Vol. 7 No. 3 441--446 A C T A SEISMOLOGICA SINICA Aug. ,1994

Expected magnitude and distance of potential source area and the estimating method" Meng-Tan GAO ( ~ )

Institute o f Geophysics, State Seismological Bureau, Beijing 100081, China

Abstract

Magnitude and distance of major potential source are needed in order to determine duration time of artificial ground motion and to determine the type of response spectrum (near field or far field) when using the seismic in- tensity zonation map. The magnitude probabilistic distribution function of seismic belt and the magnitude and space joint distribution function for given intensity of the site in a potential source are provided. Then the basic formula of calculating expected magnitude and expected distance are developed. Several examples for ealculating expected magnitude and expected distance in northern China are discussed. These results show that expected magnitude and expected distance are related not only to geometry of potential source and magnitude but also to the intensity of the site with certain exceeding probability.

Key words: seismic potential source area, seismic belt, magnitude distribution function, expected magnitude, expected distance

1 InWoduction

It is necessary to know the magnitudes and distances of major contribution potential sources

for given intensity value (or given acceleration) of the site, when the artificial ground motion

time history is made. These parameters are also needed when using GBJ11-89 building code in

China. The potential source occupied certain area in space and some potentials are very large. In

some studies, the nearest distance from the site to the potential source or the distance from the

site to the center of the potential source is defined as the distance from potential source to the

site. It may be reasonable only when the scale of the potential source is relatively small and the site is far away from any point of the potential source. But if the potential source is very large and the site is close to the boundary or even in the potential source, this definition is very ridiculous.

The shortcomings of this definition are. (1) the contribution of every segment of the potential source is not considered~ (2) this definition is independent with the intensity of the site. The

shortcomings about the magnitude are similar.

We think that the contribution of every segment of the potential source to the intensity of

the site must be considered to determine the distance for potential source to the site. The contri-

bution is also related to the magnitude. In the probabilistie hazard analysis method, magnitude is

a random variable, and follows certain probabilistic distribution. Thus the joint probabilistic dis-

tribution of space and magnitude in the potential source must be determined so that the magni-

tude and distance can be reasonably evaluated. We think the definition of expected magnitude

and expected distance (Campos-Costa et al. , 1992) from potential source to the site is suitable.

* Received March 17, 1993; Accepted April 24, 1993. Contribution No. 94A003, Institute of Geophysics, SSB, China.

442 ACTA SEISMOLOG1CA SINICA Vol. 7

The expected magnitude and expected distance can be easily evaluated from the joint distribution function. These two parameters can be used to determine the duration time and whether the ba- sic intensity belongs to "the near field problem" or "the far field problem".

2 Joint probabilistie distribution magnitude and space for given intensity value

In the currently used seismic hazard analysis model, spacial distribution of earthquakes and distribution of magnitude in the potential source are independent, and they are homogeneous dis- tributed in the potential source. First, we divide potential source into a series small segment. Suppose that the area of each segment is dxdy, the area of the potential source is A,. Thus the probability of earthquakes occurred in the segment is dxdy/A, . The probabilistic distribution of magnitude in seismic belt is the truncated exponential function. Because of the inhomogeneous distribution of earthquakes with different magnitudes, magnitude distribution in potential source will be different. We will deduce this function from the probabilistic distribution of magnitude in seismic province and spatial distribution of earthquakes in the belt. For the concise and applica- tion, we divide magnitudes into intervals. The total number of the interval N,,, will be determined by the interval ,',M, the maximum magnitude M .... and the minimum magnitude m0 of the seis- mic belt :

M ..... -- m0 (1) N,,,-- AM

According to the currently used seismic hazard analysis method in China, the discrete proba- bilistic distribution function for each magnitude interval and spatial distribution of earthquakes can be determined (Gao, 1988; Shi, 1991; State Seismological Bureau, 1991). The two func- tions were evaluated refer to the Chinese seismic intensity zonation map (1990). From the marginal distribution and conditional distribution ~ (Gao, 1993) , it is not difficult to deduce the /-th probabilistic distribution function of magnitude of potential source:

P~,,,(rnj)J~.,,,s (2) Pb,,(mj) = N , ,

P~,,, (~,)Z.o,, j = l

The joint probabilistic distribution function of the location k and the magnitude for more than a given intensity value Iu can be deduced by using the equation (2) and definite attenuation law. The steps will be:

1) Separate the distribution function into two functions: marginal distribution function of magnitude for given site intensity value Id[independent with spatial location ( x , y ) , evaluate from magnitude distribution of potential source] and spatial distribution function for given magni- tude and intensity value >~Id. For the convenience of illustration, the series number of the small segment k is taken as the variable of spatial distribution, the central coordinate of segment k is (xk,yk). Thus the joint probabilistic distribution can be simplified as two dimensional discrete joint probabilistic distribution. For a certain segment, if the affected intensity of an earthquake with certain magnitude is equal to or greater than the given intensity value Ia, the occurrence probability of such earthquake is 1/N~¢ , or else is 0. N,,, is the total number of segments within

* Gao, M. T. , 1991. Te application of random field in the study of seismicity, dissertation of doctor degree, Institute of Geophysics, State Seismological Bureau.

