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Hierarchical network approach to modeling natural complexities. Ilya Zaliapin. Department of Mathematics and Statistics University of Nevada, Reno, USA. Co-authors: Yehuda Ben-Zion (USC), Michael Ghil (UCLA), Efi Foufoula-Georgiou (UM), Andrew Hicks (UNR), Yevgeniy Kovchegov (OSU). - PowerPoint PPT Presentation
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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 =