Prediction of Modified Mercalli Intensity From...

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J Seismol (2012) 16:489–511DOI 10.1007/s10950-012-9291-x

ORIGINAL ARTICLE

Prediction of modified Mercalli intensity from PGA, PGV,

moment magnitude, and epicentral distance using severalnonlinear statistical algorithms

Diego A. Alvarez · Jorge E. Hurtado ·Daniel Alveiro Bedoya-Ruíz

Received: 8 March 2011 / Accepted: 24 February 2012 / Published online: 21 March 2012© Springer Science+Business Media B.V. 2012

Abstract Despite technological advances in seis-mic instrumentation, the assessment of the inten-sity of an earthquake using an observational scaleas given, for example, by the modified Mercalliintensity scale is highly useful for practical pur-poses. In order to link the qualitative numbers ex-tracted from the acceleration record of an earth-quake and other instrumental data such as peakground velocity, epicentral distance, and momentmagnitude on the one hand and the modifiedMercalli intensity scale on the other, simple sta-tistical regression has been generally employed.In this paper, we will employ three methodsof nonlinear regression, namely support vectorregression, multilayer perceptrons, and geneticprogramming in order to find a functional depen-dence between the instrumental records and themodified Mercalli intensity scale. The proposedmethods predict the intensity of an earthquake

D. A. Alvarez ( B ) · J. E. Hurtado ·D. A. Bedoya-RuízUniversidad Nacional de Colombia,Apartado 127, Manizales, Colombiae-mail: [email protected]

J. E. Hurtadoe-mail: [email protected]

D. A. Bedoya-Ruíze-mail: [email protected]

while dealing with nonlinearity and the noiseinherent to the data. The nonlinear regressionswith good estimation results have been performedusing the “Did You Feel It?” database of theUS Geological Survey and the database of theCenter for Engineering Strong Motion Data forthe California region.

Keywords Modified Mercalli scale · Seismicintensity · Multilayer perceptron · Geneticprogramming · Support vector regression · Modelidentification · Ground motion · California

1 Introduction

The macroseismic intensity is an essential para-meter of earthquake ground motion that allowsa simple and understandable description of earth-quake damage on the basis of observed effects at agiven place. It is measured, for example, using theEuropean Macroseismic Scale, the China SeismicIntensity Scale, Mercalli–Cancani–Sieberg scale,the Modified Mercalli Intensity (MMI) scale, orthe Japan Meteorological Agency (JMA) seismicintensity scale (see e.g., Grüntha l 2011).

The intensity scales quantify the effects of astrong motion on the Earth’s surface, humans,objects of nature, and man-made structures at agiven location based on detailed description of indoor and outdoor effects that occur during the


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shaking. For example, the modified Mercalli in-tensity (MMI) scale (Wood and Neumann 1931;Richter 1958) measures the effects of an earth-quake using 12 degrees, with 1 denoting not feltand 12 as total destruction (take into accountthat the US Geological Survey (USGS) no longerassigns intensities higher than 10 and has not as-signed 10 in many decades; see e.g., Stover andCoffman ( 1993), Dengler and Dewey ( 1998); inaddition, the current practice of using the MMIscale does neither follow the wording of the ver-sion by Wood and Neumann ( 1931) nor by Richter(1958) (Grüntha l 2011)). The values will differbased on the distance to the earthquake, withthe highest intensities being usually around theepicentral area, and based on subjective data thatare gathered from individuals who have felt thequake at that location. Take into considerationthat all modern scales have 12 degrees with theexception of the JMA scale which was upgradedfrom 7 to 10 degrees in 1996 (see e.g., Musson et al.2010).

The assessment of the intensity of a quake ata given location used to be a slow process, as itwas usually performed by means of personalizedsurveys; however, with the advent of macroseis-mic intensity internet surveys (a list of some of these services can be found in Table 1), this isnot the fact anymore. One can see, for exam-ple, that when a strong shaking happens in the

California region, the USGS receives thousands of responses from people after the event (they can bein excess of 10,000 responses in less than half anhour). In any case, it is convenient to have an au-tomatized intensity measure based on instrumen-tal data, so that distribution intensity maps canbe drawn almost immediately after the shaking.With that information, emergency service systemscould prioritize their attention to those placeswhere the largest damage is expected. In addition,the automatic shut-off of natural gas supplies af-ter an earthquake in certain localities could beplanned.

In this paper, three methods are analyzed,namely support vector regression, multilayer per-ceptrons, and genetic programming. The accuracyof the algorithms have been tested using the “DidYou Feel It?” (DYFI) database of the USGS andthe database of the Center for Engineering StrongMotion Data (CESMD) for the California region.

The plan of the paper is as follows: first, we willbegin in Section 2 with a succinct review on previ-ous studies on the topic and then in Section 3 wewill explain the mathematical background of thenonlinear regression methods employed. With thedatabase described in Section 4, we will introduceand then perform some numerical experiments inSection 5 and thereafter, the analysis of resultswill be done in Section 6. The paper finalizes inSection 7 with the conclusions.

Table 1 Some web-based macroseismic intensity questionnaires

Institution or agency URLAmateur Seismic Centre (India) http://asc-india.org/menu/felt-india.htmBritish Geological Survey http://www.earthquakes.bgs.ac.uk/questionnaire/EqQuestIntro.htmlCentral Institute for Meteorology and http://www.zamg.ac.at/erdbeben/bebenbericht/index.php

GeodynamicsEuropean-Mediterranean Seismological http://www.emsc-csem.org/Earthquake/Contribute/testimonies.php

CenterGeoNet (New Zealand) http://www.geonet.org.nz/earthquake/Istituto Nazionale di Geofisica e Vulcanologia http://www.haisentitoilterremoto.it/Le Bureau Central Sismologique Français http://www.seisme.prd.fr/english.phpNatural Resources Canada http://earthquakescanada.nrcan.gc.ca/dyfi/Royal Observatory of Belgium http://seismologie.oma.be/index.php?LANG=EN&CNT=BE&LEVEL=0Servicio Geológico Colombiano http://seisan.ingeominas.gov.co/RSNC/index.phpSwiss Seismological Service http://www.seismo.ethz.ch/eq/detected/eq_form/index_ENU.S. Geological Survey (USGS) http://earthquake.usgs.gov/earthquakes/dyfzi/

http://asc-india.org/menu/felt-india.htmhttp://www.earthquakes.bgs.ac.uk/questionnaire/EqQuestIntro.htmlhttp://www.zamg.ac.at/erdbeben/bebenbericht/index.phphttp://www.emsc-csem.org/Earthquake/Contribute/testimonies.phphttp://www.geonet.org.nz/earthquake/http://www.haisentitoilterremoto.it/http://www.seisme.prd.fr/english.phphttp://earthquakescanada.nrcan.gc.ca/dyf/i/http://seismologie.oma.be/index.php?LANG=EN&CNT=BE&LEVEL=0http://seisan.ingeominas.gov.co/RSNC/index.phphttp://www.seismo.ethz.ch/eq/detected/eq_form/index_ENhttp://earthquake.usgs.gov/earthquakes/dyfzi/http://earthquake.usgs.gov/earthquakes/dyfzi/http://www.seismo.ethz.ch/eq/detected/eq_form/index_ENhttp://seisan.ingeominas.gov.co/RSNC/index.phphttp://seismologie.oma.be/index.php?LANG=EN&CNT=BE&LEVEL=0http://earthquakescanada.nrcan.gc.ca/dyf/i/http://www.seisme.prd.fr/english.phphttp://www.haisentitoilterremoto.it/http://www.geonet.org.nz/earthquake/http://www.emsc-csem.org/Earthquake/Contribute/testimonies.phphttp://www.zamg.ac.at/erdbeben/bebenbericht/index.phphttp://www.earthquakes.bgs.ac.uk/questionnaire/EqQuestIntro.htmlhttp://asc-india.org/menu/felt-india.htm


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2 Previous studies

Several attempts have been made in order torelate the intensity scale with some earthquakeparameters such as peak ground acceleration(PGA), peak ground velocity (PGV), peak grounddisplacement (PGD), the magnitude scale, andthe epicentral distance among others. For in-stance, Trifunac and Brady ( 1975), Murphy andO’Brien ( 1977) have tried to correlate intensitywith PGA, PGV, and PGD. In particular, for theCalifornia region, we can list the works of Wald et al . (1999b), Atkinson and Sonley ( 2000),Atkinson and Kaka ( 2007). For example, Waldet al . (1999b) suggested the following piecewiselinear relationships between PGA, PGV, andMMI:

MMI =1.00+2.20 log 10 (PGA ) for log10 (PGA ) ≤1.82−1.66+3.66 log 10 (PGA ) for log10 (PGA )> 1.82

(1)

and

MMI =3.40+2.10log 10 (PGV ) for log10 (PGV ) ≤ 0.762.35+3.47log 10 (PGV ) for log10 (PGV ) > 0.76

(2)

while Atkinson and Kaka ( 2007) proposed:

MMI =2.65+1.39 log 10 (PGA ) for log10 (PGA ) ≤1.69

−1.91+4.09 log 10 (PGA ) for log10 (PGA ) > 1.69(3)

and

MMI =4.37

+1.32log

10(PGV ) for log

10(PGV )

≤0.48

3.54+3.03log 10 (PGV ) for log10 (PGV )> 0.48(4)

Recently, Worden et al. ( 2012) suggested:

MMI =1.78+1.55 log 10 (PGA ) for log10 (PGA ) ≤1.57

−1.60+3.70 log 10 (PGA ) for log10 (PGA ) > 1.57(5)

and

MMI =3.78+1.47 log 10 (PGA ) for log10 (PGA ) ≤0.532.89+3.16 log 10 (PGA ) for log10 (PGA )> 0.53

(6)

In all the above equations, the PGA is ex-pressed in centimeter per square second, while thePGV is given in centimeter per second.

Karim and Yamazaki ( 2002) performed a sim-ilar research relating PGA, PGV, PGD, and theso-called spectrum intensity to the JMA seis-mic intensity scale. Other authors like Atkinsonand Sonley ( 2000), Sokolov ( 2002), Kaka andAtkinson ( 2004) have developed empirical rela-tionships between response spectra or Fourier ac-celeration spectra and modified Mercalli intensity.Shabestari and Yamazaki ( 2001) developed someexpressions that related the JMA intensity scaleand the MMI scale. Tselentis and Danciu ( 2008)derived empirical regression equations for MMIand for various ground motion parameters such asduration, cumulative absolute velocity, Housner’sspectrum intensity, and total elastic input energyindex.

Note that most of the aforementioned relation-ships were performed using univariate and mul-tivariate linear regression analysis showing largescatter. It is the authors’ belief that those studieswould have been better using robust regressionmethods instead of standard linear regression inorder to account for possible outliers, that is, someobservations that do not follow the pattern thatthe other observations have and that usually oc-cur when large measurement errors are present.Our preference for robust regression methods isbased on the fact that, in particular, least squaresestimates for regression models are highly biasedby the outliers that dominate the recordings fromfew earthquakes, since the regression might bydragged towards them.

It seems, however, that the linear model chosenin the previously mentioned papers comes justfrom mathematical convenience. The relationshipthat links the MMI scale and the instrumental datais complicated enough that nonlinear methodsseem to be the best choice for a model. This isthe reason that motivated us to propose a newapproach in Section 5.


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Nonlinear regression methods have alreadybeen used. For example artificial neural net-works have been employed by Tung et al. ( 1993),Davenport ( 2004), Tselentis and Vladutu ( 2010)in order to relate MMI and PGA, PGV, PGD,response spectral acceleration, and response spec-tral velocity for selected frequencies, Arias inten-sity, and JMA intensity. All of them have reportedsatisfactory results. In the present paper, we willcompare this approach to other nonlinear regres-sion methods.

Finally, it is convenient to mention that re-cently, Faenza and Michelini ( 2010); Kuehn andScherbaum ( 2010); Worden et al . (2012) proposedmethods that treat the relationship of intensityand instrumental information not using an ordi-nary least squares regression methodology.

Commonly employed strong motion features

There are several numbers that are either instru-mental or are obtained after the processing of the acceleration records of the shaking and whichhave a preponderant role in the estimation of theseismic intensity. Some of the most popular andthat were used in the investigations listed aboveare mentioned in the following:

– PGA, PGV, and PGD defined as the geomet-ric mean of the two horizontal componentsin the directions originally recorded. The firstparameter is specially important because ac-cording to Newton’s second law, the PGAmultiplied by the mass provides the maximuminertial force that affects the structure duringthe shaking.

– duration of the shaking;– spectral content (characteristic periods)

at some specific frequencies; according toSokolov ( 2002), the frequency range of theamplitude spectrum | X ( f )|of the accelerationrecord that is of interest from an engineeringpoint of view lies between 0.3 and 14 Hz.In fact, the frequency band 0.78–2.0 Hz isrepresentative for MMI greater than 8, whilethe 3.0–6.0 Hz range represents best MMIfrom 5 to 7 and the 7.0–8.0 Hz correlates bestwith the lowest MMI.

– magnitude (moment magnitude, local magni-tude, surface wave magnitude, or body wavemagnitude);

– epicentral distance;– amplitudes of acceleration, velocity, or dis-

placement response spectra;– regional propagation path (geological condi-

tions) and local soil conditions among others.

Some other features that can be extracted fromnumerical processing of the acceleration recordare as follows: the spectrum intensity, the Ariasintensity, the cumulative absolute velocity, andthe Japan Meteorological Agency seismic inten-sity reference acceleration value among others.It is the authors’ opinion that more informativefeatures can be extracted from the accelerationrecords by using the different techniques that thefield of digital signal processing presents us, eventhough there is not a clear physical interpretationof those numbers. Some of these features mightbe extracted by means of the wavelet and shorttime Fourier transforms (see e.g., Mallat 2008),and the time-varying ARMA process (see e.g.,Poulimenos and Fassois 2006) among others. Thiswill be the topic of a future article.

3 Some mathematical background on nonlinearregression

It was said in Section 2 that the relationship thatlinks the MMI scale and the instrumental data iscomplicated enough that nonlinear methods seemto be the best choice for a model. In the following,we will introduce briefly three methods of nonlin-ear regression, namely support vector regression,multilayer perceptrons, and genetic programmingin Section 5, that will help us find a functional de-pendence between the instrumental records andthe modified Mercalli intensity scale.

3.1 Support vector regression

In the following, we will make a succint overviewof the theory behind the support vector regres-sion (SVR); for a more thorough coverage of the


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algorithm, the reader is referred to the excellenttutorial by Smola and Schölkop f (2004).

Suppose we are given a training dataset D ={( x1 , y1) , ( x2 , y2) , . . . , ( xn , yn )}, where xi ∈R d isthe i-th input vector and yi ∈R is its correspond-ing output vector; the idea of SVR is to fit the datato a function of the form as follows:

f ( x;β, b ) =n

i=1β i K ( x, xi) +b , (7)

where K is a so-called kernel function . Thereare several options for a kernel function, andhere, we considered the popular radial basiskernel function , which is given by K ( x, y) =exp −γ x − y 2 , where γ > 0 is a scalar parame-ter that defines the kernel function. The weightsβ i, the bias b , and the parameter γ are chosen, sothat the deviation from each target yi is at most εfor all training data, and at the same time, is as flatas possible. In this way, the data will lie aroundan ε-sensitive tube as seen in Fig. 1. Here, ε > 0is the width of an insensitive zone that controlsthe amount of noise that the algorithm can handle.For instance, when a training sample has no noise,one may set ε = 0.

The flatness of the function f can be ensured by

minimizing the fitting error using a ε-insensitiveloss function which gives zero error if the absolute

f x

x

f x ε

f x

f x ε ξ 0, ξ̂ 0

ξ 0, ξ̂ 0

Fig. 1 An SVR, showing the regression curve togetherwith the ε -insensitive “tube”. Examples of slack variablesξ and ξ̂ are also shown

difference between the prediction f ( x) and thetarget y is less than the constant ε . This loss func-tion is given by the following:

E ( x;β, b ) = 1n

n

i=1E ε ( f ( xi;β, b ) − yi) +λ(β)

(8)

where λ is a regularization term and,

E ε ( f ( x) − y) =0 if | f ( x) − y| < ε| f ( x) − y| −ε otherwise

(9)

as illustrated in Fig. 2. The choice of this errorfunction makes the SVR rather different fromtraditional error minimization problems, and veryrobust to outliers.

