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1 次の地震のマグニチュード予測と評価 Magnitude forecasts of the next earthquake and evaluation 統計数理研究所 The Institute of Statistical Mathematics CSEP の地震予測検証実験が始まって 10 年以上経つその主な取組みは空間領域 ( 例えば 3 ヶ月 1 年 5 年間における予測 ) および時空領域 ( 日々の予測 ) における確率的予測を行いそれらの性能を評価することである CSEP の主な目的は様々な地震活動モデルの開発を促進し各地の通常の地震活動の標準的な相場を確立することで異常現象に基づいた大地震の予測の各種提案に対する客観的評価のインフラを整備することであるこれまでのところ CSEP の殆どの提案モデルの地震マグニチュード ( 以下 M と記す ) 予測は実験全域および全期間にわたって同一の b 値の Gutenberg-Richter(G-R) 則に基づく独立分布系列を仮定しているこれは実際には二重の意味で単純であると考える第 1 に G-R 則の b 値は地域性があるこのような b 値モデルは CSEP で唯一検証中である 1) 第 2 に G-R 則の b 値や一般の M 分布は地震活動の履歴に依存する可能性がある本報告では前震群, 群発地震群, 本震余震群の統計的判別による方法 2) を参考に CSEP の検証規格に則って過去の震源データから逐次次の地震の M の確率予測を試み b = 0.9 の G-R 則 ( 以下基準 G-R 則と記す ) と比較し検証した 1

2 先ず M 4 の気象庁地震カタログから Single-link 法 3) で群分けを行い, 第 1 図にある様に群内の M 列がそれまでの最大 M より 0.5 以上の飛躍 ( M 0.5 ) がある毎にリセットし予測 M 確率分布を再計算するすなわち先頭の地震 ( 孤立地震を含む ) に関しては基準 G-R で予測し群の 2 番目の地震が M 0.5 の大きな地震である確率は p 2 c = µ ( x, y ) 1 1 n 3 番目以降の地震が M 0.5 の大きな地震である確率 pnc は図 107 頁中の式で計算するそして各時点での M の予測確率密度分布は第 112 頁図の式 Ψ ( M M1,, ) で与えられている各時点での M 予測の性能は図 11 頁 8 の最下行にある対数尤度比で比べることができる大規模なクラスターの地震は殆ど負の情報利得スコアが得られ小さいサイズのクラスターは一般に正の利得を取る 85% 以上のクラスターは高々 4 つの地震しか含まないのでクラスターの 5 番目以降の地震については基準 G-R で予測する事にすると全体としてこの予測は基準 G-R より優位であることが分かる ((14 第 3 頁図 ) ) この様に様々な前震型アルゴリズムに対応する M 配列の分布を単一の G-R 型から適切に広げることは大地震の確率利得を高め有用である M n 10 頁 2 参考文献 1)Ogata, Y., 2011, Earth, Planets Space, 63, )Ogata, Y., Utsu, T. and K. Katsura, 1996, Geophys. J. Int., 127, 17. 3)Ogata, Y., Utsu, T. and K. Katsura, 1995, Geophys. J. Int., 121, 233.

3 3 地震マグニチュードの予測と評価尾形良彦統計数理研究所

4 All Japan California Italy { an event in a bin t

yj, M j); tj t } Pr [, ] [, ] [, ] [, ], λ(, txym,,

< 地震の発生履歴 ; Ft その他データ { } Pr an event in[ t, t + t)

