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- Editorial board
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**Research article**
12 Jun 2019

**Research article** | 12 Jun 2019

Taylor's power law in the Wenchuan earthquake sequence with fluctuation scaling

^{1}Co-Innovation Centre for Sustainable Forestry in Southern China, College of Biology and the Environment, Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China^{2}China Earthquake Networks Center, China Earthquake Administration, Beijing 100045, China^{3}Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812, USA

^{1}Co-Innovation Centre for Sustainable Forestry in Southern China, College of Biology and the Environment, Bamboo Research Institute, Nanjing Forestry University, Nanjing 210037, China^{2}China Earthquake Networks Center, China Earthquake Administration, Beijing 100045, China^{3}Department of Mathematics and Statistics, University of Minnesota Duluth, Duluth, MN 55812, USA

Abstract

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Taylor's power law (TPL) describes the scaling relationship between the temporal or spatial variance and mean of population densities by a simple power law. TPL has been widely testified across space and time in biomedical sciences, botany, ecology, economics, epidemiology, and other fields. In this paper, TPL is analytically reconfirmed by testifying the variance as a function of the mean of the released energy of earthquakes with different magnitudes on varying timescales during the Wenchuan earthquake sequence. Estimates of the exponent of TPL are approximately 2, showing that there is mutual attraction among the events in the sequence. On the other hand, the spatio-temporal distribution of the Wenchuan aftershocks tends to be nonrandom but approximately definite and deterministic, which highly indicates a stable spatio-temporally dependent energy release caused by regional stress adjustment and redistribution during the fault revolution after the mainshock. The effect of different divisions on estimation of the intercept of TPL straight line has been checked, while the exponent is kept to be 2. The result shows that the intercept acts as a logarithm function of the time division. It implies that the mean–variance relationship of the energy release from the earthquakes can be predicted, although we cannot accurately know the occurrence time and locations of imminent events.

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How to cite.

Shi, P., Li, M., Li, Y., Liu, J., Shi, H., Xie, T., and Yue, C.: Taylor's power law in the Wenchuan earthquake sequence with fluctuation scaling, Nat. Hazards Earth Syst. Sci., 19, 1119-1127, https://doi.org/10.5194/nhess-19-1119-2019, 2019.

1 Introduction

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The Wenchuan *M*_{S} 8.0 earthquake on 12 May 2008 was the result of the
intensively compressive movement between the Qinghai–Tibet Plateau and the
Sichuan Basin. It ruptured the middle segment of the Longmenshan (LMS)
thrust belt (Burchfiel et al., 2008), with a total length of the fault trace of
approximately 400 km along the edge of the Sichuan Basin and the eastern
margin of the Tibetan Plateau, in the middle of the north–south seismic
belt of China. Millions of aftershocks occurred after the main event.
Up to now, the focus zone has tended to be quiet, with only small ones occurring
occasionally. A complete Wenchuan earthquake sequence has been attained.

Statistical seismology applies statistical methods to the investigation of seismic activities, and stochastic point process theory promotes the development of statistical seismology (Vere-Jones et al., 2005). After some improvement, most of the point process theories and methods can be used to analyze spatio-temporal data of earthquake occurrence and to describe active laws of aftershocks. The term “aftershock” is widely used to refer to those earthquakes which follow the occurrence of a large earthquake and aggregately take place in abundance within a limited interval of space and time. This population of earthquakes is usually called an earthquake sequence. In seismological investigations, one important subject has long been the statistical properties of the aftershocks. The spatial and temporal distribution of aftershocks after a destructive earthquake is usually performed in a general survey (Utsu, 1969). In seismology, one of the most famous theories describing the activities of aftershocks is the Gutenberg–Richter law (Gutenberg and Richter, 1956), which expresses the relationship between the magnitude and the total number of earthquakes with at least that magnitude in any given region and time interval. Another one is Omori's law, which was first depicted by Fusakichi Omori in 1894 (Omori, 1894) and shows that the frequency of aftershocks decreases roughly with the reciprocal of time after the mainshock. Utsu (1969) and Utsu et al. (1995) developed this law and proposed the modified Omori formula afterwards. Since the 1980s, with the development of nonlinear theory, an epidemic-type aftershock sequence (ETAS) model has been proposed by Ogata (1988, 1989, 1999), which is based on the empirical laws of aftershocks and quantifies the dynamic forecasting of the induced effects. This model has been used broadly in earthquake sequence study (Kumazawa and Ogata, 2013; Console, 2010).

