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c Nico Sneeuw 2002 2006, These are lecture notes in progress Please contact me sneeuw gis uni stuttgart de. for remarks errors suggestions etc,1 Introduction 6. 1 1 Physical Geodesy 6,1 2 Links to Earth sciences 6. 1 3 Applications in engineering 8,2 Gravitation 10. 2 1 Newtonian gravitation 10,2 1 1 Vectorial attraction of a point mass 11.

2 1 2 Gravitational potential 12,2 1 3 Superposition discrete 13. 2 1 4 Superposition continuous 14,2 2 Ideal solids 15. 2 2 1 Solid homogeneous sphere 15,2 2 2 Spherical shell 19. 2 2 3 Solid homogeneous cylinder 22,2 3 Tides 26,2 4 Summary 27. 3 Rotation 28, 3 1 Kinematics acceleration in a rotating frame 29.

3 2 Dynamics precession nutation polar motion 32, 3 3 Geometry defining the inertial reference system 36. 3 3 1 Inertial space 36,3 3 2 Transformations 36,3 3 3 Conventional inertial reference system 38. 3 3 4 Overview 40,4 Gravity and Gravimetry 42,4 1 Gravity attraction and potential 42. 4 2 Gravimetry 47, 4 2 1 Gravimetric measurement principles pendulum 47. 4 2 2 Gravimetric measurement principles spring 51. 4 2 3 Gravimetric measurement principles free fall 55. 4 3 Gravity networks 57,4 3 1 Gravity observation procedures 58.

4 3 2 Relative gravity observation equation 58,5 Elements from potential theory 60. 5 1 Some vector calculus rules 61,5 2 Divergence Gauss 62. 5 3 Special cases and applications 65,5 4 Boundary value problems 68. 6 Solving Laplace s equation 71,6 1 Cartesian coordinates 71. 6 1 1 Solution of Dirichlet and Neumann BVPs in x y z 73. 6 2 Spherical coordinates 75, 6 2 1 Solution of Dirichlet and Neumann BVPs in r 78.

6 3 Properties of spherical harmonics 80, 6 3 1 Orthogonal and orthonormal base functions 80. 6 3 2 Calculating Legendre polynomials and Legendre functions 85. 6 3 3 The addition theorem 88, 6 4 Physical meaning of spherical harmonic coefficients 89. 6 5 Tides revisited 92,7 The normal field 93,7 1 Normal potential 94. 7 2 Normal gravity 97,7 3 Adopted normal gravity 98. 7 3 1 Formulae 98,7 3 2 GRS80 constants 100,8 Linear model of physical geodesy 102.

8 1 Two step linearization 102,8 2 Disturbing potential and gravity 103. 8 3 Anomalous potential and gravity 108,8 4 Gravity reductions 111. 8 4 1 Free air reduction 112,8 4 2 Bouguer reduction 113. 8 4 3 Isostasy 115,9 Geoid determination 120,9 1 The Stokes approach 121. 9 2 Spectral domain solutions 123,9 2 1 Local Fourier 124.

9 2 2 Global spherical harmonics 125,9 3 Stokes integration 126. 9 4 Practical aspects of geoid calculation 129,9 4 1 Discretization 129. 9 4 2 Singularity at 0 130,9 4 3 Combination method 132. 9 4 4 Indirect effects 134,A Reference Textbooks 136. B The Greek alphabet 137,1 Introduction,1 1 Physical Geodesy.

Geodesy aims at the determination of the geometrical and physical shape of the Earth. and its orientation in space The branch of geodesy that is concerned with determining. the physical shape of the Earth is called physical geodesy It does interact strongly with. the other branches though as will be seen later, Physical geodesy is different from other geomatics disciplines in that it is concerned with. field quantities the scalar potential field or the vectorial gravity and gravitational fields. These are continuous quantities as opposed to point fields networks pixels etc which. are discrete by nature, Gravity field theory uses a number of tools from mathematics and physics. Newtonian gravitation theory relativity is not required for now. Potential theory,Vector calculus,Special functions Legendre. Partial differential equations,Boundary value problems. Signal processing, Gravity field theory is interacting with many other disciplines A few examples may.

