Bulletin of the Seismological Society of America

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1 Bulletin of the Seismological Society of America This copy is for distribution only by the authors of the article and their institutions in accordance with the Open Access Policy of the Seismological Society of America. For more information see the publications section of the SSA website at THE SESMOLOGCAL SOCETY OF AMERCA 400 Evelyn Ave., Suite 201 Albany, CA (510) ; FAX (510)

2 Bulletin of the Seismological Society of America, Vol. 74, No. 5, pp , October 1984 TRANSTON FROM AFTERSHOCK TO NORMAL ACTVTY: THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE BY YOSHKO OGATA AND KUNHKO SHMAZAK ABSTRACT A transition from aftershock to normal activity is identified 2200 days after the occurrence of the 1965 Rat slands earthquake of M, = 8.7. After this time, we observe a significant departure of the seismicity in the focal region from the extrapolated trend of the aftershock activity predicted by the modified Omori formula proposed by Utsu in The maximum likelihood estimate of the parameters for the formula and the selection of a statistical model based on the Akaike nformation Criterion show that the whole aftershock sequence is best modeled by the combination of an ordinary aftershock sequence and secondary aftershock activity associated with the largest aftershock. Fitting of the activity after the transition period to the modified Omori formula results in an unusually low value of the decay constant. We interpret that this activity represents normal or background seismicity and not another secondary aftershock sequence. Oefining background activity in a focal region of a large earthquake is essential for detection of seismic quiescence prior to the next major event. NTRODUCTON Although several qualitative descriptions of the seismicity in focal regions of large earthquakes have been made (e.g., Kelleher and Savino, 1975; Mogi, 1977, 1981; Kanamori, 1981), most considered only the seismicity prior to the occurrence of the major earthquake. Only a few attempts have been made to describe quantitatively the seismicity in the focal region over the whole seismic cycle (Fedotov, 1968; Perez, 1983). This is partly because the period of instrumental observation is too short to cover the whole seismic cycle in most regions, but also because of a lack of proper statistical methods to treat seismicity problems. This study marks the first step toward a quantitative description of seismicity over the whole seismic cycle and will focus on the transition from aftershock to normal or background seismicity on the basis of recent progress in the application to seismicity problems of statistical techniques in point processes (e.g., Vere-Jones, 1970; Ogata, 1984). t is widely accepted that some time after the occurrence of a major earthquake, the aftershock activity dies off and normal, or background seismicity surpasses the aftershock activity. However, it is not obvious when and how the transition occurs from aftershock to normal activity. Prior to the next major earthquake, seismic quiescence (Habermann et al., 1983) is expected to appear in the focal region. However, the quiescence cannot be defined objectively unless the background activity is well established, since the quiescence can be considered a mere result of the decaying activity of aftershocks from the last major earthquake (Lomnitz, 1982; Lomnitz and Hava, 1983). n this study, we analyze the aftershock sequence of the 1965 Rat slands earthquake (moment magnitude Mw = 8.7; Kanamori, 1977) and try to show how a transition from aftershock sequence because this is the largest and longest aftershock sequence in the NOAA Hypocenter Data File according to Habermann (1983). 1757

