arXiv:1312.7712v1 [stat.ME] 30 Dec 2013 iyadcmlxt fteerhuk phenomenon. earthquake the con- of diver- complexity been of and because sity have remain though uncertainties discussions large even ducted, and However, analysis it. detailed with associated earthquake, anisms mech- major seismic remark- important every elucidated progressed have as After researchers has geophysics. increased in earthquakes steadily ably of has study data the relevant avail- of The of understanding significantly. ability our increased 1960s, has late earthquakes the since science rseto atqaePrediction Ogata Yosihiko Earthquake of Research Prospect A 03 o.2,N.4 521–541 4, DOI: No. 28, Vol. 2013, Science Statistical er-u oy 5-55e-mail: Komaba, 153-8505 4-6-1 Tokyo Professor, Megro-Ku, Visiting Science, Tokyo, Industrial of of University Institute and 190-8562 Tachikawa,Tokyo Midori-cho, 10-3 Organization, System Research and Information Mathematics, Statistical

oiioOaai rfso mrts nttt of Institute Emeritus, Professor is Ogata Yosihiko c ttsia Science article Statistical original the the of by reprint published electronic an is This ern iesfo h rgnli aiainand pagination in detail. original typographic the from differs reprint hog eakbedvlpet nsldEarth solid in developments remarkable Through nttt fMteaia Statistics Mathematical of Institute 10.1214/13-STS439 .INTRODUCTION 1. nttt fMteaia Statistics Mathematical of Institute , rhclsaetm TSmdl,poaiiyfrcss p forecasts, probability changes. stress models, gains, ETAS mo (ETAS) space–time sequence archical aftershock epidemic-type constraints, eerht elz rcia prtoa oeatn nt phrases: and in words forecasting Key operational practical realize conside earthquake to of is research prospect the risk describes manuscript of earthqua this modeling fore, the quantitative earthquak to unless for urgency information incomplete t assigning useful potentially precursors in of as are discovery well phenomena unc as abnormal of earthquakes, nece whether amount which large identifying a considered, in particular, be In compl must prediction. prediction, elements stochastic effective uncertain determinist For and apparent, difficult. tions the readily is of not prediction most are quake Because mechanisms crust. their Earth’s and the in stress accumulated Abstract. 03 o.2,N.4 521–541 4, No. 28, Vol. 2013, 2013 , atqae cu eas farp lp nfut u to due faults on slips abrupt of because occur Earthquakes [email protected] bomlpeoea simcsi,Bayesian slip, aseismic phenomena, Abnormal This . . in 1 o h td fErhuk Predictability Earthquake of as Study known the study for interna- cooperative An research predictability. tional organized earthquake an on develop program to seismologists for manner. this should in power models evolve forecasting predictive Earthquake whether improved. be is determine should models to seismicity over evaluated standard information in claim used useful those that potentially models incorporate prediction to New merit. lack presented may arguments otherwise, evaluation; such for a mn onre rn omjrearthquakes major to prone countries among way http://www.cseptesting.org/ (Jor- controversial effec- ( is al. the techniques et however, dan these basis types; the of various on tiveness of proposed anomalies been of have earthquakes ing effective for considered pro- be prediction. the earthquake to reflect earthquakes of faithfully and cesses must diverse scenarios all because complex determin- prediction in earthquake challenges unachievable istic to leads This eety hr a engoigmomentum growing been has there Recently, nteohrhn,svrltcnqe o predict- for techniques several hand, other the On 2011 ) hrfr,ojciiyi required is objectivity Therefore, )). ena future. near he rdcinis prediction e predictability ranylies ertainty e.There- red. es hier- dels, robability xcondi- ex cearth- ic efaults se e Any ke. ssitates large o scretyunder currently is ) Collaboratory (CSEP; 2 Y. OGATA for exploring possibilities in earthquake prediction has been referred to as (relative) entropy, which is (e.g., Jordan (2006)). An immediate objective of the essentially similar to the log-likelihood. project is to encourage the development of statisti- 2.2 Space–Time-Magnitude Forecasting of cal models of seismicity, such as those subsequently Earthquakes discussed in Section 2, and to evaluate their predic- tive performances in terms of probability. Baseline models should be set to compare with In addition, the CSEP study aims to develop a and evaluate all predictability models. Based on em- scientific infrastructure to evaluate statistical signif- pirical laws, we can predict standard reference prob- icance and probability gain (Aki (1981)) of various ability of earthquakes in a space–time-magnitude methods used to predict large earthquakes by using range on the basis of the time series of present and observed abnormalities such as seismicity anomaly, past earthquakes. The framework of CSEP, which transient crustal movements and electromagnetic has evaluated performances of submitted forecasts anomaly. Here probability gain is defined as the ra- of respective regions (Jordan (2006); Zechar, Ger- tio of probability of a large earthquake estimated stenberger and Rhoades, 2010; Nanjo et al. (2011)), based on an anomaly to the underlying probability is similar to that of the California Regional Earth- without anomaly. Section 3 describes this important quake Likelihood Models (RELM) project for spatial concept, and then discusses statistical point-process forecast (Field (2007); Schorlemmer et al. (2010)). models to examine the significance of causality of Different space–time models were submitted to the anomalies and also to evaluate the probability gains CSEP Japan Testing Center at the Earthquake Re- conditional on the anomalous events. search Institute, University of Tokyo, for the one- For prediction of large earthquakes with a higher day forecast applied to the testing region in Japan probability gain, comprehensive studies of anoma- (Nanjo et al. (2012)). This means that the model lous phenomena and observations of earthquake forecasts the probability of an earthquake at each mechanisms are essential. Several such studies are space–time-magnitude bin. However, the CSEP pro- summarized in Sections 4–6. Particularly, I have tocols have to be improved to those in terms of been interested in elucidating abnormal seismic ac- point-processes on a continuous time axis for the tivities and geodetic anomalies to apply them for evaluation including a real-time forecast (Ogata promoting forecasting abilities, as described in these et al. (2013)). sections. Almost all models incorporated the Gutenberg– 2. PROBABILITY FORECASTING OF Richter (G–R) law for forecasting the magnitude BASELINE SEISMICITY factor, and take different variants of the space–time epidemic-type aftershock sequence (ETAS) model 2.1 Log-Likelihood for the Evaluation Score of (Nanjo et al. (2012), and Ogata et al. (2013)). In Probability Forecast the following sections, I will outline these models. Through repeated revisions, CSEP attempts to es- 2.2.1 Magnitude frequency distribution Guten- tablish standard models to predict probability that berg and Richter (1944) determined that the num- conform to various parts of the world. Here I mean ber of earthquakes increased (decreased) exponen- the prediction/estimation of probability as predict- tially as their magnitude decreased (increased). De- ing/estimating conditional probabilities given the scribing this theory in terms of point processes, the past history of earthquakes and other possible pre- intensity of magnitude M is cursors. The likelihood is used as a reasonable measure of prediction performance (cf. Boltzmann 1 λ0(M) = lim Prob(M < Magnitude ≤ M + ∆) (1878); Akaike (1985)). The evaluation method for ∆→0 ∆ (1) probabilistic forecasts of earthquakes by the log- = 10a−bM = Ae−βM likelihood function has been proposed, discussed and implemented (e.g., Kagan and Jackson (1995); for constants a and b. In other words, the magnitude Ogata (1995); Ogata, Utsu and Katsura, 1996; Vere- of each earthquake will obey an exponential dis- −β(M−Mc) Jones (1999); Harte and Vere-Jones (2005); Schor- tribution such that f(M)= β e , M ≥ Mc, lemmer et al. (2007); Zechar, Gerstenberger and where β = b ln 10, and Mc is a cutoff magnitude Rhoades, 2010; Nanjo et al. (2012); Ogata et al. value above which all earthquakes are detected. Tra- (2013)). In some such studies, the evaluation score ditionally, the b-value had been estimated graph- EARTHQUAKE PREDICTION 3

