Review Article Characteristic Changes of Scale …downloads.hindawi.com/journals/ijge/2013/391637.pdfReview Article Characteristic Changes of Scale Invariance of Seismicity Prior to

Hindawi Publishing CorporationInternational Journal of GeophysicsVolume 2013 Article ID 391637 11 pageshttpdxdoiorg1011552013391637

Review ArticleCharacteristic Changes of Scale Invariance of Seismicity Prior toLarge Earthquakes A Constructive Review

Qiang Li and Gui-Ming Xu

Research Center for Earthquake Prediction Earthquake Administration of Jiangsu Province No 3 Wei Gang Nanjing 210014 China

Correspondence should be addressed to Qiang Li lqdzjybzx126com

Received 7 February 2013 Revised 2 May 2013 Accepted 10 June 2013

Academic Editor Filippos Vallianatos

Copyright copy 2013 Q Li and G-M XuThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Recently research on the characteristic changes of scale invariance of seismicity before large earthquakes has received considerableattention However in some circumstances it is not easy to obtain these characteristic changes because the features of seismicity indifferent regions are various In this paper we firstly introduced some important research developments of the characteristic changesof scale invariance of seismicity before large earthquakes which are of particular importance to the researchers in earthquakeforecasting and seismic activity We secondly discussed the strengths and weaknesses of different scale invariance methods such asthe local scaling property themultifractal spectrum theHurst exponent analysis and the correlation dimensionWefinally cameupwith a constructive suggestion for the research strategy in this topic Our suggestion is that when people try to obtain the precursoryinformation before large earthquakes or to study the fractal property of seismicity by means of the previous scale invariance meth-ods the strengths and weaknesses of these methods have to be taken into consideration for the purpose of increasing research effi-ciency If they do not consider the strengths andweaknesses of thesemethods the efficiency of their researchmight greatly decrease

1 Introduction

It is a well-known fact that the natural seismic system andthe rock fracture system in laboratory have the properties ofscale invariance [1ndash13] Study on the characteristic changes ofscale invariance of seismicity before large ruptures has beenan intriguing subject to geophysicists recently So far greatprogresses have beenmade in this topic For instance analysisresults of the temporal and spatial multifractal characteristicof seismicity indicate that there are anomalous changes of thesingularity spectrum and generalized dimension spectrumbefore some large earthquakes [14ndash20] study results of theearthquakes [21 22] rock mechanics experiments [23 24]and rock burst [25] indicate that the there are anomalousvariations in the dimension of fractal objects prior to themajor ruptures research by Li andXu [26] indicates that thereis the possible correlation between the featuring change of thelocal scaling property and the process of seismogeny a studyby Zhao andWang [27] shows that theHurst exponent for thesequence of the interval time between earthquakes decreasesprior to some large inland earthquakes In addition somestudy results proposed that the decrease of fractal dimension

and Hurst exponent as well as the characteristic change ofgeneralized dimension spectrum is deemed as precursoryphenomena before main avalanches or ruptures [16 28ndash35]

However different scale invariance methods have theirown strengths and weaknesses Some methods can only givethe description of themonofractal property of seismicity [36]while others can sufficiently give the description of heteroge-neous properties of fractal seismic system [37 38] Besidesthe research results of some scale invariance methods (suchas the multifractal spectrum the correlation dimension thefractal dimension andHurst exponent analysis of seismicity)are relatively mature because there have been many studiedcases which show characteristic changes of scale invarianceof seismicity prior to large earthquakes while the researchresults of local scaling property are only tentative [26 39ndash41] because the observed cases which show characteristicchanges of local scaling property of seismicity prior to largeearthquakes have been quite few so far

In this paper we introduce some important researchdevelopments of characteristic changes of scale invariance ofseismicity before large earthquakes and discuss the strengthsand weaknesses of different methods of scale invariance

2 International Journal of Geophysics

for the purpose of improving the study validity for theresearchers in earthquake forecasting and seismic activity

2 Multifractal Spectrum of Seismicity

The multifractal spectrum includes the generalized dimen-sion spectrum and singularity spectrum Because the singu-larity spectrum is closely related to the generalized dimensionspectrum we mainly focus on generalized dimension spec-trum (the curve graph of multifractal dimension119863119902 versus 119902is called the generalized dimension spectrum) in this paper

The generalized dimension spectrum of seismicity isstudied by an algorithm based on the correlative integral[38] Computation ofmultifractal dimension119863119902 applying thecorrelative integral algorithm [18] is made using

119863119902 = lim119903rarr0

log119862119902 (119903)log 119903

(1)

where119862119902(119903) is the 119902th-order generalized correlation functiondefined as

119862119902 (119903) =

1

119873

119873

sum

119895=1

[[

[

1

119873

119873

sum

119894=1119894 = 119895

Θ(119903 minus10038161003816100381610038161003816119883119894 minus 119883119895

10038161003816100381610038161003816)]]

]

119902minus1

1(119902minus1)

(2)

with119873 as the number of data points 119903 as the scaling radiusΘ(119904) as the Heaviside step function and |119883119894 minus 119883119895| as thedistance between the two data points 119883119894 and 119883119895 If the serieshas the property of fractals the 119862119902(119903) is indicated by a powerlaw Thus the value of 119863119902 can be settled by calculating thelinear segment slope in the graph of log 119862119902(119903) versus log 119903

Roy and Padhi [19] studied the generalized dimensionspectrum of seismicity in Iran and adjacent regions Theyfound that the precursory clustering pattern in the short timeperiod before three large earthquakes (119872119908 = 78 on 1691978119872119908 = 68 on 26122003119872119908 = 77 on 10597) is perceived byanalyzing the multifractal property of seismicity in this areaThey believe that the probability for future large earthquakescan be evaluated by studying the spatial and temporal cluster-ing pattern of earthquakes They conclude that the clusteringpattern study based on seismic catalogue for majority of theknown seismic fault systems may be finally helpful to theprevention and mitigation of earthquake hazards

Caruso et al [14] made a study on the multifractal pro-perty of Mount St Helens seismicity during the period of1980ndash2002 in which the temporal distribution of seismicityrelated to the eruptive activity primarily characterized bythe major explosive eruptions in 1980 and the eruptionsduring the period of 1980ndash1986 is analyzed They found thatthe generalized dimension spectrum which is calculatedfrom the data of seismicity can help us to recognize twomain temporal distribution patterns of seismicity The firstpattern shows a multifractal clustering related to the strongearthquake swarm of the dome building activity The secondpattern is typified by very small variation value of 119863119902 closeto 1 as for a stochastic uniform distribution The temporalchange of 119863119902 calculated from an invariant window length of

seismic events and at different depths shows that the fragilemechanical response of the shallow zones to swift magmaticintrusions through the whole course of the eruption isindicated by its rapid variations in a short period and thesmallest values of119863119902 (about 03)

