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Bulletin of the Seismological Society of America, Vol. 95, No. 6, pp. 22592271, December 2005, doi: 10.1785/0120040145

Stochastic Modeling of the 30 September 1999 Mw7.5 Earthquake,

Oaxaca, Mexico

by Raul R. Castro and Euclides Ruz-Cruz

Abstract We used the stochastic method proposed by Beresnev and Atkinson(1997, 1998a) for finite faults to model the 30 September 1999 Mw 7.5 Oaxaca,

Mexico earthquake. This large intraplate event was located close to the coast and

caused important damage in the state of Oaxaca. We modeled acceleration records

from 10 strong-motion stations located near the rupture and at regional distances.

The site response of the stations used was determined using more than 100 additional

records from other events recorded at the sites of interest. We estimated average

spectral ratios between horizontal and vertical components of ground motion ( HVSR

method), and we incorporated the site response estimates in the stochastic simula-

tions. We also analyzed the decay of the observed spectral amplitudes with hypo-

central distance and estimated the attenuation relation to be Qs 416.5 f0.7. The

main event had a normal-faulting mechanism with a fault plane 90 km long and 45km wide. We divided the fault plane into 9 5 subfaults to apply the point-source

formalism. Specific slip weights were prescribed on the individual subfaults using

the slip distribution obtained by Hernandez et al.(2001). Then, we looked for values

of the radiation-strength factor (sfact) and the stress parameter (r) that gave the

minimum model bias of the acceleration response spectra. We found that sfact 1

and r 90 bars provided the best fit to the observed response spectra and peak

ground acceleration (PGA). These results will be useful to estimate the regional PGA

generated by earthquakes with similar source characteristics as the 30 September

1999 event.

Introduction

The Oaxaca, Mexico, earthquake of 30 September 1999

(Mw 7.5) was located below the coast at a depth of 40 km

(16.00N, 97.02E), near the aftershock area of the 29

November 1978 (Mw7.7) thrust event (Singh et al., 2000).

The Oaxaca 1999 event was a normal-faulting intraplate

earthquake. The Harvard centroid moment tensor (CMT) fo-

cal mechanism solution gives a nodal plane dipping north-

east that can be considered as the fault plane (dip, 50; strike,

295; rake, 82). Based on local and regional data, Singh

et al. (2000) suggested that the rupture propagated toward

the northwest. They estimated a total rupture duration of

14.5 sec and a seismic moment M0of 2.0 1027 dyne cm.

Hernandezet al. (2001) also studied this earthquake using

near-source strong-motion records. They determined the slip

distribution on the fault and found that the rupture propa-

gated from east-southeast to west-northwest with an average

rupture velocity of 3 km/sec.

In the epicentral area, near the coast, intensities accord-

ing to the Mercalli Modified scale (MM) of VIIVIII were

reported in San Pedro Mixtepec and Puerto Escondido. In

the city of Oaxaca, intensities of VIVII were reported.

Models to simulate the ground motion generated by intra-

plate events in the subducted slab are important because

these earthquakes tend to cause more damage than other

events with comparable magnitude. For instance, Singh et

al.(1980) found that for earthquakes of the same magnitude,

the area experiencing MM intensity VI is about four times

greater for intraslab events than for interplate earthquakes.

They observed that in Mexico, interplate events show larger

attenuation than intraplate events. They suggested, as a pos-

sible explanation, the higher scattering and viscous losses

from interplate earthquakes, which occur near the trench of

the subduction zone.

In this article, we use strong-motion records from 10

stations azimuthally well distributed to find the necessary

source and propagation parameters for the stochastic model

proposed by Beresnev and Atkinson (1997, 1998a, 1998b).

Data

Figure 1 shows the distribution of the 10 stations se-

lected for the analyses and the location of the earthquake


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2260 R. R. Castro and E. Ruz-Cruz

Figure1. Epicentral location of the 30 September, 1999 (Mw 7.5) earthquake anddistribution of stations used. The focal mechanism and the rupture area are also shown.At the bottom of the figure a section along AA (modified from Sing et al. 2000) isshown.


