# Fractal plate reconstructions with spreading asymmetry

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## Abstract

Information theory and fractal analysis are the basis of a novel fitting criterion for simultaneous plate tectonic reconstructions of magnetic isochrons and fracture zone crossings of a range of ages, rather than a single isochron age. Accretionary boundaries are modeled as two-dimensional fractal structures including both contemporary spreading boundaries and reconstructed magnetic isochron and fracture zone crossings. Each model incorporates reconstruction parameters which describe the full accretionary history, including asymmetry. The reconstruction parameters are derived by spline interpolation in time of trial rotation pseudovectors, including variable asymmetric spreading between ridge segments. Iterative algorithms, without partial derivative constraints, converge on a nominally optimal model by minimizing the sum of two-dimensional fractal bins, over the range of bin-spacings, and produce thereby progressively refined fractal spectra. The new method can incorporate all isochron identifications from the selected plates and age range in the iterative calculation set. The solution set also provides continuous instantaneous rotation parameters, including asymmetries. An example data set illustrates the methodology and model results.

The rationale for an optimal fractal criterion is rooted in recent developments in information theory: fractal structures maximize Shannon information entropy distributed over a range of scales. The fractal measure is the sum of bins occupied by reconstructed data points for each bin spacing. The fitting criterion utilized in this work is, in turn, the grand sum of the fractal measures over all calculated bin spacings. The optimal fractal measure for the grand sum has *minimal* integrated “fractality” relative to non-optimal sets while maximizing entropy for the optimal parameters for each bin spacing.

## Keywords

Plate tectonics Plate reconstructions Seafloor spreading Asymmetric spreading Fractals Information theory Maximum entropy## Supplementary material

