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arXiv:0902.4331v1 [physics.geo-ph] 25 Feb 2009

Turbulent-Like Behavior of Seismic Time Series

P. Manshour,1S. Saberi,1Muhammad Sahimi,2

J. Peinke,3Amalio F. Pacheco,4, M. Reza Rahimi Tabar,1,3,5

1Department of Physics, Sharif University of Technology, Tehran 11155-9161, Iran

2Mork Family Department of Chemical Engineering & Materials Science,

University of Southern California, Los Angeles, California 90089-1211, USA

3Carl von Ossietzky University, Institute of Physics, D-26111 Oldenburg, Germany

4Department of Theoretical Physics, University of Zaragoza,

Pedro Cerbuna 12, 50009 Zaragoza, Spain

5CNRS UMR 6202, Observatoire de la Cˆ ote d’Azur, BP 4229, 06304 Nice Cedex 4, France

Abstract

We report on a novel stochastic analysis of seismic time series for the Earth’s vertical velocity, by

using methods originally developed for complex hierarchical systems, and in particular for turbulent

flows. Analysis of the fluctuations of the detrended increments of the series reveals a pronounced

change of the shapes of the probability density functions (PDF) of the series’ increments. Before

and close to an earthquake the shape of the PDF and the long-range correlation in the increments

both manifest significant changes. For a moderate or large-size earthquake the typical time at which

the PDF undergoes the transition from a Gaussian to a non-Gaussian is about 5-10 hours. Thus,

the transition represents a new precursor for detecting such earthquakes.

PACS number(s): 05.45.Tp 64.60.Ht 89.75.Da 91.30.Px

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A grand challenge in geophysics, and the science of analyzing seismic activities, is devel-

oping methods for predicting when earthquakes may occur. Any accurate information about

an impending large earthquake may reduce the damage to the infrastructures and the number

of casualties. Although there are many known precursors, experience over the last several

decades indicates that reliable and quantitative methods for analyzing seismic data are still

lacking [1].

A large number of stations around the world currently perform precise measurements

of seismic time series. Several concepts and ideas have been advanced over the past few

decades in order to explain important aspects of such seismic time series and what they imply

for earthquakes, ranging from the Gutenberg-Richter law [2] for the number of earthquakes

with a magnitude greater than a given value M, to the Omori law [3] for the distribution of

the aftershocks, the concept of self-organized criticality [4], and the percolation model of the

epicenters’ spatial distribution [5]. At the same time, there has also been much interest in

investigating the precursors to, and the predictability of, extreme increments in time series

[6] associated with disparate phenomena, ranging from earthquakes [7,8], to epileptic seizures

[9], and stock market crashes [10-12].

In this Letter we provide compelling evidence for the existence of a novel transition in the

probability density function (PDF) of the detrended increments of the stochastic fluctuations

of the Earth’s vertical velocity Vz, collected by broad-band stations (resolution 100 Hz). As

an important new result, we demonstrate that there is a strong transition from a Gaussian

to a non-Gaussian behavior of the increments’ PDF as an earthquake is approached. We

characterize the non-Gaussian nature of the PDF of the fluctuations in the increments of Vz,

and the time-dependence of the PDF of the background fluctuations far from earthquakes.

The results presented in this Letter are based on the detailed analysis of the data ob-

tained from Spain’s and California’s broad-band networks for three earthquakes of large and

intermediate magnitudes: the May 21, 2003, M = 7.1 event in Oran-Argel, detected in Ibiza

(Balearic Islands); the 2004, M = 6.1 event in Alhucemas, and the M = 5.4 earthquake that

occurred in California on April 30, 2008. Due to our recent discovery of localization of elastic

waves in rock, both experimentally [13] and theoretically [14], we choose the distance of the

detectors from the epicenters to be three times less than 300 km and one time 400 km. We

have also analyzed many other earthquakes around the world, the results of which will be

described briefly.

The data are first detrended over different time scales in order to remove the possible

trends in the time series x(t) ≡ Vz(t). To do so, the time series is divided into semi-overlapping

subintervals [1 + s(k − 1),s(k + 1)] of length 2s and labeled by the index k ≥ 1. Next, we

fit x(t) to a third-order polynomial [15-17] to detrend the original series in the corresponding

time window. The detrended increments on scale s are defined by Zs(t) = x∗(t + s) − x∗(t),

where t ∈ [1 + s(k − 1),sk], with x∗(t) being the detrended series, i.e., the deviation of x(t)

from its fitted value.

