Geophysical Journal International

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1 Geophysical Journal International Geophys. J. Int. (2013) 194, Advance Access publication 2013 June 17 doi: /gji/ggt194 Interevent times in a new alarm-based earthquake forecasting model Abdelhak Talbi, 1,2 Kazuyoshi Nanjo, 1 Jiancang Zhuang, 3 Kenji Satake 1 and Mohamed Hamdache 2 1 Earthquake Research Institute, University of Tokyo, Yayoi, Bunkyo-ku, Tokyo , Japan. talbi@eri.u-tokyo.ac.jp 2 Département Etude et Surveillance Sismique, Centre de Recherche en Astronomie Astrophysique et Géophysique (CRAAG), BP 63 Route de l observatoire, Bouzaréah 16340, Algiers, Algeria 3 Institute of Statistical Mathematics, 10 3 Midori-Cho, Tachikawa, Tokyo , Japan Accepted 2013 May 9. Received 2013 May 8; in original form 2012 October 12 SUMMARY This study introduces a new earthquake forecasting model that uses the moment ratio (MR) of the first to second order moments of earthquake interevent times as a precursory alarm index to forecast large earthquake events. This MR model is based on the idea that the MR is associated with anomalous long-term changes in background seismicity prior to large earthquake events. In a given region, the MR statistic is defined as the inverse of the index of dispersion or Fano factor, with MR values (or scores) providing a biased estimate of the relative regional frequency of background events, here termed the background fraction. To test the forecasting performance of this proposed MR model, a composite Japan-wide earthquake catalogue for the years between 679 and 2012 was compiled using the Japan Meteorological Agency catalogue for the period between 1923 and 2012, and the Utsu historical seismicity records between 679 and MR values were estimated by sampling interevent times from events with magnitude M 6 using an earthquake random sampling (ERS) algorithm developed during previous research. Three retrospective tests of M 7 target earthquakes were undertaken to evaluate the long-, intermediate- and short-term performance of MR forecasting, using mainly Molchan diagrams and optimal spatial maps obtained by minimizing forecasting error defined by miss and alarm rate addition. This testing indicates that the MR forecasting technique performs well at long-, intermediate- and short-term. The MR maps produced during longterm testing indicate significant alarm levels before 15 of the 18 shallow earthquakes within the testing region during the past two decades, with an alarm region covering about 20 per cent (alarm rate) of the testing region. The number of shallow events missed by forecasting was reduced by about 60 per cent after using the MR method instead of the relative intensity (RI) forecasting method. At short term, our model succeeded in forecasting the occurrence region of the 2011 M w 9.0 Tohoku earthquake, whereas the RI method did not. Cases where a period of quiescent seismicity occurred before the target event often lead to low MR scores, meaning that the target event was not predicted and indicating that our model could be further improved by taking into account quiescent periods in the alarm strategy. Key words: Persistence, memory, correlations, clustering; Probabilistic forecasting; Earthquake dynamics; Earthquake interaction, forecasting, and prediction; Statistical seismology. GJI Seismology 1 INTRODUCTION One of the main purposes of seismology is to enable the prediction of the timing, location and magnitude of future earthquakes, although predictions of magnitude are not necessary if we focus on future seismic events with magnitudes above a given threshold; consequently, earthquake forecasting can focus on spatial and temporal constraints. Mogi (1969) undertook an early attempt to define seismicity as a possible precursor for large earthquakes, suggesting that areas subjected to large earthquakes underwent a period of low seismicity for several to 20 yr prior to the large seismic event, while contemporaneously the region surrounding this area of low seismicity was markedly active, with widespread foreshock activity (in the broad sense) prior to the large seismic event. This suggests that earthquakes have a doughnut-shaped spatial distribution, with the inner part of the Mogi doughnut being an area of low seismicity surrounded by a ring of increased seismicity, with an eventual very large earthquake in the very centre of the doughnut. More recently, a number of researchers have investigated periods of seismic quiescence and discussed the possibility of earthquake forecasting as C The Authors Published by Oxford University Press on behalf of The Royal Astronomical Society. 1823

2 1824 A. Talbi et al. part of a resurgence in seismicity-based precursory studies (e.g. Habermann 1988; Wyss & Habermann 1988; Kagan & Jackson 1991; Wyss & Martirosyan 1998; Wyss & Wiemer 1999; Hainzl et al. 2000; Shearer & Lin 2009; Katsumata 2011). In comparison, seismic activation prior to large earthquakes has been reported as an increase in intermediate-size earthquakes (e.g. Sykes & Jaume 1990; Knopoff et al. 1996; Rundle et al. 2000). In addition, previous studies have identified precursory foreshock activity prior to large earthquakes (e.g. Papazachos 1975; Kagan & Knopoff 1978; Jones & Molnar 1979; Papadopoulos et al. 2010), suggesting that seismic activation prior to large events and foreshocks may represent a unique precursor process involving either a temporally short cluster of intermediate-size earthquakes, or alternatively an increase in seismicity distributed over a relatively long period of time. In practice, a number of precursors may be observed prior to some earthquakes, whereas other earthquakes have no known precursor events. Foreshocks, pre-shocks and quiescence are among seismicity precursors that have been evaluated and been considered as significant (Wyss 1991, 1997a,b), with the presence of these precursor events suggesting that changes in spatio-temporal seismicity patterns before large earthquakes could be identified. In particular, a statistical analysis of the