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Stochastic Modeling of the 30 September 1999 Mw 7.5 Earthquake, Oaxaca, Mexico 2261

Table 1Station Coordinates (from the Mexican Strong Motion Database, 2000)

Stat. Code Name

Lat N

(Deg)

Long W

(Deg)

r

(km)

PGA

(cm/sec2)

Mean PGA

(cm/sec2)

fmax

(Hz)

Instrument

Type

LANE Las Negras 15.948 97.187 43.1 251.8 245.4 7.94 Altus-Etna

RIOG Ro Grande 16.014 97.439 59.5 307.5 299.0 10.0 Altus-Etna

HUIG Huatulco 15.768 96.108 108.3 146.6 124.9 20.0 Q880; FBA-23

PNIG Pinotepa 16.392 98.127 133.1 40.44 38.0 25.0 Q680; FBA-23

OXIG Oaxaca 17.072 96.733 134.2 196.5 154.1 10.0 Q680; FBA-23

VIGA Las Vigas 16.757 99.236 255.1 68.3 60.1 10.0 Altus-Etna

TEMC Temascal 18.228 96.415 263.5 96.6 73.5 10.0 DCA-333

MEZC Mezcala 17.930 99.591 352.2 12.3 11.1 5.0 Altus-Etna

YAIG Yautepec 18.862 99.067 390.2 17.5 15.7 10.0 Q680; FBA-23

PENB Penitas 17.433 93.450 417.5 9.4 8.5 6.5 DCA-333

epicenter. All of the available records were taken from the

Mexican Strong Motion Database (2000) (MSMD), except

those from LANE and RIOG, which were provided by the

Instituto de Ingenieria, Universidad Nacional Autonoma de

Mexico (UNAM). We selected records from stations located

on free-field sites with clear P- and S-wave arrivals and a

signal-to-noise ratio greater than 1.64 dB. Table 1 lists the

coordinates and site characteristics of these stations. Figure1 also shows the rupture area (gridded rectangle), the focal

mechanism, and a cross section showing the location of the

hypocenter with respect to the trench, the coast, and the city

of Oaxaca. This normal fault event was located at about

40 km depth (Hernandezet al., 2001) inside the Cocos plate.

The acceleration records were baseline corrected, and

then we selected time windows starting a few seconds before

theS-wave arrivals and extending until the window reaches

90% of the energy of the record. We calculated the 5%

damping response acceleration spectra for the selected win-

dows, and we use these as the observation input to model

the 30 September 1999 earthquake.

The hypocentral distances of the records range from43.1 to 417.5 km, and the average horizontal peak ground

acceleration (PGA) ranges from 8.5 to 299.0 Gal. All the

stations used are free-field sites, and most are located on hard

rock. For this reason, previous studies (Singh et al., 2000;

Hernandez et al., 2001) neglected the near-site amplification.

Nevertheless, we estimated the seismic site response of the

10 stations using horizontal-to-vertical spectral ratios

(HVSR).

Site Response Estimates

In general, site amplification is expected to occur at sta-

tions located on soft soils or sediments; however, importantamplifications have also been observed at rock sites (e.g.,

Tuckeret al., 1984; Castro et al., 1990; Humphrey and An-

derson, 1992). Although the stations selected for this study

are located on rock (see Table 1), we estimated the site re-

sponse of the 10 stations selected for the modeling using the

HVSRtechnique.

Site amplification has been estimated using HVSR of

earthquake records in the past (Lermo and Chavez-Garca,

1993; Field and Jacob, 1995; Castro et al., 1996, among

others), and the technique continues to be useful for analyz-

ing the seismic response of recording sites to S-wave inci-

dence. The fundamental assumption of the HVSRmethod is

that site amplification on the vertical component of motion

can be neglected. This assumption has recently become con-troversial because the presence of surface waves may induce

vertical amplification (Castroet al., 1997, 2004; Bindiet al.,

2004) and because S-to-P conversions can invalidate that

assumption (e.g., Bonillaet al., 2002). However, these prob-

lems can be overcome if the time window used to calculate

the spectral amplitudes contains mainly Swaves, since the

vertical amplification is minimized.