We then develop a new approach, originally proposed for fully-developed turbulence

[18-21], in order to describe the cascading process that determines how the fluctuations in the

series evolve, as one passes from the coarse to fine scales. For a fixed t, the fluctuations at

scales s and λs are related through the cascading rule,

Zλs(t) = WλZs(t),∀s, λ > 0 ,(1)

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where ln(Wλ) is a random variable. Iterating Eq. (1) forces implicitly the random variable Wλ

to follow a log infinitely-divisible law [22]. One of the simplest candidates for such processes

is represented [17] by, Zs(t) = ζs(t)exp[ωs(t)], where ζsand ωs(t) are independent Gaussian

variables with zero mean and variances σ2

on the variance of ωs, and is expressed by [21],

ζand σ2

ω. The PDF of Zs(t) has fat tails that depend

Ps(Zs) =

?

Fs

?Zs

σ

?1

σGs(lnσ)dlnσ ,(2)

where Fsand Gsare both Gaussian with zero mean and variances σ2

e.g., Gs(lnσ) = 1/(√2πλs)exp(−ln2σ/2λ2

converges to a Gaussian distribution as λ2

log-normal cascade model, originally introduced to study fully-developed turbulence [23,24],

it also describes approximately the non-Gaussian PDFs observed in a broad range of other

phenomena and systems, such as foreign exchange markets [17,24,25] and heartbeat interval

fluctuations [17,18] (see also [26-28]).

To carry out a quantitative analysis of the seismic times series, we focus on two aspects,

namely, deviations of the PDF of the detrended increments from a Gaussian distribution, and

the dependence of correlations in the increments on the scale parameter s. We begin with

the time series for the largest (M = 7.1) earthquake for two distinct time intervals: (i) data

set (I) representing the background fluctuations far from the time of the earthquake, and (ii)

data set (II) close (less than 5 hours) to the earthquake.

To fit the increments’ PDF to Eq. (2), we estimate the variance λ2

squares method, with the error bars estimated by the goodness of the fit method. Deviation

of λ2

find an accurate parametrization of the PDFs by λ2

of Zsfor the data set (I) becomes essentially Gaussian as s increases to 800 ms, whereas it

deviates from the Gaussian distribution for the data set (II). The time scale s = 800 ms for λ2

within a moving window was estimated by the plot of λ2

fluctuations), and selecting s such that λs→ 0 (see also below).

The scale-dependence of the parameter λ2

set (I) of the M = 7.1 earthquake, shown in Fig. 2(a), and times 200 ms < s < 500 ms,

a logarithmic decay, λ2

extends to 300 ms < s < 2000 ms. Figure 2(b) presents similar behavior for the M = 6.1

earthquake. We note that, for the data set (II) of the M = 6.1 earthquake, there is a crossover

time at which λ2

The importance of the results shown in Fig. 2 is that, they indicate that the increments’

PDFs for s > 2000 ms and s > 1500 ms are almost Gaussian (λ2

M = 6.1 earthquakes [for the data set (II)], respectively. Transforming the time scales to

length scales via the velocity of the elastic waves in Earth, ∼ 5000 m/sec, the corresponding

length scales are about 10 km and 7.5 km, for the same earthquakes, respectively, implying

that larger earthquakes have larger characteristic length scales, and that for the M = 6.1

event, there is a smaller active part in the fault.

We note that as one moves down the cascade process from the large to small scales, one

expects the statistics to increasingly deviate from Gaussianity (see above and Fig. 2), in order

sand λ2

s, respectively,

s). In this case, Ps(Zs) is expressed by Eq. (2) and

s→ 0. Although Eq. (2) is equivalent to that for a

s(s), using the least-

s(s) from zero is a possible indicator of non-Gaussian statistics. As shown in Fig. 1, we

s(s) for both data sets. Moreover, the PDF

s

svs s for the data set (I) (background

sof the PDF is shown in Fig. 2. For the data

s∝ logs, is obtained. For the data set (II), the logarithmic regime

schanges from a ∼ log(s) behavior to having a finite value, ≃ 0.3.

s→ 0) for the M = 7.1 and

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to derive Eq. (2). Note that, a non-Gaussian PDF with fat tails on small scales indicates an

increased probability of the occurrence of short-time extreme seismic fluctuations.