Fig. 1. Top panel shows Delaunay tessellation upon which the piecewise linear function is defined. The smoothness con- straint is posed in the sum of squares of integrated slopes. Bottom panel shows b-value estimates of the G–R formula in equation (1) estimated from the data for earthquakes of M ≥ 5.4 from the Harvard University global CMT catalog (http: // www. globalcmt. org/ CMTsearch. html ). One conspicuous feature is that b-values are large in oceanic ridge zones, but small in plate subduction zones. ically, however, more efficient estimation is per- ther varies with time. Temporal and spatial b-value formed by the maximum likelihood method. Utsu changes have attracted the attention of many re- (1965) derived it by the moment method. Later, searchers ever since Suyehiro (1966) reported a dif- Aki (1965) demonstrated that this is a maximum- ference between b-values of foreshocks and after- likelihood estimate (MLE) and provided the error shocks in a sequence. estimate. It should be noted that the magnitudes in Here, we consider that β can vary with time, most catalogs are given in the interval of 0.1 (dis- space and space–time according to a function such crete magnitude values), hence, care should be taken as β(t), β(x,y,z) or β(t,x,y). Various nonparamet- for avoiding the bias of the b estimate in likelihood- ric smoothing algorithms such as kernel methods based estimation procedures (Utsu (1969)). have been proposed (Wiemer and Wyss, 1997). Al- Although the coefficient b in a wide region is ternatively, the β value can be parameterized by generally slightly smaller than 1.0, Gutenberg and smooth cubic splines (Ogata, Imoto and Katsura, Richter (1944) further determined that the b-value 1991; Ogata and Katsura (1993)) or piece-wise lin- varies according to location in smaller seismic re- ear expansions on Delaunay tessellated space (Ogata gions. The b-value varies even within Japan and fur- (2011c); see also Figure 1, e.g.). In such a case, a 4 Y. OGATA penalized log-likelihood (Good and Gaskins (1971)) where M is the magnitude of an aftershock and t is is used whereby the log-likelihood function is asso- the time following a main shock of magnitude M0; ciated with penalty functions in which the coeffi- the parameters are independently estimated by the cients are constrained for smoothness of the β func- maximum-likelihood method for respective empiri- tion. For the optimal estimation of the β function, cal laws. the weights in the penalty function are objectively After a large earthquake occurs, the Japan Me- adjusted in a Bayesian framework, as suggested by teorological Agency (JMA) and the United States Akaike (1980a). Geological Survey (USGS) have undertaken opera- 2.2.2 Aftershock analysis and probabilistic fore- tional probability forecast of the aftershocks. How- casting Typical aftershock frequency decays accord- ever, the forecast is announced after the elapse of ing to the reverse power function with time (Omori 24 h or more. This is due to the deficiency of after- (1894); Utsu, 1961, 1969). First, let N(s,t) be the shock data due to overlapping of seismograms after number of aftershocks in an interval (s,t). Then, the the main shock. In particular, the parameter a is occurrence rate of aftershocks at the elapsed time t crucial for the early forecast, but difficult to estimate since the main shock is in an early period, whereas the other parameters can 1 be default values for the early forecast [Reasenberg ν(t) = lim P {N(t,t + d∆) ≥ 1| and Jones (1994); Earthquake Research Committee ∆→0 ∆ (ERC), 1998]. The difficulty is because the parame- (2) Mainshock at time 0} ter a can substantially differ even if the magnitudes K of the main shocks are almost the same: for exam- = (t + c)p ple, the numbers of the aftershocks of M ≥ 4.0 of for constants K, c and p. This is known as the two nearby main shocks of the same M6.8 differ by Omori–Utsu (O–U) law. 6–7 times (JMA (2009)). Traditionally, estimates of the parameter p have It is notable that the strongest aftershocks oc- been obtained since the study of Utsu (1961) in curred within 24 h in most sequences (JMA (2008)). the following manner. The numbers of aftershocks Therefore, despite adverse conditions during data in a unit time interval n(t) are first plotted against collection, probabilistic aftershock forecasts should elapsed time on doubly logarithmic axes, and then be delivered as soon as possible within 24 h after are fit to an asymptotic straight line. The slope of the main shock to mitigate secondary disasters in this line is an estimate for p. The values of c can affected areas. be determined by measuring the bending curve im- For this purpose, it is necessary to estimate time- mediately after the main shock. Such an analysis is dependent missing rates, or detection rates, of af- based on the time series of counted numbers of af- tershocks (Ogata and Katsura, 1993, 2006; Ogata, tershocks. By such a plot, we can find aftershock 2005c) because they enable probabilistic forecasting sequences for which the formula (2) lasts a long pe- immediately after the main shock (Ogata, 2005c; riod, more than 120 years, for example (Utsu (1969); Ogata and Katsura (2006)). The detection rate of Ogata (1989); Utsu, Ogata and Matsu’ura, 1995). earthquakes is described by a probability function To efficiently estimate the three parameters di- q(M) of magnitude M such that 0 ≤ q(M) ≤ 1. The rectly on the basis of occurrence time records of intensity λ(M) for actually observed magnitude fre- aftershocks, assuming nonstationary Poisson pro- quency is described by λ(M)= λ0(M)q(M), corre- cess with intensity function (2), Ogata (1983) sug- sponding to thinning or random deletion. An ex- gested the maximum-likelihood method, which en- ample of the detection rate function is the cumu- abled the practical aftershock forecasting. Reasen- lative of Gaussian distribution or the so-called er- berg and Jones (1989) proposed a procedure based ror function q(M) = erf{M|µ,σ}. The parameter on the joint intensity rate of time and magnitude µ represents the magnitude at which earthquakes of aftershocks (Utsu (1970)) according to the G–R are detected at a rate of 50%, and σ represents law (1), a range of magnitudes in which earthquakes are partially detected. Let a data set of magnitudes λ(t, M)= λ0(M)ν(t) (3) {(ti, Mi); i = 1,...,N} be given at a period imme- 10a+b(M0−M) diately after the main shock. Assume that the pa- = (a, b, c, p; constant), (t + c)p rameters are time-dependent during the period such EARTHQUAKE PREDICTION 5 that exogenous variables. The ETAS model is applied to 10a+b(M0−M) data from the target time interval [S, T ]. The occur- (4) λ(t, M)= q{M|µ(t),σ} rence history H during the precursor period [0,S] (t + c)p S is used for sustaining stationarity of the process af- with an improving detection rate µ(t). An additional ter the time S. parametric approach proposed by Omi et al. (2013) Then, the model’s effectiveness in fitting an earth- uses the state–space representation method for real- quake sequence can be evaluated by comparing the time forecasting within the 24 h period. cumulative number N(S,t) of earthquakes with the 2.2.3 Epidemic-type aftershock sequence (ETAS) rate predicted by the model model The epidemic-type aftershock sequence t (ETAS) model describes earthquake activity as a (7) Λ(S,t)= λ(u|Hu) du point process (Ogata (1986), 1988) and includes the ZS O–U law for aftershocks as a descendant process. in the time interval S

Although several alternative versions to the spa- tial factor given by the bracket of (8), as described by Ogata (1998), are available, the form in (8) fits best in terms of the Akaike information criterion (AIC; Akaike (1974)) for Japanese earthquake data sets. All extensions of the temporal ETAS model are referred to as space–time ETAS models (e.g., Nanjo et al. (2012)). 2.2.5 Hierarchical space–time ETAS model When a region becomes wide or the number of earthquakes becomes sufficiently large, spatial heterogeneity of seismicity becomes conspicuous. For example, many studies have been conducted on regional variation of seismicity-related parameters such as the b-value of the G–R law and p-values of the O–U law (Utsu, 1961, 1969; Mogi (1967)). Regarding space–time ETAS models, the af- Fig. 2. Iso-surface plot of the estimated conditional in- tensity function (8) of the space–time ETAS model (Ogata, tershock productivity K may differ significantly 1998) to the JMA hypocenter data of shallow earthquakes among locations, even if magnitudes of triggering (depth ≤ 100 km) of magnitude 5.0 or larger from the period earthquakes are similar (see Section 2.2.2). More- 1926–1995; in addition, Utsu’s earthquake catalog for the 40 over, the main shock–aftershock and swarm-type years period before 1926 (1885–1925) was used as the preced- clusters exhibit significantly different activity pat- ing occurrence history of the space–time ETAS model. Space means longitude and latitude, whereas the depth data are ne- terns. Therefore, we applied an extension to the glected. above space–time model to earthquakes in the en- tire region for developing a hierarchical space–time ETAS (HIST–ETAS) model, which is a space–time where Sj is a normalized positive definite symmetric ETAS model in which parameter values µ, K, α, matrix for anisotropic clusters such that p and q can vary depending on location, such as (x,y)S(x,y)t µ(x,y), K(¯xj, y¯j), α(¯xj , y¯j),p(¯xj, y¯j) and q(¯xj, y¯j). (9) Thus, coefficients of parameter functions of the 1 σ σ space–time ETAS model in equation (8) must be = 2 x2 + 2ρxy + 1 y2 . 