Fractal and Morlet-wavelet analyses of M ≥ 6 earthquakes in the South-North Seismic Belt, China

The M ≥ 6 earthquakes occurred in the South-North Seismic Belt, Mainland China (longitudes from 98 107°E and latitudes from 21 41°N) during 190

A positive relation between the b-value of the frequency-magnitude law proposed by Gutenberg and Richter (1944) and the D value, which is the fractal dimension as described below, were: D = 3b/c (c = 1.5) by Aki (1982); b = D/3 by Turcotte (1986a); and b = D/2 by Turcotte (1986b).Terr. Atmos. Ocean. Sci., Vol. 28, No. 6, 799-814, December 2017 However, Hirata (1989) reported a negative correlation between the two parameters for earthquakes in Japan.He also argued the positive relation given by Aki as mentioned above.From simulation results based on the 1-D dynamic spring-slider model, Wang (1991) found weak dependence of the b-value upon the D value.This conclusion is different from the theoretical postulation made by Aki (1982) and Turcotte (1986a, b) and from the observations (Hirata 1989).A possible reason for the difference might be that the D value used by Wang (1991) concerns the distribution of the fault breaking strengths while the D value used by the others is related to the fault geometry.Wang and Lin (1993) and Wang and Lee (1997) observed that the D value is negatively correlated with the b-value as pointed out by Hirata (1989).Hence, in order to understand the fractal characteristics of spatial distribution and temporal variation in earthquakes it is significant to study the fractal properties of seismicity.
It is important to explore the dominant recurrence periods in an earthquake sequence not only for earthquake physics but also for seismic hazard assessment.Although Fourier analysis is commonly applied to evaluate the dominant periods (or frequency) of a time series, it cannot provide the temporal variations in the dominant periods.The wavelet transform can be used to analyze time series that contain nonstationary power at many different frequencies (Daubechies 1990).In this study, the wavelet analysis (Combes et al. 1989;Pyrak-Nolte and Nolte 1995), also known as the multiresolution analysis, is taken into account.For this technique a series of scaled and delayed oscillatory functions are used to decompose a time-varying signal into its non-stationary spectral components.Hence, the key advantage of wavelet analysis over traditional Fourier analysis is that the wavelet analysis provides information on how the spectral content varies with time delay.Wavelets are also advantageous over so-called windowed Fourier methods because with wavelets the relative accuracy of the delay and frequency remain constants cover all of the delay-frequency parameter space.The wavelet analysis application to geophysical problems can be seen in Torrence and Compo (1998).Here, a nonorthonormal Morlet wavelet analysis (Morlet et al. 1982;Chen et al. 2015) is taken into account.
Active orogeny due to the collision between the Indian and Eurasian plates (Molnar and Tapponnier 1975;Tapponnier and Molnar 1977;Tapponnier et al. 1982Tapponnier et al. , 2001;;Avouac and Tapponnier 1993;Royden et al. 2008) causes complex geological features and high seismicity in Mainland China.Complex geological conditions result in a nonuniform temporal-spatial distribution of earthquakes (Xu and Shen 1981;Huan and Shi 1987;Tan et al. 1987;Wang and Zhong 1987;Chen et al. 1994).Ma et al. (1992) delineated three highly seismic belts in Mainland China.The first is the South-North Seismic Belt (SNSB), with longitudes from 98 -107°E and latitudes from 21 -41°N, in the middle part of Mainland China.The earthquakes occurring in the SNSB should be inter-plate events.Historically, numerous large earthquakes occurred in the region.For example, two M ≥ 6 destructive earthquakes were located in the SNSB, i.e., the 3 February 1996 M 6.6 Lijiang earthquake, Yunnan (YSB 1998) and the 12 May 2008 M 7.9 Wenchung earthquake (Cheng et al. 2009;Zhang et al. 2009).The two events caused severe damage in the respective areas.Seismic hazard assessment is obviously an important issue in the region.Fractal analyses of spatial distribution of earthquakes will lead to geometric structures of earthquakes associated with active faults.Fractal and Morlet-wavelet analysesof earthquake sequence will provide the temporal characteristics and predominant recurrence periods for earthquakes.Active fault geometrical structures and earthquake recurrence times will influence the seismic hazard potential in a region.The present studies are helpful for understanding seismic hazards in the study region.
For the M ≥ 6 earthquakes that occurred from 1900 to 1990 in the region, Wang and Lee (1997) measured multifractal dimensions of earthquake sequence and Wang and Shen (1999) measured the multifractal dimensions of epicentral distributions.For the M ≥ 6 earthquakes that occurred in the region during 1901-2008, Wang (2013) ) studied the memory effect in the earthquake magnitude sequences and inter-event time using the fluctuation analysis technique.The calculated results show that the exponents of scaling law offluctuation versus window length are less than 0.5 for the earthquake magnitude sequences and interevent time.Together with all kinds of given information, he concluded that the earthquakes are short-term correlated, in other words, the short-term memory effect is operative.
After 1990, numerous M ≥ 6 earthquakes occurred in the region.Hence, it is significant to re-investigate the multifractal properties of spatial distribution and earthquake sequence for M ≥ 6 earthquakes that occurred over a longer time period from 1900 to 2016.In addition, the Morlet wavelet analysis will also be taken to analyze the dominant earthquake sequence period.

