Geodesy

Fundamental problems of geodesy

By basing us on the definition of Helmert and by supplementing it, we can state the essential problem which arises for geodesy in the following way: to determine the “figure”, i.e. dimensions, the form and the field gravific external of the Earth (and possibly of other planetary bodies) according to time; moreover, to determine the “average ellipsoid” terrestrial or planetary starting from parameters observed on terrestrial or planetary surface, or outside this surface.

It acts-there of a problem of values fixed at the borders called geodetic problem of values in extreme cases (“geodetic boundary-been worth problem” or, in the form of initials: “GBVP”). This one incorporates at the same time a geometrical formulation (for the form) and a physical formulation (for the field of gravity). In fact, these two aspects, geometrical and physical, are dependant in an inextricable way. To be convinced some, it is enough to remember that the determination of the shape of the Earth necessarily will call upon the measurement of angles to the means of instruments (for example of the Théodolite S) which it is initially appropriate “to put in station”, in other words to use the Verticale as direction of reference. However, the vertical in a place is defined by means of the field of gravity, and thus of the field of gravity.

Under the term “forms” or “appears” of the Earth hide several possible meanings. With the most elementary direction, one can include/understand by this word the topographic surface , which represents the material border between the Lithosphère (the whole of the rock masses) and the Hydrosphère (the whole of the liquid masses) or the atmosphere (the whole of the gas masses). It is only during second half of the century that one included the oceanic bottom in the formulation of the geodetic problem. This last constitutes the border between the lithosphere and the oceanic masses of water. Extension of the geodetic problem to the oceans is treated within the framework of the geodesy marinades .

Topographic surface

The topographic surface of the solid Earth is a very irregular surface on all the scales because of the atomic and molecular forces of cohesion which ensure solidity of it, and thus lends itself badly to a mathematical or parametric description, except if one is interested in the very smoothed representation of great structures such as contours of the continents or the principal assembly lines. For this reason one describes it by means of a whole of check-points located by means of coordinates in a well defined system.

Geoid

Such a discrete representation is not what one would like to call the “shape of the Earth”, for which one would rather wish to have a precise mathematical representation. However, the surface of the oceans - which only constitutes with it approximately 70% of total terrestrial surface - can be conceived like a equipotential surface , i.e. like an equipotential surface of the field of gravity. Indeed, the principle of formation of the surface of the oceans and the seas and much simpler than that of topographic surface, since it are controlled primarily by the force of gravity, made abstraction of disturbing phenomena the such marine currents, the tides, the swell caused by the winds, the variations of the atmospheric pressure, etc These phenomena disturbing are at first approximation of periodic nature in time, the periods being able to be more or less long. By recording the sea level in many places using marigraphs during long time intervals, then by calculating at the various places the average of the values thus measured, one could define the mean level of the sea. This last corresponds, essentially, on an equipotential surface which can be used as reference to the establishment of charts of datum lines being to as many meters above or below the sea level. At present, the marigraphs were advantageously replaced by geodetic satellites particularly dedicated to the total observation heights of oceanic water during time.

Consequently, realizing certain simplifying assumptions, oceanic surface represents with balance part of surface on which the potential of gravity is constant, in other words one (surface) equipotential . By the thought, we can prolong this one under the continents and then identify it with the mathematical figure of the Earth, by opposition with the topographic figure. It is this mathematical figure which German engineer-geodesist J.B. Listing called in 1873 the geoid . Of course, the geoid had already been used as surface of reference before being named. Thus, in 1828, c.f. Gauss refers explicitly in the following terms to the geoid, without him to allot particular name: “What we call terrestrial surface with the geometrical direction is nothing more than the surface which intersects everywhere the direction of gravity with right angle, and part of this surface coincides with the surface of the oceans”.

The majority of the total parameters measured in geodesy refer to the external field of gravity. This way, the detailed study of the properties of this field is paramount for a good comprehension of physical geodesy. The space limit concerned is that provided by the geodetic use of artificial satellites or space probes, like by the Moon, our natural satellite. This field of external gravity can conceive like an infinity of encased equipotential surfaces the ones in the others, located entirely or only partially outside the Earth limited by its topographic surface. The physical aspect of the geodetic problem is thus conditioned by the consideration of topographic surface on the one hand, geoid on the other hand, surface-borders of the field of gravity.

