A non-stationary epidemic type aftershock sequence model for seismicity prior to the December 26, 2004 M 9.1 Sumatra-Andaman Islands mega-earthquake

Authors


Corresponding author: A.R. Bansal, CSIR—National Geophysical Research Institute, Uppal Road, Hyderabad 500 007, India. (abhey_bansal@ngri.res.in)

Abstract

[1] We study temporal changes in seismicity in Sumatra-Andaman Islands region before the M 9.1 earthquake of December 26, 2004. We applied the epidemic type aftershock sequence (ETAS) models to the seismicity. The two-stage non-stationary ETAS model with a single change-point provides a better statistical fit to the seismicity data than the stationary ETAS model throughout the whole period. We made further change-point analysis of data sets by dividing into two sub-regions. The best fitted models suggest that the seismic activation relative to the ETAS rates started in the middle of July 2000 (about 4.5 years before the M 9.1 earthquake). This includes an increase in the background seismicity rates, particularly in the southern part of the seismogenic zone near the epicenter. A space-time ETAS model also suggests that the background seismicity throughout the entire Sumatra-Andaman Islands area had increased after the change-point time.

1 Introduction

1.1 Seismicity Anomalies

[2] The analysis of changes in seismicity prior to large earthquakes is an active area of current research, with many competing and often contradictory claims, largely based on retrospective case studies. For example, seismic quiescence (a reduction in seismicity rate) has been suggested as a potential intermediate-term earthquake precursor [Kisslinger, 1988]. Alternately, seismic activation (an increase in seismicity rate) has also been studied extensively, for example, in the M8 algorithm, in accelerating seismic “moment” release (AMR) models, and precursory localized swarms [Keilis-Borok and Malinovskaya, 1964; Bufe and Varnes, 1993]. Some models combine quiescence and activation (e.g., the pattern informatics method, Wu et al., [2008]).

[3] Seismic activation in the form of localized swarms, without an obvious large main event prior to the main shock, occasionally precedes larger earthquakes. For example, a swarm of activity that included two moderate earthquakes of magnitude 6.6 preceded the Kuril Island earthquake of November 15, 2006 of Mw 8.3 [Ammon et al., 2008]. Similarly, the April 6, 2009 Mw 6.3 L'Aquila earthquake was also preceded by a swarm, but the vast majority of such swarms did not end in a large earthquake, so specific actions such as evacuation of the population were rarely justified [van Stiphout et al., 2010]. There is no way of determining in real time which swarms or activations/quiescence will end in a large earthquake, and many large earthquakes occur without any obvious warning. Nevertheless, seismic anomalies do have demonstrable skill in terms of probabilistic earthquake forecasting, relative say to a stationary long-term hazard model based on Poisson recurrence [see Jordan et al., 2011].

[4] Before studying the seismic anomalies, we have to examine the homogeneity and completeness of catalogs particularly in case where the period is long. Magnitude shifts and incomplete detection of earthquakes around the threshold magnitude have been statistically studied [e.g., Habermann, 1991; Mulargia and Tinti, 1985]. We must be cautious to such artefacts that mask genuine seismic activity. Also, some researchers have argued that activation and quiescence may be contaminated by the earthquake clusters in space and time such as aftershocks and swarms. In the past, these anomalies were generally studied by first removing the aftershocks data from the catalog [Habermann and Wyss, 1984; Wyss et al., 1997]. However, in the absence of “parent” and “daughter” events being clearly and uniquely “marked” in natural data, it is very difficult to determine which individual events belong to a given cluster, and hence to de-cluster accurately — some correlated events remain.

[5] Ogata [1992] suggested accounting for aftershocks directly by using a stationary statistical model, the epidemic type aftershock sequence (ETAS) model, as a null hypothesis for detecting activation or quiescence as a non-stationary process. Ogata [2005] defined relative seismic quiescence and activation based on the deviation from the rate of seismicity predicted by the stationary ETAS model. The non-stationary ETAS models associated with change-points in its parameters have been associated both with main shocks and large aftershocks in some individual case studies [Ogata, 2011a]. Less work has been done on periods that do not end in large earthquakes, so we can hardly assign probabilities of a later large event occurring, not occurring (false alarm) or being missed. Nevertheless, such a possibility is shown in the comprehensive case studies of aftershock sequences in Japan [Ogata, 2001]. Also, the ETAS model proved effective in forecasting event rate in real time (on a daily scale) during the aftershock sequence of the recent L'Aquila earthquake [Marzocchi and Lombardi, 2009].

[6] Here we formally compare stationary and non-stationary ETAS models of the Sumatran seismicity to determine the significances of potential change-points occurring prior to the Sumatran mega-earthquake of December 26, 2004.