An increasing number of investigations show that there is an interaction effect for the occurrence of aftershocks in a given area. A stress-triggering model is usually used to depict interaction between larger earthquakes in the view of physics (Harris, 1998; Stein, 1999). More and more results show that obvious increases in Coulomb stress not only promote the occurrence of upcoming middle or strong events of an earthquake sequence but also affect their spatial distribution to some degree (Robinson and Zhou, 2005).

The goal of this paper is to introduce a different statistical method called Taylor's power law (TPL) into the statistical seismology field by analyzing the Wenchuan earthquake sequence from the point of view of energy distribution or energy release. We aim to find out whether or not the energy distribution or energy release of the Wenchuan earthquake sequence complies with a specific power-law function as described by the TPL for different scaled samples and what the spatial and temporal properties are.

In statistics, there are two important moments in a distribution, the mean
(*μ*) and the variance (*σ*^{2}). It is common to describe the
types of the distributions using the relationship between these two
parameters. For instance, we have *σ*^{2}=*μ* for a
Poisson distribution. In nature, however, the variance is not always equal
to or proportional to the mean. Mutual attraction or mutual repulsion for
individuals in natural populations, e.g., the intra-specific competition of
plants, makes variance different from the mean. After examining many sets of
samples of animal and plant population spatial densities, Taylor (1961)
found that the variance appears to be related to the mean by a power-law
function: the variance is proportional to the mean raised to a certain power,

$$\begin{array}{}\text{(1)}& {\mathit{\sigma}}^{\mathrm{2}}=a{\mathit{\mu}}^{b},\end{array}$$

or equivalent to a linear function when the mean and variance are both logarithmically transformed,

$$\begin{array}{}\text{(2)}& {\mathrm{log}}_{\mathrm{10}}\left({\mathit{\sigma}}^{\mathrm{2}}\right)={\mathrm{log}}_{\mathrm{10}}\left(a\right)+b\times {\mathrm{log}}_{\mathrm{10}}\left(\mathit{\mu}\right)=c+b\times {\mathrm{log}}_{\mathrm{10}}\left(\mathit{\mu}\right),\end{array}$$

where *a* and *b* are constants and *c*=log _{10}(*a*). Equations (1) and (2) are called
Taylor's power law (henceforth TPL) or Taylor's power law of fluctuation
scaling (Eisler et al., 2008).

Equations (1) and (2) may be exact if the mean and variance are population moments
calculated from certain parametric families of skewed probability
distributions (Cohen and Xu, 2015). TPL describes the species-specific
relationship between the spatial or temporal variance of populations and
their mean abundances (Kilpatrick and Ives, 2003). It has been verified for
hundreds of biological species and abiotic quantities in biomedical
sciences, botany, ecology, epidemiology, and
other fields (Taylor, 1961, 1984; Kendal, 2002; Eisler et al., 2008; Cohen
and Xu, 2015; Shi et al., 2016, 2017; Lin et al., 2018). Most of the
scientific investigations of TPL mainly focus on the power-law exponent *b*
(or slope *b* in the linear form), which has been believed to contain
information on aggregation in space or time of populations for a certain
species (Horne and Schneider, 1995).

In this study, we also concentrate on the parameter *b* of TPL. We expect that
*b* is independent of the temporal block size *A*, which is used to divide the
Wenchuan sequence into different temporal blocks because the aftershock area
is invariable during this period.

2 Wenchuan earthquake sequence

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A large earthquake of magnitude *M*_{S} 8.0 hit Wenchuan in the Sichuan province
of China, at 14:28:01 CST (China Standard Time) on 12 May 2008, with an
epicenter located at 103.4^{∘} N and 31.0^{∘} E and a depth of
19 km.