1 − ρ2 σ1 σ2 evaluated. Coefficients of each parameter function      are defined by values at epicenter locations of earth- Here, (¯xjp, y¯j) is an average location of earthquakes quakes and a number of points on the region bound- that are placed in the same cluster as (xj,yj). Both ary. Hence, each function is uniquely defined by lin- (¯xj, y¯j) and coefficients of Sj for a selected set of ear interpolation of values at three nearest points large earthquakes j are identified by fitting a bi- (earthquakes) determined by Delaunay tessellation variate normal distribution to spatial coordinates that is constructed by all the earthquake locations of the cluster occurring within a square of 3.33 × and additional points on the boundary of the region 100.5Mj −2 km side-length and within 100.5Mj −1 days (see Figure 1). after the large event of magnitude Mj , according to For a stable optimal estimation, the freedom of Utsu (1969); but I use 1 h in prediction stage. The lo- coefficients of parameter functions needs to be con- cations (¯xj, y¯j) of all other events, including cluster strained to assign penalties against roughness of the members, remain the same as the epicenter coordi- functions. The coefficients that maximize the pe- nates of the original catalog; and they are associated nalized log-likelihood are then sought, which is the with the identity matrix for Sj, namely, σ1 = σ2 = 1 equivalent of attaining the maximum posterior dis- and ρ = 0. See Figure 2 for an illustrative view of tribution. Here, we adjusted the optimal prior func- the conditional intensity (8). Further details of the tion for the parameter constraints in terms of the algorithm can be found in studies of Ogata (1998, penalty function by an empirical Bayesian method 2011a, 2011b). (Akaike (1980a)). Further details can be found in EARTHQUAKE PREDICTION 7

This equation considers the potential of further model extensions by useful physical concepts in the elastic rebound theory, such as stress interaction from neighboring earthquake ruptures. This phys- ical concept will be subsequently described in Sec- tions 4 and 5. BPT renewal process is based on the following Brownian perturbation process: (11) S(t)= λt + σW (t), t ≥ 0, which includes linearly increasing drift for stress ac- cumulation and diffusion rate σ. An earthquake oc- curs when the path S(t) attains the critical stress level sf , and the accumulated stress is released down to the ground state s0 based on elastic rebound the- ory of earthquakes (Reid (1910)). Random fluctua- Fig. 3. The optimal solution of the µ-values for background tions represent the transient stress changes due to seismicity of the space time ETAS model (8) in terms of min- the effect of other earthquakes in close proximity imum ABIC priors. The model is estimated from the JMA (see Section 4). This model includes four parame- data with earthquakes of M5.0 or larger for the target period ters: the stress accumulation rate λ, perturbation 1926–1995. In addition, Utsu’s earthquake catalog for the 40 rate σ, failure state s , and ground state s . If we years period before 1926 (1885–1925) was used as the precur- f 0 sory occurrence history of the space–time ETAS model. Con- assume that failure and ground states sf and s0, re- tours are equidistant in the logarithmic scale. Stars indicate spectively, are constant, the interval of earthquakes locations of earthquakes of magnitude 6.7 or larger that oc- is independent and identically distributed with the curred during 1996–2009, which mostly occurred in high back- BPT distribution, in which parameters are related ground rates. Note however that there are several large earth- by µ = (sf − s0)/λ and α = σ/ λ(sf − s0). quakes occurred at very low seismicity rates, the issue of which Because of very small sample size available from lead to Section 2.3. each fault, the mean parameterp µ has been esti- mated using methods other than the MLE. The the studies of Ogata, Katsura and Tanemura (2003) ERC (2001) uses a common α value of 0.24 through- and Ogata (2004a, 2011b). Figure 3 shows the opti- out Japan. This is because better fit of the same α mal solution of background seismic activities µ(x,y), value was shown by the AIC comparison than the which appear useful for long-term prediction of large different α estimates for respective active faults, for earthquakes in and near Japan. Moreover, stochastic a set of occurrence data with moderate sample sizes declustering using the space–time ETAS model can from four active faults (ERC, 2001). Also, the ERC make realizations of background seismicity (Zhuang, has estimated µ in two ways: as the mean of past Ogata and Vere-Jones, 2002; Zhuang et al. (2005b); recurrence intervals and as expected intervals esti- Bansal and Ogata (2013)). mated from the slip data of the fault plane. The 2.3 Long-Term Probability Forecasts of latter estimate is expressed by T = U/V , where U is Characteristic Earthquakes the slip size per earthquake and V is the deforma- tion rate per year, observed from the escarpment of A characteristic earthquake is a repeating large the fault. The ERC selects and applies one of these earthquake that is traditionally defined from pale- estimates for µ of each active fault according to re- oseismology observations. The estimation is made liability of the data. by using recurrence times of a large earthquake on Alternatively, Nomura et al. (2011) propose fol- an active fault or a particular seismogenic region on lowing Bayesian estimation procedure assuming a plate boundary. The Earthquake Research Com- a common prior distribution for fault segments mittee of Japan (ERC, 2001) adopted the Brown- throughout Japan. Consider historical occurrence ian Passage Time (BPT; Matthews, Ellsworth and X j data j = {Xi ; i = 1, 2,...,n} in the jth segment of Reasenberg, 2002) renewal process, in which the m fault segments. Consider a posterior density inter-event probability density function is given by posterior(µj, αj|Tj, φµ, φα) µ (x − µ)2 (12) (10) f(x|µ, α)= exp − . X 2πα2x3 2µα2x = L(µj, αj| j)π1(µj|Tj, φµ)π2(αj|φα), r   8 Y. OGATA where likelihood L is based on the renewal process Bayesian framework can also be used when the taking account of the forward and backward recur- occurrence times are uncertain, especially when we rence times (Daley and Vere-Jones (2003)) and Tj are dealing with geological data (Ogata (1999b); No- is the above-mentioned geologically estimated slip mura et al. (2011)) in addition to the magnitude deformation ratio from slip data. Furthermore, the dependent model (Ogata (2002)) based on the time- values of the hyperparameters φµ and φα character- predictable model (Shimazaki and Nakata (1980)). izing the prior densities of µ and α are common to all considered fault segments. We obtain their esti- 3. PRACTICAL EARTHQUAKE mates by maximizing the integrated posterior dis- FORECASTING tribution Probability gain refers to the ratio of predicted m ∞ ∞ conditional probability relative to baseline earth- Λ(φ)= posterior(µj, αj| quake probability. As far as I know, most probabil- 0 0 j=1 Z Z ity gains of predictions are not very high, even rela- (13) Y tive to the stationary Poisson process model. There- Tj, φµ, φα) dµj dαj, fore, predictions based on a single anomaly data set where the subscript j represents the jth segment alone are not satisfactory for disaster prevention be- of m fault segments. This maximizing procedure cause baseline probability of a large earthquake it- is called the Type II maximum likelihood method self is very small according to the G–R law. Also, (Good (1965)). The selection of the best combi- the BPT renewal process model has been applied nation of the prior distribution factors and the to active fault segments to estimate time-dependent optimal values of the hyper-parameters in (12) probability on the basis of the last earthquake and are carried out to attain the