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Schedule of the Course^{1}

**GEODES****Y**

**UNIVERSI****TY OF ALMERIA**

**Theoretical Contents**

**Blo****ck I. Geometrical Geodesy**^{1}

### Chapter I. Definition, division and objective of Geodesy

· What is Geodesy?

· Problems of Geodesy.

· Division of Geodesy.

· Historical and technical data. The Spanish Geodetic Network.

The incorporation of the Spanish Geodetic Network to others European networks.

### Chapter II. Basic concepts of classic differential geometry

Representation of parametric curves.

Representation of parametric surfaces.

First differential form.

Second differential form. Meusnier’s theorem.

Principal directions. Lines of curvature.

First and second specie Christoffel’s symbols.

· Egregium theorem and Gauss’ curvature.

### Chapter III. General notions about Geodesy

Geodesic curvature.

· Geodesic torsion.

__Geodesic lines.__

· Gauss-Bonnet’s theorem. Applications.

Surfaces of revolution.

· Clairaut’s theorem.

### Chapter IV. Geometry of the revolution ellipsoid

· Elements of the meridian ellipse.

Differential geometry of the revolution ellipsoid.

Radii of curvature.

· Longitude of the meridian and parallel arcs.

__Geodesic lines over the ellipsoid.__

· Clairaut and Laplace formulas.

· Astronomical and geodetic coordinate systems.

· Relations between astronomical and geodetic coordinates.

Coordinate transformation and datum change.

### Chapter V. Representation of a surface over another one: charts

Representations of a surface over another one.

Conformal representation.

Isothermal parameters.

Local scale of a representation.

· Calculation of the corrections in the charts.

Isothermal parameters in the revolution surfaces.

· Mercator projection. Loxodromical and orthodromical lines.

· Projections more used in Geodesy and Topography. UTM projection.

### Chapter VI. Basic applications of the geometrical geodesy

· __Measure of distances__.

· __Measure of azimuths__.

· Measure of cenital distances.

Reciprocal normal sections. Relations.

· Longitude of an arc of normal section.

· Difference of azimuths between normal sections.

__B____ASIC REFERENCES__

· D. J. Struik. *Classic Differential Geometry*.

Addison-Wesley Publishing Company. Massachusetts, 1961.

· P. Vanicek and E. Krakiwsky. *Geodesy: The Concepts*.

Elsevier Science Publishers. Amsterdam, 1986.

· W. Torge. *Geodesy*.

Walter de Gruyter. Berlin, 1991.

**Blo****ck II. Physical Geodesy**^{1}

### Chapter VII. The normal gravity field

· The field of terrestrial gravity.

Level surfaces.

· Plumb lines and their curvature.

Natural coordinates.

· Potential of the earth gravity field. Harmonics of lower degree.

The gravity field of the level ellipsoid.

· Normal gravity.

· The normal potential in spherical harmonics.

· Expansions for the normal gravity field.

Numerical values. International ellipsoid. GRS80 reference system.

### Chapter VIII. The anomalous gravity field

Geoid undulations and vertical deviations.

Expansion of the disturbing potential in spherical harmonics.

Stokes’ formula.

· Expansion of the Stokes’ function in spherical harmonics.

Generalization to an arbitrary reference ellipsoid.

Generalization of the Stokes’ formula.

· Determination of the Earth’s physical constants.

· Terrestrial medium ellipsoid.

· Vertical deviations.

· Vertical gradient of the gravity.

Evaluation of integral formulas by means of the Fast Fourier Transform (FFT).

### Chapter IX. Gravity reduction

Gravity reduction: statement of the problem.

· Formulas to calculate gravity reductions.

Bouguer reduction. Free-air reduction. Correction of the topography.

Helmert reduction.

· Others gravity reductions.

· The indirect effect of the gravity reduction.

The indirect effect in the Helmert reduction.

__C____hapter X. Determination of the geoid__

Operational procedure: pre-processing y post-processing.

Validity of the spherical approximations.

The Stokes’ integral in the spectral domaine.

Correction of the topography: pre-processing y post-processing.

Atmospheric correction.

### Chapter XI. Precise levelling

Geometrical levelling.

