strong ground-motion relations for mexican interplate...

J Seismol (2010) 14:769–785DOI 10.1007/s10950-010-9200-0

ORIGINAL ARTICLE

Strong ground-motion relations for Mexicaninterplate earthquakes

Danny Arroyo · Daniel García · Mario Ordaz ·Mauricio Alexander Mora · Shri Krishna Singh

Received: 26 May 2009 / Accepted: 13 June 2010 / Published online: 2 July 2010© Springer Science+Business Media B.V. 2010

Abstract We derive strong ground-motion re-lations for horizontal components of pseudo-acceleration response spectra from Mexican inter-plate earthquakes at rock sites (NEHRP B class)in the forearc region. The functional form is ob-tained from the analytical solution of a circularfinite-source model. For the regression analysiswe use a recently proposed multivariate Bayesiantechnique. The resulting model has similar accu-racy as those models derived from regional andworldwide subduction-zone databases. However,there are significant differences in the estima-tions computed from our model and other mod-els. First, our results reveal that attenuation inMexico tends to be stronger than that of world-wide relations, especially for large events. Sec-ond, our model predicts ground motions for largeearthquakes at close distances to the source thatare considerably larger than the estimations ofglobal models. Lack of data in this range makes

D. Arroyo (B)Departamento de Materiales,Universidad Autónoma Metropolitana, Azcapotzalco,Mexico DF, Mexicoe-mail: [email protected]

D. García · S. K. SinghInstituto de Geofísica, UNAM, Mexico DF, Mexico

M. Ordaz · M. A. MoraInstituto de Ingeniería, UNAM, Mexico DF, Mexico

it difficult to identify the most appropriate modelfor this scenario. Nevertheless, according to theavailable data at the city of Acapulco, our modelseems to estimate seismic hazard more adequatelythan the other models. These new relations maybe useful in computing seismic hazard for theMexican forearc region, where no similar equa-tions had been yet proposed.

Keywords Ground motion model · SA spectra ·Subduction-zone interplate earthquakes

1 Introduction

Earthquake engineering in Mexico has been tra-ditionally focused on estimating ground motionat the Valley of Mexico, located in the volcanicbelt, caused by subduction-zone interplate earth-quakes (e.g., Singh et al. 1987; Castro et al. 1988;Rosenblueth et al. 1989; Ordaz et al. 1994). Thisinterest is motivated by the large population den-sity on this area, and its particular propagationand site effects. Unfortunately, these effects makethe referred relations of no use for sites outsidethe volcanic belt.

In contrast, in the forearc region only a fewrelations for peak ground acceleration (PGA) andModified Mercalli Intensity have been proposed(Anderson and Quaas 1988; Chávez and Castro1988; Ordaz et al. 1989; Anderson and Lei 1994;

Author's personal copy

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Anderson 1997). These models are only valid forthe state of Guerrero. Due to the proximity ofmoderate to large (Mw 6.0–8.0) earthquake focialong the Pacific coast, there is an urgent needof ground-motion models in the forearc region.These models are crucial to estimate accuratelyseismic hazard at coastal areas, where much of thepopulation in the region concentrates.

In the last decade there has been a consid-erable improvement in the seismic networks in

Mexico. Here we take advantage of the largerdatabase currently available to develop pseudo-acceleration (SA) strong ground-motion relationsfor rock sites in the Mexican forearc region. Toachieve this goal we use a function based on thesolution of a circular finite-source combined witha recently proposed Bayesian regression tech-nique. We compare our results with previous PGAmodels for Guerrero and with other SA relationsbased on worldwide subduction-zone datasets.

Table 1 Interplateearthquakes used in thisstudy

aNumber of three-component records used

Event no. Date (yymmdd) Lat ◦N Lon ◦W H(km) Mw Recordsa

1 850919 18.14 102.71 17 8.0 102 850921 17.62 101.82 22 7.6 83 880208 17.45 101.19 22 5.8 94 890310 17.45 101.19 20 5.4 65 890425 16.61 99.43 16 6.9 106 890502 16.68 99.41 15 5.5 57 900113 16.82 99.64 16 5.3 88 900511 17.12 100.87 21 5.5 69 900531 17.12 100.88 18 5.9 910 930515 16.47 98.72 16 5.5 611 931024 16.65 98.87 26 6.6 1212 950914 16.48 98.76 16 7.3 1513 960313 16.59 99.12 25 5.1 1014 960327 16.36 98.30 18 5.4 915 960715 17.33 101.21 27 6.6 1716 960718 17.44 101.21 25 5.4 1017 970121 16.42 98.21 28 5.4 1218 971216 16.04 99.41 27 5.9 719 980509 17.50 101.24 23 5.2 1220 980516 17.27 101.34 28 5.2 1021 980705 16.81 100.14 25 5.3 1522 980711 17.35 101.41 29 5.4 1223 980712 16.85 100.47 26 5.5 1424 010904 16.29 98.37 20 5.2 1025 011110 16.09 98.32 17 5.4 1126 020607 15.99 96.92 20 5.2 1027 020607 15.96 96.93 19 5.5 1228 020619 16.29 98.02 20 5.3 1229 020805 15.94 96.26 15 5.4 730 020827 16.16 97.54 15 5.0 1031 020830 16.76 100.95 15 5.2 632 020925 16.80 100.12 12 5.3 1533 021108 16.28 98.12 16 5.2 1034 021210 17.36 101.25 24 5.4 835 030110 17.01 100.35 28 5.2 1536 030122 18.62 104.12 10 7.5 837 040101 17.27 101.54 17 6.0 1538 040101 17.32 101.47 27 5.6 1139 040206 18.16 102.83 12 5.1 840 040614 16.19 98.13 20 5.9 18


