Ilya ZaliapinDepartment of Mathematics and Statistics

University of Nevada, Reno, USA

ENHANS Workshop, Hatfield, Pretoria, South Africa17-20 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

Natural disasters in Africa

Networks & trees: A unified approach to modeling natural complexities

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3 Seismic clustering vs. physical properties of the crust

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44 Conclusions

EarthquakesAlgeria, 2003, 2266 killed

Data according to AON Re

VolcanoesCongo, 2002, 200 killed

StormsMagadascar, 2004, 363

killed

WildfiresMozambique, 2008, 49

killed

DroughtsMalawi, 2002, 500 killed

Heat wavesNigeria, 2002, 60 killed

FloodsAlgeria, 2001, 921 killed

Cold wavesSouth Africa, 2007, 22 killed

Botanical trees Blood/Lungs systems River basins

Valleys on Mars Snowflakes Neurons

1. Networks & trees = non-Eucledian metric

Noyo basin, Mendocino county, California, US

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]

Primary branches

Side branches

r rN M Power law relationship between size Mr and number Nr of objects. A counterpart of statistical “self-similarity”. Notably: a weak constraint on the hierarchy.

ij ij jT N N j iij j iT T ac

Provides a complete description of the hierarchy.Defines the “true”, structural self-similarity.

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Noyo basin, Mendocino county, California, USSee [Sklar et al., Water Resor. Res, 2006] for basin details

A: Naturally connects topology and geometry/physics of a hierarchy

A1: Very simple, two-parametric class of trees…

A2: 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]

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.

Baiesi and Paczuski, PRE, 69, 066106 (2004)Zaliapin et al., PRL, 101, 018501 (2008)

Zaliapin and Ben-Zion, GJI (2011)

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

/210 imij ijT t

/210 imdij ijR r

Theoretical prediction for a Poisson field

[Zaliapin et al. 2008]

World seismicity, USGS/NEICm ≥ 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 SeismoLabm > 1.0; 75,351 events

weak link

strong link

Cluster #3

Cluster #2

Cluster #1

Identification of clusters: data driven

Foreshocks

Aftershocks

Mainshock

Identification of event types: problem driven

Time

Joint distribution of the number of fore/aftershocks

Thick cold lithosphere in subduction and collision environments:

(i)high proportion of isolated events, (ii)enhanced aftershock production

Transform DivergentMOR, rift valleys

Convergentsubduction, 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

Peru-Chile trench

Philippine trenchManila trench

Middle America trench

Carlsberg ridgeOrogenic belt,Tethyan Zone

Mid-Atlantic Ridge

Red Sea rift + Aden ridge

East Pacific rise

Carlsberg ridge

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

Thin hot lithosphere enhanced clustering, more foreshocks

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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 complexitiesHorton-Strahler,Tokunaga indexing

Tokunaga self-similarity

Earthquake clustering vs. physical properties of the crust

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

Cluster separation is time- & space-dependent

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 Platethe Peru-Chile Trench

Cocos Plate -- Caribbean Platethe Middle America Trench

Pacific Plate -- Eurasian and Philippine Sea Platesthe Mariana Trench

Pacific Plate -- North American Plate the Aleutian Trench.

Philippine Sea Plate -- Philippine Mobile Beltthe Philippine Trench + the East Luzon Trench

Eurasian Plate -- the Philippine Mobile Beltthe Manila Trench

Sunda Plate -- Philippine Mobile Beltthe 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

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 =