Geophys. J. Int.-1994-Ordaz-335-44

Geophys. J . Int. (1994) 117, 335-344

Bayesian attenuation regressions: an application to Mexico City

M. Ordaz,'t2 S. K. S ingh '~~ and A. Arciniega4 'Instituro de Ingenieria, UNA M , Ciudud Universitaria, Coyoacdn 045 10, DF, Mexico 2Cenrro Nacional de Preuencidn de Desustres, Delfiri Madrigal 665, Coyoacdn 04650, D F, Mexico 'Imtituro de Geofisica, UNAM, Ciudad Universitaria, Coyoacan 04510, DF, Mexico 4Centro de Invesfigucih .%mica, Camino a1 Ajusco 203, Tlalpan 14200. DF, Mexico

Accepted 1993 September 14. Received I993 August 25; in original form 1993 March 29

S U M M A R Y We describe the application of a bayesian linear regression technique to the problem of deriving strong-motion attenuation relations. This approach provides a concep- tual framework for the formal incorporation of knowledge about the involved phenomena that comes from sources other than the observed data (prior information, according to the bayesian terminology). The procedure produces numerical solutions that are more stable and rational than those obtained from conventional regression schemes. W e illustrate the use of the proposed technique with the derivation of attenuation laws for the Fourier acceleration spectrum, as a function of magnitude and distance, at a hill-zone station in Mexico City.

Key words: bayesian regression, ground motion, Mexico City.

INTRODUCTION

Historically, Mexico City has frequently been struck by earthquakes, which have produced loss of human lives and severe damage to and collapse of buildings. Ground motions produced by these events have been recorded in the city since the early 1960s. At present, the Mexico City Accelerographic Network is formed by more than 110 three-component digital instruments, operated by three institutions: Instituto de Ingenieria. UNAM, Fundacidn Javier Barros Sierra, and the National Center for Disaster Prevention. The availability of hundreds of accelerograms has given rise to detailed studies concerning local soil amplification. The seismic response of soil has successfully been characterized by empirical transfer functions (Singh et al. 1988) between soft sites and a reference hill-zone station (station CU, located in the University City). Since empirical transfer functions are reasonably well estimated for several 10s of sites within the Valley of Mexico, Fourier acceleration spectra at these sites can be computed for a future earthquake if the Fourier spectrum at the reference station is known. Once the Fourier spectrum at a soft site is known, other measures of seismic intensity, such as the response spectrum, can also be estimated.

The simplest way to predict the Fourier spectrum at the reference station for a given earthquake is to construct attenuation relations using regression techniques and the strong-motion recordings obtained at station CU in the last

30 years. Conventional analyses of this type usually find the unknown parameters of the attenuation relation by carrying out a linear or non-linear least-squares (LS) fit to the observed data. However, there is a major disadvantage in using the LS approach: it is impossible to include properly the enormous amount o f knowledge that we have from seismological theory and from observations of other regions. Suppose, for instance, that Q values for a region have been derived from sources other than strong-motion data. How does this information influence our results when deriving attenuation relations for the same region with strong-motion data?

We present a bayesian estimation technique that allows for inclusion of the knowledge not directly derived from observed data. In accordance with bayesian usage, we shall call this knowledge prior information. We will show how this information, combined by means of Bayes' theorem with that contained in the data, gives more stable and reasonable predictions than those obtained with the LS technique. Bayesian regression is a well-established technique (see, e.g. Broemling 1985); to the authors' knowledge, it was used for the first time in deriving attenuation relations by Veneziano & Heidari (1985). They imposed bayesian constraints on the parameters controlling the scaling of several intensity measures with magnitude. The underlying theory, however, was not discussed. In this paper we present the theory in some detail and discuss how theoretical seismological knowledge can be translated into prior information.

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336 M . Ordaz, S . K . Singh and A . Arciniega

THE LEAST-SQUARES APPROACH

The strong-motion data base for this study consists of 23 accelerograms from Mexican subduction zone earthquakes obtained at station CU since 1965. It includes recordings from the 1985 September 19 (M=8.1) Michoacan earthquake. Relevant information about the earthquakes whose recordings are analysed is given in Table 1. Fig. 1 depicts a map of southern Mexico with epicentre locations. Data cover the magnitude and distance ranges 5.0 < M < 8.1 and 260 < R < 466 km, respectively.

Let A ( f ) be the horizontal north-south acceleration amplitude at station CU for a given frequency f. Consider the following attenuation model

where R is the closest distance to rupture area, R,,, is the average radiation pattern, p is the mass density, p is the shear-wave velocity, = 2 (free-surface amplification), P = 0.707 (partition of energy in two horizontal com- ponents), Q ( f ) is the quality factor and C ( R ) is the geometrical spreading term, given by

Table 1. Earthquakes whose recordings were used in this study

Event No.