clarify the importance of physical geodesy to those disciplines The Earth sciences. disciplines are rather operating on a global scale whereas the engineering applications. are more local This distinction is not fundamental though. 1 2 Links to Earth sciences, Oceanography The Earth s gravity field determines the geoid which is the equipo. tential surface at mean sea level If the oceans would be at rest no waves no currents. no tides the ocean surface would coincide with the geoid In reality it deviates by. 1 2 Links to Earth sciences, up to 1 m The difference is called sea surface topography It reflects the dynamical. equilibrium in the oceans Only large scale currents can sustain these deviations. The sea surface itself can be accurately measured by radar altimeter satellites If the. geoid would be known up to the same accuracy the sea surface topography and conse. quently the global ocean circulation could be determined The problem is the insufficient. knowledge of the marine geoid, Geophysics The Earth s gravity field reflects the internal mass distribution the de. termination of which is one of the tasks of geophysics By itself gravity field knowledge. is insufficient to recover this distribution A given gravity field can be produced by an. infinity of mass distributions Nevertheless gravity is is an important constraint which. is used together with seismic and other data, As an example consider the gravity field over a volcanic island like Hawaii A volcano by. itself represents a geophysical anomaly already which will have a gravitational signature. Over geologic time scales a huge volcanic mass is piled up on the ocean sphere This. will cause a bending of the ocean floor Geometrically speaking one would have a cone. in a bowl This bowl is likely to be filled with sediment Moreover the mass load will. be supported by buoyant forces within the mantle This process is called isostasy The. gravity signal of this whole mass configuration carries clues to the density structure. below the surface, Geology Different geological formations have different density structures and hence.

different gravity signals One interesting example of this is the Chicxulub crater partially. on the Yucatan peninsula Mexico and partially in the Gulf of Mexico This crater. with a diameter of 180 km was caused by a meteorite impact which occurred at the K T. boundary cretaceous tertiary some 66 million years ago This impact is thought to. have caused the extinction of dinosaurs The Chicxulub crater was discovered by careful. analysis of gravity data, Hydrology Minute changes in the gravity field over time after correcting for other. time variable effects like tides or atmospheric loading can be attributed to changes in. hydrological parameters soil moisture water table snow load For static gravimetry. these are usually nuisance effects Nowadays with precise satellite techniques hydrology. is one of the main aims of spaceborne gravimetry Despite a low spatial resolution the. results of satellite gravity missions may be used to constrain basin scale hydrological. parameters,1 Introduction, Glaciology and sea level The behaviour of the Earth s ice masses is a critical indicator. of global climate change and global sea level behaviour Thus monitoring of the melting. of the Greenland and Antarctica ice caps is an important issue The ice caps are huge. mass loads sitting on the Earth s crust which will necessarily be depressed Melting. causes a rebound of the crust This process is still going on since the last Ice Age but. there is also an instant effect from melting taking place right now The change in surface. ice contains a direct gravitational component and an effect due to the uplift Therefore. precise gravity measurements carry information on ice melting and consequently on sea. level rise,1 3 Applications in engineering, Geophysical prospecting Since gravity contains information on the subsurface density. structure gravimetry is a standard tool in the oil and gas industry and other mineral. resources for that matter It will always be used together with seismic profiling test. drilling and magnetometry The advantages of gravimetry over these other techniques. relatively inexpensive, non destructive one can easily measure inside buildings. compact equipment e g for borehole measurements, Gravimetry is used to localize salt domes or fractures in layers to estimate depth and.