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A complete catalogue is needed to study the properties of earthquake temporal variation, especially for that made from instrumentally-recorded data.The earliest documented earthquake and damage data in Mainland China were found in 780 B.C. (ASIG 1975).Li (1960) compiled the first catalogue of Chinese earthquakes.Several Chinese earthquake catalogues were compiled by Chinese and foreign seismologists.Some of them were completed before 1978 and described by Lee et al. (1978).Ku (1983a, b) compiled a catalogue for earthquakes occurring from 1831 B.C. to 1979 A.D. Hsieh et al. (1989) compiled a catalogue for M ≥ 4.7 earthquakes.Ma et al. (1992) completed a complete catalogue for M ≥ 6 earthquakes occurring in the SNSB and other two seismic belts as mentioned above during 1901 and 1990.After 1990, two M ≥ 6 destructive earthquakes were located in the SNSB, i.e., the 3 February 1996 M 6.6 Lijiang earthquake, Yunnan (YSB 1998) and the 12 May 2008 M 7.9 Wenchung earthquake (Cheng et al. 2009;Zhang et al. 2009).The two events caused severe damage in the respective surrounding areas.In addition to these two destructive earthquakes, several M ≥ 6 earthquakes occurred in the region after 1990.The earthquake data from those events were taken directly from the Earthquake Determination Report by US Geological Survey (USGS).The magnitude of the Wenchung earthquake was given as 8.0 by the China Earthquake Administration (CEA).Nevertheless, the magnitude value (M = 7.9) determined by USGS is taken in this study.Since Ma et al. (1992) corrected the earthquake magnitudes in their catalogue as the surface-wave magnitudes used by USGS.The magnitude scale is uniform for all data used in this study.
One hundred eighteen earthquakes with magnitudes from 6.0 -8.5 are used in this study.Among them, 25 events had magnitudes ≥ 7. The occurrence dates, epicenters, and magnitudes of those earthquakes are listed in Table A1 in the appendix.The focal depths were not given by Ma et al. (1992) and thus not shown in Table A1.The pre-1967 earthquakes had higher uncertainties of epicentral location than the post-1967 events.However, at present it is difficult to accurately estimate the difference between the uncertainties of epicentral location in the two time periods.From Table A1, we can see five pairs of events, i.e., Events 3 and 4, Events 76 and 77, Events 90 and 91, Events 92 and 93, Events 95 and 96, and Events 104 and 105.For each pair, the two events happened close to each other in a short time span.Meanwhile, there was only a small difference between their magnitudes.The two events of a pair form an earthquake "doublet".An earthquake doublet is a pair of similarly sized earthquakes located relatively close together within short time duration.In addition, four sequent earthquakes, i.e., events 27, 28, 29, and 30, occurred in 1930 and 1931.The four events form an earthquake "quadruplet."An earthquake quadruplet includes four similarly sized earthquakes occurring relatively closely within short time duration.However, each doublet or quadruplet event is considered an individual one in this study.According to Bath's law (cf.Richter 1958), the magnitude of largest aftershock is, on the average, about 1.2 smaller than that of the mainshock.Hence, an M ≥ 8 earthquake will be followed by several M ≥ 7 aftershocks and an M ≥ 7 earthquake will be followed by several M ≥ 6 aftershocks.Hence, some events with magnitudes < 7 could be the aftershocks of relevant M ≥ 8 earthquakes.Figure 1 shows the plot of log(N) versus M and the straight line represents the least-squared equation: log(N) = (7.56 ± 0.19) -(0.90 ± 0.03)M, which is the frequencymagnitude law proposed by Gutenberg and Richter (1944).
Clearly, the data points are all around the straight line, thus suggesting data set completeness.Nevertheless, the data points for are slightly below the straight line.Hence, the fluctuation analysis will be conducted for three magnitude ranges with different lower-bound magnitudes, M low , i.e., M low = 6, 6.5, and 7.
The epicenters are displayed in Fig. 2. The epicentral distance between two events ranges from 0 -2141.0km  for M ≥ 6 events, from 0 -1934.3km for M ≥ 6.5 events, and from 22 -1934.3km for M ≥ 7 events.Figure 3 shows the earthquake sequences represented by magnitudes.The shortest inter-event time is smaller than 1 year for M ≥ 6, M ≥ 6.5, and M ≥ 7 events, and the shortest and longest interevent times are, respectively, 0 and 12.27 years (with an average of 0.96 years) for M ≥ 6 events, 0 and 12.27 years (with an average of 1.93 years) for M ≥ 6.5 events, and 0 and 19.52 years (with an average of 4.15 years) for M ≥ 7 events.Since a few events occurred in a short time interval, the line segments representing them are close to one another and thus cannot be clearly separated.