1.2 Great Earthquakes in the Sumatra-Andaman Islands Region

[7] The Sumatra-Andaman Islands region experienced a mega-thrust earthquake determined by the NEIC (National Earthquake Information Centre) to be magnitude Mw 9.1 at 30 km depth with hypocenter at 3.3 °N, 95.98 °E and origin time of 00:58:53 GMT on December 26, 2004. This was followed by the Mw 8.6 Nias-Simeulue earthquake with hypocenter 2.09 °N, 97.11 °E, at 16:09:36.53 GMT on March 28, 2005. Both earthquakes ruptured the boundary between the Indo-Australian and South-Eastern portion of the Eurasian plate. Its rupture length (~1500 km), average slip (~20 m), and duration (8 min) of December, 2004 are greater in term of duration than any other earthquake recorded in the digital instrumental period [Subarya et al., 2006]. The 2004 rupture stopped along strike beneath Simeulue Island, just at the northern edge of the 2005 Nias earthquake rupture (Figure 1), implying a structural barrier between the two ruptures at that region [Subarya et al., 2006]. The proximity in space and time of the two events and the prior identification of enhanced Coulomb stress change in the area of the 2005 event suggest that this event is likely to have been triggered by the 2004 event [McCloskey et al., 2005], although this can be hard to prove with just a single pair of events.

Figure 1.

The epicentral distribution of the earthquakes (m ≥ 4.7) in the Sumatran region. The red polygon denotes selection of the region for the ETAS modelling. The region is further divided into two sub-regions I and II by the dotted red segment. The slip distribution of 2004 and 2005 is shown as blue polygons (http://www.tectonics.caltech.edu/slip_history/index.html).

2 Data: Catalog Homogeneity and Completeness

[8] For the ETAS model, it is essential that the catalog should be homogeneous, i.e., without any systematic shifts in the magnitude scale due to anthropogenic effects that could propagate into apparent temporal variations in the model parameters. Many catalogs exhibit inhomogeneous shifts in the magnitude scale associated with changes in instrumentation or processing methodology. Using a single catalog, it is not easy to identify such systematic magnitude shifts uniquely. Instead, they are often found pragmatically by comparing different catalogs.

[9] Here we compared the NEIC magnitude (mb) with the ISC (International Seismological Centre) magnitude (Mb) for identical earthquakes in the Sumatra-Andaman Islands region. It is known that there is a systematic difference in magnitudes between the two catalogs [Utsu, 1982], but we are concerned with the fact whether such differences are time dependent or not. The difference in magnitudes for both the catalog is presented in Figures 2a and 3a, which show a transient shift takes place in the magnitude differences around 1996. Figures 2a and 3a suggest either the NEIC over-estimates or the ISC under-estimates sizes of earthquakes after that time.

Figure 2.

Time series of magnitude differences between NEIC (mb), ISC (Mb), and JMA (MJ) catalogs for the same earthquakes: (a) mb − Mb for the Sumatra region, (b) mb − Mb for the Japan region, (c) MJ − Mb for Japan region, and (d) MJ − mb for the Japan region. The horizontal red lines are mean differences during the earlier period till 8500 days. The vertical lines correspond to the time of the possible magnitude shift.

Figure 3.

Normalized cumulative functions of the magnitude differences (in order from the smallest to the largest) between NEIC (mb), ISC (Mb), and JMA (MJ) catalogs for the same earthquakes: (a) mb − Mb for the Sumatra region, (b) mb − Mb for the Japan region, (c) MJ − Mb for Japan region, and (d) MJ− mb for the Japan region. Gray and black cumulative function indicates those for the earthquakes before and after 8500 days. Black and grey stars indicate the corresponding averages of the magnitude differences.

[10] To establish which has changed, a third catalog is required. In the Sumatra region, we do not have any third catalog to investigate the magnitude shift. Therefore, we choose the Japan region as an analog, where the earthquake magnitudes are given by catalogs of NEIC, ISC, and the Japan Meteorological Agency (JMA). These data sets are downloaded separately from each site for the magnitude computed by individual agency. The JMA catalog has been revised (JMA, 2004) to modify the magnitude shift [e.g., Ogata, 1998a] that was mainly caused by the introduction of velocity-type seismographs for smaller earthquakes.

[11] The NEIC and ISC catalogs have the same magnitude-difference shift around 1996 for the Japan region also (Figures 2b and 3b), consistent with a similar change for the Sumatra region (Figures 2a and 3a). The comparison of the ISC and JMA magnitudes in Figures 2c and 3c show the same shift whereas comparison of the NEIC and JMA in Figures 2d and 3d does not show such a shift. Therefore, the ISC catalog is very likely to have a magnitude shift after 1996, whereas the NEIC catalog maintains the same magnitude differences on average from the JMA magnitudes throughout the whole period. We conclude that the NEIC is less susceptible to magnitude shift issues for all the time period and hence used it in this study.

[12] To further explore the reasons of this change, we studied the magnitude of completeness of the ISC with time and found that the magnitude of completeness of the ISC catalog decreased from 4.5 to 4.2 in the period from 1994 to 1999. The lower magnitude of completeness falls in the period of observed magnitude shift, and it is likely that underestimation of the magnitudes took place during this time. The lowering of the ISC magnitudes in the year 1996 is due to the use of IDC (International Data Centre) amplitude data of the Comprehensive Nuclear Test Ban Treaty Organization (CTBTO) [James W. Dewey, April 1, 2011, personal communication]. The NEIC discontinued the use of the IDC data because the IDC amplitudes were obtained with a different amplitude-measuring procedure than used at the NEIC and implied systematical lowering of the magnitude using IDC amplitudes.