1 2 3 4 5 6 7 8 9

10 11 1 2a 13' 14' 15 16 17 18 19 20 21 22 23

Date

Aug 23 1965 Feb 03 1968 Aug 02 1968 Feb01 1976 Jun07 1976 Mar 19 1978 Nov 29 1978 Mar 14 1979 Oct 25 1981 Jun 07 1982 Jun 07 1982 Sep 19 1985 Sep 19 1985 Sep 19 1985 Sep21 1985 Sep21 1985 Apr 30 1986 Feb 08 1988 Apr 25 1989 Oct 08 1989 Jan 13 1990

May 11 1990 May 31 1990

bicentre

Lat (N) 16.28 16.67 16.25 17.15 17.45 16.85 16.00 17.46 17.75 16.35 16.45 18.14 18.14 18.14 17.62 17.62 18.42 17.00 16.00 17.92 16.84 17.15 17.15

Long (w) 96.02 99.39 98.08 100.23 100.65 99.90 96.69 101.46 102.25 98.37 98.54 102.71 102.71 102.71 101.82 101.82 102.99 101.00 99.00 100.21 99.65 100.85 100.85

M 7.8 5.9 7.4 5.6 6.4 6.4 7.8 7.6 7.3 6.9 7.0 8.1 8.1 8.1 7.6 7.6 7.0 5.8 6.9 5.1 5.0 5.3 6.1

Distance (W 466 297 326 282 292 285 414 281 339 304 303 295 295 295 318 318 409 289 304 260 282 295 295

Notes. "Three recordings were obtained in CU during this earthquake, in instruments located in an area of roughly 2 X 2 k m . The first 11 events were recorded with analogue, low-sensitivity instruments. All events are shallow, with reported focal depth of 16 km.

where R, is a reference distance. This form of G ( R ) leaves room for different rates of decaying of amplitude with distance. For instance, a2 = -1 implies attenuation of body waves. Note that in our study, R > 260 km, so surface waves should predominate and the amplitudes should attenuate as (l/R)"2, hence a2= -1/2. In eq. (l), S(f) is assumed to be Brune's (1970) w2 source spectrum:

(3)

where M,, stands for seismic moment and fc is the corner frequency given by Brune's formula:

f, = 4.91 X 106(Au/Mo)"3 (4)

(Mo in dyne-cm, Au, the stress drop, in bars, in km s-I). M,, and M, the moment magnitude, relate through (Kanamori 1977)

M = 4 log Mo - 10.7.

After taking natural logarithms of both sides of eq. (l), it can be approximately linearized so that

In A ( f ) = d f ) + a , ( f )M + %(f) In R + 4 f ) R + E , (6)

where al(f), i = 0, . . . , 3 are the regression parameters, and E is a random error. a&) includes moment and stress-drop related constants (frequency independent), plus some frequency-dependent terms such as site effects, while a,(f) controls the scaling with magnitude. a2, which is assumed to be frequency dependent, controls the geometrical spreading (see eq. 2). a 3 ( f ) is related to Q(f) through

(7)

This model, or variations of it that include terms to remove the singularity at R = 0, or a term in M2, have been extensively used to relate magnitude and distance with peak ground acceleration and velocity, response spectral ordin- ates and Fourier spectra (see, e.g. Esteva & Villaverde 1974; Campbell 1985; Joyner & Boore 1988; Castro, Singh & Mena 1988).

We estimated a = (ao, aI, a2, a,)T, where T stands for transpose, and the standard error, by applying an unconstrained LS fit to our data base. For simplicity, we have temporarily dropped the dependence of at on f from the notation. Results are displayed in Fig. 2, where they are compared with the values obtained using the model of eq. (1) with R,,@ =0.55, p = 2 . 8 g ~ m - ~ , /3=3.5kms-', and Au= 100 bar (Singh et al. 1989). Since the LS solution minimizes the standard error, it certainly approximates the observed spectra. However, the values of some of the parameters are so far away from those expected by theory than one would feel uncomfortable when predicting spectra for future earthquakes. For instance, the geometrical spreading term cu, is positive, with values up to 30. This means that, in the absence of anelastic attenuation, amplitudes would increase as R3". To compensate for this, the computed Q-related term a3 is unrealistic. The LS solution presented in Fig. 1 is clearly unacceptable: some restrictions have to be imposed on the regression. This has been done in the past by fixing one or more parameters, which has resulted, very likely, not from certainty about

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19

n Z W 0 18

0 W 3 4 .- + 17 0 -I

16

15

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Bayesian attenuation regressions

OAXACA

EPICENTRE p ,50 100 km

I I I I I I I I

- - LS solution

337

-

-104 -103 -102 -101 -100 -99 -98 -97 -96 -95 Longitude (OE)

Figure 1. Map of southern Mexico showing the epicentres of the earthquakes used in this study.