Multifractal dimensions of epicentral distributions
A fractal set is defined as one in which the Hausdorff-Besicovitch dimension strictly exceeds the commonly-used topological dimension (Mandelbrot 1983).The fractal dimension is a characteristic fractal set index.However, it is not easy to apply the Hausdorff-Besicovitch's definition to estimate the fractal dimension in the real world.Hence, a single fractal dimension value is not enough to characterize the multifractal properties.Therefore, the fractal dimension has been extended to the generalized fractal dimension or multifractal dimension, D q (Grassberger 1983;Hentschel and Procaccia 1983).Generalized fractal dimension D q is defined by the following expression: where p i is the probability that the events fall into a box with a length δ (Grassberger 1983;Hentschel and Procaccia 1983).The parameter q can take any real number in the range from -∞ to ∞. D q of large, positive q shows the fractal property of dense regions, where p i is large, and D q of large, negative q displays that of thin regions, where p i is small.D q for negative q can take a value larger than the spatial topological dimension d, thus calling D q a dimension make no geometric sense for D q > d (Mandelbrot 1989).For q > 0, the largest D q is D 0 , and D q decreases with increasing q.In Fig. 3. Time sequences of M ≥ 6 earthquakes displayed in Fig. 1.
the cases of q = 0, 1, and 2, D q relates to the capacity dimension D CA , information dimension D I , and correlation dimension D C , respectively.The probability p i can be estimated by the box-counting method from the observed data.However, the box-counting method requires a large number of data.An alternative correlation integral method was suggested by Kurths and Herzel (1987).A local density function n j (r) is defined by the following expression: where the value of Θ(s) is 1 if δr = r -|x j -x k | ≥ 0 and 0 if δr < 0. Conventionally, both r and |x j -x k | were defined as the distance between two points (x j , x k ).
A generalized correlation integral C q (r) for the epicentral distance, r, is defined by C q (r) scales with r in the following form: (5) The D q will be evaluated from the linear portion between log(r 1 ) and log(r 2 ) of the log-log plot of C q versus r.In this study the D q value at positive q is measured to establish the D q -q relation for investigating multifractal behavior of epicentral distribution of earthquakes in use.

Multifractal dimensions of earthquake time Sequences
In order to study the multifractal behavior of earthquake sequences, Wang and Lee (1995) replaced the two quantities r and |r i -r k | by time interval t and inter-event time |t i -t k | between two events, respectively.Wang (1996a) measured the generalized fractal dimensions at positive q for M ≥ 7 earthquake sequence in Taiwan.Wang and Lee (1995) and Wang (1996a) defined the local density function n j (t) to be: where the Θ(s) value is 1 if δt = t t t j k --≥ 0 and 0 if δt < 0. Hence, the generalized correlation integral C q (t) for the inter-event time, t, is C q (t) scales with t in the following form: The D q will be evaluated from the linear portion between log(t 1 ) and log(t 2 ) of the log-log plot of C q versus t.In this study the value of D q at positive q is measured to establish the D q -q relation for investigating multifractal behavior of earthquake sequences in use.

Morlet Wavelet Analysis for earthquake Sequences
Assume that there is a time series, x i , with an equal time spacing δt and n = 0 … N -1 and ( ) t } is a wavelet function of time t.To be "admissible" as a wavelet, ( ) t } must have zero mean and be localized in both time and frequency space (Farge 1992).An example is the Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud et al. (1984).The main function Ψ(t) is composed of a harmonic wave, as a constant k c subtracted from a plane wave, modulated by a Gaussian envelope.A detailed description about the technique can be found in Torrence and Compo (1998).
The wavelet is written as: where A c = [1 + exp(-ω 2 ) -2exp(-3ω 2 /4)] 1/2 and k c = exp(-ω 2 /2) and ω is the angular frequency (in Hz or s -1 ).Pyrak-Nolte and Nolte (1995) reported ω = 2π/T 0 where T 0 is the (characteristic) period of oscillations.In order to avoid some problems caused by small ω, ω > 5 is usually taken into account.Farge (1992) took ω to be 6 to satisfy the admissibility condition.As ω = 6, T 0 = ~1 s.As ω >> 1, Eq. ( 9) can be re-written as The continuous wavelet transform of a discrete sequence x i is defined as the convolution of x i with a scaled and translated version of ( ) t } : where the (*) indicates the complex conjugate, δt is the time shift, and s is the wavelet scale.By varying the wavelet scale s and translating along the localized time index n, a picture can be constructed to show both the amplitude of any features versus the scale and how this amplitude varies with time.Although it is possible to calculate the wavelet transform using Eq. ( 11), it is considerably faster to do the calculations in Fourier space.
Because the wavelet function ( ) t } is usually complex, the wavelet transform W n (s) is also complex.The transform can then be divided into the real part R[W n (s)] and the imaginary part I[W n (s)].Hence, the amplitude and phase are, In the continuous limit, the Fourier transform of a function ( / ) t s } is given by ( ) F s } 6 @.By the convolution theorem, the wavelet transform is the inverse Fourier transform of the product: To ensure that the wavelet transforms at each scale s are directly comparable to each other and to the transforms of other time series, the wavelet function at each scale s is normalized to have unit energy: where

