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ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS

ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS. Sten Claessens and Will Featherstone Western Australian Centre for Geodesy Curtin University of Technology Perth, Australia. IGFS 2006, ISTANBUL. Introduction (1/2). EXISTING ELLIPSOIDAL CORRECTION METHODS.

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ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS

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  1. ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS Sten Claessens and Will Featherstone Western Australian Centre for Geodesy Curtin University of Technology Perth, Australia IGFS 2006, ISTANBUL

  2. Introduction (1/2) EXISTING ELLIPSOIDAL CORRECTION METHODS • Many methods to compute ellipsoidal corrections to geoid heights exist • all rely on approximations to the order of the square of the eccentricity of the ellipsoid • many are limited to the use of only one or two choices of reference radius R • many can not be applied if the Stokes kernel is modified • many are complicated and/or computationally inefficient • they generally don’t agree with one another

  3. Introduction (2/2) REPRESENTATION OF ELLIPSOIDAL CORRECTIONS • Ellipsoidal corrections to geoid heights can be represented by: • an integration over the sphere or ellipsoid • a spherical harmonic expansion • The spherical harmonic representation is preferred, because: • computation of corrections is practical and efficient, due to the domination of long wavelengths • The spherical harmonic coefficients beyond degree 20 only contribute 10% of the total ellipsoidal correction

  4. Formulation (1/3) DEFINITION OF ELLIPSOIDAL CORRECTIONS ‘ellipsoidal’ geoid height ‘spherical’ geoid height

  5. Formulation (1/3) DEFINITION OF ELLIPSOIDAL CORRECTIONS ‘ellipsoidal’ geoid height ‘spherical’ geoid height ellipsoidal correction

  6. Formulation (2/3) COMPUTATION OF CORRECTION COEFFICIENTS • Spherical harmonic synthesis and analysis

  7. Formulation (2/3) COMPUTATION OF CORRECTION COEFFICIENTS • Spherical harmonic synthesis and analysis • or • Spherical harmonic coefficient transformation

  8. Formulation (3/3) RECAPITULATION • Ellipsoidal corrections can easily be described by surface spherical harmonic coefficients • computation of the coefficients is straightforward, application of the coefficients even more so • no approximations to the order of the eccentricity of the ellipsoid are required (even though all existing methodologies rely on them)

  9. Choice of reference sphere (1/5) INFLUENCE OF THE REFERENCE SPHERE RADIUS Ellipsoidal corrections depend upon the choice of the reference sphere radius R Many existing formulations only allow for one or two choices of R

  10. Choice of reference sphere (2/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  11. Choice of reference sphere (2/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  12. Choice of reference sphere (3/5) A VARIABLE REFERENCE SPHERE RADIUS The reference sphere radius can be set equal to the ellipsoidal radius for each computation point The ellipsoidal correction coefficients can still be found:

  13. Choice of reference sphere (4/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  14. Choice of reference sphere (5/5) SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL CORRECTIONS

  15. Choice of reference sphere (5/5) SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL CORRECTIONS

  16. Choice of reference sphere (5/5) SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL CORRECTIONS

  17. Choice of reference sphere (5/5) SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL CORRECTIONS

  18. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: is equivalent to the simple Stokes integral

  19. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: is equivalent to the simple Stokes integral  ellipsoidal corrections are also the same, unless an additional approximation is applied

  20. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: is equivalent to the simple Stokes integral  ellipsoidal corrections are also the same, unless an additional approximation is applied

  21. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: is equivalent to the simple Stokes integral  ellipsoidal corrections are also the same, unless an additional approximation is applied

  22. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: is equivalent to the simple Stokes integral  ellipsoidal corrections are also the same, unless an additional approximation is applied

  23. Modified kernels (1/5) ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS The combined geoid solution with modified kernel: The ellipsoidal correction becomes:

  24. Modified kernels (2/5) THE SPHEROIDAL STOKES KERNEL Wong and Gore (1969) modification: 

  25. Modified kernels (2/5) THE SPHEROIDAL STOKES KERNEL Wong and Gore (1969) modification:  global absolute maximum of ellipsoidal corrections (excluding first degree term)

  26. Modified kernels (3/5) THE MOLODENSKY-MODIFIED SPHEROIDAL STOKES KERNEL Vaníček and Kleusberg (1987) modification: 

  27. Modified kernels (4/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  28. Modified kernels (4/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  29. Modified kernels (4/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS

  30. Modified kernels (5/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n  2)

  31. Modified kernels (5/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n  2)

  32. Modified kernels (5/5) ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n  2)

  33. Summary and Conclusions • Ellipsoidal corrections can easily be computed using surface spherical harmonic coefficients of the disturbing potential and gravity anomalies

  34. Summary and Conclusions • Ellipsoidal corrections can easily be computed using surface spherical harmonic coefficients of the disturbing potential and gravity anomalies • Ellipsoidal corrections to modified kernels can be found using the same set of correction coefficients

  35. Summary and Conclusions • Ellipsoidal corrections can easily be computed using surface spherical harmonic coefficients of the disturbing potential and gravity anomalies • Ellipsoidal corrections to modified kernels can be found using the same set of correction coefficients • Choosing the reference radius equal to the ellipsoidal radius significantly reduces the high-frequent power of the ellipsoidal corrections • The spherical harmonic coefficients beyond degree 20 only contribute 2% of the total ellipsoidal correction (less than 1 cm anywhere on Earth)

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