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2 Data and processing

We used a subset of the Mexican interplate earth-quakes database by García et al. (2009). Theseauthors compiled all the interplate, thrust-faultingevents with Mw ≥ 5.0 occurred between 1985 and2004 in the Pacific coast between the states ofColima and Oaxaca, only excluding those near-trench events whose high-frequency radiation isanomalously low (Shapiro et al. 1998; Iglesiaset al. 2003). The dataset contains free-field, hard-rock (NEHRP B class; BSSC 2004) recordingsavailable from permanent networks. Any stationwith known, significant site amplification (Castroet al. 1990; Humphrey and Anderson 1992; Castroand Ruiz-Cruz 2005), as well as those locatedin the volcanic belt, was excluded. In addition,García et al. (2009) applied the H/V spectral ra-tio technique (Lermo and Chávez 1993) to verifythat all stations included in their dataset satisfiedthe criteria of a generic rock station. For a moredetailed description of the database the reader isreferred to García et al. (2009).

This study is focused on the magnitude–distance range of most engineering interest (Mw ≥6.5 and R < 150–200 km). In order to increase theinfluence of this data range, we excluded thosesmall events (Mw ≤ 5.5) with few records and onlycollected at distant stations (R > 100 km). Forevents with magnitude Mw ≥ 6.0, we estimated

the minimum distance from the station to the faultplane; for the smaller events we took the hypocen-tral distance. We limited the distance range to400 km. This upper limit takes into account theslow decay of the ground motion toward the con-tinent reported by several authors (e.g., Singhet al. 1988; Cárdenas et al. 1998; Cárdenas andChávez 2003; García et al. 2009), which forces usto consider larger distances than usual. To reducethe potential variability of the data, for any eventrecorded at two or more stations less than 5 kmapart we selected only one of them, based onvisual inspection of the traces.

The resulting subset consists of 418 recordsfrom 40 interplate earthquakes obtained at 56 sta-tions located at distances between 20 and 400 km(Table 1; Fig. 1). The selected data were recordedat 80 to 250 sps by 12–19 bit digital accelerographs(66% of the data), which present a flat responsefor acceleration down to less than 0.1 Hz, and 24-bit broadband seismographs (34%), with a flat re-sponse for velocities between 0.01 and 30 Hz. Boththe instrumental responses and the sampling ratesensure that reliable information can be retrievedfrom the records for frequencies up to nearly30 Hz. Figure 2 shows the magnitude–distancedistribution of the data. Note that roughly 45% ofthe data come from 20–100 km distance. The wavepaths are shown in Fig. 3. Though the majority ofrecords come from Guerrero, a noticeable amount

Fig. 1 Map of centralMexico showingepicenters (circles) andstations (triangles) used inthis study. Filled symbolsrepresent strong-motionstations (accelerographs)and open symbolsrepresent broadbandstations. The grey squarerepresents ACAP station(see text). MVB MexicanVolcanic Belt (shadedgrey). MAT MiddleAmerican Trench. Statesmentioned in the text arelabeled


772 J Seismol (2010) 14:769–785

Fig. 2 Magnitude versus distance plot summarizing thedata used in this study. Symbols indicate the type of dataavailable. Circles accelerograms, open diamonds broad-band velocity data

of data also comes from other regions, especiallyOaxaca.

From the acceleration records, we read at thehorizontal components PGA values and com-puted 5% damped SA spectra at 56 periods be-tween 0.04 and 5 s (0.2–25 Hz).

3 Functional form

We chose a functional form based on the solu-tion of a circular finite-source. According to the

Fig. 3 Epicenters of the earthquakes (circles), stations(triangles), and ray paths used in this study. Symbols arethe same as in Fig. 1

point-source model, the far-field approximationof the Fourier acceleration spectral amplitude ofthe most intense part of the ground motion, A( f ),assuming an ω−2 source model (Brune 1970), isdefined by Eqs. 1 to 3:

A ( f ) = CF ( f )M0 f 2

1 + (f/

fc)2 e− π f R

βQ( f ) e−πκ f /R (1)

C = (2π)2 RP F1 P4πρβ3

(2)

fc = 4.91 × 106β

(�σ

M0

) 13

(3)

where f is frequency, F( f ) is a factor that correctsfor the amplification of S waves as they propagateupwards through material of progressively lowervelocity and it is roughly 2 for f ≥ 1 Hz (Boore1986), M0 is the seismic moment (in dyne·cm), fc

is the corner frequency, R is a measure of distanceto the source (see below), β = 3.2 km/s is theshear-wave velocity, Q = Q0·f (Q0 = 100 s) is thequality factor that accounts for anelastic attenua-tion for the Pacific coast (Singh et al. 1989), κ is aparameter that corrects for the site effect (Singhet al. 1982; Anderson and Hough 1984), RP =0.6 is the average radiation pattern (Boore andBoatwright 1984), F1 = 2 is a factor that accountsfor the free-surface amplification, P = 0.707 is afactor that takes into account the equal partition-ing of energy in the two horizontal components,ρ = 2.8 g/cm3 is the density in the focal region,and �σ is the Brune stress drop (in bars).