3 1758 YOSHKO OGATA AND KUNHKO SHMAZAK DATA We use the NOAA Hypocenter Data File as the source of the data in this study. The completeness of the catalog for events in the Aleutian island arc is discussed by Habermann (1983). The minimum magnitude of homogeneity introduced by him is the level of magnitude above which some constant portion of the events occurring is consistently reported through time. According to his study, the minimum magnitude of homogeneous reporting is mb= 4.7 for the Aleutians during a period from 1963 through All events above this level of magnitude with depths less than 100 km are used in a region bounded by the 170 E and 180 E meridians, and the 48 N and 55 N parallels. Figure 1 shows these events during the period from ( l E w 55 o o o O~oo ~ 0 0 o o 0~ooO o o oo ~o ~ o o~ o ~ o _~V~o o ~o o~' o oooo-~oo~,~_'5 o ~2 E o o o o-oo c ::~o~ o o Q ~ o o o 8 ~ o~3~ N FG. 1. Seismicity map in the study area along the Rat slands in the western Aleutians for 1963 through The size of the circles depend on magnitude, mb. The closed diamond shows the epicenter of the Rat [slands earthquake of 4 February through The minimum magnitude of complete reporting is estimated to be mb = by Habermann (1983). METHOD The frequency of aftershocks per unit time interval (1 day, 1 month, etc.) n(t) is well represented by the modified Omori formula (Utsu, 1961) n(t) = K(t + c) -p, (1) where t is the elapsed time since the occurrence of the main shock and K, c, and p are constant parameters. Assuming that all events in an aftershock sequence are distributed according to a nonstationary Poisson process (Lomnitz and Hax, 1966), Ogata (1983) proposed the maximum likelihood method to estimate the parameters of the above formula. f we denote by Pa(t), the probability that an earthquake

4 THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE 1759 occurs in a small time interval between t and t + 5, then an intensity function X(t) of the Poisson process can be defined as X(t) = lim Pa(t)/A. (2) A--4) Then the log-likelihood function of the aftershock sequence {ti} during a time interval between t = S and t = T is given as follows lnl= Y~ lnx(ti)- ~S 7' X(t) dt. S<ti<T (3) We can write the intensity function corresponding to the modified Omori formulas as X(t;0) = K(t + c) -~, O= (K,c,p), (4) and the maximum likelihood estimates (MLE) of the parameters K, c, and p are obtained numerically by using the gradients of the log-likelihood function (3) and a standard nonlinear optimization technique such as that in Fletcher and Powell {1963). An integration of the intensity function, X(t) is defined as A(t) = f0 t X(s) ds. The integration gives a transformation to a frequency-linearized time r; on this time axis the occurrence of earthquakes becomes the standard stationary Poisson process (i.e., the intensity is just 1), if the choice of the intensity function X(t) is correct. This time axis will be used to detect a deviation of seismic activity from a decaying trend of aftershock sequence. A similar transform of the time axis for the original Omori formula (Omori, 1894) was introduced by Lomnitz and Hax (1966). The frequency-linearized time for a single aftershock sequence can be defined as r = A(t) = f0 t K(s + c) -p ds. (5) t is often observed that a sequence of aftershocks contains secondary aftershocks, aftershocks of a major aftershock (Utsu, 1970). f a secondary aftershock sequence starts at time t = t2, then the intensity function X(t) can be given as X(t; O) = K~(t + c,) -~ + H(t - t2)k2(t - t2 + c2) -~, 0 = (K, ci, p,, K2, c2, p2; t2), (6) where H(t) is the Heaviside unit step function, and MLE of 0 is also obtained by maximizing (3). The frequency-linearized time for a sequence including the secondary aftershock sequence which starts at time t = t.,, can be defined as r = K,(s + cl) -J~ ds + H(t - t2) K2(s - t~ + c2) -p~ ds. (7)

1760 YOSHKO OGATA AND KUNHKO SHMAZAK t now becomes necessary to judge whether the model (6) including the secondary aftershock activity fits to the observation better than model (4) without secondary