[13] Therefore, we use data from the NEIC catalog from January 13, 1973 to December 25, 2004 (Figure 1). We consider the red polygon region for the ETAS modelling. We used higher (commonly accepted) magnitudes from the NEIC catalog. The body wave magnitude (mb) and surface wave magnitudes (Ms) can be analytically converted to Mw. However, Kagan [1991] found inconclusive results, and Kagan and Jackson [2010] used maximum magnitude for long and short term earthquake forecast in California and Nevada. We also use maximum magnitude without converting mb/Ms to Mw for this study.

[14] There are many methods to detect the magnitude of completeness of a catalog which can be divided into two classes: (1) based on the earthquake catalogs and (2) wave-form [Schorlemmer and Woessner, 2008]. We computed the Mc based on the maximum curvature method [Wiemer and Wyss, 2000] and stability of b values [Cao and Gao, 2002] and found the magnitude of completeness as 4.7. Woessner and Wiemer [2005] presented a comparative study of different methods for estimation of Mc and found that a stable Mc can be found for the larger size ≥ 200 whereas, in this study, the number of events M ≥ 4.7 is greater than 500.

[15] We also estimated the magnitude of completeness with time for checking the magnitude of completeness throughout the catalog period. The data set is divided almost in five years intervals to study the magnitude of completeness. The magnitude of completeness has improved from 4.7 to 4.5 during 1996–2000. Therefore, for consistency, we used an Mc of 4.7 with the completeness. Dewey et al. [2007] also find the magnitude of completeness of 4.7 from the NEIC catalog since 1978.

[16] We also looked at the magnitude frequency distribution in the divided periods as shown in Figure 4. This is because we will investigate the change of seismic activity before and after the middle of year 2000. Figure 4 confirms that we can apply the ETAS model for the earthquakes with the threshold Mc = 4.7. From Figure 5, we see that the proportion of the cumulative number of earthquakes of each threshold magnitude or larger appears to be nearly invariant with time throughout the whole period. Also, regardless of the threshold magnitude, we see that the occurrence rate became higher after July 2000.

Figure 4.

Circles indicate the earthquake frequencies for period 1973–1999, and gray dots indicate the earthquake frequencies for period from 2000 till December 25, 2004. The black and gray step functions are the cumulative numbers of the earthquakes in the corresponding period. The vertical lines show the magnitude of completeness.

Figure 5.

Cumulative numbers of the earthquakes with the different cut-off magnitudes plotted against the time. The colors of the cumulative curves are indicated in the legend. The inset diagram shows the normalized cumulative curves corresponding to those of the same colors as in the main panel.

3 The ETAS Model

3.1 Definitions and Implication of the Parameters

[17] The ETAS model describes earthquake activity as a point process [Ogata, 1988]. The model assumes that the background seismicity is a stationary Poisson process with a constant occurrence rate or number of earthquakes per day μ. The conditional intensity function of the process is described by

display math(1)

where Ht = {(ti, Mi); ti < t} is the history of the occurrence times and magnitudes of earthquakes before time t, and K, α, c, and p are constants.

[18] The parameter K is a “productivity” factor for the aftershock event rate, conditional on the occurrence of a triggering earthquake or main shock. It is proportional to the expected number of aftershocks of magnitude greater than the threshold magnitude. Physically, the factor c (day) is a characteristic time, allowing a finite rate of aftershocks at the origin time of the triggering event. In practice, the fitted values for c are more likely to be caused by under-reporting of small events hidden in the wave train of larger ones close to the origin time [Utsu, et al., 1995; Helmstetter et al., 2006]. The p is the power-law exponent of the event rate decay in the Omori-Utsu law. The magnitude sensitivity parameter α (magnitude−1) accounts for the efficiency of an earthquake of a given magnitude in generating aftershocks. The parameter α allows a small but finite chance of a smaller event occasionally triggering a larger event in this model, so the term “aftershock” here is not used in its conventional sense — rather it means any triggered event, irrespective of the sign of the magnitude difference.

[19] The temporal ETAS model is also very useful for quantifying the activation and quiescence in the different regions of the world [Kumazawa et al., 2010; Ogata, 2011a; Bansal and Ogata, 2010; Bansal et al., 2012].

