with Large Rupture Dimension
Thesis by
Masumi Yamada
In Partial Fulllmentof the Requirements
for the Degree of
Dotor of Philosophy
California Instituteof Tehnology
Pasadena, California
2007
(Defended February15, 2007)
2007
Masumi Yamada
Aknowledgements
I would liketothank my advisor, TomHeaton,for hisgreatguidane onthis projet
overthe past 4years. He wasalways enthusiasti and supportive of hisstudents like
a real father. Whenever I asked simple questions, he answered really sinerely and
taught me something from hisenormous knowledgeof seismology.
The members of my advisory and defense ommittees, Jim Bek from Civil En-
gineering, Hiroo Kanamoriand Rob ClaytonfromSeismolab, and Yih-MinWufrom
Taiwan National University, were all very generous with their time and ideas. I
espeially thank Dr. Bek for giving me a guidane of probability theory. His on-
tributionwasveryimportantforinorporatingtheBayesianapproah. Many thanks
for Kanamori sensei, who talked me in a friendly and frank way whenever I visited
hisoÆe. He gaveme alot of helpful advie for the physis of earthquakes.
I also appreiate people who provided me with seismi data and their researh
results,BradAagaard attheUSGS, ChenJiatUC SantaBarbara, HirooKanamori,
Jing Liu-Zeng at Institute of Tibetan Plateau Researh, David Wald at the USGS,
and Yih-MinWu.
My thanks go to my fellow graduate students, Sai Hung Cheung, Anna Olsen,
and Alexandros Taanidis for helping me with researh, exams, ourse work, and
making the life in Thomas very enjoyable. GeorgiaCua, who developed the Virtual
Seismologist method, provides a lot of advie and support. I learned many things
from her high-quality Ph.D. thesis. Support from Carolina Oseguera and all the
members inCivilEngineeringand Seismolab were alsoindispensable. Speial thanks
for allmembers in Calteh Japanese Assoiation, I enjoyed lifeCaltehwith them.
always support my deisions. It was a big adventure for me to go to US and get a
Ph.D. degree. My dream is aomplishedthanks to their understanding.
Abstrat
Earthquake early warning systems have beome popular these days, and many seis-
mologists and engineers are making researh eorts for their pratial appliation.
The existingearthquakeearlywarning systems provideestimates ofthe loationand
size of earthquakes, and then ground motions at a site are estimated as a funtion
of the epientral distane and site soil properties. However, for large earthquakes,
the energy is radiated from a large area surrounding the entire fault plane, and the
epienter indiates onlywhere rupture starts.
In this projet,we fous onanearthquake earlywarningsystem onsideringfault
niteness. Weprovideanewmethodologytoestimaterupture geometryandslipsize
ona nite faultin real time for the purpose of earthquake early warning.
We propose a new model to simulate high-frequeny motions from earthquakes
with large fault dimension: the envelope of high-frequeny ground motion from a
large earthquake an be expressed as a root-mean-squared ombination of envelope
funtionsfromsmallerearthquakes. Weparameterizethe faultgeometrywithanepi-
enter, afaultstrike,and twoalong-strikerupture lengths,andnd theseparameters
by minimizingthe residual sum of squares of errors between ground motion models
and observed ground motionenvelopes.
To providethe informationonthe spatialextent of rupture geometry, we present
a methodology toestimatea faultdimension of anearthquakein real time by lassi-
fying seismireordsintonear-soureorfar-soure reords. Weanalyze peakground
motions and use Bayesian model lass seletion to nd a funtion that best lassi-
es near-soure and far-sourereords basedonthese parameters. This disriminant
large earthquakes.
In order to haraterize slip on the fault in real time, we onstrut ananalytial
funtion toestimatesliponthe faultfromnear-soure grounddisplaementobserva-
tions. Inreal-timeanalysis,webak projetthe reorded displaementdata ontothe
faultlinetoestimatethesizeofthesliponthefault. Thesimulationresultsshowthat
the slip size estimation predits the observed GPS stati displaement on the fault
quite well. This urrent slip size on the fault is used for a probabilisti predition
of additional rupture length in the near future. We haraterize the distribution of
additionalrupture lengthonditionedonthe urrentsliponthe faultfor theongoing
rupture fromthe simulationwith a 1-D slip model. The probability density of addi-
tional rupture length an be approximated by a lognormal distribution onditioned
onthe urrent slipsize.
Contents
Aknowledgements iii
Abstrat v
1 Introdution 1
1.1 Motivation . . . 1
1.2 Bakground onseismi early warningsystem . . . 3
1.2.1 Historyof researh eorts inseismi earlywarning system . . 3
1.2.2 Seismiearly warning systems inthe world . . . 5
1.2.2.1 Earthquake early warning system inJapan . . . 6
1.2.2.2 SeismiAlert System (SAS) of Mexioity. . . 7
1.2.2.3 Earlywarningsystem inTaiwan . . . 8
1.2.2.4 Earlywarningsystem inthe United States . . . 8
1.2.2.5 Earlywarningsystems in other ountries . . . 9
1.3 Objetives and road map for this thesis . . . 10
2 General Virtual Seismologist Method 11 2.1 Bayes' theorem for seismiearly warning system . . . 12
2.2 Dening the prior prob(M;R) . . . 14
2.2.1 Loation of known faults . . . 14
2.2.2 Previously observed seismiity . . . 15
2.2.3 Geometri onsiderationof stations . . . 17
2.2.4 Gutenberg-Rihter law . . . 20
2.3.1 Magnitude ground motionrelationships . . . 21
2.3.2 P-wave and S-wave disriminant . . . 23
2.3.3 Groundmotion models . . . 24
2.3.4 Complete formof the likelihoodfuntion . . . 28
2.4 Findingthe best estimates . . . 30
2.5 Summary . . . 31
3 Extended Virtual Seismologist Method 32 3.1 Road map for Virtual SeismologistFinite-Soure method . . . 32
3.2 Statistisof observed high-frequeny and low-frequeny ground motions 36 3.2.1 Data . . . 37
3.2.2 Statistisof observed high-frequeny groundmotions . . . 38
3.2.3 Statistisof observed low-frequeny groundmotions . . . 40
3.2.4 Comparisonofhigh-frequeny and low-frequenygroundmotions 43 3.2.5 Denitionsof the horizontalomponent . . . 45
3.3 Summary . . . 49
4 Estimating the Loation of Fault Rupture Using Envelopes of A- eleration 50 4.1 Groundmotion models for large earthquakes . . . 51
4.2 Findingthe best estimates . . . 58
4.3 Example fromthe Chi-Chi earthquake . . . 60
4.3.1 Dataused forthe VS-FSmethod . . . 60
4.3.2 Results fromthe analysis of the VS-FS method . . . 67
4.3.3 Comparisonbetween predited envelopes andobserved envelopes 68 4.3.3.1 Result of model 1 (horizontaland vertial data) . . . 77
4.3.3.2 Resultofmodel 2(horizontaldata)and model3(ver- tial data) . . . 78
4.3.3.3 Result of model 4 (eet of area weight) . . . 78
4.3.3.4 Result of model 5 and model 6 (the eet of station
4.3.4 Geometryof the parameter spae . . . 79
4.3.5 Eets of dierent mistfuntions . . . 85
4.4 Summary . . . 87
5 Near-Soure versus Far-Soure Classiation Analysis 89 5.1 Strongmotion data . . . 90
5.1.1 Datasoures. . . 91
5.1.2 Dataproessing . . . 92
5.1.3 Datalassiation . . . 94
5.2 Near-soure versus far-souredisriminantfuntion . . . 99
5.2.1 Fisher's linear disriminantanalysis . . . 100
5.2.2 Bayesian approah . . . 103
5.2.2.1 Asymptoti approximation. . . 107
5.2.2.2 Stohasti simulationusing Metropolisalgorithm . . 109
5.2.3 Comparisonbetween traditionalLDA and Bayesian approah 113 5.3 Bayesian model lass seletion . . . 116
5.3.1 Method . . . 116
5.3.2 Results of Bayesian model lass seletion . . . 118
5.3.3 Eet of the hoie of prior . . . 120
5.4 Results and disussion . . . 122
5.5 Summary . . . 123
6 Estimating the Slip on the Fault from Low-Frequeny Ground Mo- tion 127 6.1 Data . . . 128
6.2 Estimatingthe slipon the faultfrom low-frequeny ground motion. . 132
6.2.1 Construtinga slipfuntion as afuntion of faultdistane . . 132