From the point of view of the increments’ PDF, the non-Gaussian noise with uncorre-

lated ωsin the process, Zs(t) = ζs(t)exp[ωs(t)], and a multifractal formulation are indistin-

guishable, because their one-point statistics at any given scale may be identical. Thus, to

understand the origin of the non-Gaussian fluctuations, we explore the correlation properties

of ωs. To do so, we use an alternative method for studying the correlation functions of the

local “energy” fluctuations [20]. We define the magnitude of local variance over a scale s by,

σ2

s

interval and, ns≡ s/∆t. The magnitude of the correlation function of ¯ ωsis then defined by

C(s)(τ) = ?[¯ ωs(t) − ?¯ ωs?][¯ ωs(t + τ) − ?¯ ωs?]? ,

where ?·? indicates a statistical average. Figure 3 shows the results for the two data sets. The

correlation function decays sharply for the data set (I) - far from the earthquakes - whereas

it is of long-range type for the set (II) - close to the earthquakes.

We emphasize that for the data set (II), the PDF deviates from being Gaussian even

for s > 800 ms. Although one might argue that the deviations might be due to an underlying

L´ evy statistics, this possibility is ruled out due to the deduced hierarchical structures that

imply that the increments for different scales are not independent; see Fig 3.

We now show that the analysis may be used as a new precursor for detecting an impend-

ing earthquake. A window containing one hour of data is selected and moved with ∆t = 15

minutes to determine the temporal dependence of λ2

variations of λ2

difference between the values of λ2is large enough for the background data and the data set

near the earthquakes. Hence, such a time scale may be used as the characteristic time for the

dynamics of the non-Gaussian indicator λ2

systematic increase in λ2

mated error of λ2

before the earthquakes values of λ2

those for the background.

We note that in defining λs, one supposes that the PDF of the increments is log-normal.

This may induce errors due to the fact that, one must fit the PDF with a predefined functional

form. An unbiased quantity that measures the intermittency and deviation from Gaussian is

the flatness, which provides an unbiased estimator of deviations from Gaussianity, without

having to adopt a priori any functional form for the PDF. In Figs. 4(c) and 4(d) we present

the flatness of the time series in the same windows (see above) for time scale s ≃ 800 ms.

They show that, consistent with λ2

earthquake.

Due to the localization of elastic waves in Earth, stations that are far from an earthquake

epicenter cannot provide any clue to the transition in the shape of the increments’ PDF. We

checked this point for several earthquakes. Shown in Fig. 5 are the results for the M = 5.4

California earthquake, occurred at (40.837 N, 123.499 W). The distance from the epicenter of

the station that does provide an alert of about 3 hours for the earthquake is about ≃ 128 km,

while the second station that does not yield any alert is about ≃ 400 km from the epicenter.

s(t)−1

?ns/2

k=−ns/2Zs(t+k∆t)2, and, ¯ ωs(i) =1

2logσ2

s(i), respectively. Here, ∆t is the sampling

(3)

s. Guided by Fig. 2, the local temporal

sfor s = 800 ms are investigated. According to Fig. 2, for s ≃ 800 ms, the

s. Figures 4(a) and 4(b) display a well-pronounced,

sas the earthquakes are approached. Taking into account the esti-

sfor the background fluctuations in Figs. 4, we see that about 7 and 5 hours

sare larger, by more than two standard deviations, than

s, the flatness also yields a clear alert for an impending

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We also analyzed several other earthquakes of various magnitudes. We found that for

earthquakes with M ≤ 5 the increase in λ2

in stations about 100 km from the epicenters. Moreover, we also analyzed seismic data for

large earthquakes in Pakistan and Iran, and found that they exhibit the same types of trends

and results, as those presented above, for the time variation of λ2

For example, for the M = 7.6 earthquake that occurred on August 10, 2005, in Pakistan, the

transition in the value of λ2

M = 6.3 earthquake that occurred in northern Iran on May 28, 2004, the transition happened

about 4 hours before the earthquake.

