ATTENUATION RELATIONSHIPS

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Hosein Rahnema

Dept. of Civil & Environmental Engg.Shiraz University of Technology

Earthquake Hazard Analysis Course

ATTENUATION RELATIONSHIPS

ground acceleration,spectral response values

etc.

magnitude,soil conditions,

site-to-source distance,etc.

earthquake properties of response quantities

Various earthquake parameters

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An attenuation expression provides a functional relationship

Variables that affect attenuation

geometric spreading, reflection, refraction,

absorption, and geologic structure

Subsurface conditions, Topographic variations,soil-structure interaction all constitute local conditions

source mechanism

travel path

local conditions

stress and strain conditions, rupture dimensions,

source depth

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Geometric spreading and absorption

Geometric Spreeding

Spherical wave-fronts occupy more area as they progress from a seismic source.

In keeping with the laws of conservation, the amplitude of the waves must decrease; hence, the mathematical expression that wave amplitudes decrease proportionally with 1/R is easily confirmed.

R is distance, and is a constant dependent on travel path and geologic conditions [Reiter, 1990].

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Geometric spreading and absorption

AbsorptionAbsorption results from friction in the medium and scattering of the waves which can cause destructive interference.

Q = Q0fn

Q is called the quality factor, f is frequency,Q0 and n are constants dependent on rock properties.Large values of Q correspond to

low absorption (and attenuation), and vice-versa.

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Types of attenuation relationships

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empirical theoretical

many theoretical type relationships incorporate empirically derived constants.

Empirical type relationships are typically derived by applying regression analysis methods to observed and recorded earthquake data

theoretical expressions attempt to model directly the physics of earthquakes and related mechanisms, with constants being determined empirically

In some cases, the distinction as to whether an expression is empirical or theoretical is difficult to discern

Empirical attenuation relationships

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Y = b1f1 (M ) f2 (R ) f 3 (M,R) f4 (P1) Y=strong motion parameter, (e.g. peak acceleration),b1= constant scaling factor,f1(M) = a function of the independent variable M, (magnitude or earthquake source size),f2(R)= a function of the independent variable R, (site-to-source distance),f3(M,R) = a joint function of the variables M and R,f4(P1) = function(s) representing possible source, site, and building effects, = an error term representing uncertainty in Y.

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lnY =ln b1+lnf1(M ) +lnf2(R ) +lnf3(M,R) +lnf4(P1) +ln

above expression to be presented in the more useful additive form:

Y = b1f1 (M ) f2 (R ) f3(M,R) f4 (P1) ,

R 1 - Distance to causative faultR2- Epicentral distanceR3 - Map distance to energy centerR4 - Hypocentral distanceR5 - Distance to energy center

Empirical attenuation relationships

Empirical attenuation relationships

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Another typical predictive relationship may have the form

Peak Horizontal Acceleration0.05g 0.48 0.51

0.1 0.46 0.490.15 0.41 0.43≥0.3 0.3 0.31

Donovan and Bornstein (1978)

102,154,000 . . . 25 . .for 8; 5 ,where

a = median peak horizontal peak ground accelerationin gals (or ) .R = distance from the site to the center of energy on the causative fault in kilometers (km), and M = magnitude .

The standard error ln and the coefficient ofvariation of PGA ( ) of the above expression vary as functionof acceleration:

exp 11

Crouse (1991)

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Crouse proposed the following expression to estimate horizontal ground motion for the Cascadia subduction zone for shallow firm sites in the Pacific Northwest:

With the appropriate values of b1-b7, applied [Crouse, 1991], the above expression reduces to:

where Y = median peak ground acceleration (gals ) ,M = Moment magnitude,R = site to center of energy release distance (km) ,h = focal depth (km),= constants, and

= standard error of ln(Y).

ln ln exp ,

ln 6.36 1.76 2.73 ln 1.58 exp 0.608 0.00916 ,0.773, . . , 0.90

Crouse (1991)

12No distance limitations are given, but the site-to-source distance used in this equation is the distance from the site to center of energy release (i.e., R5 in Figure 3.1). In the analysis of the earthquake records of M <= 7.5, Crouse assumed the site to center of energy release distance to be approximately equal to the hypocentral distance, and for larger earthquakes this distance was assumed to be the site to centroid-of-fault-plane distance

Boore, Joyner, and Fumal(1993)

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Boore, Joyner and Fumalhave recently proposed the following relationship to estimate horizontal ground motion for shallow earthquakes in western North America:

Boore, Joyner, and Fumal(1993)

14In the use of this expression, a site is classified into one of four categories (A, B, C, and D) depending on the average shear-wave velocities of the upper 30m of geologic material. Classes A, B, C, and D include sites where the average shear-wave velocity are:

greater than 750 m/s;between 360 m/s and 750 m/s; between 180 m/s and 360 m/s; and less than 180 m/s, respectively.