· Geopotential numbers and dynamic altitudes.

· Poincaré and Prey gravity reductions.

Orthometric altitudes.

· Normal altitudes.

Comparison between different altitude systems.

· Practical procedure to perform a precise levelling.

### Chapter XII. Utilization of GPS in Geodesy

· Determination of the point position over the earth surface using GPS.

· Relation between ellipsoidal and astronomical coordinates: the vertical deviation.

Relation between the vertical deviation components and the geoid model.

· Determination of orthometric heights with GPS.

GPS/levelling procedure: precise measure of the geoid undulation.

· An example of the geoid calculation by means of GPS/levelling: ALGESTAR project.

__BAIC REFERENCES__

· Bürki B. and Marti U. (1990). *The Swiss Geoid Computation: A Status Report. *International Association of Geodesy Simposia. Symposia nº 106: Determination of the Geoid. Springer Verlag.

· Corchete V., Chourak M. and Khattach D. (2005). *The high-resolution gravimetric geoid of Iberia: IGG2005**.* Geophys. J. Int., 162, 676–684.

· Haagmans R., de Min E. and von Gelderen M. (1993). *Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral*. Manuscripta Geodaetica, 18, 227-241.

· W. A. Heiskanen and H. Moritz. *Physical Geodesy*.

W. H. Freeman. San Francisco, 1967.

· Kuroishi Y. (1995). *Precise determination of geoid in the vicinity of Japan*. Bulletin of the Geographical Survey Institute, 41, 1-94.

· H. Moritz. *Advanced Physical Geodesy*.

Herbert Wichmann Verlag. Karlsruhe, 1980.

· Pavlis N. K. (1988). *Modeling and estimation of a low degree geopotential model from terrestrial gravity data*. Rep. 386, Dep. Geod. Sci. Surv., Ohio State University.

· Rapp R. H. (1971). *Methods for the computation of geoid undulations from potential coefficients*. Bull. Géod., 101, 283-297.

· Schwarz K. P., Sideris M. G. And Forsberg R. (1990). *The use of FFT in physical geodesy*. Geophys. J. Int., 100, 485-514.

· Sideris M. G. (1990). *Rigorous gravimetric terrain modelling using Molodensky’s operator*. Manuscripta Geodaetica, 15, 97-106.

· Strang van Hess G. (1990). *Stokes formula using Fast Fourier Techniques*. Manuscripta Geodaetica, 15, 235-239.

· Wichiencharoen C. (1982a). *The indirect effects on the computation of geoid undulations*. Rep. 336, Dep. Geod. Sci. Surv., Ohio State University.

· Wichiencharoen C. (1982b). *FORTRAN programs for computing geoid undulations from potential coefficients and gravity anomalies*. Internal Rep., Dep. Geod. Sci. Surv., Ohio State University

**Pr****actical Contents**

__Blo____ck I. Geometrical Geodesy__^{1}

**Practical**** 1. Calculation of the area of a revolution surface and its volume enclosed **:

In this practical the students must perform a computer program written in BASIC o FORTRAN, to determine the area and volume of the *revolution surface* obtained from a digitized contour. For it, the students can digitize any contour and save it in a data file. Later, they must perform the computer program that calculates the area and volume, of the *revolution surface* obtained rotating 180 degrees each one of the four quadrants of this digitized contour.

**Practical**** 2. Transformation of coordinates and datum change **:

In this practical the students must perform a computer program written in BASIC o FORTRAN, to perform the *transformation of coordinates* from geodetics given in a datum to coordinates given in another datum. This program must also convert geodetic coordinates in cartesians coordinates and vice versa.

**Practical**__ 3. Use of different projections: Mercator, Lambert and UTM__:

In this practical the students will use a computer program to perform the conversion between geodetic coordinates and UTM coordinates and vice versa. Also, they will perform other *coordinate transformations* as the Lambert or Mercator transformations.

**Practical**** 4. Measure of distances by the Väisälä method**

This method is based on the physical phenomena of the *light interference *and it requires the use of a well-known distance d_{0}. Then, it can be measured distances, multiples of d_{0}, by means of a suitable combination of lenses and mirrors.