Iso-contour of λ(t, x, y Ht) Space-Time ETAS model

p + M j { jt : }( ) j t t tj c e α < + x x 1 j

4 4 All Japan California Italy { an event in a bin t t + t x x + x y y + y M M + M Ht Ft} Ht { ( tj, xj, yj, M j); tj t } Pr [, ] [, ] [, ] [, ], λ(, txym,, Ht, Ft ) t x y M t 時刻 ; (x, y) 経度緯度 ; M マグニチュード ; = < 地震の発生履歴 ; Ft その他データ { } Pr an event in[ t, t + t) [ x, x + x) [ y, y+ y) Ht λ(, t xy, Ht ) t x y Iso-contour of λ(t, x, y Ht) Space-Time ETAS model K Qj( x xj, y yj) λ θ (, txy, Ht ) = µν ( xy, ) + d p + M j { jt : }( ) j t t tj c e α < + x x 1 j where Qj( xy, ) = ( x xj, y yj) Sj y yj q latitude longitude

0 (, ) 1 K x y x x j j j y yj S x xj y y j j (, t x, y

5 4/14 00:00 M7.3 4/15 01:03 (, ) (, ) (d) M6.

5 0 (, ) 1 K x y x x j j j y yj S x xj y y j j (, t x, y Ht ) = ( x, y) + p( xj, yj) ( xj, yj)( M j M c ) { j; tj < t} ( t tj + c) e α λ µ Rates of M 4 event during the 2016 Kumamoto sequence M 4.0 (a) (b) (c) M6.5 4/14 00:00 M7.3 4/15 01:03 (, ) (, ) (d) M6.4 4/14 22:26 4/16 13:25 t + d q 5 event/day/100 km^2 event/day/100 km^2

6 地震マグニチュードの予測モデル 6 基準モデル : Gutenberg-Richter 則 (b= 定数 ~ 0.9) GR GR a b( M M ) c ( M ) = 10 Gutenberg-Richter 則 (b= 位置依存, Ogata, 2011 EPS) axy (, ) bxy (, )( M M ) c ( M xy, ) = 10 履歴に依存するマグニチュード分布但し. ( H ) Γ( M Ht ) dm = P M < Magnitude M + dm t { (,,, ); } H = t x y M t < t t j j j j j

1926-1993 M 4 d = + ( c ) 2 2 ST space time 0.3 (or 33.

earthquake 7 x First earthquake Probability of the first event of the

7 M 4 d = + ( c ) 2 2 ST space time 0.3 (or 33.33km) c = 1 o / month 1 km / day x x Isolated or the first M 4 earthquake 7 x First earthquake Probability of the first event of the cluster or isolated event will be FORESHOCK 1% Probability 群れの先頭 ( 孤立地震を含む ) が前震である確率の地域性 10%

8 8 d = + ( c ) 2 2 ST space time 0.3 (or 33.33km) M 4 Ordinary time (days) Order in number (events)

9 SPACE ln % 時間経過ク0% Probability of ラthe first event Aftershocks of the cluster スor isolated event タwill be Swarms FORESHOCK ーForeshocks F c10% の時間差 ( 日 ) 前µ ( x 震1, y1) A 10% 確率1% S F 予F 測震央間距離 (km) p c 0.1% A S F 単位立方体への変換 ( τ 0.01% i, j, ρi, j, γi, j) マグニチュード差 pc 1 k k k = 1 µ ( x1, y1) ln + a1 + bkγi, j + ckρi, j + dkτi, j pc #{ i< j} µ ( x, y ) i< j k= 1 k= 1 k= Ogata et al. (1995, 1996; GJI ) Ogata & Katsura (2012, GJI ; 2014, JGR) (1 ケ月 ) 予測と実際の結果 M 4 実際の前震型その他クラスター内の地震の順番 9

10 Segmentation of Single Link Clusters ΔM 0.5 ΔM < 0.5 ΔM 0.5 ΔM p = cn Sub-clusters Sub-clusters 地震群 c のn 番目の地震でマグニチュードが0.5 以上の更新確率 1 p 1 µ ( x, y ) 1 ln p < cn k k k = ln + µ ( x a1 bkγi, j ckρi, j dkτi, j cn #{ i j} , y1) i< j< n k= 1 k = 1 k= 1 1% 地震群の先頭が前震である確率 µ ( x, y ) % SPACE 1 Sub-cluster 時間経過 5 係数 from Ogata, Utsu and Katsura, 1996, GJI k ak bk ck dk

