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Although we have only one extant fragment from the sixth-century BC Milesian philosopher, Anaximander, a number of doxographies enable us to reconstruct his ideas concerning the origin and evolution of the present order of things, that is, what one may call an historia (or investigation) of the nrepi 0oiaws' type. What is distinctly important about the Milesian's historia is that, being the first rational account of this type to have come down to us, it helps us to understand just what is involved in the movement from a mythopoeic to a speculative account. Certainly one of the most interesting facets of Anaximander's system is his cosmological model, which places an immobile earth at the center of the celestial sphere. The reason given for this is that, the earth being equidistant from all the points of the celestial circumference, there is no reason for it to move up rather than down, or left rather than right. In sum, the Milesian's reasoning behind the position of the earth appears to be mathematical. It is therefore not surprising that this is often considered Anaximander's greatest achievement in cosmology; for it liberated the mind (or should have) from the idea that the earth needed a material support. This is not the only role that the earth played in his cosmology. The earth is also the most important element in determining the sizes and distances of the other celestial bodies; that is, their sizes and distances are analogous to the dimensions of the earth. This may be deduced from a certain number of doxographies which tell us that the Milesian conceived the shape of the earth as that of a column drum three times as broad as it is high and that the distance of the stars, of the moon and of the sun (or their respective rings) from the center of the earth are in a ratio of 1:2:3. It appears that Anaximander realized his universe according to a mathematical plan following the series 3. This hypothesis, first formulated by Paul Tannery in the late nineteenth century, has, in spite of its conjectural na

The stochastic inversion method in ionospheric radiotomography is reviewed with a special emphasis on regularization used in the inversion process. Regularization is used both for preventing vigorous point-to-point oscillations and for controlling the peak altitude and thickness of the inversion result. The latter usually means importing a priori information on the layer height and thickness to the solver. In this paper it is pointed out that due to the curvature of the Earth and of the ionosphere, the measurements contain some information on the ionospheric altitude and the profile shape even in the case of a purely horizontally stratified layer. If this information could be used in choosing an appropriate regularization, no additional information would be needed. Simulation tests are presented which indicate that the altitude of a horizontally stratified layer can be determined with a reasonable accuracy without any a priori information. An attempt is also made to use the data for determining the shape of a proper regularization profile. Although some success is achieved in this effort, it is concluded that available a priori information, for example, ionosonde or incoherent scatter measurements, should be used in choosing the regularization profile. The ideas are tested with true data obtained from difference Doppler measurements carried out in Scandinavia, and the results are compared with simultaneous observations made by the European incoherent scatter radar. The comparison shows a reasonable agreement, although clear discrepancies also occur, for instance, in the shape of the bottomside profile.

INTRODUCTION In recent years, data from geophysical and laboratory physics gravity experiments suggested the existence of an anomalous gravitational or “fifth” force (Stacey et al. , 1987; Boynton et al. , 1987; Thieberger, 1987; Fischbach and Talmadge, 1992). However, it appears that these data are better explained by improving the models of the Earth's gravitational field instead of making major adjustments in the basic theories of physics (Kim, 1988; Kim, 1989; Braswell, 1992). Detailed modeling of the Earth's gravitational field requires a knowledge of variations in density within the Earth as well as a knowledge of the shape of the Earth (Chulick et al. , 1990). This in part is facilitated by the inclusion of a detailed model of the three-dimensional density structure of the Earth's crust. Such a model can be constructed by using the velocity and layer thickness data collected from various types of seismic surveys (Christiansen and Mooney, 1995). This...