According to the earthquake catalogue of the China Earthquake Networks
Center (CENC; http://www.csi.ac.cn/, last access: 30 June 2017), there have been 54 554
earthquakes of magnitudes *M* > 0 recorded for the Wenchuan sequence
by 31 December 2016. Figure 1 shows the frequency of aftershocks with
different magnitudes. Here, aftershocks with *M* < 2.0 account for
77.9 % of the total sequence due to the fact that only weak ones occur
after a long period of time after the mainshock. In addition, except for
the mainshock, the number of aftershocks is 733 for magnitudes 4.0 ≤ *M* < 5.0 and 86 for 5.0 ≤ *M* < 8.0. They
account for a very small percentage of the total.

Figure 2 displays the fluctuation variability of the Wenchuan earthquake
sequence with *M*≥3.0 from 12 May 2008 to 31 December 2016. The
temporal distribution of the magnitudes of aftershocks attenuates quickly
after the mainshock. The three larger aftershocks all occurred in 2008, with
*M* 6.4 on 25 May, *M* 6.1 on 1 August and *M* 6.1 on 5 August; 85 % of aftershocks with *M*≥3.0 occurred by the end of
2011, about 2.5 years after the mainshock.

Figure 3a shows the spatial distribution of epicenters of the Wenchuan
earthquake sequence with *M* > 0 from 12 May 2008 to 31 December 2016. The aftershocks are distributed in the region with latitude
102–107^{∘} E and longitude 30–34^{∘} N, mainly along the Longmenshan thrust fault, which is a
junction region of the Songpan–Garze block and South China block and extends
along the north–east–east (NEE) direction for more than 400 km. The size of
the aftershocks on different scales is characterized by a population density
of the events distributed in space and time after the Wenchuan *M*_{S} 8.0
earthquake, but we neglect the variations of the aftershock area in the next
step. The distribution of strong aftershocks is of different segment
characteristics. Earthquakes with a magnitude *M*≥5.0 are mainly spread in
the southern Miaoxian and Mianzhu area and northern Pingwu area. There are no strong
aftershocks occurring in the middle areas such as Beichuan and Anxian (see
Fig. 3b). According to the primary investigation results of the Wenchuan
rupture process conducted by Chen et al. (2008), the rupture of the Wenchuan
*M*_{S} 8.0 earthquake originated from Wenchuan thrust fault with a little right
lateral slip component and extended mainly in the northeast (NE) direction.
The whole process formed two areas with larger dislocations. One is the
southern area of Miaoxian located in the bottom section in Fig. 3b. The other
one lies near Beichuan area (the middle segment in Fig. 3b), but no strong
shocks happened there.

3 Data processing method

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For the complete Wenchuan earthquake sequence, we denote the number of all
earthquakes by *N*, i.e., *N*= 54 554, and use $q=\mathrm{1},\mathrm{\dots},N$ to index
each earthquake. For each earthquake with magnitude *M*_{q}, its
corresponding energy release is labeled by *E*_{q}, and it can be attained in
the light of the following relationship (Xu and Zhou, 1982):

$$\begin{array}{}\text{(3)}& {\mathrm{log}}_{\mathrm{10}}\left({E}_{q}\right)=\mathrm{11.8}+\mathrm{1.5}{M}_{q}.\end{array}$$

We use *t*_{q} to index the time lag of the *q*th aftershock from the mainshock (in days), i.e., *t*_{1}=0 for the main event. The last aftershock
occurred at 18:05:57 CST (China Standard Time) on 31 December 2016, and its
*t*_{q} value is 3155.

