Earthworms: multiscale (generalized wavelet?) analysis of the EGM96 gravity field model of the Earth

1
Earthworms multiscale (generalized wavelet?)
analysis of the EGM96 gravity field model of the
Earth

• Frank Horowitz1, Peter Hornby1, Gabriel
  Strykowski2, Fabio Boschetti1
• 1CSIRO Exploration Mining, Perth, Australia
• 2National Survey and Cadstre (KMS), Copenhagen,
  Denmark
• http//www.ned.dem.csiro.au/HorowitzFrank

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Overview

• Edge detection on the EGM96 global geodetic
  gravity field
• EGM96 is the geodetic communitys spherical
  harmonic model of the gravity field of the Earth,
  valid to degree and order 360 (roughly 30 minutes
  of arc)
• For the spherical earth, the technique does not
  yield a "traditional" wavelet transform.
• Still working on demonstrating whether
  construction preserves desirable properties of
  flat earth wavelet.

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Overview (cont.)

• Uses
• Visualisation
• Big picture tectonics
• Corrections to global gravity models including
  error distributions
• Source distributions?
• Method of images (Kelvin transform)

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Sources? Severity of problem
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Theory

• Flat earth wavelets from Green's function
• potential
• vertical acceleration

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Theory (cont.)

• Flat earth wavelets from Green's function
• convolutional form
• convolution kernel (Greens function)

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Theory (cont.)

• Flat earth wavelets from Green's function
• smoothing/scaling function
• on the line
• On the plane

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Theory (cont.)

• Flat earth wavelets from Green's function
• "mother" wavelet from derivative of smoothing
  function

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Theory (cont.)

• Flat earth wavelets from Green's function
• different scales
• Upward continuations are wavelet scale changes
  (Hornby et al. 1999)

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Inverse wavelet transform

• An interpretation of the wavelet transform as
  proportional to a possible source distribution

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Spherical theory (1)

• Green's function on sphere
• Radial acceleration
• Field due to mass distribution (in spherical
  harmonics)

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Spherical theory (2)

• Green's function on sphere
• Radial acceleration
• Field due to mass distribution (in spherical
  harmonics)

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Wavelet scaling comparison

• On flat earth
• On spherical earth

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Scaling on the sphere
Our gravity construction does NOT do this!
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Spherical Smoother
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Spherical wavelet?
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Computing multiscale edges (flat earth)

• Upward continue field to height z
• Calculate horizontal derivatives as (vector
  valued) wavelet transform
• Magnitude of slope is g(x,y)
• Locations of maxima of g(x,y) are positions of
  edges
• These values of g(x,y), scaled by z/z0 , are
  magnitudes assigned to edges

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Computing multiscale edges on the sphere

• Upward continue anomalous vertical acceleration
  field and horizontal derivatives to height h
  above reference ellipsoid using GRAVSOFT
  (Tscherning, et al. 1992) to evaluate T,zx T,zy
• T is the anomalous potential, a subscripted comma
  denotes differentiation, and x, y and z are an
  orthonormal basis for a local tangent space
  denoting East(erly), North, and zenith
  respectively

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Computing multiscale edges on the sphere (cont.)

• Magnitude of slope is
• Multiscale edges are defined as local maxima of g
  
• These values of g, scaled by (R0h)/R0 , are
  magnitudes assigned to edges (mapped to color in
  the following)

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Example Skeletonizations
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Interlude Live demonstration
For one-on-one interactive demonstrations, see
Frank Horowitz and one of his Laptops of Doom
at the meeting. Or, play with the VRML model
yourself!
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Supplementing Formal Error Statistics of EGM96 in
Local Area

• Each spherical harmonic coefficient has
  associated distribution
• Build cost function consisting of mismatch
  between edges from EGM96 (or successor) and edges
  from local survey (e.g. the AGSO dataset for
  Australia)
• Search within distribution for best fitting set
  of coefficients
• independent analysis of field
• should ameliorate field translation problems

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Example Australia Perturbed and Unperturbed

• Worms derived from EGM96 mean valued coefficients
  
• displayed in green
• Worms derived from EGM96 coefficients, randomly
  drawn from within error distributions
• displayed in red
• Coincident worms
• displayed in yellow

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Conclusion

• While we can prove our construction on the sphere
  does not yield a "traditional wavelet", the
  results exhibit the main practical
  characteristics of a spherical multiscale edge
  analysis.
• We're still trying to recapture some of the
  useful properties of wavelets from our
  construction.
• Problem Inverse transform might not be simple.
• We might have lost the "horizontal gradient of
  magnitude is proportional to a dipole source
  distribution" property
• Rats!
• Signal processing operations?
• Clearly the skeletonization has utility in
  qualitative and quantitative analysis of the
  gravity field itself.