Table Of ContentAntonio Marussi
Intrinsic Geodesy
Translated by W. I. Reilly
With 7 Figures
Springer-Verlag
Berlin Heidelberg New York Tokyo
Professor ANTONIO MARUSSI, Via C. Battisti 31,1-34125 Trieste
Translator: Dr. W. IAN REILLY, Department of Scientific
and Industrial Research, Geophysics Division, P.O.B. 1320,
Wellington/New Zealand
ISBN-13: 978-3-642-70245-7 e-ISBN-13: 978-3-642-70243-3
001: 10.1007/978-3-642-70243-3
Library of Congress Cataloging in Publication Data. Marussi, Antonio. Intrinsic
geodesy. Bibliography: p. Includes indexes. I. Geodesy. 2. Spaces, Generalized. 3. Co
ordinates. I. Title. QB283.M38 1985 526'.1 85-2737
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Softcover reprint of the hardcover 15t edition 1985
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A te Lori che mi sei stata vicina fiduciosa nella carriera scien
tifica e che sei ora if mio solo sostegno.
Con molto affetto
Antonio
Antonio Marussi
Antonio Marussi bestrode the world like a colossus. From Chile to
China, from the roots of the Himalayas to the orbits of artificial
satellites, his questing mind probed deeply and ranged widely to
understand the inwardness of the Earth. At once sceptical and imag
inative, he was not content with the conventional view, and in his
reformulation of three-dimensional geodesy, as in his explorations
of the Himalayan regions, set off on different and more fundamental
paths than his contemporaries had trodden. His geodetic theory re
quired the agility of a mathematical scholar, but he was also a per
son who confronted Nature in practice and overcame her, whether
on the peaks of the high Himalayan mountains or in making delicate
apparatus in the laboratory. He was a forceful organiser, who took
advantage of the special characteristics of the Grotta Gigante to set
up the pendulums with which he saw the signatures of the spectrum
of the Earth, detected earthquakes before they happened and found
that the Moon does not shield us from the gravity of the Sun.
Thinker, observer, organiser, all these he had to be to bring to fru
ition his studies of those high regions north of the Himalayas which
we must know if we are to understand the tectonics of the con
tinents, yet where still so few observations have been made. Some
25 years after his first expeditions to the Hindu Kush, he pro
posed the first long seismic line from the Pamirs to the plains of
India. Thought, observations, organisation, all indeed were essential,
but still more was called for, and still more he gave. Fortified by the
high reputation he already enjoyed in the East, Antonio Marussi be
came diplomat, and brought the scientists of four nations to co-oper
ate with him.
Antonio Marussi left us thinking of the gravity field of the Earth
and of geodesy in ways very different from those he found. That was
a major intellectual achievement, and its value became outstand
ingly clear when geodesy came to use artificial satellites. He es
tablished his own observatory on his beloved Carso and pursued
collaboration with his neighbours of mountainous countries. His last
work, like some of his earliest at Trieste, took him to the wastes of
Asia and to the immense high plateaux that hide secrets of the
origins of the great mountains he loved and conquered.
VII
We lament indeed the untimely, unlooked-for passing of Antonio
Marussi, a proud Triestino, whose appreciation of other countries
was based firmly on his loyalty to that peculiar city, but we shall
treasure with gratitude these works with which he has enriched us,
we are proud to have been of his company and grateful for the privi
lege of his acquaintance.
The Masters' Lodge March 1985 AH.Cook
Selwyn College
Cambridge
VIII
Author's Introductory Remarks
This book is the outcome of the attempt to approach the phenom
enological problems of Geodesy, using, unlike many classical trea
tises based on fictitious reference systems, exclusively geometrical
and physical objects and quantities embodied in the gravity field of
the Earth itself, these being therefore accessible to actual observa
tion. This approach leads directly to the three-dimensional domain
which is the natural seat of the phenomena forming the object of the
study of Geodesy.
The study was undertaken from the local point of view according
to the Nahewirkungsgesetz (principle of action in the vicinity) of
Hermann Weyl, using the method of vector and tensor analysis.
The collection of papers presented covers the period from 1950
to 1981 and has been ordered into six chapters.
The first chapter, entitled Fundamentals of Intrinsic Geodesy
comprises a set of papers in which the foundations of Intrinsic Geo
desy are given, making use of the natural observable coordinates
latitude, longitude, and geopotential for which the fundamental
metric tensor, the coefficients of connection, and the structure of the
coordinate lines and surfaces are given.
The so-called first fundamental problem of Geodesy, of transfer
ing the coordinates from one given point to another, is solved in
three-dimensional space.
Application of the methods of intrinsic geodesy is also made to
the study of the microgravitational field (the tidal field) ofa satellite,
or of a spacecraft in inertial motion, including the derivation of
Ricci's coefficients of rotation which connect the eigenvectors of the
tensor surfaces describing the field.