In order to study the relationship between the variance and mean of the
energy sequence *E*_{q}, we first divide it into equally spaced short
temporal blocks with size *A* (in days). For example, if *A*=10, then the
number of blocks is $N/A=\mathrm{3155}/\mathrm{10}=\mathrm{315.5}$, which is rounded to the nearest
integer. Now the complete energy sequence *E*_{q} is partitioned into *n*=316
blocks of short energy subsequences. We use *i* to index each block, i.e., $i=\mathrm{1},\mathrm{\dots},n$, and *h*_{i} to denote the number of data points in each
block, which is variable because earthquakes occurred stochastically in the
sequence. Now we can calculate the mean (*μ*) and variance (*σ*^{2}) for each block using

$$\begin{array}{}\text{(4)}& {\displaystyle}& {\displaystyle}{\mathit{\mu}}_{i}={\displaystyle \frac{\sum _{j=\mathrm{1}}^{{h}_{i}}{E}_{i,j}}{{h}_{i}}},\text{(5)}& {\displaystyle}& {\displaystyle}{\mathit{\sigma}}_{i}^{\mathrm{2}}={\displaystyle \frac{\sum _{j=\mathrm{1}}^{{h}_{i}}({E}_{i,j}-{\mathit{\mu}}_{i}{)}^{\mathrm{2}}}{{h}_{i}-\mathrm{1}}},\end{array}$$

where *E*_{i,j} denotes the energy of the *j*th earthquake in the *i*th
block.

4 Results

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The data processing procedure has been performed with different block sizes
$A=\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{\dots},\mathrm{100}$. The number of sample points in each block
decreases as the block size increases. The relationships between the mean
and variance of the released energies from earthquakes in six representative
temporal blocks are shown in Fig. 4 on a log–log scale. The red line
stands for the fitted linear function of TPL
${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\sigma}}^{\mathrm{2}}\right)=c+b{\mathrm{log}}_{\mathrm{10}}\left(\mathit{\mu}\right)$ using the least
squares method. The 95 % confidence intervals (CIs) of the slope and the
coefficients of determination *R*^{2} are shown in Table S1 in the Supplement. For instance,
Fig. 4a shows the variance as a function of the mean for 316 time
intervals when *A*=10. The estimated intercept is 0.702, the estimated
slope is 2.060, with a 95 % CI (1.989, 2.076), and *R*^{2}=0.963. The
root-mean-square error (RMSE) was also calculated to exhibit the feasibility
of using TPL with the exponent 2 to approximate that with the exponent to
be estimated (unknown).

Figure 4 and Table S1 show that there is an apparent linear relationship
between the common logarithm of the variance and that of the mean for all
earthquakes occurring within different temporal blocks, characterized by a
property of aggregation on different timescales. The estimated value of the
intercept, *c* (or log _{10}(*a*)), which is mainly influenced by the number of
samples, increases overall with *A*, from 0.016 to 3.249 (Table S1). The
estimates of slope *b*, on the other hand, are roughly 2 for all block sizes
used in the study. All *R*^{2} values are greater than 0.96, showing a
very strong linear relationship. These results indicate that the energy
release of aftershocks of the Wenchuan sequence complies well with a
temporal TPL.

Next, we divide the Wenchuan earthquake sequence into nine time stages in
years: 2008–2008, 2008–2009, 2008–2010, 2008–2011, 2008–2012,
2008–2013, 2008–2014, 2008–2015, and 2008–2016. For each stage, we use a similar procedure, which has been used in Fig. 4, to generate Fig. 5. We first transform
all earthquakes into their energy forms using the relationship between
earthquake magnitude *M* and energy *E*. Then the energy sequences are
partitioned into temporal blocks with a fixed block size *A*=10 d. The
calculated variances and means are plotted on a log–log scale, as shown in
Fig. 5. Again, TPL comes into play for all time stages. The estimates of
the parameters in Eq. (2) for the data in different stages were listed in
Table S2.

Figure 5 shows a strong linear relationship between the variance and mean of
the earthquake energy populations on a log–log scale, especially for those
large samples. The estimates summarized in Table S2 (red fitted lines in
Fig. 5) show similar results to Table S1. The intercept gradually
increases as the total number of samples increases, but with a little more
fluctuation. Meanwhile, the estimate of slope *b* is still roughly constant
at around 2.