Dimitriu et al [15] studied the generalized dimensionspectrum in a seismically active area of Northern Greecewhere several large earthquakes containing the May 1995Arena sequence with a maximum119872119908 53 earthquake onMay4 1995 occurred They found that multifractality enhancesbefore the major seismicity and that the clusterization isreplaced by the declusterization not long before its initializa-tion They believe that this might help us to approximatelyevaluate the occurrence time of the main slip if the multi-fractality getting to its peak before the percolation makes theprevious gradual increase in multifractality

Teotia and Kumar [20] studied the generalized dimen-sion spectrum by analyzing the seismic catalog data in theNorth-Western Himalaya area which primarily contain theseismogenic zone of 1905 Kangra great earthquake Theseismic catalogue data come from USGS catalogue duringthe period of 1973ndash2009 which contains the Muzaffarabad-Kashmir earthquake (119872119908 = 76) ofOctober 2005They foundthe temporal variations in generalized dimension spectrumbefore the Muzaffarabad-Kashmir earthquake in the areaThey believe that this study may be of important value inrecognizing the seismogenic zone of large seismic events indissimilar tectonic regions

Li [17] studied the variation of multifractal characteristicbefore and after four mid-strong earthquakes (magnitudesrange from 55 to 62) in Jiangsu province and its adjacentarea in the east of China by analyzing seismicity datasince 1970 in this area where the frequency and strength ofseismicity belong to the middle level and the completenessmagnitude of the catalog estimated by using Gutenberg-Richter formula is 24 I found that the generalized dimen-sion spectrum and singularity spectrum 119891(120572) of seismicityhave three variation stages which separately corresponded tothe time far before mid-strong earthquakes the time whenthe anomalous change of the spectrum appeared and thetime when the anomalous change of the spectrum disap-peared Figure 1 shows the three variation stages of thegeneralized dimension spectrum and singularity spectrumbefore 22 April 1974 Liyang M55 earthquake (the data setincludes 311 earthquakes) Among the figures Figures 1(a)and 1(d) show the first variation stage of the generalizeddimension spectrum and singularity spectrum (two yearsprior to this earthquake) respectively in which the level dif-ference between the left end and right ends of the generalizeddimension spectrum is small (smaller than 044) Figures1(b) and 1(e) show the anomalous change stage (ie thesecond variation stage) of generalized dimension spectrumand singularity spectrum (seven months prior to this earth-quake) respectively in which the curve of the generalizeddimension spectrum is steeper than the two others (thelevel difference between the left end and right ends ofthe generalized dimension spectrum is larger than 058)and the value range of the singularity spectrum is broaderthan the two others Figures 1(c) and 1(f) show the third

International Journal of Geophysics 3

15

10

05

minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5

q

10

05

00

f(a)

a

00 10 20

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05


q

10

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00

f(a)

a

00 10 20

15

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Dq


q

10

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00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Dq

Dq

Figure 1The three variation stages of the generalized dimension spectrum and singularity spectrum before 22 April 1974 Liyang earthquake(M55) (a) and (d) show the generalized dimension spectrum and singularity spectrum two years prior to this earthquake respectively (b)and (e) show the generalized dimension spectrum and singularity spectrum seven months prior to this earthquake respectively (c) and(f) show the generalized dimension spectrum and singularity spectrum four months prior to this earthquake respectively The curve of thegeneralized dimension spectrum in Figure 1(b) is steeper than the two others and the value range of the singularity spectrum in Figure 1(e)is broader than the two others

variation stage of the generalized dimension spectrum andsingularity spectrum (four months prior to this earthquake)respectively in which the level difference between the leftend and right end of the generalized dimension spectrumbecomes small again Figure 2 shows the three variationstages of the generalized dimension spectrum and singularityspectrum before 9 July 1979 Liyang M60 earthquake (thedata set includes 827 earthquakes) The time in which theanomalous change appeared is fifteen months before thisearthquake (see anomalous change curves in Figures 2(b)and 2(e)) Figure 3 shows the three variation stages of thegeneralized dimension spectrum and singularity spectrumbefore and after 21 May 1984 Wunansha M62 earthquake(the data set includes 937 earthquakes) The time in whichthe anomalous change appeared is eight months before this

earthquake (see anomalous change curves in Figures 3(b)and 3(e)) Figure 4 shows the three variation stages of thegeneralized dimension spectrum and singularity spectrumbefore 9 November 1996 Yellow Sea M61 earthquake (thedata set includes 679 earthquakes) The time in which theanomalous change appeared is seventeen months before thisearthquake (see anomalous change curves in Figures 4(b)and 4(e)) According to the previously mentioned anoma-lous changes of the generalized dimension spectrum andsingularity spectrum in the studied cases I believe that it issignificant to apply the multifractal characteristics to mid-strong earthquake forecasting in this area

Kiyashchenko et al [16] studied the changes in theabscissa of the top of multifractal spectrum (ie the singu-larity spectrum) by carrying out the simulation of the crack


15

10

05

Dq

minus5 minus4 minus3 minus2 minus1 0 1 2 3 4 5q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 The three variation stages of the generalized dimension spectrum and singularity spectrum before 9 July 1979 Liyang earthquake(M60) (a) and (d) show the generalized dimension spectrum and singularity spectrum two years prior to this earthquake respectively (b)and (e) show the generalized dimension spectrum and singularity spectrum fifteen months prior to this earthquake respectively (c) and(f) show the generalized dimension spectrum and singularity spectrum eight months prior to this earthquake respectively The curve of thegeneralized dimension spectrum in Figure 2(b) is steeper than the two others and the value range of the singularity spectrum in Figure 2(e)is broader than the two others

network evolution in elastic body forced by outside forcesThey found that the changes in the abscissa of the top ofmultifractal spectrum contain some useful information ofthe development of the system towards main fracture Theirsimulation results were also upheld by the studies on someinstances of the seismicity in Japan and Southern California

The characteristic changes of multifractal spectrum ofseismicity have been studied for many years The observedcases in which there are characteristic changes of multifractalspectrum of seismicity prior to large earthquakes are notfew The advantage of the multifractal spectrum is that it cansufficiently give the description of heterogeneous propertiesof a fractal system However it is not good at presenting

the property of self-affine fractal and is incapable to describelocal features of the fractal seismic system because it can onlyprovide the description of global properties of fractal objects