Combining Eq. 1 and a high frequency approx-imation, Singh et al. (1989) derived, through ran-dom vibration theory, expressions for the Fourierspectral amplitude (Eq. 4a) and the root meansquared ground acceleration (Eq. 4b) for a circu-lar finite-source model:

A( f )2 = 2C2 F ( f )2 (M0 f 2

c

)2e−2πκ f

×

⎡

⎢⎢⎣

E1 (αR) − E1

(α

√R2 + r2

0

)

r20

⎤

⎥⎥⎦ (4a)


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arms = 2√

2CM0 f 3c√

πκ fc

×

⎡

⎢⎢⎣

E1 (αR) − E1

(α

√R2 + r2

0

)

r20

⎤

⎥⎥⎦

12

(4b)

where α = 2π /βQ0, E1(x) is the well-known expo-nential integral function that is defined in Eq. 5and r0 is the radius of a circular fault based on theBrune’s model, given by Eq. 6:

E1 (x) =∞∫

x

e−t

tdt (5)

r0 = 2.34β

2π fc(6)

In practice, the exponential integral function canbe computed using numerical methods (see, forexample, Abramowitz and Stegun (1972) for fur-ther details). Equations 4a and 4b were obtainedconsidering that the source intensity, (M0 · f 2

c )2, isuniformly distributed over the rupture area.

Inserting Eqs. 2 and 3 in Eq. 4b, using the re-lationship between moment magnitude, Mw, andM0 (Kanamori 1977), and taking the natural loga-rithm of Eq. 4b results in

ln arms = α1 + α2 Mw + α3 ln

⎡

⎢⎢⎣

E1 (α4 R)−E1

(α4

√R2 + r2

0

)

r20

⎤

⎥⎥⎦

(7)

In the case of PGA, for example, the coefficientswould be α2 = 0.576, α3 = 0.5, and α4 = α.However, these values should be viewed as ap-proximations, since there are several assumptionsinvolved in Eq. 4b.

In accordance with Eq. 7, we set the functionalform of the ground-motion relations at each pe-riod, T, as

SA (T)= α1 (T) + α2 (T) Mw + α3 (T) ln⎡

⎢⎢⎣

E1(α4 (T) R)−E1

(α4 (T)

√R2+r2

0

)

r20

⎤

⎥⎥⎦

(8a)

where αi(T) are the coefficients determinedthrough regression analysis, R is the closest dis-tance to fault surface (according to the circularfinite-source model), and r0 is given by

r20 = 1.4447 × 10−5e2.3026Mw (8b)

Equation 8b was obtained from Eq. 6 using astress drop equal to 100 bars. In theory, thecoefficients of Eq. 8b should be also obtainedthrough regression. However, in order to keep thefunctional form as simple as possible we decidedto fix the coefficients of Eq. 8b. As it will be shownlater, the resulting function yields satisfactory re-sults, thus regression coefficients αi(T) adequatelycorrect the error introduced by the assumption ofa stress drop of 100 bars.

The third term in Eq. 8a accounts simul-taneously for geometrical spreading, anelas-tic attenuation and near-source saturation. AsR approaches to infinity this term approachesto e−α4(T)R

2R2 , thus the geometrical spreadingand anelastic attenuation of SA are given by(− [α3 (T) α4 (T) R + 2α3 (T) ln (R)]).

When R becomes comparable to r0 the near-source effect is controlled by the coefficient α4.As α4 increases, saturation of SA increases; onthe other hand, as α4 becomes zero, saturationvanishes. We note that the proposed function al-lows for oversaturation (i.e., the value of SA startsto decrease when Mw exceeds a certain thresh-old). Although the existence of this effect is ques-tionable, we proceeded with this functional form,since saturation level will be finally determinedin the regression by the available data. Finally,we note that, although the function is undefinedfor R = 0 km, this is not a problem for Mexicaninterplate earthquakes, since their rupture areasnever reach the surface (e.g., Singh et al. 1989).


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4 Regression model

We performed the regression analysis through aBayesian scheme recently developed by Arroyoand Ordaz (2010a, b). The model is able toinclude, in the framework of Bayesian analy-sis, the correlation between: (1) observationsfor a given earthquake (intra-event correlation),(2) the coefficients of the model, and (3) ordi-nates of different periods. This level of general-ity, however, is only achieved if the function islinear.

Although the function defined in Eq. 8a is non-linear, it becomes linear once α4(T) is set to acertain value. For this reason, we performed theregression analysis as follows: for a given pe-riod and a given value of α4 for that period, wecompute the coefficients α1, α2, and α3 throughBayesian analysis (i.e., considering them as ran-dom variables with prior and posterior probabilitydensity functions). Repeating iteratively this stepwe set the value of α4 for that period as theone which yields the best fit to data. This impliesthat the regression analysis in not fully Bayesian,since coefficient α4 is not considered as a randomvariable. The same procedure is repeated for eachperiod independently, thus disregarding correla-tion between ordinates.