Here with the exponent *b*=2 fixed, the possible relationship between
the estimate of the intercept (namely *d*) in equation ${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\sigma}}^{\mathrm{2}}\right)=d+\mathrm{2}{\mathrm{log}}_{\mathrm{10}}\left(\mathit{\mu}\right)$ and the temporal block size *A* is also
examined. The estimated intercepts of the Wenchuan sequence as *A* increases
from 4 to 100 d in 1 d increments are shown in Fig. 6. At the
same time, a logarithm function and an exponential function are employed
respectively to fit the data (i.e., $d=\mathit{\alpha}+\mathit{\beta}\times {\mathrm{log}}_{\mathrm{10}}\left(A\right)$ and $d=m\times {A}^{n}$, where *α*, *β*, *m*, and *n* are
constants), and the results show that the logarithm function has a higher
goodness of fit (namely a lower residual sum of squares). The estimate of
parameter *α* is equal to 0.7398, with a 95 % CI (0.7246, 0.7581), and
the estimate of parameter *β* is equal to 0.9121, with a 95 % CI
(0.9004, 0.9229). Because ${\mathrm{log}}_{\mathrm{10}}\left(a\right)=d=\mathit{\alpha}+\mathit{\beta}\times {\mathrm{log}}_{\mathrm{10}}\left(A\right)$, we will have the following:

$$\begin{array}{}\text{(6)}& {\mathit{\sigma}}^{\mathrm{2}}=a{\mathit{\mu}}^{\mathrm{2}}={\mathrm{10}}^{\mathit{\alpha}}{A}^{\mathit{\beta}}{\mathit{\mu}}^{\mathrm{2}}.\end{array}$$

This shows that the variance of energy releases from aftershocks depends on two factors: (i) the squared mean and (ii) the size of the temporal block. Up to now, we confirm that the mean–variance relationship of energy releases from an earthquake sequence can be quantified, although the accurate prediction of the time and location of an imminent event is still not attainable.

5 Discussion

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The evolutionary process of a large earthquake is characterized by some complex features from stochastic to chaotic or pseudo-periodic dynamics (McCaffrey, 2011). On the one hand, there is a long-term slow strain of accumulation and culmination of rocks in the rigid lithosphere prior to the event, with a sudden rupture and displacement of blocks. On the other hand, there is another long-term slow strain of redistribution and energy release with a large number of aftershock occurrences in an extensive area, which generally lasts for several months, sometimes even years, after the mainshock.

It has been statistically established that in populations, if individuals
distribute randomly and are independent of each other, then the variance is
equal to the mean, i.e., *σ*^{2}=*μ*; individuals show mutual
attraction if the variance is proportional to the mean to a power more than 1, and individuals
mutually repel each other if the variance is proportional to the mean to a
power less than 1 (Taylor, 1961; Horne
and Schneider, 1995). The results obtained in this study show that the
exponent of TPL is around 2 in the Wenchuan energy sequence, either with
a different time span $A=\mathrm{4},\mathrm{5},\mathrm{6},\mathrm{\dots},\mathrm{100}$ d or with a fixed
time span *A*=10 d, but for nine time stages between 2008 and 2016. This
means that earthquakes in the Wenchuan sequence are not distributed at random and
independent of each other but have a mutual attraction. It also indicates
that there are possible interactions among different magnitudes in the
earthquake sequence. Cohen and Xu (2015) showed analytically that
observations randomly sampled in blocks from any single skewed frequency
distribution with four finite moments give rise to TPL because the covariance
of the sample mean and the sample variance is proportional to the skewness of the underlying distribution.

There are various types of interpretations for the value of parameter *b*. Ballantyne and Kerkhoff (2007) suggested that individuals' reproductive correlation
determines the size of *b*, while Kilpatrick and Ives (2003) proposed that
inter-specific competition could reduce the value of *b*. Above all,
empirically, *b* usually lies between 1 and 2 (Maurer and Taper, 2002).
However, it is expected that TPL holds with *b*=2 exactly in a population
with a constant coefficient of variation (CV) of population density. This
expectation is derived from the well-known relationship: SD (standard
deviation) equals the square root of variance (*σ*^{2}), i.e.,
SD=*σ*, and the coefficient of variation CV = SD$/\mathit{\mu}=k$, here *k*, is a constant. Then we can obtain *σ*^{2}=(*k**μ*)^{2}.
The relationship between log _{10}(*σ*^{2}) and log _{10}(*k**μ*) is
a straight line with a slope of 2 on a log–log scale.