3 Correlation Dimension andFractal Dimension

The calculation of correlation dimension 1198632 is the specialcircumstance of generalized dimension calculation If weallocate 119902 = 2 in (1) and (2) the correlation dimension 1198632can be obtained

The fractal dimension 119863 of seismicity can be computedby applying the correlation integral algorithm [38 42]


15

10

05

Dq


q

10

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f(a)

a

00 10 20

15

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Dq


q

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f(a)

a

00 10 20

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05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Figure 3The three variation stages of the generalized dimension spectrum and singularity spectrum before and after 21May 1984Wunanshaearthquake (M62) (a) and (d) show the generalized dimension spectrum and singularity spectrum two years prior to this earthquakerespectively (b) and (e) show the generalized dimension spectrum and singularity spectrum eight months prior to this earthquakerespectively (c) and (f) show the generalized dimension spectrum and singularity spectrum ten months after this earthquake respectivelyThe curve of the generalized dimension spectrum in Figure 3(b) is steeper than the two others and the value range of the singularity spectrumin Figure 3(e) is broader than the two others

The correlation integral 119862(119903) is expressed by the followingformula [21]

119862 (119903) =2119873 (119877 lt 119903)

119873 (119873 minus 1) (3)

Here119873 is the number of data points used for analysis119873(119877 lt119903) is the quantity of data points within a distance 119877 that is lessthan 119903 If the seismic distribution has the property of fractalsthe following relational expression can be got

119862 (119903) prop 119903119863 (4)

The 119863 in the relational expression is defined as the fractaldimension [38]

Lei and Satoh [24] studied the statistic characteristic ofprefailure harm on the basis of acoustic emission events

(AE) observed through the whole course of the great fractureof representative rock samples under dissimilar compres-sion They found that the prefailure harm evolution beforegreat fracture in several representative rocks is typified bya reduced correlation dimension They believed that thisprecursory change in correlation dimension may promoteshort-term prediction for the critical point behavior prior torock failure

Lu et al [25] studied the change in the fractal dimensionduring the process of a damage development especiallyduring the process of impending critical failure They foundthat the fractal dimension of the spatial distribution ofmicro-cracks reduces as the damage develops Their conclusion isthat an abrupt decrease in fractal dimension can be servedas an indicator of a possible precursor which presages animpending catastrophic rupture


15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

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Dq


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00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Figure 4 The three variation stages of the generalized dimension spectrum and singularity spectrum before 9 November 1996 Yellow Seaearthquake (M61) (a) and (d) show the generalized dimension spectrum and singularity spectrum two years prior to this earthquakerespectively (b) and (e) show the generalized dimension spectrum and singularity spectrum seventeen months prior to this earthquakerespectively (c) and (f) show the generalized dimension spectrum and singularity spectrum five months before this earthquake respectivelyThe curve of the generalized dimension spectrum in Figure 4(b) is steeper than the two others and the value range of the singularity spectrumin Figure 4(e) is broader than the two others

Roy andNath [22] studied the variation of the correlationdimension before and after the Great Sumatra earthquake of26 December 2004 (119872119908 = 9) by analyzing the seismic events(119898(119887) ge 4) occurring in the period of 1996ndash2005 in Sumatraregion They found that there is a decrease of correlationdimension in the short time period before this earthquakeSimilar precursory changes were also found ahead of theIzmith earthquake of 17 August 1999 (119872119904 = 78) and the3 November 2002 Dennali earthquake of 3 November 2002(119872119904 = 85)

Roy andPadhi [19] also studied the correlation dimensionof seismicity in Iran and neighbouring areasThey found thatthe precursory clustering pattern in the short time periodbefore three large earthquakes (119872119908 = 78 on 1691978119872119908 =68 on 26122003 119872119908 = 77 on 10597) can be recognizedby analyzing the change in the correlation dimension of

seismicity in this areaTheir conclusion is that the probabilityfor future large earthquakes can be evaluated by studyingcorrelation dimension of seismicity which is related to thetemporal and spatial clustering pattern of earthquakes

Enescu and Ito [43] studied the changes of the correlationdimension in a large region near the epicenter of the 1995Hyogo-kenNanbu (Kobe) earthquake and in the focal regionThey found that the anomalous changes of the correlationdimension turn up about two years prior to the occurrenceof the 1995 Hyogo-Ken Nanbu (Kobe) earthquake and theseanomalous changes turn up in a relatively broad area near theepicenter of the 1995 Hyogo-ken Nanbu (Kobe) earthquakeand in the focal region

Murase [21] studied the variations of the patternof hypocenter distribution before the 2003 Tokachi-okiearthquake (119872119869 = 80) by analyzing the temporal change


on the spatial fractal dimension He found that the valueof fractal dimension began to reduce in 1998 and had keptits low value for about a year prior to the occurrence of theprevious great earthquake He believes that the reductionof fractal dimension prior to the main shock is a typicalfeature of some large earthquakes and may be considered asa precursor of the large earthquakes His conclusion is that itis beneficial to capture precursory information of seismicityprior to a large earthquake by analyzing the temporal changeof fractal dimension

Kiyashchenko et al [16] analyzed the changes of fractaldimension in a study in which they carried out the simulationof the crack network evolution in elastomer acted on byoutside forces They found that the crack network developsfrom the condition in which the distributions of cracks andseismicity are random and chaotic to the state in whichthe patterns are fractal and clustered Not only that thestudy results of some instances of the seismicity in Japanand Southern California upheld the simulation results Theirconclusion is that the changes on fractal dimensions comprisesome useful information on the development of the systemtowards main fracture

Teotia and Kumar [20] studied the correlation dimen-sion by analyzing the seismic catalog data for the periodof 1973ndash2009 containing the Muzaffarabad-Kashmir earth-quake (119872119908 = 76) of October 2005 in the North-WesternHimalaya area They found the significant temporal vari-ations of correlation dimension before the occurrence ofMuzaffarabad-Kashmir earthquake in relation to epicenterdistribution in the area They concluded that the observedchange of the correlation dimension prior to the large earth-quake may help us to recognize the seismogenic zone of largeseismic events in different tectonic areas

So far there have been many studied cases ranging frompetrophysical experiments rock burst to large natural earth-quakes in which the correlation dimension or fractal dimen-sion decreased before the main raptures Therefore theresearch in this respect is relativelymature However becausethe method of correlation dimension and fractal dimensioncan only describe the monofractal property of seismicity itis inefficient to describe the heterogeneous property of theseismicityThus the useful information about heterogeneousproperty of fractal seismic system will be lost when we usethese methods