.6

- . lo -

4 t

.2 u

-2 b- 1 .o .9

.8

.7

.6

.5

.4

.3

-6 -4 L -

-

- -

-

- -

2.2 2.0 1.8

1.6 1.4

1.2

1 .o .8

-.02

-.04

-.06

-.08

-

-

-

-

a2 40 I

35 t

Figure 2. Parameters cr , ( f ) , i = 0, . . . , 3, and u obtained with an unconstrained least-squares fit (LS: solid lines). They are compared with those computed from the standard attenuation model (SAM: dotted lines). The difference between the LS solution for a;, and the corresponding value of the SAM (middle top frame) can be interpreted as site effects. Note the unrealistic LS values obtained for cr2 (the geometrical spreading term) and cr7 (the anelastic attenuation term). Uncertainty is measured by the standard error for the LS fit.

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338 M . Ordaz, S. K . Singh and A . Arciniega

their values, but to remove the instability that LS solutions usually show. We will present a better alternative to restrict the regression.

THE BAYESIAN REGRESSION MODEL

Consider again eq. (6), from which E, , the j th value of random error E is

E,. = a(, + M, + a, In R, + a,Rj - In A,. (8)

Note that in our case each earthquake produces only one recording, so each value of M and A is associated with a single value of R. We assume that given a and u, the random (independent) errors are normally distributed with zero mean and unknown variance u2. The most important difference between our approach and the classical estimation techniques is that we also regard at, i =0 , . . . , 3, as random variables, in the sense that our state of knowledge about them can be described using probability theory.

We are interested, as a first step, in the probability distributions of a and u, including our initial state of knowledge about them and the information contained in the data. These are the posterior distributions, which can be found from direct application of Bayes' theorem:

p(a, u I c )ap(a , 1 a, 01, (9)

where E = (e l , . . . , €,,)I is the vector of observed errors in n independent observations, p(a, u IE) is the postcrior joint density of a and u, p(a, a) is the prior joint density function of these variables, and L(E 1 a, IT) is the likelihood of the observed values of E as a function of a and u. The missing proportionality constant in eq. (9) is such that the integral of the right-hand side is one. An analytical expression for ~ ( r I a, a) foIIows from our assumption of normality of E 1 a and from the hypothesis of independent observations:

h L(E I a, a) = h"" exp ( - E ~ E ) , (10)

where h = l /u2 is the precision. For the sake of mathematical simplicity, in what follows we deal with h rather than with u.

The prior joint density p(a, h ) reflects our state of knowledge about the unknown parameters before our examination of the data. This function should include all we know about these parameters, except for information present in the observed values of A , M and R. Since historically prior information has been the most controver- sial issue of bayesian theory, we discuss this function in detail later. To describe our prior state of knowledge, we adopt a prior density function that is said to be a natural conjugate of the process. These densities have the property that the functional form of the posterior density function is the same as that of the prior, and only its parameters are updated in view of the data. In this case, the natural conjugate for a and h is a multivariate normal-gamma density function (Broemling 1985). This density, as shown in Appendix A, is completely defined by parameters a', R', r ' , and A', which relate t o the statistical moments of a and h through: E(a) =a', COV(a) = A'R-'/(r' - l ) , E ( h ) = r ' / A ' , and c2(h) = l /r' , where E(.) . denotes expectation, COV(.) stands for covariance matrix, and c ( . ) for coefficient

of variation. Application of Bayes' theorem yields the posterior expectations and second-order moments of a and h (see Appendix A):

E ( a I E) = (R' + XTX)-'(R'a' + XTy)

COV(a 1 E) = __ (R' + X"'X)- I

(11)

(12) A"

r" - 1

Efh I E) = ,"/A"

c2(h I E) = l / r "

where

n

2 r" = r r + -

,I" + A' + $[a'TR'a' - a"7 R a " + yTy]

y = (In A , , In A,, . . . , In A,,)" (I 1 M, 1' In.& InR, R,

X =

1 M,, In R" R,, Note from eq. (11) that if R'=O, then E(a 1 E ) =

(XTX)-'X'y equals the LS solution. To assign R' a null value amounts to disregarding all prior knowledge, because it implies that COV(a) is infinite. We can interpret terms in eq. (11) additional to the LS solution as the 'bayesian corrections' that account for prior information. These terms, as will be shown later, make the numerical process very stable, and resemble the damping terms that are sometimes added to the appropriate matrices for this purpose.

We have so far obtained the posterior distributions of a and h. We are, however, interested in the distribution of InA given the prior information, the observed data, and a new set of variables MI and R , , that define the earthquake for which ground motion needs to be predicted. Let zT = (1, M I , In R I , R,). Application of Bayes' theorem (see Appendix A) results in In A having Student's t distribution with

E(ln A I E, 2) = z T E ( a I E) (19)

and

r (r" - 1)[1 - z ' ~ ( R ' + XTX + Z Z ~ ) - ' Z ] '

VAR (In A I E , z ) =

(20)

where VAR( ) denotes variance.

APPLICATION TO MEXICO CITY

Prior information

In what follows we present our reasoning to arrive at the values we used to describe our prior state of knowledge about the unknown parameters.