6.2.2 Estimating the slip on the fault and prediting the additional rupture extent . . . 134
6.3 Prediting the probability of the additionalrupture extent . . . 136
6.3.2 Charateristis of the 1-D slipmodels . . . 139
6.3.3 Statistialdistribution of the additionalrupture length . . . . 142
6.4 Summary . . . 153
7 Conlusions 154
A An Artile in the San Franiso Dail y Evening Bul l etin 169
B Peak Ground Motion Database 171
List of Figures
2.1 Ablokdiagramtoomputetheposteriorpdffromthepriorinformation
and ground motiondata. . . 13
2.2 An example of the prior pdf for the known faults. . . 15
2.3 An example of the prior pdf for the previously observed seismiity. . . 16
2.4 An example of the prior pdf forthe geometri onsiderationof stations
(at the rst P detetion). . . 17
2.5 An example of the prior pdf forthe geometri onsiderationof stations
(at 3 seonds afterthe rst Pdetetion). . . 18
2.6 An example of the prior pdf forthe geometri onsiderationof stations
(at the seondP detetion). . . 19
2.7 Histogram of the magnitude of the earthquakes in Southern California
during 20002006. . . 20
2.8 Linear disriminantanalysis ofP-wave log(a)andlog(disp)as india-
tors of magnitude. . . 22
2.9 P/S wave disriminant using vertial and horizontal groundmotion a-
eleration and veloity. . . 24
2.10 Observed envelope for aelerogram and P-wave and S-wave envelopes
for the ground motionmodel. . . 26
3.1 The algorithmof the VS methodfor nite soure (VS-FSmethod). . . 33
3.2 A plot of near-soure PGA asa funtion of moment magnitude. . . 39
3.3 A histogram of the near-sourePGA. . . 39
3.4 A plot of near-soure PGD as afuntion of moment magnitude. . . 41
3.6 A omparison of near-soure PGA and PGD as a funtion of moment
magnitude. . . 44
3.7 A histogram of the near-sourePGA and PGD. . . 44
3.8 Comparison of the srss horizontal PGA and magnitude of horizontal aelerations asa funtion of magnitude. . . 46
3.9 Comparison of the srss horizontal PGD and magnitude of horizontal displaements as afuntion of magnitude. . . 47
3.10 A histogram of the srss horizontal PGA and magnitude of horizontal aelerations. . . 47
3.11 A histogram of the srss horizontal PGD and magnitude of horizontal displaements. . . 48
4.1 Near-soure aelerations inthe vertial omponent. . . 52
4.2 Envelopesof near-soure aelerations inthe vertial omponent. . . . 53
4.3 Near-soure aelerations inthe EWomponent. . . 54
4.4 Envelopesof near-soure aelerations inthe EWomponent. . . 55
4.5 Shemati diagramof the multiple soure model. . . 56
4.6 Envelopes of vertial aeleration reorded at the station C024 for the Chi-Chi earthquake. . . 58
4.7 ThefaultgeometryandthestationdistributionoftheChi-Chiearthquake. 60 4.8 Topographi map of Taiwan. . . 63
4.9 Strong motionstations inthe southern part of Taiwan. . . 64
4.10 Strong motionstations inthe northern part of Taiwan. . . 65
4.11 Strong motionstations inthe entral part of Taiwan. . . 66
4.12 The estimated parameters, N1 and N2, for dierent rupture veloities. 68 4.13 Predited and observed envelopes in the horizontalomponent. . . 70
4.14 Predited and observed envelopes in the vertial omponent. . . 71
4.15 Predited andobserved envelopes inthe horizontalomponentwithdif-
4.16 Predited andobserved envelopesinthevertialomponentwith dier-
ent saling. . . 73
4.17 Enlargedmap of gure4.13. . . 74
4.18 Enlargedmap of gure4.14. . . 75
4.19 Time series of the estimated parameters, , N1,and N2, for eah model. 76 4.20 Error surfae of and N1 for model 1 . . . 80
4.21 Error surfae of N1 and N2 for model 1. . . 81
4.22 Contour maps of the error surfae of N1 and N2 for model 1 . . . 82
4.23 Posterior probability for the parameter . . . 83
4.24 Posterior probability for the parameter N1. . . 83
4.25 Posterior probability for the parameter N2. . . 84
4.26 Two-dimensionalposteriorprobability for the parameters N1 and N2. . 84
4.27 Eets of dierent error funtions. . . 86
5.1 AnexampleofbaselineorretionforaveloityreordfromtheChi-Chi earthquake. . . 94
5.2 Maps of the faultprojetions and stationdistributions. . . 96
5.2 Maps of the faultprojetions and stationdistributions (ontinued). . . 97
5.3 Histogramsand Gaussian densitiesof the logof groundmotionsfor the near-soure and far-soure reords. . . 98
5.4 Histogram of the near-soure and far-soure data towhihthe disrim- inant funtionobtained from traditionalLDA isapplied. . . 104
5.5 Logisti sigmoidfuntion (x)=1=(1+e x ). . . 106
5.6 SamplesgeneratedbytheMetropolisalgorithmplottedintheparameter spae. . . 112
5.7 Mean and standard deviationof samples plottedagainst the numberof samples inluded. . . 112
5.8 Distribution of samples for 3 parameters generated by the Metropolis
5.9 Correlationplotofposteriorsamplesofthemodelparametersgenerated
by the Metropolis algorithm. . . 114
5.10 Probabilities of near-soure basedon the optimaldisriminantfuntion obtained by the Bayesian approah. . . 124
5.10 Probabilities of near-soure basedon the optimaldisriminantfuntion obtained by the Bayesian approah (ontinued). . . 125
5.11 Snapshotsoftheprobabilitiesofnear-sourefortheChi-Chiearthquake, based onthe optimaldisriminantfuntion fromthe Bayesian approah. 126 6.1 DistributionofthestatidisplaementsfortheChi-Chiearthquake(EW omponent).. . . 129
6.2 Distributionofthe statidisplaementsfor theChi-Chiearthquake(NS omponent).. . . 130
6.3 Distributionof thestatidisplaementsfortheChi-Chiearthquake(UD omponent).. . . 131
6.4 Apeakdisplaementperunitslipasafuntionoffaultdistaneobtained from groundmotion simulations. . . 133
6.5 A slipon the faultan beobtained by bakprojeting the displaement data onto the faultline. . . 135
6.6 Slipdistribution for the Chi-Chiearthquake. . . 136
6.7 Cross setion of the slipdistribution ingure 6.6. . . 137
6.8 An example of the 1-D slipmodels. . . 138
6.9 The eet of the low-passlter withdierentorders onthe slipmodels.139 6.10 Plotfor the averageslip ( D) and rupture length(L)for the slipmodels with dierent . . . 140
6.11 Plot for the average slip ( D) and rupture length (L) for the observed earthquake data. . . 141
6.12 Plotfortheslopeoflog
D=logLand . Thevalueofthatorresponds
6.13 Plot for the average slip (D) and rupture length (L) for the observed
data and simulation results fromthe slipmodel with =1.33. . . 142
6.14 Plotfor the averageslip (
D) and rupture length(L)for the slipmodels
with dierentparentseries size n. . . 143
6.15 3-D histogram of the additional rupture length (L
a
) as a funtion of
urrent slip(D). . . 143
6.16 Histogramofthe additionalrupture length(L
a
)asafuntionofurrent
slip (D). . . 147
6.17 Probability density of the additional rupture length (L
a
) as a funtion
of urrent slip(D) by the kernel smoothing method. . . 147
6.18 The parameters and for the lognormal distribution whih is an
approximationoftheprobabilitydensityoftheadditionalrupturelength
(L
a
). . . 148
6.19 Probabilitydensityoflognormaldistributionwhihistheapproximation
of the additionalrupture length(L
a
). . . 149
6.20 Probability density of lognormal distribution with mean from the for-
mula (D)=1:16ln(D)+4:94 and onstant . . . 149
6.21 Histogramofthe additionalrupture length(L
a
)asafuntionofurrent
slip (D). The bin size of the histogram is10 km. . . 150
6.22 Probability density of the additional rupture length (L
a
) as a funtion
of urrent slip(D) by the kernel smoothing method. . . 150
6.23 Probabilitydensityoflognormaldistributionwhihistheapproximation
of the additionalrupture length(L
a
). . . 151
6.24 Probability density of lognormal distribution with mean from the for-
mula (D)=1:16ln(D)+4:94 and onstant . . . 151
6.25 The probability that the additional rupture length exeeds a ertain
value onditionedonthe urrent slipsize D. . . 152
List of Tables
1.1 On-site andregionalapproahes for the earthquake earlywarning system. 5