#
, that is, the function has been normalized to have unit energy.
Using the normalization in Eq. ( 14), and referring to Eq. ( 13), the expectation value for |W n (s)| 2 is equal to N times the expectation value for k 2 | .For a white-noise time series, this expectation value is σ 2 /N, where σ 2 is the variance.Thus, for a white-noise process, the expectation value for the wavelet transform is ( ) W s n 2 = σ 2 at all n and s.It is necessary to explain the significance levels of the calculated values.The null hypothesis is defined as follows: It is assumed that the time series has a mean power spectrum; when a peak in the wavelet power spectrum is significantly above this background spectrum, then it can be considered to be a true feature with a certain percentage of confidence.The 95% confidence level implies a test against a certain background level, while the 95% confidence interval refers to the range of confidence about a given value.The normalized Fourier power spectrum is given by 2 N v , where N is the number of data points, k | is from Eq. ( 12), and σ 2 is the variance of the time series.
If x n is a normally distributed random variable, both the real and imaginary parts of k | are normally distributed (Chatfield 1989).Since the square of a normally distributed variable is chi-square distributed with one degree of freedom (DOF), then k 2 | is chi-square distributed with two DOFs, denoted by 2 2 | (Jenkins and Watts 1968).To determine the 95% confidence level (significant at 5%), one multiplies the background spectrum by the 95th percentile value for 2 2 | (Gilman et al. 1963).The 95% Fourier confidence spectrum will be displayed by a dashed curve below.Note that only at few periods, the power will be above the 95% line.The dominant period is significant when its peak value is higher than the related 95% confidence level.The period associated with such a peak is taken to be the dominant period.
In order to meet the equal time interval requirement, we consider the number of events occurring in a year to represent the degree of earthquake occurrences.Hence, the dominant periods (with a unit of year) of time sequence for the number of yearly events are evaluated in the following.

Multifractal dimensions of epicentral distributions
The generalized correlation integral functions, C q , versus the epicentral distance, r, in kilometers between two earthquakes at q = 2, ..., 15 are calculated for three data sets with M ≥ 6, M ≥ 6.5, and M ≥ 7 events in the study region.Since a few pairs of events have very similar epicenters, C q (r) at q = 0 and 1 cannot be evaluated.For simplification, only the log-log plots of C q (r) versus r at q = 2 (triangle), 6 (square), 10 (circle), and 14 (star) are plotted in Fig. 4a for M ≥ 6 events, in Fig. 4b for M ≥ 6.5 events, and in Fig. 4c for M ≥ 7 events.It can be seen from Fig. 4 that when r l < r < r u , the data points are well distributed around a linear trend; while when r < r l or r > r u , the data point pattern bends and departs from the linear trend.Bending or departure of the pattern indicates that the logC q values for r < r l or r > r u are different from those estimated from the linear regression equation deduced from the data points with r l < r < r u .The r 1 and r 2 values are, respectively, 120 km [log(r l ) = 2.08] and 560 km [log(r u ) = 2.8] for M ≥ 6 events, 100 km [log(r l ) = 2.0] and 560 km [log(r u ) = 2.8] for M ≥ 6.5 events, 65 km [log(r l ) = 1.8] and 560 km [log(r u ) = 2.8] for M ≥ 7 events.The D q value is evaluated from the data points in the linear range.The least-squared method is applied to infer a linear regression equation for the D q estimation from the linear portion.The D q values are between 1.618 and 1.426, with standard deviations from 0.001 -0.003, for M ≥ 6 events, between 1.562 and 1.108, with standard deviations from 0.001 -0.003, for M ≥ 6.5 events, and between 1.365 and 0.841, with standard deviations from 0.001 -0.003, for M ≥ 7 events.The D q -q relation is displayed in Fig. 5: solid circles for M ≥ 6 events, solid squares for M ≥ 6.5 events, and solid triangles for M ≥ 7 events.The D q error for each q is less than 0.003 for M ≥ 6, M ≥ 6.5, and M ≥ 7 events.