smallest value of stress accumulation rate. In Californian, probability the Akaike Bayesian information criterion (ABIC ; gain showed an improvement of approximately 1.7 Akaike (1980a)) that is defined by ABIC = times over Poisson process model predictions (Jor- −2maxφ log Λ(φ)+2dim(φ), where dim{φ} denotes dan et al. (2011)). the number of hyperparameters. The key for research progress in practical proba- The most common forecast technique is a plug-in bility earthquake forecasting is to use a multiple pre- method, which is a probability forecast that uses a diction formula (Utsu (1979)) such that total prob- distribution or conditional intensity function with ability gain is approximately equal to the product of a parameter set to its estimated value, such as individual probability gains (Aki (1981)). The rate MLE. This method works well if the estimation er- of probability gain that an individual anomaly was ror is sufficiently small. However, its predictive per- actually a precursor to an earthquake may be calcu- formance can be inadequate when the sample size lated as its success rate of the anomaly divided by is small. Hence, ERC adopts the plug-in method precursor time (Utsu (1979)). Success rate can only only for µ and uses a common α value of 0.24 be determined from accumulation of actual earth- throughout Japan. Alternatively, Rhoades, Van Dis- quake occurrences; and precursor time can be stud- sen and Dowrick (1994), Ogata (1999b, 2002) and ied experimentally and theoretically (Aki (1981)). In Nomura et al. (2011) propose the Bayesian predic- this section I review important suggestions by Utsu tion (Akaike (1985)) (1979) and Aki (1981) and provide some examples ∞ ∞ of causality modeling toward improved accuracy for ˜ X h(y| )= h(y|µ, α){1 − F (y|µ, α)} probability gain. 0 0 (14) Z Z n 3.1 Abnormal and Precursor Phenomena × f(Xi|µ, α) dµ dα, The continuing pursuit of possible algorithms used i Y=1 to predict large earthquakes should consider specific where F (y|µ, α) is the cumulative distribution of the developmental patterns listed in the seismic catalog. density f(y|µ, α) in (10), and h(y|µ, α) is the hazard So far, an alarm-type method of earthquake pre- rate function. This prediction is shown to provide diction (Keilis-Borok et al., 1988; Keilis-Borok and a systematically better performance in the sense of Malinovskaya (1964); Rundle et al. (2002); Shebalin expected entropy criterion (Akaike (1985)) than the et al. (2006); Sobolev (2001); Tiampo, et al., 2002) plug-in predictor in case of very small sample sizes based on seismicity patterns has been operationally of data (Nomura et al. (2011)). implemented, in which predictions are conveyed to EARTHQUAKE PREDICTION 9 seismologists through e-mail. Some predictions in greater than M in a specified area, under the con- each year are published in an official document such dition that N anomalies A,B,C,...,S appeared si- as the Center for Analysis and Prediction, State multaneously. Assuming that anomalies are condi- Seismological Bureau, China (1990–2003). In addi- tionally independent on EM and the complement of tion, many papers have been published on earth- EM , Aki (1981) derived the following equation of quake predictions. Some of these predictions may be Utsu (1977, 1979) using Bayes’ theorem: statistically significant against the stationary Pois- son process that is assumed for normal seismicity. P (EM |A,B,C,...,S) Such alarm-type predictions have been further eval- 1 1 1 (15) = 1+ − 1 − 1 − 1 · · · uated by Zechar and Zhuang (2010), Jordan et al. PA PB PC (2011), Zhuang and Ogata (2011) and Zhuang and      1 1 N−1 −1 Jiang (2012). · − 1 − 1 , Comprehensive studies of anomalous phenomena P P  S   0   and observations of earthquake mechanisms are es- . where P = P (E ), P = P (E |A), P = P (E |B), sential for predicting large earthquakes with high 0 M A M B M ...,P = P (E |S). Note that this formula can be probability gains. However, it is difficult to deter- S M written as a linear relation of logit functions of prob- mine whether detected abnormalities are precursors abilities [see equation (23) in Section 3.4.2]. These to large earthquakes. Nevertheless, we aspire to be- probabilities for a short time interval become very come able to say that the probability of occurrence of a large earthquake, in a certain period and a cer- small so that (15) can be approximated by tain region, has increased a certain extent as com- PA PB PC PS (16) P (EM |A,B,C,...,S) ≈ P0 · · · . pared with the reference probability. Therefore, it is P0 P0 P0 P0 necessary to estimate uncertainty of the nature and urgency of abnormal phenomena relative to their The above relation shows that for multiple inde- roles as precursors to major earthquakes. For this pendent precursors, the conditional rate of earth- purpose, it is necessary to study a large number of quake occurrence can be obtained by multiplying anomalous cases for potential precursory links to the unconditional rate P0 with ratios of conditional large earthquakes. Thus, incorporation of this in- probability to unconditional probability P0. Each formation in the design of a prediction model for ratio is defined as the probability gain of a precursor. probability that exceeds the underlying probability Utsu (1979) retrospectively reported a high proba- is important. bility forecast of the 1978 Izu–Oshima–Kinkai earth- quake of M7.0 using the multiple independent pre- 3.2 Conditional Probability of an Earthquake for cursor formula. This is based on each probability as- Multiple Independent Precursors sessment of the anomalous phenomena consisting of As previously described, although an individual uplift in the Izu Peninsula, swarm, and a composite precursory anomaly is insufficient for providing a of a radon anomaly, anomalous water table change forecast of an earthquake with a high probabil- and volumetric strain anomaly. Each of such prob- ity, forecasting probability can be enhanced if sev- abilities was not very high. Aki (1981) summarized eral anomalies are simultaneously observed (Utsu, the Utsu report and further explained similar possi- 1977, 1979, 1982; Aki (1981)). The probability of ble calculations for successful prediction of the 1975 an anomaly being a precursor of a large earth- Haicheng earthquake of M7.3 in China by consid- quake should be estimated through comprehensive ering long-term, intermediate-term, short-term and observations. Then, it provides medium- or short- imminent precursory phenomena. term probability forecasts depending on the time 3.3 Improving Probability Gains by Seeking scale of enhanced earthquake probability following Statistically Significant Phenomenon the anomaly. For example, identification of fore- shocks (Section 3.4.2) and seismicity quiescence Here I would like to describe several point process (Section 5.1) belongs to short- and medium-term models which can enhance the probability gains. To forecasting, respectively. examine whether certain abnormal phenomena af- Let us find the probability P (EM |A,B,C,...,S) fect changes in the baseline rate of earthquake occur- of occurrence of an earthquake, with a magnitude rences, Ogata and Akaike (1982), Ogata, Akaike and 10 Y. OGATA

Katsura (1982) and Ogata and Katsura (1986) an- An example is the data of unusual intensities of alyzed causal relationships between earthquake se- ground electric potential, which were observed in ries from two different seismogenic regions, A and the vicinity of Beijing, China, during a 16-year pe- A B B. Let Nt and Nt be the number of earthquakes riod beginning in 1982. The issue was whether or above a certain magnitude thereshold in the time in- not these factors were useful as precursors to strong terval (0,t) in regions A and B, respectively. Then, earthquakes of M ≥ 4.0. Electricity anomalies could consider a model of the intensity function of point have been aftereffects of strong earthquakes. How- A process Nt for earthquake occurrences in the region ever, by comparing the goodness of fit of models (17) A, conditional on the history of earthquake occur- by AIC, anomalies were deemed statistically signif- A B rences Ht and Ht in both regions: icant as precursors to earthquakes (Zhuang et al. (2005a)). Moreover, the conditional intensity rate of J t A B j A declustered earthquakes M ≥ 4.0 or larger within a λA(t|Ht ,Ht )= µ + ajt + g(t − s) dNs radius of 300 km from the Huailai ground-electricity j=1 Z0 (17) X station was given by t B t + h(t − s) dNs , a 0 λ(t|Ht)= µ + h(t − s)ξ(s) ds Z S where the first two terms on the right-hand side (19) Z t of the equation represent the Poisson process of a −0.142(t−j) 0.69 trend, the third term represents