As stated before, the SVR regression algorithmtries to position the ε-insensitive tube aroundthe data as shown in Fig. 1. This is achievedby minimizing the error (Eq. 8). In order to al-low points to lie outside the ε-insensitive tube,

let us define the so-called slack variables ξ i, ˆξ ithat represent the distance from actual values yi to the corresponding boundary values of theε -insensitive tube. For each point xi, we need twoslack variables ξ i ≥ 0 and ξ̂ i ≥ 0. If xi lies abovethe ε -tube, then ξ i ≥ 0 and ξ̂ i = 0; if xi lies below

E ε z

zε ε 0

Fig. 2 Plot of an ε -insensitive error function ( continuousline ), in which the error increases linearly with distancebeyond the insensitive region. Also shown for comparisonis the quadratic error function ( dashed line )


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the ε-tube, then ξ i = 0 and ξ̂ i ≥ 0. Finally, if xilies inside the ε-tube, then ξ i = ξ̂ i = 0. Introducingthe slack variables allows points to lie outside thetube, provided the slack variables are nonzero,and in this way:

yi ≤ f ( xi;β, b ) +ε +ξ i (10) yi ≥ f ( xi;β, b ) −ε − ξ̂ i (11)

By substituting Eq. 9 into Eq. 8, the minimizationof the error (Eq. 8) becomes (see e.g., Smola andSchölkop f 2004):

minβ, b

C n

i

=1

ξ i + ξ̂ i +λ(β) (12)

subject to C > 0, ξ i ≥ 0, ξ̂ i ≥ 0 and the constraints(10) and (11). Here, C is another parameterthat is used to control noise and that determinesthe trade-off between the flatness of f and theamount up to which deviations larger than ε aretolerated. The optimization problem ( 12) can beexpressed using its dual Lagrangian formulationas the quadratic optimization problem (see e.g.,Smola and Schölkopf 2004):

maxα,α ∗

12

n

i=1

n

j =1β iβ j K xi, x j −

n

i=1β i yi

+n

i=1(α i +α∗i )ε (13)

with β i = α i −α∗i and subject to the restrictions0 ≤ α i ≤ C , 0 ≤ α∗i ≤ C for i = 1, 2, . . . , n , andn

i

=1 β i

= 0. Here, the α i-s and α∗

i are Lagrange

multipliers that appeared when including the con-straints mentioned above into the Lagrangianformulation.

Usually, ε is set a-priori, and the parametersC and γ are found by means of an optimizationthat produces the minimum error in Eq. 8 af-ter performing statistical cross-validation. Cross-validation is a technique for assessing how theresults of a statistical analysis will generalize to

an independent dataset. One round of cross-validation involves partitioning the dataset intocomplementary subsets, performing the analysison one subset and validating the analysis onthe other subset. To reduce variability, multiplerounds of cross-validation are performed usingdifferent partitions, and the validation results areaveraged over the rounds.

It can be shown that Eq. 13 is a convexquadratic optimization problem, and in conse-quence, it has a unique solution. Those Lagrangemultipliers β i which are different from zero arethe so-called support vectors . The support vectorregression has the property that the solution issparse, that is, most of the β i are zero.

The parameter b is found for a data point forwhich 0 < α i < C or 0 < α ∗i < C by solving theequation:

b = yn −ε −n

i=1β i K ( xn , xi) (14)

In practice, it is better to average over all suchestimates of b . In short, the steps to train the SVRare as follows:

1. Choose ε > 0, γ > 0, and C > 02. Estimate the generalization error for ε , γ , and

C using only the training set and the leave-one-out methodology. On each step of theleave-one-out, one has to solve the optimiza-tion problem ( 13) in order to find the β i-s andthen find b from Eq. 14. The generalizationerror is estimated as the average error (Eq. 8)obtained with the training set during all stagesof the leave-one-out.

3. Choose new values of ε > 0, γ > 0, and C > 0and repeat step 2, until a low generalizationerror is found.

Once appropriate values of ε, γ , and C arechosen, the predicted value of the regression isfound when x is used as a input to the Eq. 7, thatuses the weights β and the bias b obtained afterperforming the previous optimizations. All thesesteps can be easily performed using the LIBSVMsoftware (Chang and Lin 2011).


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3.2 Multilayer perceptrons

An artificial neural network is a mathematicalmodel that is inspired by the structure and/orfunctional aspects of biological neural networks.Modern neural networks are nonlinear statisticaldata modeling tools. They are usually used tomodel complex relationships between inputs andoutputs or to find patterns in the data.

The most popular neural network is the so-called multilayer perceptron (MLP). A MLP (seeFig. 3) consists of interconnected layers (an input,hidden, and output layer) of processing units orneurons. For example, the network shown in Fig. 3has d units in the input layer, p neurons in thehidden layer, and a single neuron in the outputlayer. As seen in Fig. 3, the flow of informa-tion comes from left to right and is altered bymeans of some parameters (the so-called weightsw 0 ji, w

1kj , and biases b

0 j and b

1k ) and the func-

tions g and h. These functions (called activation functions in neural network terminology) usuallyare set to g(a) = tanh (a) and h(a) = a . A MLPis a parameterized, adaptable vector functionwhich may be trained to perform classificationor regression tasks. Given a training dataset D =

{( x1 , y1) , ( x2 , y2) , . . . , ( xn , yn )}, where xr ∈R d isthe r -th input vector and yr ∈R is its correspond-ing output vector, the idea of a MLP with a singleoutput unit is to estimate a function f :R d →Rof the form:

f x;w 0 , w 1 , b 0 , b 1

= h p

j =1w 11 j g

d

i=1w 0 ji xi +b 0 j +b 11 (15)

The number of neurons in the hidden layer mustbe chosen so that the fitting of the network tothe data is adequate: if too few neurons are used,the model will be unable to represent complexdata, and the resulting fit will be poor; if too manyneurons are used, the network may overfit thedata; when overfitting occurs, that is, the networkfits the training data extremely well but it gen-eralizes poorly to new, unseen data. Therefore, avalidation set must be used to find the appropriatenumber of neurons in the hidden layer.

Note that the parameters w0 ji, w11 j , b

0 j , and b

11

were grouped as the matrices w0 , w1 , b 0 , and b 1

respectively. They are found by minimizing the

Fig. 3 Topology of a multilayer perceptron with a sin-gle output unit. This network has d inputs, p neurons inthe hidden layer, and a single output. w0 ji represents theweights between the j -th neuron of the hidden layer and

the i-th input, while w11 j stands for the weights between theoutput neuron and the j -th neuron of the hidden layer. b 0 j represents the bias weight of the j -th neuron of the hiddenlayer and b 11 symbolizes the bias weight of the output layer


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mean squared error function (see dashed line of Fig. 2):

E x;w 0 , w 1 , b 0 , b 1

=

1

n

n

r =1 yr

− f xr

;w 0 , w 1 , b 0 , b 1

2(16)

with respect to the weights and biases. For moredetails of neural network training, see for exampleHaykin ( 1998), Bishop ( 2007).

Several authors have shown that under someassumptions, MLPs are universal approximators ;that is, if the number of hidden nodes p is allowedto increase towards infinity, they can approximateany continuous function with arbitrary precision(see e.g., Hornik et al . 1989).

However, there are a number of problems withMLPs: (a) There is no theoretically sound way of choosing the network topology. (b) For a givenarchitecture, learning algorithms often end up in alocal minimum of E instead of a global minimum.(c) They are black box solutions to the problem.In any case, these drawbacks do not imply that thefittings performed in this paper affect the reliabil-ity of the results. They simply indicate that othernonlinear models like SVR have better theoreticalproperties for regression than MLPs.

3.3 Genetic programming

Genetic programming (GP) is a problem-solvingapproach inspired by biological evolution inwhich computer programs (mathematical formu-las, computer programs, logical expressions, etc.)are evolved in order to find solutions to problemsthat perform a user-defined task. The solutionmethod is based on the Darwinian principle of “survival of the fittest” and is closely related to thefield of genetic algorithms (GA). There are threemain differences between GA and GP: (a) Struc-ture: GP usually evolves tree structures, while GAevolvebinary or real number strings. (b) Programsvs. binary strings: GP usually evolves computerprograms while GA typically operate on codedbinary strings. (c) Variable vs. fixed length: Intraditional GAs, the length of the binary string isfixed before the solution procedure begins. How-ever, a GP tree can vary in length throughout the

execution. The theory behind genetic program-ming is large. Here, just a brief review of its mainconcepts will be given. The interested reader isreferred to Koza ( 1992) for an ample discussionon the topic.

Genetic programming uses the following stepsto solve problems:

1. Generate an initial population of computerprograms

2. Iteratively perform the following sub-steps onthe population until the termination criteria issatisfied:

a. Execute each program in the populationand assign it a fitness value according tohow well it solves the problem

b. Create a new population by executing thethe following evolutionary operators withcertain probability:– Reproduction: it selects an individual

from within the current population sothat it can have an offspring. Thereare several forms of choosing whichindividual deserves to breed includ-ing “fitness proportionate” selection,“rank” selection, and “tournament”selection.