4 119877119878 Hurst Analysis

Themethod of119877119878Hurst analysis is introduced byHurst whodeveloped this method to aim at studying the flow of theNile River [44] Yet this method is also applied to analyzeobserved time series including the interevent time series ofthe seismic sequences [27] and the time series of earthquakefrequency [30 45]

The method of 119877119878Hurst analysis is as follows [46 47]Suppose that a typical time series is expressed by 120585(119905) (119905 =

1 2 ) For a positive integer 120591 the average value of 120585(119905) is

⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)

The accumulated deviation119883(119905 120591) is expressed as

119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)

Corresponding to the time 119905 that varies from 1 to 120591 the valuesof accumulated deviation can be obtained The differencebetween the maximal accumulated deviation and the mini-mal deviation is called as range 119877(120591) and it is expressed by

119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)

The range 119877(120591) value is the maximal fluctuation of the accu-mulated deviation in the time interval 1 le 119905 le 120591The standarddeviation 119878(120591) in the same time interval is defined as

119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)

For the dissimilar time length 120591 the dimensionless value 119877119878is represented by the following empirical relationship

119877 (120591)

119878 (120591)prop 120591119867 (9)

Equation (9) is called as the 119877119878 empirical relationalexpression which shows that the ratio of range to standarddeviation changes with the power of time length The expo-nent119867 is defined Hurst exponent that ranges from 0 to 1

Zhao and Wang [27] studied the temporal variation ofHurst exponent for the sequence of the interval time betweenearthquakes several years before and after some large earth-quakes (119872119904 ge 70) in ChinaMainland since 1970They foundthat the anomalous change of decrease of Hurst exponentappeared two years or so before the large earthquakes Theybelieve that this variation feature of anomaly of Hurst expo-nent can be considered as a medium-short-term earthquakeprecursor and the anomalous process of Hurst exponentreveals the property of seismicity from disorder to order ina large earthquake generating system

Guo [30] studied the temporal variation of Hurst expo-nent for the earthquake frequency before some mediumand strong earthquakes (magnitudes range from M50 to78) in north China He found that there is an anomalouscharacteristic that the value of Hurst exponent is less than087 one or two years before some earthquakes and theanomalous time of Hurst exponent ranges from threemonthsto two years He believes that anomalous characteristic ofHurst exponent for earthquake frequency can be served asa medium-short-term indicator for the medium and strongearthquake forecasting

Li and Wang [48] studied the temporal variation ofHurst exponent for the earthquake frequency before and aftertwelve strong earthquakes (magnitudes range from M57 to66) in Qinghai-Tibet Plateau in China They found that thevalue of Hurst exponent for seven earthquake cases decreasesbefore the occurrence of the strong earthquakes and thisdecrease continues more than three months He believes thatsuch anomalous change of Hurst exponent for earthquakefrequency is related to seismicity in Qinghai-Tibet Plateau


116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year

Figure 5The temporal variation of Hurst exponent before and afterthe 22 April 1974 Liyang earthquake (M55)

109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year

Figure 6The temporal variation of Hurst exponent before and afterthe 26 July 1994 Yellow Sea earthquake (M53)

Wang et al [49] studied the temporal change of Hurstexponent for the earthquake frequency before and afterfourteen earthquakes (119872119904 ge 72) in China Mainland Theyfound that the decrease changes ofHurst exponent formost ofearthquake cases (about 78 percent of the earthquake cases)appeared several months to one and half years before theoccurrence of the strong earthquakes He believes that suchdecrease change of Hurst exponent for earthquake frequencyis meaningful for the earthquake forecasting

Li and Xu [45] studied the temporal variation of Hurstexponent for earthquake frequency by analyzing the seis-micity data in Jiangsu and adjacent area where several mid-strong earthquakes (magnitudes ranging from M50 to 62)occurred We found that the anomalous changes in the Hurstexponent for some earthquake cases appear severalmonths toabout a little over a year before the mid-strong earthquakesFigure 5 shows the temporal variation of Hurst exponentbefore and after the 22 April 1974 Liyang earthquake (M55)As can be seen from Figure 5 the Hurst exponent fluctuateswithin small range and shows no feature change beforeMarch1973 However it begins to decrease in March 1973 AfterLiyang earthquake it increases back to the state of fluctuatingbefore March 1973 Figure 6 shows the temporal variationof Hurst exponent before and after the 26 July 1994 YellowSea earthquake (M53) From Figure 6 we see that the Hurstexponent fluctuates within small range before September1993 The Hurst exponent begins to decrease on September1993 After Yellow Sea earthquake it gradually returns to thelevel before September 1993 Figure 7 shows the temporalvariation of Hurst exponent before and after the 10 February1990 Changshu earthquake (M51) As can be seen fromFigure 7 the Hurst exponent fluctuates within small limitand shows no characteristic variation before March 1989

111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year

Figure 7The temporal variation of Hurst exponent before and afterthe 10 February 1990 Changshu earthquake (M51)

106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year

Figure 8The temporal variation of Hurst exponent before and afterthe 21 May 1984 Wunansha earthquake (M62)

The Hurst exponent begins to decrease in March 1989 AfterChangshu earthquake it increases gradually back to the stateof fluctuating beforeMarch 1989 Figure 8 shows the temporalvariation of Hurst exponent before and after the 21 May 1984Wunansha earthquake (M62) From Figure 8 we see that theHurst exponent fluctuates within small range before June1983 The Hurst exponent begins to decrease on June 1983After Wunansha earthquake it quickly returns to the levelbefore June 1983 Based on our study we believe that theseanomalous changes in the Hurst exponent can be of referencesignificance in earthquake forecasting in this area

Because theHurst exponent analysis is good at presentingthe property of self-affine fractal of seismicity it is efficientfor analyzing temporal characteristic variation of self-affinefractal of seismicity prior to large earthquakes However theHurst exponent analysis is incapable to describe the heteroge-neous property of the fractal objectsThus the heterogeneouscharacteristic of the seismicity will not be demonstrated if weuse this method

5 Local Scaling Property of Seismicity

Themethod of local scaling property is a typical method thatfocuses on the local property of fractal bodies It is as follows[50ndash52]

A particular characteristic of fractal bodies is that theyhave the feature of asymptotical self-similarity at small lengthscales Assume that a fractal is represented by a real function119891 Viewing near a discretionary point 1199090 of function 119891 at dif-ferent scales we can invariably observe the similar functionup to a scaling factor Letting

1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)

Here 120572(1199090) is defined as the local scaling exponent (alsocalled the singularity exponent) indicating the singularitystrength at point 1199090

