Download Zaliapin and Kovchegov, 2011

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)
ENHANS Workshop, Hatfield, Pretoria, South Africa
17-20 January, 2011
The research is supported by NSF grants
DMS-0620838 and EAR-0934871
1
Natural disasters in Africa
2
Networks & trees:
A unified approach to modeling natural complexities
3
Seismic clustering vs. physical properties of the crust
4
Conclusions
Floods
Algeria, 2001, 921 killed
Volcanoes
Congo, 2002, 200 killed
Droughts
Malawi, 2002, 500 killed
Earthquakes
Algeria, 2003, 2266 killed
Wildfires
Mozambique, 2008, 49 killed
Heat waves
Nigeria, 2002, 60 killed
Cold waves
South Africa, 2007, 22 killed
Storms
Magadascar, 2004, 363 killed
Data according to
AON Re
Botanical trees
Valleys on Mars
Blood/Lungs systems
Snowflakes
River basins
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]
N r  M r 
Power law relationship between size Mr and number Nr of objects. A counterpart
of statistical “self-similarity”. Notably: a weak constraint on the hierarchy.
Tij  N ij N j
11
Tij  T j i  ac j i
11
11
11
11
22
22
11
Provides a complete description of the hierarchy.
Defines the “true”, structural self-similarity.
11
11
12
22
22
11
12
11
33
Side branches
33
23
11
23
Primary branches
11
A: Naturally connects topology and geometry/physics of a hierarchy
Noyo basin, Mendocino county, California, US
See [Sklar et al., Water Resor. Res, 2006] for basin details
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 GaltonWatson 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)
 mi /2
Rij  r ij 10
d
Clustered part: events are
much closer to each other than
in the homogeneous part
Theoretical prediction
for a Poisson field
[Zaliapin et al. 2008]
Tij  tij 10 mi /2
World seismicity, USGS/NEIC
m ≥ 4.0; 223,600 events
California, Shearer et al. (2005)
m ≥ 2.0; 70,895 events
Nevada, Nevada SeismoLab
m > 1.0; 75,351 events
Parkfield, Thurber et al. (2006)
m > 0.0; 8,993 events
Identification of clusters: data driven
Cluster #3
Cluster #1
Cluster #2
weak link
strong link
Identification of event types: problem driven
Foreshocks
Mainshock
Aftershocks
Time
Joint distribution of the number of fore/aftershocks
Thin hot lithosphere
in transform and especially divergent boundaries:
(i) high clustering,
(ii) enhanced foreshock production
Transform
Divergent
Convergent
MOR, rift valleys
subduction, orogenic belts
Thick cold lithosphere
in subduction and collision environments:
(i) high proportion of isolated events,
(ii) enhanced aftershock production
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
Philippine trench
Manila trench
Middle America trench
Peru-Chile trench
Orogenic ridge
belt,
Carlsberg
Tethyan Zone
Red Sea rift + Aden ridge
East Pacific rise
Carlsberg ridge
Mid-Atlantic Ridge
Extremely hot places, with
abnormally high foreshock
productivity, similar to
mid-oceanic
ridges
=>
enhanced possibility for
earthquake forecast
1
Network approach to understanding natural complexities
Horton-Strahler,Tokunaga indexing
Tokunaga self-similarity
2
Earthquake clustering vs. physical properties of the crust
A unified approach to study aftershocks, foreshocks, swarms, etc.
Notable deviation from self-similarity
Objective non-parametric declustering
3
Thin hot lithosphere 
enhanced clustering, more foreshocks
Thick cold lithosphere 
depressed clustering, more aftershocks
4
Possibility for region-based forecasting strategies
Regions & catalogs analyzed
World-wide (1973-present, m ≥ 4.0 )
USGS/NEIC
http://earthquake.usgs.gov/earthquakes/eqarchives/epic/epic_global.php
California (1984-present, m ≥ 2.0)
ANSS, http://www.ncedc.org/anss/catalog-search.html
Southern California (1981-2005, m ≥ 2.0)
Shearer et al. (2005), BSSA, 95(3), 904–915.
Lin et al. (2007), JGR, 112, B12309.
Parkfield (1984-2005, m > 0.0)
Thurber et al. (2006), BSSA, 96, 4B, S38-S49.
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.
Nevada (1990-present, m ≥ 1.0)
Nevada Seismological Laboratory
http://www.seismo.unr.edu/Catalog/search.html
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 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
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