– Crossover: mimics sexual combina-tion in nature, where two parents arechosen and parts of their trees areswapped in a form that each crossoveroperation should result in a legalstructure.

– Mutation: it causes random changesin an individual before it is intro-duced into the subsequent popula-tion. During mutation it may happenthat all functions and terminals are

removed beneath an arbitrarily de-termined node and a new branch israndomly created or a single node isswapped for another.

3. The best computer program that appears inany generation is designated as the result of genetic programming.

One of the main uses of genetic programmingis to evolve relationships between variables: this


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is called symbolic regression . Symbolic regressionvia genetic programming is a branch of empiricalmodeling that evolves summary expressions foravailable data. For a long time, symbolic regres-sion was a domain only for us, humans; how-ever, over the last few decades, it has becomethat of computers as well. Unique benefits of symbolic regression include human insight andin some cases, interpretability of model results,identification of key variables and variable com-binations, and the generation of computationallysimple models for deployment into operationalmodels.

4 Earthquake data

Two earthquake datasets have been used in thisstudy. The first one is the USGS’s Did You FeelIt? database (U.S. Geological Survey 2011) whichcollects, by means of internet surveys, informationabout how people actually experienced the earth-quakes. The form of the questionnaire employedin the DYFI database and the method for as-signment of intensities are based on an algorithmdeveloped by Dengler and Dewey (1998).

In this dataset, one can find for a given earth-quake a table of modified Mercalli intensities ag-gregated by city or postal code, number of re-sponses for that region and epicentral distance,and a representative latitude and longitude of thesurveyed region. In addition, it is possible to findin the same database the depth of the earthquakeand the latitude and longitude of the epicenter.The second employed database is the one of the Center for Engineering Strong Motion Data(CESMD - Center for Engineering Strong MotionData 2011). Here, one can find for some represen-tative earthquakes the actual accelerograms of theshakings measured at different stations. For eachstation, one can find its code and name, its latitudeand longitude, and for the given earthquake, itsepicentral distance, magnitude, PGA, PGV, PDG,and the amplitudes of acceleration response spec-tra for the 0.3, 1, and 3 s. However, one musttake into consideration that, usually, not all of theabove-mentioned data are available at the sametime in the CESMD database.

The records of the earthquakes that happenedin the California region after 2000, plus therecords of the Loma Prieta earthquake of October17, 1989, and the Petrolia earthquake of April 25,1992 were employed. All of the available recordsof the CESMD database were used, but it wasfound out that the records prior to 2000, exceptthe two formerly mentioned, gave incorrect MMIpredictions in the regressions performed (as wecould find out in our initial numerical experi-ments). This is in agreement to a comment madeby Vince Quitoriano of the USGS who told usin an email, “We [the USGS] did not start col-lecting internet responses [in the DYFI database]until mid-1998. In that time period, Hector Minewas the only large event that people respondedto immediately. Most of the pre-2000 data areactually entries from people recalling historicalevents (Northridge, Whittier Narrows, etc.) muchlater, rather than responding to a current earth-quake”. And later he said, “Note that all thedata in DYFI are from internet questionnaires; wedid not process any paper surveys. The question-naire itself and the underlying algorithm have notchanged from Wald et al . (1999a).” We allowedthe records of the Loma Prieta and the Petroliaearthquake in any case because the records withMMI intensity 7 and above were scarce, and thoseearthquakes provided us with that information.

100 200 300 400 500 600 7000

20

40

60

80

100

120

number of responses

f r e q u e n c y

Fig. 4 Histogram of the number of responses for a singleMMI reading the training dataset


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Fig. 5 Locations where the MMI readings used in thisstudy were reported. All of the 843 readings were locatedin California

A MATLAB program was written in orderto automatically download, match, and parse thedata from the databases using the Event ID of the earthquake. We choose to correlate only thoserecords where the strong motion was near a MMIobservation. For each station, the nearest obser-vation intensity with at least four reports was cho-sen, with the intention of decreasing the internalvariability of the MMI readings (remember that

the standard deviation of the mean of a normallydistributed and independent sample is given byσ/ √ n where σ 2 is the variance of the populationand n is the number of samples used to estimatethe mean). Figure 4 shows the histogram of thenumber of responses for a single MMI readingin the training dataset. It can be seen from thishistogram that most of the MMI observations arethe average of a large number of responses. Inconsequence much of the inherent variability of the MMI readings has been removed from thedataset. We used only the station readings thatwere within 1.0 km to the MMI reading whenthe modified Mercalli intensity was less than 6and within 3 km when then MMI reading wasgreater or equal to 6. For measuring that distance,we used the latitude and longitude position of the stations/MMI readings. All other data weredisregarded. Using those criteria, we came up witha database composed of 843 station record—MMIobservation pairs coming from 63 earthquakes.Figure 5 shows a map that indicates the locationswhere the MMI readings that were used in thisstudy were reported. Since most of the readingswere done in places with high concentration of population, the reported MMI data in the DYFIdatabase are the average of the reported MMI of small ZIP regions. This fact explains as well thesmall variability of the reported MMI readings inthe DYFI database.

From the above database, four representa-tive features were coupled to the MMI reading,namely moment magnitude, epicentral distance,PGA, and PGV. We intended to include depth aswell, but the high variability of this random vari-able refrained us from including it in the analysis.Sometimes, there was conflicting information, andin that case, we deferred to the DYFI database.

Table 2 Spearman correlation coefficients between the analyzed variables

MMI Epicentral PGA PGV Moment Depthdistance magnitude

MMI 1.00 −0.32 0.79 0.73 0.26 0.14Epicentral distance −0.32 1.00 −0.52 −0.01 0.66 −0.23PGA 0.79 −0.52 1.00 0.72 0.01 0.30PGV 0.73 −0.01 0.72 1.00 0.55 0.15Moment magnitude 0.26 0.66 0.01 0.55 1.00 −0.17Depth 0.14 −0.23 0.30 0.15 −0.17 1.00


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0 50 100 150 200 250 300 350 4002

3

4

5

6

7

8

epicentral distance (km)

m o

d i f i e d M e r c a

l l i i n t e n s i

t y

Fig. 6 Plot of epicentral distance vs. MMI for the trainingdataset

In order to gain insight into the employeddataset, Table 2 shows the Spearman correlationcoefficients between the analyzed variables. Thisinformation shows not only the contribution of each component in order to explain the MMIbut also shows the redundance of informationbetween variables. It can be seen from this tablethat the MMI tends to grow when the epicentraldistance decreases and the PGA, PGV, or themoment magnitude increases. Figures 6, 7, and 8

10 1 10 2

2

3

4

5

6

7

8

9

PGA (cm/s 2) [log10

scale]

m o


l l i i n t e n s

i t y

Wald et. al. (1999)Atkinson and Kaka (2007)Worden et. at. (2012)Equation (23)

Fig. 7 Plot of PGA vs. MMI for the training dataset. Thelines represents the relationship between these variablesshown, given by Eqs. 1, 3, 5, and 23

10−1

100

101

2

3

4

5

6

7

8

PGV (cm/s) [log10

scale]

m o


l l i i n t e n s

i t y

Wald et. al. (1999)Atkinson and Kaka (2007)Worden et. at. (2012)Equation (24)

Fig. 8 Plot of PGV vs. MMI for the training dataset. Thelines represents the relationship between these variables

shown, given by Eqs. 2, 4, 6, and 24

confirm this fact. On the other hand, the depthdoes not seem to have a strong relationship withMMI. We have included in Fig. 7 the linear re-gressions ( 1), ( 3), and ( 5) that associate PGA andMMI; while in Fig. 8, we have included the Eqs. 2,4, and 6. Note that some of these regressions tendto overestimate the MMI given with the actualdatabase.

Figure 9 shows a plot relating moment mag-nitude and MMI. It can be seen from this plotthat the correlation between these two variables is

4 4.5 5 5.5 6 6.5 7 7.52

3

4

5

6

7

8

magnitude

m o

d i f i e

d M e r c a

l l i i n t e n s i

t y

Fig. 9 Plot of moment magnitude vs. MMI for the trainingdataset


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1 2 3 4 5 6 7 80

10

20

30

40

50

60

70

80

modified Mercalli intensity

f r e q u e n c y

Fig. 10 Histogram of the modified Mercalli intensity forthe training dataset

weak. However, we opt to keep this information inour database because this variable, together withthe epicentral distance, provides a better impres-sion of the intensity of the quake.

