Turbulencelike behavior of seismic time series.
ABSTRACT We report on a stochastic analysis of Earth's vertical velocity time series 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 transition in their probability density function from Gaussian to non-Gaussian. The transition occurs 5-10 hours prior to a moderate or large earthquake, hence representing a new and reliable precursor for detecting such earthquakes.
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ABSTRACT: We investigate the predictability of extreme events in time series. The focus of this work is to understand under which circumstances large events are better predictable than smaller events. Therefore we use a simple prediction algorithm based on precursory structures which are identified using the maximum likelihood principle. Using the receiver operator characteristic curve as a measure for the quality of predictions we find that the dependence on the event size is closely linked to the probability distribution function of the underlying stochastic process. We evaluate this dependence on the probability distribution function analytically and numerically. If we assume that the optimal precursory structures are used to make the predictions, we find that large increments are better predictable if the underlying stochastic process has a Gaussian probability distribution function, whereas larger increments are harder to predict if the underlying probability distribution function has a power-law tail. In the case of an exponential distribution function we find no significant dependence on the event size. Furthermore we compare these results with predictions of increments in correlated data, namely, velocity increments of a free jet flow. The velocity increments in the free jet flow are in dependence on the time scale either asymptotically Gaussian or asymptotically exponential distributed. The numerical results for predictions within free jet data are in good agreement with the previous analytical considerations for random numbers.Physical Review E 02/2008; 77(1 Pt 1):011108. · 2.31 Impact Factor
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ABSTRACT: We study non-Gaussian probability density functions (PDF's) of multiplicative log-normal models in which the multiplication of Gaussian and log-normally distributed random variables is considered. To describe the PDF of the velocity difference between two points in fully developed turbulent flows, the non-Gaussian PDF model was originally introduced by Castaing [Physica D 46, 177 (1990)]. In practical applications, an experimental PDF is approximated with Castaing's model by tuning a single non-Gaussian parameter, which corresponds to the logarithmic variance of the log-normally distributed variable in the model. In this paper, we propose an estimator of the non-Gaussian parameter based on the q th order absolute moments. To test the estimator, we introduce two types of stochastic processes within the framework of the multiplicative log-normal model. One is a sequence of independent and identically distributed random variables. The other is a log-normal cascade-type multiplicative process. By analyzing the numerically generated time series, we demonstrate that the estimator can reliably determine the theoretical value of the non-Gaussian parameter. Scale dependence of the non-Gaussian parameter in multiplicative log-normal models is also studied, both analytically and numerically. As an application of the estimator, we demonstrate that non-Gaussian PDF's observed in the S&P500 index fluctuations are well described by the multiplicative log-normal model.Physical Review E 10/2007; 76(4 Pt 1):041113. · 2.31 Impact Factor
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ABSTRACT: Some properties of a model of intermittency in fully developed turbulence due to She and Lévêque [Phys. Rev. Lett. 72, 336 (1994)] are explored. The probability functions solution of the model is shown to be simply related to the log-Poisson statistics of local nondimensional energy dissipation. It is also shown that the intermittency obtained by She and Lévêque can be interpreted as the consequence of the scale covariance of the energy dissipation. Based on these observations, a new picture of turbulence is presented, in which scale covariance plays a central role.Physical Review Letters 09/1994; 73(7):959-962. · 7.94 Impact Factor
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
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
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
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  for the number of earthquakes
with a magnitude greater than a given value M, to the Omori law  for the distribution of
the aftershocks, the concept of self-organized criticality , and the percolation model of the
epicenters’ spatial distribution . At the same time, there has also been much interest in
investigating the precursors to, and the predictability of, extreme increments in time series
 associated with disparate phenomena, ranging from earthquakes [7,8], to epileptic seizures
, 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  and theoretically , 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
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)
where ln(Wλ) is a random variable. Iterating Eq. (1) forces implicitly the random variable Wλ
to follow a log infinitely-divisible law . One of the simplest candidates for such processes
is represented  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 ,
ω. The PDF of Zs(t) has fat tails that depend
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
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
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
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
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 . We define the magnitude of local variance over a scale s by,
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
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.
k=−ns/2Zs(t+k∆t)2, and, ¯ ωs(i) =1
s(i), respectively. Here, ∆t is the sampling
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
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 , 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
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.
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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.
Close to EQ
Far from EQ
Close to EQ
Far from EQ
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
Close to EQ
Far from EQ
s = 800 ms
s = 700 ms
s = 600 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.
-20 -15-10 -50
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
sand the flatness for s = 800 ms, over a one-hour period,
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.