As a result of lack of data presently available for site class D, Boore, Joyner, and Fumal excluded it from their analysis.When the constants derived for estimating the peak acceleration for the larger of two horizontal components are substituted, the above equation reduces to:

0 for site class A 0 for site class A

GB = 1 for site class B Gc 0 for site class B

0 for site class C 1 for site class C

Theoretical attenuation relationships

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The remaining discussion on theoretical attenuation relationships is limited to expressions based on Random Vibration Theory-Band Limited White Noise (RVT-BLWN). (Other theoretical type expressions such as those based on Green's function are not discussed in this report.) The use of RVT-BLWN came from the observation that acceleration time histories are, to a very good approximation, band-limited white Gaussian noise within the S-wave arrival window. The upper and lower band limitations are the spectral corner frequency (fo) and the highest frequency passed by the earth or accelerograph

Hanks and McGuire model (1981)

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Hanks and McGuire model (1981)

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Peak Acceleration

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In 1981, Campbell (1981) used worldwide data to develop an attenuation relationship for the mean PHA for sites within 50 km of the fault rupture in magnitude 5.0 to 7.7 earthquakes:

In 1994, Campbell and Bozorgnia (1994) used worldwide accelerograms from earth-quakes of moment magnitude ranging from 4.7 to 8.1 to develop the attenuation relationship

Peak Acceleration

20Toro et al. (1994) developed an attenuation relationship for peak horizontal rock acceleration

Peak Acceleration

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Youngs et al. (1988) used strong-motion measurements obtained on rock from 60 earthquakes and numerical simulations of Mw>= 8 earthquakes to develop a subduction zone attenuation relationship:

Peak Velocity

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Joyner and Boore (1988), for example, used strong-motion records from earthquakes of moment magnitude between 5.0 and 7.7 to develop the attenuation relationship

Table 3-7 Coeffecients for Joyner and Boore (1988) Peak Horizontal Velocity Attenuation Relationship

Variation of peak horizontal acceleration

with distance23

Figure 3.22 Variation of peak horizontal acceleration with distance for M = 5.5, M= 6.5, and M =7.5 earthquakes according to various attenuation relationships: (a) Campbell and Bozorgnia (1994), soft rock sites and strike-slip faulting; (b) Boore et al. (1993), site class B; (c) Toro et al. (1994); and (d) Youngs et al, (1988), intraslab event.

Estimation of Frequency Content Parameters

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Predominant Period

Figure 3.23 Variation of predominant period at rock outcrops with magnitude and distance. (After Seed et al., 1969.)

Estimation of Frequency Content Parameters

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Fourier Amplitude SpectraBased on Brune's (1970, 1971) solution for instantaneous slip of a

circular rupture surface, the Fourier amplitudes for a far-field event at distance R can be expressed (McGuire and Hanks, 1980; Boore, 1983) as

Where fc is the corner frequency, fmax, the cutoff frequency , Q(f) is the frequency-dependent quality factor and C is a constant given by

where vs is in km/sec, Mo is in dyne-cm, and is referred to as the stress parameter or stress drop in liars. Stress parameters of 50 bars and 100 bars are commonly used for sources in western and eastern North America, respectively

Estimation of Frequency Content Parameters

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Figure 3.24 (a) Variation of Fourier amplitude spectra at R =10 km for different moment magnitudes ( = 100 bars); (b) accelerograms generated from the M = 4 and M = 7 spectra. (After Boore, 1983.)

Estimation of Frequency Content Parameters

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Ratio Vmax/amax

Estimation of Duration28

Epicentral distance (km)

Figure 3.27 Variation of bracketed duration (0.05g threshold) with magnitude and epicentral distance: (a) rock sites; (b) soil sites. (After Chang and Krinitzsky, 1977.)

Attenuation relationshops

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RMS AccelerationHanks and McGuire (1981) used a database of California earthquakes of local magnitude 4.0 to 7.0

to develop an attenuation relationship for rms acceleration for hypocentral distances between 10 and 100 km (6.2 and 62 mi):

Kavazanjian et al. (1985) used the definition of duration proposed by Vanmarcke and Lai (1980) with a database of 83 strong motion records from 18 different earthquakes to obtain

Attenuation relationshops

30Arias Intensity

Campbell and Duke (1974) used data from California earthquakes to predict the variation of Arias intensity within 15 to 110 km (9 to 68 mi) of magnitude 4.5 to 8.5 events.

Wilson (1993) analyzed strong motion records from California to develop an attenuation relationship which, using the Arias intensity definition of equation (3.17), can be expressed as

D is the minimum horizontal distance to the vertical projection of the fault plane, h is a correction factor