It is important, as well, to know the distributionof the training data, so that we can know in whichregions there is enough information to estimatethe MMI with the relationships that will be pro-posed in Section 5. In this sense, Figs. 10, 11, 12,

13, and 14 show the histograms of the differentvariables. One can deduce that the regressionsperformed are reliable when the expected MMI

0 50 100 150 200 250 300 350 4000

20

40

60

80

100

120

epicentral distance (km)

f r e q u e n c y

Fig. 11 Histogram of epicentral distance for the trainingdataset

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

50

100

150

200

250

300

PGA (g)

f r e q u e n c y

Fig. 12 Histogram of PGA for the training dataset

lies in the range 2–6.5, the epicentral distance isless than 400 km, the PGA is less than 0.2 g,the PGV is less than 20 cm/s, and when the mo-ment magnitude is less than 7.2. In principle, onecould use a regression model to prognose beyondthe extreme values found in the dataset in orderto predict higher intensities (in this case, MMIsgreater than 6.5 in the current database) or use,for example, the model for PGAs larger than 0.2 g.Even though this is tempting, this not advisable

because extrapolation of a model outside the pa-rameter boundaries of its underlying dataset canbe dangerous (see for example Bommer et al .2007, for a discussion about the extrapolation of

0 5 10 15 20 25 30 35 400

50

100

150

200

250

300

PGV (cm/s)

f r e q u e n c y

Fig. 13 Histogram of PGV for the training dataset


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4 4.5 5 5.5 6 6.5 7 7.50

20

40

60

80

100

120

140

moment magnitude

f r e q u e n c y

Fig. 14 Histogram of moment magnitude for the trainingdataset

ground motion prediction equations). This is akind of curse for all prediction equations basedon data, and in this case, the only safe solution isto employ a regression model fitted to a databasethat covers the range of information in which theextrapolation is to be performed. In other words,regression algorithms, in general, are expected toperform well in regions of the space of variableswhere there exists a set of points modeling similar

characteristics, but are not so good at extrapo-lation, inasmuch as these results are subject togreater uncertainty.

It is necessary to note that the differences inearthquake parameters such as source mecha-nism, regional tectonics, propagation path prop-erties, and geological and geotechnical conditionsare not taken into account in the present study.Therefore, these factors are considered to be ran-dom variables affecting the ground motion para-meters for a given location and intensity level.

In the next lines, we will present the resultsof our numerical experimentation performed withthose 843 observations.

5 The proposed approach and its numericalexperimentation

Using the algorithms described in Section 3,we related the MMI to the moment magnitude,

epicentral distance, PGA, and PGV measured atthe closest station to the observation. Even thoughthe MMI is reported as a roman natural number,we will use it here as a real continuous number (inarabic notation), so that when it is rounded to theclosest integer, it coincides with its correspondingmeasurement in the modified Mercalli intensityscale. In fact, the MMI in the DYFI database isexpressed by a real number with one decimal digitof approximation (see Wald et al . 2006).

In order to validate the training, we randomlysplit the 843 observations into three sets: a train-ing, validation and testing set with 506 (60 %), 126(15 %), and 169 (25 %) elements, respectively.

Before using the algorithms that we will de-scribe below, each variable in the training set waseither normalized or standardized.

In the first case (which was applied beforetraining the MLP), each variable in the trainingset was normalized (so that it had a value in theinterval [−1, 1]) by means of the equation:

zk = 2 X k −min ( X k )

max ( X k ) −min ( X k ) −1 (17)

and employing the minimum and maximum valuesthat can be found in Table 3.

In the second case (which was applied be-fore training the SVR and GP algorithms), thestandardization was performed by subtracting themean and dividing by the standard deviation usingthe equation:

zk = X k −mean ( X k )

std ( X k )(18)

and employing the means and standard devia-tions that can be found in Table 3. Even thoughthis procedure is inspired in the normalization of Gaussian random variables, it is applicable to anykind of distribution, since the idea is to reducethe spread in the data. Both methods are popu-lar in nonlinear regression for making the inputvariables rather small in order to improve thenumerical stability of the employed algorithms,regardless of the distribution of the data. In otherwords, this process tends to make the training


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Table 3 Mean, standard deviation, minimum, and maximum of each variable of the dataset

Variable Mean Standard deviation Minimum MaximumMMI 3.6425 0.9669 2.0 7.5 x1 = epicentral distance (km) 87.4656 102.9749 1.6 393.7 x2 = PGA (g) 0.0460 0.0619 0.002 0.588 x3 = PGV (cm/s) 3.4752 5.4888 0.080 62.88 x4 = moment magnitude 5.2261 0.9574 4.0 7.2

The mean and standard deviation correspond to the training set and the minimum and maximum values correspond to thewhole database

process better behaved by improving the numeri-cal condition of the underlying optimization algo-rithms employed and ensuring that various defaultvalues involved in initialization and termination of the algorithms are appropriate (see for instance,Sarle 2002).

We performed the nonlinear regressions withthe algorithms described in Section 3 in the fol-lowing way:

Training with support vector regression

The quadratic optimization (Eq. 13) is convex,and therefore, its solution is unique. Hence, therewas no need of using a validation set, and in con-sequence, the training and the validation set were

merged. After setting a priori ε to 0.5 (we choosethis value because 0.5 is half the distance betweentwo MMI degrees), the constants C = 494 .559 andγ = 0.0625 were found by means of an optimiza-tion that selected the parameters which producedthe minimum least squares error in a leave-one-out cross-validation. The training was performedusing the LIBSVM software (Chang and Lin2011), obtaining 147 support vectors, which can befound together with their corresponding weightsin Appendix 1. Normalization by means of Eq. 17

was only performed on the inputs, since it wasfound in this case that without normalization inthe outputs, the algorithm provided slightly betterresults.

In Appendix 2.1, we have included the MAT-LAB code that calculates an estimation of the MMI from the epicentral distance, PGA,PGV, and moment magnitude using the SVRalgorithm.

Training with multilayer perceptrons

Using the neutral network toolbox of MATLAB(specifically the utility nftool ), we trained sev-eral times a multilayer perceptron with threehidden units with the help of the Levenberg–Marquardt training method (see e.g., Bishop2007). The number of hidden units, p, was cho-sen so that neither the model was underfittingor overfitting the validation set. In this case, weset p = 3. The training basically tried to minimizeEq. 16, and it halted when the error on the valida-tion set began to increase (this is called an early- stopping strategy). The MLP that produced thesmallest error (according to Eq. 16) on the testingset was chosen for the results reported below. Theweights of that network are as follows:

w 0 =−4.2849 0 .1396 0 .8048 1 .60881.3383 −1.1684 −3.4321 −0.8322−0.1812 5 .4063 −0.0084 0 .3572

w 1 = 0.4873 −0.2759 1 .4386

and the biases are the following:

b 0 = −3.9342 −3.0765 6 .5198 T

and

b 1 = −1.1465

Remember that variable normalization usingEq. 17 before using the aforementioned weights


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is mandatory when using Eq. 15. Then, the MMIis calcutated from Eq. 19 using the equation:

MMI

=(outputNN +1)×(max (MMI )−min ( MMI ))

2+ min (MMI )

= 2.75 ×outputNN +2.0

and the values that appear in Table 3.In Appendix 2.2, we have included the

MATLAB code that estimates the MMI fromthe epicentral distance, PGA, PGV, and mo-ment magnitude using the multilayer perceptronalgorithm.

Training with genetic programming

We run the genetic programming method usingthe GPTIPS software (Searson 2010), a popu-lation size of 400 individuals, 3,000 generations,Luke and Panait ( 2002) plain lexicographic tour-nament selection method (choosing from a poolof size 7), a mean squares fitness function, a max-imum depth of trees of 3, using multigene individ-

uals with a maximum number of four genes in anindividual, 5 % of elitism (that is, the fraction of population to copy directly to the next generationwithout modification), a probability of mutationof 10 %, a probability of crossover of 85 % anda probability of direct tree copy of 5 %. We runthe algorithm several times, and the symbolic re-gression that produced the minimum error on thetesting set was as follows:

y=0.3339 max (

z1,

z4)

+0.5415

+0.4488 tanh (

z2)

−0.578 z1+0.1507 ln (|z3 +0.5909 |)+0.3339 ifte (z4 ≤ z2 , z4 , z3)+0.1604 ifte ifte (z3 ≤4.234 , −0.571 , z4) ≤z1 ,ifte (z4 ≤ −0.499 , −3.29 , −1.231 ) ,ifte (z4 ≤ −0.6676 , −1.823 , z2) (19)

where ifte() stands for the if-then-else functionand z1 , z2 , z3 , and z4 are the standardized epi-central distance, PGA, PGV, and magnitude, re-spectively, which were obtained by means of Eq. 18.