In summary, the temporal dependence of the fat tails of the PDF of the increments of

the vertical velocity Vz(t) of Earth exhibits a gradual, systematic increase in the probability

of the appearance of large values on approaching a large or moderate earthquake, which is

interpreted as an alert for the earthquake. To estimate the alert time one must, (i) utilize the

time series Vz(t), collected at broad-band stations near the epicenters. Due to localization of

elastic waves in rock [15], data from far away stations cannot provide any clue to the transition

in the shape of the increments’ PDF. The station’s distance (300 km) is not universal and

depends on the geology, but it is of the correct order of magnitude. (ii) The time scale s for

moving λ2

that λ2→ 0. On this scale the difference between λ2

large enough to obtain a meaningful alert for the earthquake. (iii) One must also estimate λ2

or the flatness in some windows (here, 1 hour windows) and move it over the time series, in

order to observe its variations with the time.

We thank R. Friedrich, U. Frisch, H. Kantz, and K. R. Sreenivasan for critical reading of

the manuscript, as well as J. G. Jim´ enez, T. Matsumoto, M. Mokhtari and S. M. Movahed for

useful discussions. M.R.R.T. would like to thank the Alexander von Humboldt Foundation,

the Associate program of the Abdus Salam International Center for Theoretical Physics, and

the Knowledge Archive funding for financial support. The data for Vz(t) were provided by the

USGS and the National Geographic Institute of Spain.

sis not large, even for the data that are collected

sclose to the earthquakes.

soccurred about 10 hours before the earthquake, while for the

s(here, 800 ms) is estimated by its plot vs s for data set (I), and a choice of s such

sfor the data sets (I) and (II) will be

s

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-6-4 -20246

10-4

10-2

100

102

Ps(Zs)

Zs/σs

(

(b)

-6 -4-20246

10-4

10-2

100

102

Zs/σs

(a)

Ps(Zs)

FIG. 1: Continuous deformation of the increments’ PDFs in log-linear scale, for the M = 7.1

earthquake across scales for, from top to bottom, s = 200, 400, 600, and 800 ms, and (a) far from,

and (b) close to, the earthquake. Solid curves are the PDFs based on Eq. (2), while dashed curves

are the Gaussian PDF.

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10002000

0

0.2

0.4

0.6

0.8

1

Close to EQ

Far from EQ

λs

2

s [ms]

M=6.1

(b)

1000 2000

0

0.2

0.4

0.6

0.8

1

Close to EQ

Far from EQ

λs

2

s [ms]

M=7.1

(a)

FIG. 2: Scale-dependence of λ2

close to [data set (II)] the earthquake. For the data set (I) and s > 700 ms, λ2

the increments’ PDF is Gaussian, whereas for the data set (II) λ2

700 ms < s < 1500 ms. (b) Same as in (a), but for the M = 6.1 earthquake. When λs→ 0 the error

bars are very small, about the same size as the symbols.

svs logs. (a) The M = 7.1 event, both far from [data set (I)] and

s→ 0, implying that

sdeviates strongly from zero for

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20004000 6000

0

0.2

0.4

0.6

0.8

Close to EQ

Far from EQ

s = 800 ms

s = 700 ms

s = 600 ms

C(s)(τ)/C(s)(0)

τ [ms]

FIG. 3: The correlation function C(s)(τ) for the data set (I) far from, and the set (II) close to, the

M = 7.1 earthquake.

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-20-15 -10-50

3

3.2

3.4

3.6

3.8

Time [hour]

Fs=800

(c)

M=7.1

-20 -15-10-50

0

0.1

0.2

0.3

0.4

Time [hour]

λ2

s=800

(a)

M=7.1

-20 -15-10 -50

3

3.2

3.4

3.6

3.8

4

4.2

4.4

Time [hour]

(d)

M=6.1

Fs=800

-20-15-10 -50

0

0.1

0.2

0.3

0.4

Time [hour]

λ2

s=800

(b)

M=6.1

FIG. 4: The local temporal dependance of λ2

for the M = 7.1 and M = 6.1 events, indicating a gradual, systematic increase on approaching the

earthquakes.

sand the flatness for s = 800 ms, over a one-hour period,

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-20-15-10-50

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [hour]

λ2

s=500

(c)

d≈400 km

-20-15-10-50

0

0.2

0.4

Time [hour]

λ2

s=500

(b)

d≈128 km

-20-15-10-50

-40000

0

40000

Time [hour]

(a)

Vz

FIG. 5: (a) The data for the M = 5.4 earthquake in California. (b) and (c) Show the local temporal

dependance of λ2

sfor s = 500 ms, collected at stations with a distance d from the epicenter. The

station at d = 128 km provides the alert, whereas the second station does not yield any alert.

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