Such paradoxes as synchronous compression and tension in extensional sedimentary basins, and synchronous deformation of intractonic basins and eustatic sea-level falls, can each be related to pulsed changes in curvature of the Earth's surface. Australia is everywhere in a state of significant, relatively uniform, upper crustal horizontal compression, approximately perpendicular to the continental margins and generally directed towards its centre. The compression commenced in the Pliocene but was absent in the Late Miocene. During the evolution of the Gippsland and Bass basins under extensional stress from the Late Jurassic to Recent, seven pulses of similarly directed compression have been identified from the Campanian to Recent. Time equivalent compressional pulses can also be demonstrated in such widely separated areas as Northwest and Northeast Australia, Northwest Europe and Central America. Nine earlier pulses of Early Triassic to Albian age are similarly synchronous global events and another twenty-three pulses inferred since the Neoproterozoic (800 Ma) are probably similarly globally synchronous. Earth expansion (with subduction) can explain these observations since it causes the surface curvature to decrease resulting in ‘flattening’ of the semi-rigid crust. Flattening results in lower crustal extension which generates sedimentary basins while contemporaneous upper crustal compression inverts and folds the sediments. Since globally synchronous compressional pulses may reflect expansion in the ocean basins there is a correlation of such pulses with major eustatic sea-level lowstands. Any site of expansion within the earth is transmitted to the semi-rigid crust as a volume increase by the whole gravitating-fluid, oblate spheroid. This results in ubiquitous synchronous compressional pulses in the upper crust detectable on seismic reflection data not only at basin scale, but also in individual petroleum-producing structures.

SUMMARY Effects of sphericity are commonly ignored in the lithospheric bending problem. In order to examine its effects, I solve a simple axisymmetric spherical-shell model. The full solution and the asymptotic solution are derived from the basic equations, and their relationship to the flat-plate solution is examined. For displacement, effects of sphericity are small, and use of the flat-plate solution produces results that are numerically indistinguishable from those of the spherical solution. The most significant effect of sphericity appears in the stress, in particular the normal stress along the strike direction of the trench. This stress is approximately given by Eu,/R, where E is Young’s modulus, u, is the vertical deformation of the shell and R is its radius of curvature. If the shell (lithosphere) is bent downwards and reaches 30 km, this stress can become about 5 kbar in the Earth. While plastic behaviour may set in under such high pressure conditions and analysis beyond elasticity theory may be required, sphericity may be a cause of large compressive stress in the trench strike direction. This stress may play an important role in forming the overall shape of the Earth’s subduction zones.

The geoid is the fundamental surface that defines the figure of the Earth. It is approximated by mean sea-level and undulates due to spatial variations in the Earth's gravity field. The use of the geoid in regional geophysics is illustrated for the North West Shelf of Australia by removing long-wavelength geoid features, due predominantly to deep-Earth mass anomalies, in order to reveal near-surface structure. After this process, the residual geoid anomalies correlate well with known geological structures. Therefore, the geoid can provide information, complementary to other geophysical data, of the Earth's internal structure.

Research on students' conceptions about the Earth and its place in the universe has been active since 1976. These years have also witnessed the development of the constructivist model of learning and a growing interest in epistemological and historical considerations among science educators. The paper presents a critical review of the research in the light of epistemological, historical and cognitive aspects. The analysis shows that, as far as the research on the shape of the Earth is concerned, the research results are valid and conclusive in giving general information about children's ideas. The same cannot be said for the research concerned with the position of the Earth in the Universe, where the Copernican model, seen as the final essence of astronomical concepts, drives the research questioning with little correlation of the model with the empirical level of observation.

According to Stokes’ approach, given the gravity anomaly on the geoid as the boundary, a disturbing potential function satisfying some boundary conditions should be solved. The basic requirement is that the disturbing potential function should be harmonic in the region outside the geoid. However, since the normal gravity potential is not defined inside the reference ellipsoid (taking the WGS84 ellipsoid as an example), when the geoid is below the ellipsoidal surface, the disturbing potential function is not harmonic in the whole region outside the geoid, and is not defined on the whole geoid itself. These are theoretical difficulties in Stokes’ approach. To remove these difficulties from Stokes’ approach, this study provides a new formulation of Stokes’ approach. An inner ellipsoid with four fundamental parameters is chosen, two of which, the geocentric gravitational constant and the rotational angular velocity, coincide with the corresponding parameters of the WGS84 ellipsoid. The other two parameters, the semi-major axis and flattening, are different from the corresponding ones of the ellipsoid. Then, the normal gravity potential generated by the inner ellipsoid is determined, by requiring that it holds a constant on the surface of the inner ellipsoid or on the surface of the ellipsoid. With this new formulation, the disturbing potential function is harmonic in the whole region outside the geoid, and the difficulties in Stokes’ approach disappear. The new formulation proposed in this study is also adequate for analogous geodetic boundary-value problems.