In order to stabilize the regression and avoidphysically unacceptable values of the coefficients,the Bayesian method requires prior informationabout the coefficients. For a given combinationof period and α4 this information was definedthrough random vibration theory and the Fourieramplitude spectrum defined in Eq. 4a by way ofcomputing SA values for several combinations ofMw and R. Following Singh et al. (1989), we used�σ = 100 bars, κ = 0.023 s−1, and Q0 = 100 s.Then, we applied the least squares method tocompute the prior value of α1, α2, and α3. Thisimplies that a priori we believe that the behaviorof SA could be properly characterized by thefinite-source model defined in Eq. 4a. The priorcovariance of α1, α2, and α3, the prior expectedvalue of the covariance of the residuals (σ 2; whereresiduals are defined as the logarithmic differencebetween observations and estimations), and theprior expected value of the inter-event correlation(γ e) were set following previous studies. Further

details can be found in Ordaz et al. (1994) andArroyo and Ordaz (2010a, b).

In order to obtain the posterior expected valuesof α1(T), α2(T), α3(T), σ 2, and γ e, the joint poste-rior density of the regression coefficients was nu-merically marginalized using the Gibbs samplingmethod with a number of terms of 150, for whichconvergence was achieved (see Arroyo and Ordaz(2010a) for further details).

Although Eqs. 4a and 4b hold only in the shortperiod range due to the high frequency approx-imation used, we decided to use these equationsto define the prior mean value of the regressioncoefficients for all periods, since prior informationwill be modified with the information contained inthe database.

5 Near-source saturation term: resultsfor the α4 coefficient

The coefficient α4 controls the near-source satu-ration effect. It might be viewed as an empiricalmodification of α = 2π /βQ0 (Eq. 4b) to take intoaccount the variation of the quality factor Q0 withfrequency.

In the first stage of the regression analysis, westudied the variation of the standard deviationof the residuals (random variability, σ ) at eachperiod: performing the Bayesian analysis for eachvalue of α4, we obtained the resultant values ofα1, α2, α3, and σ . Figure 4a shows random vari-ability at four representative periods. For longperiods (T ≥ 1 s) σ increases monotonically withα4, due to the absence of near-source saturationfor SA in that range (the peak displacement ofa long-period oscillator should be equal to thepeak ground displacement). On the other hand,for short periods (T < 1 s) minimum random vari-ability is attained at a certain value of α4, whichdepends on period. The shorter the period, thelarger value of α4 for which the minimum is found.In other words, near-source saturation effect ismore pronounced at short periods, as expected.For certain periods it is not clear which valueof α4 corresponds to the minimum σ ; in thosecases, we selected the value of α4 which resultedin the minimum mean value of residuals (bias,b). In Fig. 4b we plot the value of coefficient α4


J Seismol (2010) 14:769–785 775

Fig. 4 a Effect of coefficient α4 on random variability, σ

for different periods. Black circles T = 0 s (PGA), whitecircles T = 0.5 s, black squares T = 1 s, and white trianglesT = 5 s. b Coefficient α4 obtained at the first stage ofthe analysis (minimum σ ; dashed curve) versus period, andsmoothed function used in the second stage of the analysis(continuous curve)

selected for each period. For the next stage of theanalysis these values were smoothed over period(continuous curve).

6 Regression coefficients

Once the smoothed values of α4 were fixed,we generated prior information according to theconsiderations stated before and performed theBayesian regression analysis to compute the finalvalues for the rest of the regression coefficients.The results (in natural logarithm units) are sum-marized in Table 2 and Fig. 5. We have includedin Table 2 the bias (b), the inter-even variability(σ e), and the intra-event variability (σ r). Theselast two values were computed from γ e and σ .

Figure 5 includes a comparison between themean prior values and the mean marginal poste-rior values obtained from the Bayesian analysis.The effect of the information in the dataset can be

clearly observed: final regression coefficients arenot over-constrained to their mean prior values.In addition, we noted that usually the prior valuesare closer to the posterior values in the shortperiod range due to the high frequency approxi-mation implicit in the prior information.

Figure 5f also shows that the proposed modelsystematically tends to overestimate the observedvalues in the whole period range. However, sincethe largest bias is nearly 5% (T = 4.5 and 5 s), weconsider this trend acceptable. Plots of residualsfor the same periods as in Fig. 4 as a functionof magnitude, distance, and depth are shown inFigs. 6, 7, and 8, respectively. There is no sig-nificant trend in the residuals, and similar resultsare obtained for the rest of periods. Despite notincluding focal depth (H) as a parameter in theregression, residuals do not exhibit significant de-pendence on it (only a slight trend can be ob-served at 5 s).