distribution of the time elapsed between successive earthquakes, hereafter referred to as interevent times, allows the derivation of useful information that can allow the development of earthquake forecasting strategies, primarily as such data enable an evaluation of the statistical probability of the time elapsed between preceding and following earthquakes. However, the number of large events is too small to allow an accurate estimation of the interevent time distribution. The failure of the Parkfield experiment (Lindh 2005; Jackson & Kagan 2006) highlights the confusion that can be caused by extrapolation from incomplete interevent data and the necessity to account for highly variable interevent times produced by interactions of earthquakes on neighbouring faults (Nature debate on earthquake forecasting, Nevertheless, a number of differing distributions have been proposed to account for interevent time variability, including exponential, lognormal, inverse Gaussian, Gamma and Weibull distributions (e.g. Hagiwara 1974; Nishenko & Buland 1987; Matthews et al. 2002; Zoller & Hainzl 2007). Seismicity-based earthquake forecasting methods are currently thought to be promising tools that could effectively tackle earthquake prediction issues (Bormann 2011). Pattern informatic (PI) and relative intensity (RI) methods (Rundle et al. 2002, 2003; Tiampo et al. 2002a,b; Holliday et al. 2005) have been tested in a number of regions worldwide (Tiampo & Shcherbakov 2013 and references therein). The formulation of seismicity forecasting methods is generally based on quantifying one or more precursory variables; for example, the PI method uses a principal component analysisderived measurement to detect seismic quiescence and activation, whereas the RI method assumes that earthquakes are likely to occur in regions with elevated previous levels of seismicity, using a simple spatial distribution of past earthquakes to forecast areas where future larger earthquakes may occur, with high forecast scores associated with areas of seismic activation. The fact that earthquakes occur in spatial and temporal clusters allows us to use clustering as a detection method to identify precursor patterns. For example, earthquake clustering statistics were used to detect Mogi-like doughnut behaviour in small earthquakes (Shearer & Lin 2009). When background events are modelled by a homogeneous Poisson process, we can use the change in background seismicity that may occur prior to large earthquake events to detect seismic quiescence and activation. Furthermore, changes in background seismicity levels are at least implicitly used by a number of earthquake forecasting methods, including pattern-recognition (Keilis-Borok et al. 1988) and M8 (Keilis-Borok & Kossobokov 1990; Kossobokov et al. 1999) algorithms, and PI and RI methods. However, each forecasting method uses differing methods to estimate the rate of background seismicity, meaning that estimating background seismicity is problematic. The majority of researchers suggest that background seismicity is described by a homogeneous Poisson process with a constant rate that can be calculated using a declustering algorithm (Gardner & Knopoff 1974; Reasenberg 1985; Zhuang et al. 2002, 2004; see van Stiphout et al for a review). However, filtering the original seismic processes to obtain a process with a Poisson distribution is a non-unique approach, meaning that this methodology is not straightforward (Luen & Stark 2012). In addition, temporal variations in background seismicity can be induced by different physical processes, such as viscoelastic relaxation, fluid flow, or afterslip (Hainzl & Ogata 2005; Lombardi et al. 2006, 2010; Lombardi & Marzocchi 2007; Marzocchi & Lombardi 2008; Llenos et al. 2009). These observations suggest that the performance of a given forecasting strategy can be enhanced by an improved integration of background seismicity change evaluation. In this study, background events are characterized by their rate (proportion) relative to all seismic events rate, that is, background fraction. Under general assumption of a homogeneous Poisson process for background events and non-homogeneous Poisson process with Omori rate for clustered events, Molchan (2005) showed that the background fraction can be estimated by the ratio of interevent time mean over variance. As background events are heterogeneous in space, we can estimate the probability of presence of background events in a given region by using the background fraction. Thus, interevent times that are close to the mean interevent time of all seismic events increase the variance and hence increase the probability of presence of background events in the corresponding region. In other words, low interevent time variance is indicative of background seismicity, whereas high interevent time variance indicates clustered seismicity. If we consider that the mean interevent time for all seismic events, or the rate of seismic events, in a given region is constant over a long timescale (stationary), and that observed fluctuations around the mean are created by transient clustering at short and long interevent time ranges (Talbi et al. 2011), then background seismic events occur quasi-uniformly with interevent times close to the mean. The proportion of background events (or the background fraction) within the entire seismic record of an area decreases rapidly after large earthquakes inside the aftershock zone, primarily because aftershocks have short interevent times. After aftershocks cease, the background fraction undergoes a continuous increase and eventually reaches an anomalously high level that may either decrease during the onset of a quiescence period with independent seismic events, or move into a state where seismic events are more frequent (activation), with the quiescence and activation occurring separately or combined before the next large earthquake. This study uses anomalous increases in background fraction values as potential precursors before large earthquakes (Talbi et al. 2012a,b). The ratio of the mean to the interevent time variance, hereafter called the moment (or mean) ratio, and abbreviated to moment ratio (MR), is used to estimate background fraction values, with the alarm function value of this approach tested by retrospective forecasting of magnitude M 7(M7+) earthquakes in Japan (Talbi et al. 2012,b). In statistics, MR is equivalent to the inverse of the index of dispersion with low values less than 1 indicating the existence of clustering and high values exceeding 1 showing

