d'apres les methodes astronomiques

Sommaire La discussion des observations des latitudes faites dans des stations astronomiques situées sur les continents divers peut permettre de déterminer des mouvements des continents. L'accroissement du nombre des stations astronomiques permet de déceler non seulement le déplacement des continents mais aussi des mouvements de rotation des continents. Le travail qui. suit a été présenté au Symposium tenu à Zurich en septembre 1974 et consacré aux “Problems of Recent Crustal Movements”. N. Stoyko est revenu gravement malade de ce symposium (il devait décéder deux ans plus tard) sans avoir pu contribuer à la rédaction de cette communication qui ainsi ne se trouve pas insérée dans les actes du Symposium de Zurich. A. Stoyko a rédigé le texte par la suite et j'ai demandé au Bulletin Géodésique en hommage à N. Stoyko s'il pouvait en assurer la publication. Je le remercie d'avoir bien voulu le faire.S. Débarbat “Problems of Recent Crustal Movements”, Fourth International Symposium, Moscow, USSR, 1971 Ed. “Valgus”, Tallinn, 1975, pp. 205–211.     

Probleme de l'implantation d'une grille sur une shphere

Résumé Dans une première partie, traitée de fa?on élémentaire, l'auteur, après avoir rappelé diverses solutions particulières, propose deux méthodes pour résoudre le problème de l'optimisation d'un réseau de points de grille sur une sphère, admettant toutes deux une symétrie équatoriale, des points équidistants sur des parallèles régulièrement espacés en latitude, une ligne de points équatoriaux et deux points polaires. Une critique permet de dégager la meilleure de ces deux solutions; cette dernière solution convient, quelque soit le nombre N de points, si N>5, mais conviendra d'autant mieux que N sera plus grand. Une application numérique est donnée pour un réseau de 1.000 points, avec la localisation de ces points sur une carte de l'hémisphère terrestre Nord. Dans une seconde partie, traitée de fa?on plus théorique, l'auteur précise la notion de distance interpoints, montre les limites de validité des solutions précédentes pour finalement proposer une troisième solution, la solution dite “en spirale”, qui lui para?t définitivement la meilleure.     

Lilliput and Blefuscu and the figure of the Earth

Is the intense controversy between French and English scientists of the late 1600 th and early 1700 th—relating to the figure of the Earth—reflected in the “most obstinate war” fought between “the two great empires of Lilliput and Blefuscu”—over the correct way of breaking eggs?     

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.     

THE THEORY AND APPLICATION OF THE STRUCTURE LIGHT ENGINEERING SURVEYING BASED ON A LASER THEODOLITE WITH THREE FREEDOMS OF ROTATION

1　ＩｎｔｒｏｄｕｃｔｉｏｎＩｎｏｕｒｃｏｕｎｔｒｙ ,ｔｈｅｓｕｒｖｅｙｉｎｇｗｏｒｋｏｎｉｎｄｕｓｔｒｉａｌｏｂｊｅｃｔｓｈａｖｅ ｇｒａｄｕａｌｌｙｉｎｃｒｅａｓｅｄ .Ｔｈｏｓｅｉｎｄｕｓｔｒｉａｌｏｂｊｅｃｔｓｉｎｃｌｕｄｅｒａｉｌｗａｙａｎｄｈｉｇｈｗａｙｔｕｎｎｅｌｓ,ｖａｒｉ ｏｕｓｏｉｌａｎｄｇａｓｃａｎｓ ,ｖａｒｉｏｕｓａｉｒｃｒａｆｔ,ｓｈｅｌｌｏｆｃａｒｓ,ｌａｒｇｅｍｉｌｉｔａｒｙａｎｔｅｎｎａｕｓｅｄｆｏｒｓｃｏｕｔｉｎｇｓｔｅａｌｔｈｐｌａｎｅ ,ｓｈｉｐ ,ｉｎｔｅｒｎａｌｓｔ…     