The MMI can be retrieved from Eq. 19 usingthe formula:

MMI = y ×std (MMI ) +mean (MMI )= 0.9669 y +3.6425 (20)

In Appendix 2.3, we have included the MAT-LAB code that retrieves the estimated MMI fromthe epicentral distance, PGA, PGV, and momentmagnitude using the symbolic regression ( 19) andEq. 20.

Take into account that the form of Eq. 19should not be used outside of the presentstudy, inasmuch as the functional form and thecoefficients of this equation depend on the em-ployed database. If this methodology is used withanother dataset, most probably the algorithm willconverge to a different functional form.

5.1 Weighted linear regression

In order to make a fair comparison of the non-linear regression algorithms with the linear case,we estimated the ordinary least squares linearregression and several robust linear regressions(varying the weighting function). The ordinaryleast squares linear regression between the inputvariables was as follows:

MMI = 2 .0303 −0.0063 x1 +0.4465 log 10 ( x2)

+0.7688 log 10 ( x3)

+0.5247 x4 (21)

while the robust linear regression that was thesmallest error in the testing set was the one withthe weighting function w( r ) = |r | < 1, that is,

MMI = 2.2733 −0.0059 x1 +0.4069 log 10 ( x2)+0.8383 log 10 ( x3) +0.4538 x4; (22)


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Here, x1 to x4 denote, respectively, the epicentraldistance in kilometers, the PGA in gravities, thePGV in centimeters per second, and the momentmagnitude.

For the purposes of completeness, we haveused robust linear regression to provide relation-ships similar to Eqs. 1 and 2 as follows:

MMI =1.78+1.29log 10 (PGA ) for log10 (PGA ) ≤1.73

−0.39+2.54log 10 (PGA ) for log10 (PGA )> 1.73(23)

and

MMI =3.45+1.38 log 10 (PGV ) for log10 (PGV ) ≤0.961.73+3.18 log 10 (PGV ) for log10 (PGV )> 0.96

(24)

Here, the PGA must be expressed in centime-ter per square second, while the PGV mustbe given in centimeter per second. These rela-tionships have been plotted in Figs. 7 and 8,respectively.

5.2 A note on the usage of the regressions

Note that according to Fig. 10, the MMI in our

database is bounded between 2 and 6.5, and inconsequence, these should be constraints of thevalues produced by the regression methods de-scribed above. Take into account that the mini-mum and the maximum values of each variableshould be set as guidelines on the interpolativepower of the presented results, inasmuch as it isnot advisable to extrapolate with them. If that is

the case, one should calculate new equations orparameters/weights with a dataset that containsrepresentative samples of those outliers.

6 Analysis of results

Although input MMI levels are discrete naturalnumbers, the output is in the form of continuousreal numbers. We define a successful predictionwhen the estimation is within ±0.5 the MMI levelreported by the DYFI database. In this sense, weevaluate the performance of the algorithm: thefirst part, the loss function ( 8) with ε = 0.5, wasevaluated on the testing set (a set of data whichwas not employed in the training phase of the

algorithm). For comparison reasons, the quadraticmean error function ( 16) is calculated as well.These numbers, together with the coefficients of correlation of the predicted MMI vs. the actualMMI and the percentage of misclassification onthe testing set are shown in Table 4.

In comparison to Eq. 23, which is only de-pendent of the PGA, the inclusion of more in-formation in the MMI assessment is beneficialfor its prediction. The best nonlinear regressionmethod seems to be the one produced by MLP,

followed closely by the genetic programming, andthe SVR. In general, the MMI estimation showsa good agreement with the reported intensity, theε -insensitive loss function, the mean square error,and the missclassification error obtained with thenonlinear algorithms are lower in comparison tothe values obtained with the linear regression ( 23).Figure 15 illustrates how well the predicted MMI

Table 4 Performance of the different nonlinear regression algorithms on the testing dataset

Algorithm ε-insensitive Mean square Correlation of predicted Percentage of loss function error vs. actual MMI misclassication (%)

SVR 0.037 0.146 0.928 19.91MLP 0.035 0.141 0.928 17.06Genetic programming 0.033 0.139 0.929 19.43Ordinary least squares regression ( 21) 0.046 0.169 0.913 22.75Robust regression ( 22) 0.046 0.170 0.914 21.33Equation 23 0.067 0.224 0.883 23.22Equation 24 0.097 0.278 0.853 31.28


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2 3 4 5 6 72

3

4

5

6

7

Actual MMI

P r e d i c t e d M M I

(

Ideal fitData fitActual,Predicted)

Regression: R=0.9280

Fig. 15 Predicted MMI vs. actual MMI in the case of theestimation done by the genetic programming algorithm.The least mean squares fit for both variables is Predicted-MMI ≈ 0.88 ×ActualMMI +0.46

corresponds to the actual MMI in the case of theestimation done by the MLP.

The nature of the misclassified points by thenonlinear algorithms was analyzed. Initially, wethought that most of those points belonged eitherto regions where there was a unique ZIP code fora large area or either to large intensities (as therewere not many data corresponding to large inten-sities to train the algorithms). However, it turnedout not to be like that in most of the cases. Weattribute the errors in the assignment of the MMIto the fact that the moment magnitude, epicentraldistance, PGA, and PGV are not enough to fully

describe the different aspects of the earthquake,and therefore, it would be convenient if otherdescriptive parameters of the earthquake like theones mentioned in Section 2 would be included inthe analysis, as well.

With respect to the variability of PGA andPGV, we are using the ones stated in the publicdatabases. It is to be expected that this randomnumber, it is the best estimate of the PGA and

PGV that is at our disposal, given the availableinformation. If we had in our disposition prob-ability density functions, possibility distributions,or interval information on those values, we coulduse that information in our proposed approach. If that were the case, simple Monte Carlo samplingor even the “extension principle” used in fuzzy settheory and in the theory of random sets wouldbe excellent tools to propagate the uncertaintythrough the nonlinear regression algorithms. Inthis case, the expected MMI would be presentedin form of a probability density function, a nor-malized fuzzy set or an interval.

7 Conclusions

In this paper, we presented three nonlinear regres-sion methods, namely support vector regression,multilayer perceptrons, and genetic programmingto model the relationship between the modifiedMercalli intensity scale and the earthquake mo-ment magnitude, epicentral distance, PGA, andPGV measured at the closest stations to the MMIreading. In general, the MMI estimation showsa good agreement with the reported intensity.The best results were obtained by the multilayerperceptron.

As seen from the results, nonlinear regressionshould be applied in order to find a relationshipbetween MMI and instrumental information in-stead of the linear regressions that are popular inthis class of studies. Our numerical experimentshave shown for example that all of the nonlinearregression algorithms employed perform betterthan the linear regressions ( 21) and (22).

Acknowledgements The authors would like to thankJohn R. Evans, David Wald, Vince Quitoriano, and BruceWorden of the USGS for their helpful advise over theinternet. Also, we would like to thank the anonymousreviewers and the associate editor, Dr. Gottfried Grünthal,for their constructive comments that have notoriouslyimproved the paper. Financial support for the realiza-tion of the present research has been received from theUniversidad Nacional de Colombia; the support is gra-ciously acknowledged.


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Appendix 1: Support vectors of the SVR method

The bias b of the SVR was b = 6.6477 and the support vectors together with their weights β i are listed inthe following:

SV β z1 z2 z3 z41

−4.4895

−0.6223 1.5550 2.7381 2.0587

2 −405.5060 −0.6048 −0.6038 −0.5537 −0.24323 −494.5590 −0.1701 −0.6708 −0.5894 −0.24324 −45.3807 −0.1353 0.6178 0.7897 1.95415 −7.2997 0.0990 1.3207 2.8716 1.95416 −494.5590 −0.0249 −0.5704 −0.5443 −0.13867 494.5590 −0.6436 −0.4700 −0.4973 −1.08038 494.5590 −0.3647 −0.5536 −0.5462 −0.34789 494.5590 0.2665 −0.6206 −0.5274 −0.347810 494.5590 0.2675 −0.5871 −0.5199 −0.347811 494.5590 0.2684 −0.6373 −0.5518 −0.347812 494.5590 0.3014