7 Comparisons with other subduction-zonemodels

In the past some ground-motion models to com-pute PGA for subduction-zone earthquakes inthe state of Guerrero have been developed(Anderson and Quaas 1988; Ordaz et al. 1989;Anderson and Lei 1994; Anderson 1997). Also,in the last decades several authors have proposedground-motion models for subduction-zone earth-quakes using worldwide data (e.g., Crouse et al.1988; Crouse 1991; Youngs et al. 1997; Atkinsonand Boore 2003; Kanno et al. 2006; Zhao et al.2006). In this section we compare the estimationsfrom our model with those from regional andworldwide studies, with special emphasis on largeearthquakes at close distances.

Among the regional models we considered thenon-parametric model proposed by Anderson(1997) and the model proposed by Ordaz et al.(1989) [hereinafter referred to as A97 and O89,respectively]. For the case of the worldwidemodels, we chose the relations for interplateearthquakes at generic rock sites (NEHRP B)proposed by Youngs et al. (1997), Atkinson andBoore (2003), and Zhao et al. (2006) [hereinafterreferred to as Y97, AB03, and Z06, respectively].


776 J Seismol (2010) 14:769–785

Table 2 Regressionparameters of theproposed strongground-motion model

T (s) α1(T) α2(T) α3(T) α4(T) γ e b σ σ e σ r

PGA 2.4862 0.9392 0.5061 0.0150 0.3850 −0.0181 0.7500 0.4654 0.58820.040 3.8123 0.8636 0.5578 0.0150 0.3962 −0.0254 0.8228 0.5179 0.63940.045 4.0440 0.8489 0.5645 0.0150 0.3874 −0.0285 0.8429 0.5246 0.65970.050 4.1429 0.8580 0.5725 0.0150 0.3731 −0.0181 0.8512 0.5199 0.67400.055 4.3092 0.8424 0.5765 0.0150 0.3746 0.0004 0.8583 0.5253 0.67880.060 4.3770 0.8458 0.5798 0.0150 0.4192 −0.0120 0.8591 0.5563 0.65470.065 4.5185 0.8273 0.5796 0.0150 0.3888 −0.0226 0.8452 0.5270 0.66070.070 4.4591 0.8394 0.5762 0.0150 0.3872 −0.0346 0.8423 0.5241 0.65940.075 4.5939 0.8313 0.5804 0.0150 0.3775 −0.0241 0.8473 0.5205 0.66850.080 4.4832 0.8541 0.5792 0.0150 0.3737 −0.0241 0.8421 0.5148 0.66640.085 4.5062 0.8481 0.5771 0.0150 0.3757 −0.0138 0.8344 0.5115 0.65930.090 4.4648 0.8536 0.5742 0.0150 0.4031 −0.0248 0.8304 0.5273 0.64150.095 4.3940 0.8580 0.5712 0.0150 0.4097 0.0040 0.8294 0.5309 0.63730.100 4.3391 0.8620 0.5666 0.0150 0.3841 −0.0045 0.8254 0.5116 0.64770.120 4.0505 0.8933 0.5546 0.0150 0.3589 −0.0202 0.7960 0.4768 0.63740.140 3.5599 0.9379 0.5350 0.0150 0.3528 −0.0293 0.7828 0.4650 0.62980.160 3.1311 0.9736 0.5175 0.0150 0.3324 −0.0246 0.7845 0.4523 0.64090.180 2.7012 1.0030 0.4985 0.0150 0.3291 −0.0196 0.7717 0.4427 0.63210.200 2.5485 0.9988 0.4850 0.0150 0.3439 −0.0250 0.7551 0.4428 0.61160.220 2.2699 1.0125 0.4710 0.0150 0.3240 −0.0205 0.7431 0.4229 0.61090.240 1.9130 1.0450 0.4591 0.0150 0.3285 −0.0246 0.7369 0.4223 0.60390.260 1.7181 1.0418 0.4450 0.0150 0.3595 −0.0220 0.7264 0.4356 0.58140.280 1.4039 1.0782 0.4391 0.0150 0.3381 −0.0260 0.7209 0.4191 0.58650.300 1.1080 1.1038 0.4287 0.0150 0.3537 −0.0368 0.7198 0.4281 0.57870.320 1.0652 1.0868 0.4208 0.0150 0.3702 −0.0345 0.7206 0.4384 0.57190.340 0.8319 1.1088 0.4142 0.0150 0.3423 −0.0381 0.7264 0.4250 0.58910.360 0.4965 1.1408 0.4044 0.0150 0.3591 −0.0383 0.7255 0.4348 0.58080.380 0.3173 1.1388 0.3930 0.0150 0.3673 −0.0264 0.7292 0.4419 0.58000.400 0.2735 1.1533 0.4067 0.0134 0.3956 −0.0317 0.7272 0.4574 0.56530.450 0.0990 1.1662 0.4127 0.0117 0.3466 −0.0267 0.7216 0.4249 0.58330.500 −0.0379 1.2206 0.4523 0.0084 0.3519 −0.0338 0.7189 0.4265 0.57880.550 −0.3512 1.2445 0.4493 0.0076 0.3529 −0.0298 0.7095 0.4215 0.57070.600 −0.6897 1.2522 0.4421 0.0067 0.3691 −0.0127 0.7084 0.4304 0.56270.650 −0.6673 1.2995 0.4785 0.0051 0.3361 −0.0192 0.7065 0.4096 0.57560.700 −0.7154 1.3263 0.5068 0.0034 0.3200 −0.0243 0.7070 0.3999 0.58300.750 −0.7015 1.2994 0.5056 0.0029 0.3364 −0.0122 0.7092 0.4113 0.57780.800 −0.8581 1.3205 0.5103 0.0023 0.3164 −0.0337 0.6974 0.3923 0.57660.850 −0.9712 1.3375 0.5201 0.0018 0.3435 −0.0244 0.6906 0.4047 0.55960.900 −1.0970 1.3532 0.5278 0.0012 0.3306 −0.0275 0.6923 0.3980 0.56650.950 −1.2346 1.3687 0.5345 0.0007 0.3264 −0.0306 0.6863 0.3921 0.56321.000 −1.2600 1.3652 0.5426 0.0001 0.3194 −0.0183 0.6798 0.3842 0.56081.100 −1.7687 1.4146 0.5342 0.0001 0.3336 −0.0229 0.6701 0.3871 0.54711.200 −2.1339 1.4417 0.5263 0.0001 0.3445 −0.0232 0.6697 0.3931 0.54221.300 −2.4122 1.4577 0.5201 0.0001 0.3355 −0.0231 0.6801 0.3939 0.55441.400 −2.5442 1.4618 0.5242 0.0001 0.3759 −0.0039 0.6763 0.4146 0.53431.500 −2.8509 1.4920 0.5220 0.0001 0.3780 −0.0122 0.6765 0.4159 0.53351.600 −3.0887 1.5157 0.5215 0.0001 0.3937 −0.0204 0.6674 0.4187 0.51971.700 −3.4884 1.5750 0.5261 0.0001 0.4130 −0.0208 0.6480 0.4164 0.49651.800 −3.7195 1.5966 0.5255 0.0001 0.3967 −0.0196 0.6327 0.3985 0.4914

