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Computer-Based Ground Motion Attenuation Modeling Using Levenberg-Marquardt

Method E. Irwan syah, Ri an Budi Lukm anto, Rokhana D. Bekti and Prisci lia Budiman

Abstr act In this paper, we presen t the resul ts of resear ch on the optimiza tion model

ing of ground motion attenuatio n of the two establis h model s by Youn gs et al. [ 25 ]

and the model of Lin and Lee [ 13 ] using the Lev enberg- Marquardt metho d.

This modeling is parti cularly imp ortan t in the case of ground motion given that it

takes a good model for p redicting the stre ngth of earthquakes in order to reduce the

risk of the imp act o f the natur al disas ter. The re are two mai n contri- bution s of this

resear ch is the opti mizatio n of ground mot ion attenuati on models with Levenberg-

Marqu ardt met hod on two models that have been extens ively used and the deve

lopment o f compu ter appli cations to help acceler ate model ing, espe- cially on large

data with an area o f extens ive resear ch.

Lev enberg- Marquardt metho d proved to give a good contribut ion to the modeling of

ground motion attenu ation that is indicated by the very small deviat ions between the

predicted values with the actual value. Keyword s Ground motion attenuati on _ Levenbe

rg-Marquar dt _ Earthqu ake E. Irwansyah ( &) _ R.B. Lukmanto School of Computer

Science, Bina Nusantara University, Jl. KH. Syahdan No. 9, Kemanggisan, Jakarta 11480,

Indonesia e-mail: eirwansyah@binus.edu URL: http://binus.ac.id/ R.B. Lukmanto e-mail:

rianbl@binusian.org R.D. Bekti Department of Statistics, Institut Sains and Teknologi

AKPRIND, Jl. Kalisahak No.

28 Kompleks Balapan Tromol Pos 45, Yogyakarta 55222, Indonesia e-mail:

rokhana@akprind.ac.id URL: http://www.akprind .ac.id/ P. Budiman Department of

Statistics, Bina Nusantara University, Jl. KH. Syahdan No. 9, Kemanggisan, Jakarta 11480,

Indonesia e-mail: priscilia@binus.ac.id URL: http://binus.ac.id/ © Springer Interna tional

Publish ing Switzerland 2016 R. Silhavy et al. (eds.), Automation Control Theory

Perspectives in Intelligent Systems , Advances in Intelligent Systems and Computing 466,

DOI 10.1007/978-3-319-33389-2_28 293 1 Introd uction Indones ia is a country in the

wor ld that is prone to natural disasters due to its position earth quake locat ed at the

con ?uence of three tectonic plat es, the Australian plate, the Asian plat e and the Paci ?c

plate, which is moving on the different direc tion.

In 2004, an earth quake and tsuna mi in Nang groe Aceh Dar ussalam as impac t of the

plate tectonic movement , which takes a lot of lives, loss of proper ty, and severe damag

e to the envir onmen t. In order to overcom e or reduce the risk of natur al disas ters, we

need a model that can predict how big consequ ences or risks caused by the earth

quake that occurr ed at the point of the epicen ter and surro unding areas.

The esti mate of ground motion attenu ation can be done through model ing the va lue

of pea k ground acceler ation (PGA). PGA is the acce leration that occurs on the earth

surfa ce from rest until exposed to shocks which in this case is an earth quake. PGA unli

ke the Richter scale or magni tude scale , PGA meas ure how stro ng the earth ’ s surface

moves in the earth quake whi ch occurr ed in a region [ 24 ].

Research that has been conduct ed in order to ? nd the nonlinea r model o f the PGA

has been do ne by the scientists but the results obtai ned every model is different and

no t necess arily compa tible and can ’ t be used in certa in area. The most comm only

used equati on to ?nd the value of PGA is the equati on [ 25 ] develo ped from the d ata

of earth quakes in Alaska, Chi le, Cas cadia, Japan, Mexico, Peru, and the Solom on Isla

nds with a magnitud e rangi ng from 5.0 – 8.2 on a Richte r scale .