**Practical**** 5. Geodimeter**

This method is based on the constancy of the *light velocity*. Based on this fact, it can be measured any distance if the travel time of the light ray is known.

**Practical**** 6. Optical and electronic theodolite**

In this practical the students will use theodolites to measure angles in a *triangulation**campaign*. In this practical the students learn to use optical and electronic theodolites, knowing the advantage and inconvenient that has both kind of instrument.

**Practical**** 7. Distance meter**

Actually, electronic instruments are used to determine small distances, with the goal of measure geodetic bases, usually. In this practical the students will learn the use of electronic distance meters, knowing the advantage and inconvenient that has the use of this kind of instrument.

**Practical**** 8. Total station operation**

In this practical the students will learn the use of a *total station* to determine* positions* and *height**differences*.

__Blo____ck II. Physical Geodesy__^{1}

**Practical**** 9. Determination of the gravitational constant ";G";**

The goal of this practical is the obtaining of estimation for the value of the *gravitational constant*, by means of a torsion balance.

**Practical**** 10. Determination of ";g"; by the free fall method**

The goal of this practical is the determination of the absolute value for the *gravity acceleration* in a point of the earth, by means of the measure of the fall time of a body, which fall distance is well known. This experiment is the operative foundation of the *absolute gravimeters*, which precision is the highest achieved up to now.

**Practical 11. The mathematical pendulum. Determination of the gravity acceleration**

One of the easiest experiments to determine the *gravity acceleration* consists on the measure of the period of a *mathematical pendulum*, constituted by a point mass suspended in an extreme of a thin thread. This is the experience that will be performed in this practical.

**Practical**** 12. Physical pendulum**

The goal of this practical is the experimental study of the *physical pendulum* and the determination of the *gravity acceleration*. This experiment is the operative foundation of the *relative**gravimeters*. These instruments are more easy to handle and to operate that the absolute gravimeters, but the precision achieved by the relative gravimeters is minor than the precision achieved by the absolute gravimeters.

**Practical 13. Kater pendulum**

The goal of this practical is the determination of the *gravity acceleration *by means of a reversible Kater’s pendulum. In this practical will be also studied the physical concepts involved in this experiment.

**Practical**** 14. Campaign gravimeter**

The goal of this practical is the determination de la *gravity acceleration* by means of a *campaign gravimeter*, which precision is minor than the precision achieved by an absolute gravimeter, but it is more easy to handle and operate in geodetic and gravimetric campaigns.

**Practical**__ 15. Calculation of free-air and Bouguer anomalies__:

In this practical the students must perform a computer program written in BASIC o FORTRAN, to calculate* free-air and Bouguer anomalies *(in the GRS80 reference system) from observed gravity data as the gravity data measured in the practical 14.

**Practical**__ 16. Computation of the geoid from a geopotential model__:

In this practical the students must perform a computer program written in BASIC o FORTRAN, to calculatethe* geoid undulation N *from the coefficients of a geopotential model, supplied in an ASCII data file. This program could be used in the practical 17 to obtain values for *N* that can be compared with the values for *N* obtained en the practical 18.

**Practical**** 17. GPS positioning**

In this practical the students must determine the *position of a point on the earth surface* by means of the GPS instrument, calculating also *orthometric**heights* with the use of a *geoid model*. For it, all __geoid models__ available for the study area will be used, including the *geopotential model* which computer program has been used in the practical 16.

**Practical**** 18. Operation of a high-precision level**

The goal of this practical is the study of the operation and handling of a *precision level*, to determine height differences and to perform a levelling process, including gravity data that can be determined by means of the practical 14. The orthometric heights obtained can be compared with the ones calculated in the practical 17. Thus, it can be proved the precision of all __geoid models__ available for the study area.

**Tutorial Contents**

### Tutorial I. Error analysis and treatment of the data obtained in the laboratory

· Concepts of precision, sensibility and accuracy.

· Classification of the errors.

· Absolute error. Relative error.

· Determination of the errors produced in direct measures.

· Determination of the error in a magnitude indirectly measured.

· Construction of graphics.

