Seismogenesis, scaling and Seismogenesis, scaling and the EEPAS modelthe EEPAS model

David RhoadesDavid RhoadesGNS Science, Lower Hutt, New ZealandGNS Science, Lower Hutt, New Zealand

4th International Workshop on Statistical Seismology, Shonan Village, Japan, 9-13 January 2006

Precursory Scale Increase (Ψ) Precursory Scale Increase (Ψ) – example– example

Dashed lines show:Dashed lines show:a. Seismogenic area b. Magnitude increase c. Rate increasea. Seismogenic area b. Magnitude increase c. Rate increase

ΨΨ-phenomenon: Predictive relations-phenomenon: Predictive relations

PAAP

PTTP

PMMm

MbaA

MbaT

MbaM

log

log

EEPAS Model - FormulationEEPAS Model - Formulation ““Every Earthquake is a Precursor According to Every Earthquake is a Precursor According to

Scale”; i.e., it is evidence of the occurrence of Scale”; i.e., it is evidence of the occurrence of the the ΨΨ-phenomenon on a particular scale .-phenomenon on a particular scale .

Every earthquake initiates a transient Every earthquake initiates a transient increment of long-term hazard. The scale (of increment of long-term hazard. The scale (of time, magnitude, location) depends on its time, magnitude, location) depends on its magnitude.magnitude.

The weight of its contribution may depend on The weight of its contribution may depend on other earthquakes around it.other earthquakes around it.

The hazard at any given time, magnitude, and The hazard at any given time, magnitude, and location depends on all previous earthquakes location depends on all previous earthquakes within a neighbourhood of appropriate scale.within a neighbourhood of appropriate scale.

EEPAS model rate densityEEPAS model rate density

0

),,,()(),,,(),,,( 0tt

ii yxmtmyxmtyxmt

where λ 0 is a baseline rate density, η is a normalising function and

),()()(),,,( 111 yxhmgtfwyxmt iiiii

wi is a weighting factor and f, g, & h probability densities:

iAiA mbA

iimb

Ai

M

iMM

M

i

T

iTti

Ti

ii

yyxxyxh

mbammg

mbatt

tt

ttHtf

102

)()(exp

102

1),(

2

1exp

2

1)(

)log(

2

1exp

2)(

)()(

2

22

21

2

1

2

1

Contribution Contribution of an of an individual individual earthquakearthquake to the e to the rate rate density density under the under the EEPAS EEPAS modelmodel

(a)(a) mmii=4=4

(a)(a) mmii=5=5

Normalised rate density under the EEPAS model relative to a reference (RTR) rate density in which one earthquake per year, on average, exceeds any magnitude m in 10m km2. The fixed coordinates are those of the W. Tottori earthquake.

Weighting strategiesWeighting strategies

1. Equal weights

2. Low weight to aftershocks

),,,(

),,,(

iiii

iiiii yxmt

yxmtw

0

where is a rate density that includes aftershocks

and ν is the proportion of earthquakes that are not aftershocks

EEPAS model – fitting & testingEEPAS model – fitting & testing Fitted to NZ earthquake catalogue 1965-2000, Fitted to NZ earthquake catalogue 1965-2000,

M>5.75M>5.75 Tested against PPE on CNSS catalogue of Tested against PPE on CNSS catalogue of

California, M > 5.75California, M > 5.75 Tested against PPE on JMA catalogue of Tested against PPE on JMA catalogue of

Japan, M > 6.75Japan, M > 6.75 Optimised for JMA catalogue M > 6.25Optimised for JMA catalogue M > 6.25 Fitted to NIED catalogue of central Japan Fitted to NIED catalogue of central Japan

M>4.75M>4.75 Tested against PPE on NZ catalogue Tested against PPE on NZ catalogue

2001-20042001-2004 Fitted to AUT catalogue of Greece, 1966-80, Fitted to AUT catalogue of Greece, 1966-80,

M>5.95, and M>5.95, and tested against SVP 1981-2002tested against SVP 1981-2002 Fitted to ANSS catalogue of southern Fitted to ANSS catalogue of southern

California, M>4.95California, M>4.95

QuestionsQuestions

Does the EEPAS model work equally well Does the EEPAS model work equally well at all magnitude scales?at all magnitude scales?

Are the parameter values universal across Are the parameter values universal across different regions and magnitude different regions and magnitude thresholds?thresholds?

