Nonextensive analysis of crustal seismicity in Taiwan

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1 Nt. Hzrds Erth Syst. Sci.,, 93 97, doi:.594/nhess--93- Author(s). CC Attribution 3. License. Nturl Hzrds nd Erth System Sciences Nonextensive nlysis of crustl seismicity in Tiwn L. Telesc nd C.-C. Chen Institute of Methodologies for Environmentl Anlysis, Ntionl Reserch Council, Tito Sclo (PZ), Itly Grdute Institute of Geophysics, Ntionl Centrl University, Jhongli 3, Tiwn Received: April Accepted: 8 My Published: June Abstrct. Using the Tiwn Centrl Wether Bureu erthuke ctlogue, the crustl seismicity of Tiwn ws nlyzed by mens of nonextensive pproch. The time spn of the nlyzed ctlogue is from Jnury 99 to 3 November 7, nd only erthukes with mgnitude M. were considered. Our findings revel tht the nonextensive sttistics furnishes very good prediction of the cumultive mgnitude distribution of crustl seismicity in Tiwn, even if the ftershocks re removed, indicting tht the pproch is robust for clustered s well s declustered seismicity. Introduction The well known empiricl Gutenberg-Richter (944) reltionship between the cumultive number of erthukes (for mgnitude lrger thn threshold ) with explins sttisticlly the seismicity occurrence, but ws not relted with generl physicl principles. Sotolongo-Cost nd Posds (4) developed generl model for erthuke dynmics, nd the Gutenberg-Richter lw could be considered prticulr cse. In this model the mechnism of erthuke triggering is bsed not only on the slippge of fult plnes nd reltive displcement due to the brek of the sperities (De Rubeis et l., 996), but is lso cused by the frgments filling the spce between fult plnes nd originted from the brekge of the tectonic pltes. Using the non-extensive Tsllis formlism (Tsllis, 988), more relistic mgnitude distribution ws deduced, providing n excellent fit to the seismicities of severl seismic regions (Silv et l., 6; Vilr et l., 7), nd from regionl seismic ctlogues down to specific seismic res (Telesc,, b). Nonextensive erthuke model An erthuke frgment-sperity interction model ws developed by Sotolongo-Cost nd Posds (4) nd, then revised by Silv et l. (6), in which the erthuke triggering mechnism is given by the interction between the irregulrities of the fult plnes with the frgments between them, originted by the locl brekge of the tectonic pltes, from which the fults re generted. In this model the reltive position of frgments filling the spce between the irregulr fult plnes cn contribute to the hindering of their reltive motion. Then, stress increses until displcement of one of the sperities, due to the displcement of the hindering frgment, or even its brekge in the point of contct with the frgment leds to reltive displcement of the fult plnes of the order of the size ρ of the hindering frgment, producing libertion of energy ε (Sotolongo-Cost nd Posds, 4). As lrge frgments re more difficult to relese thn smll ones, this energy is ssumed to be proportionl to the volume of the frgment, so tht the energy distribution of erthukes generted by this mechnism cn reflect the volumetric distribution of the frgments between pltes (Silv et l., 6; Vilr et l., 7). The use of nonextensive sttistics is suited tool to describe the volumetric distribution function of the frgments. Applying the mximum entropy principle for the Tsllis entropy (Tsllis, 988), the Tsllis entropy for our problem is given by p (σ ) ( p(σ ) ) dσ S =, () where p(σ ) is the probbility of finding frgment of surfce σ nd is rel number (Vilr et l., 7). Correspondence to: C.-C. Chen (chencc@erth.ncu.edu.tw) Published by Copernicus Publictions on behlf of the Europen Geosciences Union.