3 New alarm-based earthquake forecasting model 1825 more regularity than Poisson process (underdispersion). In signal processing, MR can be viewed as a kind of signal-to-noise ratio. In the context of interevent times, the MR statistic was initially used for fitting local interevent times to a Gamma distribution (Corral 2003), then it was found to provide a non-parametric estimate of the proportion of background events within the entire seismic catalogue used (Molchan 2005; Hainzl et al. 2006). The second-order properties of interevent times, here described by the variance, are clearly associated with clustering and could be used to characterize precursory patterns prior to large earthquakes. The establishment of an earthquake forecasting model requires at least two steps: formulation of a model hypothesis and statistical testing of the model. Our hypothesis is based on an increase in background seismicity relative to the entire seismic record in an area prior to large earthquakes, hereafter termed the background fraction, and uses MR values as a proxy for background fraction values. This hypothesis differs from a seismic quiescence approach in which MR values would decrease (see Section 4). The objective in this study is to test MR forecasts, and to compare the performances of the MR-based approach and RI methods, which is an appropriate comparison given the absence of significantly improved performances in other models (Zechar & Jordan 2008; Nanjo 2010). MR forecast performance in this study is evaluated using mainly Molchan error diagrams (Molchan & Keillis-Borok 2008; Molchan 2010). In addition, receiver operating characteristic (ROC) diagrams are used in the preliminary assessment of MR forecasts despite the fact that they use an unrealistic uniform spatial distribution of earthquakes as a reference model (Zechar 2010). This study does not discuss time dimension of the MR forecasts in application. Molchan diagrams show the link between alarm thresholds and forecast counts during a fixed time period and study region. MR maps show spatial details of forecasts for a given alarm threshold, whereas time details of forecasts can be studied using time variation of the precursory MR signal inside grid cells. Such analysis called precursory signal cell analysis is accomplished as part of ongoing research work to reveal clues and information about the time dimension of our forecasts. In this paper, Section 2 introduces the data and forecasting model used in this study together with a short description of the testing methodology. Section 3 discusses the results of three retrospective performance tests used to evaluate the long- and intermediate-term forecasting ability of the MR model, and to discuss short-term forecasting of the 2011 M w 9.0 Tohoku earthquake. At the end of Section 3, the properties of the MR signal defined by the time variation of the MR scores are discussed. In the conclusion (Section 4), we show that the MR forecast method retrospectively performs well in our tests, outscoring the RI forecasting method over short, intermediate and long terms. In particular, the MR approach successfully forecasted the occurrence region of the Tohoku earthquake, whereas the RI method did not. 2 METHODS AND DATA A composite Japan-wide seismic catalogue for the period was compiled by combining Utsu historical seismicity records between 679 and 1922 with data from the Japan Meteorological Agency (JMA) catalogue between 1923 and 2011, providing a complete catalogue for events with magnitudes greater than the threshold m c = 6, starting from approximately 1890 January 1. Fig. 1 shows variation of m c with time, estimated using the maximum curvature method (MAXC; Wiemer & Wyss 2000). The resulting subcatalogue for the period is used in the long-term forecasting test 1, whereas tests 2 (intermediate-term forecasting) and 3 (shortterm forecasting for the Tohoku earthquake) used catalogue data that were updated to the end of 2012 March. In this study, the terms long-, intermediate- and short-terms are used for periods of tens to hundreds of years, few years up to one decade, and few days or months up to 1 yr, respectively. Although the JMA catalogue used during this study contains preliminarily determined epicentres from 2011 September onwards, Nanjo et al. (2012) showed that the preliminarily determined catalogue for M5+ events is nearly identical to the finalized catalogue. The learning process is defined by the calculation of the MR statistics from the data collected during the Figure 1. Magnitude of completeness m c as function of time calculated using the maximum curvature method (MAXC; Wiemer & Wyss 2000). Events reported during the period can be considered roughly complete above magnitude 6. Dashed lines are the bootstrap error limits. We used the standard parameter predefined in the Zmap free Matlab code that is a sample window size of 500, a binning of 0.1 and 200 bootstrap samples to calculate uncertainties.