Several other resear chers have done a case study to develo p a more general model of

PGA as [ 2 ] which uses the same area with [ 24 ] to get a PGA equati on but the streng

th of the earthqu ake was raised to 5.0 – 8.3 on the Richte r scale . Some others model

are develo ped for calculatin g the value of PGA among others such as [ 10 , 12 , 14 , 20

].

Pa per [ 14 , 25 ] using nonlinear least square (NLS) for model ing PGA. Research

published in [ 15 , 8 ], conduct ed model ing which aim s to predict e arth- quakes at

vario us location. Pa per [ 15 ] improved the speed and stabili ty of back propaga tion

neural netw ork (BPNN) using LMM and paper [ 7 ] conduct ed LMM and ANN for predi

cting seismi c-induced damag e using PGA data.

Along wi th the develo pment of Levenberg- Marqu ardt method for esti mation and

model ing in vario us ?elds, on the other hand, the compu ter as a compu ting tool is

also experiencin g rapid develo pment both hardware and soft ware. Comput er techno

logy is alrea dy widely used in vario us ?elds be cause it provi des ease of use and speed

up the time to complete a job mainly related to the modeling proces s. PGA computer-

based model ing will resul t in the model an d the predictiv e value of the PGA of an

area more easily an d quickly.

In the condition s of a shif t or change in the value of a varia ble, then the value of the

PGA of an area would be faster counte d. In this study , conducted modeling PGA v

alues using nonlinear models with Lev enberg- Marquardt opti mizatio n method that u

ses computer- based seis mic data in the regio n of Aceh and surrounding areas, Aceh

Provin ce, Indones ia. 294 E. Irwansyah et al.

2 Ground Motion Attenuation Model Attenua tion relations hip is one of the key c

omponents of earth quake or seis mic hazard asses sment an area [ 17 ]. Currently vario

us attenuati on function has no good funct ion for shallo w seis mic source s, seismi c

source with deep backgro und and attenu ation funct ion to the source of the

earthquake due to the earthquake in sub-du ction zone as publi shed by [ 2 , 5 , 10 , 14 ,

20 , 25 ].

Paper [ 25 ] has propose d an attenuati on funct ion regres sion using the data c atalog

inter-pl ate earth quakes with magni tude variation s of 5 – 8.2 recorded in the area of

sub-du ction in the area of Alaska, Chile, Cascadi an, Japan, Mexi co, Peru a nd the

Solom on Islands. Thi s attenuatio n function modi ?ed by [ 20 ] by comparing the

results of observ atio ns and predi cted using data catalog of earthquakes in 1991 and

2001 in the wider a rea include New Irel and, New Brita in, Kamchat ka, Sa nta Cruz Is, Pe

ru, Is Kurile, Japan and Su matra.

Modi ? cations attenuation funct ion per- formed mainly in cases with earth quake

source within more than 200 km with earth quake magni tude 6.8 – 8.3 Mw . Paper [ 10 ]

to develop an atte nuation funct ion to the locat ion in the Cascadi an sub-du ction of

the same funct ions as propos ed by [ 25 ] used a stochastic model of ?nite -fault

ground mot ion from [ 3 ] wi th a varia tion on the high magnitud e of 8.0 – 9.0 Mw.

The adv antage of using this model is that unli ke the empi rical attenu ation relat

ionships, which require ?eld samp les and geome try based on strong -motion seri es of

data avail able, the effect of such ?nite-faul t ruptu re prop- agatio n, direc tion and

resour ces to sit e geome try, stoch astic model s ?nite - fault can be systemat ically

calcul ated. Maxi mum simila rity regression method with moment magnitud e 5.0 – 8.3

Mw in vario us sub-du ction areas in the wor ld such as Alaska, Japan, Mexico and Cen

tral Ameri ca are used by [ 2 ] to develo p the ground motion atte nuation function.

Regional varia bility analysis resul t of the ampl itude of the ground motion using a

global databa se available to suppor t the fact that there are signi ? cant difference s

between regio nal as shown by the ampl itude difference of more than tw o factors

among the Cascadi an area and the area of Japan.

This model uses only the shortest distance from the source of the ea rthquake at a

distance of 10 – 500 km as used by [ 10 , 25 ]. In the same year, [ 5 ] using a combi

nation of empi rical model s that use the estimat ed value of both the stochastic ground

mot ion and theor etical, to develo p typica l regression model to be implem ented in

the zone of Eas tern North Ameri ca (ENA) using a relationa l model that has previously

be en develo ped using the data seismi c data for the West ern part of Nor th Ameri ca

(WNA) .