5 1760 YOSHKO OGATA AND KUNHKO SHMAZAK t now becomes necessary to judge whether the model (6) including the secondary aftershock activity fits to the observation better than model (4) without secondary aftershocks. For this purpose, we adopt the Akaike nformation Criterion (AC) (Akaike, 1974) as a measure for selecting the best among competing models under a fixed data set. This is a measure to see which model most frequently reproduces similar features to the given observations and is defined by AC = (-2)Max(log-likelihood) + 2(number of used parameters) (8) where log denotes the natural logarithm. The model with the smaller AC shows the better fit to the data. t is sometimes useful to note that the log-likelihood ratio statistic takes the form (-2)log(Lo/L~) = AC(Ho) - AC(H) + 2k, where the model H1 contains the model Ho as a restricted family, and k denotes the difference of dimensions of parameters in Ho and H1. For example, in the context mentioned above, Ho corresponds to the model (4) without secondary aftershocks. Under the null hypothesis H0, the statistic (-2)log(LolL1) is expected to distribute according to 2 with degree of freedom k. The comparison of the minimum AC procedure with the conventional likelihood ratio test is discussed in Akaike (1977, 1983). RESULTS We analyzed an aftershock sequence from 3 hr to 1000 days after the 4 February 1965 Rat sland earthquake by fitting it to the modified Omori formula. Assuming there was a low detection capability of small events just after this great earthquake, we removed the first 3 hr of data from the analysis. Also, we did not use the data after 1000 days to avoid possible contamination from background activity. The obtained parameters and AC are as follows K = c = p = AC = By using these parameters, the cumulative number of events are plotted against the frequency-linearlized time r defined in equation (5) and is shown in Figure 2. f the modeling of the aftershock sequence is appropriate, the cumulative number of events analyzed, i.e., events from 3 hr to 1000 days after the main shock, should increase linearly with r. The figure shows a bump in the plot which is caused by a rapid increase of events roughly 50 days after the main shock. The time of rapid increase coincides with the occurrence of the largest aftershock, the 30 March 1965 event. This is a normal faulting event within the subducting oceanic lithosphere and its moment magnitude is Mw = 7.6 (Abe, 1972). Thus, we reconstruct a model which takes into account the effect of the secondary aftershock activity associated with the 30 March event. By assuming the same p

6 THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE 1761 and c parameters for the secondary sequence as those for the main aftershock sequence, i.e., p] = p2 and cl = c2, we get K1 = K2 = cl = c2 = Pl = P2 = AC = TME, DAYS (/) ll < O "r - n" < UJ t. O.J m =! Z > am,..- <,. :f :) O O.l To T1 t f 0 lo BOO FREQUENCY-LNEARZED TME FG. 2. Plot of the cumulative number of events versus the frequency-linearized time 7, which is calculated with parameters K = , c = 0.204, p = These parameters are obtained for the time period between % = 3 hr and T1 = 1000 days after the main shock. Note that the AC in this case is much smaller than that obtained previously. Thus, the model with secondary aftershock sequence fits better to the data than does the model without it. A plot of the cumulative number similar to Figure 2 is constructed by substituting the above parameters into equation (7). The result is shown in Figure 3. A nearly linear trend of aftershock decay now continues up to 2200 days, or about 6 hr after the main shock, although there remains a slight S-shaped deviation from the linear trend. Around 2200 days after the main shock, that is in February 1971,

7 1762 YOSHKO OGATA AND KUNHKO SHMAZAK the cumulative number starts to increase rapidly with T and shows significant deviation from the prior trend. There is no large earthquake occurring in the Rat slands region in this time period. The largest event which occurred within 2 yr before or after the time of the change in the trend is the 11 June 1971 event with mb = 5.9. We attempted to fit the activity after this time period to the modified Omori formula. Only a poor fit can be achieved with an unusually small decay constant, p = 0.62, by assuming c = 0.3. Although there is some decreasing tendency of the activity with time, we interpret the activity as the background or normal seismicity ll <C 0 -r. -. < JJ t. O rr ill m TME, DAYS l l l l l l To T T1 Z > m -..J O 0 lo FREQUENCY-LNEARZED TME F6. 3. Plot of the cumulative number of events versus the frequeney-linearized time r, which is calculated with parameters K~ = , K~ = 6.089, cl = c2 = 0.176, pl = p2 = 1.079,/2 = 7'2 = 53.9 days. These parameters are obtained for the time period between To = 3 hr and 7'1 = 1000 days after the main shock. The occurrence time of the largest aftershock which is followed by the secondary aftershocks is denoted by Te. in the focal region of the 1965 Rat slands earthquake. Thus, we consider that the change in slope clearly seen around 2200 days in Figure 3 marks the transition from aftershock to normal activity. One might misunderstand that the end of the aftershock sequence is marked by an increase in the seismicity rate. However, the upward deviation from the linear trend of aftershock decay in Figure 3 does not necessarily mean an increase in the seismicity rate, but shows that more events are occurring than expected for a decaying aftershock sequence [note that the time axis is compressed as indicated by equations (5) or {7)].