Global plate tectonics and the secular motion of the Pole

Astronomical data compiled during the last 70 years by the international organizations (ILS/IPMS, BIH) providing the coordinates of the instantaneous pole, clearly shows a continuous drift of the “mean pole” (≡barycenter of the wobble cycle with respect to the Conventional International Origin (CIO). This study was undertaken to investigate the possibility of an actual secular motion of the barycenter (approximated by the earth's maximum principal moment of inertia axis or axis of figure) due to differential mass displacements from lithospheric plate rotations. The method assumes the earth's crust modeled as a mosaic of 1°×1° blocks, each one moving independently with their corresponding absolute plate velocities. The differential contributions to the earth's second-order tensor of inertia were computed, resulting in no significant displacement of the earth's axis of figure. In view of the above, the possibleapparent displacement of the “mean pole” as a consequence of station drifting due to absolute plate motions was also analyzed, again without great success. As a further step the old speculation of the whole crust possibly sliding over the upper mantle is revived and the usefulness of the CIO is questioned. Presented at the IAU Symposium No. 78, “Nutation and the Earth Rotation”, Kiev, 22–29, May, 1977.     

＂Digital Region＂ in the Context of the ＂Grid Computing＂

This paper is to construct a “digital local, regional, region“ information framework based on the technology of “SIG“ and its significance and application to the regional sustainable development evaluation system. First, the concept of the “grid computing“ and “SIG“ is interpreted and discussed, then the relationship between the “grid computing“ and “digital region“ is analyzed, and the framework of the “digital region“ is put forward. Finally, the significance and application of “grid computing“ to the “region sustainable development evaluation system“ are discussed.     

The Abel-Poisson kernel and the Abel-Poisson integral in a moving tangent space

The upward-downward continuation of a harmonic function like the gravitational potential is conventionally based on the direct-inverse Abel-Poisson integral with respect to a sphere of reference. Here we aim at an error estimation of the “planar approximation” of the Abel-Poisson kernel, which is often used due to its convolution form. Such a convolution form is a prerequisite to applying fast Fourier transformation techniques. By means of an oblique azimuthal map projection / projection onto the local tangent plane at an evaluation point of the reference sphere of type “equiareal” we arrive at a rigorous transformation of the Abel-Poisson kernel/Abel-Poisson integral in a convolution form. As soon as we expand the “equiareal” Abel-Poisson kernel/Abel-Poisson integral we gain the “planar approximation”. The differences between the exact Abel-Poisson kernel of type “equiareal” and the “planar approximation” are plotted and tabulated. Six configurations are studied in detail in order to document the error budget, which varies from 0.1% for points at a spherical height H=10km above the terrestrial reference sphere up to 98% for points at a spherical height H = 6.3×10⁶km. Received: 18 March 1997 / Accepted: 19 January 1998     

Toute la Geodesie sans ellipsoide Les reperes laplaciens Application plus particuliere aux travaux a latitude elevee

Resume Après de nombreuses années d’hésitation, on a finalement reconnu, au Congrès de Florence, en 1955, que dans le repérage des altitudes, seule la notion depotentiel était claire et sans ambigu?té, l’altitude au sens courant du terme étant conventionnelle. De la même fa?on, pour le repérage géométrique des points à la surface de la Terre, les coordonnées (X Y Z) des points, dans letrièdre cartésien terrestre général, sont les inconnues fondamentales; les coordonnées géodésiques couramment utilisées (longitude, latitude altitude H au-dessus de l’ellipso?de) sont conventionnelles. Mais pratiquement, afin d’écrire commodément les relations d’observation, il para?t intéressant de passer par l’intermédiaire detrièdres locaux (trièdres laplaciens), liés de fa?on invariable au système cartésien général, et de repérer toutes les grandeurs dans ces trièdres locaux. Toutes les observations utilisées en Géodésie s’expriment de fa?on simple et sans singularités dans ces trièdres locaux. La jonction des triangulations classiques, l’Astrogéodésie, la synthèse des Géodésies classique et spatiale sont facilitées. En astronomie de position, les grandeurs longitude, latitude, azimut, sont avantageusement remplacées par: déviation Est-Ouest, déviation Nord-Sud, azimut de Laplace. Les relations d’observation s’écrivent sans difficulté, même dans les régions polaires. L’application pratique des nouvelles formules obtenues a été réalisée avec succès par L.F. Gregerson (Service Géodésique du Canada).

