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 Empirical Relationships For Estimating Peak Ground Acceleration 

Commonly used empirical relationships for estimating peak ground acceleration (PGA) values from hypocentral parameters and magnitude data are listed below. Here, the seismological nomenclature of magnitudes ML, mb, MS and MW have been used for the local magnitude, the body wave magnitude, the surface wave magnitude and the moment magnitude, respectively. M has been used where the magnitude scale is not specified, explicitly. Re, Rh, and Rf have been used for epicentral distance, hypocentral distance and distance from the causative fault (on which the hypocenter of the earthquake is located), respectively, has been used. R has been used when no distinction is made between the different representations for the term distance.  H has been used for focal depth in kilometers, a is used for the peak ground acceleration (horizontal component), v is used for the peak ground velocity and d for the peak ground displacement. Subscript h and z have been used for characterizing the horizontal and vertical components of ground motion respectively. The symbol s represents the standard deviation of the relationship, wherever available. Unless specified differently, all distances are in kilometers (kms), a in units of 'g', v in cms/s and d in  cms.

 1.          Milne and Davenport (1969). The peak ground acceleration in units of g is given by:

 a = 0.0069 exp (1.64 M)/{1.1 exp (1.1 M) + Re2}

 The database used in the derivation of this relationship pertained to the Western United States, Central America and Chile. Scatter of the data is not reported.

  2.  Mickey (1971). Peak value of particle acceleration is related to the hypocentral distance and body wave magnitude of the earthquake. (Expressions for velocity and displacement have also been derived.)

 a = (3.04.100.74 mb - 4)/Rh1.4

 3.  Davenport (1972). The value of the peak ground acceleration in units of g is given by:

 a = 279 x 10-6 Exp (1.8 * M)/Re1.64

4.  Donovan (1973). World wide average values of the peak ground acceleration in units of g are given by:

a = 1.320 Exp (0.58M)/(R + 25)1.52

 The relationship does not distinguish between the epicentral distance, the hypocentral distance or the distance from the causative fault. Standard deviation of the estimated value is given by:

s(loge a) = 0.84

5.  Donovan (1974). Donovan derived several relationships to describe the values of the peak ground acceleration from the San Fernando earthquake of 1971, recorded under different site conditions. Magnitude did not appear explicitly in these relationships. A magnitude dependent relationship, which may be applied for other geographical areas and for earthquakes of different magnitudes, is based on data from Western North America, Japan and Papua New Guinea. In this relationship the peak ground acceleration is given by:

 a = 1.080 Exp (0.5M)/(R + 25)1.32

 The standard deviation is given by:

 s(loge a) = 0.707

 6.  McGuire (1974). Values of peak ground acceleration is given in terms of earthquake magnitude and hypocentral distance

 a = 0.472 x 100.28M/(Rh + 25)1.3

s(log10 a) = 0.222

7.  Orphal and Lahoud (1974). The relationship gives the values of peak ground acceleration in terms of the magnitude and hypocentral distance.

a = 0.066 x 100.4M/Rh1.39

 8.  Esteva and Villaverde (1974). Values of the peak ground acceleration and velocity can be estimated in terms of the magnitude of the earthquake and the Hypocentral distance

 a = 5.6 Exp(0.8M)/(Rh + 40)2

s(loge a) = 0.64

 9.  Trifunac and Brady (1976). Using data in the magnitude range 4.8 < M < 7.5 a relationship in the following form was derived

 log a = M + log Ao(R) - log {yo (M, p, s', v')} -log(980)

 p specifies the confidence limit, s' specifies the site classification (s'= 0 for alluvium, = 1 for intermediate and = 2 for rock). The expression for yo is written as follows:

 log yo = AA p + AB M + AC + ADs' + AE V' + AF M2

 AA, AB, AC, AD, AE and AF are coefficients derived from regression analysis for acceleration values. V' = 0 for horizontal motion and V' = 1 for vertical motion. p = 0.5 for the mean and p = 0.8 for the 84th percentile values. log Ao(R) is a function which accounts for the attenuation with distance R, and has been tabulated by Trifunal at discrete values of R. Within 100 kilometers distance the values can be computed using the following polynomial:

 log10 Ao(R) = Co + C1X + C2X2 + C3X3 + C4X4

 where X = log10(R) ; Co = -1.399793 ; C1 = 1.3615414;