Celestial and terrestrial frames of reference

One introduces frames of reference to describe the earthmoving in space (“celestial system”), as well as the geometry of surface and the field of gravity of the Earth (“terrestrial system”). The choice of the best frames of reference, taking into account spectacular progress of the current Metrology, became one of the broad topics of discussion of the moment. In fact, for certain applications, one is now obliged to locate the problem within the framework of the Theory of relativity, as well restricted as general, and one can speak about relativistic geodesy , which essentially is four-dimensional. However, within the framework of this article, we restrict ourselves to expose the traditional geodesy, which is treated in a physical space with three dimensions, Euclidean space indicated E₃. To deal with problems of total geodesy, the use of Cartesian coordinates in this three-dimensional Euclidean space is generally appropriate. On the other hand, on a surface of nonplane reference unspecified, which constitutes a two-dimensional mathematical object intrinsically, the use of curvilinear Coordonnées is essential. The natural choice of these last rises from the particular geometry of surface. Thus, the coordinates adapted to a Sphère are the spherical Coordonnées, those adapted to a Ellipsoïde are ellipsoidal Coordonnées, those adapted to a Cylindre are the cylindrical Coordonnées, etc

As fundamental system of terrestrial coordinates one introduces a space frame of reference Cartesian X , Y , Z anchored in the Earth (“English Earth-fixed”), and turning with this one. It is named Conventional Terrestrial System , CTS (“conventional terrestrial system”). It is about a relative frame of reference , for which it is advisable to consider accelerations of Coriolis, of drive and axifuge. The origin O is the center of mass of the Earth, or géocentre . The mass M of the Earth contains by definition the mass of the atmosphere. The axis O Z coincides with the average axis of rotation of the Earth. The plan of the average equator is perpendicular to this axis O Z , and thus contained in the plan O XY . By convention, the plan O XZ contains the average meridian plan of Greenwich. This last is defined by the average axis of rotation and the fundamental meridian adopted by the International office of Hour (BIH) like origin of longitudes. It corresponds to the “average” longitude of the Observatory of Greenwich, in the suburbs of London.

The axis O X is thus at the same time in the plan of the average equator and the plan of the fundamental meridian line. It points géocentre towards the point of intersection G' of the average equator with the fundamental meridian line. The G' point defines the origin of the geographical coordinates (latitude φ, longitude λ), and is in the Golfe of Guinea, to a thousand of kilometers to broad of Libreville (Gabon). The axis O Y is perpendicular to O X so that (O X , O Y , O Z ) form a trihedron dextrorsum. O Y is also contained in the plan of the average equator and point towards a point Y located at 90° in the east of G'. This one is in the Indian Ocean, to a good thousand of kilometers in the west of Sumatra (Indonesia). The axis O Z is normal in the plan of the average equator and point towards the geographical north pole NR.

Movement of the pole

The introduction of the average axis of rotation proves to be necessary, because terrestrial rotation is variable in time. This is true so much for the orientation of the terrestrial axis of rotation compared to the figure of the Earth (movement of the pole) that for the angular velocity of rotation of the Earth on itself (variation in the length of the day). The movement of the pole contains several components, in particular an annual or quasi-annual component, a component having one period of approximately 430 days (approximately 14 month), and a secular component. The fourteen month old component is the movement of Chandler. It is about a movement of the pole quasi-circular of an amplitude ranging between 0,'' 1 and 0,'' 2, which is done in the opposite direction of the needles of a watch when one observes it starting from north.

This movement is caused by the fact that the Earth turns and that the axis of greater inertia does not coincide exactly with the instantaneous axis of clean rotation. If the Earth were perfectly indeformable (= rigid), one would observe a precession of the axis of rotation compared to the axis of figure with one 305 days period, called “period of Euler”. The lengthening of the period of Chandler compared to the period of Euler is due to the fact that the Earth is actually deformable. Thus, according to the Principle of Châtelier, the deformation produced by a primarily elastic force of recall is made so as to be opposed to this force of recall which disturbs initial balance, and it results a lengthening from it from the period.

In addition to the component of Chandler, there exists in the movement of the pole another periodic component or quasi-periodical with a annual period , in general having the amplitude ranging between 0,'' 05 and 0,'' 1, therefore definitely lower than that of Chandler. It is made in the same direction that the fourteen months movement and has due seasonal displacement to masses of air in the atmosphere or water masses in the hydrosphere. Processes weather, oceanologic and hydrological complex are at the base of these displacements of great volumes of matter which are reflected by seasonal variations of the tensor of inertia I . In the absence of one external moment of force, the total kinetic moment must be preserved. That results in the fact that the quantity I · Ω is constant. Thus, if I varies, the vector Ω describing instantaneous rotation must vary in opposite direction.