We selected all of the available records reported by the

MSMD. Most of the events used have hypocentral distances

equal or less than that from the 30 September event at the

corresponding site. For stations LANE, RIOG, HUIG, and

OXIG, we also included events with greater hypocentral dis-

tances to increase the number of spectral records and to makethe average HVSR meaningful. Table 2 lists the number of

earthquakes used for each station and the range of magnitude

and hypocentral distance of the events.

We calculated the acceleration spectra of the three com-

ponents of motion using the same windows as for the re-

sponse spectra. The spectra were smoothed using a variable

frequency band of25% of 20 predefined frequencies from

0.25 to 20.0 Hz. The length of the time window used to

calculate the spectra defines the lower limit of the frequency

band useful for further analysis, and the noise level defines

the upper limit. For all records the lower and upper limits

are before 0.25 and after 20 Hz, respectively. The time win-

dows used to calculate the spectra contain 90% of the energyof the records, starting with the first S-wave arrival. A 5%

cosine taper was applied to the beginning and end of the

record section. The length of the windows used varied be-

tween 10 and 32 sec.

To estimate the site amplification Z(f) at each station,

we calculated the arithmetic average of all events:


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M1 N(f) E(f)

Z(f) , (1) M 2V(f)i1 iwhere Mis the number of earthquakes and N(f), E(f), and

V(f) are the spectral amplitudes of the northsouth, east

west, and vertical components, respectively.

Figure 2 shows the amplification functions obtained

(solid lines) and the standard deviation of the average (dotted

lines). Most of the stations show peak amplification of less

than a factor of 3.0. Two sites have significant amplifica-

tions, but in a narrow frequency band: LANE, with a peak

amplification of 8.8 at 0.5 Hz, and TEMC, with amplification

of 8.4 at 3.0 Hz. The frequencies at which the peak ampli-

fication occurs are likely related to the natural frequency of

vibration of the sites.

Simulation Method

Stochastic Model

We used a finite-fault model and the stochastic method

proposed by Beresnev and Atkinson (1997, 1998b) to sim-

ulate the observed records of the 30 September 1999, (Mw7.5) Oaxaca, Mexico, earthquake. In this model the fault

plane is divided into equal rectangular subfaults. Then, the

subfaults are treated as point sources with x2 spectra, and

the contribution of each subfault is summed, assuming that

the rupture propagates radially from the hypocenter. To cal-

ibrate the input parameters of the model, we defined the

bias as

n1 PSA(f)obs

E(f) log , (2) n PSA(f)i1 isimwhere nis the number of stations and PSA(f) are the response

acceleration spectra. The simulated PSA(f)sim was obtained

using the code FINSIM (Beresnev and Atkinson. 1998a).

This code has been validated in diverse tectonic environ-

ments for ground-motion prediction (Hartzell et al., 1999;

Berardiet al., 2000; Castro et al., 2001; Beresnev and At-

kinson. 2002; Hough et al., 2002, among others). We also

defined an average error (e) for the frequency band used as

m1

e |E(f)| , (3) jm j1

wherem is the number of frequencies used to calculate the

average.

Fault Discretization

Singh et al. (2000) estimated the rupture area of the

main event from the aftershock distribution and found a fault

length of 90 km and a width of 45 km. In order to apply the

point source approximation used by the code, we divided the

fault plane into subfaults. We calculated the size of the sub-

faults using the magnitudelength relation obtained by Be-

resnev and Atkinson (1999):

logDl 2 0.4M , (4)

where Dlis the subfault length and M(7.5) is the magnitude.

Assuming that Dl Dw, where Dw is the subfault width,

we divided the rupture area into 9 5 subfaults. Note that

this assumption makes the simulated fault width 5 km

greater than that reported by Singh et al. (2000). However,

this amount is within the expected error of the aftershock

area estimation.

We weighted the contribution of the individual subfaults

to the total seismic moment (M02.0 1027 dyne cm) using

the slip distribution obtained by Hernandez et al. (2001)(Fig. 3). We also used the fault geometry corresponding to

the Harvard CMT solution (dip, 50; strike, 295; rake,

82).