the cluster com- = 0.00702 + 0.000117e ξj j S ponent within region A including aftershocks and X= swarms, and the last term represents the effect of (event/day) in the study of Ogata and Zhuang earthquake occurrences in region B. (2001), in which successively occurring M ≥ 4 earth- Here, it must be noted that even if a significant quakes within five days and 30 km distance were re- correlation is observed between the two series of moved from the data to account for the self-exciting events, it is insufficient from the standpoint of pre- effect in equation (18). According to this model, the diction, and it is necessary to identify causality. rate of M ≥ 4 earthquakes varies from a half to 10 Thus, we must examine the opposite causality by times the average occurrence of 0.0126 event/day. interchanging A and B in equation (17). If both Furthermore, the time series of electric anomaly direction models hold, this process is mutually ex- records were available from three other stations near citing (Hawkes (1971)). Furthermore, the correla- Beijing. If we assume that the four sets of the time tion between A and B regions may be indirect such series are approximately independent, we may con- that activities in both regions may be affected by sider the following conditional intensity rate by ex- additional factors, for which the trend term may tending the multiple precursor in equation (16): be useful if the polynomial can efficiently capture 4 4 m such an effect. According to our analysis of seismic- λAm (t|Ht ) (20) λ t Hm ≈ λˆ ity in two seismogenic regions along the subducting A t A ˆ m=1 ! m=1 λAm Pacific plate interface beneath the central Honshu, \ Y 4 Japan, seismicity causality was found as a one-way for the common region A = m=1 Am among four effect from the deeper to the shallower. The maxi- circular regions Am of 300 km radii from the four mum probability gain of the causal effect was several stations. Retrospective total probabilityT gain varies times the average occurrence rate. in the range 1/10–100 times of the average occur- Similarly, we can examine the causal relationship rence rate λA in the common region (Ogata and from some observed geophysical time series of ξs Zhuang (2001)). (Ogata and Akaike (1982); Ogata, Akaike and Kat- Here, we considered declustered earthquakes near sura (1982)) as follows: Beijing, but if we can consider the original data and model that take the triggered clustering effect into J t A ξ j A account (cf. Zhuang et al., 2005a, 2013), the corre- λA(t|H ,H )= µ + ajt + g(t − s) dN t t s sponding model would become j 0 X=1 Z (18) 4 4 m t m λ(t|Ht ) + h(t − s)f(ξ ) ds. (21) λ t Ht ≈ λ0(t|Ht) . s ! λ0(t|Ht) 0 m=1 m=1 Z \ Y

EARTHQUAKE PREDICTION 11

A similar but more general model is applied in 3.4 Probabilistic Identification of Foreshocks foreshock forecasting discussed in the following sec- The study of foreshocks should lead to a short- tion (Section 3.4.2). term forecasting. Although a considerable number Moreover, we examined periodicity, or seasonal- of foreshocks are observed, most are recognized af- ity, of earthquake occurrences (Ogata and Katsura, ter occurrence of a large earthquake. Nevertheless, 1986; Ma and Vere-Jones (1997)). Although such when earthquakes begin to occur in a local region, issues have been frequently discussed in statistical its residents should determine whether or not such seismology, it was difficult to analyze correlations in movement is a precursor of a significantly larger the conventional method because the clustering fea- earthquake. The probability of foreshock type can ture of earthquakes frequently leads to incorrect re- be determined statistically from the data of ongo- sults (Aki, 1956). On the other hand, we found it ef- ing earthquakes in a particular region. Moreover, by fective to apply statistical models of stochastic point using composite identification data of magnitude se- processes that incorporate a clustering component; quence and degree of hypocenter concentration, the further details can be found in the study of Ogata probability gain of prediction is heightened. (1999a), and the references therein. These models can also be applied to examine whether or not var- 3.4.1 Working definitions for foreshock discrimi- ious geophysical anomalies are statistically signifi- nation When an earthquake of M4.0 or larger oc- cant as precursors of an approaching large earth- curs, it must first be determined whether or not the quake: movement is a continuation of nearby earthquakes. J Precisely, the connection to past earthquakes is de- j λθ(t|Ht)= µ + ajt termined by the single-link clustering (SLC) algo- j=1 rithm of Frohlich and Davis (1990). X The largest earthquake in a cluster is designated K 2πkt 2πkt as the main shock. Pre-shocks refer to all earth- (22) + c2k−1 cos + c2k sin quakes preceding the main shock of a cluster. All T0 T0 Xk=1  pre-shocks in a cluster become foreshocks when the t magnitude gap or magnitude difference between the + g(t − s) dNs. largest pre-shock and main shock is 0.45 or greater. 0 Z If the magnitude gap is smaller than 0.45, the cluster From estimated amplitudes of the one-year peri- with pre-shocks is defined as a swarm. An additional odic term with T = 365.24 days, it is evident that 0 type of cluster is the main shock–aftershock type, probability gains vary around the average occur- in which the main shock occurs first in the clus- rence rate of corresponding M ≥ 4.0 and M ≥ 5.0 ter. A magnitude gap of 0.45 or larger between the earthquakes. More extensive studies were reported main shock and largest pre-shock occurs in less than by Matsumura (1986) by using the above model, in approximately 20% of pre-shock clusters in Japan. which the trend-term (first two terms) was used for This 0.45 borderline of foreshock- and the swarm- artificial nonstationarity due to an increasing num- types has been determined by a trade-off between ber of observed earthquakes in the long-term global achievement of a larger magnitude gap, which re- catalog. He detected periodic effects for many mid- sults in better discrimination of the foreshock, and latitude seismic inland regions, whereas the season- a greater number of foreshock clusters, which re- ality was rarely observed in tropical seismic regions sults in better statistics. Here, to characterize these and ocean seismogenic zones. Correlations with pre- features, we note that clusters of foreshock-type cipitation variations were common in these results exclude main shock and subsequent aftershocks, and are most probably due to pore fluid pressure whereas other cluster types include all events in each changes in faults (see Section 4.1 for the physical cluster. This designation is made because real-time mechanism). An extension of the above periodic- recognition of the main shock, which is preceded by ity model, reported by Iwata and Katao (2006), in- foreshocks, is easy owing to the large magnitude gap, cludes a combination of (lunar) synodic and semi- whereas main shocks of other cluster types are dif- synodic periods to examine whether or not and how ficult to recognize until the end of the cluster. certain seismicity is affected by Earth tides. Statis- tical models, applications to validate data from the 3.4.2 Probability forecast by discrimination of earthquake-induced phenomena and their references foreshocks Using the location (x,y) of the first were reviewed by Ogata (1999a). earthquakes from clusters or isolated single earth- 12 Y. OGATA quakes, by the empirical Bayesian logit model, (23) 3 k Ogata, Utsu and Katsura (1996) obtained a map × µ0 + bkγi,j µ(x,y) of a probability that the earthquake will be a i 0, The earth crust and upper mantle lithosphere can respectively, where σ1 = 0.6709 and σ2 = 0.4456 be approximately considered as an elastic body. (km). These become distorted under stress, which in- Suppose that, at the current time, c|n shows the creases steadily in a particular direction. Fault stage where the nth earthquake (n = 2, 3, 4,..., #c) planes are cracks within the lithosphere or plate has occurred in a cluster c, where #c is the number boundary interfaces. Earthquakes occur through of all earthquakes in the cluster c. We propose the distortion of subsurface rocks and both sides of the forecasting probability pc|n by using the following fault plane moving out of alignment. The earth- logistic transformation: Set f = logit p = (1 − p)/p, quake location listed in hypocenter catalogs is lo- or p = 1/(1 + ef ); then, cation at which a fault displacement started, and earthquake magnitude represents eventual size of logit pc|n = logit µ(x1,y1) the displacement. Moreover, some catalogs record 1 the orientations and slips of fault planes of relatively + #(i < j ≤ n) large earthquakes. EARTHQUAKE PREDICTION 13