Multifractal dimensions of time Sequences
The generalized correlation integral functions, C q , versus inter-event time, t, in years between two earthquakes at q = 2, ..., 15 are calculated for two data sets with M ≥ 6 and M ≥ 7 earthquakes in the study region.Since a few pairs of events have very similar occurrence times, the C q (r) values at q = 0 and 1 cannot be evaluated.For simplification, only the log-log plots of C q (t) versus t at q = 2 (triangle), 6 (square), 10 (circle), and 14 (star) are plotted in Fig. 6a for M ≥ 6 events, in Fig. 6b for M ≥ 6.5 events, and in Fig. 6c for M ≥ 7 events.It can be seen from Fig. 6a that when t l < t < t u , where t l = 5 years [log(t l ) = 0.7] and t u = 50.1 years [log(t u ) = 1.8], the data points are well distributed around a linear trend; while when t < t l or t > t u , the data point pattern bends and departs from the linear trend.The bending pattern indicates that the logC q value for t > t u is less than that estimated from the linear regression equation deduced from the data points with t l < t < t u .From Fig. 6b for M ≥ 6.5 events, we can see that when t l < t < t u , where t l and t u are, respectively, 5 years [log(t l ) = 0.7] and 50.1 years [log(t u ) = 1.8], the data points are well distributed around a linear trend; while when t < t l or t > t u , the pattern of data points bends and departs from the linear trend.From Fig. 6c for M ≥ 7 events, we can see that when t l < t < t u , where t l and t u are, respectively, 16 years [log(t l ) = 1.2] and 63 years [log(t u ) = 1.8], the data points are well distributed around a linear trend.When t < t l or t > t u , the data point pattern bends and departs from the linear trend.
The least-squared method is applied to infer a linear regression equation for the D q estimation from the linear portion.The D q values are between 0.830 and 0.703, with a standard deviation of 0.001.For M ≥ 6 events between 0.835 and 0.820, with standard deviation of 0.001, for M ≥ 6.5 events, and between 0.786 and 0.685, with standard deviations from 0.002 -0.004, for M ≥ 7 events.The D q -q relation is displayed in Fig. 7: solid circles for M ≥ 6 events, solid squares for M ≥ 6.5 events, and solid triangles for M ≥ 7 events.The D q error for each q is less than 0.001 for M ≥ 6 and M ≥ 6.5 events and less than 0.004 for M ≥ 7 events.

Morlet Wavelet Analysis for earthquake Sequences
The Morlet wavelet analysis results are displayed in Figs. 8 -10: Fig. 8 for M ≥ 6 events, Fig. 9 for M ≥ 6.5 events, and Fig. 10 for M ≥ 7 events.Each figure consists of three panels: (a) for the time sequence of the number of yearly events; (b) for the wavelet power spectrum; and (c) for the global wavelet spectrum.In panel (b) the logarithmic wavelet power spectrum values for different periods (in year) at a certain time span are displayed using distinct colors (from dark red to dark blue).The thick contour is the 95% confidence level using a white-noise background spectrum.The black net region is the cone of influence, where zero padding has reduced the variance.The wavelet power spectrum values inside the net have high uncertainties and thus cannot be taken into account.In each panel (b), the local maximums and local minimums are colored, respectively, by dark red and dark blue.The period related to the local maximum is the local dominant period in a time span.In order to examine the dominant local maximum and related dominant period it is necessary to calculate the average wavelet power spectra from panel (b) over time at a certain period.This average is named the global wavelet spectrum.The results are demonstrated in panel (c), where the solid line represents the global wavelet spectrum and the dashed line denotes the 95% confidence level using a white-noise background spectrum.
Figure 8a shows the time sequence for the yearly number of events for M ≥ 6 earthquakes.Figure 8b shows the wavelet transform power spectrum for the yearly events.The local maximums at several periods in different time spans can be seen.Figure 8c shows that the global wavelet spectrum value obviously increases at the large period around 32 years.Meanwhile, except at a period of 2.92 years the solid line is to the left of the 95% confidence level dashed line.
Figure 9a shows the time sequence of the number of yearly events for M ≥ 6.5 earthquakes.Essentially, there was only one spike in 1975.The wavelet power spectra of the time sequence is quite abnormal, as shown in Fig. 9b.  .The plot of D q versus q from Fig. 4: solid circles for M ≥ 6 events, solid squares for M ≥ 6.5 events, solid triangles for M ≥ 7 events, and open circles for the data from Wang and Shen (1999).
(a) (b) (c) Fig. 6.The log-log plots of C q (t) versus t at q = 2 (triangle), 6 (square), 10 (circle), and 14 (star) for the time sequences in the study region: (a) for M ≥ 6 events; (b) for M ≥ 6.5; and (c) for M ≥ 7 events.The solid lines represent the regression lines inferred from the data points with log(t 1 ) ≤ log(t) ≤ log(t 2 ), where t 1 and t 2 are explained in the text.Fig. 7.The plot of D q versus q from Fig. 6: solid circles for M ≥ 6 events, solid squares for M ≥ 6.5 events, solid triangles for M ≥ 7 events, and open circles for the data from Wang and Lee (1995).In (b) the thick contour is the 95% confidence level, using a white-noise background spectrum.The black net is described in the text.The dashed line is the 95% confidence level for the global wavelet spectrum, using a white-noise background spectrum.
Although there are several peaks in Fig. 9c, their spectral values are lower than the 95% confidence level and the solid line is to the left of the dashed line.Hence, there is no dominant period for the M ≥ 6.5 earthquake sequence.
Figure 10a shows the time sequence of the number of yearly events for M ≥ 7 earthquakes.Essentially, there was only one spike in 1975.The wavelet power spectra of the time sequence is quite abnormal, as shown in Fig. 10b.Although there are several peaks in Fig. 10c, their spectral values are lower than the 95% confidence level and the solid line is to the left of the dashed line.Hence, there is no dominant period for the M ≥ 7 earthquake sequence.