Lastly, there exists inside the Earth of the matter movements on very large space scales (movements of Convection in the coat and core, Subduction of the Tectonic plates, etc). These movements are very slow, but give place on geological time intervals with considerable displacements, implying considerable variations of the tensor of inertia. These secular variations induce a drift , or migration , pole (“whodunnit wander” in English, “German Polwanderung”). Thus, of 1900 to 1996, one notes a drift from approximately 0,'' 003 per annum, roughly along the 80e meridian West. By superimposing these three components, the instantaneous pole follows a spiral curve whose central point advances slowly during time. The deviations of the instantaneous position of the pole compared to the central point remain lower than 0,'' 3 over one year.

Matérialisation of the CTS, conventional terrestrial system

The CTS is materialized by a certain number of distributed observatories the best possible one on the surface of the sphere, taking into account various geographical constraints, political and financial. These observatories control terrestrial rotation permanently, so as to provide the reductions necessary compared to the average axis of rotation. The average axis of rotation of the CTS is defined by international convention using a fictitious average pole, the conventional international origin (“Conventional International Origin”, or CIO). This last represents the average of the positions of the average pole given between 1900,0 and 1906,0 (i.e. between on January 1st, 1900 at zero hour and on December 31st, 1905 at midnight).

The position of the pole (Northern) instantaneous is determined by an international service. Until 1988 this service was ensured by astronomical observatories carrying out of the precise determinations of the latitude and time. The movement of the pole was given since 1899 by five dedicated observatories, all located on the 35e parallel North. These five observatories, of which that located in Europe is in Carloforte, close to Cagliari in Sardinia, constituted the International Service of the Latitudes (“International Latitude Service”, or THEY). By associating with THEY it International Service of the Movement of the Pole (“International Whodunnit Motion Service”, or IPMS), and by also considering the participation of the International office of Hour (BIH), some fifty astronomical observatories finally contributed to the determination of the movement of the pole and the rotation of the Earth, in other words astronomical time. The results were published like averages over five days with a precision of approximately ±0,'' 02 on the coordinates of the pole, of ±1 ms over the length of the day. In particular, the average meridian line of Greenwich is defined through geographical longitudes of the observatories which provided their data to the Service of the Time of the BIH. For this reason, one often calls this conventional meridian line the meridian zero of the BIH (“BIH zero meridian”). Let us notice that the instantaneous position of the north pole is provided by Cartesian coordinates ( X _P, there _P), called “coordinated pole”. The origin of the frame of reference of the pole is the CIO, and the plan X _P _P is tangent there on the surface of the conventional figure of the Earth with the CIO. The axis of the X _P is directed along the average meridian line of Greenwich, and _P centers it is there along the meridian line of 90°W.

From 1967, the precise determination of the rotation of the Earth was not any more the exclusive prerogative of the astronomers. Indeed, following the fast development of the techniques of space geodesy, the very precise determination of the orbits of the artificial satellites became possible, and these orbits could thus be used as reference to which to bring back the movement of the pole. It is initially thanks to the satellites belonging to “US Navy Navigation Satellite System” (NSS), whose orbits were established by means of a method based on the Doppler effect, that the measurement of the movement of the pole quickly became routine for the geodesists. The precision obtained by space geodesy started initially by competing with, then in very little time by largely exceeding best measurements than one could obtain by astronomical way. Consequently, it was necessary no more to maintain in service the astronomical observatories with very expensive and heavy operation, incompetents to provide the positions of the pole in real-time. Thus, since 1988, the International Service of the Rotation of the Earth (“International Earth Rotation Service”, IERS) replaces the IPMS and the section of the BIH devoted to terrestrial rotation. The IERS is an international service established jointly by the International Astronomical Union, UAI (“International Astronomical Union”, IAU) and the Geodetic and Geophysical Union International, UGGI (“International Union off Geodesy and Geophysics”, IUGG).