Regions of surveillance

(a) New Zealand

(b) California

(c) Japan

(d) Greece

Evolution Evolution of of

performanperformance factor = ce factor = LL(EEPAS)/(EEPAS)/LL(PPE)(PPE)(a-c),(a-c),

or or LL(EEPAS)/(EEPAS)/LL(SVP)(SVP)

(d)(d)

Regions of surveillanceRegions of surveillance

Kanto: M > 4.75 S. California: M > 4.95Kanto: M > 4.75 S. California: M > 4.95

ObservationsObservations

For low magnitude applications in S. California and For low magnitude applications in S. California and Kanto regions:Kanto regions:

Spatially varying models are more informative Spatially varying models are more informative with respect to SUP.with respect to SUP.

Equal weights version of EEPAS is better than Equal weights version of EEPAS is better than version with aftershocks down-weighted.version with aftershocks down-weighted.

Information rate of EEPAS with respect to Information rate of EEPAS with respect to spatially varying model is similar to applications spatially varying model is similar to applications at higher magnitude.at higher magnitude.

Fitted distributions for time, magnitude & location, given mi in applications of EEPAS model.

Modified magnitude distributionModified magnitude distribution

Present model appears to be Present model appears to be compromising between forecasting compromising between forecasting mainshocks and aftershocks for low mainshocks and aftershocks for low magnitude application in S. Californiamagnitude application in S. California

Change magnitude distribution to Change magnitude distribution to allow for aftershocksallow for aftershocks

Modified magnitude distribution (2)Modified magnitude distribution (2)

)()](exp[)|( yxHxyxyf

y

ii dxxgxyfyf )()|()(

where where HH((ss) = 1 if ) = 1 if ss > 0 and 0 otherwise. > 0 and 0 otherwise. (Density integrates to expected number of aftershocks).(Density integrates to expected number of aftershocks).Then magnitude distribution of aftershocks predicted byThen magnitude distribution of aftershocks predicted by ith earthquake isith earthquake is

Let Let xx denote magnitude of mainshock, and denote magnitude of mainshock, and yy that of an that of an aftershock. Assumeaftershock. Assume

If we set γ = α σM2, and δ′ = δ - α σM

2/2, then

)(exp)()( iMMii mbayyGyf

where Gi(y) is the survivor function of gi(y).

Modified magnitude distribution (3)Modified magnitude distribution (3)

Then the combined magnitude distribution (for mainshocks and their aftershocks) is

iMMiii mbammGmgmg exp)()()(

• If α > β, then g′i(m) can be normalized so that the forecast magnitude distribution follows the G-R relation with slope parameter b=βln10.

• If bM = 1, then the normalising function reduces to a constant (i.e., is independent of m).

Individual earthquake contribution to rate Individual earthquake contribution to rate densitydensity

a. Original magnitude distribution b. Modified magnitude distributiona. Original magnitude distribution b. Modified magnitude distribution

ResultsResults For S. California dataset, lnFor S. California dataset, lnLL of model is of model is

hardly improved.hardly improved. Equal weight version of EEPAS still Equal weight version of EEPAS still

prevails.prevails. Optimal value of Optimal value of δδ′ ′ ~1.3.~1.3. ffii((tt) parameters not changed much, but if ) parameters not changed much, but if

σσMM and and σσTT are constrained not to be small, are constrained not to be small, then then ffii((tt) is similar to other datasets, with ) is similar to other datasets, with only a small reduction of lnonly a small reduction of lnLL..

Fitted distributions for time, magnitude & location, given mi in applications of EEPAS model.

Modified magnitude distributionApplied to S. California with σT<0.5& σM<0.5.

ConclusionsConclusions

EEPAS model works similarly well at higher and EEPAS model works similarly well at higher and lower magnitudes, but with some parameter lower magnitudes, but with some parameter differences, that may indicate deviations from differences, that may indicate deviations from scaling in the long-term seismogenic process.scaling in the long-term seismogenic process.

Superiority of equal-weights version at low Superiority of equal-weights version at low magnitudes is unexplained.magnitudes is unexplained.

Effect of aftershocks on the fitting and Effect of aftershocks on the fitting and performance of the model needs further performance of the model needs further investigation. investigation.

When When σσMM and and σσTT are constrained, the optimal are constrained, the optimal time, magnitude and location distributions differ time, magnitude and location distributions differ little between regions.little between regions.