11 { } ( n c) Magnitude Gap : M = max Mk ; k = 1,, n in cluster c Probability of M Mmax+0.5 of the next magnitude; n c = > If ( tn+ 1, xn+ 1, yn+ 1)is connected to c, 1 ( M ) Ψ ( M M1,, M n) = (1 pnc ) ( nc ) + p dm bm bm ( nc ) ( n c) ( Mc, M ) ( M, ) ( ) M nc M bm bm 10 dm ( nc ) 10 M M c 11 ( n c) { n+ 1 } p P M M in c Probability density Probability density Ψ ( M ) M,, 1 M n log Probability distribution p n c p n c M ( n c) Magnitude Otherwise, the reference model M ( n c) Magnitude M ( ) ( ) bm Ψ M = ( Mc, ) M M c ( n c) bm dm Magnitude log likelihood-ratio = information gain: log LL 0 = # c log Ψ ( n) c( Mn+ 1 Mc ) Ψ ( M ) c n= 1 c n+ 1

12 12 All Japan M 4 log Ψ ( n) c( Mn+ 1 Mc ) Ψ c ( M ) n+ 1 = Information gain score per earthquake (+ signs) + = score/event (x500) magnitude All clusters c

13 Single-linked clusters used for the learning Log cumulative number of clusters 100% Number of cluster members Forecasts 13 Single-linked clusters used for the experiments Probability of foreshocks in log scale 10% 1% 0.1% 0.01% M M c = 3.95 Actual foreshock cluster Other type cluster Order n of earthquake in a cluster c

14 Information gain scores; All Japan , M = score/event (x500) magnitude All clusters c + = score/event (x100) magnitude Cumulative Information gains Only for thefirst 4earthquakesin each cluster cumulative scores (x1) Order in number (events)

15 Field et al. (2017, BSSA) ETAS (no fault) 15 UCERF3-ETAS

16 まとめと提案 16 (1) CSEP プロジェクトの次の課題は地震発生履歴の特徴および関連地球物理的異常現象に関係するマグニチュード予測モデルを探求することである地震発生特徴には地震マグニチュード列の変化前震判別に有効な時空間クラスタリングの集中性の強さ地震の静穏化と活発化および先駆的群発地震活動などが含まれる (2) 警報型の大地震の予測は経験的な成功率の統計を考慮してマグニチュードの分布でモデル化することもできるこれらは前駆的異常情報に基づくマグニチュードの予測アルゴリズムとして提案すればそれらを独立 G-R 分布を基準モデルとして情報利得を比較できる (3) 既存の CSEP の時間空間マグニチュードの対数尤度スコアを用いて試験を総合的に実施すべきであるしかし CSEP で採用されている従来のマグニチュードテストではマグニチュード予測の体系的な違いには関係していないテストはモデルを改善するための診断目的で使用する必要があるためマグニチュード頻度に関する対数周辺尤度の局所的なスコアまたは対数の条件付き尤度によるテストを実行できる

17 All Japan M 4 log Ψ ( n) c( Mn+ 1 Mc ) Ψ c ( M ) n+ 1 = Information gain score per earthquake (+ signs) + = score/event (x500) cumulative scores (x1) Cumulative Information gains magnitude

Algorithm of foreshock probability calculations in case of plural earthquakes in a cluster For plural earthquakes in a cluster, time differences, (days),epicenter separation (km),magnitude difference

