Time (s) from event
Model 1 Model 2
Model 3 Model 4 Model 5 Model 6
0 10 20 30 40 50 60
−15
−10
−5 0 5 10 15
V r =2.0 km/s
(south) N2 N1 (north)
time(s)
Model 1 Model 2 Model 3 Model 4 Model 5 Model 6
Figure4.19: Time seriesof the estimatedparameters,,N1, andN2, foreahmodel.
The model numbers orrespond to the numbers in table 4.2. Time is relative to the
origin. The parametersare omputed ateah seondusingonlythe data availableat
thattime. Thebroken linesare thebest estimatesbasedonthefaultmodelproposed
byJietal.(2003). Top: timeseriesestimationsfor. Bottom: timeseriesestimations
for for N1 and N2. The solid thin lines are the upper limits for N1 and N2 for the
Figure 4.19 shows the estimation results of three parameters (azimuthal angle
of fault line (), number of the point soures to the north (N1) and to the south
(N2)). Three parametersare omputed at eahseond using only the data available
atthat time. Theestimation isupdated every seondasthe groundmotiondata are
observed.
4.3.3.1 Result of model 1 (horizontal and vertial data)
Model 1inludesallofthedataonsidered inthisstudy. Althoughitdoesagoodjob
atharaterizingthe rupture lengthandtiming,we see thatitisdiÆulttoresolve
until15seonds afterthe event onsetsinethe eventan beapproximated asapoint
soure at the beginning. The estimated at 15 seonds is about -20 degrees and it
inreases graduallyafter20seonds due toaimpulsiveaelerationarrivalatstation
C080whihisloatedatthesouth ofthe epienter. Estimatesof stabilizeatabout
13 degrees with respet to additional data after 26 seonds. There is an additional
small shift at 44 seonds, at whih point the inversion ahieves its nal solution of
15 degrees, whih ompares favorably with the observed average fault strike of the
Chelungpu faultrupture.
Sine the subsoures are equally spaed, the length of the faultis represented by
the number of the point soures to the north (N1) and to the south (N2). Figure
4.19(bottom)shows values ofN1andN2 asafuntionof timeaftertheorigin. From
the gure, wean see the fault lengthgrows bilaterally alongthe dashedblak lines.
At 26 seonds, the rupture stops growing to the south. It also stops to the north
temporarily,but it grows again around40 seonds. This is due to the delayed high-
frequeny radiation at stations north of the Chenlungpu surfae rupture and may
have been aused by rupture on the Shihtan fault. Even though the result of the
simulation ts the atual loation of the faultaurately, the multiple soure model
does not onsider \rupture jumping disloations" (i.e., the rupture at the adjaent
ative faults triggered by the main shok) (Shin and Teng, 2001). The nal result
shows7pointsourestothe northand4pointsourestothesouth. Thisfaultlength
gure 4.7.
4.3.3.2 Result of model 2 (horizontal data) and model 3 (vertial data)
Model 2 only uses the horizontal aelerationdata for the analysis whereas model 3
onlyuses the vertialaeleration data. The azimuthalanglesof thefaultfor models
2and 3 are not signiantlydierentfrom model 1. The estimation ofthe angle, N1
and N2 from the horizontal omponent data (model 2) is similar to the estimation
of model 1. However, the estimation of rupture length from the vertial omponent
data (model 3) is a little smaller than that of model 1. In partiular, the inversion
indiates unilateral rupture to the north (i.e., N2 is zero) until 18 seonds after the
origin. Thereasonisthatthepreditedenvelopesoverestimatetheobservedenvelopes
intheepientralregionfortherst10seonds(seegure4.14). Overall,thepredited
envelope islarger thanthe observed envelopefor the vertialomponentand smaller
for the horizontalomponent.
4.3.3.3 Result of model 4 (eet of area weight)
Model 4onsiderstheheterogeneityofstationdistributionand appliesanareaweight
when weharaterizethe mist funtion. The area weight isa oeÆient appliedfor
eahstation. SinethestationdistributionisnotuniformfortheChi-Chiearthquake
dataset, we attempt to normalizethe eet of eah station. We assume a station in
a sparse area is more important than a station in a dense area. Therefore, when we
ompute the mist funtioninequation 4.2, the mist ofeah stationis weighted by
the area weight, whihis proportionalto the area of the Voronoi ellof eah station
(shown in gure4.7).
There are quite a few dierenes between the estimates for N1 and N2 of model
1 and model 4. The real-time estimation of the azimuthal angle has unique hara-
teristis. It stays around -20 degrees at the beginning of the rupture, and it jumps
to 35 degrees suddenly at 36 seonds. The angle estimation is very unstable even
after 40 seonds. Moreover, the estimate for N1 and N2 are a lot smaller than that
area weighting (e.g., T088,T074,C074)ontrolthe parameters. When the envelopes
of those stations are weighted, the residual sum of squares hanges greatly, and the
Neighborhood Algorithm hooses the parameter to redue the residuals. We would
liketo obtainaurate informationof the faultloation assoon aspossible. For this
purpose, model 1 is more robust than model 4. In a larger sense though, it means
that itbeomesdiÆulttodetermine thefaultgeometryif the stationdistributionis
sparse and uneven.