Campbel l atten uation funct ion 2003, was develo ped especi ally for earthqu akes with

a magni tude of varia tion 5.0 – 7.5 with the stra ight line dist ance neares t to the locat

ion of the source of the earthqua ke was at 1 – 1000 km. Paper [14 ] with the refere nce

of the previ ous atte nuation function developed by [ 6 , 25 ], de velop model s of other

regres sion a ttenuation funct ion using the record ing seismi c movement s in the

bedrock in the area between the plates at sub-du ction zone in the Nor theast Taiw an

and other regio ns of the value of the low magnitud e Computer-Based Ground Motion

Attenuation Modeling … 295 of about 4.1 – 8.7 on the Richte r Scale. The use of a low

magni tude value

Newto n, Brute- Force and Lev enberg-Marq uardt Meth od. Levenberg- Marqu ardt met

hod is a method of combi nation between Gauss-N ewton method and gradient decreas

e met hod (gradie nt descent ).

Levenberg- Marqu ardt method [ 13 , 16 ] c ommonly known as dampe d least square s

met hod (DLS) whi ch produce s a numer ical solution to min imize a non- linear

function of the pa rameters in the funct ion. Levenberg- Marqu ardt method is the resul

t of interpola tion between the Gauss-New ton met hod and met hod Gradient -Descent

.

The mai n appli cation of the Lev enberg-Marq uardt met hod is the least square s

problem that aims to optimiz e the param eter ß from the model f( x i ; b), so that the

RSS in ( 1 ) be min imal value. S ð bÞ¼ X n i ¼ 1 _ 2 i ¼ X n i¼ 1 ð y i _ f ð x i ; bÞÞ 2 ð 1

Þ where _ is resi duals, y is depend ent varia ble, x i is indepe ndent varia ble, ß is param

eter model, i is observ atio n ð i ¼ 1 ; 2 ; 3 ; ... n Þ . Levenberg- Marqu ardt method is

using the iterative procedu re.

To begin the proces s o f min imization with the ?rst step is to estimat e the value of the

parameter vector , ß . At each stage of iterati on, the vector param eter, ß , wi ll be repla

ced with a new estimat ed value, i.e. b þ d to ?nd the value of d , the funct ion f ð x i ; b

þ d Þ is approac hed in the way making it linear as in ( 2 ).

f ð x i ; b þ d Þ_ f ð x i ; bÞþ J i d ð 2 Þ where d is incre ment on ß . J i is the gradi ent

(row vector ) of f to the parameter ß . It calcula te by ( 3 ), 296 E. Irwansyah et al. J i ¼ d f

ð x i ; b Þ db ð 3 Þ Approxim ation of f ð x i ; b þ d Þ wi ll produce new funct ion ( 4 ), S ð

b þ d Þ_ X n i¼ 1 ½ y i _ f ð x i ; bÞ_ J i d ? 2 ð 4 Þ or in vector notation becom es ( 5 ), S

ð b þ d Þÿ_ÿjj y _ f ð bÞ_ J d jj 2 ð 5 Þ Levenberg- Marqu ardt met hod is modi fying the

Gaus s-New ton step into ( 6 ) ð J 0 J þ k I Þ d ¼ J 0 ½ y _ f ðb Þ?ÿð 6 Þ where J is the

Jacobi an mat rix that has rows J i and where f and y is a vector with compo nents f ðx i ;

bÞ and y i is dependen t variable and much as i . d values are values that give direc tion

down (descent direc tion) of the vector parameter ß .

? value is the dampi ng param eter that must not be negative and it will be ad justed in

each iter- ation. Dampin g param eter ?, will be adjus ted in each iter ation. If S is drast

ically declin ed or fast, we can use a small ? value, whi ch would make this met hod

becom es simil ar to the Gaus s-New ton method, which iterations will give small residu

al resul ts.