For E[a( , ( f ) ] and E [ a l ( f ) ] we took directly the values obtained from linearization of the attenuation model of eqs (1)-(4) , with Re, = 0.55, p = 2.8 gcm-', p = 3.5 km s-' and A u = 100 bar. This implies that a priori we believe that

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Bayesian attenuation regressions 339

real values of a&) and a,(f) are likely to be close to those predicted by Brune’s model. Although site amplifications at CU are well documented (Ordaz & Singh 1992), we have ignored them at this point because they were quantified using many of the strong-motion recordings we are analysing, and the information contained in the data must not be used to fix the prior densities as well. Note that the large site amplifications at CU make almost useless the spectral attenuation relations derived for other sites.

We must also assign prior uncertainties to ag(f) and al(f). This quantity controls the scaling with magnitude, and is better constrained by theory than cuO(f), which includes site effects. Thus, we should assign cul(f) a smaller prior variance than to cu,,(f). It is very unlikely that we should find a region in which cu,(f) departed more than 0.3 from the value predicted from Brune’s model; it is almost certain that most seismologists would regard as suspicious an attenuation law in which, for instance, scaled as exp (1.1M) or exp (0 .5M) . We adopted as the most likely region for each parameter the one that includes 90 per cent of the total area under the prior density function, that is, the mean plus minus 1.7 times the standard deviation. For this reason, we took SD[a,(f)] = 0.3/1.7 = 0.18, where SD denotes standard deviation.

For q , ( f ) , our prior state of knowledge is vaguer. We believe that values predicted by eqs (1)-(4) can not be in error by more than a factor of 20 (e’) either way. W e choose this number because we know of at least one case (the lake-bed sites in Mexico City) where waves are amplified much more than 20 times with respect to reasonable values. We therefore adopted SD[cu,,(f)] = 3/1.7 = 1.8, which implies a very vague knowledge of a(,(f).

Parameter aZ( f ) is also relatively well constrained by theory. One possibility would be to make cuZ(f) a deterministic quantity (hence E [ a&”)] = -0.5, S D [ a 2 ( f ) ] = 0 ) . This does not describe our prior knowledge, because this is the way surface waves should attenuate, not the way they actually attenuate. What we believe is that for our range of distances, a2(f) should not be far from -0.5, and this is the information we must describe using a probability density function. Non-negative values of a 2 ( f ) are untenable on physical grounds. If we dismiss positive values for a#), then a Gaussian distribution is inadequate to describe our beliefs about this parameter, unless, for the sake of simplicity, one assigns very small prior probabilities to the range a&) 2 0. In order to keep mathematics simple, we adopted E[a , ( f ) ] = -0.5 and SD[a2( f ) ] = 0.5/1.7 = 0.29. We leave room for a2(f) to be greater than 0 or smaller than -1.0, but with small prior probabilities.

For a3(f) , we have a valuable source of information in previous studies of Q values for our region of interest carried out using strong-motion recordings at sites different from the one we are examining (Castro, Anderson & Singh 1990; Ordaz & Singh 1992). We shall adopt for E[a , ( f ) ] the values that can be inferred from the Q values reported by Ordaz & Singh (1992). According to these authors, Q(f) = 273f”-(*, so

(21) nf -34 q a 3 ( f ) l = - ~ = -3.29 x 1 0 - 3 p 3 4 . 2738

These Q values can not be considered totally correct

since, while they include regional information, they fail t o include the attenuation characteristics of the crust in the vicinity of station CU. However, in view of the values that Q takes in other regions of the world with similar geological conditions, we think that the reported values could not be in error by more than 100 per cent. We then assigned

Parameter h = l/a2 must also be assigned a prior probability distribution. Inspection of attenuation relations derived for other regions shows that typical values of u-the standard deviation of InA-are usually of the order of 0.7 (note that we are using natural logarithms; in the decimal scale, u is about 0.3). We adopted this value as E ( a ) , and decided that we would consider values outside the range (0.1, 1.3) as unlikely, in view of which we set SD(o)= 0.6/1.7 = 0.35. These moments are associated with r’ = 2.05, A ’ = 0.647 (see Appendix A).

In order to define completely the prior joint distribution of a and a, we must fix the entire matrix R’. This amounts to fixing a correlation scheme between pairs of as. Our prior knowledge does not allow us t o estimate second-order crossed moments, so we set all the prior correlation coefficients to 0.

Note that the procedure employed to derive E ( a ) is much the same we would have carried out to derive attenuation relations for a site where no instrumental information was available.

S D [ ~ ~ ) I = - ~ [ a ~ ( f ) 1 / 1 . 7 .

Posterior values

Application of eqs (11)-(18) yields the posterior expected values that are shown in Fig. 3, where they are compared with their respective prior expectations. In Table 2 we give the prior and posterior expectations of the as for some selected frequencies, and we compare them with the values obtained with the LS fit. In Fig. 4 we compare the observed spectra of some of the earthquakes used in the study with the median values predicted by the bayesian attenuation relation. We show the four earthquakes for which the prediction is in better agreement with the observation, along with the four worst predictions. We also computed median A ( f ) for M = 8 and different distances; these results are depicted in Fig. 5 , where we also show the predicted median spectrum for a hypothetical M = 8.2 earthquake that would take place in the Guerrero gap (see Fig. l), with R = 280 km.