18 Algorithm of foreshock probability calculations in case of plural earthquakes in a cluster For plural earthquakes in a cluster, time differences, (days),epicenter separation (km),magnitude difference are transformed into the unit cube r i, j g i, j t i j ( t, r, g ) ( τ, ρ, γ ) [0,1] i, j i, j i, j i, j i, j i, j Probability p c is calculated sequentially 3 p 1 = f 1+ e f 1 p logit( p) ln p k k k ( p ) { µ c ( x1, y1) } a1 bk i, j ck i, j dk i, j #{ i j} γ ρ τ logit = logit < i< j k= 1 k= 1 k= 1 µ (x, y) indicates probability of initial earthquake at location (x,y). Arithmetic mean of polynomials of the normalized space-time magnitude variables for all pairs of earthquakes (i < j) in a cluster. The coefficients a, b, c, d are estimated by the maximum likelihood method together with the AIC. Ogata, Utsu and Katsura, 1996, GJI) probability k ak bk ck dk

19 Probability of isolated or first earthquakes will be foreshock probability Forecasted results for 1994 Mar 2011

20 Multiple earthquakes in a cluster Measuring inter-events concentrations in a cluster and magnitude increments Aftershocks Swarms Foreshocks F A S F F A S F

21 Normalized time, distance & magnitude difference in unit cube (t, r, g) (τ, ρ, γ) in [0,1] 3 Time Interval Transformation Epicenter Separation Transformation ρ = 1 exp{ min( r,50) / 20} Magnitude Difference Transformation where σ = 6709, σ =

22 Forecasted sequence and evaluation ( Mar ) # F? #C Pc ENTRPY CU~ENT P1 P2 P3 P4 P5 P6 P7 P8 P9 P % % % % 12.66% % % % % % % % % % % 11.17% 7.87% 9.82% % % % % % % M7.3 Foreshock of 9 Mar % % 27.8% 27.7% 20.1% 14.0% 14.2% 13.6% 11.6% 15.7% 11.9% 10.1% 8.2% 10.1% 11.7% 10.9% 10.6% 11.5% 11.1% 9.9% 8.2% 7.2% 6.8% 7.6% 7.3% 7.4% 6.7% 7.0% 7.0% 8.0% 8.5% 8.6% 8.2% 8.0% 8.1% 8.4% 7.8% 7.3% 7.5% 7.8% 8.1% 8.1% 7.8% 7.4% 7.7% 7.8% 7.6% 7.2% 7.2% 6.9% 6.8% 6.7% 7.4% 8.0% 7.8% 7.6% 7.7% 8.3% 9.0% 8.7% 8.5% 8.6% 8.3% 8.4% 8.2% 8.2% 8.0% 7.9% 7.9% 8.4% 8.4% 8.6% 8.5% 8.6% 8.4% 8.2% 8.4% 8.3% 8.3% 8.1% 7.9% M % % 4.77% 6.21% 3.42% 1.74% 1.24% 1.04% 0.90% 0.83% 0.97% 1.03% % % 0.25% 0.51% 0.83% 2.77% 2.21% 2.02% 3.19% 2.78% 2.50% 2.43% 3.07% 2.92% 2.74% 2.84% 2.68% % % 0.79% 1.70% 2.06% 1.90% 1.90% 1.88% % % % % % % 7.42% 4.88% 3.98% 3.56% 4.05% 4.49% % % % % % % % % *Entropy0 = ; 2*Entropy = : 2* Entropy = 63.68

24 λ ETAS (, txy, ) Conditional intensity function of the ETAS model n φ ( txy,, ) = apν( t t) ρ( x x, y y), n # c, t t, a = 1 cn k kc k k k n k k= 1 k= 1 where, in Ogata et al. (GJI,1995); ν(t) is normalized density of foreshock survival function of foreshocks in Fig. 5a, and ρ(x,y) is normalized density of foreshock survival function of foreshocks in Fig. 5b. Moreover, pk n is defined in the paragraph including equation (18) of Ogata et al. (GJI,1996), Manitude frequency for the next event after the n-th earthquake in the cluster c small l arge ( ) ( ) ( ) GRdensity m = ψ m M + ψ m M ψ ( m M ), ψ ( m M ); normalized small l arge 0 0 { } = max, = 1,, ( n) M Mk k n Ψ ( m M ) = p ψ ( m M ) + (1 p ) ψ ( m M ) ( n) l arg e ( n) small ( n) nc 0 nc 0 GRdensity( m) otherwise If ψ(t) is normalized density of magnitude-differences between foreshocks in Fig. 5c of Ogata et al. (GJI,1995), { j } ψ ( m) = GRdensity m max( M, j = 1,, k) k k = 1 { ψ } Ψ ( m n + 1) = GRdensity( m) (1 a ) ( m) n k k n if ( t, x, y)is connected to cn