3.2 Parameter Estimation and the Time Transformation

[20] We estimate the ETAS parameters using maximum likelihood estimation where the log-likelihood function,

display math(2)

is maximized with respect to the parameters θ = (μ, K, c, α, p). Here “ln” is a natural logarithm and {(ti, Mi), Mi ≥ Mc; i = 1, 2, …} are the data consisting of occurrence times and magnitudes of earthquakes in the target time interval [S, T]. Then we can see how well or how poorly the model fits an earthquake sequence by comparing the cumulative number N (S, t) of earthquakes with the rate predicted by the model

display math(3)

in the time interval S < t < T. If the earthquakes in the catalog are described well by the ETAS model, then transformed times τi (which stretches the time to correct for the Omori law decay) of the earthquakes will be distributed according to a stationary Poisson process, and a plot of the actual cumulative number of events versus the transformed time should be close to linear [Ogata, 1988]. The transformed time {τi} can be defined as τi = Λ(ti) and useful to judge goodness-of-fit of the ETAS model because it gives a visual check of the fit to a stationary Poisson process. Anomalous seismicity not explained by the stationary ETAS model will appear as systematic deviations from this trend.

3.3 Error Analysis in ETAS Parameters

[21] Error analysis in the ETAS parameters is non-trivial, and the optimal method is the subject of some debate [Kagan et al., 2010]. The Hessian matrix of the negative log-likelihood can be used as a first order estimate of the parameter error. A full Monte-Carlo simulation is technically a better solution in that a full probability density function is recovered [Wang et al., 2010a], but the computational time can be prohibitive. Hence, in this study, we calculated the standard errors in the ETAS parameters from the Hessian matrix. In this approach, standard errors are the square roots of the diagonal elements of the covariance matrix of parameter estimates.

[22] However, the ETAS parameters are not independent of each other; therefore, the joint distribution of their errors is correlated. Hence, the corresponding standard error of the ETAS parameters in Tables 1-3 are of the marginal distributions and each of these is considerably larger than the standard error under fixing other parameters. For a comparison, we add another type of error estimates in Tables 1 to 3, which is the square-root of the reciprocal of each diagonal element of the Hessian matrix. Namely, this is conditional estimate on that the remaining other parameters are fixed at their maximum likelihood estimate (MLE) values.

Table 1. Comparison of Stationary and Change-Point Models in the Pre-main Shock Period for the Selected Polygon Region Before and After the Change-point
Periodμ (per day)Kc (days)αpAIC
  1. ΔAIC = 3934.468 + 1004.555 + 2×5.46 − 4952.682 = −2.739. This negative value indicates that the change-point model is preferred by the information criterion as defined in Appendix. Here, 1280, 10,060, and 11,681 days correspond to July 4, 1976, July 18, 2000, and December 25, 2004, respectively. The two types of error estimates are given. The upper row gives by the square-root diagonal elements of the inverse Hessian matrix of the negative log-likelihood, and the values with brackets in the lower row gives the square-root of the reciprocal of each diagonal element of the Hessian matrix. See Section 3.3 for the reason of displaying the two types of errors.

1280–11,6810.034920.016340.005221.413600.946404952.682
 ±0.00891±0.00587±0.00288±0.13228±0.05944 
 (±0.00240)(±0.00106)(±0.00038)(±0.06748)(±0.00884) 
1280–10,0600.048680.012080.011501.187101.149003934.468
 ±0.00634±0.00474±0.00816±0.19471±0.17695 
 (±0.00253)(±0.00140)(±0.00145)(±0.12714)(±0.03486) 
10,060–11,6810.066130.012480.008341.781701.055401004.555
 ±0.01583±0.00679±0.00632±0.18536±0.10589 
 (±0.00756)(±0.00168)(±0.00121)(±0.09278)(±0.02683) 
Table 2. Comparison of Stationary and Change-Point Models in the Pre-mainshock Period for Sub-region I
Periodμ (per day)Kc (days)αpAIC
  1. ΔAIC = 3292.566 + 772.5368 − 4068.438 = −3.3352.

1280–11,6810.030850.014850.007601.594801.008404068.438
 ±0.00535±0.00516±0.00380±0.14067±0.06114 
 (±0.00210)(±0.00114)(±0.00064)(±0.07264)(±0.01280) 
1280–10,0600.035660.013880.010631.293101.092603292.566
 ±0.00550±0.00578±0.00707±0.22762±0.12893 
 (±0.00223)(±0.00160)(±0.00132)(±0.13604)(±0.02738) 
10,060–11,6810.038260.017800.009091.677201.04910772.5368
 ±0.01228±0.00963±0.00665±0.17445±0.08784 
 (±0.00620)(±0.00224)(±0.00122)(±0.08967)(±0.02307) 
Table 3. Comparison of Stationary and Change-Point Models in the Pre-main Shock Period for Sub-region II
Periodμ (per day)Kc (days)αpAIC
  1. ΔAIC = 1202.079 + 407.382 − 1613.786 = −4.325. The model of the latter period is stationary Poisson.