dISCuSSIoN
From Fig. 2 we can see that the epicentral spatial distribution is not uniform for both M ≥ 6 and M ≥ 7 events in the whole area.It seems that the number of events per unit area is higher in the southern area than in the northern.Figure 3 shows that the temporal variation in events is not uniform in the whole time period.The frequency of events is relatively low before 1920 and after 1986.
Figure 4 reveals that the log-log plots of C q versus r show a linear portion when r l < r < r u .The pattern bends when log(r) < log(r l ) and bends as well as becomes flat when log(r) > log(r u ).The value of r 1 = 120 km for M ≥ 6 events is longer than r 1 = 100 km for M ≥ 6.5 events and r l = 64 km for M ≥ 7 events.The value of r 2 = 560 km is almost the same for M ≥ 6, M ≥ 6.5, and M ≥ 7 events.This means that the range of epicentral distances exhibiting multifractal properties is narrower for M ≥ 6 events than for M ≥ 6.5 and M ≥ 7 events.
Clearly, the number of data points with r < r 1 is larger than that with r > r 1 for M ≥ 6 events.This might be because some M ≥ 6 events, especially for M < 6.5 events, are aftershocks of M ≥ 7 earthquakes.Meanwhile, the data points with log(r) < log(r l ) for M ≥ 6 events are more dispersive than those for M ≥ 6.5 and M ≥ 7 events.The D q values from q = 2 to q = 15 are between 1.618 and 1.426 for M ≥ 6 events, between 1.562 and 1.108 for M ≥ 6.5 events, and between 1.365 and 0.841 for M ≥ 7 events.Clearly, the D q values are smaller for M ≥ 7 events than for M ≥ 6 events.This means the epicentral distribution is sparser for M ≥ 7 events than for M ≥ 6.5 and M ≥ 6 events.This is reasonable, because the density of events is higher for M ≥ 6 and M ≥ 6.5 events than for M ≥ 7 events as displayed in Fig. 2.
Figure 5 shows that the D q -q relations of epicenter for In (b) the thick contour is the 95% confidence level, using a white-noise background spectrum.The black net is described in the text.The dashed line is the 95% confidence level for the global wavelet spectrum, using a white-noise background spectrum.
the three magnitude ranges.Obviously, D q decreases with increasing q.The difference in the D q values for the three magnitude ranges increase slightly with q.From the epicentral distributions and numbers of events, the spatial distribution of M ≥ 6 events is denser than those of M ≥ 6.5 and M ≥ 7 events.Hence, based on fractal geometry D q at positive q should be larger for M ≥ 6 events than those of M ≥ 6.5 and M ≥ 7 events.The D q values for the three magnitude ranges are smaller than the topological dimension d = 2 of two-dimensional space.This indicates that the spatial distributions of those earthquakes are sparse.For the purpose of comparison, the D q -q relation for M ≥ 6 events occurred from 1900 to 1990 in the study region obtained by Wang and Shen (1999) is also plotted in Fig. 5 using stars.Their data are the same as those from 1900 -1990 used in this study.Their D q -q relation is close to that in this study; while their D q values are slightly smaller than those in this study.
The addition of events after 1990 only slightly increases the D q values, in other words, only slight increases the density of spatial distribution of earthquakes.Figures 6 displays that the log-log plot of C q (t) versus t show a linear distribution when log(t l ) ≤ log(t) ≤ log(t u ), where t l and t u are, respectively, 5 and 50.1 years for M ≥ 6 events, 5 and 50.1 years for M ≥ 6.5 events, and 16 years and 63.1 years for M ≥ 7 event, thus suggesting that the earthquake sequences in the study region are multifractal.For M ≥ 6 events, the pattern bends and becomes flat when log(t) > log(t u ); while for M ≥ 6.5 and M ≥ 7 event, the pattern bends when log(t) < log(t l ) and bends as well as becomes flat when log(t) > log(t u ).The D q values from q = 2 to q = 15 are between 0.830 and 0.703 for M ≥ 6 events, between 0.850 and 0.820 for M ≥ 6.5 events, and between 0.786 and 0.685 for M ≥ 7 events.This means that the range of inter-event times for the existence of multifractal properties is wider for M ≥ 6.5 events than for both M ≥ 6 and M ≥ 7 events.Although the D q value for M ≥ 6, M ≥ 6.5, and M ≥ 7 events are smaller than the topological dimension d = 1 of one-dimensional time series, they do not depart from d = 1 too much.This indicates that within the time period in consideration, the earthquake sequences are only slightly sparse.
Figure 7 shows that the D q -q relations of inter-event times for the three magnitude ranges.Obviously, D q decreases with increasing q.Unlike the epicentral distributions, the D q values are larger for M ≥ 6.5 events than both M ≥ 6 and M ≥ 7 events.Comparing Figs.7a with b exhibits that the log(t) ranges for the D q estimates are almost the same for both M ≥ 6 and M ≥ 6.5 events and the log[C(t)] values at small t are higher for the former than for the latter.This leads to larger D q for M ≥ 6 events than for M ≥ 6.5 events.On the other hand, D q is evaluated from a narrower range of log(t) for M ≥ 7 events.The difference in the D q values between M ≥ 6.5 events and M ≥ 6 events as well as between M ≥ 6.5 events and M ≥ 7 events increases with q.The difference in the D q values between M ≥ 6 events and M ≥ 7 events slightly decreases with increasing q.Although the temporal variations and numbers of events for M ≥ 6 and M ≥ 7 are different and the temporal variation of M ≥ 6 events is denser than that of M ≥ 7 events, the difference in D q between the two magnitude ranges is small.For comparison, the D q -q relation for M ≥ 6 events occurred from 1900 -1990 in the study region obtained by Wang and Lee (1995) are also plotted in Fig. 7 using the solid square.Their data are the same as those from 1900 to 1990 used in this study.Their D q -q relation is nearly parallel to that of this study; while their D q values are smaller than those in this study.The addition of events after 1990 only slightly increases the D q values.
For the New Hebrides seismicity between mid-1978mid- and mid-1984mid- , Smalley et al. (1987) ) found that the fractal dimension varies from 0.126 -0.255 and the earthquake occurrences deviate significantly from random or Poisson behavior.Kagan and Jackson (1991) stated that for 1-D processes, if the correlation dimension equals 1 over all time periods from zero to infinity, the process is Poissonian.For global seismicity, they found that long-term, weak clustering characterizes all mainshocks and is governed by a power-law temporal distribution.They also mentioned that the fractal dimension of the set of earthquakes on the time axis is of the order of 0.8 -0.9, thus, mainshock occurrence is closer to a stationary Poisson process.Gardner and Knopoff (1974) stated that the M ≥ 2.8 mainshock time series in Southern California, USA, during 1932 -1971 is Poissonian.The fractal dimension used by these authors is the correlation dimension and equivalent to D 2 in this study.Their results suggest that the time sequence for earthquakes with a higher D 2 has a stronger Poissonian process component.The D 2 value is 0.830 for M ≥ 6 events, 0.835 for M ≥ 6.5 events, and 0.785 for M ≥ 7 events.Hence, in comparison with the global seismicity, the Poissonian process component in the earthquake sequences of the seismic belt in this study is strong.For M ≥ 6 earthquakes occurring in the 1900 -1990 period in the study region, Wang and Kuo (1995) found that both the exponential function and gamma function can describe the distributions and the former is more appropriate than the latter.This indicates that the earthquake sequence in the study region has a significant Poissonian process component.Wang (2013) also found that the earthquake sequence in the study region shows a weak memory effect and thus has a strong Poissonian process component.Moreover, the M ≥ 7 earthquake sequence is more Poissonian than the M ≥ 6 and M ≥ 6.5 ones.This is also consistent with the conclusion made by Wang (2013).
Figure 8a shows the time sequence for the number of yearly events for M ≥ 6 earthquakes.There were several large spikes in the time period before 1990.This means that seismicity was higher before 1990 than after 1990.Figure 8c shows that for the M ≥ 6 earthquake sequence, peaks exist at several periods in the solid line.However, only the global wavelet spectral value at 2.92 years is slightly to the right of the dashed line representing the 95% confidence level.Hence, there is only one dominant period of 2.92 years for the M ≥ 6 earthquake sequence.This dominant period is about three times longer than the average inter-event time of 0.96 years.
Figure 9a shows the time sequence for the number of yearly events for M ≥ 6.5 earthquakes.There was only one spike in 1975.Figure 9c shows that for M ≥ 6.5 earthquakes there are peaks in the solid line at several periods.However, their global wavelet spectral values are all to the left of the dashed line representing the 95% confidence level.Hence, there is no dominant period for the M ≥ 6.5 earthquake sequence.As mentioned previously, the number of M ≥ 6.5 earthquakes is relatively small and thus their wavelet power spectrum (Fig. 9b) and the global wavelet spectrum (Fig. 9c) are not normal.Hence, M ≥ 6.5 earthquakes in the study region are not dominant.
Figure 10a shows the time sequence for the number of yearly events for M ≥ 7 earthquakes.There was only one spike in 1975.Figure 10c shows that for M ≥ 7 earthquakes there are peaks in the solid line at several periods.However, their global wavelet spectral values are all to the left of the dashed line representing the 95% confidence level.Hence, there is no dominant period for the M ≥ 7 earthquake sequence.As mentioned previously, the number of M ≥ 7 earthquakes is relatively small and thus their wavelet power spectrum (Fig. 10b) and the global wavelet spectrum (Fig. 10c) are not normal.Hence, M ≥ 7 earthquakes in the study region are not dominant.