? value can be e nlarged that will have an imp act on the direc tion of decreas ing gradi

ent with a g radient S of the ß equal to _2 ð J T ½ y _ f ðb Þ?Þ T . Therefor e, for large ?

value, the stage s will be c arried out in the direc tion of the gradi ent approxi mation.

Iter ations stop if the nu mber of stages, d, or reduct ion Sum of Squares of the last

param eter vector , b þ d , is below a predeterm ined limit.

Based on [ 19 ], the last parame ter, ß , is a solution of the Lev enberg- Marquardt met

hod can be written in as ( 7 ) b ð j þ 1 Þ ¼ b ðj Þ þð J 0 J þ k j I Þ _ 1 J 0 ð y _ f ð bÞÞÿð 7

Þ Acco rding wi th [9 ], iteration in the Lev enberg- Marqu ardt met hod also be deter

mined by tw o limit s, namel y by: 1. First converg ence test .

Iteratio n will be stop if it meet s ( 8 ), j fvec j\ ð 1 þ ftol Þjfvec 0 jð 8 Þ wher e fvec are

residuals and ftol is a non-neg ative numbers. Iteratio n will stop if both the relative

reduct ion (actu al and forecas t) the sum of square s over ftol value. Computer-Based

Ground Motion Attenuation Modeling … 297 2. Second converg ence test .

Iter ation will be stop if it meets ( 9 ), jD ð par _ par 0 Þj \ ptol j Dpar 0 jð 9 Þ wher e par

is the best param eter obtained and ptol is a non-neg ative numbe rs. Iteratio n will stop

if the value of the relative error between two c onsecutive iterati ons over ptol value. 4

Me thodology 4.1 Ground Motion Attenuation Model Optimization Research conducte d

using seconda ry earth quake data c onsist of dist ance of the location of the epicen ters

(Km), the depth of the earthquake (Km), magnitud e (Mw), and the value of the PGA

(gals ) from year 2005 throu gh 2007 are deriv ed from the met eorology, climato logy,

and geophys ical (BMK G) centr al government of?ce.

Po pulation and samp le used was the West Coast of Sumat ra, Indones ia in the area in

a radiu s of 500 km from the center of Banda Aceh municipal ity, Aceh provi nce,

Indones ia. The study consi sted of two mai n stages, opti mizatio n modeling and

computer applic ation develo pment stage . Optim ization modeling consi sts of four

stages (Fig.

1 ), namely: (1) To test the linearity of the data wi th the test Ramsey ’ s RESE T [ 22 ], (2)

determin ing the PGA nonli near model s for Youn gs et a l. [ 25 ] and the Lin and Lee [

14 ] model : 1. Youn gs et al. [ 25 ] model in ( 10 ): In ðPGA Þ¼ C 1 þ C 2 M þ C 3 In½ R

þ e c 4 _ð c 2 c 3 Þ M ?þ C 9 H ð 10 Þ 2. Lin dan Lee [ 14 ] model in ( 11 ): In ð PGA Þ¼ C

1 þ C 2 M þ C 3 In ð R þ C 4 e C 5 M Þþ C 6 H ð 11 Þ (3) d oing param eters model

estimat ion for Young s et al.

[ 25 ] and Lin and Lee [ 14 ] using the Leve nberg-Marqu ardt method. Model ing

conduct ed with the stage (a) deter mining the starting value for e ach param eter from

each model , (b) d eter- mine the boundar ies of iterations (c) perfor m iterations to

obtain param eters that minim ize the residual sum of square s (RSS) and (d) testing the

resi dual assum ptions (iden tical, independe nt, and norm al distrib ution) for the model

is formed. In detail, to four stage s of modeling is as show n in the pict ure (4) compa re

the modeling result. 298 E.

Irwansyah et al. 4.2 Application Development Deve lopment of computer applicati ons

aimed at assisting the model ing of ground motion atte nuation was done in two stage

s: (1) the desig n of a compu ter progra m with the waterfal l model is a serie s of stages

de ?n ing progra m requi rements, desig ning syst ems and soft ware, imp lementat ion

an d unit testing, integrati on and test of the system operational and (2) the desig n of

the inte rface (gradient descent) [ 21 ]. 5 Result and Discussion 5.1

Nonlinearity Test Nonline arity test in this study was condu cted to determin e whet her

the data used to follow the pattern of non linear model s o r not. Nonl inearity test

conduc ted by plottin g the data between seis micity variable such as distance from

earth quake center , de pth of the earth quake and earthquake magni tude with the PGA

and the nonlin earity pattern. Nonline arity test can also be done by the met hod of

Ramsey ’ s RESET .