1280–11,6810.010480.007720.000531.284200.843451613.786
 ±0.00459±0.01241±0.00139±0.35788±0.16404 
 (±0.00117)(±0.00160)(±0.00013)(±0.19058)(±0.02131) 
1280–10,0600.011040.005070.001301.435500.991571202.079
 ±0.00189±0.00753±0.00286±0.34808±0.18406 
 (±0.00118)(±0.00160)(±0.00047)(±0.23291)(±0.05493) 
10,060–11,6810.02714407.3816
 ±0.00409     

[23] In any case, our analysis relies on the likelihood ratio (or the AIC difference) which is most sensitive to discriminate the difference between such models with many parameters. The application of ETAS model to the Sumatran seismicity before the mega event is presented below for the red polygon shown in Figure 1.

4 Seismic Activity Preceding the Mega-event

[24] The cumulative number of events versus ordinary time and the transformed time for polygon region is presented in Figures 6a and 6b, respectively. The MLEs of the parameters inferred from the ETAS model from fitting the rate data are also shown on the Figure 6a, as well as listed in the first row of Table 1. From Figure 6 we can see that the fitted ETAS curve one-sidedly biased from the observed curve, indicating but not proving a non-stationary change in seismicity parameters. Accordingly, we then examine the data formally for another possible change-points at some points between S and Tend on this plot.

Figure 6.

(a) Cumulative number of events and a plot of earthquake magnitude versus ordinary time with fitted ETAS model pre-main shock period for the whole polygonal region. The red curve represents the theoretical cumulative function calculated by the best fitted stationary ETAS model. (b) The same plots with the transformed time in the horizontal axis.

[25] Here we took S = 1280 days (corresponding to July 4, 1976) to include the M 7.0 earthquake of June 20, 1976 in the period (0, S). We need a proper period (0, S) for a history HS of the occurrences, including a large earthquake at least, to maintain stationarity of the estimated ETAS model. Otherwise, applying the ETAS model to all the data in the [0, T] period without occurrence history, the process becomes unnatural nonstationary in the early days.

[26] First, we examined the likelihood of candidate change-points Tc (see Appendix) for every one month (30 days) in normal time for the whole polygon region. The green colored function in Figure 7 shows the normalized likelihood function with respect to different change-point candidates, which implies that the MLE of the change-point for the whole region data is inline image = 10,060 days that corresponds to July 18, 2000, and further that the maximum of the likelihood of the change-point is beyond the green horizontal dashed line of the significance. This change-point model is preferred over the best fit stationary model with ΔAIC = −2.74 (see Table 1). The method for calculating ΔAIC is presented in Appendix. The estimated ETAS parameters are presented in Table 1. The first, second, and third rows of Table 1 give the ETAS parameters for the whole time period (S, Tend), up to the MLE of change-point (S, inline image), and from the MLE to the end of time period (inline image, Tend), respectively.

Figure 7.

The normalized likelihood function against time of change-point candidates for the whole region (green), Region 1 (red) and Region 2 (blue). The horizontal dashed lines are the respective significance levels of the change-points for the corresponding regions with the same colors, which correspond to q(N) presented in Appendix. The vertical dashed line indicates the maximum likelihood estimate of the change-point for the whole region.

[27] Thus, we found the best fitting single change-point model at July 18, 2000 (see Table 1), indicating a seismicity rate change there. Hence, we compare the theoretical and empirical cumulative function based on the ETAS for the period [S, inline image] = [June 24, 1976 to July 18, 2000] days, using the MLE parameters in the second row in Table 1, which is then extended for the remaining period [inline image, Tend] = [July 18, 2000 to December 25, 2004] days using the function (3) that integrates the rate of the estimated ETAS model of the first interval.

[28] In Figures 8a and 8b, the ETAS model appears to be well adapted during the period up to the MLE change-point inline image, but after that, the empirical cumulative curves deviate upward from the two-fold error parabola. Here the parabolic confidence limits in the figure are shown as two-fold standard deviations of the Brownian process from which we see that the 95% error bounds, for example, is given by((τ − 2σ, τ + 2σ) at time τ since Λ (S, inline image) where inline image taking the model estimation error into consideration [see Ogata, 1992, Appendix B].

Figure 8.

(a): Top panels show the estimated cumulative curves of the ETAS model to the data from the whole region for the period up to the MLE for change-point (red vertical dashed line) and the extrapolated ones to the rest of the period, against (a) ordinary time and (b) the transformed time. The 95% confidence parabola is shown in (b). The middle (c, d) and bottom (e, f) panels are for the data from sub-region I and II (see Figure 1), respectively; these are the same as the top panels except that the change-points are set to the same change-point as the MLE for the whole region data.

[29] One may argue that selection of a large area may create a bias, producing an artificial non-stationary model. Ogata [2011a] suggested selection of area based on the clustering of the earthquakes containing sufficient events having less interaction with other areas. We divided the region in Figure 1 into the two sub-regions marked as I and II, taking account of the rupture zones of the 2004 Sumatra and 2005 Nias earthquake.