## References

- Aviles CA, Scholz CH, Boatwright (1987) Fractal analysis applied to characteristic segments of the San Andreas fault. J Geophys Res 92:331–344CrossRefGoogle Scholar
- Ballmer MD, van Hunen J, Ito G, Tackley PJ, Bianco TA (2007) Non-hotspot volcano chains originating from small-scale sublithospheric convection. Geophys Res Lett 34:L23310. doi: 10.1029/2007GL031636 CrossRefGoogle Scholar
- Barnsley MF (1993) Fractals everywhere. Morgan Kaufmann, BurlingtonGoogle Scholar
- Bird P (2003) An updated digital model of plate boundaries. Geochem Geophys Geosyst 4, doi: 10.1029/2001GC000252. Data accessed July, 2009, at http://peterbird.name/oldFTP/PB2002/.
- Cande SC, Kent DV (1995) Revised calibration of the geomagnetic polarity timescale for the Late Cretaceous and Cenozoic. J Geophys Res 100:6093–6095CrossRefGoogle Scholar
- Cande SC, Stock JM (2004) Pacific-Antarctic-Australia motion and the formation of the Macquarie plate. Geophys J Int 157:399–414CrossRefGoogle Scholar
- Chopard B, Herrmann HJ, Vicsek T (1991) Structure and growth mechanism of mineral dendrites. Nature 353:409–411CrossRefGoogle Scholar
- DeMets C, Gordon RG, Argus DF, Stein S (1994) Effect of recent revisions to the geomagnetic reversal timescale on estimates of current plate motions. Geophys Res Lett 21:2191–2194CrossRefGoogle Scholar
- DeMets C, Gordon RG, Argus DF (2010) Geologically current plate motions. Geophys J Int 181:1–80. doi: 10.1111/j.1365-246X.2009.04491.x CrossRefGoogle Scholar
- Feder J (1988) Fractals. Plenum Press, New YorkGoogle Scholar
- Francheteau J (1970) Paleomagnetism and plate tectonics, Ph.D. Dissertation, University of California, San DiegoGoogle Scholar
- Gradstein FM, Ogg JG, Smith AG (2005) A geologic time scale-2004. Cambridge University Press, CambridgeCrossRefGoogle Scholar
- Grandy WT (1992) The origins of entropy and irreversibility. Open Syst Inf Dyn 1:183–196CrossRefGoogle Scholar
- Grandy WT (2008) Entropy and the time evolution of macroscopic systems. Int Ser Mon Phys. Oxford University Press, New York, vol 141Google Scholar
- Hanna MS, Chang T (1990) On graphically representing the confidence region for an unknown rotation in three dimensions. Comput Geosci 16:163–194Google Scholar
- Hellinger SJ (1981) The uncertainties of finite rotations in plate tectonics. J Geophys Res 86B:9312–9318CrossRefGoogle Scholar
- Hey RF, Martinez Á, Höskuldsson, Benediktsdóttir Á (2010) Propagating rift model for the V-shaped ridges south of Iceland. Geochem Geophys Geosyst 11, doi: 10.1029/2009GC002865
- Jaynes ET (1957) Information theory and statistical mechanics. Phys Rev 106:620–630CrossRefGoogle Scholar
- Kanasewich ER (1974) Time sequence analysis in geophysics, 3rd edn. University of Alberta Press, AlbertaGoogle Scholar
- Kirkwood BH, Royer J-Y, Chang TC, Gordon RG (1999) Statistical tools for estimating and combining finite rotations and their uncertainties. Geophys J Int 137:408–428CrossRefGoogle Scholar
- Le Pichon X, Francheteau J, Bonnin J (1973) Plate tectonics. Elsevier, AmsterdamGoogle Scholar
- Livermore R, Nankivell A, Eagles G, Morris P (2005) Paleogene opening of Drake Passage. Earth Plan Sci Lett 236:459–470CrossRefGoogle Scholar
- Mandelbrot BB (1953) An information theory of the statistical structure of language. In: Jackson W (ed) Communication theory. Academic Press, New York, pp 503–512Google Scholar
- Mandelbrot BB (1967) How long is the coast of Britain? Science 156:636–638CrossRefGoogle Scholar
- Mandelbrot BB (1975) Stochastic models for the Earth’s relief, the shape and the fractal dimension of the coastlines, and the number-area rule for islands. Proc Natl Acad Sci USA 72:3825–3828CrossRefGoogle Scholar
- Mandelbrot BB (1982) The fractal geometry of nature. W. H. Freeman, New YorkGoogle Scholar
- McKenzie D, Sclater JG (1971) The evolution of the Indian Ocean since the Late Cretaceous. Geophys J R Astr Soc 25:437–528Google Scholar
- McKenzie D, Molnar P, Davies D (1970) Plate tectonics of the Red Sea and East Africa. Nature 226:243–248CrossRefGoogle Scholar
- Merdan Z, Bayirli M (2005) Computation of the fractal pattern in manganese dendrites. Chin Phys Lett 22:2112–2115CrossRefGoogle Scholar
- Minster JB, Jordan TH, Molnar P, Haines E (1974) Numerical modelling of instantaneous plate tectonics. Geophys J R Astr Soc 36:541–576Google Scholar
- Müller R, Sdrolias M, Gaina C, Roest, Walter R (2008) Age, spreading rates, and spreading asymmetry of the world’s ocean crust. Geochem Geophys Geosyst 9, P. NIL_18-NIL_36, pp 1525–2027Google Scholar
- Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313Google Scholar
- Okubo PG, Aki K (1987) Fractal geometry in the San Andreas fault system. J Geophys Res 92:345–355CrossRefGoogle Scholar
- Pastor-Satorras R, Wagensberg J (1998) The maximum entropy principle and the nature of fractals. Phys A 251:291–302CrossRefGoogle Scholar
- Pilger RH (1978) A method for finite plate reconstructions, with applications to Pacific-Nazca Plate evolution. Geophys Res Lett 5:469–472CrossRefGoogle Scholar
- Pilger RH (2003) Geokinematics: prelude to geodynamics. Springer, BerlinGoogle Scholar
- Pitman WC III, Talwani M (1972) Sea-floor spreading in the North Atlantic. Geol Soc Am Bull 83:619–646CrossRefGoogle Scholar
- Powell MJD (1974) Unconstrained minimization algorithms without computation of derivatives. Bollettino della Unione Matematica Italiana 9:60–69Google Scholar
- Press WH, Teukolsky SA, Vetterling WA, Flannery BP (1998) Numerical recipes, 2nd edn. Cambridge University Press, CambridgeGoogle Scholar
- Richardson LF (1922) Weather prediction by numerical process. Cambridge University Press, CambridgeGoogle Scholar
- Richardson LF (1951) The problem of contiguity: an appendix to Statistic of Deadly Quarrels. General systems: yearbook of the Society for the Advancement of General Systems Theory (1961) 6:139–187Google Scholar
- Rodriguez-Iturbe I, Rinaldo A (2001) Fractal river basins: chance and self-organization. Cambridge University Press, Cambridge 564 pGoogle Scholar
- Schroeder M (1991) Fractals, chaos, power laws—minutes from an infinite paradise. W. H. Freeman, New YorkGoogle Scholar
- Seton M, Müller RD, Zahirovic S, Gaina C, Torsvik T, Shephard G, Talsma A, Gurnis M, Turner M, Chandler M (2012) Global continental and ocean basin reconstructions since 200 Ma. Earth-Science Rev (in press)Google Scholar
- Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423, 623–656Google Scholar
- Shannon CE (1951) Prediction and entropy of written English. Bell Syst Tech J 30:50–64Google Scholar
- Smith EGC (1981) Calculation of poles of instantaneous rotation from poles of finite rotation. Geophys J R Astron Soc 65:223–227CrossRefGoogle Scholar
- Sornette D, Pisarenko VF (2003) Fractal plate tectonics. Geophys Res Lett 30:1105. doi: 10.1029/2002GL015043 CrossRefGoogle Scholar
- Turcotte DL (1997) Fractals and chaos in geology. Cambridge University Press, CambridgeGoogle Scholar
- Walsh JJ, Watterson J (1993) Fractal analysis of fracture patterns using the standard box-counting technique: valid and invalid methodologies. J Struct Geol 15:1509–1512CrossRefGoogle Scholar