The nonli nearity test generate resul t of F value ¼ 10 : 5256 wi th df 1 ¼ 3 and I n i t i a

l i z a t i o n s t e p s : 1 . D e t e r m i n e th e s t a r t i n g va l u e f o r e a ch p a r a m e

t e r 2 . C h o os e a s t a r t i n g p o i nt , x 1 3 . D e t e r m i n e _< 0 4 . D e t e r m i n e

k =1 5 . D e t e r m i n e th e va l u e f o r t h e b o u n d a r y i t e r a t i o n fÿtol 6 .

M a ke a mo d e l o r f o r mu l a ||? f (x k )||

[ 4 ]. In Table 1 we can see the details of each param eter estimat ion in the second

model in the opti mizatio n. It use Eq. ( 7 ) to get the parame ter equation based on

Youn g model ( 10 ) and Lin and Lee model (11 ). After e ntering the data throu gh the

funct ion Impo rt Data, the user can star t the proces s of model ing in compu ter

application s throu gh Analyze function, both to test Ramsey ’ s RESE T, Lev enberg-

Marqu ardt Regress ion (Fig. 2 a, b).

After model ing the nonline ar regression hav e conduct ed, then perfor med the assum

ption that residuals must meet with _ _ IIND , namely resi dual must meet identical test

using Glejse r test, indepe ndent test with Durbin- Wats on and lag1- plot, and the

normal distrib ution test using Kolmog orov-Sm irnov. Res idual assum ption test

conduct ed show ed that both model s tested had ful ? lled all residual assump- tions, as

can be seen in the test results using compu ter appli cations summari zed in Table 2 .

Knowi ng the best model is generated by compa ring the predi cted results o f the two

model s was o ptimized.

Compar isons wer e perfor med in this research is to compa re the de scriptive statistics

value of data such as average, variance, and residu al standard error value. In Table 3 , it

can be seen that the two models a re built have bee n able to predi ct the PGA with bett

er value, especi ally for the ne arest average value and residual standard error . The

resultin g average value of the actual PGA compa res Youn gs et al. [ 25 ] model . Table 1

Parameter estimation of Youngs et al.

[25 ] and Lin and Lee [14] model with Levenberg-Marquardt method Youngs et al. [25 ]

model Lin and Lee [14 ] model Parameter Estimation Parameter Estimation C1 - 1.101 C1

-0.7644 C2 - 0.0008244 C2 -0.4649 C3 0.005139 C3 0.177 C4 11.83 C4 0.1986 C5

0.0000283 C5 2.638 C6 0.00001767 300 E. Irwansyah et al. Fig. 2 Window display of

Levenberg-Marquardt regression for Youngs et al.

[25]( a) and Lin and Lee [14] model (b) Table 2 Residual assumption of Young et al. [25]

and Lin and Lee [14 ] model Residual assumption Levenberg-Marquardt model Young et

al. [25] model Lin and Lee [14 ] model ($) Identical Yes Yes Independent Yes Yes Normal

distribution Yes Yes Computer-Based Ground Motion Attenuation Modeling … 301

Plotting PGA predi ctions agains t the PGA actual value for each of the model as in Fig. 3

, shows that the Youngs et al.

[ 25 ] model whi ch in the esti mation of the Leven berg-Marqua rdt met hod produce s

PGA predi ctive value which is closer to the actual PGA value as average va lues in Table

3 are show simil arities and difference s Table 3 Descriptive statistics comparison of

Young et al. [ 25] and Lin and Lee [14 ] Descriptive statistics Actual PGA

Levenberg-Marquardt model Young et al. [25 ] model Lin and Lee [14] model Averages

0.35383 0.35379 0.35473 Variance 0.0000002 0.00000001 0.00000016 Residual standard

error 0.001251 0.001102 Fig. 3 Comparison between PGA actual and PGA prediction 302

E.