2 94 L. Telesc nd C.-C. Chen: Nonextensive nlysis of crustl seismicity in Tiwn The mximum entropy formultion for Tsllis entropy implies tht two conditions hve to be introduced (Tsllis et R l., 998): ) the normliztion of p(σ ), p(σ )dσ = ; ) the d hoc condition bout the -expecttion vlue, σ = R hσ i = σ p (σ )dσ. For, the lst condition becomes the definition of men vlue. Applying the stndrd method of conditionl extremiztion of the entropy functionl S, we obtin the following expression for the frgment size distribution: ( ) p(σ ) = σ σ ( ) () which corresponds to the re distribution for the frgments of the fult plnes (Silv et l., 6; Sotolongo-Cost nd Posds, 4). Assuming the energy scle ε r 3, the proportionlity be tween the relesed energy ε nd r 3 becomes σ σ = ε 3, where σ scles with r nd (the proportionlity constnt between ε nd r 3 ) hs dimensions of volumetric energy density. This scle is in full greement with the stndrd theory of seismic moment scling with rupture length (Ly nd Wl lce, 995). Thus, using the trnsformtion σ σ = ε 3, the following form for the energy distribution function of the erthukes follows: C ε 3 dε. p(ε)dε = h i/ ( ) + C ε 3 with C = 3 3 nd C = (3) ( ). The probbility of energy ( ) 3 p(ε)=n(ε)/n, where n(ε) is the number of erthukes of energy ε nd N the totl number of erthukes. Therefore, the cumultive number of erthukes cn be clculted s the integrl of E. (3): N (ε > εth ) = N Z p(ε)dε, This not trivil result incorportes the chrcteristics of nonextensivity into the distribution of erthukes by mgnitude. The -vlue is untittive mesure of the length scle of the sptil interctions: indictes short-rnged sptil correltions; s increses, the physicl stte becomes much more unstble; high vlues of men tht the fult plnes re not in euilibrium nd more erthukes cn be expected (Sotolongo-Cost nd Posds, ). 3 (4) εth Where N (ε > εth ) is the number of erthukes with energy lrger thn the threshold εth. Due to the reltionship m = 3 log(ε), where m is the mgnitude, the following formul for the distribution of the number N of erthukes whose mgnitude m is lrger thn the threshold Mth normlized to the totl number of events is obtined: N (m>mth ) log N!# " Mth.= log (5) / 3 Nt. Hzrds Erth Syst. Sci.,, 93 97, Fig.. Sptil distribution of the Tiwnese seismicity from Jnury 99 to 3 November 7. The dt were extrcted from the Centrl Wether Bureu ctlogue. Dt nlysis The Tiwnese seismicity from Jnury 99 to 3 November 7 ws investigted. The dt were relesed from the Centrl Wether Bureu (CWB) erthuke ctlogue (Fig. ). The minimum mgnitude M for which the ctlogue cn be considered complete is. (Telesc et l., 9). Figure shows the normlized cumultive mgnitude distribution (NCMD) of the Tiwnese crustl seismicity. The fitting of NCMD with E. (5) ws performed by mens of the Levenberg-Mrudt nonliner lest sure method (Levenberg, 944; Mrurdt, 963). When using this method, the initil vlues nd re firstly fixed, then the lgorithm furnishes the vlues for nd tht best fit the dt nd lie within dmissible rnges tht, in this study, were [, ] for nd [ 3, ] for. The best fitting is estblished fter the minimum lest sure fitting hs been reched.