Though the database of AB03 comprises thewhole dataset of Y97 and many more records, weselected both formulations since they use different

approaches to model near-source saturation. Bothdatasets contain several Mexican earthquakes, butonly a small fraction of them are included in our


J Seismol (2010) 14:769–785 777

Table 2 (continued) T (s) α1(T) α2(T) α3(T) α4(T) γ e b σ σ e σ r

1.900 −4.0141 1.6162 0.5187 0.0001 0.4248 −0.0107 0.6231 0.4062 0.47262.000 −4.1908 1.6314 0.5199 0.0001 0.3967 −0.0133 0.6078 0.3828 0.47212.500 −5.1104 1.7269 0.5277 0.0001 0.4302 −0.0192 0.6001 0.3936 0.45303.000 −5.5926 1.7515 0.5298 0.0001 0.4735 −0.0319 0.6029 0.4148 0.43753.500 −6.1202 1.8077 0.5402 0.0001 0.4848 −0.0277 0.6137 0.4273 0.44054.000 −6.5318 1.8353 0.5394 0.0001 0.5020 −0.0368 0.6201 0.4393 0.43764.500 −6.9744 1.8685 0.5328 0.0001 0.5085 −0.0539 0.6419 0.4577 0.45005.000 −7.1389 1.8721 0.5376 0.0001 0.5592 −0.0534 0.6701 0.5011 0.4449

database. The reason is twofold: (1) some of theMexican events classified as ‘interplate’ in Y97and AB03 sets are actually reverse-faulting inslabevents (H 30–45 km) mislocated by global cat-

alogues, thus having considerably higher stress-drops than interplate events and (2) some of therecords used by those authors do not satisfy ourquality criteria. On the other hand, the database

Fig. 5 Prior values (whitedots) and final valuesobtained from Bayesianregression analysis (blackdots) of a coefficient α1,b coefficient α2,c coefficient α3, d randomvariability, σ ; e parameterγ e, and f bias, b , for theproposed model versusperiod. All values are innatural logarithmic units(Eq. 8a)


778 J Seismol (2010) 14:769–785

Fig. 6 Residuals (innatural logarithmic units)for the model as afunction of distance, R,for periods: a 0 s (PGA),b 0.5 s, c 1 s, and d 5 s.Continuous line showslinear regression ofresiduals

of Z06 is mainly comprised of Japanese strong-ground-motion records. Although the Z06 data-base contains a small number of near-source (R <

40 km) records of worldwide earthquakes from

shallow crustal active regions, its authors did notinclude records of Mexican earthquakes becauseof the differences in the subduction characteristicsbetween Mexico and Japan (Kanamori 1986).