According to Friedrich Robert Helmert geodesy is the science of the measurement and mapping of the figure of the Earth. With the advent of the space age it was possible to establish a global reference frame, which can relate different locations across the entire Earth with an accuracy of currently about 1 cm with respect to each other. A rapid development of highly precise and stable frequency standards took place at the same time and is an important prerequisite for the actual quality of this global terrestrial reference frame. In fact all measurement techniques in modern space geodesy are based on precise time and frequency. Growing demands for the monitoring of global change as one of several examples of the importance of a highly resolved reference frame require more than one order of magnitude improvement over the existing level of quality. Optical clocks, optical frequency combs and near lossless time and frequency transfer have the potential to provide significant improvements of the individual techniques of space geodesy. This talk reviews how time and frequency impacts space geodesy and how the observation techniques may benefit from the recent progress in this field.

The Earth's gravity field is a basic physical parameter, which reflects mass transport and re‐ distribution in the Earth System. It not only contributes to study the Earth's interior physical state and the dynamic mechanism in geophysics, but also provides an important way to re‐ search the Earth's interior mass distribution and characteristics. The gravity field and its changes with time is of great significance for studying various geodynamics and physical processes, especially for the dynamic mechanism of the lithosphere, mantle convection and lithospheric drift, glacial isostatic adjustment (GIA), sea level change, hydrologic cycle, mass balance of ice sheets and glaciers, rotation of the Earth and mass displacement [33; 37; 7; 39; 17 and 18]. For Geodesy, the gravity field is an important parameter to study the size and shape of the Earth. Meanwhile the Earth’s gravity field is very important to determine the trajectory of carrier rocket, long-range weapons, artificial Earth’s satellites and spacecrafts. In addition, the gravity field could provide some signals of pre-, co-, and post-earthquake with mass transport following earthquakes [25; 14]. Therefore, precisely determining Earth’s gravity field and its time-varying information are very important in geodesy, seismology, oceanography, space science and national defense as well as geohazards.

If a layer of air with unusually rapid decrease of water vapor with height and/or increase in temperature with height exists near the earth, electromagnetic refractivity decreases very quickly with height. This condition, called superrefraction or ducting, causes the curvature of radio rays to exceed the curvature of the earth's surface. This effect can trap or reflect radio waves. If refractivity increases with height near the earth, a positive refractive gradient condition called subrefraction exists. In this case, there is upward bending of the electromagnetic wave rather than the normal, slightly downward bending. This bending may become so extreme that the radio wave cannot reach the receive antenna. This phenomenon is known as obstruction fading. The performance of terrestrial fixed point to point microwave radio paths is influenced by three propagation phenomena: rain, multipath, ducting, and obstruction.

Many empirical models have been developed over the years to describe the average location and shape of the Earth's geophysical boundaries; i.e., the bow shock and magnetopause. Several of these models are parameterized according to the external conditions of the solar wind. Because of the comparatively simple nature of these models, they are able to be run quickly and provide reasonable though crude estimates of the shapes of the geophysical boundaries into the near future based upon real-time solar wind observations taken far upstream of the Earth. This work describes the strengths and weaknesses of using such empirical models in a dynamic manner to nowcast and forecast the locations of the geophysical boundaries, and how they are being used for the purposes of space weather.

In the Earth gravity-field model, potential coefficients are projections of the Earth gravitational potential on the corresponding spherical harmonics by degree and order. These coefficients reflect the spectral composition of the Earth gravitational potential in the whole space, but they cannot reflect the spectral composition in a local area. In this study, the Earth gravitational potential was projected on spherical harmonics in a local area using a window function introduced to facilitate the mathematical expressions. Then a model for the spectral structure of the Earth gravitational potential in the local area was established. The model can reflect the signal strength of any gravity-field spectrum component in the local area, which adds information for the description of the gravity field of the Earth, and which has great significance for Earth-sciences research and satellite gravity measurements. Taking the third-degree coefficients, for example, the signal-strength distributions of potential coefficients were computed. The data show that the signal-strength distributions of third-degree coefficients on the sphere surface are independent of the longitude. Among the third-degree coefficients, the zero-order, first-order and second-order coefficients are stronger near the two poles of the Earth, while the third-order coefficient is stronger near the equator.

