ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS

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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
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EXISTING ELLIPSOIDAL CORRECTION METHODS
Introduction (1/2)

• 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 dont agree with one another

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REPRESENTATION OF ELLIPSOIDAL CORRECTIONS
Introduction (2/2)

• 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

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DEFINITION OF ELLIPSOIDAL CORRECTIONS
Formulation (1/3)
ellipsoidal geoid height
spherical geoid height

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DEFINITION OF ELLIPSOIDAL CORRECTIONS
Formulation (1/3)
ellipsoidal geoid height
spherical geoid height

ellipsoidal correction
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COMPUTATION OF CORRECTION COEFFICIENTS
Formulation (2/3)

• Spherical harmonic synthesis and analysis

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COMPUTATION OF CORRECTION COEFFICIENTS
Formulation (2/3)

• Spherical harmonic synthesis and analysis

• or
• Spherical harmonic coefficient transformation

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RECAPITULATION
Formulation (3/3)

• 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)

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INFLUENCE OF THE REFERENCE SPHERE RADIUS
Choice of reference sphere (1/5)
Ellipsoidal corrections depend upon the choice of
the reference sphere radius R Many existing
formulations only allow for one or two choices of
R
method radius
Moritz (1980)
Fei Sideris (2000)
Heck Seitz (2003)
Sjöberg (2003) separate scale factor
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Choice of reference sphere (2/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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Choice of reference sphere (2/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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A VARIABLE REFERENCE SPHERE RADIUS
Choice of reference sphere (3/5)
The reference sphere radius can be set equal to
the ellipsoidal radius for each computation
point The ellipsoidal correction coefficients
can still be found

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Choice of reference sphere (4/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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Choice of reference sphere (5/5)
SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL
CORRECTIONS
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Choice of reference sphere (5/5)
SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL
CORRECTIONS
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Choice of reference sphere (5/5)
SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL
CORRECTIONS
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Choice of reference sphere (5/5)
SPHERICAL HARMONIC SPECTRA OF ELLIPSOIDAL
CORRECTIONS
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ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS
Modified kernels (1/5)
The combined geoid solution with modified
kernel is equivalent to the simple Stokes
integral

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ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS
Modified kernels (1/5)
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

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ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS
Modified kernels (1/5)
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

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ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS
Modified kernels (1/5)
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

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ELLIPSOIDAL CORRECTIONS FOR MODIFIED KERNELS
Modified kernels (1/5)
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

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

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

64.4 mm 41.5 mm 6.6 mm
44.4 mm 29.5 mm 0.6 mm
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THE MOLODENSKY-MODIFIED SPHEROIDAL STOKES KERNEL
Modified kernels (3/5)
Vanícek and Kleusberg (1987) modification
?
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Modified kernels (4/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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Modified kernels (4/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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Modified kernels (4/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS
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Modified kernels (5/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n ? 2)
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Modified kernels (5/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n ? 2)
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Modified kernels (5/5)
ELLIPSOIDAL CORRECTIONS TO GEOID HEIGHTS (n ? 2)
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Summary and Conclusions

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

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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

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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)