The second chapter, entitled Structure of the Gravity Field and
Laplace's Equation, comprises two papers dealing with the curvature
and torsion of the gravity field and a generalization of the famous
Dalby's theorem which expresses, in an absolute form, Laplace's
equation.
In the third chapter, entitled Principles of Intrinsic Geodesy Ap
plied to the Normal Reference Field, the general equations established
previously are applied to the case of the reference field endowed
with rotational symmetry, e.g., Somigliana's ellipsoidal field. In this
case the integrability conditions furnish the equations for the con-
IX
tinuation of the field in space starting from the values assigned on
the boundary surface.
The first fundamental problem of Geodesy for the transfer of the
geographical coordinates and the potential along a given curve is
solved. The fundamental parameters for the ellipsoidal field are
computed.
In the fourth chapter, entitled Mapping of the Actual Gravity
Field onto the Normal Reference Field, the correspondence between
points of the surface of the Earth and the surface of an ellipsoid is
generalized in three dimensions by establishing a one-to-one cor
respondence between points of the actual gravity field and points of
the normal ellipsoidal reference field, assuming that the centre of
mass of the Earth coincides with the centre of figure of the ellipsoi
dal field. Such correspondence is of Hirvonen's telluroidal type
in which the potential is conserved. The aim of the mapping is the
introduction of coordinates endowed with a simple metric tensor for
easy computation; all the irregularities of the actual field are there
fore removed to the modulus of deformation proper to the mapping.
A further aim of the mapping is to establish a rigorous definition
of the anomalies based on analytical principles. The integrability
conditions furnish, furthermore, a rigorous expression for the fun
damental equation of Physical Geodesy and for the generalized Vil
larceau's equations relating the derivatives of the deflections of the
vertical among themselves and the derivatives of the gravity anoma
ly. The conditions of harmonicity for the anomaly of the potential
are also given.
A procedure for the adjustment of geodetic networks in the
three-dimensional ellipsoidal model space is also given which
generalises the method of variation of coordinates as used in the ad
justment of bidimensional networks applying conformal represen
tations.
One paper is devoted to the problem of conformal represen
'tations in the three-dimensional space. It is shown that it is impos
sible to introduce in conformal space a system of orthogonal coor
dinates having the transforms of the equipotential surfaces as one of
the families of coordinate surfaces.
In the fifth chapter, entitled Mapping Between Surfaces, the map
ping problem is approached from the local point of view by as
suming that the quadratic form determining the modulus of de
formation is assigned. The various types of representations are
classified accordingly and the alterations induced in the curvatures
are determined.
Some integral properties of the conformal representations relat
ing the variation of the integral curvature with the flux of the gradi
ent of the logarithm of the modulus of deformation are given.
In the sixth chapter, entitled Propagation of a Light Path in Con
tinuous Isotropic Refracting Media, the geometric laws of propa-
x
gation of a light ray in a continuous isotropic refracting medium are
compared with the properties of conformal mapping in three dimen
sions.
The ideas expounded in the present collection of papers were first il
lustrated in the course of the General Assembly of the International
Association of Geodesy in Oslo in 1948, where I had the good for
tune to meet General Martin Hotine, who became greatly interested
in the subject and developed independently a wealth of ideas that
are condensed in his book Mathematical Geodesy, published in 1969.
The strong friendship that arose from the intimate collaboration
with Martin Hotine was at the root of a number of Symposia origi
nally held on Three-dimensional Geodesy and, then later, on Mathe
matical Geodesy. All Symposia were held, according to Martin's
wish, in Italy (Venezia, 1959; Cortina d'Ampezzo, 1962; Torino,
1965; Trieste, 1969; Firenze, 1972; Siena, 1975; Assisi, 1978; Como,
1981). After Martin Hotine's death in 1968, the Symposia were de
dicated to his memory.
The Symposia gave me the opportunity to become acquainted
with a number of colleagues who greatly contributed to the devel
opment and extension of the original idea of Intrinsic Geodesy. I
particularly wish to mention the most important contributions by
Prof Erik Grafarend of the University of Stuttgart and by Prof
Evangelos Livieratos of the University of Thessaloniki and their
schools. I also wish to mention especially the fundamental consider
ations developed on the subject by Professor Nathaniel Grossman of
the University of California, Los Angeles.
It was further my great good fortune to meet Dr. W. I. Reilly,
who also became interested in the theory of Intrinsic Geodesy and
has made important contributions to its application in practice.
Being highly proficient in the Italian language, Dr. Reilly spon
taneously undertook the task of translating my earliest papers, most
of them written in my mother tongue, into English. I do not have suf
ficient words to thank him.
Special thanks are also due to Mrs. Maria Luisa Princivalli, my
former student and now professor in the University of Trieste, who
assisted me in carefully preparing the manuscripts and in the proof
reading; and to Mrs. Ida C. Sbona, librarian of the Institute of
Geodesy and Geophysics in the University of Trieste, who kindly of
fered to complete the bibliography from books in the Institute.
Antonio Marussi
XI
