Bowman, D. D.

Jimenez, A.

Klein, W.

Rundle, J. B.

Tiampo, K. F.

Earth-prints Repository

Earthquake Statistics in Models and Data
K.F. Tiampo
W. Klein 
A. Jiménez
S. Kohen-Kadosh
C. Ferguson
J.B. Rundle
J. Sá Martins
Motivation
 Most models of earthquake 
systems today focus, as a first 
order parameter, on reproducing 
the Gutenberg-Richter distribution.
 Most of these models then focus on 
reproducing various phenomena 
that we see in nature. 
 As a result of both the growing 
complexity of these models and the 
expansion of the catalog date, 
integrating data analysis techniques 
with model validation today requires 
additional measures of evaluation.
 Here I introduce a statistical 
mechanics concept used to quantify 
model behavior, ergodicity, to 
investigate the behaviour of the 
natural system from seismicity data.
Cumulative number of earthquake, N, 
with magnitude greater than m for 
various years in southern California, a) 
1980-1984 and b)1990-1994 (Turcotte 
1997).
Ergodicity
 A system is determined to be ergodic if it visits every possible state 
in phase space over the course of time. It can be shown that, given 
enough sampling time, the temporal averages of a particular 
observable must equal the ensemble average.
 Recently, we employed a measure of ergodicity, the Thirumalai-
Mountain (TM) metric (Thirumalai et al., 1989) to study both models 
and data from various tectonic locations (Tiampo et al., PRL, 2003; 
Tiampo et al., in press, PRE).
 Notice that the TM metric is the spatial variance of the temporal 
mean, and if those are equivalent, it not only goes to zero with time, 
it does so as 1/t, from the Central Limit Theorem. 
∑ = −[=Ω i ie ttNt 1 )]()()( εε∫= t ii dttEtt 0 ')'(1)(ε ∑= i i tNt )()( εε
∑
=
−[=Ω N
i
ie ttN
t
1
2)]()(1)( εε
∫=
t
ii dttEt
t
0
')'(1)(ε
∑= N
i
i tN
t )(1)( εε
Ergodicity in Fault Models
Loader plate update
etDte
=Ω
Ω
)(
(0)
 Therefore, if the system is effectively ergodic at long times, the TM 
metric varies with time as
 A number of earthquake 
system parameters can be 
formulated for analysis 
using the TM metric.  At the 
right is shown a plot of the 
inverse TM metric for 
energy in a slider block 
model, from Ferguson et 
al., 1999.
Ergodicity
 An ergodic system is said to be in equilibrium, at least for 
some period of time, because it has no significant (large) 
excursions from the mean  energy state for that same time 
interval. 
 In addition, a necessary condition for ergodicity is that the 
system be stationary.  Stationarity implies that a system is 
well-behaved, and linear analyses such as Karhunen-Loeve 
(KL), or principal component, decomposition are applicable for 
those same spatial and temporal parameters.
 Note that the corollary of this is true – a linear KL analysis  
only contains a complete set of orthonormal basis functions if 
the system is ergodic (Holmes et al., 2006; Glösmann and 
Kreuzer, 2005).
 Finally, De is a diffusion constant, proportional to 1/t, and 
related to the sampling rate of the system.  Therefore, 1/ De is 
the mixing time, or the time to reach effective ergodicity.  
∑ = −[=Ω i ie ttNt 1 )]()()( εε∫= t ii dttEtt 0 ')'(1)(ε ∑= i i tNt )()( εε
Ergodicity in Fault Models
 At the left is shown the 
inverse TM metric for 
numbers of events, in a 
slider block model with 
precursory slip (Tiampo et 
al., 2003).
 Note that, while the linear 
regions here indicate 
ergodicity, or punctuated 
ergodicity, there are also 
certain ranges of 
parameters, not shown for 
which these models are not
ergodic (Tiampo et al., in 
press, 2007).
 Other parameters can be related to the energy of the earthquake fault 
system.
Ergodicity in Natural Catalogs
 Here, we bin two very different 
tectonic regions, California and 
eastern Canada, into a set of 
locations, and use the number
of events as our parameter of 
interest, in order to investigate 
under what conditions the 
system is, or is not, ergodic
(Tiampo et al., in press, PRE). M ≥ 4
M ≥ 3
California
M ≥ 2
Seismicity Data
 Southern California Earthquake Center (SCEC) and Northern 
California Seismic Network (NCSN) earthquake catalogs, or the 
ANSS catalog, for the period 1932-2004.  
 Events are binned into areas 0.1° to a side (approximately 11 
kms).
 Analysis is performed for an area ranging from 32° to 39° latitude, 
-123° to -115° longitude, or some subset thereof.  No declustering
is performed, except for a particular magnitude cutoff.
 A matrix is created consisting of the seismicity time series (n time 
steps) for each location (p locations).
⎥⎥
⎥⎥
⎥
⎦
⎤
⎢⎢
⎢⎢
⎢
⎣
⎡
==
p
nnn
p
p
p
yyy
yyy
yyy
yyyT
...
...
...
],...,[
21
2
2
2
1
2
1
2
1
1
1
21 MOMM
Ergodicity in Natural Catalogs
 Again, we bin the California 
region into a set of locations, 
and use the numbers of events 
as our parameter of interest
M ≥ 4
M ≥ 3
California
M ≥ 2
Karhunen-Loeve Analysis
 A Karhunen-Loeve (KL) expansion analysis is a method for decomposing 
large data sets into their orthonormal eigenvectors and associated time 
series based upon the correlations that exist in the data. 
 The vector space is spanned by the eigenvectors, or eigenpatterns, of an 
N-dimensional correlation matrix, C(xi,xj).  The elements of C are obtained 
by cross-correlating this set of location time series, T.
 The eigenvalues and eigenvectors of C are computed, such that they 
represent the correlations in the seismicity data in space and time.
 These eigenvectors are ordered from largest to smallest percent of the 
correlation.  This also results in an ordering from largest to smallest scales. 
 This method can be used to study those modes most responsible for 
these correlations and their sources (Savage, 1988), to remove the 
uninteresting modes from the system (Preisendorfer, 1988), or project their 
trajectories forward in time (Penland and others).
Computer Simulations, Virtual California
Eigenpatterns 1 – 4
Note that there are no small events (M ≤ 6) in this model.
Correlated Patterns in Historic Seismicity Data
Southern California seismicity, all events, 1932 through 1999
KLE1 KLE2
Southern California Seismicity, 1932 through 1991
KLE1 KLE2
Again, all events M ≥ 0
Spatial Eigenvectors, 1932 through 1991
Southern California seismicity, M ≥ 3
KLE1
KLE3
KLE4
KLE11
1932 through 1982
KLE1
KLE3
KLE7M ≥ 3
44
Decomposition of Annual Seismicity - 1982
Mode
EIGENVALUE POWER
1982
1980
1981
1976
39
1976
1973
The PI Index:
As a measure of 
seismicity, is a 
linear combination 
of linear 
eigenvectors
1982
1932 through 2003
KLE2
KLE1
KLE7
Decomposition of Annual Seismicity - 2003
2003
1997
2000
1994
EIGENVALUE POWER
Mode25
19
The PI Index
20021997
1995
Ergodicity in Natural Catalogs
Tiampo et al., PRE, in press
Eastern Canada: 1900 – present.  
Catalog courtesy of J. Adams, GSC
M ≥ 3I
n
v
e
r
s
e
 