25 Algorithm of foreshock probability calculations in case of plural earthquakes in a cluster For plural earthquakes in a cluster, time differences (days),epicenter separation r (km),magnitude difference gij are transformed into the unit cube ij Probability p c is calculated sequentially k k k logit( p ) logit { µ c ( x1, y1) } a1 bk i, j ck i, j dk i, j #{ i j} γ ρ τ = < i< j k= 1 k= 1 k= 1 Here µ (x, y) indicates probability of initial earthquake at location (x,y),and the 2 nd term calculates arithmetic mean of polynomials of the normalised space-time magnitude variables for all pairs of earthquakes (i < j) in a cluster, where the coefficients a, b, c, d are as follows. t ij ( t, r, g ) ( τ, ρ, γ ) [0,1] i, j i, j i, j i, j i, j i, j Ogata, Utsu and Katsura, 1996, GJI) k ak bk ck dk

Plural earthq. 1994-2011 Single earthq. Probability of foreshocks in log scale 100% 10% 1% 0.1% Forecasts and results 1994-2011 M 4 2011 March 9 M7.3 largest foreshock M main M 4 M9.0 6.5 0.

26 Plural earthq Single earthq. Probability of foreshocks in log scale 100% 10% 1% 0.1% Forecasts and results M March 9 M7.3 largest foreshock M main M 4 M % Actual foreshock cluster Other type cluster Actual foreshock cluster Other type cluster Predicted probability Foreshock Others Order of earthquake in a cluster Relative Frequency

Forecast Evaluation for 1994-2011 Mar. Actual foreshock cluster Other type cluster 100% 100% M7.3 Foreshock M7.0 Ibaragi-Ken of May 2008 Probability forecast (%) 10% 1% 0.

27 Forecast Evaluation for Mar. Actual foreshock cluster Other type cluster 100% 100% M7.3 Foreshock M7.0 Ibaragi-Ken of May 2008 Probability forecast (%) 10% 1% 0.1% Relative frequencies Probability forecast (%) 10% 1% Foreshocks. Others 0.1% M % M main Probability forecast (%) 4.5, Mc 4.0 Earthquake number in a cluster 0.01% M 6.5 Mc 4.0 main Earthquake number in a cluster

28 Earthquake number in a cluster 100% 100% M7.3 Foreshock of 9 Mar 2011 Probability forecast (%) 10% 1% 0.1% Mc 4.0 Probability forecast (%) 10% 1% 0.1% M main 6.5 Mc 4.0 M % Earthquake number in a cluster 0.01% Earthquake number in a cluster 1994 年年 3 月

Southern California 1932 2006 M>=3.5 d Single-link-clustering = + ( c ) 0.3 (or 30km) 2 2 ST space time ent0 = 235.0 ent = 236.8 Predicted Foreshock probability 0-2.5 2.5-5. 5.

-------+---------+----------+---------+---------+ ratio% 1.2 3.1 9.2 4.