For each fault plane, stress tensor in lithosphere 4.2 Predicting Seismicity in the Peripheral Area is decomposed into two perpendicular components. by Abrupt Stress Changes The shear stress acts in a direction parallel to fault To explain earthquake induction or suppression shifting, and the normal stress acts perpendicular to of seismicity, it is useful to determine whether or the fault plane. The orientation of each fault plane not ∆CFS was due to a rapid faulting event that determines shear and normal stress, which define caused an earthquake. When a large earthquake oc- Coulomb failure stress (CFS): curs, low-frequency seismic waves and global posi- CFS = (Shear Stress) − (friction coefficient) tioning system (GPS) crustal displacement are ob- (24) served. From such observations, source fault mecha- × (Normal Stress − pore fluid pressure). nisms of the earthquake can be solved, such as size, orientation and vector of the fault slip. Such source CFS increases at a constant rate over time. When parameters are input into a computer program de- CFS exceeds a particular threshold, the fault slips signed by Okada (1992) to calculate ∆CFS in a re- dramatically (an earthquake). Then, the stress re- ceiver fault system on the basis of source fault data. duces to a certain value and accumulates again Thus, studies on induction of earthquakes, based on over decades to result in large earthquakes on plate ∆CFS, have become popular. Special issue volumes boundaries, and over thousands of years to result in on this subject have been edited by Harris (1998) slips on inland active faults (see Section 2.3). and Steacy, Gomberg and Cocco (2005). When an earthquake occurs, the displacement For example, Ogata (2004b) examined regional of the source fault causes sudden Coulomb stress ∆CFS in southwestern Japan by analyzing M7.9 To- changes (∆CFS) on the peripheral receiver faults. nankai and M8.1 Nankai earthquakes in 1944 and The ∆CFS of each receiver fault plane either de- 1946, respectively. Conventionally, some seismic qui- creases or increases depending on its orientations escence in this period was either considered as a relative to the slip angles of the source fault. On the genuine precursor or suspected as an artifact be- faults with increased ∆CFS, earthquakes occur ear- cause of incomplete detection of earthquakes during lier than expected, whereas on those with decreased the Second World War. Positive and negative ∆CFS ∆CFS, forthcoming earthquakes are delayed. When correlated strongly with seismic activation and qui- faults of similar orientations dominate a region, ei- escence, respectively. In particular, this study classi- ther seismic activation or quiescence is expected in fied seismicity anomalies into pre-seismic, coseismic the region. and post-seismic before, during and after massive In equation (24), the pore fluid pressure of the earthquakes, respectively. These scenarios may be gap fault related to CFS is generally a constant. helpful in interpreting seismicity in western Japan However, its changes may be important. For exam- prior to occurrences of expected subsequent large ple, pressure changes in the fluid magma gap affect earthquakes along the Nankai Trough. swarm activity in volcanic areas (Dieterich, Cayol and Okubo, 2000; Toda, Stein and Sagiya, 2002). 4.3 Physical Implication of the ETAS Model and In addition, earthquakes can be induced through Seismicity increased pore fluid pressure in a fault system In general, interactions among earthquakes are (Hainzl and Ogata (2005); Terakawa, Hashimoto fairly complex. Once an earthquake occurs in a par- and Matsu’ura (2013)), which is occasionally due to ticular location, CFS of the fault system adjacent heavy rainfall or shaking of the earth crust owing to to the source fault is considerably increased, and propagated strong seismic waves; the latter causes many earthquakes are induced. Traditionally, these dynamic triggering (Steacy, Gomberg and Cocco, earthquakes are called as aftershocks. Some of them 2005, and papers included in the same volume). are induced outside the aftershock region; these are Relevantly, the seasonal nature of seismicity or an- also called as off-fault aftershocks, or aftershocks in nual periodicity has been discussed in Section 3.3 a broad sense. Significant changes in stress result [cf. equation (22)]. Moreover, the ETAS model ap- in many aftershocks; even small changes can induce plications to seismicity changes that were induced aftershocks to some extent. Furthermore, any after- by dynamic triggering or injection of water were shock can change stress, too, causing their after- reported (Lei et al., 2008, Lei, Xie and Fu, 2011). shocks. Because of such complex interactions among 14 Y. OGATA invisible fault segments in the crust, detailed calcu- studies such as aftershocks in Japan and the time lations of such stress changes are difficult and im- evolution of seismicity rate. Because regional charac- practical. teristics of earthquake occurrences can be captured Therefore, statistical models designed to describe and considered as typical seismicity in this model, the actual macroscopic outcome of these stress inter- it has been accepted by seismologists as a standard actions are required. For example, the ETAS model model of ordinary seismicity. The ETAS model is in equation (5), which consists of empirical laws used as a “barometer” for detecting significant de- of aftershocks, quantifies dynamic forecasting of in- viations from normal activities as demonstrated in duced effects. By fitting to the selected data from Figure 4 in Ogata (2005a). the catalog earthquake, this ETAS model deter- mines the parameters by the maximum-likelihood 5. SEISMICITY ANOMALIES FOR method. Thus, prediction of earthquakes conform- INTERMEDIATE-TERM FORECASTS ing to regional diversity is possible. 5.1 Seismicity Quiescence Relative to the ETAS On the other hand, the friction law of Dieterich Model (1994), which was developed on the basis of rock fracture experiments with controlled stress, can be The deviation of actual cumulative number of linked to statistical laws of earthquake occurrences. earthquakes is measured relative to the theoretical In particular, this law reproduces temporal and spa- cumulative function of the earthquake that serves tial distribution of the attenuation rate of after- as an indefinite integral in time for the predicted shocks, such as that determined by the O–U law rate function (7) of the ETAS model. Relative qui- in equation (2). However, because of seismicity di- escence occurs when actual earthquake occurrence versity, predictions adapting well to development of rates are systematically lowered in comparison with seismicity appear to be difficult. the predicted incidence that is determined by the Seismicity anomalies can hardly be detected by ETAS model (Ogata (1992)). Relative quiescence observing conventional plots of earthquake series lasting for many years was observed in a broad because they show a complex generation due to region before great earthquakes of M8 class and successive occurrences of earthquakes or clustering. larger occurred in and near Japan (Ogata (1992), The clustering nature is also the main difficulty 2006b). Similar phenomena were observed before for traditional statistical test analysis. Complexity M9-class large earthquakes in other regions of the due to the clustering feature creates difficulties in world. revealing anomalies of seismicity caused by slight Since 2001, I have reported 25 agendas of var- stress changes, hence, various anomalous signals are ious seismicity anomalies and forecasting propos- missed. als in Japan at the Coordinating Committee for Therefore, some seismologists have devised vari- Earthquake Prediction of Japan (CCEP). Except ous declustering methods that include only isolated the agenda that reported seismic quiescence of af- and largest earthquakes in a clustering group or the tershock activity before the largest aftershock (see main shock, and other earthquakes are excluded. Section 6.2 for detail), all were ex-post analysis re- On the basis of declustered data, statistical signif- port; the agendas were summarized in Ogata (2009). icance of seismic quiescence was tested against the In addition, among 76 aftershock cases in Japan Poisson process. Occasionally, however, analysis re- that I have investigated, relative quiescence was ob- sults depend on the choice of criteria of the adopted served in 34 (see Ogata (2001b), and its appendix declustering algorithm (Van Stiphout, Zhuang and for details of the case studies). Moreover, Section 5.4 Marsan, 2012). Hence, results could be due to arti- includes a discussion on the manner in which re- ficial treatment. In addition, declustering methods sults of this aftershock study will be used for space– result in a significant loss of information because time probability prediction of a neighboring large they discard a large amount of data from the origi- earthquake with a size similar to that of the main nal catalog. shock. The ETAS model uses original earthquake data Here, I will note the results on the aftershock without declustering. As mentioned in Section 2.2.3, research of inland earthquakes of M6.0 or larger the ETAS model is a point process model configured in southwestern Japan that occurred 30 years be- to conform to empirical laws derived from various fore and after the M8.1 Nankai earthquake in 1946. EARTHQUAKE PREDICTION 15