No. 3 GAO,M. T. : EXPECTED MAGNITUDE AND DISTANCE OF POTENTIAL SOURCE 443

the potential source for given magnitude interval j and affected intensity of such earthquake is e- qual to or greater than Id.

2) Multiply these two functions to get the wanted probabilistic distribution function. Ac- cording to the assumption that the earthquakes uniformly distribute within the potential source, the marginal distribution function of magnitude for given intensity value I~ is:

1 N,, Pl,,,(m~lI ~ Ia) = -~ N/Pl,,,(mi) (3)

Where Q is the normalized parameter:

N Nm / Q = ~ ~P,, , , (mi) (4)

Where N, is total number of segments of the potential source. The equation of the joint probabilistic distribution will be deduced for different attenuation

models : 2. 1 Circle attenuation model

The so-called circle attenuation model means that the isoseism can be approximated as the shape of circle. This model is correct for small and moderate earthquakes. If the magnitude is given, the distance Rku from the site to the segment k for the given intensity Id can be evaluated through the attenuation model (shown in Figure 1). It is c~ident that I~I , l is e- qual to the condition R~Rka. It is not difficult to develop the joint probabilistic distribution of spatial location k and mag- nitude considering the above condition and the assumption that magnitude is in- dependent with spatial variable within the potential source:

m Site

(a) Circle attenuation model

Site

@ (b) Ellipse attenuation model

Figure 1 Sketch of geometric relation for two models to deduce Pj (mj ,k [I~Iu)

[Plm(mj) Ps(mj,k[I ~ I~) = -~- ~ R ~ Rka (5)

[o R ~> R~ 2 . 2 Ellipse attenuation model

The shape of the isosesim of the ellipse attenuation model is ellipse. The coordinate system is transferred into a new coordinate system according to the orientation of the long axis of isosesim, the site coordinate and the central coordinate of the segment k. In the new system, the orienta- tion of the x axis is along the long axis, the origin is in the source, the site coordinate is (Xk, YD. The half long axis Ra and half short axis RB of the isoseism can be evaluated from the given

magnitude rnj and given intensity value Ia through the attenuation model. For given X,, Ykd can be written as:

Y~ = RB R~ (6)

444 ACTA SEISMOLOGICA SINICA Vol. 7

It can be seen from Figure lb that if the Yk is smaller than or equal to the value Yk,~, the site affected intensity will be greater than la. So the joint probabilistic distribution for such case will

be.

[ P~,,, (mj) P . , (m, ,k l I ~ L,) = -~- -N~

[o Y ~ Yk~

Y ~ Ykd (7)

3 Definition of expected magnitude and expected distance and some ex- amples

According to the above results, the expected magnitude and expected distance are definite

by the following two formulas: N N m

fi~ = ~_2 ~_2R~,'P,'(m,'kl I >~ I,') (8) l - - i j 1

N., N m

M = z...a \~ ~_2miP.'(rn,'k] I ~ I+~) (9) / = 1 j = l

Where R is the expected distance from potential source to the site when the site affected intensity

is greater than the given intensity value Id; ~ ' is the expected magnitude under the same condi-

tion.

Site No. 5

Potent ia l seismic source area

Site No. 1

Site No. 2

Site No. 4

Figure '~ Sketch map of the relative locamm of the

potential seismic source and five sites

According to the definition of the joint

probabilistic distribution function, the expect- ed magnitude can be simplified as:

"~ rniN,,, Pt,,,(mj) M = N,Q (10)

j = l

and expected distance can be simplified as:

:L RjN,,,PI,,,(mj) = ~ N,Q (11)

j ~ l

Where Rj is the average of the distance from the center of the segment to the site where site affected intensity is greater than or equal

to Id when magnitude is in j th magnitude-in-

terval.

Formula (10) and formula (11 are the

basic formulas to calculate expected magnitude

and expected distance. So, the basic steps to

evaluate expected distance and epxected mag- nitude are •

1) Divide magnitude into intervals, the total number of the intervals is Nm ; 2) Determine the probabilistic distribution function of magnitude for the potential source P~m

(rn,)

3) Discrete the potential source into a series of segments, the total number of the segment is

No. 3 GAf).M.T. : EXI'I:'.CTI:~I.) MAGNITUI)E ANI) I)ISIAN(.'I~ ()t: I'I)TI.;NTIAI. S()Uta,('E d15

,i ) According to the concrete a t tenuat ion model , evaluate the nun lbe r of segments N,,, of

different n-tagnitude interwtls which cause the affected intensi ty in lhe site is equal to or greater

t t ,an I,:, ils related average distance R,;

5) Calculate expemcd nmgni lude and expecled distance by using forlnubl (1 0) and fornmla

( l l ) .