−0.4198

−0.5462

−0.3478

13 −494.5590 −0.6561 −0.1520 −0.4165 −0.871014 −494.5590 −0.6358 0.0823 −0.2211 −0.871015 −494.5590 0.3140 −0.5202 −0.4936 0.175316 −494.5590 0.3875 −0.5536 −0.5593 0.175317 10.7370 1.8688 −0.6541 −0.4823 1.326318 131.7847 1.8978 −0.5202 −0.1948 1.326319 83.2952 2.1534 −0.6708 −0.3451 1.326320 494.5590 −0.7113 0.6346 0.0382 −0.347821 277.7219 −0.6348 0.0656 −0.3527 −0.347822 494.5590 −0.6232 0.2664 −0.3395 −0.347823 494.5590

−0.5893 0.3835

−0.0783

−0.3478

24 494.5590 −0.5777 0.1492 −0.3132 −0.347825 494.5590 −0.3328 0.0990 −0.2794 −0.347826 −388.3332 −0.0646 −0.1353 −0.4372 −0.347827 494.5590 0.2065 −0.2691 −0.3959 −0.347828 494.5590 0.2539 −0.2859 −0.4729 −0.347829 494.5590 −0.7936 1.1701 −0.0182 −1.080330 494.5590 −0.6813 −0.3361 −0.4729 −1.080331 −249.7611 −0.6736 −0.4532 −0.4071 −1.080332 494.5590 −0.6600 −0.5536 −0.5199 −1.080333 494.5590 −0.6048 −0.6206 −0.5537 −1.080334 494.5590 −0.5932 −0.5704 −0.5537 −1.080335 494.5590 −0.5341 −0.6875 −0.6044 −1.080336 494.5590 −0.5003 −0.6541 −0.5744 −1.080337 494.5590 −0.7791 −0.4867 −0.4898 −1.080338 274.6066 −0.7752 2.9440 0.8085 −1.080339 494.5590 −0.7462 −0.1185 −0.3771 −1.080340 494.5590 −0.7239 −0.1353 −0.4259 −1.080341 494.5590 −0.6639 −0.6541 −0.6044 −1.080342 494.5590 −0.6329 −0.5704 −0.5086 −1.080343 −494.5590 −0.3366 −0.5704 −0.5499 −0.5571


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SV β z1 z2 z3 z444 494.5590 −0.7045 1.5717 0.3463 0.384645 −494.5590 −0.6358 0.0488 0.1171 0.384646 −116.8932 −0.6329 0.0656 0.0757 0.384647 −494.5590 −0.6116 0.3668 0.0532 0.384648

−494.5590 0.8890

−0.7043

−0.6044 0.1753

49 126.2576 −0.7433 1.4713 1.6597 0.175350 489.1067 −0.6978 0.4672 0.7503 0.175351 −494.5590 −0.6600 1.3040 0.5981 0.175352 −24.8224 −0.6552 4.0318 3.7020 0.175353 −242.1119 −0.6436 1.9567 0.4553 0.175354 −494.5590 −0.5738 1.1199 0.6807 0.175355 −494.5590 −0.5012 0.8187 −0.2531 0.175356 −209.7443 −0.4916 1.1032 1.0509 0.175357 −494.5590 −0.4402 0.5676 0.0551 0.175358 −494.5590 −0.3831 0.0656 0.0720 0.175359

−494.5590

−0.3802

−0.0851 0.0062 0.1753

60 494.5590 −0.3260 −0.3361 −0.3959 0.175361 494.5590 −0.3202 −0.3696 −0.3921 0.175362 −494.5590 0.2104 −0.2691 −0.2380 0.175363 −494.5590 −0.8323 1.6052 0.4515 −0.766464 494.5590 −0.7162 0.0321 −0.4616 −0.766465 −494.5590 −0.2602 −0.6373 −0.5894 −0.766466 494.5590 −0.0326 −0.6206 −0.5744 −0.766467 494.5590 0.0206 −0.6038 −0.5669 −0.766468 10.5189 −0.8139 1.0864 0.2298 −1.289569 −494.5590 −0.7045 −0.6206 −0.5631 −1.289570

−494.5590

−0.3153

−0.6541

−0.6063

−0.8710

71 −23.3037 −0.8101 2.5424 0.7390 −0.557172 7.0743 −0.7384 2.6763 0.9006 −0.557173 494.5590 −0.7239 2.0069 0.7052 −0.557174 494.5590 −0.6900 0.9860 0.0250 −0.557175 494.5590 −0.6852 2.0403 0.2016 −0.557176 −494.5590 0.0768 −0.5704 −0.5631 −0.557177 −494.5590 −0.7839 −0.0014 −0.3357 −1.289578 −178.8668 −0.7588 0.1492 −0.3827 −1.289579 −494.5590 −0.5361 −0.4365 −0.5481 −1.184980 −123.2760 −0.4548 −0.6206 −0.6025 −1.184981 10.9674 −0.3231 4.7347 4.8181 1.326382 −494.5590 −0.4490 −0.5704 −0.5988 −0.975683 −66.9874 −0.8120 3.9649 0.6751 −0.871084 −494.5590 −0.7278 0.0321 −0.4691 −0.871085 −494.5590 −0.7084 0.1158 −0.4015 −0.871086 −494.5590 −0.6949 1.9064 0.3407 −0.871087 −494.5590 −0.6949 2.1240 0.3839 −0.871088 −494.5590 −0.6910 0.4505 −0.1516 −0.871089 −494.5590 −0.6871 0.0321 −0.2907 −0.871090 −494.5590 −0.6803 −0.1185 −0.2474 −0.8710


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SV β z1 z2 z3 z491 −494.5590 −0.6542 −0.3528 −0.4823 −0.871092 −494.5590 −0.6223 −0.3696 −0.4466 −0.871093 −494.5590 −0.6155 −0.6038 −0.5932 −0.871094 −494.5590 −0.5913 −0.4867 −0.5481 −0.871095

−494.5590

−0.5680

−0.3863

−0.4992

−0.8710

96 −494.5590 −0.5554 −0.2691 −0.4278 −0.871097 −0.0167 −0.2534 2.3918 5.5809 2.058798 −12.2687 0.8823 0.0823 0.9720 2.058799 −159.9334 2.2667 −0.4867 0.1039 2.0587100 −289.0971 2.4081 −0.6206 −0.0032 2.0587101 −494.5590 2.4894 −0.6206 0.0344 2.0587102 494.5590 2.5833 −0.6206 0.1434 2.0587103 494.5590 2.5852 −0.6206 −0.0577 2.0587104 494.5590 2.6346 −0.6038 0.0551 2.0587105 −415.6563 0.5240 −0.6038 −0.5781 −0.7664106 494.5590 0.2781

−0.5704

−0.4259 0.4892

107 −494.5590 1.2656 −0.6708 −0.5781 0.4892108 494.5590 1.6171 −0.6373 −0.5199 0.4892109 494.5590 −0.6252 0.6513 0.4121 0.1753110 427.5960 0.1106 −0.4198 −0.3489 0.1753111 494.5590 0.1116 −0.5871 −0.4842 0.1753112 494.5590 0.1164 −0.4867 −0.4391 0.1753113 −10.2066 0.4292 −0.6206 −0.5499 0.1753114 −494.5590 0.6170 −0.4030 −0.3827 0.1753115 −494.5590 0.7438 −0.6373 −0.4616 0.1753116 −392.9353 1.2540 −0.6373 −0.4710 0.1753117

−494.5590

−0.5554

−0.6875

−0.6082

−1.1849

118 −494.5590 −0.4296 −0.6373 −0.5932 −1.1849119 494.5590 0.2830 −0.6206 −0.5650 −0.3478120 494.5590 −0.2727 −0.5704 −0.5687 0.1753121 −494.5590 0.4718 −0.5704 −0.5293 0.1753122 494.5590 −0.7171 0.4839 −0.0595 −0.3478123 494.5590 −0.5787 0.9191 −0.2662 −0.3478124 494.5590 0.2249 −0.2859 −0.4992 −0.3478125 494.5590 0.2597 −0.5536 −0.5255 −0.3478126 494.5590 −0.5496 −0.6373 −0.5744 −1.0803127 494.5590 −0.7849 −0.3026 −0.3733 −1.0803128 494.5590

−0.7646 0.4672

−0.0633

−1.0803

129 494.5590 −0.7626 −0.3361 −0.4541 −1.0803130 −494.5590 −0.4170 −0.6206 −0.5875 −0.6617131 −494.5590 −0.2844 −0.4532 −0.4691 −0.7664132 −494.5590 −0.2350 −0.6373 −0.5838 −0.4525133 −331.6516 −0.2147 −0.6206 −0.6007 −0.8710134 494.5590 −0.7462 −0.2859 −0.5274 −1.0803135 −494.5590 −0.7926 1.2203 −0.1742 −0.5571136 494.5590 −0.2253 0.7684 0.5022 1.3263137 −494.5590 −0.1585 0.1492 −0.2662 0.6985


