Earthquake recurrence as a record breaking process
Jörn Davidsen
Collaborators:
M. Paczuski, Dept. of Physics and Astronomy, University of Calgary
P. Grassberger, John-von-Neumann Institute for Computing, FZ Jülich
References: Phys. Rev. Lett. 94, 048501; Geophys. Res. Lett. 31, L21612, and 33, L11304;
Earthquake recurrence
Jörn Davidsen
Outline
I. Earthquakes & seismicity
☞ Introduction and definitions
☞ General relevance and physical questions
II. Recurrences & record breaking processes
☞ Network of record breaking events
☞ Results for Southern California
III. Discussion & open questions
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I. Earthquakes & seismicity
Jörn Davidsen
I. Earthquakes & seismicity
June 30th 2006
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I. Earthquakes & seismicity
Jörn Davidsen
Elastic rebound theory
year
Definition.
Seismic moment M :
M = µ A δe
Magnitude m:
m = (log10 M − d)/c
Epicenter (hypocenter):
Center of earthquake in 2D (3D)
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I. Earthquakes & seismicity
Jörn Davidsen
Examples of empirical laws
Gutenberg-Richter law:
log10 N (m > mth) ∝ −b mth with b ≈ 1
Omori law:
n(t) =
C
(K+t)p
for t < tcutof f with p ≈ 1
Further characteristics
☞ Long-range spatiotemporal correlations
and spatiotemporal clustering
☞ Complex, fractal fault structures
and epicenter distribution
☞ Scale-free plate area distribution
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I. Earthquakes & seismicity
Jörn Davidsen
Clustering and causal connections
3
?
2
?
?
1
Underlying microscopic dynamics is generally not observable
Challenge: Measure correlation structure in an ”unbiased” way
Compare with null hypothesis of uncorrelated events
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6
II. Recurrences
Jörn Davidsen
magnitude
II. Recurrences & record breaking processes
1
time
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II. Recurrences
Jörn Davidsen
magnitude
II. Recurrences & record breaking processes
T1
T2
m3
m2
m1
time
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II. Recurrences
Jörn Davidsen
Extension to spatiotemporal point processes
2
6
4
7
1
8
5
3
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II. Recurrences
Jörn Davidsen
Extension to spatiotemporal point processes
2
6
4
7
1
8
5
3
Record breaking process: Cascade of earthquake recurrences!
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II. Recurrences
Jörn Davidsen
Extension to spatiotemporal point processes
2
6
4
7
1
8
5
3
directed network of earthquakes (no predefined scales)
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II. Recurrences
Jörn Davidsen
Southern California (1984-2002)
comparison with null hypothesis of uncorrelated events
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II. Recurrences
Jörn Davidsen
PDF of distances
P m(l) ∼ l−1.05F (l/l∗ )
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l∗ ≈ 0.012km × 100.45m ≈ LR
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II. Recurrences
Jörn Davidsen
First recurrences
P1m(l) ∼ l−0.6 F̃ (l/l∗ )
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II. Recurrences
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Distance ratios (m=2.5)
Pi∆(li+1/li) ∼ (li+1/li)−0.6
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II. Recurrences
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Temporal ordering of seismicity
δuncorrelated ' −0.14
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D2 = 1 − δuncorrelated ≈ 1.14
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II. Recurrences
Jörn Davidsen
Directed network of earthquakes
2
6
4
7
1
8
5
3
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II. Recurrences
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Network topology: Degree distribution (m=2.5)
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II. Recurrences
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Network topology: Degree distribution (m=2.5)
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II. Recurrences
Jörn Davidsen
Network topology: Degree distribution
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Earthquake recurrence
Jörn Davidsen
Discussion
☞ Spatiotemporal point processes can be characterized
by their cascade of recurrences
☞ For seismicity, cascade of recurrences shows nontrivial and robust features
☞ Rupture length emerges as a fundamental scale for
the length of links
☞ Distribution of relative separations for the next record
in space ∼ r −δr , with δr ≈ 0.6
☞ Network of recurrences shows large deviations from
random topologies
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Earthquake recurrence
Jörn Davidsen
Waiting time distribution
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B1
Earthquake recurrence
Jörn Davidsen
Waiting time ratios (m=2.5)
Pi(Ti/Ti+1) ∼ (Ti/Ti+1)−0.6
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B2
Earthquake recurrence
Jörn Davidsen
Gutenberg-Richter law: California
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B3
Earthquake recurrence
Jörn Davidsen
Epicenters in California: Generalized dimensions
D0 = 1.6 ± 0.13, D1 = 1.38 ± 0.13, D2 = 1.22 ± 0.05, D∞ ≈ 1
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Earthquake recurrence
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Epicenters in California: Generalized dimensions
Dq = limL→0
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1 ln Mq (L)
q−1 ln L
with Mq (L) =
P
i Pi(L)
q
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Earthquake recurrence
Jörn Davidsen
Waiting time distribution (BAK et al. 2002)
-125˚
45˚
-120˚
-115˚
-110˚
-105˚
-100˚
45˚
-125˚
45˚
-120˚
-115˚
-110˚
-105˚
-100˚
45˚
40˚
40˚
40˚
40˚
35˚
35˚
35˚
35˚
30˚
30˚
30˚
30˚
25˚
25˚
25˚
25˚
20˚
-125˚
-120˚
-115˚
-110˚
-105˚
20˚
-100˚
20˚
-125˚
-120˚
-115˚
-110˚
-105˚
20˚
-100˚
Waiting time distribution depends on m and L!
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B6
Earthquake recurrence
Jörn Davidsen
Jump distribution: California (JD & Paczuski 2005)
Pm,L(∆r) =
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f (∆r/L)
L
f (x) =
x−0.6,
x < 0.5
fast decay, x > 0.5
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Earthquake recurrence
Jörn Davidsen
Prediction & hazard assessment
No coherent predictive framework — simple cause and effect relations
cannot be identified — and seismic hazard assessment remains elusive
despite many decades of efforts (Nature debate, 1999)
Example: Parkfield, California
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B8
Earthquake recurrence
Jörn Davidsen
Random graph theory
Definition. A graph is a pair of sets G = {P, E}, where P is a set of N
nodes P1, P2, . . . PN and E is a set of n edges that connect two elements
of P . A random graph is an undirected graph for which the edges are
chosen randomly from the N (N2−1) possible edges.
Some results of random graph theory (Erdös & Renyi):
Mean degree
Degree distribution
Average shortest path length
Clustering coefficient
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hki =
2n
N
P (k) = e
−hki hki
k
k!
ln N
l∼
ln hki
2Ei
hki
C :=
=
ki(ki − 1) i N − 1
B9