DISCUSSION

The computed attenuation relations

Note from Figs 2 and 3 that, regardless of the statistical analysis technique, uncertainties at low frequencies are substantially higher than at high frequencies. This is due, in part, to the fact that the older accelerograms (see Table 1) were recorded with low-sensitivity analogue instruments, SO

the quality of data at longer periods is not very good. But perhaps the main reason is that, although all earthquakes have a reported depth of about 16 km, slight differences in source depth might excite different modes of surface waves. This is of great importance for the prediction of future

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340 M. Ordaz, S. K . Singh and A. Arciniega

1.0 .9 .a .7 .6 .5 .4 .3

54

52

50

48

46

44

42

- - -

- - - -

-

., __ Boyes E(a,)=-0.5 - - - - Prior values I - - Boyes E(a2)=-l .O

, I

I

3.0

2.0 1.5 1 .o

.002

.ooo -.002 -

-.004 -

-.006 -

-

-

-.ooa -

lo-' 100 10' 10-1 100 10' -1.2- -.010 I I 1 1 I l l l l I I 1 1 1 1 1

Frequency ( H z ) Frequency (Hz)

.u lo-' 100 10'

4 . 2 ' I I I ' l ( I I I

lo-' 100 10' Frequency (Hz)

Figure 3. Solid lines: parameters obtained with the bayesian regression technique, assuming E( (yz) = -0.5. Dashed lines: bayesian solutions with E(cu,) = -1.0. Dotted lines: values of the parameters as predicted by the standard attenuation model described in the text. These values are taken as prior expectations in the bayesian analyses.

Table 2. Expected prior and posterior values of a;, i = 0, . . . , 3 computed using the bayesian regression technique (BAY) with E((Yz) = -0.5, for some selected frequencies. They are compared with the corresponding parameters computed with the unconstrained least-squares (LS) fit.

Frequency (Hz)

(Yo Prior cl, BAY a. LS

0, Prior Q1, BAY a, LS

a, Prior

a, L S BAY

CU, Prior

a3 LS Q13 BAY

5.0OD+OO 1.67D+OO 1 .OOD+OO 7.14D-01 5.56D-01 4.550-01 3.85D-01 3.330-01 2.940-01 2.630-01

4.861)+01 4.78D+01 4.67D+01 4.560+01 4.460+01 4.37D+01 4.290+01 4.22D+01 4.16D+01 4.llD+O1 4.991)+01 4.890+01 4.83D+O1 4.77D+01 4.640+01 4.580+01 4.55D+O1 4.480+01 4.35D+O1 4.300+01 4.590+01 4.66D+01 4.71D+01 4.71D+01 4.460+01 4.59D+01 4.45D+O1 4.54D+01 4.08D+01 3.82D+01

?.17D+OO 1.27D+OO 1.41D+00 1.55D+00 1.67D+OO 1.78~+00 1.88D+OO 1.96D+OO 2.04D+00 2.flD+00 9.77D-01 1.25D+00 1.39D+OO 1.50D+00 1.660+00 1.72D+00 1.76D+00 1.81D+00 1.970+00 2.01D+00 5.82D-01 1.13D+O0 1.25D+00 1.33D+00 1.490+00 1.52D+00 1.47D+OO 1.490+00 1.66D+OO 1.63D+00

-5.00D-01 -5.00D-01 -5.000-01 - 5. OOD - 01 - 5. OOD - 01 - 5 , OOD - 0 1 -5. OOD - 0 1 - 5. OOD - 0 1 -5. OOD - 01 -5.00D-01 -4.74D-01 -4.94D-01 -4.851)-01 -5.090-01 -4.580-01 -4.71D-01 -4.89D-01 -4,730-01 -4.31D-01 -4.36D-01

2.25D+O1 1.34D+01 9.740+00 1.090+01 1.15~+01 6.080+00 1.410+01 7.23D+OO 1.71D+01 3.00D+01

-5.68D-03 -3.91D-03 -3.29D-03 -2.93D-03 -2.69D-03 -2.51D-03 -2.38D-03 -2.26D-03 -2.17D-03 -2.09D-03 -4.97D-03 -4.21D-03 -3.48D-03 -3.510-03 -2.58D-03 -2.571)-03 -2.65D-03 -2.381)-03 -1.910-03 -1.90D-03 -6.690-02 -4.420-02 -3.36D-02 -3.9OD-02 -3.631)-02 -2.21D-02 -4.580-02 -2.51D-02 -4.97D-02 -8.860-02

0 Prior 6.990-01 6.990-01 6.99D-01 6.990-01 6.99~)-01 6.99D-01 6.99D-01 6.990-01 6.991)-01 6.991)-01 U BAY 5.34D-01 4.360-01 5.390-01 6.330-01 6.00D-01 7.05D-01 9.00D-01 1.01D+00 8.20D-01 8.460-01 u LS 3.260-01 3.42D-01 4.630-01 5.170-01 5.370-01 6.381)-01 8.15D-01 9.460-01 7.528-01 7.50D-01