CoNCluSIoNS
The generalized fractal dimensions are measured for M ≥ 6 earthquakes in the South-North Seismic Belt, Mainland China (longitudes from 98 -107°E and latitudes from 21 -41°N) during 1900 -2015 based on the spatial distribution (using the epicentral distances between two events, r) and the earthquake sequence (using the inter-event time between two events, t).The epicentral distance between two events ranges from 0 -2141.0km for M ≥ 6 events, from 0 -2141.0km for M ≥ 6.5 events, and from 0 -1934.0km for M ≥ 7 events.Multifractal measures are made from the log-log plot of C q (r) versus r and that of C q (t) versus t.The plots show a linear distribution of data points when log(r l ) ≤ log(r) ≤ log(r u ) and roll-over when log(r) > log(r u ).The r l and r u values are, respectively, 120 and 560 km for M ≥ 6 events, 100 and 560 km for M ≥ 6.5 events, and 65 and 560 km for M ≥ 7 events.The D q value, which is the slope of the linear portion, is between1.618and 1.426 for M ≥ 6 events, between 1.562 and 1.108 for M ≥ 6.5 events, and between 1.365 and 0.841 for M ≥ 7 events.For the three magnitude ranges, D q monotonically decreases with increasing q, thus indicating that the epicentral distributions show multifractal behavior.The D q values are smaller than the topological di-mension d = 2 for the two-dimensional surface.
The log-log plot of C q (t) versus t show a linear distribution when log(t l ) ≤ log(t) ≤ log(t u ), where t l and t u are, respectively, 5 and 50.1 years for M ≥ 6 events, 5 and 50.1 years for M ≥ 6 events, 16 and 63.1 years for M ≥ 7 event, thus suggesting that the earthquake time sequences in the study region are multifractal.The D q values are between 0.830 and 0.703 for M ≥ 6 events, between 0.835 and 0.820 for M ≥ 6.5 events, and between 0.786 and 0.685 for M ≥ 7 events.For the three magnitude ranges, D q monotonically decreases with increasing q, thus indicating that the earthquake sequences show multifractal behavior.The D q values are all smaller than the topological dimension d = 1 of the one-dimensional time series.The M ≥ 7 earthquake sequence is more Poissonian than the M ≥ 6 and M ≥ 6.5 ones.
The Morlet wavelet technique is applied to analyze the dominant temporal variation periods in numbers of yearly events for the three magnitude ranges, i.e., M ≥ 6, M ≥ 6.5, and M ≥ 7.In the time domain the average inter-event time is 0.96 years for M ≥ 6 events, 1.93 years for M ≥ 6.5 events, and 4.15 for M ≥ 7 events.The resultant dominant period is 2.92 years, which is about three times of 0.96 years, for M ≥ 6 events and cannot be evaluated for M ≥ 6.5 and M ≥ 7 events.
Several alternatives have been suggested to estimate the fractal dimension (cf.Takayasu 1990; Korvin 1992; Chen and Koyama 1995).Similarity dimension D S is defined for an exact self-similar set as D S = logL/logN, where L is the linear size and N is the number of similar daughters.Capacity dimension D CA is defined as D CA = logN(r)/log(1/r), where N(r) is the smallest number of coverings of a set with a size of r.Information dimension D I is defined as D I = Σp i (r)log[p i (r)]/ log(r), where r is the distance between two points, based on the probabilistic distribution.Correlation dimension D C is defined from the correlation integral C(r) in the following relation: C(r) ~ r Dc-d , where d = 2 is the spatial topological dimension D S = D CA ≥ D I ≥ D C (Takayasu 1990), and D C is the smallest value of the four fractal dimensions.The equality D S = D CA = D I = D C holds only for a homogeneous fractal set.Most natural fractals are not completely self-similar and actually multifractal.For such fractals, we have D S = D CA > D I > D C .
respectively, |W n (s)| and θ = tan -1 {R[W n (s)]/I[W n (s)]}.Finally, one can define the wavelet power spectrum as |W n (s)| 2 .Let the discrete Fourier transform (DFT) of x i be k