4 1826 A. Talbi et al. learning period. The estimates of Interevent time mean and variance are directly calculated from the data. The learning period starts on 1890 January 1 and ends 7 d before a reference M7+ earthquake, with this period being varied for each test to maximize data quality and quantity. In particular, the learning period used in our study ensures somehow complete catalogue for events with magnitude M m c (i.e. M6+). Earthquakes that occurred during the learning period were used to calculate MR and RI alarm functions that are in turn used to forecast the occurrence region of M7+ target earthquakes during a fixed testing period. It should be noted that the majority of other studies used smaller size earthquakes within the learning period and as target earthquakes, to increase the amount of data used and improve the resulting statistics; however, it is not clear if the statistics for intermediate and small earthquakes can be extrapolated to large events. Here, we attempt a more objective approach by retrospectively forecasting M7+ earthquakes, and testing our methodology with, in addition the statistics derived from M6+ earthquakes. Interevent times were estimated using the earthquake random sampling (ERS) algorithm (Talbi & Yamazaki 2009, 2010) with a fixed sampling radius r of 100 km, determining a series of interevent times, {ξ i } n i=1, n 2, for each sampling disk centred on a point x. Namely, the main steps of the ERS algorithm applied with a target radius R = 2r > 0, can be resumed as follows, Step 1. An earthquake location x (0) is selected at random from the epicentre distribution map. Step 2. A radius r is chosen such that at least one earthquake is located outside the disk D(x (0),2r) withradius2r centred on x (0).A second earthquake location x (1) is selected as the nearest neighbour of x (0) from earthquakes occurred outside the disk D(x (0),2r). Step 3. x (2) is selected as the nearest neighbour of x (1) from earthquakes occurred outside the disks with radius 2r and centred on x (0) and x (1). Step 4. x (i 1) is selected as the nearest neighbour of x (i 2) from earthquakes occurred outside the set of disks with radius 2r and centred on x (0), x (1),...,x (i 2). For a finite catalogue of earthquakes, our algorithm stops when no location can be selected. ERS is used iteratively to provide interevent time samples from sampling disks. In each iteration, a set of locations (control or target points) (x (i) ) i = 0,1,...,m, is selected. Then, for each sampling disk D(x, r) centred on x with radius r,the corresponding interevent time-series {ξ i } n i=1 is calculated. When applying ERS iteratively using the above steps, dense network of intersecting sampling disks is obtained and earthquakes are used more than once to calculate interevent times. The number and density of the ERS control points is linked to the choice of the ERS radius r and to the minimum sample size n t required to calculate interevent times inside each sampling disk (here number of M6+ interevent times). The radius r acts as smoothing parameter for maps obtained using our ERS algorithm. Small radiuses produce denser ERS control points in areas of high seismicity but sparse ERS control point distribution in areas with low seismicity. Big radiuses produce sparse distribution of ERS control points. In this study, the moment ratio (MR) score was calculated for each location, or control point, x, and sampling disk radius r using, MR (x, r) = MR(x) = ξ σ 2 ξ, (1) Figure 2. Sketch showing a grid cell holding three ERS control points. Earthquake interevent times are sampled inside the three sampling disks centred on those control points. The maximum MR value shown in red is considered in releasing the alarm. where ξ and σ ξ 2 are the arithmetic mean and the variance of the time-series {ξ i } n i=1, respectively. A minimum sample size n t = 10 is set to ensure a minimum accuracy of MR estimates and to support the stability of results for different radiuses r. Consequently, only MR scores calculated from at least 10 interevent times are considered in our calculation (n 10). In eq. (1), the dependence of MR on r is omitted because the r value is fixed at 100 km. For our forecasts to be testable using Molchan or ROC diagrams, it is essential to calculate the number of successes and failures inside grid cells. For this reason, the MR scores obtained from all sampling disks are plotted on a regular grid with cell size l; in this study, we use an l value of 0.5, or approximately 55 km, to reflect the linear size of magnitude M 7 target earthquakes (Wells & Coppersmith 1994). Using a regular grid cells with size l l, l = 0.5 can be considered as a second level l-smoothing applied of the first level ERS r-smoothing. A set of grid cells that contains at least one observed MR score value defines the testing region G, with the MR alarm function P MR for a given cell C defined as the maximum observed MR score located inside the cell C (Fig. 2), P MR (C) = 1 P max MR max [MR(x)]. (2) x C Division by the maximum observed score PMR max = max (max [MR(x)]) is applied to normalize MR scores so C G x C that all P MR (C) values vary between 0 and 1. Similarly to the mean ratio, the RI score was calculated for each location x and sampling disk radius r using, RI (x, r) = RI(x) = n c + 1 c G (n c + 1), (3) Then, the RI value-derived alarm function P RI is calculated using the relative frequency of M6+ events, where the number n c + 1of M6+ events occurring inside cell C during the learning period is divided by the total number of M6+ events in all cells of the testing region G,

5 New alarm-based earthquake forecasting model 1827 P RI (C) = 1 P max RI max [RI(x)], (4) x C where n c is the number of interevent times in cell C and PRI max = max (max [RI(x)]) is a normalization factor. C G x C The next step is retrospective binary forecasting, where the testing region G is divided into m subregions C i, with i = 1, 2,..., m,(here G = Ui=1 m C i where C i is the ith cell or the testing region i), whereas the testing period [0, T] is divided into s sub-periods of equal length t (i.e. T = s t). Inside each space time region C i [t, t + t], a strategy π i is defined as follows (Molchan 2010): { 1 if an alarm is declared in the region Ci [t, t + t] π i (t) =. 