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No.9, M=7.3. R=339 No.21, M=5.0, R=282 No.12,13,14, M = 8 . 1 , 1 0 2 __ Observed

V e, v)

10’

10’ 0 v

Q

100 10-2 100 10-1 1 0 0 10’ 10-1 100 10’ 10-1 100 10’ 10-1 100

No.16, M=7.E, R = 3 1 8 No.19, M = 6 . 9 , R = 3 0 4 No.4, M = 5 . 6 . R=282 No.2, M = 5 . 9 , R = 2 9 7

Q

1 or’ lo-’ lo - ’ 100 10’ 10-1 100 10’ 10-1 100 10’ 10-1 100 10‘

Frequency (Hz) Frequency (Hz) Frequency (Hz) Frequency (Hz)

Figure 4. Observed (solid lines) and predicted median (dashed lines) spectra using the bayesian attenuation relation with E(cu,) = -0.5 for the four best (top frames) and the four worst predicted (bottom frames) earthquakes. In each frame we indicate the event number (from Table 1) and its magnitude and distance. Cases 12, 13 and 14 correspond to the same earthquake, so they are plotted together.

1 0 2

n 0 a, 10’ -? E 0

a, -0 3

v

.w .- - E“ 100 a

10-1

R=300, several magnitudes 1 0 2

10’

1 0 0

R=280, M=8.2

t

__ Predicted

- - - - Observed Sept. 19, 1985

I I \

I, M=8.1. R=295 \

\ , lo-’ 100 101,- lo-’ 100 10‘

Frequency (Hz) Frequency (Hz)

Figure 5. Left: median spectra predicted using the bayesian regression with !?(a2) = -0.5, for R = 300 kin and several magnitudes. Right: median spectrum predicted for a M = 8.2, R = 280 km, hypothetical earthquake in the Guerrero gap. For comparison, we also plot the spectrum of the 1985, M = 8.1, Michoacan earthquake.

ground motion in Mexico City. Further research should be for the predictions of Fig. 5 . Site effects, contained in devoted to this topic. parameter q,, are independent of magnitude. Therefore,

Note also in the left-hand frame of Fig. 5 that the peak the shift reflects the faster increase of spectral amplitudes spectral amplitude shifts from about 1 Hz to 0.5 Hz as the with magnitude at low frequencies. magnitude increases. This shift can not be attributed to the The predicted median spectrum of the hypothetical effect of Q or geometrical spreading, since distance is fixed M = 8.2, R = 280 km, Guerrero earthquake (Fig. 5 ,

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right-hand frame), shows higher amplitudes than the spectrum of the 1985 September 19 Michoacan earthquake. This is not unexpected, since the future earthquake has larger magnitude and shorter distance. The result implies that the ground motions experienced in Mexico City during the 1985 event are not an upper limit.

The bayesian technique

The first question that comes to mind after going through a bayesian analysis is how dependent are the final results on the prior parameters. In Fig. 3 we show the posterior expected values of a and u computed assuming E [ a , ( f ) ] = -1.0, instead of -0.5, and leaving the rest of the parameters unchanged. Comparison of both bayesian solutions shows that, indeed, the posterior expected values of the parameters depend on our prior choice of E(cu,(f)]. This means that two persons with different prior states of knowledge would have arrived at different final estimations of the unknown parameters. But the reason why some of the final parameters depend so much on the prior values is because the evidence is not enough to decide for a particular value.

It has been argued that, because of the inherent subjectivity of the bayesian approach, it lends itself to the perpetuation of unrecognized error. In Appendix B we present a numerical example which demonstrates that this objection is not justified.

From comparison of results from the LS and bayesian techniques (Figs 2 and 3) we note that their predictive powers, measured with the standard error for the LS approach and with E ( u I r) (eq. A8 in Appendix A) for the bayesian case, respectively, are not very different. Thus, the prior restrictions we imposed to the data in the bayesian approach did not result in a much larger variance, so the solutions are not overconstrained. But since the bayesian solutions for a are closer to those expected by theory, they should be preferred.

CONCLUSIONS

We have presented the application of a bayesian regression technique to derive attenuation relations for the Fourier spectral accelerations at station CU, Mexico City. This approach is especially useful when the data are not conclusive about the values of certain parameters, so we use prior information to place probabilistic constraints on their values. After combining prior information and empirical evidence, we end the analysis with stable and physically plausible solutions, and with a clear description of our state of knowledge about the unknown parameters.

The bayesian approach provides a framework to incorporate rationally a large amount of knowledge that comes from sources other than the observed data. Although theoretical knowledge has not been completely ignored in the past when deriving attenuation relations with conventional techniques, the approach we have outlined permits a more formal use of this information and provides the mathematical basis to combine it with observed data. Numerical work is greater in the proposed technique than in conventional LS estimation. However, in view of the

advantages of the bayesian undoubtedly worth doing.