It is well established that there is a specific property on the population
either in space or in time when *b* equals 2. Ballantyne (2005) proposed that
*b*=2 is a consequence of deterministic population growth, while Cohen (2013) showed that *b*=2 arose from exponentially growing, noninteracting
clones. Furthermore, using the Lewontin–Cohen (LC) model of stochastic
population dynamics, Cohen et al. (2013) provided an explicit, exact
interpretation of its parameters of TPL. They proposed that the exponent of
TPL will be equal to 2 if and only if the LC model is deterministic; it will
be greater than 2 if the model is supercritical (growing on average) and will be
less than 2 if the model is subcritical (declining on average). This
property indicates that parameter *b*=2 in our investigation on the
Wenchuan earthquake sequence depends exactly on its specific distribution of
aftershocks. In other words, the law of occurrence of all events or energy
release in space and time is deterministic following the mainshock on 12 May 2008.

Although various empirical confirmations suggest that no specific
biological, physical, technological, or behavioral mechanism can explain all
instances of TPL, there has been some improvement in understanding the
distribution and duration time of aftershocks after the main event. Jiang et
al. (2008) studied the Wenchuan earthquake sequence using the Gutenberg–Richter
law (Gutenberg and Richter, 1956) and Omori's law (Omori, 1894). Their
investigation attained a specific relationship between the magnitude and the
total number of earthquakes for a stable *b* value, which indicates that the
frequency of aftershocks decreases roughly with the reciprocal of time after
the mainshock. One of the models with physical parameters is the stress-triggering mechanism put forward by Dieterich (1994) with Dieterich and
Kilgore (1996). Shen et al. (2013) achieved a good fit between the observed
Wenchuan aftershocks and the analytic solution of the modified Dieterich
model. Their results suggested that the generation of earthquakes is
actually related to the state of the fault and can quantitatively describe the
temporal evolution of the aftershock decay. In this sense, the Wenchuan
energy sequence satisfies TPL with the slope *b*=2, indicating a stable
spatio-temporal dependent energy release caused by regional stress
adjustment and redistribution during the fault revolution after the mainshock. These results are of high coherence with what has been attained by
Christensen et al. (2002), who proposed a unified scaling law linking
the Gutenberg–Richter law, Omori's law of aftershocks, and the
fractal dimensions of the faults. Their results show that a nonzero driving
force in the crust of the Earth leads to an earthquake as a sequence of
hierarchical correlated processes, and this mechanism, responsible for small
events, is also responsible for large events. In other words, a mainshock
and an aftershock are consequences of the same process.

It is possible that there are some interactions among earthquakes with
different magnitudes in an earthquake sequence. This kind of interaction is
probably derived from the medium-stress state of the focus zone where
earthquakes happen. The stress field in the aftershock area is in a rapidly
adjusting state when a lager earthquake has occurred. It is probable that a
light stress adjustment caused by a small earthquake most likely induces an
obvious event in its surroundings in the near future. This process can lead
to aggregation of aftershocks in space and time in extensive areas, causing
TPL to hold for the Wenchuan earthquake energy sequence. However, whether
TPL accords with all earthquake sequences and complies with specific
parameters, e.g., *b*=2, needs further investigation. Up to now, one thing
we can confirm is that the missing events can lead to the exponent in TPL
increase. For example, the estimated *b* is approximately 2.1 to 2.2 if only the
events with magnitude *M* > 1.0 are used. It indicates that missing
events can change the state of energy release from a stable (deterministic)
state to an unstable (supercritical) state, as Cohen et al. (2013)
proposed.

