Statistical Models for Earthquake Occurrences and Residual Analysis for Point Processes

[Ogata, 1988]

Apr. 17, 2017 Yuta Yamaguchi (hatano lab. M2)�

地震発生論セミナー2017夏

[Ogata, 1988]

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Introduction Section 1Statistical Models Section 2Analysis Section 3Seismic Quiescence Section 4Conclusion and Some Remarks Section 5

Agenda

[Ogata, 1988]

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1. Introduction

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

[Ogata, 1988]

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Introduction Section 1Statistical Models Section 2Analysis Section 3Seismic Quiescence Section 4Conclusion and Some Remarks Section 5

Agenda

[Ogata, 1988]

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2. Statistical Models

•  Main shock à completely random in time

•  After shock à each of main shocks may generate aftershocks

•  ξis average number of aftershocks triggered by a primary event at t0

the conditional probabilities that an aftershock will occur in the interval (t, t+dt), triggered by a main shock at time t0

Trigger models [Lomnitz and Nava, 1983]

Epidemic-Type model

λ(t) is the conditional intensity rate

•  First term à back ground seismicity

•  Second term à all events (main shocks /after shocks)

elastic aftereffect

modified Omori

[Ogata, 1988]

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Analysis

Describe the model using the parameterized conditional intensity

Restricted trigger models Epidemic-Type model

•  First term à back ground seismicity

•  Second term à all events (main shocks /after shocks)

•  First term à main shock distribution

•  Second term à secondary events triggered by a main shock at ti

c with magnitude mic

[Ogata, 1988]

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Introduction Section 1Statistical Models Section 2Analysis Section 3Seismic Quiescence Section 4Conclusion and Some Remarks Section 5

Agenda

[Ogata, 1988]

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3. Analysis

(42˚N, 142˚E)

(39˚N, 142˚E)

(38˚N, 141˚E)

(35˚N, 140.5˚E)

(42˚N, 146˚E)

(35˚N, 144˚E)

(39˚N, 146˚E)

3.1 The Data and Their Features

Using dataset compiled by Utsu (1982)

•  M ≥ 6.0 ( = Mr )

•  483 shocks occurred from 1885 through 1980

•  A part of the northwestern Pacific seismic belt

•  A swarm of large earthquake around 1938 at Shioya-Oki

[Ogata, 1988]

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3. Analysis

Fig3. Density and cumulative distribution of magnitudes

Dataset supports GR law

cumulative

Fig4. Cumulative number of time interval

Exponential distribution of time intervals à (Stationary Poisson)

logP{X > x}

density

[Ogata, 1988]

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3. Analysis

Fig6. Histogram for estimating mf(s)

Main shock

Aftershocks

Background seismicity

|mf (s)� �0|

This data supports modified Omori’s law

[Ogata, 1988]

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3. Analysis

3.2 Comparison of restricted trigger models and epidemic models

The Akaike information criterion (AIC)

λ(t ;θ) : the parameterized conditional intensity rate {ti} : the set of occurrence times of earthquakes

AIC = (-2)max(log-likelihood) + 2(number of used parameters)

•  (log-likelihood) is larger

•  Number of used parameters is fewer

AIC is smaller

The model can fit the data better

[Ogata, 1988]

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3. Analysis

Eq(15) Eq(14)

Triggermodel

Epidemic-typemodel

�(t) = µ+X

ti<t

g(t� ti)e�(mi�Mr)

c

AIC minimum

It is necessary to check whether the major features can be reproduced by the estimated model

[Ogata, 1988]

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3. Analysis

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)

Fig9Fig2

the transformed time

A deviation from {τi}

= “residuals”

[Ogata, 1988]

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3. AnalysisGraphic test of complete randomness for residual analysis•  The transformed interarrival times,

•  Yk are iid (independent and identically distributed) exponential random variables

•  Uk = 1 – exp(-Yk) are iid uniform random variables on [0,1)

Fig11Fig10

Within the 99% error bounds of the KS test

à uniform distribution

The neighboring intervals have no correlation

à random distribution?

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3. Analysis

Fig15

h = 8

h = 5

•  Consider the number of points ∆N = N(τ-h, τ)

•  ∆N is a Poisson random variable

•  Use below transformation [Shimizu and Yuasa(1984)]

•  ξ is a normal random variable with mean 0 and variance 1

•  Histogram behaves like a Gaussian, except for the around 1938

ξ

ξ

ξ

swarm

[Ogata, 1988]

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Introduction Section 1Statistical Models Section 2Analysis Section 3Seismic Quiescence Section 4Conclusion and Some Remarks Section 5

Agenda

[Ogata, 1988]

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4. Seismic Quiescence

Fig15

Quantitatively evaluation of the seismic quiescence

h = 8

h = 5

ξ = - 2

•  (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 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

•  S à major earthquake occurred

within a year after quiescence

•  F à major earthquake not occurred

within a year after quiescence ξ = -1.5

[Ogata, 1988]

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4. Seismic Quiescence•  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 Mi is independent from the past event {(tj, Mj); tj<ti}

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4. Seismic Quiescence

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

M≥7.4 occurred 19 times à 19/96 per year

M≥7.0 occurred 47 times à 47/96 per year

ETAS

Simulation using

•  S à major earthquake occurred

within a year after quiescence

•  F à major earthquake not occurred

within a year after quiescence

•  Small probability events à the magnitude distribution is dependent on the past occurrence time, and seismic quiescence is useful for predicting a major event

[Ogata, 1988]

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Introduction Section 1Statistical Models Section 2Analysis Section 3Seismic Quiescence Section 4Conclusion and Some Remarks Section 5

Agenda

[Ogata, 1988]

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5. Conclusion

•  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 eβM

(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