ACKNOWLEDGMENTS

We thank E. Rosenblueth

analysis, the extra work is

for his encouragement on applying bayesian statistics to attenuation relations; he, E. Faccioli, W. B. Joyner, K. L. McLaughlin, S. Ward and an anonymous reviewer read the original manuscript and made important constructive suggestions. M. Feriche provided inspiration, even if unaware of it at that time; R. Rueda gave valuable advice on the use of bayesian statistics. Data used in this study have been carefully collected for more than 25 years by the Seismic Instrumentation Group of the Institute of Engineering, National Autonomous University of Mexico (UNAM); this work and that by many others would not have been possible without their valuable assistance. This study was partially supported by National Council for Science and Technology (CONACYT) and DGAPA, UNAM.

REFERENCES

Brocmling, L.D., 1985. Bayesian Analysis of Linear Models, Marcel Dekker, Inc., New York.

Brune, J.N., 1970. Tectonic stresses and spectra of seismic waves from earthquakes, 1. geophys. Res., 75, 4997-5009.

Campbell, K.W., 1985. Strong motion attenuation relations: A ten-year perspective, Earthq. Spectru, 1, 759-804.

Castro, R., Singh, S.K. & Mena, E., 1988. An empirical model to predict Fourier amplitude spcctra of horizontal ground motion, Earthq. Spectru, 4, 675-686.

Castro, R., Anderson, J.G. & Singh, S.K., 1990. Site response, attenuation and source spectra of S waves along the Guerrero, Mexico, subduction zone, Bull. seism. SOC. A m . , 80,

Davis, P.J., 1972. Gamma function and related functions, in Handbook of Mathematical Functions, pp. 253-294, eds Abramowitz, M. & Stegun, I., Dover Publications Inc., New York.

Esteva, L. & Villaverde, R., 1974. Seismic risk, design spectra and structural reliability, in Proc. P h World Conf. on Earthyuake Engineering, pp. 2586-2597, Rome, Italy.

Joyner, W. B. & Boore, D.M., 1988. Measurement, characteriza- tion, and prediction of strong ground motion, in Proc. ConJ on Earthquake Engineering and Soil Dynamics 11 Recent Advances in Ground-Motion evaluation, pp. 43-102, ed. Von Thun, J.L., Geotechnical Special Publication No. 20, American Society of Civil Engineers, New York.

Kanamori, H.. 1977. The energy release in great earthquakes, J . geophys. Res., 82, 2981-2987.

Ordaz, M. & Singh, S.K., 1992. Source spectral and spectral attenuation of seismic waves from Mexican earthquakes, and evidence of amplification in the hill zone of Mexico City, Bull. seism. SOC. A m . 82, 24-43.

Singh, S.K., Lermo, J., Dominguez, T., Ordaz, M., Espinosa, J.M., Mena, E. & Quaas, R., 1988. A study of amplification of seismic waves in the Valley of Mexico with respect to a hill-zone site, Earthq. Spectra, 4, 653-674.

Singh, S.K., Ordaz. M., Anderson, J.G., Rodriguez, M., Quaas, R., Mena, E., Ottaviani, M. & Almora, D., 1989. Analysis of near-source string motion recordings along the Mexican subduction zone, Bull. seism. Soc. A m . , 79, 1697-1717.

Veneziano, D. & Heidari, M., 1985. Statistical analysis of attenuation in the eastern United States, in Methods of Earthquake Ground-Motion Estimation for the Eastern United States, EPRI Research Project RP2556-16, Palo Alto, CA.

1481- 1503.

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Hence, if E ( u ) and VAR(u) are known, simultaneous solution of eqs (A8) and (A10) yields the corresponding values of r' and A'.

The marginal distribution of a can be obtained by integration of eq. (A5):

APPENDIX A: THEORY

Consider the linear model of eq. (6) and the vector of observed values of InA y = (In A , , In A,, . . . , In A,)T. The vector of observed errors, E = ( E ~ , E , , . . . , can be written as

E = y - X a , (Al l

where a = (a,,, a', a,, aj)T is the vector of unknown coefficients of the linear model and

/ 1 MI InR1 R , \

1 . . Mz InR, 'Iz

\ 1 M, InR, R , /

is the matrix of observed values of the non-random variables M and R. Under the assumption of the random error E

being normally distributed with zero mean and variance u2, the likelihood of n independent observations of E is

Of

h L ( E I a, h ) = ~2 exp [ - 5 ( y - Xa)T(y - x a ) ] ('44)

where h = l/u2. The natural conjugate prior density function, p(a , h ) , associated with this likelihood is the normal-gamma (Broemling 1985):

P(aJ h ) = p ( a I hlP(h), (-45)

where

p ( a I h ) = K,hkl2 exp c z " - - ( a - a')TR'(a - a') ] ('46)

and

p ( h ) = K,,h"-' exp (-A'h). (A7)

