1. Ilya Zaliapin Department of Mathematics and Statistics University of Nevada, Reno, USA ENHANS Workshop, Hatfield, Pretoria, South Africa 17-20 January, 2011 Co-authors: Yehuda Ben-Zion (USC), Michael Ghil (UCLA), Efi Foufoula-Georgiou (UM), Andrew Hicks (UNR), Yevgeniy Kovchegov (OSU) The research is supported by NSF grants DMS-0620838 and EAR-0934871

2. Natural disasters in Africa Networks & trees: A unified approach to modeling natural complexities 1 2 3 Seismic clustering vs. physical properties of the crust 1 2 3 44 Conclusions

4. Earthquakes Algeria, 2003, 2266 killed Data according to AON Re Volcanoes Congo, 2002, 200 killed Storms Magadascar, 2004, 363 killed Wildfires Mozambique, 2008, 49 killed Droughts Malawi, 2002, 500 killed Heat waves Nigeria, 2002, 60 killed Floods Algeria, 2001, 921 killed Cold waves South Africa, 2007, 22 killed

8. Botanical treesBlood/Lungs systemsRiver basins Valleys on MarsSnowflakesNeurons

9. 1. Networks & trees = non-Eucledian metric Noyo basin, Mendocino county, California, US

10. 1. Networks & trees = non-Eucledian metric Branching structures (rivers, drainage networks, etc.) [Horton, 1945; Shreve, 1966; Tokunaga, 1978, Peckham, 1995; Rodrigez-Iturbo & Rinaldo, 1997] Interaction of climate system components [Tsonis, 2006, Donges et al., 2009] Structural organization of Solid Earth [Turcotte, 1997; Keilis-Borok, 2002] Spread of epidemics, diseases, rumors [Newman et al. 2006] Evolutionary relationships (phylogenetic trees) [Maher, 2002] etc. 2. Networks & trees = branching and aggregation (coalescence) Environmental transport of rivers and hillslopes [Zaliapin et al., 2010] Fracture development is solids [Kagan, 1982; Lawn, 1993; Baiesi, 2005; Davidsen et al., 2008] Percolation phenomena [Yakovlev et al., 2005] Food webs [Power, 2000] Systems of interacting particles [Gabrielov et al., 2008]

11. Primary branches Side branches Power law relationship between size M r and number N r of objects. A counterpart of statistical “self-similarity”. Notably: a weak constraint on the hierarchy. Provides a complete description of the hierarchy. Defines the “true”, structural self-similarity. 12 11 12 11 22 11 22 33 23

12. Noyo basin, Mendocino county, California, US See [Sklar et al., Water Resor. Res, 2006] for basin details A: Naturally connects topology and geometry/physics of a hierarchy

13. A 1 : Very simple, two-parametric class of trees… A 2 : Very flexible class of trees, observed in unprecedented variety of modeled and natural systems: Numerical studies river stream networks hillslope topography earthquake aftershock clustering vein structure of botanical leaves diffusion limited aggregation percolation nearest-neighbor aggregation in Euclidean spaces level-set tree of fractional Brownian motion Theoretical results critical Galton-Watson branching process [Burd at al., 2000] Shreve random river network model [Shreve, 1966] SOC-type general aggregation model [Gabrielov et al., 1999] regular Brownian motion [Neveu and Pitman, 1989 + Burd at al., 2000] symmetric Markov chains [Zaliapin and Kovchegov, 2011]

14. Theorem 1 [Burd, Waymire, Winn, 2000] Critical Galton-Watson binary branching process corresponds to a Tokunaga self-similar tree (SST). Theorem 2 [Neveu and Pitman, 1989] The level set tree of a regular Brownian motion correspond to the critical Galton- Watson process. Theorem 3 [Zaliapin and Kovchegov, 2011] The level set tree of a symmetric homogeneous Markov chain is a Tokunaga SST. Conjecture [Webb2009; Zaliapin and Kovchegov, 2011] The level set tree of a fractional Brownian motion is a Tokunaga SST. Conjecture [Zaliapin et al., 2010; Zaliapin and Kovchegov, 2011] Nearest-neighbor aggregation in Euclidean space corresponds to a Tokunaga SST.