29 Southern California M>=3.5 d Single-link-clustering = + ( c ) 0.3 (or 30km) 2 2 ST space time ent0 = ent = Predicted Foreshock probability all Other Fore All ratio% # #fore % (+/-) #sw % (+/- ) #Maft #f+#s All (0.4) (0.5) (1.1) (2.0) (1.7) (2.9) (2.3) (3.8) (2.8) (4.5) (2.6) (5.2) (3.3) (6.1) (3.3) (7.0) (3.8) (7.5) (3.4) (7.7) Probability forecast (%) M main 4.0, Mc 3.5 M 5.5, Mc 3.5 main aic0 = aic1 = Earthquake number in a cluster Earthquake number in a cluster

30 Global Forecast Result using NEIC-PDE catalog (M 4.7) 1973 ~ 1993: learning period, calibrating the forecasting parameters in Ogata et al. (1993, GJI) 1994 ~ 2013 April: forecasting period Isolated or 1 st quake in a cluster Plural earthquakes within a cluster Relative frequency Actual foreshock cluster Other type cluster APR Relative frequency Actual foreshock cluster Other type cluster APR Forecast probability (%) Forecast probability (%) Predicted probability 2.5% 5.0% + all Foreshock Others Frequency ratio All 2* LL = aic = 129.6

1998-2010 Relative frequency Actual foreshock cluster Other type cluster 1994 2013 APR M 2.0 Forecast probability (%) 2.5 5.0 10.0 15.0 All 2* LL = 195.

31 Relative frequency Actual foreshock cluster Other type cluster APR M 2.0 Forecast probability (%) All 2* LL = aic =

(1993, GJI) 1994 ~ 2013 April: forecasting period d Foreshock probability for isolated or the 1 st quake estimated from the NEIC data from

32 Global Forecasting using NEIC-PDE catalog (M 4.7) Single-link-clustering by connecting the space-time distance = + ( c ) 0.45 (or 50km) 2 2 ST space time 1973 ~ 1993: learning period, calibrating the forecasting parameters in Ogata et al. (1993, GJI) 1994 ~ 2013 April: forecasting period d Foreshock probability for isolated or the 1 st quake estimated from the NEIC data from Given location of a future earthquake, probability is calculated by the interpolation using the including Delaunay triangle April probability probability

33 Global Forecast Result using NEIC-PDE catalog (M 4.7) 1973 ~ 1993: learning period, calibrating the forecasting parameters in Ogata et al. (1993, GJI) 1994 ~ 2013 April: forecasting period Isolated or 1 st quake in a cluster Plural earthquakes within a cluster Relative frequency Actual foreshock cluster Other type cluster APR Relative frequency Actual foreshock cluster Other type cluster APR Forecast probability (%) Forecast probability (%) Predicted 2.5% 5.0% + all probability Predicted probability 5% 10% 20% 30% + all Foreshock Foreshock Others Others Frequency ratio Frequency ratio

確率予測(%対数スケール)前震の確率予報 M>=4 孤立地震または群れの先頭の地震 1926-1993 1926-1993 M>=4 の地震群発型前震型本震余震型 F 100% 10%

2011 年 3 月 9 日の M7.3 最大の前震確率予測最初の地震の予測の結果複数個の地震の群れの場合 1% 0.

実際に前震だったその他の場合 Logit { µ ( x, y) } Logit{ p c } 複数地震の予測の結果 1994-2011 10% 1% 0.1% 0.01% M9.0 M 本震 6.

34 確率予測(%対数スケール)前震の確率予報 M>=4 孤立地震または群れの先頭の地震 M>=4 の地震群発型前震型本震余震型 F 100% 10% 推定孤立地震または群れの先頭の地震が前震である予報確率非線形変換群れ内の地震の時間間隔 ( 日 ) A S F 群れ内の地震同士の距離 (km) F A S F 100% 2011 年 3 月 9 日の M7.3 最大の前震確率予測最初の地震の予測の結果複数個の地震の群れの場合 1% 0.1% 0.01% M 4 群れの中の地震の順番複合確率予測群れ内の地震同士のマグニチュード差確率予測(%対数スケール) 実際に前震だったその他の場合 Logit { µ ( x, y) } Logit{ p c } 複数地震の予測の結果 % 1% 0.1% 0.01% M9.0 M 本震 6.5 M 4 実際に前震だったその他の場合 Ogata, Y. and K. Katsura (2012) Prospective foreshock forecast experiment during the last 17 years, Geophys. J. Int. (in press)

35 Summary and suggestions It is conceivable that the b value of the G-R rule depends on the earthquake location when the earthquakes are small. When the earthquakes are small, such location-dependent b-value model performs a slightly better forecast performance than the reference model of b = 0.9 through out entire regions. But, there are many outlyingly negative information gain score which causes total predictive performance worse; this is clearly seen inland Japan experiments. We need to pursue the physics of aftershocks and elaborate the magnitude frequency models.