2.1 CoeÆientsforthe envelopeattenuationrelationshipsforhorizontaland
vertial aelerationon asoilsite inequation 2.16. . . 28
3.1 Earthquakedata set used for the near-soure groundmotion analysis. . 38
4.1 P-wave and S-wave veloity model inentralTaiwan. . . 62
4.2 Model parameters for estimatinga fault geometry. . . 67
5.1 The earthquake datasetused for the lassiation analysis. . . 90
5.2 Eight measurements of peak groundmotions. . . 95
5.3 Estimated model parameters. . . 102
5.4 Theonfusionmatrixfortheross-validationanalysiswiththeBayesian methodwith asymptotiapproximation. . . 109
5.5 The onfusion matrix for near-soure versus far-soure lassiation. . 115
5.6 Results of leave-one-out ross-validationfor LDA and Bayesian approah.116 5.7 Results for Bayesian model lass seletionwhen fteen ombinationsof the ground motionparameters are examined. . . 118
5.8 The best ve model lasses in the Bayesian model lass seletionwhen 255 ombinationsof the groundmotionparameters are examined. . . . 119
5.9 Theposteriorprobabilityofthemodellassseletionwithdierenttypes of prior distribution for parameters.. . . 121
5.10 The estimated parameters fromBayesian approahwith dierent types
B.1 Earthquakedata set used for the near-soure groundmotion analysis. . 171
B.2 Peak values of the strong motionreords forten earthquakes. . . 172
B.2 Continued. . . 173
B.2 Continued. . . 174
B.2 Continued. . . 175
B.2 Continued. . . 176
B.2 Continued. . . 177
B.2 Continued. . . 178
B.2 Continued. . . 179
B.2 Continued. . . 180
B.2 Continued. . . 181
B.2 Continued. . . 182
B.2 Continued. . . 183
B.2 Continued. . . 184
B.2 Continued. . . 185
B.2 Continued. . . 186
B.2 Continued. . . 187
B.2 Continued. . . 188
B.2 Continued. . . 189
B.2 Continued. . . 190
B.2 Continued. . . 191
B.2 Continued. . . 192
B.2 Continued. . . 193
B.2 Continued. . . 194
B.2 Continued. . . 195
B.2 Continued. . . 196
B.2 Continued. . . 197
Chapter 1
Introdution
1.1 Motivation
Reently, with advanes in dataanalysis and inreased awarenessof the seismi haz-
ard, the topi of earthquake early warning has attrated more researh attention,
and various early warningmethodshave been proposed fromseismologists and engi-
neers (Nakamura and Tuker, 1988; Allen and Kanamori, 2003; Odaka et al., 2003;
Wuand Kanamori, 2005a). Currently, the most ambitious system is the earthquake
early warning system provided by the Japan Meteorologial Ageny, whih is in a
testing phase. The news of the system was broadasted widely and attrated on-
siderable publi attention in Japan. The goal of seismi early warning is to initiate
optimalmitigatingations based onthe arrivaltime and amplitude of seismi waves
predited at a given loation. To ahieve this, an earthquake early warning system
mustolletandquikly analyzeseismidatainamannerthatanbeusedtopredit
future shaking. In priniple, this ould be ahieved by using the present value of an
approximately known waveeld as a boundary ondition topredit future waveelds
using Navier's equation (Baker et al., 2005). However, from a pratial viewpoint,
there are advantages to data analysis shemes that involve haraterization of the
earthquake soure. Preditions of future shaking an be ahieved by utilizing the
extensive existing work on prediting ground shaking from seismi soures. Ideally,
an early warning system would provide the best estimate of slip in time and spae
CuaandHeatondeveloped theVirtualSeismologist(VS)method(Cua,2005;Cua
and Heaton, 2006). It is a Bayesian approah to seismi early warning designed for
modern seismi networks, and is proposed for small to moderate earthquakes with
ruptures that an be approximately modeled as a point soure. The VS algorithm
uses an envelope attenuation relationship and the predominant frequeny ontent
fromthe rst fewseonds afterthe P-wavearrival. The advantage of the VSmethod
is itsapaity toassimilate dierent types of informationthat may be useful to nd
quik and reliable estimates of magnitude and loation (Cua, 2005). It gives the
best estimate of an earthquake property in terms of a probability density funtion.
The Bayesian approah isa sheme to emulate human apabilities to judge omplex
informationby modelingunertainty in aprobabilisti way.
Our goal is to extend the VS method to large earthquakes where fault niteness
is important. Most other earthquake early warning systems fous on estimating
epienters and magnitudes of earthquakes, not the fault geometry (Nakamura and
Tuker, 1988; Allen and Kanamori, 2003). However, for large earthquakes, rupture
lengthan beonthe orderof tens tohundreds ofkilometers,and the inhomogeneous
slip distribution signiantly aets the ground motion amplitude at a site. For
example, the fault rupture in the 1999 Chi-Chi earthquake was longer than 80 km,
andthelargestslipwasreordedatthenorthernendofthefault. ItwouldbediÆult,
if not impossible,to predit suh large shaking atlarge distanes from the epienter
when using asheme that only haraterizes the earthquake asa point soure.
Early warning for large earthquakes provides two types of preditions: 1) At
a given instant, it reognizes the present geometry of an ongoing earthquake, and
predits the shaking from waves that are traveling to another site; 2) Given the
present dimensions of a rupture, what is the probability distribution for the nal
dimensions of the rupture?
We introdue a two-step strategy to aomplish the rst type of preditions; 1)
we determine the spatial and temporal extent of an ongoing rupture by analyzing
waveformenvelopesofhigh-frequenyshaking,2)wedetermineapproximateslipfrom
the rupture. Based on the urrent onguration of the fault, the seond type of
predition an be aomplished.
1.2 Bakground on seismi early warning system
1.2.1 History of researh eorts in seismi early warning sys-
tem
Lee and Espinosa-Aranda (2003) and Kanamori (2005) provide a reent review and
history of researh eorts in seismi early warning. Aording to the quotation in
Nakamura (1988), the onept of a seismi early warning system dates as far bak
as 140 years ago. Cooper (1868) proposed to \arrange a very simple mehanial
ontrivaneatvariouspoints from10to100 milesfromSan Franiso," and\instan-
taneously ring an alarm bell, whih should be hung in a high tower near the enter
of the ity" when the \very simple mehanial ontrivane" detets an earthquake.
This artile explains the fundamentals of a seismi early warning system. It refers
tothe automation of the system, danger offalse alarms,and weakness of the system
forvery near-soureearthquakes (see Appendix Afor the quotation). Unfortunately,
Cooper's oneptwasneverimplemented. Ahundredyears later,arailwayompany,
Japan Railways (JR) designed an earthquake warning system in 1965 and started
operationthe next year (Nakamura and Tuker, 1988; Nakamura, 1988).
IntheUnitedStates,Heaton(1985)developedamodelforaseismiomputerized
alertnetwork(SCAN),whihisasystemtoprovideshort-termwarningforimminent
strong ground motion from large earthquake in southern California. By using this
model, the relationship between the size of the ground motions, warning time, and
area where the warning is issued was analyzed. Aording to the results, although
warning times are likely tobe short for areas greatly damaged by relativelysmall to
moderate earthquakes, large areas that experiene very strong shaking during large
earthquakes would reeivelonger warningtimes. Healsoomments thatlarge earth-
times. Toksoz et al. (1990) desribed a prototype earthquake warning system for
strike-slip earthquakes whose slip an be approximated by only horizontal displae-
ment. As the rst pratialappliation in US, aprototype early warning system for
aftershoks wasoperated by the United StatesGeologialSurvey (USGS) inthe San
Franiso Bay area after the 1989 Loma Prieta earthquake, M
w
=6.9 (Bakun et al.,
1994).