Itmay be confirmed that thewavelet transform coefficientof 119891(119909) close to the point 1199090 shows the same property of scaleinvariance as well We treat

119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)

as the wavelet transform close to the point 1199090 and

119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)

as the form of the wavelet transform coefficient close to thepoint 1199090 when the scale varies This transformation canbe served as a mathematical microscope the position andmagnification of which are 119887 and 119897119886 respectively and theoptics of which are determined by the selection of the specificwavelet 120595 Then we obtain

119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)

Corresponding to the different positions 119887 where the fractalis asymptotically self-similar at small scales the plot ofln |119879(119886 119887)| versus ln 119886 shows an approximate straight linewhose slope 119870 can be obtained by using a least squares fitwith a check value of 119865 test Thus the local scaling exponent120572 at position 119887 can be got from the following expression

119870 = 120572 +1

2 (15)

Changing parameter 119887 the relationship between theposition and the local scaling exponent 120572 can be got

In the practical process of calculation it is necessaryto select suitable wavelet transform parameters which cor-respond to the seismicity features of different regions byusing trial-and-error method The plots of ln |119879(119886 119887)| versusln 119886 which we get by calculating the series of intereventtimes between successive earthquakes point by point showtwo dissimilar types one is the proximate straight line theother is the fluctuating line For the proximate straight linethe slope 119870 can be got by performing the computation ofleast square fit with the help of test value (if there are theoscillations which attach to the proximate straight line thefitting calculation should be performed after the oscillationsare deleted for the purpose of avoiding the erroneous fittingresult)Therefore the local scaling exponent 120572 at such pointscan be calculated using (15) and such points are defined assingular points For the fluctuating line both the slope 119870and local scaling exponent 120572 cannot be obtained by doing fitwhich means that singularity does not exist at these pointsThus by analyzing temporal distribution features of singularpoints we can obtain the temporal variation of local scalingproperty of seismicity

Li and Xu [26 39ndash41] studied the temporal variationof local scaling property for the series of interevent timesbetween successive earthquakes by using the seismic activitydata in several regions of China We found that there arecharacteristic changes of local scaling property prior tosome large earthquakes We believe that such characteristicchanges of local scaling property might be useful for usto get precursory information about the scale invariance ofseismicity before large earthquakes

The strengths of the method of local scaling propertyare that it cannot only give the description of heterogeneousproperties of seismicity but also give the description of thelocal features of it However there are also some weaknessesin thismethod Firstly because the calculation of thismethodis done point by point and the process of calculation iscomplex thus if we do not control the cumulative error inthe calculation process the deviation of calculation resultswill be large Secondly it is the complicated process to choosesuitable wavelet transform parameters which correspond tothe seismicity features of different regions by using trial-and-error method If we do not choose suitable parametersthe useful information before large earthquakes will beconcealed Thirdly the observed cases in which there arecharacteristic changes of local scaling property of seismicityprior to large earthquakes have been quite few and theresearch results have not widely been examined so farTherefore the research results obtained are only tentativeand cannot be treated as the final results Nevertheless thestudy on the characteristic changes of local scaling propertyof seismicity is meaningful because it is carried out from anew theoretical viewpoint

6 Conclusions

In this paper we presented some important research devel-opments of characteristic changes of scale invariance ofseismicity before large earthquakes These studied cases areof particular importance to the researchers in earthquakeforecasting and seismic activity Meanwhile the strengthsand weaknesses of different methods of scale invariance arediscussed There have been many studied cases in whichthere are characteristic changes of the multifractal spectrumthe correlation dimension the fractal dimension and Hurstexponent analysis of seismicity prior to large earthquakesindicating that the research results obtained by using themultifractal spectrum the correlation dimension the fractaldimension andHurst exponent analysis are relativelymaturewhile there have only been few observed cases in which thereare the characteristic changes of local scaling property of seis-micity prior to large earthquakes indicating that the researchresults obtained by using themethod of local scaling propertyare tentative and not final The advantage of the methodof multifractal spectrum is that it can sufficiently give thedescription of heterogeneous properties of a fractal systembut it is not good at presenting the property of self-affinefractal and is incapable to describe local features of the fractalseismic system The peculiarity of the correlation dimensionand fractal dimension is that they can only describe themonofractal property of seismicity The advantage of the


Hurst exponent analysis is that it is good at presentingthe property of self-affine fractal of seismicity however itis incapable to describe the heterogeneous property of thefractal objects The peculiarity of local scaling property isthat it can not only give the description of heterogeneousproperties of seismicity but also give the description of thelocal features of it

Due to the previously mentioned strengths and weak-nesses of different methods of scale invariance we suggestthat when people try to obtain the precursory informationbefore large earthquakes or to study the fractal property ofseismicity bymeans of the previous scale invariancemethodsthe strengths and weaknesses of these methods have tobe taken into consideration for the purpose of increasingresearch efficiency If they do not consider the strengths andweaknesses of these methods the efficiency of their researchmight greatly decrease

Acknowledgments

The authors thank Professor S S Dong for helpful conversa-tion This work is supported by the Natural Science Founda-tion of Jiangsu province China (BK2008486)

References

[1] T Chelidze and T Matcharashvili ldquoComplexity of seismic pro-cess measuring and applicationsmdasha reviewrdquo Tectonophysicsvol 431 no 1ndash4 pp 49ndash60 2007

[2] B Enescu K Ito M Radulian E Popescu and O BazacliuldquoMultifractal and chaotic analysis of Vrancea (Romania) inter-mediate-depth earthquakes investigation of the temporal dis-tribution of eventsrdquo Pure and Applied Geophysics vol 162 no 2pp 249ndash271 2005

[3] Y Y Kagan ldquoEarthquake spatial distribution the correlationdimensionrdquo Geophysical Journal International vol 168 no 3pp 1175ndash1194 2007

[4] D Kiyashchenko N Smirnova V Troyan and F VallianatosldquoDynamics of multifractal and correlation characteristics of thespatio-temporal distribution of regional seismicity before thestrong earthquakesrdquoNatural Hazards and Earth System Sciencevol 3 no 3-4 pp 285ndash298 2003

[5] R JMittag ldquoFractal analysis of earthquake swarms ofVogtlandNW-Bohemia intraplate seismicityrdquo Journal of Geodynamicsvol 35 no 1-2 pp 173ndash189 2003

[6] A O Oncel and T H Wilson ldquoSpace-time correlations of seis-motectonic parameters examples from Japan and from Turkeypreceding the Izmit earthquakerdquo Bulletin of the SeismologicalSociety of America vol 92 no 1 pp 339ndash349 2002