3 5 53 Fig. L. Telesc nd C.-C. Chen: Nonextensive nlysis of crustl seismicity in Tiwn 95 log(n >Mth /N) b= , * Fig.. Normlized cumultive mgnitude distribution of the crustl Tiwnese seismicity (open circles) nd fitting curve (solid line). The Fig. vlues of the prmeters re 79 nd 7. The dotted line indictes the Gutenberg-Richter reltionship, with b= Since priori the initil vlues nd giving the minimum estimtion re not known, the ws clculted vrying the initil vlues of nd. The is given by the sure- norm of the residul, which is the difference between the originl dt nd the vlues evluted by the fitting nonliner function. Figure 3 shows the, the prmeters nd, respectively, vrying the vlues of nd. The vlue of the is stble for 4 ; in this 56 rnge the obtined vlues of nd re lso uite stble, with round.68 nd of the order of The minimum error is.868 nd ws obtined for nd = 7 ; then, these initil vlues were used to find the best fitting prmeters 79 nd The fitting curve is lso shown Fig. in 3() Fig. s solid line. The greement between57 the NCMD with E. (5) is very good, indicting tht the nonextensive frgment-sperity interction model is well suited to describe the crustl seismicity of Tiwn. In the Tiwnese territory mny strong erthukes occur nd the ftershock ctivity is, therefore, lso uite intense. In order to verify if ftershocks could influence the vlues of nd, the nonextensive nlysis ws lso performed on the ftershock-depleted ctlogue. For the lgorithm of removing ftershock, we hd pplied the method of sptiotemporl double-link cluster nlysis to eliminte ftershocks in the ctlogue. This method is similr to the single-link cluster nlysis proposed by Dvis nd Frohlich (99). Given 58 mgnitude threshold of min shocks, the declustering lgorithm specifies two linking prmeters in the time nd 59 spce scles, 3 dys nd 5 km for exmple. An event would be identified s n ftershock when its epicenter nd occurrence time ly within the prescribed sptiotemporl window of some min shock. Then, the procedure itertively serches 58 for secondry ftershocks, i.e. the ftershock of n erlier ftershock. The ue turns out to be seprted59 by () Fig. 3() (b) Fig. 3(b) (c) Fig. 3(b).4x 7.x 7.x 7 6.x 6.4x 7 4.x 6.x 7.x 6.x 7 4.x x 7 8.x.6x x 6 -.x 6 6.x Fig. 3..x The 6 () nd the prmeters (b) nd (c) versus Fig. initil3(c) vlue., clculted for the whole seismicity. The different colored curves re obtined 4 with different vlues of. The vlue of the error -.xe 6 is stble for 4 ; in this rnge the obtined vlues of nd re lso uite stble, with round.68 nd of the order of 7. The minimum error is.868 nd ws obtined for nd = 7 ; then, these initil vlues were used to find the best fitting prmeters 79 nd 7 for the fitting curve, Fig. shown3(c) in Fig Nt. Hzrds Erth Syst. Sci.,, 93 97,

4 6 96 L. Telesc nd 6 C.-C. Fig. Chen: 4 Nonextensive nlysis of crustl seismicity in Tiwn declustered ctlogue log(n >Mth /N) b= , * Fig. 4. Normlized cumultive mgnitude distribution of the Fig. declustered 4 crustl Tiwnese seismicity (open circles) nd fitting curve (solid line). The vlues of the prmeters re 85 nd 7 6. The dotted line indictes the Gutenberg-Richter reltionship, with b= mny seuences of min shock nd ftershocks. By using the temporl nd sptil linking prmeters of 3 dys nd 5 km, we hd removed the ftershock events generted from min shocks with M L >4.5. Those linking prmeters were usully used for declustering the CWB ctlogue (Wu nd Chen, 7). 