In eqs (A6) and (A7), K , and K , are normalization constants and k is the dimension of a, 4 in our case. It can be noted from eq. (A6) that, conditioned on h, a has a k-variate normal joint distribution with E ( a 1 h ) = a' and C O V ( a ) h ) = ( h R ' ) - ' , where E( ) and COV( ) denote, respectively, expected value and matrix of covariances; R' is a known, symmetric, and positive definite matrix of order k , and a' is a k-dimensional vector. Eq. (A7) implies that h is gamma-distributed with parameters r' and A' such that E(h) = r'/k' and c2(h) = l /r' , where c( ) denotes coefficient of variation. It can be shown that

kt r ' - 1 '

E(a2) = E ( h - ' ) -

where r( ) is the gamma function (Davis 1972). From eqs (A8) and (A9), an expression for the variance of u, VAR(u) can be found:

where K, is a normalization constant. This function is a k-variate Student's t density with 2r' degrees of freedom, E(a) = a' , and precision matrix r'R'/A' , which implies that

Bayes' theorem states that

p ( a , h I E ) a p ( a , ~ ) L ( E 1 a, h) . (A14)

Replacing eqs (A4)-(A7) in eq., (A14), it is found that

(A151

which is also a normal-gamma distribution for a and h, with r" = r' + n/2, R" = R' + X ' r X , and

The predictive density function of y , given E and a new set of non-random variables zT= (1, M,,, I, In R, ,+ , , R , ,+ ' ) , is the distribution of y given parameters a and h, averaged with respect to the posterior distribution p ( a , h 1 t):

Xp(a, h 1 E ) da, . . . da,, dh. (A18)

From eq. (A18) is found that y has a Student's t distribution with 2r" degrees of freedom and

A" VAR(y I E , Z) = ____

(r" - 1)d '

where

d = 1 - Z"(R + zz')- 'z.

Note that VAR(y 1 E , z) includes the uncertainty due to randomness (measured with h or (I, that is, with r" and A") and the uncertainties in the unknown parameters of the linear model (measured with coefficient d ) .

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APPENDIX B: S Y N T H E T I C E X A M P L E

Consider the attenuation model of eq. (6):

In A = a,, + a , M + a2 In R + a,R + 6, (B1)

where E is a normally distributed random error with zero mean and variance u2. We fixed q , = O , a , =0.8, a2= -1.0, and a3= -0.005 and generated a random sample of 1000 values of M, R and InA. The range of magnitudes is reasonably broad (4-7) but the range of distances is narrow (200-350 km), to try to reproduce the case of subduction earthquakes that affect Mexico City. We then selected prior expected values for a;, i = 0, . . . , 3, which are far from the real ones: E(ao)=5 , E(a , )=2.0 , ,!?(a2) = -0.5 and E(a3) = -0.01 (twice as large as the real one, but at least negative, as we know from theory it should be). We assigned very flat distributions to these prior values, that is, we assigned them very large prior standard deviations, implying that we do not feel very confident about their prior expected values: SD(a,,) = 5 , S D ( a , ) = 2 , SD(a2) = 1, and SD(cr3) = 0.01. This means, for instance, that we believe a priori that, with high probability, a j would fall in the range -0.01(f)0.01 = (0, -0.02). Note that our prior knowledge about the as is very vague. We applied both the bayesian (BAY) and the least-squares (LS) regression techniques to the same data set. First to the first 10 samples, then SO, and so on, up to 1000, trying to reproduce the process of data gathering with time. The corresponding estimators for the as are given in Table B1.

Table B1. Least-squares (LS) and bayesian (BAY) solutions for the simulated example.

N 10 uo LS -19.17

BAY 1.734 a, LS 0.768

BAY 0.715 u2 LS 3.342

BAY -1.065 a, LS -0.024

BAY -0.009 u LS 0.530

BAY 0.687

so 100 57.59 -2.77 1.075 0.904 0.832 0.826 0.839 0.827 -13.72 -0.326 -1.126 -1.109 0.045 -0.009 -0.007 -0.007 0.659 0.729 0.694 0.741

500 -8 23 0 212 0 830 0 829 0 767 -1 094 -0 012 -0 004 0 715 0 719

loo0 Real -14 7 0000 -0 094 0811 0 800 0 812 2244 -loo0 -1 028 -0 010 -0 00.5 -0 004 0 690 0 700 0 692

Note that even with N = 1000, the LS approach is unable to get the correct values of the as, with the exception of a,. Results for several sample sizes with the LS approach are unstable. It turns out, after observing 1000 values, that In A grows as R2. Even when D is well predicted, and the solution minimizes the standard error, the LS attenuation relation is unacceptable.

On the other hand, the bayesian solution converges to the correct values, even when all the prior values are wrong. After not many observations, the weight of the empirical evidence corrects the initial mistakes. The bayesian procedure finds a solution that is not the one of minimum standard error, but is very close, and leads to final estimates that are physically reasonable. As we have shown in this example, possible initial errors are rapidly corrected by empirical evidence, so they do not go unrecognized.

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