Fig. 7 Residuals (innatural logarithmic units)for the model as afunction of magnitude,Mw, for periods: a 0 s(PGA); b 0.5 s; c 1 s, andd 5 s. Continuous lineshows linear regression ofresiduals


J Seismol (2010) 14:769–785 779

Fig. 8 Residuals (innatural logarithmic units)for the model as afunction of depth, H, forperiods: a 0 s (PGA),b 0.5 s, c 1 s, and d 5 s.Continuous line showslinear regression ofresiduals

In Fig. 9 we compare the scaling of SA withmagnitude in the near-source region from ourmodel and those by Y97, AB03, and Z06. For

PGA we include the curve derived from a finite-source model (Eq. 4a). Also plotted are all ourdata at distances shorter than 25 km. Note that

Fig. 9 Scaling of SA withMw for distances close to20 km. Curves correspondto our proposed modelfor Mexico (continuous),models for Guerrero(only for PGA) by A97(dashed and dotted) andO89 (medium dotted),Japanese model by Z06(line and double dotted),and worldwide models byY97 (dotted) and AB03(short dashed). Opencircles Mexican data atdistances shorter than25 km (R 17.3–24.7 km;average of 20.8 km). ForPGA (frame a) largedashed curve representsthe estimation from afinite-source model with�σ = 100 bars, κ = 0.023,and Q0 = 100 s (Eq. 4b)


780 J Seismol (2010) 14:769–785

only 16 records are available at this range, andonly five of them correspond to large earthquakes(three events of Mw > 7.0). Thus, any anom-alous characteristic on these large earthquakesmay strongly influence the regression results.

There are significant differences between themodels. In particular, for large magnitudes ourmodel predicts considerably larger motions thanthose estimated by A97, Y97, and AB03. For ex-ample, for an Mw 8.0 event at 20 km the estimated

Fig. 10 Observed (opencircles) and estimated SA(curves; same symbols asin Fig. 9) as a function ofdistance for the sameperiods as in Fig. 9and magnitudes Mw6.0 and 8.0


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PGA values are roughly 1.5 and three times largerthan those by Y97 and AB03, respectively. Wenote that AB03 model is in closer agreement withthe observed values during the 1985 MichoacanMw 8.0 earthquake than the other models andwith the non-parametric A97 model. Therefore,according to AB03 and A97 models Michoacanvalues are representative of the average valuesof SA for Mw 8.0. However, this assumption isquestionable, since there is a consensus on theanomalously low SA values observed during thatearthquake (e.g., Cohee et al. 1991; Crouse 1991).Moreover, the PGA values predicted by the A97model start to decrease for Mw larger than 6.Also, we note that the O89 model yields largervalues of PGA than the other models especiallyfor Mw larger than 7. Interestingly, our model issimilar to the Z06 model, even more similar thanthe regional models for Guerrero, despite the Z06model was developed with a completely differentdatabase and regression technique. For this dis-tance range the maximum differences betweenour model and the Z06 model for PGA and SAat 0.1 s, 1 s, and 3 s are roughly 10%, 20%, 15%,and 35%, respectively.

Figure 10 compares the decay of SA with dis-tance for Mw 6.0 and 8.0. The adopted functionalform is based on the theory for far-field bodywaves, so the question if the ground motion modelis able to properly describe the decay of groundmotion for long distances naturally arises for dis-tances where the influence of surface waves mightbe important. However, Fig. 10 suggests that, onaverage, the proposed model seems to reproducethe attenuation of Mexican data better than othermodels. Differences in attenuation are more pro-nounced for large earthquakes, where our modeltends to predict a faster decay, especially for shortperiods. Furthermore, the decay of ground motionwith distance for our model is similar to the decayproposed by Z06, despite the shorter distancerange considered in the latter model. However,our model clearly departs from the data for Mw

8.0 at 3 s of period, where it predicts larger valuesof SA than the observed ones. In the case ofthe regional models for Guerrero, we note thatthe attenuation of our model is comparable to theobserved in the O89 model and it decays fasterthan the A97 model. Nevertheless, the A97 model

seems physically unacceptable since in some casesthe predicted values increase with distance.

8 Discussion

The differences observed in the previous sectionmay be due to differences in the datasets, theregression technique, and the adopted functionalform. Although there are few coincidences in thedata used by each model, for the most interest-ing range (large magnitudes at close distances)all databases are nearly equal, except for theZ06 model, mainly derived from Japanese data.However, it is precisely at this range where ourmodel significantly differs from the Y97 and AB03models, and where the Z06 model and the pro-posed model are more similar. Closer inspectionof random variability and residuals of the mod-els does not reveal any noticeable difference be-tween them. Therefore, from a statistical point ofview, we cannot determine which model fits databetter.

Certainly, more data than currently availableare needed to define statistically the mean valueof SA in the near-field of large Mexican interplateearthquakes. In spite of this limitation, to gainfurther insight into this question we examined thescaling of PGA with Mw at close distances in abroader range of magnitudes than that used by thestudies considered. For this purpose we included115 PGA values obtained from Mexican interplateevents with Mw 2.5–4.9 recorded at hypocentraldistances between 16 and 37 km (Gaite 2005). AllPGA values were reduced to a common distanceof 16 km by correcting them with the factor (Singhet al. 1989):

fR =(

R16

)eπ (R−16)

β Q0 (9)

where the values of β and Q0 have been previ-ously defined.

The reduced PGA values and the estimationsfrom the considered models are shown in Fig. 11.For magnitudes where enough data are available,there is a large scatter in the data, spanning frommore than an order of magnitude. Therefore, sta-tistical significance of the scarce data from large


782 J Seismol (2010) 14:769–785

Fig. 11 Near-source PGA data (open circles) from Mexi-can interplate earthquakes reduced to a distance of 16 km(see text for details) as a function of Mw. The differentcurves (same symbols as in Fig. 9) represent the estimationsfrom O89, A97, Y97, AB03, and Z06 models, and this study

events is very low. However, the following obser-vations can be drawn:

A97 and AB03 models tend to systematicallyunderestimate Mexican near-source PGA, evenfor the magnitude range considered by the models(Mw 5.0–8.0). For Mw < 5 our model is closer tothe mean value of the data than O89, Y97, andZ06 models, though those data were not includedin the regression. For Mw > 5 it is difficult todiscern if O89, Y97, Z06, or our model gives morerational estimates of near-source PGA.