Irwansyah et al. were very small wi th PGA predic tions va lue rangi ng around _ 0 :0001.

Lin and Lee [ 14 ] model in whi ch esti mation by the same met hod, showed a signi

?cant differ- ence that is suppor ted by the fact that the PGA predi ctive values that

range _ 0 :00 09. Both of model s were esti mated by Lev enberg- Marqu ardt met hod,

Youn gs et al.

[ 25 ] model, is a model that can predict the PGA value more bett er and this is show n

by the close ness point in scatterpl ot between the PGA predictio ns and PGA actual .

The se ?ndings woul d sugges t that the model develo ped by Youn gs et al. [ 25 ] is

more suita ble for use as a model for the predictio n of ground motion attenu ation a s

in infer red by [ 11 ] especi ally for data which is based on the e arth- quake data in

subduct ion zon es on the West Coa st of Sumat ra, Indones ia.

Deve lopment of local atte nuation funct ion that uses the da ta in the same loca- tion,

has b een carried out by [ 17 ] with the attenuati on funct ion develo ped by [ 18 ] in the

form of syntheti c regres sion movem ent of bedrock in an earthquake zone as a result

of subduct ion of the plates using a kinem atic model of ?nite fault as adopte d from [

10 ]. The size of the d ata from the centr al radiu s distance varie s between 200 and

1500 km.

Model validati on is carried out using the data Sumatra, Indones ia megat hrust that

includes a very large earthq uake with strength of up to 9.0 Mw. 6 Concl usion Model

ing of ground motion atte nuation generat ed for Youn gs et al. [ 25 ] model and the Lin

and Lee [ 14 ] model which are estimat ed by Lev enberg- Marqu ardt method is as foll

ows 1. Youn gs et al.

[ 25 ] model : In ð PGA Þ¼_ 1 : 101 _ 0 :000 82 M þ 0 :00514 ln ½ R þ _ 11: 83_ð _0 :

00082 0 :00514 Þ M ? 2. Lin and Lee [ 14 ] model: ln ð PGA Þ¼_ 0 : 7644 _ 0 :4649 M þ 0

:177 ln ð R _ 0 : 1986 _ 2: 638 M Þ_ 0 :00001 H Levenberg- Marqu ardt meth od proved

to give a good contribut ion to the mod- eling of ground mot ion atte nuation as

indicated by the very smal l deviat ions both in average value, varia nce and resi dual

standard error betw een the predicted values with the actual value.

In addit ion, all assumpti ons are met and the results o f residual plots showed simil

arities predi cted results with actual value. Deve lopment of compu ter appli cations c an

be acc elerate the proces s o f modeling the ground motion atte nuation by the Lev

enberg- Marqu ardt met hod, especi ally for the amoun t of data and a lot more

extensive research sites. Computer-Based Ground Motion Attenuation Modeling … 303

References 1. Ahumada, A.,

Altunkaynak, A., Ayoub, A.: Fuzzy logic-based attenuation relationships of strong motion

earthquake records. Expert Syst. Appl. 42 (3), 1287 – 1297 (2015) 2. Atkinson, G.M.,

Boore, D.M.: Empirical ground-motion relations for subduction-zone earthquakes and

their application to cascadia and other regions. Bull. Seismol. Soc. Am. 93 (4), 1703 –

1729 (2003) 3. Atkinson, G.M., Silva, W.: Stochastic modeling of california ground

motions. Bull. Seismol. Soc. Am.

90 (2), 255– 274 (2000) 4. Boore, D.M., Joyne r, W.B., Fumal, T.E.: Equations for estimating

horizontal response spectra and peak acceleration from western north american

earthquakes: a summary of recent work. Seismol. Res. Lett. 68(1), 128– 153 (1997) 5.

Campbell, K.W.:

Prediction of strong ground motion using the hybrid empirical method and its use in the

development of ground-motion (attenuation) relations in eastern north america. Bull.

Seismol. Soc. Am. 93(3), 1012 –1033 (2003) 6. Crouse, C.: Ground-motion attenuation

equations for earthquakes on the cascadia subduction zone. Earthq. Spectra 7 (2), 201–

236 (1991) 11. de Lautour, O.R., Omenzetter, P.: Predict ion of seismic-induced structural

damage using arti ?cial neural networks. Eng. Struct.