The onept of amplitude-based loation estimate was introdued by Kanamori
(1993). In hismethod, anattenuation relationship is t to the observed peak ael-
eration data, and parameters of magnitude, latitude, and longitude are determined
by minimizingthe error between observations and preditions. This tehnique is the
fundamentalprinipleusedinVS method. Kanamori etal.(1997)desribeexamples
of seismi early warning system developed in several parts of the world. They dis-
ussed the urrent onguration of the seismi network in California and tehnial
issues for providing real-time information. In the paper, they pointed out an issue
that the energy is radiatedfrom a large area for major earthquakes, and estimating
the epienter loation is not enough to determine the ground motion at a site. It is
proposed to loate not only the traditionalhypoenter, but the enter of the energy
radiation,whih is referred toas the ground motionentroid.
Kanamori (2005) lassiesearlywarningapproahes as either on-siteor regional.
Anon-siteapproahusesavailablegroundmotionsatagiven sitetopreditthelater-
arriving main shok at the same site. This method is suitable for the region lose
to the epienter. The regional approah predits the ground motion at a site based
on anestimate of the size and magnitude of the event from the near-soure reords.
This approah is more reliable and provides more aurate information for stations
relatively distant from the epienter. The on-site approah an make a more rapid
warning for the region lose to the epienter, sine there is no need to ompute the
magnitude or loation of the earthquake. On the other hand, the regionalapproah
isuseful forissuing a regionalwarning forthe relatively distant stations. The merits
and demeritsof these approahes are shown in table 1.1.
Table 1.1: On-siteand regionalapproahes for the earthquake earlywarning system.
Examples of eahapproah are explained inSetion 1.2.2.
Type On-site EWS Regional EWS
Datato be used Reords of a station whose
ground motion isestimated.
All the urrent available
reords.
Outputinformation
Peak groundmotion at asite. Soure information.
(additionaly, magnitude and
epienterloation)
(groundmotionat asitean be
estimated from attenuation re-
lationships)
Merits Simpleand quik. Reliableand aulate.
Demerits Large unertainty. Taking timefor data olletion
and omputation.
Suitablefor Regions loseto theepienter. Relativelydistantregions.
Examples
-UrEDAS (Nakamura, 1988) -Mexio itySAS
-ElarmS (Espinosa-Aranda et al.,1995)
(Allen and Kanamori,2003) -Japan EWS
-Taiwan EWS (Odaka et al.,2003)
(Wu and Kanamori,2005b) -VSmethod (Cua,2005)
whih is an on-site approah for the California Integrated Seismi Network (CISN).
This algorithm determines the magnitude of events from the predominant period of
the rst few seonds of the P-wave, based on the assumption that the seismi mag-
nitude has a linear relationship with the predominant period of the ground motion.
Wu and Kanamori (2005a) introdued an approah based on a predominant period
and displaement amplitude for the Taiwan early warning system. The regional ap-
proah for seismi early warning is employed in Japan and Mexio (Odaka et al.,
2003;Espinosa-Arandaetal.,1995, respetively). TheVSmethodisalsoategorized
as aregional approah.
1.2.2 Seismi early warning systems in the world
Wereview earthquake early warning systems that are urrently in operationaround
1.2.2.1 Earthquake early warning system in Japan
1) Urgent Earthquake Detetion and Alarm System (UrEDAS)
The Bullet Train, or Shinkansen, of the Japan Railways (JR) started operation in
1964. The next year, Shizuoka earthquake (M6.1) hit the route of the train and
damaged the train trak. From the onern for the potential of serious damage
from large earthquakes, the earthquake early warning system began operation in
1966 (Nakamura and Tuker, 1988). The system onsists of aelerometers installed
at the transforming stations along the train route, eah separated by about 20 km
(NakamuraandTuker,1988;SaitaandNakamura,2003). Whenaelerationexeeds
40gals, theeletripowertothe BulletTrainisautomatiallyshutoandthebrakes
are applied (Nakamura and Tuker, 1988; Saita and Nakamura, 2003).
Starting from 1983, an intelligent earthquake warning system alled UrEDAS
(Urgent Earthquake Detetion and Alarm System) was implemented (Nakamura,
1996b,a). Inthisupgradedsystem,theaelerometersareinstalledontheoastalline,
whihislosertotheJapanesesubdutionzone,toprovidemorewarningtime(Naka-
mura and Tuker, 1988). When the aelerometers reord a strong ground motion,
eah station estimates the epientral azimuth, magnitude, and hypoentral distane
oftheearthquakefromtherst fewseonds ofthereords(Nakamura,1996a). Based
on this information, it then issues an alarm and automatially shuts o the eletri
power for trains whih are running at high speed. The system worked during the
NiigataChuetsu earthquake in2004. It immediatelydeteted the P-wavearrivaland
shut o the train'spower inless than 3seonds (Nakamura etal., 2006).
2) Early Warning System in Japan (extended Nowast system)
The Japan MeteorologialAgeny (JMA) and NationalResearh Institute for Earth
SieneandDisasterPrevention(NIED)reentlyimplementedaprototypeemergeny
earthquake warning system in Japan (Doi, 2003; Odaka etal., 2003; Horiuhi et al.,
2005).
ina short amountof time (Odakaetal.,2003). They t afuntion Btexp ( At) to
the initialpart of the waveform envelopes of the past earthquakes and determine A
andBbytheleast-squaresmethod. ItisfoundthatthelogB isinverselyproportional
tologof epientraldistane. Therefore, inreal-timeanalysis, the observed envelopes
are t tothe empirialfuntion to estimatethe epientral distane.
Afterdeidingdistaneestimate,they estimatethemagnitudefromthemaximum
amplitudeobservedwithinagivenshorttimeintervalaftertheP-wavearrivalbyusing
an empirial magnitude-amplitude relation that inludes the epientral distane as
a parameter. Using epientral loation, depth, and magnitude as input data, the
amplitude of the maximum veloity on loal site bedrok and the arrival time are
estimated from a veloity attenuation relationship (Si and Midorikawa, 1999). In
order to obtain the peak ground veloity estimate from the site bedrok veloity
estimate, the latter is multiplied by a site ampliation fator from an available
database alled the digital national land information. Currently, this early warning
system is under going a testing phase, and the distribution of the early warning
informationis limitedto the people in harge of emergeny servies.
1.2.2.2 Seismi Alert System (SAS) of Mexio ity
SeismiAlertSystem(SAS)isaseismiearlywarningsystemforMexioity(Espinosa-
Aranda et al.,1995, 1996; Lee and Espinosa-Aranda, 2003). From the lesson of the
aftermath of the 1985 Mihoaan earthquake, the SAS was implemented to detet
subdution earthquakes ouring in the Mexian subdution zone loated several
hundred kilometers south-west of Mexio ity. The system onsists of a seismi de-
tetor on the Pai oast, teleommuniations, entral ontrol, and radio warning.
The loalmagnitude is estimated from anempirial relation embedded in eah seis-
mi detetor, and a warning message is sent via the teleommuniations unit if the
estimated magnitude is greater than 6. The system is eetive sine Mexio ity is
loated 300 km from the oast lineand it takes about 1minute for seismiwaves to
travel from the oast to the entral ity. The harateristis of the seismi damage
struture. The SAS would be more useful if the warning information is eetively
used for those high-risebuildings.
1.2.2.3 Early warning system in Taiwan
Taiwan has established several researh programs that are atively pursuing earth-
quake early warning and rapid reporting systems (e.g., Teng et al., 1997; Wu et al.,
1998). The earlywarningsystem establishedby the TaiwanCentralWeather Bureau
(CWB)usesareal-timestrong-motionaelerographnetworkonsistingof86stations
distributed aroundTaiwan(Wuand Kanamori, 2005b). The system takes anon-site
approah and the predominant period (
) and peak amplitude of displaement in
the rst 3 seonds after the P-wave arrival (Pd) determine the seismi magnitude
(Wu etal.,2006). Wuand Kanamori(2005a)alsofound thatPdorrelates wellwith
the peak ground displaement (PGD) and peak ground veloity (PGV)at the same
site. Therefore, P-wave arrivaltime,
, and Pd an jointlybe used todetermine the
hypoenter, magnitude, and the ground motion intensity at the site. For an event
withthesameloationasthe1999Chi-Chiearthquake,theTaipeimetropolitanarea,
at 145 km from the epienter, would have more than 20 se of early warning time
with this early warningsystem (Wu and Kanamori, 2005b).