0 if not (5) If N target events occur in the testing region G [0, T], the following statistics can be calculated (Shcherbakov et al. 2010): (1) Number of target earthquakes that occurred in alarm cells. (2) Number of alarm cells with no target earthquakes. (3) Number of target earthquakes occurring outside alarm cells. (4) Number of non-alarm cells with no target earthquakes. Two diagrams are used to evaluate our forecasts: ROC and Molchan diagrams. The former plots the hit rate H = a/(a + c) against the false alarm rate F = b/(b + d), obtained for different alarm thresholds, with points above the diagonal H = F outscoring the random guessing strategy (Joliffe & Stephenson 2003). Although simple to apply and interpret, ROC diagram suffers from a serious deficiency when applied to evaluate earthquake forecasting. ROC diagram compares forecasts against a random distribution of earthquakes in space, which is clearly unrealistic reference model (Zechar 2010). The Molchan diagram (Molchan 1997) plots the miss rate ν = c/(a + c) = 1 H against the space time alarm rate τ = (a + b)/(a + b + c + d), obtained for different alarm thresholds (Harte & Vere-Jones 2005; Shcherbakov et al. 2010). In this diagram, a τ +ν = 1 diagonal indicates a trivial random guess strategy, with any points significantly below this diagonal outscoring the random guessing strategy, defined here as P U. This use of the Molchan diagram demonstrates the difference between random guessing and predictions resulting from a given algorithm, and is used in this study to evaluate MR and RI forecasts, and to compare their performance by testing the null hypothesis H 0 :MRand random guessing strategy are comparable (respectively H 0 :RIand the random guessing strategy are comparable) against the alternative H 1 : MR outscores the random guessing strategy (respectively H 1 : RI outscores the random guessing strategy). Recently, generalized Molchan diagrams have been used to evaluate the difference between a prediction and a given referential model (Molchan & Keillis-Borok 2008; Molchan 2010). In all cases, miss rates are plotted against the following weighted space time alarm rate m τ w = w i τ i, (6) i=1 where τ i is the time alarm rate in the testing region i, andw i a weight coefficient depending on the spatial prior distribution taken as a reference for comparison. τ i = 1 s 1 1 {πi ( j t)=1}. (7) s j=0 The logical function 1 A equals 1 if A is true, and 0 otherwise. In this study, eqs (6) and (7) are adopted in the calculation of the space time alarm rate used to plot Molchan diagram. A uniform spatial prior distribution that assigns equal weights w i = w U i = 1/m to all cells C i is used in eq. (6) to obtain a diagonal corresponding to the random guessing strategy. In this case, the notation is simplified as τ w = τ. Alternatively, RI weighted spatial prior are used by replacing the unknown normalized rate of target events w i = w RI i = n i /N in eq. (6), with n i equalling the number of target events that occurred in the testing region C i. In such case all RI reference strategies are projected onto the diagonal ν + τ w = 1. However, the RI method uses normalized rate estimates derived from learning events that have magnitudes smaller than the target events, suggesting that RI strategies define a domain around the diagonal. In this case, the Molchan diagram is used to test the null hypothesis H 0 :MRandRI methods are comparable against the alternative H 1 : the MR method outscores the RI method. Molchan diagrams are plotted for a given test using uniform and RI spatial prior distributions, represented by the weights w U i and w RI i in eq. (6), respectively. Then, they are used to evaluate the MR forecasting performance compared with both random guessing and the RI method, respectively. The confidence intervals of Molchan diagram can be obtained by considering the binomial distribution for the number of hits by chance in the alarm region (Kossobokov 2006; Zechar & Jordan 2008). Namely, the curve Ɣ α associated with the confidence level 1 α is obtained as follows, Ɣ α = {(τ,ν α (τ)) [0, 1] {1, 2,...,N}} (8) [ ] ν α (τ) = 1 min {k/p(b (N,τ) = k 1) > 1 α} /N, 1 k N (9) where N is the number of target events occurred in the testing region G and k is the number of hits. To test the applicability of MR maps in earthquake forecasting, we use optimal MR forecast maps obtained by threshold optimization (Shcherbakov et al. 2010; Tiampo & Shcherbakov 2013). Namely, these optimal maps are obtained by plotting MR scores exceeding the alarm function threshold corresponding to the minimum forecasting error e (τ,ν) = τ + ν. The minimal forecasting error e (τ,ν) is obtained by maximizing the Peirce Skill score SS p (Peirce 1884; Joliffe & Stephenson 2003; Tiampo & Shcherbakov 2013), SS p (τ,ν) = 1 ν τ, (10) ( min {e(τ,ν)} = Max SSp (τ,ν) ). (11) (τ,ν) [0,1] 2 (τ,ν) [0,1] 2 To compare MR and RI forecast maps, optimal RI maps were compiled by plotting RI scores with a space time alarm rate equivalent to the corresponding optimal MR maps. 3 RESULTS AND DISCUSSION 3.1 Test 1 (long-term forecasting) The testing of long-term MR forecasting used a learning period that started on 1890 January 1 and ended on 1993 January 8, with a testing period between 1993 January 8 and 2011 December 31. The learning period acquisition stopped 7dpriortothe1993 Kushirooki earthquake in order to evaluate the imminent prediction of this earthquake by our MR forecasting algorithm. A total of 22 M7+ target earthquakes (Table 1) occurred in the testing region during