· Fit to a curve by least square method.

· Interpolation in tables of simple input.

· Interpolation in tables of double input.

### Tutorial II. Determination of astronomical latitude, longitude and azimuth

· General considerations.

· Determination of the astronomical latitude.

· Determination of the clock correction.

· Calculation of the astronomical longitude.

· Azimuth determinations.

### Tutorial III. Resolution of spherical triangles

· General concepts.

· Resolution of spherical triangles using the Legendre’s theorem.

· Resolution of spherical triangles using the aditaments theorem.

· Spherical excess. Close error of a triangle.

### Tutorial IV. La equation of Laplace and the spherical harmonics

· Equation of Laplace in spherical coordinates.

· Spherical harmonics.

· Surface spherical harmonics.

· Functions of Legendre.

· Functions of Legendre of second class.

· Theorem of representation and orthogonal relations.

· Spherical harmonics fully normalized.

### Tutorial V. The motions of Earth

· Motion of rotation.

· Precession y nutation.

· Chandler’s wobble or free nutation.

· Observations of the polar motion. International Latitude Service (ILS).

### Tutorial VI. Spatial Geodesy

· Methods of observation.

· Determination of the earth size by observations of Moon.

· Dynamic effects of the earth flattering.

· Determination of the earth flattering from the precession.

· Orbit of an artificial terrestrial satellite (SAT).

· Determination of zonal harmonics.

· Rectangular coordinates of a SAT and their perturbations.

· Determination of tesseral harmonics and station positions.

· Photographic determination of the SAT direction.

· Very long base-line Interferometry (VLBI).

· Distance measurements to a SAT.

· Satellite altimetry.

· Measurements using the Doppler Effect.

### Tutorial VII. Theory of tides and tide corrections in geodetic measurements

· Moon-Sun potential and attraction.

· Tide potential and tide force.

· Vertical deviation and geoid tide.

· Doodson’s constant. Tide potential and tide force.

· Expansion of the tide potential as a time function.

· Doodson’s expansion.

· Cartwright and Tayler expansion.

· Tamura expansion.

· Harmonic expansion of the tide potential.

· Tide friction and tide momentum.

· Love numbers and description of the tide deformations.

· The load tide.

· Tide corrections in the determination of geodetic heights.

**B****ASIC REFERENCES**

· Cartwright D. E. (1967). *A unified analysis of tides and surges round north and east Britain*. National Institute of Oceanography, Wormley, Godalming, Surrey, 263, 1-55.

· Cartwright D. E. and Tayler R. J. (1971). *New computations of the tide-generating potential*. Geophys. J. R. astr. Soc., 23, 45-74.

· Cartwright D. E. and Tayler R. J. (1973). *Corrected tables of tidal harmonics*. Geophys. J. R. astr. Soc., 33, 253-264.

· Chiaruttini C. (1976). *Tidal loading in the Italian Peninsula*. Geophys. J. R. astr. Soc., 46, 773-793.

· Doodson H. R. (1921). *The harmonic development of the tide-generating potential*. Proc. R. Soc. A, 100, 305-329.

· Heck B. (1991). *Tidal corrections in geodetic height determination*. International Association of Geodesy Symposia, Springer-Verlag, 108, 11-24.

· W. A. Heiskanen and H. Moritz. *Physical Geodesy*.

W. H. Freeman. San Francisco, 1967.

· P. Melchior. *The tides of the planet Earth.*

Pergamon Press. Oxford, 1983.

· Rapp R. H. (1989). *The treatment of permanent tidal effects in the analysis of Satellite altimeter data for sea surface topography*. Manuscripta Geodatetica, 14, 368-372.

· F. D. Stacey. *Physics of the Earth. Third Edition*.

Brookfield Press. Kenmore, Brisbane (Australia), 1994.

· Tamura Y. (1983). *A harmonic development of the tide-generating potential*. International Latitude Observatory of Mizusawa, 6813-6855.

· P. Vanicek and E. Krakiwsky. *Geodesy: The Concepts*.

Elsevier Science Publishers. Amsterdam, 1986.