[7] L Telesca G Hloupis I Nikolintaga and F Vallianatos ldquoTem-poral patterns in southern Aegean seismicity revealed by themultiresolutionwavelet analysisrdquoCommunications inNonlinearScience and Numerical Simulation vol 12 no 8 pp 1418ndash14262007

[8] V Uritsky N Smirnova V Troyan and F Vallianatos ldquoCriticaldynamics of fractal fault systems and its role in the generation ofpre-seismic electromagnetic emissionsrdquo Physics and Chemistryof the Earth vol 29 no 4ndash9 pp 473ndash480 2004

[9] F Vallianatos G Michas G Papadakis and P Sammonds ldquoAnon-extensive statistical physics view to the spatiotemporal

properties of the June 1995 Aigion earthquake (M62) after-shock sequence (West Corinth rift Greece)rdquo Acta Geophysicavol 60 no 3 pp 758ndash768 2012

[10] F Vallianatos G Michas G Papadakis and A Tzanis ldquoEvi-dence of non-extensivity in the seismicity observed during the2011-2012 unrest at the Santorini volcanic complex GreecerdquoNatural Hazards and Earth System Sciences vol 13 pp 177ndash185

[11] F Vallianatos ANardi R Carluccio andMChiappini ldquoExper-imental evidence of a non-extensive statistical physics behaviorof electromagnetic signals emitted from rocks under stress upto fracture Preliminary resultsrdquo Acta Geophysica vol 60 no 3pp 894ndash909 2012

[12] F Vallianatos andA Tzanis ldquoOn the nature scaling and spectralproperties of pre-seismic ULF signalsrdquo Natural Hazards andEarth System Science vol 3 no 3-4 pp 237ndash242 2003

[13] A Zamani and M Agh-Atabai ldquoTemporal characteristics ofseismicity in the Alborz and Zagros regions of Iran using amultifractal approachrdquo Journal of Geodynamics vol 47 no 5pp 271ndash279 2009

[14] F Caruso S Vinciguerra V Latora A Rapisarda and S Mal-one ldquoMultifractal analysis of Mount St Helens seismicity as atool for identifying eruptive activityrdquo Fractals vol 14 no 3 pp179ndash186 2006

[15] P P Dimitriu E M Scordilis and V G Karacostas ldquoMulti-fractal analysis of the Arnea Greece Seismicity with potentialimplications for earthquake predictionrdquo Natural Hazards vol21 no 2-3 pp 277ndash295 2000

[16] D Kiyashchenko N Smirnova V Troyan E Saenger and FVallianatos ldquoSeismic hazard precursory evolution fractal andmultifractal aspectsrdquo Physics and Chemistry of the Earth vol 29no 4ndash9 pp 367ndash378 2004

[17] Q Li ldquoThe multifractal characteristics of the seismic tempo-ral series in Jiangsu Province and adjacent areas and theirapplication to earthquake predictionrdquo Journal of SeismologicalResearch vol 25 pp 257ndash261 2002 (Chinese)

[18] S Nakaya ldquoFractal properties of seismicity in regions affectedby large shallow earthquakes in western Japan implicationsfor fault formation processes based on a binary fractal fracturenetwork modelrdquo Journal of Geophysical Research B vol 110 no1 Article ID B01310 2005

[19] P N S Roy and A Padhi ldquoMultifractal analysis of earthquakesin the Southeastern Iran-Bam Regionrdquo Pure and Applied Geo-physics vol 164 no 11 pp 2271ndash2290 2007

[20] S S Teotia and D Kumar ldquoRole of multifractal analysis inunderstanding the preparation zone for large size earthquakein the North-Western Himalaya regionrdquo Nonlinear Processes inGeophysics vol 18 no 1 pp 111ndash118 2011

[21] K Murase ldquoA characteristic change in fractal dimension priorto the 2003 Tokachi-oki Earthquake (MJ = 80) HokkaidoNorthern Japanrdquo Earth Planets and Space vol 56 no 3 pp401ndash405 2004

[22] P N S Roy and S K Nath ldquoPrecursory correlation dimensionsfor three great earthquakesrdquo Current Science vol 93 no 11 pp1522ndash1529 2007

[23] X L Lei K Kusunose T Satoh and O Nishizawa ldquoThe hierar-chical rupture process of a fault an experimental studyrdquo Physicsof the Earth and Planetary Interiors vol 137 no 1ndash4 pp 213ndash2282003

[24] X L Lei and T Satoh ldquoIndicators of critical point behaviorprior to rock failure inferred from pre-failure damagerdquo Tectono-physics vol 431 no 1ndash4 pp 97ndash111 2007


[25] C Lu Y W Mai and H Xie ldquoA sudden drop of fractal dimen-sion a likely precursor of catastrophic failure in disorderedmediardquo Philosophical Magazine Letters vol 85 no 1 pp 33ndash402005

[26] Q Li and G M Xu ldquoRelationship between the characteristcvariations of local scaling property and the process of seis-mogeny the revelation of a new physical mechanism of seis-micityrdquo Fractals vol 18 no 2 pp 197ndash206 2010

[27] C P Zhao and H TWang ldquoAnomalous features of Hurst expo-nent before some large earthquakes in ChinaMainlandrdquo InlandEarthquake vol 15 pp 331ndash337 2001 (Chinese)

[28] A Carpinteri G Lacidogna and S Puzzi ldquoFrom criticality tofinal collapse evolution of the ldquob-valuerdquo from 15 to 10rdquo ChaosSolitons and Fractals vol 41 no 2 pp 843ndash853 2009

[29] C Goltz Fractal and Chaotic Properties of Earthquakes Sprin-ger Berlin Germany 1997

[30] D K Guo ldquoApplication of Hurst exponent of earthquake fre-quency to the earthquake predictionrdquo North China EarthquakeSciences vol 20 pp 44ndash50 2002 (Chinese)

[31] Y T Lee C C Chen Y F Chang and L Y Chiao ldquoPrecursoryphenomena associated with large avalanches in the long-rangeconnective sandpile (LRCS) modelrdquo Physica A vol 387 no 21pp 5263ndash5270 2008

[32] T Matcharashvili T Chelidze and Z Javakhishvili ldquoNonlinearanalysis of magnitude and interevent time interval sequencesfor earthquakes of the Caucasian regionrdquoNonlinear Processes inGeophysics vol 7 no 1-2 pp 9ndash19 2000

[33] M Radulian and C I Trifu ldquoWould it have been possible topredict the 30 August 1986 Vrancea earthquakerdquo BulletinmdashSeismological Society of America vol 81 no 6 pp 2498ndash25031991