15. Baiesi and Paczuski, PRE, 69, 066106 (2004) Zaliapin et al., PRL, 101, 018501 (2008) Zaliapin and Ben-Zion, GJI (2011)

17. Separation of clustered and homogeneous parts: NEIC, 1973-2010, M4 Homogeneous part (as in Poisson process) Clustered part: events are much closer to each other than in the homogeneous part Theoretical prediction for a Poisson field [Zaliapin et al. 2008]

18. World seismicity, USGS/NEIC m ≥ 4.0; 223,600 events Parkfield, Thurber et al. (2006) m > 0.0; 8,993 events California, Shearer et al. (2005) m ≥ 2.0; 70,895 events Nevada, Nevada SeismoLab m > 1.0; 75,351 events

20. weak link strong link Cluster #3 Cluster #2 Cluster #1 Identification of clusters: data driven

21. Foreshocks Aftershocks Mainshock Identification of event types: problem driven Time

23. Joint distribution of the number of fore/aftershocks

24. Thick cold lithosphere in subduction and collision environments: (i)high proportion of isolated events, (ii)enhanced aftershock production TransformDivergent MOR, rift valleys Convergent subduction, orogenic belts Illustration by Jose F. Vigil from This Dynamic Planet -- a wall map produced jointly by the U.S. Geological Survey, the Smithsonian Institution, and the U.S. Naval Research Laboratory. http://pubs.usgs.gov/gip/earthq1/plate.html Thin hot lithosphere in transform and especially divergent boundaries: (i)high clustering, (ii)enhanced foreshock production

25. Peru-Chile trench Philippine trench Manila trench Middle America trench Carlsberg ridge Orogenic belt, Tethyan Zone

26. Mid-Atlantic Ridge Red Sea rift + Aden ridge East Pacific rise Carlsberg ridge

27. Extremely hot places, with abnormally high foreshock productivity, similar to mid-oceanic ridges => enhanced possibility for earthquake forecast

28. Thin hot lithosphere  enhanced clustering, more foreshocks 1 2 3 4 1 2 3 4 A unified approach to study aftershocks, foreshocks, swarms, etc. Notable deviation from self-similarity Objective non-parametric declustering Thick cold lithosphere  depressed clustering, more aftershocks Possibility for region-based forecasting strategies Network approach to understanding natural complexities Horton-Strahler,Tokunaga indexing Tokunaga self-similarity Earthquake clustering vs. physical properties of the crust

30. California (1984-present, m ≥ 2.0) ANSS, http://www.ncedc.org/anss/catalog-search.html Parkfield (1984-2005, m > 0.0) Thurber et al. (2006), BSSA, 96, 4B, S38-S49. Southern California (1981-2005, m ≥ 2.0) Shearer et al. (2005), BSSA, 95 (3), 904–915. Lin et al. (2007), JGR, 112, B12309. 25 individual fault zones in CA (1984-2002) Powers and Jordan (2009), JGR, in press. Hauksson and Shearer (2005), BSSA, 95 (3), 896–903. Shearer et al. (2005), BSSA, 95 (3), 904–915. World-wide (1973-present, m ≥ 4.0 ) USGS/NEIC http://earthquake.usgs.gov/earthquakes/eqarchives/epic/epic_global.php Nevada (1990-present, m ≥ 1.0) Nevada Seismological Laboratory http://www.seismo.unr.edu/Catalog/search.html Regions & catalogs analyzed

31. Cluster separation is time- & space-dependent

32. East African Rift Mid-Atlantic Ridge East Pacific Rise Red Sea Rift Aden Ridge Carlsberg Ridge Gorda Ridge Explorer Ridge Juan de Fuca Ridge Chile Rise Nazca Plate -- South American Plate the Peru-Chile Trench Cocos Plate -- Caribbean Plate the Middle America Trench Pacific Plate -- Eurasian and Philippine Sea Plates the Mariana Trench Pacific Plate -- North American Plate the Aleutian Trench. Philippine Sea Plate -- Philippine Mobile Belt the Philippine Trench + the East Luzon Trench Eurasian Plate -- the Philippine Mobile Belt the Manila Trench Sunda Plate -- Philippine Mobile Belt the Negros Trench + the Cotobato Trench Pacific Plate -- Indo-Australian Plate Juan de Fuca, Gorda and Explorer -- North American plate South American Plate -- South Sandwich Plate the South Sandwich Trench

33. Measures of seismic clustering 1) Prop. of multiple-event clusters No. of clusters with fore/aftershocks Total no. of clusters = 2) Prop. of aftershocks No. of aftershocks No. of foreshocks + aftershocks =