6 1828 A. Talbi et al. Table 1. List of target earthquakes used in tests 1 and 3. The list of target earthquakes used in each test is designed by the last column Test n. ID Longitude Latitude Date Magnitude Name/Region Cluster Test n January Kushiro-oki C October Hokkaido-touhou-oki C October Hokkaido-touhou-oki aftershock C December Sanriku-haruka-oki C August Chichi jima C January Nemuro-oki C August Chichi jima C May Miyagi-ken-oki C September Tokachi-oki C September Tokachi-oki aftershock C September Kii-hanto-oki November Kushiro-oki C August Miyagi-ken-oki C May Ibaraki-ken-oki C June Iwate-Miyagi nairiku September Tokachi-oki aftershock C November Chichi jima C March Tohoku foreshock C 1 1and March Tohokuchiho-Taiheiyo-oki C 1 1and March Tohoku aftershock March Tohoku aftershock C 4 1and April Tohoku aftershock C 1 1and July Tohoku aftershock C 1 1and , January Torishima Kinkai 3 the testing period, with ID (event numbers in Table 1) 1, 5, 7 and 17 being events with depths exceeding 100 km. The fact that the ERS sampling disks extended outside the testing region (Figs 3a and c) meant that 1500 M6+ events were used to calculate MR and RI alarm functions with only 1115 of these events occurred in the testing region. For this test, the alarm time step t is the entire testing period of 19 yr. The testing region is composed of 282 spatial cells as shown in Figs 3(c) and (d). In order to calculate the total number of space time cells, the number of space cells should be multiplied by the number of time steps. This test considers one time step t, so that the total number of space time cells is 282 cells. Figs 3(a) and (b) shows the location of sampling disks determined from 100 ERS simulation runs and the preliminary MR map obtained by plotting MR scores as small circles centred on ERS control points (circle centres in Fig. 3a). ERS control points are dense in high seismicity areas and less dense in low seismicity areas, thus they reproduce the observed earthquake distribution and provides estimates of MR close to earthquake locations. The preliminary map shown in Fig. 3(b) was converted to the MR map in Fig. 3(c) using eq. (2). For a given threshold, this conversion allows the calculation of the a, b, c and d values defined in the preceding section using a regular grid calculation method. The same steps were used to formulate the RI map in Fig. 3(d). Hotspots delineated in red in the MR map are broader than those in the RI map, and include regions south of 35 N, whereas the RI map has a smaller hotspot area with only two large hotspots. The hotspots in the MR map allow the visual identification of five clusters (Table 1), as follows: (1) The central cluster C 1 (ID 8, 13, 18, 19, 22 and 23), including the Miyagi-ken-oki and Tohoku earthquakes. (2) The cluster C 2 formed by the Tokachi-oki earthquake and associated aftershocks (ID 9, 10 and 16), and the Sanriku-harukaoki earthquake (ID 4). (3) The cluster C 3 located east of Hokkaido formed by the Kushiro-oki, Hokkaido-touhou-oki, and Nemuro-oki earthquakes (ID 1, 2, 3, 6 and 12). (4) The cluster C 4 formed by the Ibaraki-ken-oki earthquake and the 2011 March 11 M7.6 largest aftershock of the Tohoku earthquake (ID 14 and 21, respectively). The maximum MR score which is located in the east of Tokyo Bay belongs to this cluster. (5) The southern cluster C 5 formed by the deep (>400 km) Chichi jima earthquakes (ID 5, 7 and 17). In addition, the Kii-hanto-oki earthquake (ID 11) is located close to a hotspot cell in our model. The close proximity of target earthquakes to hotspot cells suggests that our forecast could be improved by Moore neighbourhood smoothing, in which case the eight cells immediately surrounding each hotspot cell would be alarm cells. The Iwate Miyagi Nairiku earthquake (ID 15) was not associated with a hotspot, primarily because the epicentre of this event is not well covered by the data used in this study. The largest aftershock of the Tohoku earthquake (ID 21) occurred in a cell immediately adjacent to the cell with the maximum MR value (Figs 3a and c). This suggests that fragile stored energy equilibrium exists in the region around the maximum MR cell, indicating that this area may host the next big seismic event. This result is consistent with the finding of Prozorov & Schreider (1990), who suggested that long-range aftershocks occur near the epicentres of future (roughly within next decade) large earthquakes. Recent study by Ishibe et al. (2011) and Toda et al. (2011) using GPS deformation data showed that seismicity in the region around the maximum MR cell activated after the Tohoku earthquake with a marked shallow depth events, and an increase in static stress measured by the high percentage of positive Coulomb Failure Function change ( CFF). In comparison with the five clusters identified from the MR data, the RI map identifies only two hotspots: the first located around the Sanriku-haruka-oki earthquake (ID 4) in the north of the study region, and the second located around the C 4 MR cluster, close

7 New alarm-based earthquake forecasting model 1829 Figure 3. (a) Map of the 539 ERS sampling disks used to obtain interevent sampling data. (b) MR map constructed by plotting MR scores as small circles centred on the control points. (c) and (d) MR and RI forecast maps for the testing period, where circles and star show M7+ events within the testing region. The MR distribution used M6+ seismic events that occurred between 1890 and 1993, up to 7 d prior to the reference earthquake (ID 1, shown as a star marker). (e) ROC diagram for the MR and RI methods. (f g) Molchan error diagram for test 1 obtained using (f) uniform and (g) RI weighted spatial priors. Solid, dashed and dotted curves indicates 1, 5 and 10 per cent critical boundaries, and arrows point to the minimum forecasting error (eqs 10 and 11) of the MR and corresponding RI forecasts. (h) Peirce Skill Score for the Molchan diagram shown in Fig. 3f. (i) and (j) Optimal MR and RI forecast maps corresponding to the arrows shown in (f), with the star and the Chichi jima cluster in the south of the map indicating deep seismic events that were not successfully forecasted. to the Ibaraki-ken-oki earthquake and an aftershock of the Tohoku earthquake (ID 14 and 21). The cluster C1, which includes the Tohoku earthquake, is not highlighted by any RI hotspots but is located at the junction between two identified RI hotspots. A ROC diagram for MR and RI forecasts is shown in Fig. 3(e), with the dashed diagonal line showing the results of random guessing. Both forecasting methods outperform the random guessing, with the MR model demonstrating a significant improvement over the random results. In addition, Figs 3(f) and (g) shows Molchan diagrams obtained using uniform and RI weighted spatial prior distributions (wi U and wi RI ), respectively. Given that points below the critical boundary values reject the null hypothesis, we can conclude from Fig. 3(f) that MR (and respectively RI) outscores random guessing at a test level α = 1 per cent for τ [0.18, 0.67] (respectively τ [0.35, 0.46] for RI) and at α = 5 per cent for almost all τ (respectively for τ [0.16, 0.80] for RI). In addition, Fig. 3(g) shows that the MR model outscores the RI at α values of 1 and 5 per cent for τ [0.25, 0.42] and τ [0.26, 0.55], respectively. To optimize the forecasting error, Peirce skill scores corresponding to Fig. 3(f) are plotted in Fig. 3(h), with a maximum SSp max value