64 Figure 4 shows the NCMD for the declustered ctlog long with its fitting curve, which ws obtined similrly65 to declustered ctlog the whole cse. Figure 5 show the,, nd vrying the initil. vlues of nd, like in the ue The shows n lmost flt behvior for 4 64 nd for ny vlue. This feture does not llow to estimte minimum, in order to find the suited initil vlues65 of nd, nd, in turn, the vlue for the fitting prmeters Fig. nd. 5() But, due to the very smll vrition of the obtined prmeters nd in the sme rnge s the, it is resonble to estimte these s the men of the obtined vlues for 4 ; thus 85 nd 7 ; such vlues re very close to those obtined for the ue, indicting tht the nonextensive model is robust with the presence of ftershocks. 4 Conclusions 66 The crustl seismicity of Tiwn territory hs been nlyzed by mens of nonextensive pproch. The ppliction67 of nonextensivity to the cumultive erthuke mgnitude distribution hs reveled tht such the frgment-sperity interction model fits very stisfctorily nd robustly the rel cse, even in the cse of ftershock declustering. 66 Nt. Hzrds Erth Syst. Sci.,, 93 97, 67 () Fig. 5() (b) declustered ctlog declustered ctlog Fig. 5(b) (c).68 Fig. 5(b).x 7 declustered ctlog Fig. 5. The () nd the prmeters (b) nd (c) versus Fig.5(c) initil vlue., clculted for the ftershock-depleted seismicity. The different colored curves re obtined with different vlues of. The shows n lmost flt behvior for 4 nd for ny vlue. The prmeters nd were 6 estimted s the men of the obtined vlues for 4, thus 85 nd 7 (see text for detils). Fig.5(c) 7 declustered ctlog.x 8.x x declustered ctlog.x 4.x.x 7.x 6 8.x 6. 6.x x 6.x

5 L. Telesc nd C.-C. Chen: Nonextensive nlysis of crustl seismicity in Tiwn 97 Acknowledgements. The efforts of the Centrl Wether Bureu of Tiwn to mintin the seismic network re highly pprecited. L. T. cknowledges the ESRPC of Ntionl Science Council (ROC) for the support of visiting Tiwn in April. C. C. C. is grteful for reserch support from both the Ntionl Science Council (ROC) nd the Institute of Geophysics (NCU, ROC). Edited by: M. E. Contdkis Reviewed by: two nonymous referees References Dvis, S. D. nd Frohlich, C.: Single-link cluster nlysis of erthuke ftershocks: decy lws nd regionl vritions, J. Geophys. Res., 96, , 99. Gutenberg, B. nd Richter, C. F.: Freuency of erthukes in Cliforni, B. Seismol. Soc. Am., 57, 34 37, 944. Ly, T. nd Wllce, T. C.: Modern globl seismology, in: Interntionl geophysics series, Sn Diego: Acdemic Press, 58, 5 pp., 995. Levenberg, K.: A Method for the Solution of Certin Non-Liner Problems in Lest Sures, Q. Appl. Mth.,, 64 68, 944. Mrurdt, D. W.: An lgorithm for lest-sures estimtion of nonliner prmeters, J. Soc. Ind. Appl. Mth.,, 43 44, 963. Silv, R., Frnç, G. S., Vilr, C. S., nd Alcniz, J. S.: Nonextensive models for erthukes, Phys. Rev. E, 73, 6, doi:.3/physreve.73.6, 6. Sotolongo-Cost, O. nd Posds, A.: Tsllis entropy: non-extensive freuency-mgnitude distribution of erthukes, vilble t: pdf (lst ccess: April ),. Sotolongo-Cost, O. nd Posds, A.: Frgment-Asperity Interction Model for Erthukes, Phys. Rev. Lett., 9(4), 485, doi:.3/physrevlett.9.485, 4. Sotolongo-Cost, O. nd Posds, A.: Frgments, Combustion nd Erthukes, rxiv:physics/5336, vilble t: org/bs/physics/5336.pdf (lst ccess: April ), 5. Telesc, L.: Nonextensive nlysis of seismic seuences, Physic A, 389, 9 94,. Telesc, L.: A non-extensive pproch in investigting the seismicity of L Auil re (centrl Itly), struck by the 6 April 9 erthuke (M L 5.8), Terr Nov,, 87 93, b. Telesc, L., Chen, C.-C., nd Lee, Y.-T.: Scling behviour in temporl fluctutions of crustl seismicity in Tiwn, Nt. Hzrds Erth Syst. Sci., 9, 67 7, doi:.594/nhess , 9. Tsllis, C.: Possible generliztion of Boltzmnn-Gibbs sttistics, J. Stt. Phys., 5, , 988. Vilr, C. S., Frnç, G. S., Silv, R., nd Alcniz, J. S.: Nonextensivity in geologicl fults?, Physic A, 377, 85 9, 7. Wu, Y.-M. nd Chen, C.-C.: Seismic reversl pttern for the 999 Chi-Chi, Tiwn, Mw 7.6 erthuke, Tectonophysics, 49, 5 3, 7. Nt. Hzrds Erth Syst. Sci.,, 93 97,