C2 = -3.315589 ; C3 = 2.729386 ; C4 = -0.601832

 The values of the coefficients AA, AB ...VA, VB,... DA, DB... are given below:

AA = -0.898 ; AB = -1.789 ; AC = 6.217

AD = 0.06; AE = 0.331; AF = 0.186

 10. McGuire (1978). For the Western United States a relationship was given distinguishing between rock and soil sites using earthquake data between magnitude 4.5 and 7.7 and distances between 10 and 200 kms. The peak ground acceleration is given by:

 a = 0.0306 [Exp(0.89M)/Rh1.17]Exp(-0.2b)

s(logea) = 0.62

 b = 0 for rock and b = 1 for soil.

 11. Donovan and Bornstein (1978). A graphical relationship valid  between magnitudes 5 and 7.7 and a distance range (distance to the  center of energy release) between 5 and 300 kilometers is given  below:

         a = A Exp (BM)/(R + 25)C

        A = 2198/R2.1 ; B = 0.046 + 0.193 loge(R)

        C = -2.515 + 0.211 loge(R)

 12. Cornell, Banon and Shakal (1979). For Western United States for earthquakes having magnitudes between 3 and 7.7 and hypocentral distance between 20 and 200 kms

 a = 0.863 Exp(0.86M)/(Rh + 25)1.80

s(logea) = 0.57

 13.  Espinosa (1980). For the western United States, relationships were derived between PGA, local magnitude, ML and epicentral distance. Separate relations were given for distances less than 10 kms, between 10 and 60 kms, between 60 and 300 kms and between 5 and 300 kms. The relationship between 10 and 60 kms distance is given below:

 a=1.776x10-5xExp(2.3ML)/Re1.59

    

The scatter of the PGA values is not available.

 14. Battis (1981). Two separate relations for the north American Continent for peak ground acceleration are given. One is for the California region:

 a = 0.348 Exp(1.21mb)/(Re + 25)2.08 ;

s(logea) = 0.71

 The other relationship is for the Central United States

 a = 0.0239 Exp(1.24mb)/(Re + 25)1.24

s(logea) = 0.71

The relationships are based on 5<mb< 6.5.

15. Campbell (1981). This relationship estimates the peak ground acceleration in terms of the closest distance to the causative fault. The peak ground acceleration is given by:

a = 0.0159 Exp(0.868M)/[R + C(M)]1.09

 C(M) = 0.0606 Exp(0.7M)

s(logea) = 0.37

 16,17. Hasegava, Basham and Berry (1981). The relationships for peak ground acceleration and velocity were derived using body wave magnitudes between 4 and 7 and hypocentral distances between 10 and 200 kms. Separate relationships were derived for the eastern and western regions of Canada.

 (16) Eastern Canada

a = 4.0 x 10-4 Exp(2.3mb)/R1.3

 (17) Western Canada

a = (1.02/100) Exp(1.3mb)/Rh1.5

 18. Joyner and Boore (1981). The closest distance to the surface projection of the fault rupture (Rs) was used for distance. Peak ground acceleration and velocity are given for soil and rock:

 a = 0.0955 Exp(0.573 M).D-1 e-0.00587D

D = (Rs2 + 7.32);

b(logea)=0.60

 

 19. Nuttli and Hermann (1984). Relationships were derived for a and v for earthquake data from Mississippi valley. The relationship for PGA is given below:

a=3.79x103xExp(1.15mb)xD-0.83Exp(-0.00159R)

D=(R2+Hmin2)0.5  and Hmin =0.0186Exp(1.05mb)

b(logea) = 0.55

 

20. Abrahamson and Lithehiser (1989). A distinction is made between intraplate and interplate earthquakes and reverse and normal faults. The peak ground acceleration for the horizontal component is given by

 

log10a={ -0.62 + 0.177M - 0.982 log10(r + e0.284M)+

0.132F - 0.0008Er}

b(log10 a) = 0.277

E = 1 for interplate earthquakes

    0 for intraplate earthquakes

    F = 1 for reverse and oblique faults 0 otherwise

 

The vertical component of ground acceleration is given by:

 log10(av)= {-1.15 + 0.245M - 1.096 log10(r + e0.256M) +  0.096F - 0.0011 Er}

 b(log10 av) = 0.296

References

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