4.3.3.4 Result of model 5 and model 6 (the eet of station distribution)
In models 5 and 6, the eet of station distribution is examined further. To sample
thestationsrandomly,weusethereordswithanevenstationodenumberformodel
5. Formodel6,the reordswith astationodeendingin6or8(e.g.,T078)are used.
Even though the station distribution is not homogeneous asshown ingure 4.7, the
averagestationdensityis214km 2
/stationformodel5,and482km 2
/stationformodel
6. The stations are loatedin anareaof about 27,000km 2
. Even thoughthe station
density is dierent, the estimated parameters are quite similar. In gure 4.19, the
time series of and N2 for models 1, 5, and 6 are almost the same. N1 for models
5 and 6 stays around 5 after 30 seonds, and the inrease observed in Model 1 due
tothe Shihtan fault rupture does not appear. The reason isthat several near-soure
stations of the Shihtan fault have an odd numberstation ode and are not inluded
in this analysis (e.g., T045, T047, and T095). Considering that the rupture of the
Shihtan fault is quite small ompared to that of the Chelungpu fault, model 5 and
model 6 an express the Chi-Chi earthquake rupture well. The VS-FS method for
largeearthquakesworkswelleven ifthe stationdensity isredued toaquarterof the
originaldensity, as long as the stationdistribution is uniform.
4.3.4 Geometry of the parameter spae
We have solved the optimization problem in parameter spae (, N1, and N2) by a
0 2
4 6
8
10 −90 −60 −30 0 30 60 90 40
42 44 46 48 50
θ (deg) N1
misfit
42 44 46 48
Figure4.20: Errorsurfae of andN1 forModel1 atthe xedN2 =5at60seonds
afterthe origintime. Sinethe surfae ispeaked around =0,it iseasytoonverge
in . However, the optimal N1 will hange easily depending on the mist funtion
(see equation 4.2).
Figure4.20shows the errorsurfaeof andN1formodel1ataxed N2of 5and
assumingthatalldataisused inthe inversion. The surfaeissmooth andhas adeep
and narrowvalley at =10. The solution easilyonverges tothis minimum. Figure
4.21 shows theerror surfae ofN1 andN2 formodel 1ataxed of 10. The surfae
is very smooth in both N1 and N2 diretions. The global minimum is very sensitive
tothe hoie of the dataset, as shown in the results of model 5 and 6.
Contour maps ofthe error surfae of N1and N2 at10seondintervalsare shown
in gure 4.22. is xed at 10 degrees whih is the optimal nal solution. At 10
seonds, the minimum of this error surfae is (N1, N2) = (0, 1). However, it is not
the global minimum inthe parameter spae sine = 10 isnot the optimalsolution
at 10 seonds. At 20 and 30 seonds, the minimum of the error surfae is at the
0 2
4 6
8
10 0 2 4 6 8 10
30 35 40 45 50 55 60
N2 N1
misfit
40 45 50 55
Figure4.21: Errorsurfae ofN1andN2formodel1atthexed =10at60seonds
after the origin time. Sine the surfae is smooth in both N1 and N2 diretion, the
optimalsolution issensitive toa small disturbane.
Thereishighpossibilitythattherupture isstillongoingatthis point. At40seonds,
the minimum of the ontour is around (N1, N2) = (6, 4) and it suggests that the
rupture has stopped rupturing toward the south. After 40 seonds, the shape of
ontour map does not hange muh, and the ellipti shape of the smallest ontour
indiatesthatN2isdetermineduniquely,butthatonsiderableunertainty aboutN1
remains.
The NeighborhoodAlgorithm generatessamples inthe parameterspae and on-
struts the posterior probability density (ppd) from the ensemble samples. (In this
simulation,the priorpdf isassumed tobeuniform.) The 1-Dmarginalposteriorppd
of parameter , N1, and N2 are shown in are shown ingures 4.23 { 4.25. The ppd
for ismorepeaked thanthoseforN1andN2,anditisonsistentwiththegeometry
of theerror surfaewhihenablesa solutionto onverge easilytothe minimum. The
0 2 4 6 8 10 0
2 4 6 8
N1
N2
T = 10sec
0 2 4 6 8 10
0 2 4 6 8
N1
N2
T = 20sec
0 2 4 6 8 10
0 2 4 6 8
N1
N2
T = 30sec
0 2 4 6 8 10
0 2 4 6 8 10
N1
N2
T = 40sec
0 2 4 6 8 10
0 2 4 6 8 10
N1
N2
T = 50sec
0 2 4 6 8 10
0 2 4 6 8 10
N1
N2
T = 60sec
Figure4.22: Contourmaps ofthe errorsurfae ofN1and N2formodel 1atthexed
= 10. The maps are shown in 10 seond intervals. The blank area in the boxes is
the region where there is no solutiondue tothe onstraint that the rupture veloity
is less than 2 km/s.
beomes for all three parameters. Figure 4.26 is the 2-D marginal of parameters N1
and N2. The dierene between gure 4.22 and gure 4.26 is as follows: gure 4.22
is the error surfae where the mist funtion (equation 4.2) is evaluated and gure
4.26 is the posterior probability density of the parameter spae. The loationof the
most probable solution is almost idential between gure 4.22 and 4.26, but gure
4.26shows theppdwhihrepresents theprobabilityforeahvalueof theparameters.
The maximum value of ppd beomes larger with time.