8 THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE 1763 The identified transition from aftershock to normal activity might be affected by any change in reporting during 1971 as a result of the installation of the Amchitka network. n order to resolve this possibility, we plot in Figure 4 the cumulative number of events with magnitude cutoffs of 5.1+ (a complete, rather than a homogeneous data set according to Habermann, 1983) and 5.5+ against the same frequency-linearized time as used in Figure 3. The time of the end of the aftershock sequence remains the same as defined in Figure 3. The stability of the ending time leads us to conclude that the obtained result is unaffected by any detection change. Finally, the parameters for the modified Omori formula are reestimated by including the data up to 2200 days after the main shock. Several models are tried including one assuming only one aftershock sequence. The best estimates of the parameters are then as follows "=' Z~.UO ~> " '~<:,. ::~ U. L) to0- O.l 1 mb_~5.1 TME, DAYS lo , :: i =, / J To ~ T2 T1 i FREQUENCY-LNEARZED TME Fie,. 4. Plots of the cumulative numbers of events with respective magnitude cutoffs of 5.1+ and 5.5+ versus the same frequency-linearized time T as Figure 3. K1 = K,, = Cl = c,, = Pl = P2 = t is rather interesting that the two aftershock sequences share the same values of both parameters p and c indicating that the decay rate was about the same for these two different types of events. The main shock represents interplate underthrusting, while the largest aftershock is an intraplate event within the oceanic lithosphere (Abe, 1972). An ordinary log-log plot of the number of events per day against time is shown in Figure 5. The curves fitted to the data points are drawn by using the parameters obtained previously.

9 1764 YOSHKO OGATA AND KUNHKO SHMAZAK DSCUSSON Haberman (1983) concluded that a decrease of activity in the west Aleutians (170 E-18O E) occurred during early 1967 and that it marks the end of the Rat slands aftershock sequence. This conclusion is different from the result of the present study which indicates the aftershock sequence lasted at least 6 yr. n looking at time-dependent seismicity patterns, he used an algorithm which should have removed all the dependent events, i.e., aftershocks. However, since the algorithm does not work very well in removing all the aftershocks for the 1965 Rat slands sequence, he interprets the decrease in seismicity around 1967 as the end of the >.,< a rr U, a. U) u,,v,< 0 ',- -,< 1,. 0 n., i,i,i m 10 a z TME, DAYS FW,. 5. Log-log plot of the number of events in one day versus time. See the text for explanation of the curves fitted to the observed data points shown by the squares. aftershock sequence. n the present study, an algorithm which can predict the aftershock sequence is used to show that the sequence lasted much longer. CONCLUSONS The techniques presented here allow straightforward and objective detection of a transition from aftershock to normal activity. Defining the normal activity is essential in a detection of a seismic quiescence which may take place prior to the next major event. The 1965 Rat sland aftershock sequence can be best modeled by a combination of the main and secondary aftershock sequences. The secondary sequence is associated with the largest aftershock, and the 30 March 1965 event. The difference in both the parameters p and c between the main and the secondary sequences is insignificant, indicating that despite the contrast of the focal mechanism and tectonic location between the main and the largest aftershock, the decay in their aftershock sequences were very similar.

THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE 1765 ACKNOWLEDGMENTS We are grateful to Ted Habermann, Karen McNally, and Jim Mori for their constructive criticism and helpful comments, and to

10 THE 1965 RAT SLANDS EARTHQUAKE AFTERSHOCK SEQUENCE 1765 ACKNOWLEDGMENTS We are grateful to Ted Habermann, Karen McNally, and Jim Mori for their constructive criticism and helpful comments, and to Koichi Katsura for his generous help in computer programming. This research was partly supported by Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture of Japan. REFERENCES Abe, K. (1972). Lithospheric normal faulting beneath the Aleutian trench, Phys. Earth Planet. nteriors 5, Akaike, H. (1974). A new look at the statistical model identification, EEE Trans. Autorn. Control AC- 19, Akaike, H. (1977). On entropy maximization principles, in Applications of Statistics, P. R. Krishnaiah, Editor, North-Holland Publishing Co., Amsterdam, Akaike, H. (1983). nformation measure and model selection. Proc. 44th Session nt. Statistical nst., Madrid, Spain, September 12-22, Fedotov, S. A. (1968). The seismic cycle, quantitative seismic zoning, and long-term seismic forecasting, in Seismic Zoning of the USSR, S. V. Medvedev, Editor, zdatel Stvo "Nauka', Moscow, Fletcher, R. and M. J. D. Powell (1963}. A rapidly convergent method for minimization, Compt. J. 6, Habermann, R. E. (1983). Teleseismic detection in the Aleutian sland Arc, J. Geophys. Res. 88, Habermann, R. E., W. R. McCann, and S. P. Nishenko {1983). A gap is... Bull. Seism. Soc. Am. 73, Kanamori, H. {1977). The energy release in great earthquakes, J. Geophys. Res. 82, Kanamori, H. {1981). The nature of seismicity patterns before large earthquakes, in Earthquake Prediction, Maurice Ewing Series, 4, Kelleher, J. and J. Savino (1975). Distribution of seismicity before large strike slip and thrust-type earthquakes, J. Geophys. Res. 80, Lomnitz, C. (1982). What is a gap? Bull. Seism. Soc. Am. 72, Lomnitz, C. and A. Hax (1966). Clustering in aftershock sequences, American Geophysical Union Geophys. Monograph 10, Lomnitz, C. and F. A. Nava (1983). The predictive value of seismic gaps, Bull. Seism. Soc. Am. 73, Mogi, K. (1977). Dilatancy of rocks under general triaxial stress states with special references to earthquake precursors, J. Phys. Earth 25 (Suppl.), $203-$217. Mogi, K. {1981). Seismicity in Western Japan and long-term earthquake forecasting, in Earthquake Prediction, Maurice Ewing Series, 4, Ogata, Y. {1983). Estimation of the parameters in the modified Omori formula for aftershock frequencies by the maximum likelihood procedure, J. Phys. Earth 31, Ogata, Y. (1984). Likelihood analysis of point process and its application to seismological data, Proc. 44th Session nt. Statistical nst., Madrid, Spain, September 12-22, Omori, F. (1894). On the aftershocks of earthquakes, J. Coll. Sci., Tokyo mp. Univ. 7, Perez, O. J. {1983). Long-term seismic behavior of the focal and adjacent regions of great earthquakes during the time between two successive shocks {abstract), EOS 64,771. Utsu, T. (1961). A statistical study on the occurrence of aftershocks, Geophys. Mag. 30, Utsu, T. (1970). Aftershocks and earthquake statistics (2). Further investigation of aftershocks and other earthquake sequences based on a new classification of earthquake sequences, J. Fac. Sci., Hokkaido Univ., Ser. V (Geophysics) 3, Vere-Jones, D. (1970}. Stochastic models for earthquake occurrence (with discussion), J. R. Statist. Soc., B 32, THE NSTTUTE OF STATSTCAL MATHEMATCS UNVERSTY OF TOKYO TOKYO 106, JAPAN (Y.O). EARTHQUAKE RESEARCH NSTTUTE UNVERSTY OF TOKYO TOKYO 113, JAPAN (K.S.) Manuscript received 14 February 1984

arxiv:physics/ v2 [physics.geo-ph] 18 Aug 2003

arxiv:physics/ v2 [physics.geo-ph] 18 Aug 2003 Is Earthquake Triggering Driven by Small Earthquakes? arxiv:physics/0210056v2 [physics.geo-ph] 18 Aug 2003 Agnès Helmstetter Laboratoire de Géophysique Interne et Tectonophysique, Observatoire de Grenoble,