1.2.2.4 Early warning system in the United States
TheU.S.GeologialSurvey(USGS)has sponsored thedevelopmentofatelemetered
earthquake monitoring system inCaliforniato providerapid earthquake information
for the benet of publi safety, emergeny response, and lossmitigation.
In Southern California, the CUBE (Calteh-USGS Broadast of Earthquakes)
projet,startedin1991, hada goaltodevelop nearreal-timeearthquake information
systems (KanamoriandHauksson, 1991). The seisminetworkinthe originalCUBE
systemuseddigitaldatafromaseisminetworkwithanalogtelemetry,whihseverely
limited the dynami range of the data. The inreasing demand of rapid earthquake
information after the 1994 Northridge earthquake led to the deployment of 24-bit
InnorthernCaliforniatheREDI(RapidEarthquakeDataIntegration)systemwas
operatedby the University ofCaliforniaatBerkeleyinollaborationwith the USGS.
Sine1994, theCUBE andREDIsystemshavebeenupgraded tothe CaliforniaInte-
gratedSeismiNetwork(CISN). Reently, Allenand Kanamori(2003) demonstrated
the feasibility of a short-term earthquake warning using the extensive data set from
CISN. The proposed system, ElarmS (Earthquake Alarm Systems), ould issue a
warning a few to tens of seonds ahead of damaging ground motion (Lokman and
Allen, 2005; Simons et al., 2006; Allen, 2006). Currently, universities, federal and
state government agenies, and the private setor are ollaborating for the pratial
implementationof an earlywarning system on CISN.
1.2.2.5 Early warning systems in other ountries
Asa resultofinreased publipereptionof thebenetsofearthquakeearlywarning
systems, suh systems are beingdeveloped alloverthe world. SouthernEuropeisan
earthquake-pronezoneandtheir nationalandloalgovernments haveagreatinterest
in mitigatingseismi damageby installingseismi earlywarning systems.
InCampaniaRegion,southern Italy,aprototypesystem forseismiearlywarning
and rapid shake map evaluationis being developed and tested (Zolloet al.,2006).
In Istanbul, Turkey, one hundred strong motion aelerometers have been plaed
in populated areas, and ten strong motion stations are sited at loations as lose as
possible tothe main fault(GreatMarmara Fault) inon-linedata transmission mode
to provide earthquake early warning information (Zshau et al., 2003; Erdik et al.,
2003).
Seismiity inBuharest,Romania,has speialproperties suh asthe invariability
of the loation of epienters and the stability of radiation patterns (Wenzel et al.,
2003). A Mexioity-type SASsystem would be adequatefor those kinds of areas.
TheityofYerevan,Armenia,isplanningtoinstall13-15seismidetetorsaround
the ity with a radius of 30 km. Approximately 3 to 8 seonds of warning time is
1.3 Objetives and road map for this thesis
In order toonstrut an earlywarning system for large earthquakes, we haraterize
the rupture extent and the slip onthe fault inreal time and preditgroundmotions
at a given site based on the urrent rupture onguration. The objetives of this
thesis are:
Charaterize the present rupture extent fromhigh-frequeny groundmotions
Charaterize the present sliponthe faultfrom low-frequeny groundmotions
Predit the rupture extent fromthe on-goingrupture.
The thesis is organized as follows: In hapter 1 we outline the researh area of
earthquake earlywarning systems and look at the previous researh in this area. In
hapter 2 we briey disuss the basi proedures of the Virtual Seismologist (VS)
method, a seismi early warning system developed by Cua and Heaton (Cua, 2005;
Cua andHeaton, 2006). In hapter3 wedisussa strategytoextend the VSmethod
to large earthquakes. To work this problem, we rst reognize the statistial obser-
vations of high-frequeny and low-frequeny ground motions for large earthquakes
with magnitude greater than 6.0. In hapters 4 and 5 we introdue two dierent
methodologies that an estimate the rupture geometry from aeleration envelopes.
In the rst method the rupture geometry an be haraterized with three parame-
ters, an azimuthal diretion, and two rupture lengths, one in the positive diretion
and one inthe negativediretion, asmeasured fromtheepienter. Theseparameters
an be estimated from aeleration envelopes in real time. In hapter 5 we propose
a tehnique to lassify near-soure and far-soure stations. In hapter 6 we propose
a methodology to determine the slipon the fault and predit the total lengthof the
rupture propagation possible onditioned on the urrent slip. Finally, in hapter 7
we provideonlusions and future work.
Chapter 2
General Virtual Seismologist
Method
Inthishapter,webrieydisussthebasiproeduresoftheVirtualSeismologist(VS)
method developed by Cua and Heaton (Cua, 2005; Cua and Heaton, 2006), whih
formsafoundationfortheworkinthisthesis. TheVSmethodisaBayesianapproah
for seismi early warning systems. The Bayesian framework provides a means to
inorporate previous experiee and judgment that is not traditionallyand expliitly
inorporated into automated deision making. When making a deision, a human
proesses many kinds of information, ombines and analyzes them simultaneously,
and makes a judgment based on the analyzed information. The Bayesian approah
is a sheme to emulate human apabilities to judge multiple piees of information
omprehensively and makejudgements from limitedinformation.
One omponent of the VS method is a method to estimate: 1) magnitude from
observed groundmotion ratiosbetween vertial aelerationand vertialltered dis-
plaement;and2)magnitudeand loationfromP-andS-waveamplitudesofvertial
and horizontal aeleration, veloity, and ltered displaement. Any seismi early
warningsystemestimates theearthquakeinformationfromthe sparsesetof available
observations immediately afterthe initialP wave detetion. What dierentiates the
VS method from other proposed seismi early warning systems is the use of prior
information. Prior information (i.e. the state of healthof the seismi network, fault
of the initialestimateof the event information.
2.1 Bayes' theorem for seismi early warning sys-
tem
Bayes' Theorem isa simplemathematialformulatoalulate onditional probabili-
ties. The probability of event A onditioned on the ourrene of event B isalled a
posteriorprobability for theeventA. This an beexpressed asa normalizedprodut
of a prior probabilitydensity funtion (pdf) and alikelihood funtion:
prob(AjB)
posterior
=
l ikel ihood
prob(BjA) prior
prob(A)
prob(B)
normal izingonstant
(2.1)
The posterior probability for the earthquake early warning system is the probability
oftheparameterwewouldliketoestimate(e.g.,magnitude,loationoftheepienter)
given observed groundmotiondata(e.g.,aelerograms, GPSdisplaement). Forthe
VS method, Bayes' Theorem an thereforebeexpressed as:
prob(M;R jA)
posterior
=
prob(AjM;R )prob(M;R )
prob(A)
/prob(AjM;R )
l ikel ihood
prob(M;R )
prior
; (2.2)
whereA isthe observed groundmotion amplitude,M isthe magnitude ofthe earth-
quake, and R is the loation(i.e., latitude and longitude) of the epienter. The pos-
teriorpdf, prob(M;R jA), isproportionaltothe produtof thepriorpdf,prob(M;R ),
andthelikelihoodfuntion,prob(AjM;R ),sinetheonstant,prob(A),isindependent
ofthemagnitudeandtheloationoftheearthquake. Theposteriorpdfrepresentsthe
onditional probability of magnitude and loation when we observe the ground mo-
tion amplitude. The best estimation of the magnitude and loationan be obtained
by maximizingthe posteriorto give the most probable values (see Figure2.1).
Likelihood of M,R
During the event
Location of known faults
Previously observed seismicity Geometric considerations Gutenberg-Richter law
Observed ground motions (A)
Before the event
prob(M,R) prob(A|M,R)
prob(M,R|A)
Least square error
6 (A- Â) 2
Ground motion model (Â=f(M,R))
Prior of M,R
Posterior
(Most likely M,R)
Figure 2.1: A blok diagram to ompute the posterior pdf of Bayes' theorem from
the prior informationand real-timeground motion data.
servation given the magnitude and distane. It is dened using a ground motion
attenuation relationship for ground motion amplitudes in terms of magnitude and
distane. The sum ofsquare errors ((A
^
A) 2
)is oftenusedto denethe likelihood
funtion whih orresponds to taking a Gaussian probability model for eah predi-
tion error, the error between the observation (A) and predition (
^
A) based on the
models. TheBayesian approahredues tosome othergeophysialinverse methodsif
thepriorinformationisnotonsidered;thenitisthesameasthemaximumlikelihood
methodandorrespondstoaleast-squareapproahintheaseofGaussianpredition
errrors.