**REFERENCES**

**B****ASIC REFERENCES**

· Bürki B. and Marti U. (1990). *The Swiss Geoid Computation: A Status Report. *International Association of Geodesy Simposia. Symposia nº 106: Determination of the Geoid. Springer Verlag.

· Cartwright D. E. (1967). *A unified analysis of tides and surges round north and east Britain*. National Institute of Oceanography, Wormley, Godalming, Surrey, 263, 1-55.

· Cartwright D. E. and Tayler R. J. (1971). *New computations of the tide-generating potential*. Geophys. J. R. astr. Soc., 23, 45-74.

· Cartwright D. E. and Tayler R. J. (1973). *Corrected tables of tidal harmonics*. Geophys. J. R. astr. Soc., 33, 253-264.

· Chiaruttini C. (1976). *Tidal loading in the Italian Peninsula*. Geophys. J. R. astr. Soc., 46, 773-793.

· Corchete V., Chourak M. and Khattach D. (2005). *The high-resolution gravimetric geoid of Iberia: IGG2005.* Geophys. J. Int., 162, 676–684.

· Doodson H. R. (1921). *The harmonic development of the tide-generating potential*. Proc. R. Soc. A, 100, 305-329.

· Haagmans R., de Min E. and von Gelderen M. (1993). *Fast evaluation of convolution integrals on the sphere using 1D FFT, and a comparison with existing methods for Stokes’ integral*. Manuscripta Geodaetica, 18, 227-241.

· Heck B. (1991). *Tidal corrections in geodetic height determination*. International Association of Geodesy Symposia, Springer-Verlag, 108, 11-24.

· W. A. Heiskanen and H. Moritz. *Physical Geodesy*.

W. H. Freeman. San Francisco, 1967.

· Kuroishi Y. (1995). *Precise determination of geoid in the vicinity of Japan*. Bulletin of the Geographical Survey Institute, 41, 1-94.

· P. Melchior. *The tides of the planet Earth.*

Pergamon Press. Oxford, 1983.

· H. Moritz. *Advanced Physical Geodesy*.

Herbert Wichmann Verlag. Karlsruhe, 1980.

· Pavlis N. K. (1988). *Modeling and estimation of a low degree geopotential model from terrestrial gravity data*. Rep. 386, Dep. Geod. Sci. Surv., Ohio State University.

· Rapp R. H. (1971). *Methods for the computation of geoid undulations from potential coefficients*. Bull. Géod., 101, 283-297.

· Rapp R. H. (1989). *The treatment of permanent tidal effects in the analysis of Satellite altimeter data for sea surface topography*. Manuscripta Geodatetica, 14, 368-372.

· Schwarz K. P., Sideris M. G. And Forsberg R. (1990). *The use of FFT in physical geodesy*. Geophys. J. Int., 100, 485-514.

· Sideris M. G. (1990). *Rigorous gravimetric terrain modelling using Molodensky’s operator*. Manuscripta Geodaetica, 15, 97-106.

· F. D. Stacey. *Physics of the Earth. Third Edition*.

Brookfield Press. Kenmore, Brisbane (Australia), 1994.

· Strang van Hess G. (1990). *Stokes formula using Fast Fourier Techniques*. Manuscripta Geodaetica, 15, 235-239.

· D. J. Struik. *Classic Differential Geometry*.

Addison-Wesley Publishing Company. Massachusetts, 1961.

· Tamura Y. (1983). *A harmonic development of the tide-generating potential*. International Latitude Observatory of Mizusawa, 6813-6855.

· W. Torge. *Geodesy*.

Walter de Gruyter. Berlin, 1991.

· P. Vanicek and E. Krakiwsky. *Geodesy: The Concepts*.

Elsevier Science Publishers. Amsterdam, 1986.

· Wichiencharoen C. (1982a). *The indirect effects on the computation of geoid undulations*. Rep. 336, Dep. Geod. Sci. Surv., Ohio State University.

· Wichiencharoen C. (1982b). *FORTRAN programs for computing geoid undulations from potential coefficients and gravity anomalies*. Internal Rep., Dep. Geod. Sci. Surv., Ohio State University

**ADDITIONAL**__ REFERENCES__ ............................................ (

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