Fig. 4. Shallow earthquakes in the inland rectangular region in (a) were analyzed from August 2002–July 2003 to investigate the effect of the M7.0 earthquake in 2003. CFS increments of this region took the largest values transferred from the M7.0 earthquake that is shown by the small rectangle fault located at (141.7◦E, 38.8◦N) at a depth of 71 km. Epicenters and latitude versus time are shown in (a) and (b), respectively. The occurrence time of the M7.0 earthquake is shown in each panel as the vertical dotted line indicated by TJ. The ETAS model was fitted to the data from the target period (TS, TJ). The panel (c) shows cumulative numbers and magnitude versus ordinary time; and the panel (d) shows these values against the transformed time determined in equation (7) by the estimated ETAS model. The black and grey cumulative function in panels (c) and (d) show the empirical cumulative function (step function) and the theoretical one (curve and straight line) estimated and then predicted by the ETAS model, respectively. At the southeastern corner of the inland rectangular region, a large M6.2 earthquake (occurrence time indicated by TM) and its largest M5.5 foreshock (indicated by TF) occurred on July 26, 2003. The panel (d) shows that the foreshock activity was more active than was expected, which is seen from the steepest slope of the cumulative function in (d). In contrast, the aftershock activity of the M6.2 earthquake, during the period TM-Tend, appears to be similar to predicted rates in (d). Dotted parabola-like envelope curves show twofold standard deviations (95% error bands) of cumulative numbers of the transformed time.

Among six earthquakes which occurred before 1946, and future aftershock activity of similar large inland relative quiescence was observed in five aftershock earthquakes. sequences. In contrast, among seven earthquakes af- 5.2 Aseismic Slip, Stress Change and Seismicity ter 1946, relative quiescence was not observed in Anomalies six aftershock sequences, and these aftershock se- quences were on track as expected. Since the ERC Since a dense GPS observation network was es- forecasts the next large earthquake for the next 30 tablished in Japan, aseismic fault motions or slow years 60–70% (see Section 2.3, also see Ogata, 2001b slips that could not be detected by seismometers and 2002), it would be worthy to monitor recent have been successively identified in the plate bound- 16 Y. OGATA

(a) (b)

Fig. 5. (a) Seismic activity for the eight years period before the M6.8 earthquake of 2004. Dominating fault slip orientations of earthquakes in this area are similar to those of the main shock and its aftershocks. The small black rectangle in the center of the magnified geographical map shows the main shock fault model determined by the GPS observations. The regions of thin and thick contours show positive and negative Coulomb failure stress (CFS) increments, respectively, assuming that slow slips in the source have occurred for some time. These define four subregions N, S, W and E used for the following ETAS analysis. (b) The four panels show the empirical cumulative curve (thin black) of the sequence of earthquakes of magnitude 2 or larger in each of the four divided subregions from 1997 until the M6.8 earthquake (downward arrows). Thick gray curves show estimated and predicted cumulative functions before and after each change-point time, respectively. Activation and quiescence relative to those predicted by the ETAS model agrees with increase and decrease in CFS, respectively. ary regions. We can take occurrence of such motion 5.3 Variation of Local Stress Deduced from into account in discussing the relationship between Spatio-Temporal Variation of Aftershocks seismicity anomalies (quiescence or activation) and Local anomalies occur in space–time locations of stress changes. most of the aftershocks. To elucidate these anoma- Specifically, it can be assumed that slow slips on a lies, we firstly apply the O–U formula (2) for after- focal fault or its adjacent part have occurred during shock decay to data of occurrence times and convert a particular period. Then, depending on dominat- these times by the estimated theoretical cumulative ing orientations of receiver faults neighboring the function (7). We then examine whether space–time focal fault, CFS could decrease or increase. Accord- coordinates on a projected line, such as longitude ingly, we expect that seismicity there decrease or and latitude, against the converted time remained increase relative to the expected occurrence rate by temporally uniform or not. If nonuniformity in a cer- the ETAS model. Such seismicity anomalies are re- tain portion of space–time conversion is observed, vealed before some recent large earthquakes (Ogata, this implies discrepancies between theoretical and 2005b, 2007, 2010a, 2011c; and Kumazawa, Ogata actual aftershock occurrences in such a place. Sev- and Toda, 2010). See Figure 5 for an example. By eral possible scenarios for such discrepancies are of- assuming slow slip on the source fault, the peripheral fered: Secondary aftershocks that follow a large af- regions were classified as either of the increasing or tershock are obvious once seen as a cluster. Such a decreasing CFS. Then, each region can theoretically cluster shows traces of a new local rupture to ex- correspond to an area that either promoted or sup- tend the peripheral portion of the fault of the main pressed seismicity. Such anomaly patterns of seis- shock. Moreover, when a nonuniform portion other micity relative to the ETAS model (5) are in good than the secondary aftershocks is observed, it is cru- agreement with those of CFS increment. cial for us to explore the reasons. EARTHQUAKE PREDICTION 17

Based on recent accurate aftershock location data, of the distance between stations is basically linear Ogata (2010b) revealed local relative quiescence and with time because the subducting plate converges activations; these can occur associated with post- with constant speed. However, several years prior to or pre-slips of a large aftershock. These anomalies large inland earthquakes of M7 class, the time series were systematically investigated assuming that they of the baseline distance variation around the fault were related to changes in the CFS rate. In addi- was observed with systematic deviation from a lin- tion, assuming several scenarios of stress changes ear trend (Ogata, 2007, 2010a, 2011c; Kumazawa, due to slow slips, Ogata and Toda (2010) and Ogata Ogata and Toda, 2010; GSI, 2009). Each deviation (2010b) performed simulations to reproduce seismic- of these baselines was consistently explained by slow ity anomalies of relative activation and quiescence slips on the earthquake source fault or on its down- within aftershocks on the basis of the rate/state fric- dip extension. These results were due to post-hoc tion law of Dieterich (1994). analysis based on knowledge of the source fault ob- 5.4 Space–Time Probability Gain of a Large tained by coseismic displacement. Earthquake Under Relative Quiescence of From a predictive perspective, it is highly desir- Aftershocks able to estimate such a fault slip in near real-time to The probability of relative quiescence being pre- that of occurrence. So far, several estimates of suf- cursor to a large earthquake must be evaluated with ficiently large slips on plate boundaries have been their likely time and location. Because these in- obtained from GPS records by inversion analysis. volve many conditions and a number of parameters, GSI has regularly reported such estimates of coseis- they cannot be easily stated. However, by statisti- mic, post-seismic and large-size habitual slips, at cal studies of aftershock sequences in Japan (Ogata, the CCEP meeting. However, it is difficult to ob- 