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

Earthworms: multiscale (generalized wavelet?) analysis of the EGM96 gravity field model of the Earth. Frank Horowitz 1 , Peter Hornby 1 , Gabriel Strykowski 2 , Fabio Boschetti 1 1 CSIRO Exploration & Mining, Perth, Australia 2 National Survey and Cadstre (KMS), Copenhagen, Denmark

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

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  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

  2. Overview • Edge detection on the EGM96 global geodetic gravity field • EGM96 is the geodetic community’s 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.

  3. Overview (cont.) • Uses: • Visualisation • Big picture tectonics • Corrections to global gravity models; including error distributions • Source distributions? • Method of images (Kelvin transform)

  4. Sources? Severity of problem

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

  6. Theory (cont.) • Flat earth wavelets from Green's function • convolutional form • convolution kernel (Green’s function)

  7. Theory (cont.) • Flat earth wavelets from Green's function • smoothing/scaling function • on the line • On the plane

  8. Theory (cont.) • Flat earth wavelets from Green's function • "mother" wavelet from derivative of smoothing function

  9. 1 unit 2 units 3 units Theory (cont.) • Flat earth wavelets from Green's function • different scales • Upward continuations are wavelet scale changes (Hornby et al. 1999)

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

  11. Spherical theory (1) • Green's function on sphere • Radial acceleration • Field due to mass distribution (in spherical harmonics)

  12. Spherical theory (2) • Green's function on sphere • Radial acceleration • Field due to mass distribution (in spherical harmonics)

  13. Wavelet scaling comparison • On flat earth • On spherical earth

  14. Scaling on the sphere Our gravity construction does NOT do this!

  15. Spherical Smoother

  16. Spherical wavelet?

  17. 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

  18. Computing multiscale edges on the sphere • Upward continue anomalous vertical acceleration field and horizontal derivatives to height h above reference ellipsoidusing 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

  19. 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 (R0+h)/R0 , are magnitudes assigned to edges (mapped to color in the following)

  20. Example Skeletonizations

  21. EGM96 Gravity Skeletonization: Indian Ocean 500 - 1000 Km 100 - 200 Km 1000 - 5000 Km 200 - 500 Km 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!

  22. 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

  23. 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

  24. 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.

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