The current study shows that the exponents of TPL for different temporal
blocks for the Wenchuan earthquake sequence are approximately equal to 2
universally. The estimated intercept could be expressed as a linear equation
of the log transformation of temporal block *A* (Fig. 6). The goodness of fit
of the nonlinear regression is fairly high (*R*^{2}=0.9940 in Fig. 6),
indicating some interesting underlying mechanism leading to the occurrence
of the aftershocks. The distribution of the energy releases from aftershocks
should be a right-skewed unimodal curve that can be reflected by magnitude
frequency distribution as shown in Fig. 1. In fact, Cohen and Xu (2015)
have demonstrated that the correlated sampling variation of the mean and
variance of skewed distributions could account for TPL under random sampling
and the estimated exponent of TPL was proportional to the skewness of the
distribution curve. For an exponential distribution, the variance equals its
mean squared. However, in our study, although the variance of energy
releases from aftershocks is similarly proportional to its mean squared, the
coefficient of proportionality (i.e., *a* in Eq. 1) does rely on the size of
the temporal block. This means that the energy releases from aftershocks
might follow a temporal block-dependent generalized exponential
distribution, which should be more complex than the generalized exponential
distribution (Gupta and Kundu, 2007). However, the distribution function for
the energy releases from aftershocks has not been well defined so far. The
existing functions for describing a skewed distribution of energy releases
or magnitudes usually belong to pure statistical models that lack a clear
physical dynamic mechanism. Our study suggests that further studies should
focus on a temporal block-dependent or a sub-region-dependent distribution.
However, providing a clear mathematical expression for this distribution
function is beyond the scope of this paper. It deserves further
investigation.

6 Conclusions

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In summary, we attempt to use a new way of investigating a spatio-temporal
distribution property of aftershocks of the Wenchuan earthquake sequence
during 2008–2016. In terms of the energy release, the variance of samples
in the earthquake population is shown to have a simple power-law
relationship as a function of the mean on different timescales, which gives
rise to a TPL, i.e., *σ*^{2}=*a**μ*^{b}, with *b*=2. On the one
hand, the results show that the intercept of the fitted line in the linear form
${\mathrm{log}}_{\mathrm{10}}\left({\mathit{\sigma}}^{\mathrm{2}}\right)=c+b\times {\mathrm{log}}_{\mathrm{10}}\left(\mathit{\mu}\right)$ on a
log–log scale increases with the number of samples, and it is reconfirmed
that parameter *c* (namely log _{10}(*a*)) predominantly depends on the size of
the sampling units (Taylor, 1961). On the other hand, if TPL holds, the
estimated values of parameters *a* and *b* support the conclusion that the
Wenchuan aftershocks mutually trigger each other and distribute in space and
time not randomly but determinately and definitely. We fix the exponent of
TPL to be 2 and check the effects of different time divisions on the
estimate of the intercept. The results show that the intercept acts as a
logarithm function of the timescale. This implies that the mean–variance
relationship of energy releases from the earthquakes can be predicted even
though we cannot accurately predict the time and location of imminent
events.

Data availability

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Data availability.

The data that can be publicly accessed are described within the text. The corresponding author, Mei Li, would like to provide the data on request of the readers who are interested in reanalyzing them but cannot successfully download them.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/nhess-19-1119-2019-supplement.

Author contributions

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Author contributions.

PS, ML, and YL analyzed the data. ML and YL wrote the paper. PS and JL further revised the paper. HS collected the data of the Wenchuan earthquake sequence. TX and CY drew the figures. All authors read and commented on the paper.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements

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Acknowledgements.

We thank Joel E. Cohen for providing valuable help during the preparation of this work. We are also thankful to the editor and three reviewers for their productive comments. The work has been funded from the NSFC (National Natural Science Foundation of China) under grant no. 41774084 and National Key R&D Program of China under grant no. 2018YFC1503506. Peijian Shi was supported by the Priority Academic Program Development of Jiangsu Higher Education institutions.

Review statement

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Review statement.

This paper was edited by Oded Katz and reviewed by Pedro Puig and two anonymous referees.

References

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Short summary

A statistical method is tentatively utilized to study distribution properties of aftershocks of the Wenchuan sequence in the view of energy release. The results show that the events in the Wenchuan sequence are not independent but have mutual attraction, their spatio–temporal distribution tends to be nonrandom but definite and deterministic, and imply it is possible for energy release to be predicted, although we cannot accurately predict the occurrence time and locations of the imminent event.

A statistical method is tentatively utilized to study distribution properties of aftershocks of...

Natural Hazards and Earth System Sciences

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