in general to get a first idea of the subsurface structure. Geotechnical Engineering In order to gain knowledge about the subsurface structure. gravimetry is a valuable tool for certain geotechnical civil engineering projects One. can think of determining the depth to bedrock for the layout of a tunnel Or making. sure no subsurface voids exist below the planned building site of a nuclear power plant. For examples see the micro gravity case histories and applications on. http www geop ubc ca ubcgif casehist index html or. http www esci keele ac uk geophysics Research Gravity. Geomatics Engineering Most surveying observables are related to the gravity field. i After leveling a theodolite or a total station its vertical axis is automati. cally aligned with the local gravity vector Thus all measurements with these. instruments are referenced to the gravity field they are in a local astronomic. 1 3 Applications in engineering, frame To convert them to a geodetic frame the deflection of the vertical. and the perturbation in azimuth A must be known, ii The line of sight of a level is tangent to the local equipotential surface So lev. elled height differences are really physical height differences The basic quantity. of physical heights are the potentials or the potential differences To obtain pre. cise height differences one should also use a gravimeter. W g dx g dh gi hi, The hi are the levelled height increments Using gravity measurements gi. along the way gives a geopotential difference which can be transformed into a. physical height difference for instance an orthometric height difference. iii GPS positioning is a geometric techniques The geometric gps heights are. related to physically meaningful heights through the geoid or the quasi geoid. h H N orthometric height geoid height,h H n normal height quasi geoid height. In geomatics engineering gps measurements are usually made over a certain. baseline and processed in differential mode In that case the above two formulas. become h N H etc The geoid difference between the baseline s. endpoints must therefore be known, iv The basic equation of inertial surveying is x a which is integrated twice to.

provide the trajectory x t The equation says that the kinematic acceleration. equals the specific force vector a the sum of all forces per unit mass acting. on a proof mass An inertial measurement unit though measures the sum of. kinematic acceleration and gravitation Thus the gravitational field must be. corrected for before performing the integration,2 Gravitation. 2 1 Newtonian gravitation, In 1687 Newton1 published his Philosophiae naturalis principia mathematica or Prin. cipia in short The Latin title can be translated as Mathematical principles of natural. philosophy in which natural philosophy can be read as physics Although Newton was. definitely not the only physicist working on gravitation in that era his name is nev. ertheless remembered and attached to gravity because of the Principia The greatness. of this work lies in the fact that Newton was able to bring empirical observations on. a mathematical footing and to explain in a unifying manner many natural phenom. planetary motion in particular elliptical motion as discovered by Kepler2. free fall e g the famous apple from the tree,equilibrium shape of the Earth. Newton made fundamental observations on gravitation. The force between two attracting bodies is proportional to the individual masses. The force is inversely proportional to the square of the distance. The force is directed along the line connecting the two bodies. Mathematically the first two are translated into,F12 G 2 2 1. Sir Isaac Newton 1642 1727, Johannes Kepler 1571 1630 German astronomer and mathematician formulated the famous laws.

of planetary motion i orbits are ellipses with Sun in one of the foci ii the areas swept out by the. line between Sun and planet are equal over equal time intervals area law and iii the ratio of the. cube of the semi major axis and the square of the orbital period is constant or n2 a3 GM. 2 1 Newtonian gravitation, in which G is a proportionality factor It is called the gravitational constant or Newton. constant It has a value of G 6 672 10 11 m3 s 2 kg 1 or N m2 kg 2. Remark 2 1 mathematical model of gravitation Soon after the publication of the Prin. cipia Newton was strongly criticized for his law of gravitation e g by his contemporary. Huygens Equation 2 1 implies that gravitation acts at a distance and that it acts. instantaneously Such action is unphysical in a modern sense For instance in Einstein s. relativity theory no interaction can be faster than the speed of light However Newton. did not consider his formula 2 1 as some fundamental law Instead he saw it as a con. venient mathematical description As such Newton s law of gravitation is still a viable. model for gravitation in physical geodesy, Equation 2 1 is symmetric the mass m1 exerts a force on m2 and m2 exerts a force. of the same magnitude but in opposite direction on m1 From now on we will be. interested in the gravitational field generated by a single test mass For that purpose. we set m1 m and we drop the indices The mass m2 can be an arbitrary mass at an. Physical geodesy is di erent from other geomatics disciplines in that it is concerned with eld quantities the scalar potential eld or the vectorial gravity and gravitational elds These are continuous quantities as opposed to point elds networks pixels etc which

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