[30] From Figure 7, we also see the likelihoods of the change-point candidates in the sub-regions I and II, and likely change-points vary around their own MLE, 10,090 days and 9850 days, respectively. However, comparing these with the cases of the reference change-point inline image = 10,060 days fixed (see Appendix) gives better fits in the sense of the AIC for both sub-regions. Hence, we adopt the common change-point for all the considered regions, and the MLE change-point of the whole region is included in the likely range of change-points for both sub-regions. Furthermore, ΔAIC of both regions improves if we assume the change-point is the same as the MLE change-point of the whole region. Therefore, we adopt this timing as the change-point for both data from the regions I and II. Tables 2 and 3 correspondingly provide the ΔAIC values and the ETAS estimates of whole period and the divided period in the respective sub-regions.

[31] Figures 8c–f show the best fitted ETAS cumulative curves for the sub-region I and II before the change-point and extrapolated ones for the period after the change-point. The computed ETAS parameters are shown in Tables 2 and 3. The change-point corresponding to July 18, 2000 provided a better fitted nonstationary ETAS model based on the ΔAIC values in the tables. In Figures 8c–f, we can see that the predicted seismicity in each sub-region is beyond the 95% confidence bound, indicating the relative activation.

[32] Finally, we can see that, in region II, the change-point is associated with an increasing background intensity rates according to the estimated μ-values before and after the change-point in Table 3.

[33] Incidentally, we should note that the 2002 earthquake of M 7.4 induced the unusually large size of aftershocks. So, some people may wonder whether the seismicity change may have been only caused by a major shock producing an anomalously vigorous aftershock sequence. Hence, we have made a similar analysis by fitting the ETAS model changing the magnitude of the 2002 earthquake (Figure 9). Note here that this magnitude change is equivalent to searching the optimal parameters K and α in the ETAS model (1) only for this particular earthquake. The maximum log-likelihood was attained by M = 8.8. Still, Figure 9 suggests the increase in activity starting at July 2000.

Figure 9.

The estimated cumulative curves of the ETAS model to the data from the whole region for the period up to the MLE for change-point (red vertical line) and the extrapolated ones to the rest of the period, against (a) ordinary time and (b) the transformed time. The 95% confidence parabola is shown in (b). Here, the magnitude of the 2002 earthquake was adjusted to M = 8.8 from M 7.4 in the original catalog to avoid the bad fit to the aftershock of the 2002 event. See text for the reason.

5 Space-Time Analysis Before the Sumatra Earthquake

[34] The focal region of the Sumatra earthquake is fairly large with 1500 km length and 150 km width, and therefore, one may be concerned with spatial heterogeneity of the seismic activity in this wide region. There have been many works on the regional variation of seismicity such as the b-value of Gutenberg-Richter [1944] and p values of the Omori-Utsu aftershock decay parameter [Utsu, 1961, 1969; Mogi, 1967].

[35] As discussed in the last paragraph, aftershock number sizes may be very different from place to place even if the magnitude of the triggering earthquakes is similar one. Also, the mainshock-aftershock-type clusters and swarm-type clusters have very different pattern of activities. Therefore, we consider the following space-time model to apply earthquakes in the whole region in Figure 1.

5.1 Space-Time Modelling and the Background Seismicity

[36] We therefore consider a space-time ETAS model in which parameter values μ, K, α, p, and q-parameters can vary, depending on the location (x, y), but we assume that c and d values are common throughout the region. Precisely, consider the space-time occurrence rate conditioned on the occurrence history up to time t such that

display math(4)

[37] where Sj is a normalized positive definite symmetric matrix for anisotropic clusters, such that

display math(5)

[38] Here (inline image) and the coefficients of Sj for a selected set of large earthquakes j (specifically, Mj ≥ 5.7) is identified by fitting a bivariate normal distribution to spatial coordinates of the cluster occurring within a square of inline image km side-length and within inline image days after the large event of magnitude Mj, according to the empirical study of Utsu [1969]. All the other events including the cluster members remain the same as the epicenter coordinate of the original catalog associated with the identity matrix for Sj, namely, σ1 = σ2 = 1 and ρ = 0. For detail of the algorithm, see Ogata [1998b, 2011b] and Ogata et al. [2003]. We call the model the hierarchical space-time ETAS (HIST-ETAS) model.

[39] Thus, we need to estimate coefficients of parameter functions in (4) that are defined at the epicenter locations and some additionally taken points on the region boundary. For a stable estimation, we need to constrain the freedom of coefficients assuming the smoothness of the functions, namely, to give penalties to the gradients against the roughness of functions. The coefficient functions are piecewise linear surfaces interpolated on the Delaunay tessellated triangles (see the last panel in Figure 10) that are uniquely determined by the epicenters. We then seek coefficients that maximize the penalized log-likelihood in which optimal weights of penalties are objectively tuned by an empirical Bayesian method that implies the maximum posterior solution of the space-time ETAS coefficients, as described in Ogata et al. [2003] and Ogata [2004, 2011b].

Figure 10.

The optimal maximum posterior estimates of respective parameter functions of the hierarchical space-time ETAS model (see text) that are applied to the earthquakes of M 4.7 or larger during the period from 1974 to December 25, 2004. The colors represent the estimated coefficient values of the parameter functions μ, K, α, and p. Here q was almost constant 2.37. The dimension of μ and K is the number of events per unit square degree and day. The last panel illustrates the Delaunay tessellation used for the smoothing penalties of the HIST-ETAS model (see text).