31(2), 600–606 (2009) 19. do Nascimento, P.F., Fran ça, G.S., Moreira, L.P., Von Huelsen,

M.G.: Application of gauss-marquardt-levenberg method in the inversion of receiver

function in central brazil. Revista Brasileira de Geofsica 30 (3) (2012) 7. Elzhov, T., Mullen,

K., Spiess, A., Bolker , B.: minpack.

lm: R interface to the Levenberg-Marquardt nonlinear least-s quares algorithm found in

minpack, plus support for bounds. R Packag version 1, 1– 8 (2013) 8. Gregor, N.J., Silva,

W.J., Wong, I.G., Youngs, R.R.: Ground-motion attenuation relationships for cascadia

subduction zone megathrust earthquakes based on a stochastic ? nite-fault model. Bull.

Seismol. Soc. Am.

92(5), 1923 –1932 (2002) 9. Irwansyah, E., Winarko, E., Rasjid, Z., Bekti, R.: Earthquake

hazard zonation using peak ground acceleration (pga) approach. J. Phys: Conf. Ser.

423(1), 012067 (2013) 10. Kanno, T., Narita, A., Morikawa, N., Fujiwara, H., Fukushima, Y.:

A new attenuation relation for strong ground motion in japan based on recorded data.

Bull. Seismol. Soc. Am. 96(3), 879– 897 (2006) 12. Levenberg, K.:

A method for the solution of certain non– linear problems in least squares. Q. Appl.

Math. (1944) 13. Lin, P.S., Lee, C.T.: Ground-motion attenuation relationships for

subduction -zone earthquakes in northeastern taiwan. Bull. Seismol. Soc. Am. 98 (1),

220– 240 (2008) 14. Ma, L., Xu, F., Wang, X., Tang, L.: Earthquake prediction based on

Levenberg-Marquard t algorithm constrained back-propagation neural network using

demeter data.

In: Knowledge Science, Engineering and Management, pp. 591– 596. Springer (2010) 15.

Marquardt, D.W.: An algorithm for least-squares estimation of nonlinear parameters. J.

Soc. Ind. Appl. Math. 11(2), 431– 441 (1963 ) 16. Megawati, K., Pan, T.C.: Ground-motion

attenuation relationship for the sumatran megathrust earthquakes. Earthq. Eng. Struct.

Dyn. 39(8), 827– 845 (2010) 17. Megawati, K., Pan, T.C., Koketsu, K.:

Response spectral attenuation relatio nships for sumatran-subduction earthquakes and

the seismic hazard implications to singapore and kuala lumpur. Soil Dyn. Earthq. Eng.

25(1), 11 –25 (2005) 18. Monahan, J.F.: Numerical Methods of Statistics. Cambridge

University Press (2011) 20. Petersen, M.D., Dewey, J., Hartzell, S., Mueller, C., Harmsen, S.,

Frankel, A., Rukstales, K.: Probabilistic seismic hazard analysis for sumatra, indonesia and

across the southern malaysian peninsula.

Tectonophysics 390(1), 141– 158 (2004) 21. Pressman, R.S.: Software Engineering: A

Practitioner ’s Approach. McGraw-Hill, NY (2010) 304 E. Irwansyah et al. 22. Ramsey, J.B.:

Tests for speci ?cation errors in classical linear least-squares regression analysis. Journal

of the Royal Statistical Society. Series B (Methodological) pp. 350– 371 (1969) 23. Ritz, C.,

Streibig, J.C.: Nonlinear regression with R. Springer (2008) 24. Santoso, E., Widiyantoro,

S.,

Sukanta, I.N.: Studi hazard seismik dan hubungannya dengan intensitas seismik di pulau

sumatera dan sekitarnya. Jurnal Meteorologi dan Geo ?sika 12 (2) (2011) 25. Youngs, R.,

Chiou, S.J., Silva, W., Humphrey, J.: Strong ground motion attenuation relationships for

subduction zone earthquakes. Seismol. Res. Lett. 68(1), 58 –73 (1997) Computer-Based

Ground Motion Attenuation Modeling … 305

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