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Summary At Florence, in 1955, it was accepted that, in the problems of levelling, the notion ofpotential was scientifically clear, and that the altitude could derive from it only through a conventional process. In the same manner, when we want to have a geometric reference of the points at the earth surface, we use the coordinates (X Y Z) in thegeneral cartesian trihedron as fundamental unknowns, the geodetic coordinates (λϕH) deriving from (X Y Z) through a conventional process. Practically, in order to set up the observation equations, it is necessary to define local trihedrons (laplacian trihedrons), deriving from the cartesian general system through a fixed transformation, and to refer all the unknowns in these local trihedrons. All the observations used in Geodesy can be expressed simply and without any singularity in these local trihedrons. The links between classical geodetic nets, the astrogeodesy, the combination between classical and spatial geodesy, become easier. In astronomical controls, “longitude, latitude, azimut” must be replaced by: W-E deflection, N-S deflection and Laplace azimuth. Thus all the observation equations can be set, even in polar regions. A practical application of the new formulae was done successfully by L.F. Gregerson (Geodetic Survey of Canada).

     

A detailed study of the duality relation for the least squares adjustment in euclidean spaces

The Euclidean spaces with their inner products are used to describe methods of least squares adjustment as orthogonal projections on finite-dimensional subspaces. A unified Euclidean space approach to the least squares adjustment methods “observation equations” and “condition equations” is suggested. Hence not only the two adjustment solutions are treated from the view-point of Euclidean space theory in a unified frame but also the existing duality relation between the methods of “observation equations” and “condition equations” is discussed in full detail. Another purpose of this paper is to contribute to the development of some familiarity with Euclidean and Hilbert space concepts. We are convinced that Euclidean and Hilbert space techniques in least squares adjustment are elegant and powerful geodetic methods.     

GNSS data management and processing with the GPSTk

We organize complex problems in simple ways using a GNSS data management strategy based on “GNSS Data Structures” (GDS), coupled with the open source “GPS Toolkit” (GPSTk) suite. The code resulting from using the GDS and their associated “processing paradigm” is remarkably compact and easy to follow, yielding better code maintainability. Furthermore, the data abstraction allows flexible handling of concepts beyond mere data encapsulation, including programmable general solvers. An existing GPSTk class can be modified to achieve the goal. We briefly describe the “GDS paradigm” and show how the different GNSS data processing “objects” may be combined in a flexible way to develop data processing strategies such as Precise Point Positioning (PPP) and network-based PPP that computes satellite clock offsets on-the-fly.     

Superconducting tensor gravity gradiometer

A sensitive gravity gradiometer can provide much needed gravity data of the earth and improve the accuracy of inertial navigation. A complete determination of all five independent components of the gravity gradient tensor is possible in principle by rotating a single in-line component gradiometer. In order to avoid dynamically induced noise sources arising from rotation, a hard-mounted assembly of component gradiometers may nevertheless be advantageous in an ultra-sensitive device. Superconductivity and other properties of materials at low temperatures can be used to obtain a sensitive, low-drift, gravity gradiometer. By differencing the outputs of accelerometer pairs using superconducting circuits, it is possible to construct a non-rotating tensor gravity gradiometer. Additional superconducting circuits can be provided to determine the linear and angular acceleration vectors. A three-axis in-line gravity gradiometer (a “vector” gradiometer) is being developed for satellite geodesy. A two-dimensional spring concept for a three-axis cross component gradiometer is discussed. The superconducting tensor gravity gradiometer constitutes a complete package of inertial navigation instruments with angular and linear acceleration readouts as well as gravity signals. Accuracy of inertial navigation could be improved by use of such a gradiometeraided inertial navigator.

