地震発生論セミナー 2017 夏 Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes [Ogata, 1988] Apr. 17, 2017 Yuta Yamaguchi (hatano lab. M2) �
Agenda [Ogata, 1988] Introduction Section 1 Statistical Models Section 2 Analysis Section 3 Seismic Quiescence Section 4 Conclusion and Some Remarks Section 5 Seminar M ay 1, 2017 2
1. Introduction [Ogata, 1988] Seismic quiescence • Had been studied by many seismologists for the purpose of earthquake prediction [e.g., Kanamori (1981)] • Lomnitz and Nava 1983 à Suggested a mere result of the decaying activity of aftershocks by comparing certain observed earthquake sequences and sequences from stochastic models Investigate seismic quiescence by comparing background seismicity with aftershock activity quantitatively using residual analysis Seminar M ay 1, 2017 3
Agenda [Ogata, 1988] Introduction Section 1 Statistical Models Section 2 Analysis Section 3 Seismic Quiescence Section 4 Conclusion and Some Remarks Section 5 Seminar M ay 1, 2017 4
2. Statistical Models [Ogata, 1988] Trigger models Epidemic-Type model [Lomnitz and Nava, 1983] elastic aftereffect modified Omori the conditional probabilities that an aftershock will occur in the interval (t, λ (t) is the conditional intensity rate t+dt), triggered by a main shock at time t 0 • • Main shock à completely random in time First term à back ground seismicity • • After shock à each of main shocks may Second term à all events (main shocks / generate aftershocks after shocks) • ξ is average number of aftershocks triggered by a primary event at t 0 Seminar M ay 1, 2017 5
Analysis [Ogata, 1988] Describe the model using the parameterized conditional intensity Restricted trigger models Epidemic-Type model • • First term à main shock distribution First term à back ground seismicity • • Second term à secondary events triggered Second term à all events (main shocks / by a main shock at t i c with magnitude m i c after shocks) Seminar M ay 1, 2017 6
Agenda [Ogata, 1988] Introduction Section 1 Statistical Models Section 2 Analysis Section 3 Seismic Quiescence Section 4 Conclusion and Some Remarks Section 5 Seminar M ay 1, 2017 7
3. Analysis [Ogata, 1988] 3.1 The Data and Their Features (42˚N, 142˚E) (42˚N, 146˚E) Using dataset compiled by Utsu (1982) • M ≥ 6.0 ( = M r ) (39˚N, 142˚E) (39˚N, 146˚E) • 483 shocks occurred from 1885 through 1980 (38˚N, 141˚E) • A part of the northwestern Pacific seismic belt • A swarm of large earthquake around 1938 at Shioya-Oki (35˚N, 140.5˚E) (35˚N, 144˚E) Seminar M ay 1, 2017 8
3. Analysis [Ogata, 1988] Fig4. Cumulative number of time interval Fig3. Density and cumulative distribution of magnitudes log P { X > x } cumulative density Exponential distribution Dataset supports GR law of time intervals à (Stationary Poisson) Seminar M ay 1, 2017 9
3. Analysis [Ogata, 1988] Fig6. Histogram for estimating m f (s) | m f ( s ) − λ 0 | Aftershocks Background seismicity Main shock This data supports modified Omori’s law Seminar M ay 1, 2017 10
3. Analysis [Ogata, 1988] 3.2 Comparison of restricted trigger models and epidemic models The Akaike information criterion (AIC) AIC = (-2)max(log-likelihood) + 2(number of used parameters) λ (t ; θ ) : the parameterized conditional intensity rate {t i } : the set of occurrence times of earthquakes AIC is smaller • (log-likelihood) is larger • Number of used parameters is fewer The model can fit the data better Seminar M ay 1, 2017 11
3. Analysis [Ogata, 1988] Trigger model AIC minimum Epidemic-type c model Eq(15) Eq(14) X g ( t − t i )e β ( m i − M r ) λ ( t ) = µ + t i <t It is necessary to check whether the major features can be reproduced by the estimated model Seminar M ay 1, 2017 12