[40] Let λ(t,x,y|Ht) be the theoretical occurrence rate of the model (4). Let the coordinates of earthquakes be (ti, xi, yi, Mi), i = 1,2,…, n. The stochastic de-clustering algorithm [Zhuang, et al., 2002, 2004] provides a stochastic realization of background earthquakes by choosing each earthquake i with probability μ(ti,xi,yi)/λ(ti,xi,yi|Hti) otherwise rejecting that earthquake. Therefore, the realized configuration of background earthquakes depends on random numbers, but the expected local density of points remains the same in space and time. In reality, given an earthquake, the identification of whether it is categorized as a background or a cluster member is stochastic, which varies depending on the probability calculated from the fitted model. We will further use the stochastic de-clustering algorithm for analysing the change in the background seismicity.

5.2 Application to the Sumatra-Andaman Islands Area

[41] We applied the HIST-ETAS model to all earthquakes of M 4.7 and larger within the rectangular area bounded by 2°S–15°N, and 90°–100°E as shown in Figures 10a and 11a. Incidentally, we can obtain contour images and color images on the lattice of these parameters covering the whole area by the interpolation of the Delaunay triangles of the nearest three earthquake locations. The variations of μ, K, α and p in space using HIST-ETAS model are presented in Figure 10. Figure 11b shows latitude versus time plots of the earthquakes. Figure 11c shows the cumulative number and magnitude of earthquakes against the ordinary time.

Figure 11.

All earthquakes of magnitudes of 4.7 and larger in Sumatra-Andaman Island region (a–c). The background earthquakes (d–f) were obtained by stochastic de-clustering (see text) from the above data using the HIST-ETAS model. Epicenters (a, d), latitude versus ordinary time (b, e), and the cumulative functions and magnitudes versus the ordinary time (c, f).

[42] As described in the last paragraph of the previous section, stochastic de-clustering is applied to the data for obtaining stochastic realization of background seismicity. Then, Figures 11d–f show the figures for the de-clustered catalog corresponding to those in Figures 11a–c, respectively. Particularly, the cumulative function of the de-clustered earthquake data in Figure 11f suggests that the rate of de-clustered events for background activity in the whole region seems to have increased after the middle of July 2000 (around 10,060 days).

[43] Therefore, we examined whether we can see similar background rates increase throughout the region. Figure 12 shows that the increases of the background activities after middle of July 2000 on the two-fold standard error parabola throughout the whole region. Here the parabolas after the change-point are given as the same way as given in Figure 8. Furthermore, the sausage-shaped error bounds curves for the period before change-point is two folds standard deviations calculated by the Brownian Bridge for the fitted period by following asymptotic theory [Ogata, 1992]. The cumulative number function for the standard stationary Poisson process converges to the standard Brownian Bridge at fixed t (0 < t < Tc) with mean and standard deviation equal to t and inline image, respectively [Billingsley, 1961; Ogata, 1992].

Figure 12.

Cumulative functions of the background earthquakes from the regions within the indicated latitude-ranges in the legend. The dashed sausage-shaped curves and solid parabolas show two-fold of the standard errors assuming the stationary Poisson process as explained in the text.

6 Discussion

[44] Some papers interpret that the variation of global seismic rate at intermediate time-scale can be explained by short-range earthquake triggering with nearly stationary Poisson background seismic rate [Kagan and Jackson, 1991; Parsons, 2002]. The space-time ETAS model [Ogata, 1998b] and its hierarchical extension in Section 5 are models that can help quantifying such evidences. On the other hand, the global and Japanese earthquake sequences over a century exhibit long-range correlation by a nonparametric statistical analysis [Ogata and Abe, 1991]. Relevant issue is discussed by Lombardi and Marzocchi [2007] and Marzocchi and Lombardi [2008] suggesting that an ETAS model with a background varying with time can be considered statistically reliable to describe the seismicity distribution over a wide space-time-magnitude window.

[45] As such a concrete example, we have objectively found a relative activation associated with non-stationary changes in the ETAS model at about 4.5 years prior to the Sumatra earthquake of 2004. The activation was associated with an increase in the background rates across the considered region, which is particularly conspicuous in the region II.

[46] In this article, we did not seek the geophysical reason for the activation, in the background rate in particular. Incidentally, Llenos et al. [2009] discussed variations in seismicity rate that are triggered by transient aseismic processes, and found that the transient stressing rates are responsible for increasing the ETAS background seismicity rate without affecting aftershock productivity. Such a study to explain the activation relative to the ETAS rates will be required in very near future using some other geophysical records.