The prior pdf expresses information known before examining waveform data for
the ongoing earthquake rupture. Station geometry, loation of faults, or previously
observedseismiityanbeexpressedasprobabilitydensityfuntionsandusedasprior
information. For example, the regions where earthquakes were observed on previous
days have a higher probability of produing additional earthquakes. Therefore, the
is also higher for areas near known faults. Other prior information (e.g., station
geometry, Gutenberg-Rihter law) an beinluded in the same way.
2.2 Dening the prior prob(M;R )
The prior pdf is a probabilityof magnitude (M) and loation (R ) based onlyon the
information obtained before an earthquake ours. If there is no prior information,
the magnitude and loation of anearthquake are treated as equally likely to be any
size and at any plae, and so a uniform prior is used. However, generally speaking,
thereis usually someinformationbeforethe initiationof anearthquake rupture, and
that information an be used to onstrain the magnitude and loation estimates in
seismi earlywarning. The followinginformationis onsidered asprior information:
Loation of known faults
Previously observed seismiity
Geometri onsiderationof stations
Gutenberg-Rihter law
2.2.1 Loation of known faults
Reognizedativefaultsaremorelikelysouresoffuturelargeearthquakethanregions
without reognized faults. Even though there are many faults hidden underground
whiharetoosmalltoextendfromearthquakedepthstogroundlevel,theinformation
of ative faults helps to onne the soure loation. The prior pdf, onsidering the
loation of known faults, an be dened as an exponential funtion of the distane
fromfault lines (Felzer and Brodsky,2006):
1:34
where
r=the shortest distane between faultlines and astation,
=onstant.
An example of the prior pdf for the known faultsis shown in Figure2.2.
Figure 2.2: An example of the prior pdf for the known faults for the 2004 Parkeld
earthquake. Solidlines indiatethe loationof the faultlines inCaliforniaanddark-
ness of the shade around the lines show higher prior pdf values. The star symbol
shows the epienter of the Parkeld earthquake.
2.2.2 Previously observed seismiity
Sine observations of foreshoks preeding large earthquakes are signiantlyrelated
to subsequent earthquakes, the regions where an earthquake was observed on the
previousdayhaveahigherprobabilityofanearthquakeourrene(Aberrombieand
Mori,1996). AberrombieandMori(1996)foundthat44%oftheearthquakesintheir
California dataset had foreshoks. Therefore, the prior pdf is higher at regions near
observed seismiity is expressed by the exponential funtion (Felzer and Brodsky,
2006):
prob(r)=r 1:34
; (2.4)
where
r=jx x
i j;
x=loation of the station;
x
i
=loation of the foreshok epienter (i =1;:::;n);
=onstant :
An example of the prior pdf for the known faultsis shown in Figure2.3.
Figure 2.3: An example of the prior pdf for the previously observed seismiity of
the 2004 Parkeld earthquake. Open irles indiate the loation of the previously
observedseismiityanddarknessofthe shadearoundtheirleshowhigherprior pdf
2.2.3 Geometri onsideration of stations
Stationgeometryalsoprovidesageometrionstrainttotheloationofanearthquake
epienter. RydelekandPujol(2004),Cua(2005),andHoriuhietal.(2005)developed
a new tehnique to onstrain the loation of an earthquake from the P-wave arrival
time using the Voronoi ell onept (Sambridge, 1999a,b). The Voronoi ell of a
station is a onvex polygon around the station, whih is a set of all points loser to
a station than to any other stations. The loation of the earthquake epienter must
be insideof the Voronoiellof the station rst triggered by aP-wave arrival(Figure
2.4).
Figure 2.4: Voronoi ells of strong motion stations for 2004 Parkeld earthquake.
Trianglesdenotestrongmotionstationloations. Theshadedregionisthatofpossible
loation of epienter when the losest station PKD detets the rst P-wave arrival.
The star symbolshows the epienter of the Parkeld earthquake.
After the rst P-wave arrivesat the rst station,not-yet-arrived data an shrink
theprobableregionoftheepienterloationinsidetheVoronoiell(Figure2.5). From
Rydelek and Pujol (2004),the regionof likely loation of the epienter based on the
rst two P-wave arrivals forms a hyperbola, whih is a set of points the dierene
Figure 2.5: Voronoi ells of strong motion stations for 2004 Parkeld earthquake.
Trianglesdenotestrongmotionstationloations. Theshadedregionisthatofpossible
loationofepienteratthe3seondsaftertherstP-wavedetetion. Thestarsymbol
shows the epienter of the Parkeld earthquake.
onstant k (Figure2.6). Furthermore, the use of not-arrived data after the rst two
P-wave arrivalsan provideontinuously evolving onstraints on the region of likely
loation.
Figure 2.6: Voronoi ells of strong motion stations for 2004 Parkeld earthquake.
Trianglesdenotestrongmotionstationloations. Theshadedregionisthatofpossible
loation of epienter at the seond P-wave detetion. The star symbol shows the
epienter of the Parkeld earthquake.
2.2.4 Gutenberg-Rihter law
The Gutenberg-Rihter law states that the number of earthquakes per year, N, of
Rihter magnitude M is statistially proportional to 10 bM
(see Figure 2.7). This
relationship ismathematially expressed as:
N(M)=10 a bM
; (2.5)
where a and b are onstant,and the size of the onstant b is typially around1.
Aording to the Gutenberg-Rihter law, there are a lot more small earthquakes
thanlargeones. Therefore,theprior pdforrespondingtotheGutenberg-Rihterlaw
is dened as:
prob(M)/10 a bM
: (2.6)
2 3 4 5 6 7
0 500 1000 1500 2000 2500 3000 3500
Magnitude (M)
Number of events (N)
N(M)=10 a−bM
Figure 2.7: Histogram of the magnitude of the earthquakes in Southern California
2.3 Dening the likelihood funtion prob(AjM;R )
Thelikelihoodfuntionistheprobabilityofthegroundmotionamplitudeobservation
(A)giventhemagnitude(M)anddistane (R ). Cua(2005)denedalikelihoodfun-
tion intermsof the ratiobetween vertial aelerationanddisplaementamplitudes,
andtheenvelopeattenuationrelationshipsforvertialaelerationandhorizontala-
eleration, veloity, and displaement. This setion desribes the magnitude ground
motionrelationships,P-waveandS-wavedisriminant,andgroundmotionmodelsas
omponents of the likelihoodfuntion.
2.3.1 Magnitude ground motion relationships
Magnitude groundmotionrelationshipisoneof themeasurements tond magnitude
of anearthquakefromthe ground motion. Many seismologistshave pointed out that
the P-wave predominant period is linearly orrelated with the ultimate magnitude
(Nakamura and Tuker, 1988; Allen and Kanamori, 2003). Cua and Heaton (2006)
use ratios of the ground motion as indiative of the predominant frequeny of the
seismograms. Sine the aeleration is equal to the square of frequeny (!
2
) times
displaement in the frequeny domain, the magnitude is proportional to the ratio
between aelerationand displaement.
M /! 1
0
(2.7)
=
1
log(aeleration)+
2
log(displaement)+
3
;
where !
0
is the predominant frequeny of the ground motion, and
1
;
2
, and
3 are
oeÆients. Cua (2005) performed a linear disriminant analysis with over 30,000
seismogramsinSouthern California todeterminethese oeÆients. Figure2.8 shows
the dataset and the most probable linear disriminant funtion whih lassies the
dataset with dierent magnitudes. The best magnitude ground motion relationship
1.72 2.48 3.09 3.53 2
3 4 5 6 7 8
Z ad =acc 0.36 /disp 0.93 =0.36 log(acc) − 0.93 log(disp)
Magnitude
Linear Discriminant Analysis of P−wave Amplitude Ratios M > 6.0
5.0 ≤ M < 6.0 4.0 ≤ M < 5.0 3 0 ≤ M < 4.0 M < 3.0
using log(acc), log(disp)
M = −1.627 Z + 8.94 1
Figure2.8: LineardisriminantanalysisofP-wavelog(a)andlog(disp)asindiators
of magnitude. Z =X
2
u=0:36log(a) 0:93log(disp) (Cua, 2005).