3. Analysis [Ogata, 1988] 3.3 Residual analysis of point process data Introduce Λ (t), which is the expected number of earthquake at time t • Λ (t) has the distribution of a stationary Poisson process of intensity 1 [Papangelou 1972] • Using the transformed time τ = Λ (t) Fig2 Fig9 A deviation from { τ i } = “residuals” the transformed time Seminar M ay 1, 2017 13
3. Analysis [Ogata, 1988] Graphic test of complete randomness for residual analysis • The transformed interarrival times, • Y k are iid (independent and identically distributed) exponential random variables • U k = 1 – exp(- Y k ) are iid uniform random variables on [0,1) Fig10 Fig11 Within the 99% error bounds of the KS test The neighboring intervals have no correlation à uniform distribution à random distribution? Seminar M ay 1, 2017 14
3. Analysis [Ogata, 1988] swarm Fig15 • Consider the number of points ∆ N = N( τ -h, τ ) • ∆ N is a Poisson random variable ξ h = 8 • Use below transformation [Shimizu and Yuasa(1984)] ξ ξ is a normal random variable with mean • ξ h = 5 0 and variance 1 • Histogram behaves like a Gaussian, except for the around 1938 Seminar M ay 1, 2017 15
Agenda [Ogata, 1988] Introduction Section 1 Statistical Models Section 2 Analysis Section 3 Seismic Quiescence Section 4 Conclusion and Some Remarks Section 5 Seminar M ay 1, 2017 16
4 . Seismic Quiescence [Ogata, 1988] Quantitatively evaluation of the seismic quiescence Fig15 • (3a) à 5 large shocks with M ≥ 7.4 occurred within a year after ξ = -2 • (3b) à 3 large shocks with M ≥ 7.7 occurred h = 8 within a year after ξ = -2 à 5 large shocks with M ≥ 7.4 occurred within a year after ξ = -1.5 • Define the quiescence as ξ = -2 or -1.5 ξ = - 2 • S à major earthquake occurred h = 5 within a year after quiescence • F à major earthquake not occurred within a year after quiescence ξ = -1.5 Seminar M ay 1, 2017 17
4 . Seismic Quiescence [Ogata, 1988] • What should we assume about the joint distribution of magnitudes and occurrence times? à the local averages of aftershock magnitudes fluctuated slightly about a mean value [Lomnitz (1966) “magnitude stability”] generalization Båth’ s law (The different mean magnitudes between the group of main shock and the maximum aftershocks) ≈ 1.2 Vere-Jones(1966, 1975) explained this law by the simple assumption that The magnitudes in an earthquake sequence form a random sample independently selected from a distribution having the exponential form • Assume that the magnitudes M i is independent from the past event {(t j , M j ); t j <t i } Seminar M ay 1, 2017 18
4 . Seismic Quiescence [Ogata, 1988] Asses the probability of the main shock after the quiescence • Using the below data In 96 years, M ≥ 7.7 occurred 6 times à 6/96 per year • S à major earthquake occurred M ≥ 7.4 occurred 19 times à 19/96 per year within a year after quiescence M ≥ 7.0 occurred 47 times à 47/96 per year • F à major earthquake not occurred within a year after quiescence ETAS Simulation using • Small probability events à the magnitude distribution is dependent on the past occurrence time, and seismic quiescence is useful for predicting a major event Seminar M ay 1, 2017 19
Agenda [Ogata, 1988] Introduction Section 1 Statistical Models Section 2 Analysis Section 3 Seismic Quiescence Section 4 Conclusion and Some Remarks Section 5 Seminar M ay 1, 2017 20
5. Conclusion [Ogata, 1988] • The epidemic-type model with the effect of the magnitude gave the best fit to the data in terms of AIC à ETAS • The model is based on the below assumptions, (a) background seismicity is generated by a stationary Poisson process (b) each shock has a risk of stimulating aftershocks proportional to e β M (c) The hazard rate of aftershocks decreases with time by the modified Omori’ s law Using the transform time τ = Λ (t), a new method of residual analysis was • developed • Seismic quiescence defined by residual analysis can be useful for predicting a coming major earthquake Seminar M ay 1, 2017 21
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