8 1830 A. Talbi et al. Figure 3. (Continued.) reached at τ = , ν = , and with an MR threshold c 0 of , with Figs 3(i) and (j) showing optimal forecast maps for MR and RI, respectively. The optimal MR map was plotted using 58 cells with P MR values exceeding or equal to c 0, producing an alarm rate τ MR of 58/282, or , with 7 of 22 earthquakes missing (ν MR = 7/22). The number of missed target earthquakes (that occurred outside alarm cells) is reduced to five if we consider successful the forecasting of target earthquakes occurring incells located immediately adjacent to an alarm cell (i.e. Moore neighbourhood). The use of this margin of error as a coarse-grained box

9 New alarm-based earthquake forecasting model 1831 size was suggested by Rundle et al. (2003), who focused on error locations; this approach was later adopted in forecasting evaluations (e.g. Holliday et al. 2006; Nanjo et al. 2006; Zechar & Jordan 2008). The fact that M7+ event ruptures can extend beyond a sized cell is a good reason to consider location errors. In addition, the optimal RI map (Fig. 3j) is plotted using 59 cells with P RI values exceeding or equal to , a spatial extent that is equivalent to the MR map shown in Fig. 3(i), and with an RI alarm rate τ RI that is similar to the former MR alarm rate τ MR,(τ RI = 59/282 = ). However, the RI miss rate ν RI is nearly double the MR miss rate ν MR, with 12 earthquakes missing from 22 (ν RI = 12/22). If only shallow earthquakes (depth less than 100 km) are considered, ν RI increases to approximately three times higher than ν MR (8/18 versus 3/18, respectively). Fig.3(c)ismapped7dbeforethe1993 M7.5 Kushiro-oki earthquake (ID 1, shown as star in Figs 3(c), (d), (i) and (j) in order to evaluate its imminent prediction. The 1993 Kushiro-oki earthquake occurred after a period of relative quiescence in the region, associated with decreasing MR signal values. Namely, the MR signal (not shown here) showed a stable trend between 1950 and 1970, and then decreased to less than half its value in This observation indicates that the low MR values associated with the quiescence some 5 7 yr before the main event meant that the Kushiro-oki earthquake was not predicted by the MR algorithm. This also explains why the star in Fig. 3(i) is not covered by the alarm region. The analysis of the MR signal showed the same behaviour before the M December 28 Sanriku-Haruka-oki (ID 4) which was also preceded by comparable quiescence. It is important to note that our MR forecasts could be improved by releasing alarms in regions undergoing long quiescent periods followed by a significant decrease in the MR signal. 3.2 Test 2 (Intermediate-term forecasting) Intermediate-term testing used a learning period that began on 1890 January 1 and ended on 1982 March 14, with a testing period between 1982 March 14 and 2012 March 31. This testing used an alarm time step t of 3 yr, with MR and RI scores updated in each step, and an alarm declaration decided according to the results of this testing, allowing the testing of the MR method performance over intermediate timescales. The obtained results should reflect the mean forecasting performance in the 3 yr following the map production. During the testing period, 29 M7+ target events occurred in the 2710 space time cell testing region, with a total of 1354 M6+ events used in MR and RI alarm function calculations, with 987 of these events occurring in the testing region during the learning period. The testing region is composed of 271 spatial cells. Since there are 10 time steps during the 30 yr testing period, the total numberofspace time cells in the testing region is = 2710 cells. Figs 4(a) and (b) shows Molchan diagrams for uniform and RI weighted spatial priors, respectively, with MR and RI outscoring random guessing for some alarm rate ranges at α values of 1 and 5 per cent, respectively (Fig. 4a), and MR outscoring RI at an α value of 10 and 5 per cent for τ [0.37, 0.75] and τ [0.54, 0.68], respectively (Fig. 4b). 3.3 Test 3 (short-term forecasting of the Tohoku earthquake) This testing aimed to determine whether MR model forecasting could predict the 2011 M w 9.0 Tohoku earthquake (ID 19), a foreshock that occurred 2 d before the mainshock (ID 18) and the M7+ earthquakes after the Tohoku earthquake, including the large M7+ aftershocks (Table 1). The learning period for this part of the study began on 1890 January 1 and ended 7dbeforethemainshock,on 2011 March 4, with a testing period that began on 2011 March 4 and ended on 2012 March 31, during which a total of 7 M7+ target events occurred in the testing region. A total of 1795 M6+ earthquakes were used during MR and RI alarm function calculations, with 1472 of these events occurring in the testing region during the learning period. An alarm time step t of 1 yr, covering the entire testing period, was used during this testing. The testing region is composed of 380 spatial cells as shown in Figs 5(a) and (b). Since this test uses one time step, the total number of space time cells is 380 cells. MR and RI forecast maps are shown in Figs 5(a) and (b), with the star and circles indicating the locations of the Tohoku earthquake and other M7+ target earthquakes, respectively. The MR map highlights the Tohoku earthquake as a hotspot, whereas the RI map fails to predict the location of the Tohoku earthquake. This is confirmed by the optimal P MR and P RI maps shown in Figs 5(c) and (d), with the RI optimal forecast map plotted with a Figure 4. Molchan error diagram for test 2 obtained using (a) uniform and (b) RI weighted spatial priors.