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Sommaire Un gradiomètre gravimétrique sensible peut foumir une grande partie des données gravimétriques terrestres et augmenter la précision de la navigation inertielle. Une détermination complète des cinq composantes indépendantes du tenseur de gradient de gravité est possible en principe par la rotation d’un seul axe d’un gradiomètre. Dans le but d’éviter les sources de bruit provenant de la rotation, un assemblage complexe de gradiomètres peut être néanmoins avantageux dans un appareil ultra-sensible. La superconductivité et d’autres propriétés des matériaux à basse température peuvent être utilisées pour obtenir un gradiomètre gravimétrique sensible et à faible dérive. Il est possible, en différentiant les sorties de paires d’accéléromètres utilisant des circuits superconducteurs, de construire un gradimètre non-rotatif qui mesure le tenseur de gravité. Des circuits superconducteurs additionnels peuvent être ajoutés pour déterminer les vecteurs d’accélération linéaires et angulaires. Un gradiomètre gravimétrique à trois exes est présentement développé pour la géodésie spatiale. Le concept d’un ressort à deux dimensions comme composante d’un gradiomètre à trois axes est discuté. Le gradiomètre de mesure du tenseur de gravité à l’aide de circuits superconducteurs constitue l’ensemble complet pour un instrument de navigation inertielle avec lecture des accélérations angulaires et linéaires et des données de gravité. La précision de la navigation inertielle pourrait être améliorée par l’emploi d’un tel gradiomètre avec les systèmes inertiels.

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Presented at the Second International Symposium on Inertial Technology of Surveying and Geodesy, Banff, Canada, June 1–5, 1981.     

The convergence problem of collocation solutions in the framework of the stochastic interpretation

The problem of the convergence of the collocation solution to the true gravity field was defined long ago (Tscherning in Boll Geod Sci Affini 39:221–252, 1978) and some results were derived, in particular by Krarup (Boll Geod Sci Affini 40:225–240, 1981). The problem is taken up again in the context of the stochastic interpretation of collocation theory and some new results are derived, showing that, when the potential T can be really continued down to a Bjerhammar sphere, we have a quite general convergence property in the noiseless case. When noise is present in data, still reasonable convergence results hold true.

“Democrito che ’l mondo a caso pone” “Democritus who made the world stochastic” Dante Alighieri, La Divina Commedia, Inferno, IV – 136     

Global height datum unification: a new approach in gravity potential space

The problem of “global height datum unification” is solved in the gravity potential space based on: (1) high-resolution local gravity field modeling, (2) geocentric coordinates of the reference benchmark, and (3) a known value of the geoid’s potential. The high-resolution local gravity field model is derived based on a solution of the fixed-free two-boundary-value problem of the Earth’s gravity field using (a) potential difference values (from precise leveling), (b) modulus of the gravity vector (from gravimetry), (c) astronomical longitude and latitude (from geodetic astronomy and/or combination of (GNSS) Global Navigation Satellite System observations with total station measurements), (d) and satellite altimetry. Knowing the height of the reference benchmark in the national height system and its geocentric GNSS coordinates, and using the derived high-resolution local gravity field model, the gravity potential value of the zero point of the height system is computed. The difference between the derived gravity potential value of the zero point of the height system and the geoid’s potential value is computed. This potential difference gives the offset of the zero point of the height system from geoid in the “potential space”, which is transferred into “geometry space” using the transformation formula derived in this paper. The method was applied to the computation of the offset of the zero point of the Iranian height datum from the geoid’s potential value W ₀=62636855.8 m²/s². According to the geometry space computations, the height datum of Iran is 0.09 m below the geoid.     

Deblurring Texture Extraction from Digital Aerial Image by Reforming “Steep Edge” Curve

Texture extract from digital aerial image is widely used in three-dimensional city modeling to generate “photo-realistic” views. In this paper, a method based on reforming “Steep edge” curve, which clearly explains how the diffraction of the sunlight makes digital aerial image blurring, is proposed to deblur the texture extraction from digital aerial image, and the experiment shows a good result in visualization and automation.     

Changes of spheroid and of datum

Summary A datum change between two geodetic systems with points in common may be derived in three stages; slight adjustments of coordinates to make the networks of common points geometrically similar in the two systems; a scale factor to make them geometrically congruent; finally, an orthogonal transformation to swing them into coincidence. The geometrical concept is developed of a “datum screw”, not arbitrarily chosen as is the “origin” or “datum point” of a geodetic survey, but intrinsic to the geometry. The conditions under which it degenerates to a simple “datum shift” are discussed. Differential and other formulae for changes of spheroid and of datum are given, together with a set of tables of coefficients.