Duration of Time Window

In the FINSIM code, the duration of the subfault time

window (Tw) is represented as the sum of its source duration

(T) and a distance-dependent term (Td)

T T T (r) (5)w d

Ruz-Cruz (2004) analyzed different functions of Td andfound that the relation determined by Atkinson (1995) from

subduction zone earthquakes provides the best approxima-

tion to the durations observed for the 30 September event.

The Radiation-Strength Factors

In the stochastic model, the level of high-frequency ra-

diation is controlled by the radiation-strength factor s, and

this parameter is related to the maximum slip velocity on the

Table 2Ranges of Magnitudes and Hypocentral Distances of Additional

Evens Used to Estimate the Site Response

Station Site Geology

Number of

Ev ents M agn itud es

Hypocentral

Distance (km)

LANE Rock 6 4.45.2 488

RIOG Rock 4 4.55.2 30170

HUIG Rock 11 4.45.2 30142OXIG Diorite 10 3.95.2 100179

PNIG Rock 8 4.15.6 17128

VIGA Rock 52 3.55.0 18255

MEZC Rock 7 4.65.9 124286

YAIG Rock 6 4.65.6 241390

TEMC Shale 7 4.24.9 82263

PENB Rock 4 5.06.5 143417


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Figure2. Site response estimated using horizontal-to-vertical spectral ratios. Thesolid line is the average value, and the dashed line the mean 1 standard deviation.


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Figure3. Slip distribution reported by Hernandezet al. (2001).

fault (Beresnev and Atkinson, 2002). We tested values ofsbetween 0.5 and 2.0.

The Stress Parameter r

The seismic moment of the subfault (m0) is related to

its length Dl (Beresnev and Atkinson, 1997) by

rm . (6)0 3

Dl

Since m0 has major influence on the low-frequency ampli-

tudes, variations in r tend to have a greater effect on the

spectral amplitudes at lower frequencies than factor s. Weperformed simulations varying the stress parameter between

30 and 200 bars.

Cutoff Frequency fmax

The value of fmax for the high-frequency filter (Boore,

1983) used by the simulation code was selected visually

from the observed acceleration spectra. Table 1 lists the val-

ues picked for each station. fmax is the frequency at which

the spectral amplitudes start to abruptly decay.

Attenuation Model

As mentioned before, the strong-motion records of the

30 September 1999 earthquake were recorded at a distance

range of 43418 km. At local distances (r 110 km) only

3 of the 10 stations analyzed recorded the main event

(LANE, RIOG, and HUIG). In this distance range, Castro

and Mungua (1993) found a Qfrequency relation (Q

56f) by using S waves from local earthquakes recorded in

the region of Oaxaca. Similar relations based on coda waves

have been reported for Oaxaca (Acosta-Chang, 1980; Rod-

riguez, 1985), but they are also valid for local distances

(r 100 km). Other studies such as that by Canas et al.

(1988) estimated the attenuation ofLgwaves in the OaxacaChiapas region in the frequency band 0.71.7 Hz. Most re-

cently, Ottemoller et al.(2002) found an average Lgquality

factor for southern Mexico, QLg(f) 204f0.85 in the fre-

quency range 1.68 Hz, using crustal earthquakes recorded

at distances greater than 200 km. This relation is similar to

the S-wave Q-frequency relation obtained by Ordaz and

Singh (1992) from a spectral attenuation study of subduction

zone earthquakes recorded in Guerrero and the Valley of

Mexico, namely Q 273f0.66.


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Figure 4. Vertical component S-wave accelerationspectra of the 30 September, 1999 earthquake. Thesolid line corresponds to station Pinotepa (PNIG), lo-cated at 133.1 km.

In order to find an attenuation relation useful for the

whole distance range and frequency band of the 30 Septem-

ber 1999 data, we used the acceleration spectra to determine

the spectral amplitude decay with hypocentral distance at

each frequency analyzed. To minimize the amplification due

to site effects, we considered only the vertical components

for these estimates. Figure 4 shows the S-wave acceleration

spectra calculated using the vertical components and thesame procedure mentioned before. The spectrum of station

PNIG (r 133.1 km), plotted as a solid line, shows signifi-

cantly low amplitudes at frequencies f 5 Hz. compared

with stations located at even greater hypocentral distance.