^
M = 8
>
<
>
:
1:627(0:36log(Za) 0:93log(Zdisp))+8:94 : if P-wave;
(2.8)
whereZaandZdisparevertialaelerationandvertialdisplaement,respetively
and standard deviations are:
= 8
>
<
>
:
0:45 : if P-wave ;
0:41 : if S-wave :
(2.9)
By using this relationship, the observed and predited ground motion ratios in
equation 2.19 are expressed asfollows:
Z
i
=0:36log(Za) 0:93log(Zdisp); (2.10)
^
Z
i
(M)= 8
>
<
>
:
( M +8:94)=1:627 : if P-wave;
( M +8:05)=1:459 : if S-wave:
(2.11)
2.3.2 P-wave and S-wave disriminant
In equation2.11, the magnitude groundmotionrelationship isdened separatelyfor
P-waveandS-wave. Althoughitisnot signiantlysensitivetowhetherthe observed
amplitudesareP-orS-wave(seeequation2.11),weanobtainbettersoureestimates
ifweanidentifyphases(Cua,2005). Cua(2005)denedadisriminantfuntionasa
linearombinationofgroundmotionmeasures,and found thebestombinationsand
oeÆients for seismograms in Southern California by linear disriminant analysis.
The result of the P/S wave disriminant is shown in gure 2.9. The most probable
disriminantfuntion is:
PS =0:44log(Za)+0:55log(Zvel) 0:46log(Ha) 0:55log(Hvel) (2.12)
=log(
Za 0:44
0:46
)+log(
Zvel 0:55
0:55 );
−1 −0.5 0 0.5 1
−4
−2 0 2
Z 1 =X u
1
Z 2 =X u 2
S−wave P−wave
Z 1 =−0.098
P/S discriminant using acceleration, velocity
S P
Figure 2.9: P/S wave disriminant using vertial and horizontal ground motion a-
eleration and veloity (Cua, 2005).
if 8
>
<
>
:
PS>0 : P-wave;
PS<0 : S-wave;
where Za, Zvel, Ha, and Hvel are vertial aeleration and veloity, and hori-
zontalaeleration and veloity, respetively.
2.3.3 Ground motion models
Cua and Heaton examined over 30,000 seismograms in Southern California and de-
distane and station orretions (Cua, 2005; Cua and Heaton, 2006). First, the en-
velopes of the ground motions are modeled as a ombination of the envelopes of
P-wave, S-wave, and ambientnoise.
E
observed (t)=
q
E 2
P
(t)+E 2
S
(t)+E 2
ambient
+; (2.13)
where
E
observed
(t)=envelope of observed groundmotion;
E
P
(t)=envelope of P-wave ;
E
S
(t)=envelope of S-wave and later-arrivingphases ;
E
ambient
=ambient noise atthe site;
=dierene between predited and observed envelope :
Theambientnoise,E
ambient
,foragiven timehistoryismodeledasastationonstant.
The P-and S-waveenvelopes, E
P
(t)and E
S
(t), are dened by arise time (t
rise
P and
t
rise
S
),aonstant amplitude(A
P
andA
S
),aduration(t
P
and t
S
),andtwodeay
parameters(
P
and
S
)and (
P and
S
) respetively. See gure 2.10for thephysial
interpretation of these parameters.
The general formof the envelope funtion is:
E
ij (t)=
8
>
>
>
>
>
>
<
>
>
>
>
>
>
:
0 ; t<T
i
;
Aij
t
rise
ij (t T
i
) ; T
i
t<T
i +t
rise
ij
;
A
ij
; T
i +t
rise
ij
t<T
i +t
rise
ij +t
ij
;
A
ij
1
(t T
i t
rise
ij t
ij +
ij )
ij
; tT
i +t
rise
ij +t
ij
;
Figure 2.10: Observed envelope for aelerogram and P-wave and S-wave envelopes
for the ground motionmodel dened inequation 2.14 (Cua, 2005).
where
i=P-, S-wave ;
T
i
=P-, S-wave arrival times;
j =horizontaland vertialaeleration, veloity,and displaement:
CuaandHeatonparameterizedeahseismogramasasetofelevenparameters(ve
forthe P-wave envelope,vefor theS-waveenvelope,and oneforthe ambientnoise).
Furthermore, eah parameter is desribed by magnitude, distane, log of distane,
and site dependent onstants based on the traditional strong motion attenuation
relationships (Campbell, 1981; Boore and Joyner, 1982; Boore et al., 1993). The
funtional forms whih desribe the P- and S- wave envelope funtions are given
log
10 A
ij
=a
ij M+b
ij (R
1 +C
ij
(M))+d
ij log
10 (R
1 +C
ij
(M))+e
ij +
ij
; (2.15)
log
10 B
ij
=a
ij M+b
ij R
1 +d
ij log
10 R
1 +e
ij +
ij
; (2.16)
where
i=P-, S-wave;
j =horizontaland vertial aeleration, veloity, and displaement;
A
ij
=ground motionenvelopeamplitude;
B
ij
=rise time (t
rise
),duration(T),and deay parameters (, );
M =loal magnitude(M
w
for M >5:0);
R =epientraldistane inkm for M <5;
losest distane to faultfor M >5:0 (whenavailable);
R
1
= p
(R 2
+9);
C
ij
(M) =(artan(M 5)+1:4)(
1ij exp (
2ij
(M 5)));
a
ij
;b
ij
;
1ij
;
2ij
;d
ij
;e
ij
=regression onstants;
ij
=statistial(or predition)error,NID(0;
2
):
The A
ij
s are the ground motion envelope amplitudes (P- or S-wave) from tting
equations 2.13 and 2.14 to the observed ground motion envelopes in the database.
The B
ij
s are the parameters haraterizing the envelope funtion (t
rise
, T, , and
). CoeÆients in equations 2.15 and 2.16 are determined by regression analysis of
the database using the NeighborhoodAlgorithm (desribed later inSetion4.2). An
exampleofsetofoeÆients(forhorizontalandvertialaelerationsonsoilsites)are
shown in table 2.1. Table 2.1 and equations 2.13 { 2.16 an determine the envelope
funtion of ground motions with magnitude M and epientral distane R . Figure
2.10 shows an observed ground motion envelope and the best P-wave, S-wave, and
Table 2.1: CoeÆients for the envelope attenuation relationships for rms horizontal
and vertial aelerationona soilsite inequation2.16. Allattenuationrelationships
model log
10
of the envelope parameter asfuntions of magnitude and distane (Cua,
2005).
CoeÆientsforrmshorizontalaeleration on soilsites
a (M) b (R) d(log(R)) 1 2 e
A
P
0.740 3:3010 3
1.26 2.41 0.95 0.90 0.29
A
S
0.840 2:3010 3
1.56 2.42 1.05 0.19 0.31
T
rise;P
0.070 1:2510 3
0.24 - - 0.38 0.26
T
P
0.030 2:3710 3
0.39 - - 0.59 0.36
P
0.087 1:8910 3
0.58 - - 0.77 0.31
P
- - - 0.07 0.21
T
rise;S
0.055 1:2110 3
0.34 - - 0.66 0.25
T
S
0.028 - 0.07 - - 0.10 0.23
S
0.056 8:3010 4
0.51 - - 0.58 0.24
S
- - - 0.07 0.13
noise - - - 2.50 -
CoeÆientsforvertialaeleration on soilsites
a(M) b(R) d(log(R)) 1 2 e
A
P
0.739 4:1310 3
1.20 2.03 0.97 0.62 0.32
A
S
0.751 2:4710 3
1.47 1.59 1.02 0.21 0.30
T
rise;P
0.057 5:8610 4
0.23 - - 0.37 0.23
T
P
0.000 1:7610 3
0.36 - - 0.48 0.41
P
0.057 1:3610 3
0.63 - - 0.89 0.28
P
- - - 0.05 0.18
T
rise;S
0.060 2:1810 3
0.26 - - 0.66 0.25
T
S
0.029 - 0.31 - - 0.31 0.24
S
0.060 1:4510 3
0.51 - - 0.54 0.22
S
- - - 0.05 0.09
noise - - - 1.96 -
2.3.4 Complete form of the likelihood funtion
Aswedisussedatthetopofthissetion,thelikelihoodfuntionisdenedintermsof
the ground motion ratio between vertial aeleration and displaement amplitudes,
and the envelope attenuation relationships for vertial aeleration and horizontal
aeleration, veloity, and displaement.
best estimate, the error between the observation and predition from the magnitude
ground motionrelationships isminimized.