[47] The computation of the ETAS parameters will also be affected by the missing events before and within the catalog because of interaction of these events with other events [Wang et al., 2010b]. We take care of the homogeneity of the catalog for fitting the ETAS model but still missing events in the mid of catalog specially after a large earthquake cannot be avoided. This would have a slight effect on the estimation of the ETAS parameters, at least the scaling parameter c [Utsu et al., 1995; Helmstetter et al., 2006; Wang et al., 2010b] but would not affect the goodness-of-fit and the change-point analysis as shown in the figures. Overall, we conclude that the step change-point model is likely to be more statistically robust, although the 30 years data are relatively short to establish general phenomena regarding prediction of large earthquakes in this region.

7 Data and Resources

[48] The National Earthquake Information Center (NEIC), USGS catalog is used for this study (Figure 1) (http://earthquake.usgs.gov/earthquakes/eqarchives/epic/). Further, this data set is compared to the international seismological center (ISC) catalog (http://www.isc.ac.uk). The Japan Meteorological Agency (JMA) data for Japan region (20°–50°N and 120°–150°E) is obtained from the JMA as well as the data are used from NEIC and ISC catalog for this region also. The ETAS model parameters are computed using the Statistical Analysis of Seismicity software package (Ogata, 2006), the Institute of Statistical Mathematics, Tachikawa. Some of the statistical analysis is carried out by using the Statistical seismology library (SSLIB) (http://homepages.paradise.net.nz/david.harte/SSLib/) and public domain software R (http://www.r-project.org/). The first figure is made using the Generic mapping tool (http://gmt.soest.hawaii.edu/) and remaining figures are made using the public domain software R (http://www.R-project.org).

Appendix A: Search for Change-Point of Seismicity

[49] The Akaike information criterion (Akaike, 1974)

display math(A1)

is useful to compare the goodness-of-fit of the competing models to a given data set. Here, dim{θ} means the number of adjusted parameters which is 5 if all of the purely temporal ETAS parameters are adjusted. The model with a smaller AIC value shows a better fit to the data. To examine whether or not the temporal seismicity pattern changed at a suspected time Tc on a time interval [S, Tend] in a given data set starting at time S, we consider the following two-stage ETAS model with a change point, applied to the occurrence data sets on the separated sub-intervals [S, Tc] and [Tc, Tend] in order to calculate the corresponding AICs,

display math(A2)

and

display math(A3)

respectively. Then, we compare AIC1 + AIC2 with

display math(A4)

for the single ETAS model for the whole period [S, Tend]. Thus, we compare AIC0 with AIC1 + AIC2 in case where the change-point Tc is given by different information from the data.

[50] To validate the significance of the seismicity change in the case where we also search for the most likely change-point, we compare AIC0 with AIC1 + AIC2 + 2q(N) where q(N) is the penalty value corresponding to the adjusted candidate change-point time Tc since we actually search for Tc that minimizes AIC1 + AIC2. Here, the penalty q(N) varies monotonically from 3.5 to 5.5 depending on the total number of events N from 10 to 2000 in the interval [S, T] (see Kumazawa et al., 2010, for the explicit function form of q(N), while the ordinary penalty in AIC for each parameter of the ETAS model is unity whereas earlier Ogata [1999] suggested q(N) value varies from 2 to 3. Kumazawa et al. [2010] have shown that the average value of the penalty function can be represented as the ratio of polynomials (Kumazawa et al., 2010, eq. A3):

display math(A5)

[51] The penalty function q(N) for this comparison is derived by Monte-Carlo simulations taking account of the over-fitting bias of the likelihood ratio statistics due to the greater complexity of the two-stage model [Ogata, 1992, Kumazawa et al., 2010]. If ΔAIC = AIC1 + AIC2 + 2q(N) − AIC0 takes a positive value, this indicates that the single ETAS model is selected for the seismicity in the whole period. If ΔAIC takes a negative value, the seismicity change is regarded to be better fitted (e.g., see Tables 1–3).

[52] To summarize the full change-point analysis, we calculate the likelihood of the two-stage ETAS model for different change-point candidate Tc, which is expressed in terms of the AIC by exp{−ΔAIC(Tc)/2} [e.g., Akaike, 1980]. The MLE of the change-point Tc maximizes the likelihood and we searched in a interval of 30 days.

Acknowledgments

[53] This work was made possible by the award of a fellowship from JSPS (Japan Society for the Promotion of Science) to A.R.B. A.R.B. also thanks Director NGRI, Hyderabad and CSIR (Council of Scientific and Industrial Research), New Delhi for formal permission to visit the Institute of Statistical Mathematics in Tokyo. A.R.B. is also supported by grant entitled HEART funded by CSIR. Y.O. is supported by the JSPS under a Grant-in-Aid for Scientific Research 20240027 and 23240039. Y.O. is also supported by the Aihara Innovative Mathematical Modelling Project, the “Funding Program for World-Leading Innovative R&D on Science and Technology (FIRST Program),” initiated by the Council for Science and Technology Policy. We are also thankful to Professor Ian Main of the University of Edinburgh for useful discussion during this study and introducing ARB to statistical seismology. Finally, we are thankful to the AE and the anonymous reviewers for critical comments and suggestions which have significantly improved the article.

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