Fig. 4 .
Fig.4.The log-log plots of C q (r) versus r at q = 2 (triangle), 6 (square), 10 (circle), and 14 (star) for the epicentral distributions in the study region: (a) for M ≥ 6 events; (b) for M ≥ 6.5; and (c) for M ≥ 7 events.The solid lines represent the regression lines inferred from the data points with log(r 1 ) ≤ log(r) ≤ log(r 2 ), where r 1 and r 2 are explained in the text.

Fig. 5
Fig.5.The plot of D q versus q from Fig.4: solid circles for M ≥ 6 events, solid squares for M ≥ 6.5 events, solid triangles for M ≥ 7 events, and open circles for the data fromWang and Shen (1999).

Fig. 8 .Fig. 9 .
Fig. 8.The figure shows (a) the time sequence for the number of yearly events with M ≥ 6;(b) the wavelet power spectrum; and (c) the average or the global wavelet spectrum of (b) over all longitudes.In (b) the thick contour is the 95% confidence level, using a white-noise background spectrum.The black net is described in the text.The dashed line is the 95% confidence level for the global wavelet spectrum, using a white-noise background spectrum.

Fig. 10 .
Fig. 10.The figure shows (a) the time sequence for the number of yearly events with M ≥ 7; (b) the wavelet power spectrum; and (c) the average or the global wavelet spectrum of (b) over all longitudes.In (b) the thick contour is the 95% confidence level, using a white-noise background spectrum.The black net is described in the text.The dashed line is the 95% confidence level for the global wavelet spectrum, using a white-noise background spectrum.