Geodesy can be defined as the science of the figure of the Earth and its gravitational field, as well as their determination. Even though today the figure of the Earth, understood as the visible Earth’s surface, can be determined purely geometrically by satellites, using Global Positioning System (GPS) for the continents and satellite altimetry for the oceans, it would be pretty useless without gravity. One could not even stand upright or walk without being “told” by gravity where the upright direction is. So as soon as one likes to work with the Earth’s surface, one does need the gravitational field. (Not to speak of the fact that, without this gravitational field, no satellites could orbit around the Earth.) To be different from the existing textbooks, a working knowledge of professional mathematics can be taken for granted. In some areas where professors of geodesy are hesitant to enter too deeply, afraid of losing their students, some fundamental problems can be studied. Of course, there is a brief introduction to terrestrial gravitation as treated in the first few chapters of every textbook of geodesy, such as gravitation and gravity (gravitation plus the centrifugal force of the Earth’s rotation), the geoid, and heights above the ellipsoid (now determined directly by GPS) and above the sea level (a surprisingly difficult problem!). But then, as accuracies rise from 10 6 in 1960 (about

An astronomically calibrated timescale has recently been established (Hilgen, 1991 a, b) for the Pliocene and earliest Pleistocene based on the correlation of dominantly pre- cession controlled sedimentary cycles (sapropels and carbonate cycles) in Mediterranean ma- rine sequences to the precession time series of the astronomical solution of Berger and Loutre ( 1991 ) (hereinafter referred to as Ber90). Here we evaluate the accuracy of this timescale by (1) comparing the sedimentary cycle patterns with 65 oN summer insolation time series of dif- ferent astronomical solutions and (2) a cross-spectral comparison between the obliquity-related components in the 65 oN summer insolation curves and high-resolution paleoclimatic records derived from the same sections used to construct the timescale. Our results show that the car- bonate cycles older than 3.5 m.y. should be calibrated to one precession cycle older than previ- ously proposed. Application of the astronomical solution of Laskar ( 1990) (hereinafter refer- red to as La90) with present-day values for the dynamical ellipticity of the Earth and tidal dis- sipation by the Sun and Moon results in the best fit with the geological record, indicating that this solution is the most accurate from a geological point of view. Application of Ber90, or La90 solutions with dynamical ellipticity values smaller or larger than the present-day value, results in a less obvious fit with the geological record. This implies that the change in the plane- tary shape of the Earth associated with ice loadine$ and unloading near the poles during the last 5.3 million years was too small to drive the precession into resonance with the perturbation term, s-gt+g 5, of Jupiter and Saturn. Our new timescale results in a slight but significant modi- fication of all ages of the sedimentary cycles, bioevents, reversal boundaries, chronostrati- graphic boundaries, and glacial cycles. Moreover, a comparison of this timescale with the as- tronomical timescales of ODP site 846 (Shackleton et al., 1995a, b) and ODP site 659 (Tiede- mann et al., 1994) indicates that all obliquity-related

While it is well known that the rotation of the Earth causes it to be flattened at the poles, one rarely finds a quantitative treatment of this phenomenon in textbooks. Newton himself gave an approximate treatment of the amount of ellipticity and the problem is important in the history of science. Here the simplest model that can account for the phenomenon is considered. The planet is modelled by four bodies that attract gravitationally but are prevented from collapsing by elastic, spring forces. The rotating equilibrium of this system can be calculated exactly and by suitable interpretations one may compare the result with the observed ellipticity. Sammanfattning. Trots att det ?r v?lk?nt att jordens rotation orsakar en avplattning vid polerna hittar man s?llan en kvantitativ behandling av fenomenet i l?rob?cker. Newton sj?lv gjorde en approximativ ber?kning av stroleken p? ellipticiteten och problemet ?r viktigt i vetenskapshistorien. H?r betraktas den enklaste modell som kan beskriva fenomenet. Planeten modelleras med fyra kroppar som attraherar varandra gravitationellt men som f?rhindras att falla samman av elastiska fj?derkrafter. J?mvikten f?r det roterande systemet kan ber?knas exakt och, med hj?lp av l?mpliga tolkningar, kan resultatet j?mf?ras med observerade v?rdet.