The final goal of deriving strong ground-motionrelations is the computation of seismic hazardfor specific sites. It is interesting to explore howdifferences between models are translated intoprobabilistic-hazard estimates for the city of Aca-pulco. Acapulco is a medium-sized (ca. 1 millionpeople) Pacific coastal city located in the middleof a seismic gap where an Mw 7.5–8.2 earthquakeis expected in the near future (e.g., Singh et al.1981). Station ACAP, placed on rock in Aca-pulco (Fig. 1), has been continuously recording fornearly 40 years.

Following Ordaz and Reyes (1999), we com-puted empirical seismic hazard curves for ACAPstation by counting the number of times that acertain value of SA was exceeded per unit oftime from the available data due to interplateearthquakes. We compare the resulting empiricalcurves with hazard curves obtained through prob-abilistic seismic hazard analysis (PSHA) basedon seismicity information of interplate-subductionsources located in the Pacific coast of Mexico.These hazard curves were computed separatelyfor the Y97, AB03, and Z06 models, as well as forthe proposed model.

For PSHA the seismicity of the Pacific coastof Mexico was divided into two groups. The first,which is responsible for events with Mw < 7,has magnitude exceedance rates that follow themodified Gutenberg-Richter model. For the sec-ond group, responsible for events with Mw > 7,we used a Gaussian distribution of magnitudesto take into account the characteristic earthquakebehavior observed by Singh et al. (1983) for theMexican subduction zone. The first group wassubdivided into four sources, while the secondgroup was divided into 14 sources. For each sourcethe parameters of the seismicity models were ob-tained from the earthquake catalog of Mexicanearthquakes prepared by Zúñiga and Guzmán(1994), with the use of the Bayesian approachdescribed in Rosenblueth and Ordaz (1987). Thegeometry of the seismic sources and their parame-ters can be found in Ordaz and Reyes (1999). Inaddition, we assume that the conditional proba-bility density function of SA given M and R is alognormal density function with median value andstandard deviation of the natural logarithm of SAgiven by the mean value and random variability ofthe considered ground motion model. It is impor-tant to underline that the purpose of this analysisis to assess the impact of the observed differencesbetween ground-motion models in the frameworkof PSHA.

Figure 12 summarizes the results of this com-parison. The analytical curve derived from thisstudy and the curve based on the Z06 modelconstitute better approximations to the empiricalcurve than those obtained from Y97 and AB03models. However, these results should be viewedwith caution, since for the observation time of


J Seismol (2010) 14:769–785 783

Fig. 12 Seismic hazardcurves for ACAP site inAcapulco computed fromdifferent strong-motionrelations (same symbolsas in Fig. 9). Open circlesare empirical datacomputed from records atACAP

ACAP (∼40 years) the smaller exceedance ratethat could be reasonably estimated with the em-pirical data is ∼0.25 year−1 (Beauval et al. 2008).Nevertheless, for small exceedance rates the haz-ard curves obtained with the proposed model andY97 model become similar.

The trends presented in Fig. 12 are only validfor ACAP station, and may not be necessarilyappropriate for other rock sites in the forearcregion. Unfortunately, there are no more avail-able stations in the area with such a long timewindow of data to compute other empirical hazardcurves.

From the results presented in this section weconsider that the proposed model is more suitablefor Mexican interplate earthquakes than othermodels based on regional and worldwide data. Onthe other hand, the use of this model in othersubduction-zones may not be adequate, especiallyfor interplate events deeper than 30 km.

9 Conclusions

We have developed SA strong ground-motionrelations for Mexican interplate earthquakes


784 J Seismol (2010) 14:769–785

through a Bayesian regression technique in whichthe functional form has been obtained fromthe analytical solution of a circular finite-sourcemodel. These equations are valid for rock sites(NEHRP B) located in the forearc region. Theresulting model shows that amplitudes from largeearthquakes in Mexico decay faster in compari-son with estimates from models based on world-wide datasets, especially at high frequencies. Ourmodel also predicts ground motions for largeearthquakes at close distances that are larger thanthose expected from worldwide studies. Due tothe paucity of data in this range, it is difficult toelucidate which model represents the best choicefor this scenario. However, near-source PGA scal-ing with magnitude and probabilistic seismic haz-ard analysis suggest that the model derived in thisstudy is a more suitable approximation for theavailable Mexican near-field data.

Acknowledgements We thank Miguel Herraiz andBeatriz Gaite for long and fruitful discussions. Figures 1and 3 were created with the GMT free software (Wesseland Smith 1998). D. García was supported by PredoctoralFellowships Program from Universidad Complutense deMadrid, Spain, and Postdoctoral Fellowships Programfrom DGAPA, UNAM.

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