10 1832 A. Talbi et al. Figure 5. (a) MR and (b) RI forecast maps plotted 7dbeforetheoccurrenceoftheTohoku earthquake (shown as a star). (c) and (d) Optimal MR and RI forecast maps, showing data associated with the minimal MR forecasting error. space time alarm rate τ RI equivalent to that of the MR optimal map τ MR, with τ MR = (22/380) = , and τ RI = (23/380) = This indicates that MR outscores RI, with a miss rate ν MR = 3/7 (events with ID 20, 23 and 24 are missing) against a miss rate ν RI = 6/7 for the RI model. 3.4 Applicability in prospective testing experiments For given data set, the MR method proposed in this study uses specific parameters that can be modified by the user to take into account limitations linked to poor data. Namely these parameters are: the minimal interevent time sample size n t used to calculate MR in ERS sampling disks and the ERS sampling disk radius r. In addition, there are parameters linked to the catalogue dataset as the magnitudes used in the learning process and the magnitude of target earthquakes. These parameters support the flexibility of our method as they can be modified according to characteristics of the region and rules of the game in testing centres. In particular, our method can be adapted to prospective testing such as the Collaboratory for the Study of Earthquake Predictability (CSEP) tests (Schorlemmer & Gerstenberger 2007; Schorlemmer et al. 2007). For these tests,

11 New alarm-based earthquake forecasting model 1833 different models are compared based on forecasting results of a preliminary defined testing region. In our case, the testing region is determined by the MR distribution which requires at least n t = 10 earthquakes inside the sampling disk. Concerning this limitation, our method can be adapted by decreasing the magnitude of earthquakes used in learning in such a way that the CSEP testing region is covered. For example, M 5 events can be used to forecast M 6 events, and check if the testing region includes the CSEP region. Decreasing the magnitude of earthquakes used in the learning and testing processes allows also our model to be testable at short term. Another way to extend the testing region is to increase the radius r of the ERS sampling disk (oversmoothing) or to reduce the interevent time sample size required to calculate MR forecasts. Earthquake likelihood model testing involves probabilities of events within a specified space time and magnitude, or equivalently earthquake rates in specified bins with location, time and magnitude (Schorlemmer et al. 2007). Japan CSEP testing centre uses these tests to compare submitted competing models (Nanjo et al. 2009, 2011). Our method provides alarm rates that cannot be used directly to calculate expected target earthquake rates. In this case, it is essential to develop a mean to transform alarm levels into probabilities. This issue is behind this study and needs further consideration in our future research. It is important to address the applicability of our method to areas with moderate to low seismicity, i.e. with poor data and small interevent samples. In such case, simulation using for example the epidemic type aftershock sequence ETAS model provides a potential solution to improve the data and cover wider testing region. Our sampling method ERS is also useful to intensify interevent time sampling. Meanwhile the MR method in its original version presented in this study is applicable to region with high seismicity records. CSEP testing centre in southern California is processing 5-yr alarm models which may be a suitable class of model for which our method can be adapted. Lower earthquake magnitudes in learning and testing processes may be essential for prospective testing in the CSEP framework. As all forecast models, our method needs to be adapted to the characteristics of the region and rules of game adopted in testing experiments, before being submitted to CSEP testing centre. 3.5 MR signal properties The MR signals reflect second-order properties of interevent times that are characteristic of earthquake clustering, with MR values being highly sensitive to variance in eq. (1). The definition of the MR score in eq. (1) and the analysis of MR signal time variations inside cells that host target events allowed us to identify the following MR signal characteristics. 1. MR values decrease sharply after a series of events located close in time and space, including M6+ aftershocks or foreshocks that occur close to the cell being considered. This decrease is produced by a shortening of interevent times compared with the mean value ξ i ξ, which increases the variance σξ 2 in eq. (1). 2. MR values decrease after very long interevent times ξ i ξ, indicating seismic quiescence that is produced by an increase in variance σξ 2 in eq. (1). 3. MR values usually increase with some fluctuations outside case 1 and 2 particularly after the cessation of all M6+ aftershocks in neighbouring regions. This behaviour reflects interevent times that move towards the mean value ξ i ξ and that can be interpreted i as an indicator of increasing background seismicity. 4 CONCLUSIONS This study proposed and tested a new statistical moment ratio (MR) approach that interprets interevent times to define an alarm function to forecast large seismic events in Japan. The MR is defined as the ratio between mean and variance values, and is integrated within an alarm-based forecasting model. The forecasting performance of the proposed model was evaluated using various tools, including ROC and Molchan error diagrams, and optimal MR maps that were obtained by plotting MR scores above a threshold corresponding to the minimum forecasting error (miss rate + space time alarm rate). Three retrospective tests over short-, intermediate- and long-term time periods were conducted to test the applicability of MR mapping to earthquake forecasting, with MR outscoring both random guessing and the RI forecasting method. The excellent performance of the MR approach is exemplified by the long-term testing, where optimal MR maps succeeded in forecasting 15 of the 18 M7+ shallow events within the testing region during the past two decades ( ), with an alarm rate equal to about 20 per cent of the total space time alarm rate. In addition, the MR forecasting approach reduces the shallow-event missing rate by 60 per cent, compared with the RI method. Over short time periods, the MR approach succeeded in forecasting the 2011 M w 9.0 Tohoku earthquake, the M7.3 foreshock that preceded the main shock by 3 d, and subsequent M7+ aftershocks, with a very small space time alarm rate of less than 6 per cent of the total space time alarm region; in comparison, the RI method failed to predict most of these events. In addition, the presence of hotspot cells with high MR values, located close to missing target earthquakes, suggests that the MR forecasting approach may be improved by smoothing. Target earthquakes preceded by periods of seismic quiescence may not be forecasted over short and intermediate timescales, primarily due to the decrease in MR score during quiescence. This result suggests that a better forecasting approach may be to integrate additional alarms when high MR scores are followed by a quiescence-induced drop in MR signal. The results presented here clearly show the importance of second-order interevent time properties in clustering structure characterization and the forecasting of earthquakes, with the MR model presented here providing an alternative earthquake forecasting method that has the potential to accurately forecast large earthquakes. ACKNOWLEDGEMENTS This work was supported by a fellowship from the Japanese Society for the Promotion of Science. The authors are grateful to Maximilian Werner and Chun-Han Chan for helpful discussions while visiting the Earthquake Research Institute at the University of Tokyo. The authors thank the editor and two anonymous reviewers for their comments that improved an earlier version of the manuscript. REFERENCES Bormann, P., From earthquake prediction research to time-variable seismic hazard assessment applications, Pure appl. Geophys., 168, Corral, A., Local distributions and rate fluctuations in a unified scaling law for earthquakes, Phys. Rev. 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