La rigidite de la pensee d'Isidore de Seville est liee a la confusion qui caracterise son approche pour etudier la sphericite de la terre. Il pensait que les cieux etaient spheriques, et avait tendance a plutot voir la terre comme un disque occupant le centre

Significant systematic errors can result in precision surveying if the curvature of the earth is ignored in the adjustment of a survey network. The errors can become significant even for networks with very limited extent. An adjustment model for small-scale surveying networks such as precision threedimensional metrology networks is presented in this paper which uses a local Cartesian coordinate system and applies curvature corrections to all the observations in a network. The adjustment model is useful especially for precise industrial metrology networks such as those established using real-time electronic theodolite systems.

On the basis of the fundamental relations of the Molodensky's Earth's figure theory (1945), admitting the inequality of the gravity potentials at the Major Vertical Datum W0 and on the surface of the reference level ellipsoid U0, and taken into account that potential W0 enters into equations directly, it is recomended, W0 should be adopted as a primary geodetic constant. Parameters of the best fitting ellipsoid are not needed for the solution of geodetic problems and for the investigation of the Earth's gravity field. The reason for requiring the reference and actual fields be close is only that the boundary-value problem can be solved in the linear approximation.

To perform a realistic 3D inversion of gravity data covering a significant proportion of the Earth’s surface, it is necessary to take into account the curvature of the Earth. We have developed an algorithm for inverting gravity data in spherical coordinates and have demonstrated this using data covering the continental mass of Australia and surrounding ocean areas. The density structures evident in the crust and uppermost mantle of the resultant 3D inversion model are in broad agreement with knowledge of the geological features for the region and with variations in the depth to the Moho that are present in the AusMoho model.

In this research, a ray tracing model is built up to simulate the propagation of signal in global positioning system (GPS) radio occultation (RO) mission. When GPS signal propagate through the Earth's atmosphere, it will be bent and delayed due to the gradient atmosphere refractive index and received by low Earth orbit (LEO) satellite. Then the parameter profiles of atmosphere can be retrieved by using the received signal. In the previous research, in order to simplify the simulation, the Earth's atmosphere is assumed as spherical symmetry and the positions of GPS and LEO satellites are not considered in simulations. In the model, the shape of the Earth is assumed as an ellipse. The information of European Centre for Medium-Range Weather Forecasts (ECMWF) is used to construct the refractive index of Earth's atmosphere. And two aiming algorithm are developed to control the initial propagating direction of GPS signal to begin from the prescribed GPS satellite position and end at the LEO satellite position. The model is tested and verified by comparing with analytical and observational data.

SUMMARY Gravitational coupling within the Earth had been dismissed until about 10 years ago. Inappropriate application of Newton's theorem to the Earth is cited as a reason for this dismissal. For the theorem a new proof is given, a by-product of which is an exact expression for the gravitational potential within a homogeneous ellipsoidal shell of different outer and inner flattening. The gravitational torque on a spheroid inside such a shell can be analytically calculated. When applied to a simplified model of the earth, the gravitational torque between the mantle and the inner core is found to be enormous*, confirming earlier calculations for realistic earth models based on integration of the Clairaut figure equation. The torque is found to be dominated by a contribution arising from a thin layer surrounding the inner core.