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SV β z1 z2 z3 z4138 −494.5590 −0.6861 −0.1687 −0.3771 −0.8710139 −494.5590 −0.6193 −0.1185 −0.4673 −0.8710140 −494.5590 −0.5613 −0.6038 −0.5706 −0.8710141 −12.4008 2.6094 −0.5704 0.8592 2.0587142

−116.9598 2.6859

−0.6206

−0.2888 2.0587

143 −494.5590 2.9134 −0.6541 0.1114 2.0587144 494.5590 −0.4625 −0.5704 −0.5481 −0.7664145 494.5590 −0.2514 −0.2859 −0.2324 0.1753146 494.5590 0.8658 −0.6541 −0.5255 0.1753147 494.5590 −0.6484 −0.5536 −0.5593 −1.1849

Appendix 2: MATLAB implementationof the MLP and the GP approach

2.1 Support vector regression

2.2 Multilayer perceptron


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2.3 Symbolic regression by genetic programming

References

Atkinson GM, Kaka SI (2007) Relationships between feltintensity and instrumental ground motion in the cen-tral United States and California. Bull Seismol SocAm 97(2):497–510

Atkinson GM, Sonley E (2000) Empirical relationships be-tween modified Mercalli intensity and response spec-tra. Bull Seismol Soc Am 90(2):537–544

Bishop CM (2007) Pattern recognition and machine learn-

ing. Springer, NYBommer JJ, Stafford PJ, Alarcón JE, Akkar S (2007) Theinfluence of magnitude range on empirical ground-motion prediction. Bull Seismol Soc Am 97(6):2152–2170

Center for Engineering Strong Motion Data (2011) Inter-net data reports. http://www.strongmotioncenter.org/ .Accessed 15 Jan 2011

Chang CC, Lin CJ (2011) LIBSVM: a library for supportvector machines. Software available at http://www.csie.ntu.edu.tw/ ∼cjlin/libsvm . Accessed 15 Jan 2011Davenport P (2004) Neural network analysis of seismicintensity from instrumental records. In: Proceedings of

the 13th world conference on earthquake engineering,paper no. 692. Vancouver, Canada

Dengler LA, Dewey JW (1998) An intensity survey of households affected by the Northridge, California,earthquake of 17 January 1994. Bull Seismol Soc Am88(2):441–462

Faenza L, Michelini A (2010) Regression analysis of

MCS intensity and ground motion parameters inItaly and its application in ShakeMap. Geophys J Int180(3):1138–1152

Grünthal G (2011) Earthquakes, intensity. In: Gupta HK(ed) Encyclopedia of solid earth geophysics. Encyclo-pedia of earth sciences series. Springer, Dordrecht,The Netherlands, pp 237–242

Haykin S (1998) Neural networks: a comprehensive foun-dation, 2nd edn. Prentice Hall PTR, Upper SaddleRiver, NJ, USA

Hornik K, Stinchcombe MB, White H (1989) Multilayerfeedforward networks are universal approximators.Neural Netw 2(5):359–366

Kaka SI, Atkinson GM (2004) Relationships between in-strumental ground-motion parameters and modifiedMercalli intensity in eastern north America. Bull Seis-mol Soc Am 94(5):1728–1736

Karim KR, Yamazaki F (2002) Correlation of JMA instru-mental seismic intensity with strong motion parame-ters. Earthq Eng Struct Dyn 31:1191–1212

Koza JR (1992) Genetic Programming: on the program-ming of computers by means of natural selection. MITPress, Cambridge, MA, USA

Kuehn NM, Scherbaum F (2010) Short note: a naive bayesclassifier for intensities using peak ground velocityand acceleration. Bull Seismol Soc Am 100(6):3278–3283

Luke S, Panait L (2002) Lexicographic parsimony pres-sure. In: Proceedings of the genetic and evolution-ary computation conference (GECCO 2002). MorganKaufmann Publishers

Mallat S (2008) A wavelet tour of signal processing: thesparse way. 3rd edn. Academic Press, San Diego,California

Murphy JR, O’Brien LJ (1977) The correlation of peakground acceleration amplitude with seismic intensityand other physical parameters. Bull Seismol Soc Am67(3):877–915

Musson RMW, Grünthal G, Stucchi M (2010) The com-parison of macroseismic intensity scales. J Seismol14(2):413–428

Poulimenos A, Fassois S (2006) Parametric time-domainmethods for non-stationary random vibration mod-elling and analysis—a critical survey and comparison.Mech Syst Signal Process 20(4):763–816

Richter CF (1958) Elementary seismology. W. H. Freemanand Co., San Francisco, pp 135–149, 650–653

Sarle WS (2002) USENET comp.ai.neural-netsneural networks FAQ, part II: “subject: shouldI normalize/standardize/rescale the data?”.ftp://ftp.sas.com/pub/neural/FAQ2.html#A_std .Accessed 24 June 2011

Searson D (2010) GPTIPS: genetic programming andsymbolic regression for MATLAB. Software available

http://www.strongmotioncenter.org/http://www.strongmotioncenter.org/http://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.csie.ntu.edu.tw/~cjlin/libsvmftp://ftp.sas.com/pub/neural/FAQ2.html#A_stdftp://ftp.sas.com/pub/neural/FAQ2.html#A_stdftp://ftp.sas.com/pub/neural/FAQ2.html#A_stdhttp://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.csie.ntu.edu.tw/~cjlin/libsvmhttp://www.strongmotioncenter.org/


23/23

J Seismol (2012) 16:489–511 511

at http://sites.google.com/site/gptips4matlab/home .Accessed 15 Jan 2011

Shabestari KT, Yamazaki F (2001) A proposal of instru-mental seismic intensity scale compatible with MMIevaluated from three-component acceleration records.Earthq Spectra 17(4):711–723

Smola AJ, Schölkopf B (2004) A tutorial on support vector

regression. Stat Comput 14:199–222Sokolov VY (2002) Seismic intensity and Fourier accel-eration spectra: revised relationship. Earthq Spectra18(1):161–187

Stover C, Coffman J (1993) Seismicity of the United States,1568–1989 (Revised). U.S. Geological Survey Profes-sional Paper 1527

Trifunac MD, Brady AG (1975) On the correlation of seis-mic intensity scales with the peaks of recorded strongground motion. Bull Seismol Soc Am (65)1:139–162

Tselentis GA, Danciu L (2008) Empirical relationshipsbetween modified Mercalli intensity and engineeringground-motion parameters in Greece. Bull SeismolSoc Am 98(4):1863–1875

Tselentis GA, Vladutu L (2010) An attempt to model therelationship between MMI attenuation and engineer-ing ground-motion parameters using artificial neuralnetworks and genetic algorithms. Nat Hazards EarthSyst Sci 10(12):2527–2537

Tung ATY, Wong FS, Dong W (1993) A neural networksbased MMI attenuation model. In: National earth-quake conference: earthquake hazard reduction in thecentral and eastern United States: a time for examina-tion and action. Memphis, Tennessee, US

US Geological Survey (2011) Did you feel it? database.http://earthquake.usgs.gov/earthquakes/dyfi/ .

Accessed 15 Jan 2011Wald D, Quitoriano V, Dengler L, Dewey JM (1999a) Uti-lization of the internet for rapid community intensitymaps. Seismol Res Lett 70(6):680–697

Wald DJ, Quitoriano V, Heaton TH, Kanamori H (1999b)Relationships between peak ground acceleration,peak ground velocity, and modified Mercalli intensityin California. Earthq Spectra 15(3):557–564

Wald DJ, Quitoriano V, Dewey J (2006) USGS “did youfeel it?” community internet intensity maps: macro-seismic data collection via the internet. In: Proceed-ings of the first European conference on earthquakeengineering and seismology. Geneva, Switzerland

Wood HO, Neumann F (1931) Modified Mercalli intensityscale of 1931. Bull Seismol Soc Am 21:277–283

Worden CB, Gerstenberger MC, Rhoades DA, Wald DJ(2012) Probabilistic relationships between ground-motion parameters and modified Mercalli intensity inCalifornia. Bull Seismol Soc Am 102:204–221
http://sites.google.com/site/gptips4matlab/homehttp://sites.google.com/site/gptips4matlab/homehttp://earthquake.usgs.gov/earthquakes/dyfi/http://earthquake.usgs.gov/earthquakes/dyfi/http://earthquake.usgs.gov/earthquakes/dyfi/http://sites.google.com/site/gptips4matlab/home

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