[34] L Telesca andV Lapenna ldquoMeasuringmultifractality in seismicsequencesrdquo Tectonophysics vol 423 no 1ndash4 pp 115ndash123 2006

[35] L Telesca V Lapenna andMMacChiato ldquoMultifractal fluctu-ations in seismic interspike seriesrdquo Physica A vol 354 no 1ndash4pp 629ndash640 2005

[36] P Grassberger and I Procaccia ldquoDimensions and entropiesof strange attractors from a fluctuating dynamics approachrdquoPhysica D vol 13 no 1-2 pp 34ndash54 1984

[37] A Chhabra and R V Jensen ldquoDirect determination of the sin-gularity spectrumrdquo Physical Review Letters vol 62 no 12 pp1327ndash1330 1989

[38] P Grassberger ldquoGeneralized dimensions of strange attractorsrdquoPhysics Letters A vol 97 no 6 pp 227ndash230 1983

[39] Q Li and G M Xu ldquoLocal scaling property of seismicity anexample of getting valuable information from complex hierar-chical systemrdquo Nonlinear Processes in Geophysics vol 17 no 5pp 423ndash429 2010

[40] Q Li and G M Xu ldquoCharacteristic variation of local scalingproperty before Puer M64 earthquake in China the presenceof a new pattern of nonlinear behavior of seismicityrdquo IzvestiyaPhysics of the Solid Earth vol 48 no 2 pp 155ndash161 2012

[41] Q Li andGM Xu ldquoScale invariance in complex seismic systemand its uses in gaining precursory information before largeearthquakes importance of methodologyrdquo Physica A vol 392no 4 pp 929ndash940 2013

[42] Y Y Kagan ldquoObservational evidence for earthquakes as a non-linear dynamic processrdquo Physica D vol 77 no 1ndash3 pp 160ndash1921994

[43] B Enescu and K Ito ldquoSome premonitory phenomena of the1995 Hyogo-Ken Nanbu (Kobe) earthquake seismicity b-value

and fractal dimensionrdquo Tectonophysics vol 338 no 3-4 pp297ndash314 2001

[44] J Feder Fractals Plenum Press New York NY USA 1988[45] Q Li and GM Xu ldquoResearch onHurst exponent of earthquake

frequency in Jiangsu and its adjacent areardquo Northwestern Seis-mological Journal vol 24 pp 247ndash250 2002 (Chinese)

[46] R Bove V Pelino and L de Leonibus ldquoComplexity in rainfallphenomenardquoCommunications inNonlinear Science andNumer-ical Simulation vol 11 no 6 pp 678ndash684 2006

[47] J Li and Y Chen ldquoRescaled range (RS) analysis on seismicactivity parametersrdquo Acta Seismologica Sinica vol 14 no 2 pp148ndash155 2001

[48] Y Q Li and P L Wang ldquoAnalysis on earthquake frequencyHurst exponent in active block of Qinghai-Xizhang PlateaurdquoPlateau Earthquake Research vol 18 pp 36ndash40 2006 (Chinese)

[49] B Q Wang H M Huang H s Fan C Z Wang and P YChen ldquoNonlinear RS method and its applicaion in earthquakepredictionrdquo Acta Seismologica Sinica vol 17 pp 528ndash532 1995

[50] A Arneodo G Grasseau andMHolschneider ldquoWavelet trans-formofmultifractalsrdquo Physical Review Letters vol 61 no 20 pp2281ndash2284 1988

[51] F Liu and J Z Cheng ldquoLocal fractal scale wavelet analysisrdquoJournal of Xirsquoan Jiaotong University vol 33 pp 14ndash34 1999(Chinese)

[52] F S Yang Application of Wavelet Transform on EngineeringAnalysis Science Press Beijing China 2003 (Chinese)

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ClimatologyJournal of

EcologyInternational Journal of


EarthquakesJournal of


Hindawi Publishing Corporationhttpwwwhindawicom

Applied ampEnvironmentalSoil Science

Volume 2014

Mining


Journal of

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

International Journal of

Geophysics

OceanographyInternational Journal of


Journal of Computational Environmental SciencesHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal ofPetroleum Engineering


GeochemistryHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Atmospheric SciencesInternational Journal of


OceanographyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in


MineralogyInternational Journal of


MeteorologyAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Paleontology JournalHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Geological ResearchJournal of


Geology Advances in


for the purpose of improving the study validity for theresearchers in earthquake forecasting and seismic activity

2 Multifractal Spectrum of Seismicity

The multifractal spectrum includes the generalized dimen-sion spectrum and singularity spectrum Because the singu-larity spectrum is closely related to the generalized dimensionspectrum we mainly focus on generalized dimension spec-trum (the curve graph of multifractal dimension119863119902 versus 119902is called the generalized dimension spectrum) in this paper

The generalized dimension spectrum of seismicity isstudied by an algorithm based on the correlative integral[38] Computation ofmultifractal dimension119863119902 applying thecorrelative integral algorithm [18] is made using

119863119902 = lim119903rarr0

log119862119902 (119903)log 119903

(1)

where119862119902(119903) is the 119902th-order generalized correlation functiondefined as

119862119902 (119903) =

1

119873

119873

sum

119895=1

[[

[

1

119873

119873

sum

119894=1119894 = 119895

Θ(119903 minus10038161003816100381610038161003816119883119894 minus 119883119895

10038161003816100381610038161003816)]]

]

119902minus1

1(119902minus1)

(2)

with119873 as the number of data points 119903 as the scaling radiusΘ(119904) as the Heaviside step function and |119883119894 minus 119883119895| as thedistance between the two data points 119883119894 and 119883119895 If the serieshas the property of fractals the 119862119902(119903) is indicated by a powerlaw Thus the value of 119863119902 can be settled by calculating thelinear segment slope in the graph of log 119862119902(119903) versus log 119903

Roy and Padhi [19] studied the generalized dimensionspectrum of seismicity in Iran and adjacent regions Theyfound that the precursory clustering pattern in the short timeperiod before three large earthquakes (119872119908 = 78 on 1691978119872119908 = 68 on 26122003119872119908 = 77 on 10597) is perceived byanalyzing the multifractal property of seismicity in this areaThey believe that the probability for future large earthquakescan be evaluated by studying the spatial and temporal cluster-ing pattern of earthquakes They conclude that the clusteringpattern study based on seismic catalogue for majority of theknown seismic fault systems may be finally helpful to theprevention and mitigation of earthquake hazards