0.25

0.2(

0.15

o 0.1(

0.05

0.00 4.0

8.5

(

8.0

.~ 7.5

~ 7.0

6,5

i 6.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5

M

60

50

30

20

Magnitude 10

-.....o-- Distance

I I I 1 I 6 7 8 9 I0 11

Intensity

Figure 3 Probabilistic distribution elf magnitude Fi,u, urc [ l'2xt)ccttd ln/,~tllttah, imtl exptctcd dis of Tancheng potenti+tl seismic source tance for different l,z at Site No. 2

The above steps can be done easily by computer program. Several Sites near or within the

T a n c b e n g potential source in Tancheng I ,uj iang seismic beh are as examples to illustrate the

above method. The potential source and sites are shown in Figure 2. These five sites in this fig-

ure represents some typical cases such as sites locates in the potential source, near the potential

source and far away from the potential source. According to the probabilistic den-

sity distr ibution function of magni tude and Table I E x p e c t e d m a g n i t u d e and d i s tance for

spatial distribution of Tancheng- l .u j iang d i f f e r e n c e intens i ty va lues of the sites

seismic belt, probabilistic dis'tribution of ~,1 ~ w Site mJmber

magni tude of the potential source is deter- R .~ R ~ ~ M

mined with magni tude interval 0 .2 (shown 1 24 6.5 24 6.9 19 7.2

in Figure 3). The potential source is divid- 2 53 6.8 43 7. 1 33 7.4 3 9,'i 7.3 !)3 8.0 85 8.3 ed into 572 segments with area 4 square 4 192 8.0 176 8.3 -- --

kilometers, the total number of magnitude 5 108 7.3 103 7.8 89 8.2

interval is 23. Ellipse at tenuation model is

adopted. Id for every site are Vl , ~]I and ~I.

The results are listed in Table 1.

The results show that the geometric relation of site with potential source is vital impor tant to

expected magni tude and expected distance. Because earthquakes occurred in any points wi thin

the potential source will not arise affected intensi ty Vi in Site No. 4, it is evident that no defini-

tion of expected magni tude and distance under such condition. In such case, Q is equal to zero.

446 ACTA SEISMOL(X.;-ICA SINICA Vol. 7

The program will recognize this case automatically. This also shows the method represented in

this paper is reasonable. In order to study the relation of L~ with expected magnitude and expect-

ed distance, expected magnitude and expected distance are calculated for different Ie and given

site (Site No. 2). The results are shown in Figure 4. The variations for different I,t are very

large. Different Iu means different exceeding probabilities for the same site. The lower probabili-

ty means the large Id. Different expected magnitudes and expected distances should be considered

when the time history of ground motion with different exceeding probability level is to be deter-

mined.

4 Discussion

Adopted the same probabilistic analysis method currently used in the seismic hazard analysis

model, the expected magnitude and expected distance and related calculation method are provided

in this paper. The equaiton has definite physical meaning. The method is reasonable and applica-

ble after analyzing the examples of five sites nearby the Tancheng potential source in Tan-I .u

seismic belt. There are several factors affect the value of expected magnitude and expected dis-

tance, such as probabilistic distribution of magnitude in the potential source, the geometric rela-

tion of site with potential source, intensity value of the site and attenuation model etc. It is not

reasonable to ignore any one of these factors. The method provided in this paper can be used to

determine the duration time of ground motion, the spectrum type (this parameter is determined

by the relation of potential source and site, and also the the intensity in the site).

This study is supported by the Department of Seismic Hazard Prevention, SSB.

References

Campos-Costa, A. , Oliveria, C. S. and Sousa, M. 1.. , 1992. Hazard-consistent studies for Portugal. Proceed- ings o f The Tenth World Conference on Earthquake Engineering, 1, 4 7 7 - 479. 19 24 July, 1992, Madrid, Spain.

Gao, M. T. , 1988. The discussion about earthquake annual occurrence rate. Recent Developments hi World Seismology, 1, 1--5 (in Chinese).

Gao, M. T. , 1993. Discussion about the relationship between the probabilistic distribution of magnitude in seis- mic province and that distribution in seismic potential area. Earthquake Research in Chhla, 9, 1, 15--19 (in Chinese).

Shi, Z. I.. , Yan, J. Q. and Gao, M. T. , 1991. The principle and methodology of seismic zonation a trial in North China, Acta Seismologiea Sinica, 13, 180--189 (in Chinese).

State Seismological Bureau, 1991. Note o f Chinese Seismic Intensity Zoning Map (1990), 1-10. Seismological Press, Beijing (in Chinese).