The fundamental geodesic stations providing their data to the IERS implement very advanced space techniques, in particular interferometry at very long base (“Very Long Baseline Interferometry”, VLBI), the system of positioning total (“Total Positioning System”, GPS), the laser shooting on the Moon (“Lunar Laser Ranging”, LR), and the laser shooting on artificial satellites (“Ranging Laser Satellite”, SLR). The precision currently reached can be estimated at ±0,'' 002 (or ±2 farmhouse) on the movement of the pole and with ±0,2 ms over the duration of the day (“length off day”, LOD or l.o.d.), for median values taken over one day. Since the end of the XXe century, the geocentric positions of the fundamental stations are provided with a precision of ±0,01 Mr. In fact, one currently tends towards a precision of ±0,01 m, and the goal to reach is to have a known geoid except for the centimetre.

In conclusion, the Conventional Terrestrial System (CTS) is currently materialized by the instantaneous space coordinates of a total whole of stations distributed in space. Those constitute the International Terrestrial Reference frame (“International Terrestrial Reference Frame”, ITRF). The passage of the ITRF with the CTS at one unspecified time requires the use of models depending on time, that the geodesists and the geophysicists continuously try to improve. These models include/understand, inter alia, the variations of terrestrial rotation (speed and orientation of the axis), displacements of the géocentre, the relative movements of the geodesic stations being used as check-points, consequence of the deformations produced by the terrestrial tides and the seismic activity and tectonics in general. An important work of the modern geodesist is thus also as well as possible to model the terrestrial tides and the plate tectonics.

Like inertial system, fixed compared to remote stars or quasars (“space-fixed system”), the astronomers employ a conventional system of inertia (“Conventional Inertial System”, CIS), based to astrometrical measures very precise Its practical realization is provided by star catalogs and quasars.

Surfaces of reference mark on the surface of the Earth

For the geodetic raised , one introduces a surface of reference. To know the position of a point on the surface of the Earth, one must then determine the curvilinear coordinates definite on the surface of reference and the altitude of the point above (positive altitude) or below (negative altitude) surface reference. Because of its particularly simple equation, an ellipsoidal of revolution flattened to the poles is appropriate better like surfaces reference that the geoid, which has “hollows and bumps” caused by the unequal distribution of the masses inside the Earth, and thus necessarily a mathematical representation more complicated than an ellipsoid.

One attaches a significance particular to an ellipsoidal terrestrial means, called “normal ellipsoid” or “normal spheroid” which is supposed to represent as well as possible the geoid. The “normal” qualifier comes owing to the fact that it is stipulated that this ellipsoid is an equipotential surface ( normal with the line of the plumbline) and that one allots kind to him not only one geometrical significance, but more especially a physical significance. The word “spheroid” has two directions: in the broad sense, it indicates any surface roughly similar to a sphere, and in this direction even the geoid would be to some extent a sphéroîde; with the restricted direction, used here, it indicates an ellipsoid of revolution too flattened nor too not lengthened.

Problems of current geodesy

The Earth and its field gravific undergo variations during the time which can be of secular nature (for example, the variations related to the braking of terrestrial rotation following the friction of the tides or those associated with the rising of the shields laurentide and fenno-scandien following the deglaciation there is approximately ten thousand years), periodical (for example, various components of tide) or abrupt (for example, tiny variations of gravity associated with the rising or the lowering of an area before and during a seism). In space, these variations can occur on scales total, regional or local, according to the cases.

The evaluation and geodetic measurement techniques are now sufficiently advanced to detect at least part of these often negligible changes. Progress of the techniques during the last decades led to a profit of resolution and in precision which in fact completely repositions total geodesy among sciences of the Earth in the sense that it contributes now in a way essential with the study of the dynamics of the Earth, instead of simply limiting itself to provide a system of location and constraints for the static modeling of the Earth. One consequently conceives the figure of the Earth and the field of gravity like geodynamic variables depending on time. It is in this direction restricted, which has nothing to do with the theory of relativity, that one also sometimes speaks about “four-dimensional geodesy”, time being perceived like the fourth dimension.

The techniques of observation improving and the active profit in precision of par, one passed from an overall known geoid to ten meters close with a geoid reaching a threshold of precision of approximately a centimetre. It is obvious that with this precision, the reduction of the data of observation becomes complicated, and of the concepts defined in agreement with a threshold of measuring accuracy given can not be appropriate more when precision increases several orders of magnitude. Thus it of the concept geoid goes from there: with the precision of measurements reached now, it is hardly more possible to regard the mean level of the seas as an equipotential surface, and it cannot consequently strictly any more represent the geoid defined as equipotential surface. It is good to understand that these complications exist and are the subject of active research, but that does not mean that the traditional approximations of geodesy became without value.

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