prob(Z
i jM)=
1
p
2
Z
i exp
(Z
i
^
Z
i (M))
2
2 2
Z
i
; (2.17)
where
i=1;:::;n, where nis the numberof stations with P detetions;
Zi
= standard deviationin equation 2:9;
Z
i
= observed groundmotion ratio inequation 2.10;
^
Z
i
= groundmotion ratiopredited by the magnitude ground motion;
relationshipin equation2.11:
The amplitude of the ground motion envelopes estimate the magnitude and lo-
ation of earthquakes. The errors between the observed envelopes and predited
envelopes from the ground motion models are alsoapproximated by a Gaussian dis-
tribution.
prob(Y
ijk
jM;R )= 1
p
2
ijk exp
(Y
ijk
^
Y
ijk
(M;R )) 2
2 2
ijk
; (2.18)
where
j =1;:::;4,for peak amplitudes of vertial veloity, and
horizontalaeleration,veloity, and displaement;
k =1;:::;nt,time in 1-seond intervalsfrom the event onset ;
ijk
=standard deviationof j hannelsand timek atstation i
Y
ijk
=log
10
of peak observed amplitude of j hannels and time k at stationi
^
Y
ijk
=log
10
of peak amplitudeof k hannelsand phasej atstation i
The vertial aeleration and displaement are used to estimate the magnitude,
andthe amplitudesofthe vertialveloityand threehorizontalomponentssolvethe
trade-o between the magnitude and loationof the epienter. From equations 2.17
and 2.18, the likelihood funtion of 1-seond-interval ground motion envelopes (A)
onditioned onthe magnitude (M)and loation(R ) is:
prob(AjM;R ) = n
Y
i=1 4
Y
j=1 nt
Y
k=1
prob(Z
i
jM)prob(Y
ijk
jM;R )
/exp
n
X
i=1
(Z
i
^
Z
i (M))
2
2 2
Z
i
+ 4
X
j=1 nt
X
k=1 (Y
ijk
^
Y
ijk
(M;R )) 2
2 2
ijk
:
(2.19)
2.4 Finding the best estimates
InordertooperatetheVSmethodinrealtime,werstassumethatseismiwaveform
dataaretransmittedtoaentralproessorbyaseisminetworkwithsuÆientstation
density to quikly haraterize the seismi wave eld. The entral proessingstation
proessesurrentlyavailableseismireordsand produesupdatesasadditionaldata
are reeived. The prior probability inorporated in the real-time Bayesian analysis
inludes informationabout magnitude likelihood(e.g., Gutenberg-Rihter frequeny
magnitude) and loation likelihood (e.g., known faults, or previously observed seis-
miity). This prior pdf has been alulated before the ourrene of any earthquake
whih the VS method is intended to provide a warning for. As the seismi data
arrives, the proessor an use it to evaluate the likelihood funtion for any loation
and size of the earthquake in order to maximize the posteriorin equation 2.2 to get
the best estimate of magnitude and loation of the earthquake; this is done using
updated information every seond. The predited ground motionat any site an be
omputed by the ground motion model in equations 2.13 and 2.14, sine a magni-
tude and distane dene the ground motion envelope uniquely in the model. This
strategy assumesa point-soure model and worksfor small tomoderate earthquakes
2.5 Summary
Inthishapter,webrieydisussedthebasiproedures oftheVSmethoddeveloped
by Cua and Heaton (Cua,2005; Cua and Heaton, 2006).
The VS method is aBayesian approah for seismi early warning systems. It in-
orporatesprior informationwhih an be obtained beforean event and a likelihood
funtion omputed from the ground motion data available after the initial P-wave
detetion,and ndsthe mostprobable estimateformagnitudeand loationby maxi-
mizingthe posterior,whih isequivalent tomaximizingthe produtof prior pdf and
likelihoodfuntion.
Wedisussed howtodeneprior pdf andlikelihoodfuntionfromavailableset of
datainthishapter. Theloationofknownfaults,andpreviouslyobservedseismiity,
geometrionsiderationof stations,andGutenberg-Rihterlawareonsidered asthe
prior information. Likelihood funtion is dened in terms of the magnitude ground
motion relationship and envelope groundmotion amplitudes. More detail about the
VS method and examples of the appliation of the VS method are shown in Cua's
Ph.D. thesis (Cua,2005).
Chapter 3
Extended Virtual Seismologist
Method
This hapter disusses astrategy toextend the Virtual Seismologistmethodtolarge
earthquakes. We obtain the nite-rupture information by inverting high-frequeny
and low-frequeny ground motions respetively. To understand this proedure, it is
importanttoreognizetheharateristisofhigh-frequenyandlow-frequenyground
motions. Thishapteralsoanalyzesthestatistialfeaturesofobservedhigh-frequeny
andlow-frequenygroundmotionsforlargeearthquakeswithmagnitudegreaterthan
6.0.
3.1 Road map for Virtual Seismologist Finite-Soure
method
The previous hapter briey disusses the general VS method. In its urrent level
of development, this methodology seems eetive for earthquakes (M < 6.5), where
ruptureanbemodelledwithapointsoure. However, forlargeearthquakes,rupture
length an be on the order of tens tohundreds of kilometers, and the heterogeneous
slipdistributionsigniantlyaets the groundmotionamplitude expetedata site.
For example, the fault rupture in the 1999 Chi-Chi earthquake was longer than 80
km,andthe largestslipwasreorded neartheend oftherupture atthenorthern end
large distanes from the epienter when using a sheme that only haraterizes the
earthquake as apointsoure.
In order toextend the VS methodtoearthquakeswith M > 6.5, weneed to on-
sider thefaultrupture geometryand the size ofsliponthe fault. Todierentiate the
VS methodonsidering the faultniteness, we allthe generalVSmethoddesribed
intheprevioushapter\VSPoint-Soure(PS) method"and theVSmethodforlarge
earthquakes \VS Finite-Soure (FS)method."
Our strategy for large earthquakes is asfollows. (See alsogure 3.1.)
01
Prediction of future shaking at a site
Acceleration Displacement
Real-time seismic data Prior
information
Displacement envelope fitting 1. Acceleration envelope fitting
2.Near-source/far-source classification
Approximate current rupture geometry Estimate current slip on the rupture
Prediction of future rupture
Figure3.1: The algorithmofthe VSmethodfornitesoure(VS-FSmethod). First,
weestimatetherupturegeometryfromtheaelerationsbythemethodsdisussedby
YamadaandHeaton(2006)andYamadaetal.(2006). Basedonthisgeometry,slipon
the faultan beestimated fromdisplaement reords. Byombiningurrentrupture
informationand priorinformation,the preditedprobabilityofrupture extentan be
1) Apply the VS-PS method
First, apply the VS-PS method to the ongoing rupture. Estimate the epienter and
magnitudeofaneventwhentheloseststationsreordtheP-waves. Ifthemagnitude
isless thanaertainthreshold (e.g. M <5.5),the estimatedloationandmagnitude
of the earthquake is aepted. If it exeeds the threshold, then there is a reasonable
possibility that the earthquake is large, and it might not be adequately modeled as
a point soure. In this ase, we apply VS Finite-Soure (FS) method to nd the
loation of the nite fault.
2) Estimating the urrent rupture extent
The VS-FS method determinesthe ongoingrupture geometryin real timefrom high
frequeny ground motions. Aeleration reords are used to estimate the temporal
and spatial evolution of the rupture front. Use of Bayes' theorem in equation 2.1 is
alsohelpful here. The posteriorpdf of the problemof estimating a rupture extent is
the probability of the rupture loation (S) given observed ground motion data (A).
Bayes' Theorem for the problem toestimaterupture geometryis:
prob(SjA)/prob(AjS)prob(S): (3.1)
The prior prob(S) is information known before examining waveform data, suh
as the loation of known faults. Large earthquakes oftenour on reognized ative
faults, and informationabout the loation and ativity of these faults is potentially
avaluableset of priorinformation. After anearthquake initiatesand groundmotion
data beomes available, the likelihoodfuntion willbe omputed.
The likelihoodfuntion prob(AjS)is the probability ofthe ground motionampli-
tude observation given the rupture loation. Two separate methodologies have been
developed toestimatethe evolvingrupture geometry:
i) themultiple souremodel desribed inhapter 4determinesthe rupture geom-
etrythatbestpredits the envelopesof high-frequenyground motions(Yamadaand