T
M
 
M
e
t
r
i
c
M ≥ 4
I
n
v
e
r
s
e
 
T
M
 
M
e
t
r
i
c
Eastern North American Seismicity, USGS
KL Decomposition, eastern Canada
KLE1
KLE2
1900 through 
2001, M ≥ 0
1900 through 
2001, M ≥ 4
KLE2
KL Decomposition, eastern Canada
KLE4
KLE1
KL Decomposition, eastern Canada
KLE35 KLE37
1900 through 2001, M ≥ 4
PI Index, eastern Canada
PI forecast for eastern Canada, 2002-2012, M ≥ 3 (no 
data is used after 2001).  
PI forecast for eastern Canada, 2002-2012.  Scale as shown earlier.
PI Index, eastern Canada
Conclusions
 The TM metric can be used to investigate those regions of 
parameter space (magnitude, region size, spatial discretization) 
for which the natural fault system, and those models that are used 
to study it, are ergodic.
 Stationarity is a necessary, although not a sufficient, condition for 
ergodicity, so that an ergodic condition implies both that the 
system is stationary and in equilibrium.
 Preliminary results suggest that earthquake fault systems are 
ergodic for various spatial and temporal regimes, in metastable
equilibrium for some period of time.
 De is of similar order for both tectonic regions, eastern Canada and 
California, but better models are needed to determine if this is
universal or related to the parameter space in some way.
 For those periods of time, linear operators can be used to analyze 
the fault system.
 Using this, and other statistical tests derived from and tested on 
appropriate models, can be used to test both the applicability of 
those models and our assumptions about the underlying physical 
properties. 


Earthquake Statistics in Models and Data

http://www.earth-prints.org/bitstream/2122/2338/1/Tiampo_20070604.pdf

Earthquake Statistics in Models and Data

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