We believe this deamplification may be due to either strong

near-site attenuation or a radiation pattern effect.

The observed spectral amplitude U(r, f) at frequency f

and hypocentral distanceris modeled as

pfr/QbU(r, f) S G(r) e , (7)

whereSis a scalar that depends on the source effects, G(r)

is the geometrical spreading, bis the S-wave velocity, andQaccounts for anelasticity effects.

The site effects are neglected in equation (7) because

we use the vertical component of motion. The geometrical

spreading functionG(r) was assumed to have the following

form:

1/r, r 100 kmG(r) . (8)1/2(100 r) , r 100 km

Equation (8) accounts for the expected amplitude decrease

of body waves atr 100 km and for the presence of surface

waves at longer hypocentral distances (Aki and Richards,

1980). Note also that atr 100 km, equation (8) preservescontinuity. This simple functional form of the geometrical

spreading has been useful for estimating Q in other regions

of Mexico (see for instance, Castro et al., 1990; Ordaz and

Singh, 1992).

For a given frequency f, the spectral amplitudes can be

modeled as a function of distance ras follows:

u(r) b m r, (9)

whereu(r) log [U(r,f)/G(r)],b logS, andm pf/bQ

log (e). We used an average shear-wave velocity bof 3.66

km/sec, calculated from the crustal model determined by

Valdes et al. (1986) for Oaxaca, which is similar to the

model by Nava et al. (1988) for the same area. Figure 5

shows the linear fit obtained for a sample of six frequencies.

We measured the slope of the fit (m) to estimate the quality

factorQ. For these estimates, we did not use station PNIG

because of the anomalously low spectral amplitudes ob-

served, particularly at f 5 Hz. However, station OXIG is

at about the same distance and permits a good definition of

the amplitude decay. Figure 6 shows the values estimated

for all frequencies analyzed and the Q-frequency relation

found (Q 416.5 1.1f0.70.1). It is interesting to note

that thisQrelation predicts lower attenuation compared with

Q 56f, obtained by Castro and Munguia (1993) using

interplate earthquakes, consistent with the contrasting atten-

uation of intensity observed by Singh et al.(1980) between

interplate and intraplate earthquakes.

Results

To analyze the effect of different combinations of the

parameterss and r on the fitting error, we constructed the

solution space shown in Figure 7, modeling the 10 stations

in the entire frequency band (0.25 f 20 Hz). First, we

calculated the model bias with equation (2), using the am-

plitude response of all stations, and then we calculated the

average error with equation (3) for all frequencies analyzed

(30 frequencies). It can be seen in Figure 7 that the best

combination corresponds to r 90 bars and s 1.0.

Figure 8 shows the model bias calculated with the pa-

rameters of the final model. Note that the average ratio be-

tween the observed and simulatedPSAis less than 1.3 within

the frequency range 210 Hz, indicating a good agreement

in that frequency band.

The model parameters that provide the best fit between

observed and simulated acceleration response spectra (PSA)

are listed in Table 3. Figure 9 compares the observed (solid

line) and simulated (dashed line) PSA. Stations PNIG and

VIGA, located at about the same backazimuth, seem to be

overestimated, especially at lower frequencies. This may be

a radiation pattern effect not accounted for in the simulation,


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Figure5. Observed spectral amplitudes versus hypocentral distance for six differentfrequencies (0.520 Hz). The amplitudes were corrected for geometrical spreading (seeequation 8 in text). The solid line is the least-squares fit of the data, shown withtriangles.

Figure6. S-wave Q estimates. The left frame shows 1/Q 1 standard deviationand the right frame the Qfrequency relation obtained (solid line). The triangles showthe individual estimates ofQ obtained from functions shown in Figure 5.