2001a), what I can say about a probability gain that tain fine images of small slips, particularly in in- a large earthquake will occur is as follows: First, if a land, even though inland GPS stations are arranged large earthquake occurred in a particular location, closely. This is attributed to high seismicity rather the probability per unit area that another earth- than GPS observation errors. Because strong earth- quake of similar magnitude will occur in the vicin- quakes occur frequently, various effects of slow slips ity is greater than that which will occur in a distant in GPS records are mixed up with such stronger area. This is the result of simple statistics regarding changes. Hence, development of statistical models the self-similarity feature (inverse-power law corre- and methods to separate such signals is crucial. To lations), and also physically suggests that the neigh- estimate slow slips more precisely, combined model- boring earthquake will be more probably induced by ing and analysis of seismicity and geodetic anomalies a sudden stress change on the periphery because of will be useful. Analyzing both seismicity and tran- the abrupt slip of the earthquake. Moreover, if after- sient geodetic movements in a number of areas and shock activity becomes relatively quiet, it becomes locating the area of aseismic slip is very important more likely that large aftershocks will occur around for increasing the probability gain of a large earth- the boundary of the aftershock area. Furthermore, if quake. relative quiescence lasts for a sufficiently long time more than a few months, the probability that an- 6.2 Considering the Scenario of an Earthquake other earthquake of similar magnitude will increase from Aseismic Slip within six years in the vicinity of the aftershock area Observed anomalies of crustal movement and seis- within 200 km distance. micity assume fault mechanisms and locations of 6. SEISMICITY AND GEODETIC ANOMALIES slip precursors for prediction probability; therefore, their uncertainty must be estimated. In addition, 6.1 Aseismic Slip and Crustal Deformation probabilities of considered scenarios must be esti- The Geological Survey Institute (GSI) of Japan mated. Such tasks are difficult. A possible method compiles daily geodetic locations of global position- is to consider the logic tree of various scenarios re- ing system (GPS) stations throughout Japan, and garding destruction of the fault system by attaching baseline distances between GPS stations can be cal- appropriate subjective or objective probabilities to culated from data in the GPS catalog. The geode- tree components, as was performed for long-term tic time series show that contraction or extension predictions in California and Japan. Hence, such a 18 Y. OGATA scenario ensemble gives a forecast probability. Simi- boundary, where the next great earthquakes are ex- larly, medium- and short-term prediction logic trees pected. of various scenarios must be considered. At the CCEP meeting on April 6, 2005, I reported 7. CONCLUSIONS relative quiescence of aftershocks of the Fukuoka– To predict the future of a complex and diverse Oki earthquake (Ogata, 2005d). In addition, I ex- earthquake generation process, probability fore- amined potential slow slip areas on nearby active casting cannot be avoided. The likelihood (log- faults that may have created a stress shadow (re- likelihood) is rational to measure the performance gion of decreased CFS) to cause relative quiescence of the prediction. To provide a standard stochastic in the aftershock sequence. Among these areas, the prediction of seismic activity in long term and short Kego fault, traversing Fukuoka City, had a large term, it is necessary to construct proper point pro- positive ∆CFS because of the main shock rup- cess models and revise those that conform to each ture, which showed evidence of possible slow-slip in- region. By the appearance of the anomaly, we need duction. Furthermore, the seismogenic zone along to evaluate the probability that it will be a pre- the Kego fault had already activated before the cursor to a large earthquake. Namely, we need to Fukuoka–Oki earthquake occurred (Ogata (2010a)). forecast that the probability in a space–time zone will increase to an extent, relative to those of the ref- Therefore, because a slow-slip scenario on this fault erence probability. For this, we make use of a point was possible, I examined the pattern causing stress process model for the causality relationship. variation in the aftershock area. However, no stress It is desired to search any anomaly phenomena shadow in the aftershock area was found. Therefore, that enhance the probability gains. Having such I determined that probability of slow slip on the anomalies, application of the multiple element pre- Kego fault was quite low. I also examined whether diction formula increases a precursory probability. other possible slow slips in neighboring active faults A comprehensive physical study between precursory could create a stress shadow that covered the af- phenomena and earthquake mechanisms is essential tershock region. However, no large earthquake has for composing useful point process models. These occurred in those faults thus far. elements must be incorporated to achieve predicted Approximately one month later, however, the probability exceeding predictions of typical statisti- largest aftershock occurred at the southeast end cal models. of the aftershock zone. Post-mortem examination Furthermore, to determine urgency and uncer- based on information of the fault mechanism of this tainty of major earthquakes against abnormal phe- aftershock and detailed aftershock data revealed a nomena, numerous research examples must be ac- detailed scenario. This means that by assuming a cumulated. On the basis of these examples, possi- slow slip into the gap between the fault of the largest ble prediction scenarios must be presented. Further- aftershock and main shock, relative quiescence of ac- more, to adapt well to diversity of earthquake gen- tivity in the deeper part of the aftershock zone can eration, it is useful to adopt Bayesian predictions be clearly explained (Ogata (2006a)). Moreover, the (Akaike (1980b); Nomura et al. (2011)) and consider slip can explain relative quiescence in the induced region-specific models. My experiences thus far confirm that the method swarm activity that occurred away from the after- of statistical science is essential to elucidate move- shock area (Ogata (2006a)). ment leading to prediction of a complex system. This setting as a prediction of future scenarios is There is a need for development of a forecasting much more vague and difficult to explain, even if it model that reflects diversity of the vast amount includes an ex-post scenario. Moreover, the time of of information on seismicity and various covariate occurrence must be predicted in addition to loca- data. I believe that these will be developed by in- tion, which is more difficult. Occurrence of slow slip venting an appropriate hierarchical Bayesian model. does not always indicate a proximate precursor of Space–time models for seismicity have become in- fault corruption. Nevertheless, it is desired to keep creasingly complicated (Ogata, 1998, 2004a, 2011b; observing GPS data to form scenarios of forthcom- Ogata, Katsura and Tanemura (2003); Ogata and ing large earthquake. For example, using Bayesian Zhuang (2006)). inversion by using GPS records, Hashimoto et al. A similar evolution is required for statistical mod- (2009) estimated the locked zones on the plate els of geodetic GPS data. EARTHQUAKE PREDICTION 19

ACKNOWLEDGMENTS Dieterich, J. (1994). A constitutive law for rate of earth- quake production and its application to earthquake clus- I am grateful to the Japan Meteorological Agency tering. J. Geophys. Res. 99 2601–2618. (JMA), the National Research Institute for Earth Dieterich, J., Cayol, V. and Okubo, P. G. (2000). The Science and Disaster Prevention, and universities use of earthquake rate changes as a stress meter at Kilauea 408 for the providing hypocenter data. I am also grate- volcano. Nature 457–460. Earthquake Research Committee (1998). Regarding meth- ful to the anonymous referee, Associate Editor and ods for evaluating probility of aftershocks. Available at the Editor for their careful reviews and suggestions, http://www.jishin.go.jp/main/yoshin2/yoshin2.htm. which led to a significant revision of the present Earthquake Research Committee (2001). Regarding meth- manuscript. This work was supported by JSPS ods for evaluating long-term probility of earthquake oc- currence. 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