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ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS

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all rely on approximations to the order of the square of the ... many can not be applied if the Stokes kernel is modified ... RECAPITULATION. Formulation (3/3) ... – PowerPoint PPT presentation

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Title: ELLIPSOIDAL CORRECTIONS TO GRAVIMETRIC GEOID COMPUTATIONS


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
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

3
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


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

5
DEFINITION OF ELLIPSOIDAL CORRECTIONS
Formulation (1/3)
ellipsoidal geoid height
spherical geoid height

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


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

  • or
  • Spherical harmonic coefficient transformation

8
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)

9
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
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
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

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

19
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

20
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

21
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

22
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

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

24
THE SPHEROIDAL STOKES KERNEL
Modified kernels (2/5)
Wong and Gore (1969) modification
?
25
THE SPHEROIDAL STOKES KERNEL
Modified kernels (2/5)
Wong and Gore (1969) modification
?
global absolute maximum of ellipsoidal
corrections (excluding first degree term)
26
THE MOLODENSKY-MODIFIED SPHEROIDAL STOKES KERNEL
Modified kernels (3/5)
VanĂ­cek 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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