James Ivory (1765–1842) contributed to the mathematical theory of attraction. I describe his efforts (1826–1830) at determining the earth's ellipticity (e) through the adjustment of pendulum observations. At the time, several dozens of such observations had already been made in various latitudes, and their adjustment presented difficulties owing to the local anomalies of gravity. The very possibility of deducing a single ellipticity for the earth remained questionable. While achieving his goal, Ivory made methodical mistakes which he gradually corrected. His final result, 0.00333 < e < 0.00338, compares favorably with the value e = 0.00335 of the so-called Krasovsky ellipsoid. Ivory's work was forgotten mainly because new data, especially on meridian arc measurements, became available rather soon after its publication.

A new model is presented for feedback between rotational and climatic variations, operative on time scales of 10(exp 4) - 10(exp 7) years. Due to the combined effect of planetary perturbations to the Earth's orbit plane and luni-solar torques on the oblate figure of the Earth, the obliquity varies by approximately 1 deg on a 4 x 10(exp 4) year time scale. Associated changes in the seasonal and latitudinal pattern of incident solar radiation cause major glaciations. Mass transport from the oceans to the polar ice sheets during these glaciations can change the gravitational oblateness of the Earth by amounts approaching 1%. As the rate of spin axis precession is directly proportional to the oblateness, the climatically forced mass transport can by dynamically significant. A simple parameterization of this coupled orbital-rotational-climatic system suggests that there is a strong tendency for the system to evolve away from climatically sensitive values of the obliquity. This may explain the mid-Pleistocene transition from an obliquity dominated regime to the present regime in which most climatic variability is concentrated at longer (10(exp 5) year) periods.

One class of surface water models is the shallow water models obtained by depth-averaging the microscale mass and momentum balances. Application of shallow water models to large scale problems (on the order of 1000's of km) requires the use of spherical coordinates. Traditionally, balance laws in spherical coordinates are derived by simply expanding the spatial operators in the standard depth-averaged equations. However, the equations themselves are based on an assumed planar surface so that an inconsistency exists between the derivation and the interpretation. In this article, a method is presented that properly accounts for the curvature of the Earth during the depth-averaging procedure. The derivation gives rise to new terms in both the continuity and momentum balances, terms that we refer to as curvature terms. A scaling analysis evaluates the magnitude of the terms. It is shown that the curvature term in the continuity balance is insignificant when the vertical velocity is small, i.e., at least four ...

In this paper, two factors — the redistribution of the density and the variation in the angular velocity of the Earth rotation, that affect the adopted value of the flattening for equidensity surface within the Earth, are discussed. The computational results show that the contribution of the redistribution of the density in the Earth interior (especially in the core) on the change of the flattening at the core-mantle boundary (CMB) is marginal, and that the calculated value of the flattening at the CMB can be in good agreement with the VLBI observed value so long as the fact that the angular velocity of the Earth rotation has undergone the tidal evolution is taken into account. As a result, this paper presents a set of recommended values of the dynamical parameters of the Earth (see Table III) for computing Earth's forced nutation series.

Christoph Reigber, Prof. Dr.-Ing. Dr.-Ing. E.h, Director of Division 1 'Kinematics and Dynamics of the Earth' of GeoForschungsZentrum Potsdam, is in particular engaged in all aspects of satellite geodesy and its relation to geotectonics, Earth rotation, Earth gravity field, oceanography and atmosphere/ionosphere. He is Director of the CHAMP mission, Co-Principal Investigator of the GRACE mission and Chair of the Coverning Board of the international GPS Service. Peter Schwintzer, Dr.-Ing., Head of Section 1.3, 'Gravity Field and Figure of the Earth' within Division 1 of GeoForschungsZentrum Potsdam, is engaged in global gravity field modelling from space and its geophysical application. He is the Science Data System manager of the CHAMP mission.

