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Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization

Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization. Rex H. Pilger, Jr. Highlands Ranch, Colorado. What’s the problem?. Current “standard” models: Plate-to-plate Great circle approximations of spreading and fracture zone segments

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Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization

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  1. Plate Kinematic Reconstruction and Restoration via Fractal Error Minimization Rex H. Pilger, Jr. Highlands Ranch, Colorado

  2. What’s the problem? • Current “standard” models: • Plate-to-plate • Great circle approximations of spreading and fracture zone segments • Fit to chron and fracture zone crossings, stationary and rotated • Plate-to-hotspot • Pacific (single plate): spline-parameterized loci • Atlantic/Indian (multiple plates): Great circle approximations of trace loci • Fit to average or oldest dates from inferred hotspot traces

  3. Current models: plate – to - plate

  4. Current models: plate – to - hotspot

  5. Plate-to-hotspot models Hawaii: paradigmatic hotspot How to evaluate fits… Hotspotting”TM” restoration

  6. Plate-to-hotspot: “hotspotting” • How to evaluate fits… • Hotspotting”TM” restoration “TM” Wessel and Kroenke (1997)

  7. Hotspotting – Hawaiian reference frame Loci: +/- 5 my Hawaiian-Emperor Hawaii Orange <= 48 Ma Green> 48 Ma* *47-48 Ma: Age of Hawaiian - Emperor Bend

  8. Hotspotting – Hawaiian reference frame Orange <= 48 Ma Green> 48 Ma Cook Macdonald

  9. Hotspotting – Hawaiian reference frame Orange <= 48 Ma Green> 48 Ma Foundation Samoa Easter Orange <= 25 Ma Blue > 25 Ma* *25 Ma: Nazca and Cocos plates form from Farallon plate

  10. Hotspotting – Tristan reference frame Tristan-St. Helena Kerguelen-Reunion

  11. Hotspotting – Tristan Reference Frame Great Meteor-Canary Tasman

  12. Hotspotting – Tristan Reference Frame Oldest dates East Australia East Africa Youngest dates (!)

  13. Hotspotting – Tristan reference frame Caribbean Arcs to Tristan

  14. Another approach: fractal measures 1 19 5 3 11 38

  15. Fractal measure: reduced by restoration 13 2 1 7 4 2

  16. Fractal synthesis: Hawaiian frame models

  17. Plate reconstructions: Monte Carlo Monte Carlo “trial and error” Linearly “random” Equal area cells, equatorially-centered for each restored trace Sum of fractal counts over range of delta-spacing for each realization Five percent variation in total rotation pseudovectors & asymmetry 50,000 realizations Retain minimum sum of restored hotspot date cells

  18. Hawaiian Hotspots

  19. Plate Reconstructions • Australia-Antarctica • Isochron crossings1 • Background gravity field2 1Cande & Stock, 2004 2Sandwell & Smith, 1997

  20. Plate reconstructions - Australia-Antarctica Reconstruction parameters: Spline-interpolated pseudovectors “Half” total rotations If spreading was symmetric, reconstructions should produce tight “linear” clustering

  21. Plate reconstructions - Australia-Antarctica • Assuming symmetrical spreading, divergent clusters indicate asymmetrical spreading or ridge-jumping

  22. Plate reconstructions – cell counting Equal area cells Sum of fractal counts over range of delta-spacing for each realization

  23. Plate reconstructions Fractal Count: Fine   Fractal Count: Coarse

  24. Plate reconstructions: cell counts

  25. Plate reconstructions: Monte Carlo Monte Carlo “trial and error” Linearly “random” Five percent variation in total rotation pseudovectors & asymmetry 40,000 realizations (6 hrs on 2 Core, 2.40 GHz, 4GB RAM) Retain minimum sum of restored chrons and fracture zones cells

  26. Plate reconstructions – “final” • “Best fit”: Minimum summed fractals • Realization 35,261 of 40,000 • Sequence of minimum interations: • 0, 343, 464, 2468, 4751, 4912, 9025, 18497, 25793, 26613, 32105, 32298, 32476, 35261

  27. Plate Reconstructions – “final” • Tighter clustering of chrons

  28. Plate reconstructions – comparison • Initial: Yellow, orange, green • “Final”: Red, pink, blue

  29. Plate reconstructions – comparison • Initial: Yellow, orange, green • “Final”: Red, pink, blue

  30. Plate reconstructions – comparison

  31. Why fractals? • A Google Search (10/25/2010) for “fractals” produces 6,780,000 results However, very few if any of these articles recognize that: • Within an iterative, scaling process fractals “maximize information entropy” with respect to persistent information content • That is, following Jaynes’ principle: • Across a range of scales maximizing: • F = –S pn log pn – l0(S pn – 1) – Slk(Ek (p,x) – <Ik>) • Produces Mandelbrot’s fractal equation: • N = a x –d • Application: Parameters for minimum sum of fractals, producing maximum entropy scaled solution

  32. What’s next… • Plate-to-plate • More iterations for Monte Carlo • Apply to full data sets • Introduce uncertainties • Provide pseudo-gradients for iterative solutions, instead of Monte Carlo • Plate circuits with uncertainties • Plate-to-hotspot • Incorporate plate-to-plate results • Include uncertainties • Pseudo-gradients for iterative solutions, instead of Monte Carlo • Hotspot & plates to paleomagnetic models

  33. Virtual worlds • GoogleEarth, World Wind, Bing… • Three roles: • Evaluating reconstruction models with data, especially if tied to “real-time” calculations • Presentations like this • Exchanging data (e.g., via xml) • Raw • Interpreted • Meta (via embedded hyperlinks)

  34. Key references • Plate reconstruction methods: • Pilger, 1978, Geophys. Res. Lett., 5, 469-472. • Hellinger, 1981, J. Geophys. Res., 86B, 9312-9318. • Wessel & Kroenke, 1997, Nature, 387, 365-369. • Maximum Entropy: • Jaynes, 1957, Phys. Rev., 106, 620-630. • Fractals: • Mandelbrot, 1967, Science, 156, 636-638. • Maximum entropy and fractals: • Pastor-Satorras & Wagensberg, 1998, Physica A, 251, 291–302. • SE Indian Ocean magnetic isochrons (digitized from map): • Cande & Stock, 2004, Geophys. J. Int., 157, 399-414.

  35. RIP Edwin Jaynes July 5, 1922 – April 30, 1998 Benoit Mandelbrot November 20, 1924 – October 14, 2010

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