Caruso et al [14] made a study on the multifractal pro-perty of Mount St Helens seismicity during the period of1980ndash2002 in which the temporal distribution of seismicityrelated to the eruptive activity primarily characterized bythe major explosive eruptions in 1980 and the eruptionsduring the period of 1980ndash1986 is analyzed They found thatthe generalized dimension spectrum which is calculatedfrom the data of seismicity can help us to recognize twomain temporal distribution patterns of seismicity The firstpattern shows a multifractal clustering related to the strongearthquake swarm of the dome building activity The secondpattern is typified by very small variation value of 119863119902 closeto 1 as for a stochastic uniform distribution The temporalchange of 119863119902 calculated from an invariant window length of

seismic events and at different depths shows that the fragilemechanical response of the shallow zones to swift magmaticintrusions through the whole course of the eruption isindicated by its rapid variations in a short period and thesmallest values of119863119902 (about 03)

Dimitriu et al [15] studied the generalized dimensionspectrum in a seismically active area of Northern Greecewhere several large earthquakes containing the May 1995Arena sequence with a maximum119872119908 53 earthquake onMay4 1995 occurred They found that multifractality enhancesbefore the major seismicity and that the clusterization isreplaced by the declusterization not long before its initializa-tion They believe that this might help us to approximatelyevaluate the occurrence time of the main slip if the multi-fractality getting to its peak before the percolation makes theprevious gradual increase in multifractality

Teotia and Kumar [20] studied the generalized dimen-sion spectrum by analyzing the seismic catalog data in theNorth-Western Himalaya area which primarily contain theseismogenic zone of 1905 Kangra great earthquake Theseismic catalogue data come from USGS catalogue duringthe period of 1973ndash2009 which contains the Muzaffarabad-Kashmir earthquake (119872119908 = 76) ofOctober 2005They foundthe temporal variations in generalized dimension spectrumbefore the Muzaffarabad-Kashmir earthquake in the areaThey believe that this study may be of important value inrecognizing the seismogenic zone of large seismic events indissimilar tectonic regions

Li [17] studied the variation of multifractal characteristicbefore and after four mid-strong earthquakes (magnitudesrange from 55 to 62) in Jiangsu province and its adjacentarea in the east of China by analyzing seismicity datasince 1970 in this area where the frequency and strength ofseismicity belong to the middle level and the completenessmagnitude of the catalog estimated by using Gutenberg-Richter formula is 24 I found that the generalized dimen-sion spectrum and singularity spectrum 119891(120572) of seismicityhave three variation stages which separately corresponded tothe time far before mid-strong earthquakes the time whenthe anomalous change of the spectrum appeared and thetime when the anomalous change of the spectrum disap-peared Figure 1 shows the three variation stages of thegeneralized dimension spectrum and singularity spectrumbefore 22 April 1974 Liyang M55 earthquake (the data setincludes 311 earthquakes) Among the figures Figures 1(a)and 1(d) show the first variation stage of the generalizeddimension spectrum and singularity spectrum (two yearsprior to this earthquake) respectively in which the level dif-ference between the left end and right ends of the generalizeddimension spectrum is small (smaller than 044) Figures1(b) and 1(e) show the anomalous change stage (ie thesecond variation stage) of generalized dimension spectrumand singularity spectrum (seven months prior to this earth-quake) respectively in which the curve of the generalizeddimension spectrum is steeper than the two others (thelevel difference between the left end and right ends ofthe generalized dimension spectrum is larger than 058)and the value range of the singularity spectrum is broaderthan the two others Figures 1(c) and 1(f) show the third


15

10

05


q

10

05

00

f(a)

a

00 10 20

15

10

05


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Dq

Dq






15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)









15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)



119862 (119903) =2119873 (119877 lt 119903)

119873 (119873 minus 1) (3)


119862 (119903) prop 119903119863 (4)






15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)
















⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


15

10

05


q

10

05

00

f(a)

a

00 10 20

15

10

05


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)

Dq

Dq






15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)









15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)



119862 (119903) =2119873 (119877 lt 119903)

119873 (119873 minus 1) (3)


119862 (119903) prop 119903119863 (4)






15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)
















⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

15

10

05

Dq


10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)









15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)



119862 (119903) =2119873 (119877 lt 119903)

119873 (119873 minus 1) (3)


119862 (119903) prop 119903119863 (4)






15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)
















⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)



119862 (119903) =2119873 (119877 lt 119903)

119873 (119873 minus 1) (3)


119862 (119903) prop 119903119863 (4)






15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)
















⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


15

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

15

10

05

Dq


q

10

05

00

f(a)

a

00 10 20

(a)

(b)

(c)

(d)

(e)

(f)
















⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










⟨120585⟩120591=1

120591

120591

sum

119905=1

120585 (119905) (5)


119883 (119905 120591) =

119905

sum

119906=1

[120585 (119906) minus ⟨120585⟩120591] 1 le 119905 le 120591 (6)


119877 (120591) = max1le119905le120591

119883 (119905 120591) minus min1le119905le120591

119883 (119905 120591) (7)


119878 (120591) = [1

120591

120591

sum

119905=1

(120585 (119905) minus ⟨120585⟩120591)2]

12

(8)


119877 (120591)

119878 (120591)prop 120591119867 (9)






116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


116

059

M55

Hur

st ex

pone

nt

1972 1973 1974 1975Year


109

095

081

067

M53

Hur

st ex

pone

nt

1992 1993 1994 1995Year




111

058

M51

Hur

st ex

pone

nt

1988 1989 1990 1991Year


106

061

M62

Hur

st ex

pone

nt

1982 1983 1984 1985Year







1198911199090(119909) = 119891 (1199090 + 119909) minus 119891 (1199090) (10)


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in


we obtain

1198911199090(120582119909) = 120582

+120572(1199090) 1198911199090(119909) (11)



119879 (119886 1199090 + 119887) =1

radic119886int119891 (119909) 120595(

119909 minus 1199090 minus 119887

119886)119889119909 (12)


119879 (120582119886 1199090 120582119887) =1

radic120582119886

int119891 (119909) 120595(119909 minus 1199090 minus 120582119887

120582119886)119889119909 (13)


119879 (120582119886 1199090 + 120582119887) = 120582+120572(1199090)+12 119879 (119886 1199090 + 119887) (14)


119870 = 120572 +1

2 (15)





6 Conclusions





Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in




Acknowledgments


References

































































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in








































Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in










Volume 2014

Mining


Journal of



Geophysics







Journal of




Advances in











Geology Advances in

Review Article Characteristic Changes of Scale …downloads.hindawi.com/journals/ijge/2013/391637.pdfReview Article Characteristic Changes of Scale Invariance of Seismicity Prior to

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