We developed and applied a novel numerical scheme for a gravimetric forward modelling of the Earth’s crustal density structures based entirely on methods for a spherical analysis and synthesis of the gravitational field. This numerical scheme utilises expressions for the gravitational potentials and their radial derivatives generated by the homogeneous or laterally varying mass density layers with a variable height/depth and thickness given in terms of spherical harmonics. We used these expressions to compute globally the complete crust-corrected Earth’s gravity field and its contribution generated by the Earth’s crust. The gravimetric forward modelling of large known mass density structures within the Earth’s crust is realised by using global models of the Earth’s gravity field (EGM2008), topography/bathymetry (DTM2006.0), continental ice-thickness (ICE-5G), and crustal density structures (CRUST2.0). The crust-corrected gravity field is obtained after modelling and subtracting the gravitational contribution of the Earth’s crust from the EGM2008 gravity data. These refined gravity data mainly comprise information on the Moho interface and mantle lithosphere. Numerical results also reveal that the gravitational contribution of the Earth’s crust varies globally from 1,843 to 12,010 mGal. This gravitational signal is strongly correlated with the crustal thickness with its maxima in mountainous regions (Himalayas, Tibetan Plateau and Andes) with the presence of large isostatic compensation. The corresponding minima over the open oceans are due to the thin and heavier oceanic crust.

The integrated approach, i.e. the simultaneous processing of GPS satellites and Low Earth Orbiters (LEOs), for satellite orbit determination and estimation of Earth system parameters, is applied to the CHAMP–GRACE–GPS constellation for the purpose of deriving the dynamic and geometric origin of the Earth. The Earth system parameters comprise the low-degree gravity field coefficients and GPS ground station positions representing the center of mass of the Earth as well as the center of figure of the Earth. Based on simulated observations and dedicated changes of the reference models the possibilities and limits of solving and separating the parameters searched for are analyzed and the inherent datum defect can be qualified. The effects of different constraints imposed on the parameters and of various constellations of GPS satellites and LEOs are studied.

Bernhard Riemann speculates on the empirical nature of geometry Our story begins with curiosity about the precise shape of the Earth, which is interesting for the direct evidence it could give that Earth rotates in space. Newton pointed out that the pull of gravity ( toward a center ) against the centrifugal force of a spinning Earth ( away from the axis of rotation) will swell the Earth at its equator, distorting it from a perfect sphere. The effect is small, and at the time no one could look at Earth from space to check its shape.

The projection errors caused by curvature of the earth and relief are two important problems in geometric correction, especially in the case of imaging with large view angles. General polynomial correction model is only effective to flat area and ineffective to correct projection errors caused by both curvature of the earth and relief. According to the generated characteristic of the projection errors, this paper proposes a self-adjustive polynomial model which adds an adjustive factor in the large view angle direction to make effective correction on distorted images with both kinds of projection errors in the absence of precise attitude parameters. This paper first analyzes the principle of projection errors caused by curvature of the earth and relief, and then proposes the improved polynomial model. Experiments show that the proposed model has greatly improved accuracy for correcting the projection errors in the large angle direction compared with general polynomial models.

Attitude control and determination along with the fix and navigation of the spacecraft is the fundamental functional elements in the space scientific research tasks. Nadir vector is usually applied in the attitude determination of spacecraft relative to the Earth, and it is also commonly used as a reference vector in autonomous navigation especially in the satellites' preliminary coarse orbit determination. In the paper, we begin with the shape of the Earth as seen from the space, which is defined by the Earth horizon as seen from the position of the observer. The geometric model of the nadir vector pointing from the spacecraft to the Earth center is established basing on three-dimension earth spheroid model; the effect of Earth flattening on the attitude determination and positioning is discussed, the flattening and irregularity of the Earth ultraviolet layer model are the major error sources resulting in poor accuracy of attitude determination and fix. An error modification method is presented to compensate for the Earth flattening error. Several crucial techniques in determining earth center and nadir vector basing on CCD sensor are stressed in the paper. Noise elimination and image processing means are introduced to restraint the influences of harmful signals in the image acquisitions; edge detection is one of the most important aspects in the process of determining the Earth center, Randomed Hough Transformation method and least square method are effective way to check the edge of earth horizon profile.

We computed the values of flattening / of the equipotentialellipsoid of the Earth's field and of coefficients of the normal gravity,according to the Earth's parameters adopted by the International Unionof Geodesy and Geophysics at its General Assembly of 1967 taking intoaccount the mass of atmosphere and using the closed formulae of the Pizzetti-Somigliana theory that appear in: M. Caputo " The Gravity Field ofthe Earth from classical and modern Methods", Academic Press, 1967.