36 FORMLATION OF THE ISSUES Prediction models are based on the conditional intensity function of point process, { an event in a bin [, tt+ t] [, + ] [, + ] [ M, M+ M] t, t} P xx x yy y H F λ(, txym,, Ht, Ft ), t x y M for calculating probability of an earthquake occurring at a time t, a location (x, y), and a magnitude M, that conditional on history of occurrence records Ht = { ( tj, xj, yj, M j); tj < t } and can further depend on relevant information as exogenous records. Then we assume the separablity between space-time and magnitude components. λ t, t t t t t (, t x, y, M H F) λ(, t x, y H, F) γ ( M, t x, yh,, F) where ( t),, γ ( M t, x, y, H F ) dm = P M < Magnitude M + dm t, x, y, H F t t F t t, t Our task is to model γ ( M txyh,,, t F) and evaluate the probability and information gains relative to the reference model, γ (,,, ) 10 0 M txyh F = t, a b( M Mc ) t

37 Field et al. (2017, BSSA)

39 2 4 SPACE 1 3 TIME 5

40 Probability density Probability density { } ( n c) Magnitude Gap : M = max Mk ; k = 1,, n in cluster c Probability of M Mmax+0.5 of the next magnitude; n c = > If ( tn+ 1, xn+ 1, yn+ 1)is connected to c, bm bm 1 ( nc ) ( ) (, ( M ) 10 1 n c 10 Mc M ) ( M, ) ( 1,, n) (1 nc ) ( nc ) ( ) M nc M Ψ M M M = p + p bm bm 10 dm ( nc ) 10 dm Otherwise M M c ( ) ( ) bm Ψ M = ( Mc, ) M ( n c) M M c M bm ( n c) dm Ψ ( M ) M,, 1 M n ( n c) { n+ 1 } p P M M in c log Probability distribution p n c M ( n c) p n c Magnitude Magnitude log likelihood-ratio = information gain: log LL 0 = # c log Magnitude Ψ ( n) c( Mn+ 1 Mc ) Ψ ( M ) c n= 1 c n+ 1

41 Ogata et al. (1995, 1996; GJI ) Ogata & Katsura (2012, GJI ; 2014, JGR) F A S Aftershocks Swarms Foreshocks time-differences (days) A S Epicenter-separations (km) F F F Magnitude-differences Probability of foreshocks in log scale 100% 10% 1% 0.1% 0.01% pc 1 k k k ln = 1 µ ( x1, y1) ln + a1 + bkγi, j + ckρi, j + dkτi, j pc #{ i< j} µ ( x, y ) i< j k= 1 k= 1 k= Probability of the first event of the cluster or isolated event will be FORESHOCK µ ( x, y ) 1 1 Transformed to unit cube ( τi, j, ρi, j, γi, j) 0% 10% Forecasts and results M 4 Actual foreshock cluster Other type cluster Order of earthquake in a cluster

2016 年熊本地震の余震の確率予測 Probability aftershock forecasting of the M6.5 and M7.3 Kumamoto earthquakes of 2016 東京大学生産技術研究所統計数理研究所東京大学地震研究所 Institute of Indus

2016 年熊本地震の余震の確率予測 Probability aftershock forecasting of the M6.5 and M7.3 Kumamoto earthquakes of 2016 東京大学生産技術研究所統計数理研究所東京大学地震研究所 Institute of Industrial Science, University of Tokyo The Institute of