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Figure7. Average error calculated for different combinations ofrand radiation strengthfactor (sfact). We used all stations and the entire frequency band (0.2520.0 Hz).

since this correction is assumed to be a constant average

value of 0.55 in the FINSIM code (Beresnev and Atkinson,

1998a). For the other stations, the simulated PSA follows

closely the observed values. Figure 10 shows a sample of

three acceleration time series simulated using the final

model. For comparison, we also plotted the observed north

component accelerogram. The observed records start with

the P arrival and reach the peak acceleration when the S

waves arrive. The simulated acceleration time series shown

in Figure 10 contain only the S-wave arrivals.

The observed average peak acceleration (PGA) and that

resulting from the simulation of all stations are displayed in

Figure 11 as a function of hypocentral distance. The circles

are the arithmetic average of the observed horizontal com-

ponents, and the crosses show the simulated peak accelera-

tion. We also plotted in Figure 11 the regression line ob-

tained by Singh et al. (2000) using data from rock sites that

recorded the 30 September 1999 earthquake, including the

10 stations used in this study. The local stations LANE (at

43.1 km) and RIOG (at 59.5 km) show a discrepancy be-

tween simulated and observed PGA. However, at longer hy-

pocentral distances, the simulated PGA values are close to

either the observed values or the regression function of

Singh et al. (2000). Overall the agreement is satisfactory

given the significant scatter.

Conclusions

We determined the site response of 10 stations to find

a stochastic model to estimate the ground motion generated

by the 30 September 1999 (Mw 7.5) earthquake. Although

most of the stations showed small site amplification, below

a factor of 3.0, two sites had significant amplifications, but

in a narrow frequency band. LANE had a peak amplification

of 8.8 at 0.5 Hz, and TEMC an amplification of 8.4 at 3.0 Hz.

We found an attenuation relation valid for the entire

distance range and frequency band of the data set analyzed,

namely Q 416.5f0.7, with an associated geometrical

spreading model of r1 at r 100 km and r0.5 at

r 100 km. With this relation and dividing the fault rupture

into 9 5 subfaults of 10 km size (Dw Dl), we find a

radiation-strength factor s 1.0 and a stress parameter

r 90 bars. These parameters, obtained from the stochastic

modeling, reproduce reasonably well the observed PSA, par-


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Figure 8. Model bias calculated using equation(2) and the final model parameters listed in Table 3.The solid line is the average, and the dotted lines theaverage 1 standard deviation.

Table 3Final Model Parameters

Parameter Value Reference

Fault orientation Strike, 295;dip, 50

Singh et al.(2000)

Fault dime nsions Length, 90 km,

Width, 50 km

Singh et al.(2000)

This study

Location of rupture

initiation point

16.00 N,

97.02 E

Singh et al.(2000)

Focal depth 40 km Hernandez et al. (2001)

Magnitude MW7.5 Singh et al.(2000)

Seismic moment Mo2.0 1027

dyne cm

Harvard CMT

catalog

Shear-wave velocity

and density

b 3.66 km/sec

q 2.8 g/cm3Valdeset al. (1986)

Q(f) 416.5 f0.7 This study

Geometrical spreading 1/R, R 100 km

1/(100R)1/2,

R 100 km

Number of subfaults 9 5 This study

Subfault corner frequency 0.16 Hz This study

Stress parameter 90 bars This study

Radiation-strength factor 1.0 This study

ticularly at intermediate (100 km) and longer (200

400 km) distances.

Acknowledgments

We thank Leonardo Alcantara from the Institutode Ingeniera, UNAM,

for providing the records from stations LANE and RIOG. One of the au-

thors (E.R.C.) was supported by a scholarship from the National Council

of Science and Technology of Mexico (CONACYT). Luis Inzunza helpedus prepare some figures. The comments and suggestions of the two anon-

ymous referees helped us improve the manuscript. We are grateful for their

careful revision.

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Figure9. Acceleration response spectra (PSA). Solid and dashed lines are observedand simulated PSA, respectively.


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Figure10. Acceleration time series: observed northsouth component on the leftand simulated on the right.

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Centro de Investigacion Cientfica y deEducacion Superior de Ensenada (CICESE)Division Ciencias de la Tierra

Departamento de SismologaKm 107 carretera Tijuana-Ensenada, EnsenadaBaja California 22860, [email protected]

Manuscript received 6 August 2004.

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