There are two different definitions of equivalent earth's radius, and the rationality of the two definitions and application scope are controversial. In this paper, the two different definitions of the equivalent earth's radius were introduced, the signification of the equivalent earth's radius was construed in detail, the mapping relationship of the ray parameters of the equivalent earth and the true earth is given, the differentia and relationship between the two definitions are discussed thoroughly, as well as the applications limitations.

I. Abstract STRONAUT Joseph Allen recently made the observation that exploring the Moon in the 1960s was never really about going to the Moon. ―With all the arguments, pro and con, for going to the Moon,‖ he commented, ―no one suggested that we should do it to look at the Earth. But that may in fact be the one important reason.‖ This observation serves as the entree point for this essay that lays out a long range research project exploring the manner in which Western Civilization has conceived of the Earth as an entity and as a home. This essay represents a sneak peek of a new book project now underway. The study will begin with ancient ideas about the Earth as a sphere or as a flat body and trace this theme to the present, as well as the themes of hollow Earth, Spaceship Earth, and Gaia. It will then explore the manner in which scientists came to understand the size and shape of the Earth and its geodesy. It also offers a discussion of the nature of Earth maps and their evolution over time, with all of the attendant issues associated with them, privileging Europe and North America over other regions, etc. Finally, it pursues the place of imagery from space and its contribution to the public understanding of how Western Civilization envisions this planet and its place in the universe. II. Introduction While researching the history of space-based instruments in understanding the Earth as a system, I came to appreciate as never before the manner in which the Earth has been understood by humans from the ancients to the present. Of course, perceptions of a spherical Earth date back to Greek philosophy in the sixth century BCE, and Hellenistic astronomy established beyond serious doubt this planet‘s spherical shape. It was accepted by all those who were educated within Western Civilization by at least the medieval era. Maps, some of them enormously complex and sophisticated, tried to depict what was known about the planet and to a greater or lesser degree communicated this to the literate of the civilization. As humanity explored more fully, especially after the European expansion that began in the fifteenth century, more comprehensive data began to coalesce around the idea of the Earth as a spherical planet like other bodies seen beyond Earth. With the scientific revolution this perception grew deeper and more refined with the result emerging in the latter twentieth century of the Earth as a system; and a new perspective on this planet becoming apparent to the population of the world.

Several sources attest that Anaximander drew a map of the earth and even that he was the first one to do so (see DK 12A1(2) and DK 12A6). His map has been lost, just like his book. Several scholars, however, have tried to reconstruct it. Studying the possibilities of such a reconstruction, I pay special attention to the astronomical implications of drawing a map when one thinks, like Anaximander, that the earth is flat. In Chap. 8, the shape of the earth according to Anaximander will be discussed more extensively.

The discovery of the spherical shape of the earth is the finishing touch of the new world picture that was introduced by Anaximander. Strictly speaking, it is not right to use the word “discovery,” since the idea of a spherical earth, as we will see in this and the next chapter, was the product of metaphysical speculations, rather than a discovery based on verifiable observations. Such observations, of which we now would say that they should have led to the conception of a spherical earth, were met with disbelief. Herodotus, for instance, tells that in the far north, there are people who sleep 6 months of the year and that sailors who navigated around Africa had the sun at their right when sailing to the west (Histories IV 25 and IV 42). Herodotus rejects these stories as typical bluff of sailor men. Panchenko has made plausible that Anaxagoras, being convinced that the earth is flat, argued with those who held that it is spherical 1997:175–178, see also above, Chap. 15). How hard it must have been to evaluate empirical evidence is proven by the fact that Anaxagoras was credited with the true explanation of eclipses but nevertheless apparently did not conclude from the round shadow cast upon the moon that the earth must be spherical (DK 59A42). When and by whom the earth for the first time was considered as a sphere has got lost in the mist of times.

Dans les annees 1730, l'Academie Royale des Sciences a organise des expe- ditions pour determiner la forme de la terre et pour mesurer les degres terrestres en allant de l'equateur vers les poles. Les resultats de ces expeditions sont discutes