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Earthquake risk and fracture statistics
222
Physics of the Earth and Planetary Interiors, 49 (1987) 222—224 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands
Earthquake risk and fracture statistics P. Kitti and G. DIaz Departamento de Ciencia e Ingenieria de Materiales (IDIEM), Facultad de Ciencias Fisicas y Matemtiticas, Universidad de Chile, Casilla 1420, Santiago (Chile) (Received May 27, 1986; revision accepted February 16, 1987)
Kittl, P. and Diaz, G., 1987. Earthquake risk and fracture statistics. Phys. Earth Planet. Inter., 49: 222—224. On the basis of fracture statistics, which establishes a. relation between the cumulative probability of fracture in a brittle body and the stress field to which it is subjected, it is possible to describe some seismological statistics. The cumulative probability of time intervals between two consecutive strong earthquakes and the cumulative probability of the strong earthquakes magnitude, appears as Weibullian. Comparison with Chilean set data gives evidence to the theory.
According to the Weibull theory (Weibull, 1939; Freudenthal, 1968; Kitti, 1984), the cumulative failure probability of a brittle body in a nonhomogeneous stress field, is given by F( a)
=
1
—
exp
(1 — -~-
f4) [a ( ,~)}
dV
}
(1)
where J’~is the volume unit, V the body volume, a is the maximum tractional stress undergone by the body of fracture, a(r) is the stress field, r is the vector position and 4) is usually the function
q(a)
=
J
~
= ~
—
=
where i is the average position vector. The volume V is the mass subjected to stress, and supposing that a uniform increase of stress produces a practically uniform movement, then it is possible to put V = At (5) where t is the time measured between earthquakes, and A is a constant. Formula (4) may be transformed into the following expression
(o_ai~~
ao
10
(2)
/
[V 1— exp~—~~~4)[a(i)]}
0031-9201/87/$03.50
in
~(t)
=in t+1n{~~-4)[a(~)]}
(6)
a
In formula (2) aL is the stress below which no fracture occurs at all, a0 and m are constants of the body. If the integral of formula (1) is evaluated where there is a substantial increase of stress with respect to the initial state when the average stress 0, and with the integral average theorem is obtained, with aL = o F(a)
and with some transformations 1 V
(3)
© 1987 Elsevier Science Publishers B.V.
which allows us to make a Weibull diagram by plotting in E(t) versus in t, which gives a straight line. To make such a Weibull plot, we used a set of 110 strong Chilean earthquakes that occurred between the latitudes of 180 S and 410 S from 1906 to 1975 (M. Pardo, personal communication, 1975; Regional Catalogue of Earthquakes) (Fig. 1). The slope of the straight line passing through the points is practically 1, which represents a strong support
223
99
0/,
eqs. 7 and 8 we get
-
900/. 80!.
-
70/. 60°!.
C=~/j7j~•10fl/2 (9) 2M where C is an additive constant that may be a=C.lOa/
-
50°!. 40•!. 30°!. 950/,
--
200/.
-
Jo.,.
-
,j/0
ignored here. In formula (4), because a = a(M) after eq. 9, E(a) is only a function of the experimental probability F(M) of a certain earthquake of magnitude M, and hence ~(a) ~(M) and formula (4) is transformed into 1 V E(M)=lnlF(M) =~j~4)(a)
-
0
o I °!.
-
______________________
1/365
I 1/36
I 1/12
1/4
I 1/2
I 1
I .1.1 2 4 6
Time between Earthquakes 1 year
Fig. 1. Weibull plot of the cumulative probability F(t) of time (1906—1975). intervals between two consecutives strong earthquakes, in Chile
j
=
74)(C’. iOa/2M) m VC 10~u,,’2M 10 2ML (10) a~ a/ where C. 10~~/2ML = GL and V is supposed to be equal for all the earthquakes. If we plot in ~(M) vs. ln(10~/2M 10a/2ML) for 64 earthquakes of magnitude M ~ 6 that occurred in Chile between the latitudes of 180 S and 410 S from 1906 to 1975 (Pardo, 1975), we get the .graph shown in Fig. 2.
(
—
—
—
)
—
for this theory. The supposition V = At is in accordance with (Lomnitz, 1974) the tectonics of the Chilean zone, and fracture statistics come from the propagation of faults that occur during seismological movements (Kanamori and Anderson, 1975). Probabilistic models of strong earthquakes are developed in different ways (Kagan and Knopoff, 1976, 1977; Patwardhan et al., 1980; Brillinger, 1982), and in some cases (Anagnos and Kiremidjian, 1984) the Weibull distribution is used. Regarding the distribution of earthquakes of magnitudes M, the relationship between the energy E dispersed in an earthquake and the magnitude M thereof, is according to Gutenberg and Richter (1956), Kanamori and Anderson (1975) as follows iogE=aM+/3 (7) where a 1.5 and fi 11.8. For a finite volume V the seismic energy liberated, according to Bullen (1955) is2kV (8) E = ~ia where ~ is the rigidity and k is a coefficient. From —
—
~
95
0/• 0/
/
.
~ ~ 70°!. 60 ~/. 50 °/~
~oY. 30 °/. 20
1.
~ ~o
I
&
o s ./. 2 •/.
—
I •/.
—
a I 60
I I 65 10 75 Earthquake Megnltude hi
I 80
I 85
Fig. 2. Weibull plot of the cumulative probability F(M) of the strong earthquakes with magnitude minor or equal to M, in Chile (1906-1975).
224
Because we have limited ourselves to the consideration of magnitudes M ~ 6 a correction may be carried~outin formula (10) and we obtain it in this way 1 E M) = ln 1 [F( M) + FL]
(
—
VCm I 10a/2M
—
10a/2ML
,n
a0
=
where FL is the cumulative probability neglected, and here ML = 5.9 in order to avoid a point at infinity. Since the graph obtained is nearly a straight line, this implies that FL 0. This means that in this seismological zone an earthquake with M < 6 does not imply a fracture of the rock. The slope of the straight line is m = 0.87. In safe granite (K.ittl et al., 1984) m 2, this difference implies that the great mass of rock subjected to stress is very dispersive, In the present work it is worthwhile to emphasize the contributions of fracture statistics to seismological risk, which consists roughly of the graphs in Figs. 1 and 2. These graphs are represented by formulas (6) and (11); it is probable that a careful research of tensions in situ and fracture properties of rock at high pressures can give more information about seismological risk. Finally if we remember the foundations of statistical mechanics (Kittl and Aldunate, 1983; Kittl, 1984) these two formulas, (6) and (11), mean the complete aleatory of the seismological events in this case. In equations this funda.ment is as follows
.P( V1 +
V2) =
.P( V1) .P( V2) .
(12)
which means that the probability of a fracture F not occurring in the volume V1 + V2 is the product of the probabilities of the fracture not occurring in the volumes V1 and V2. This is an application of the theorem of probability of independent events. The same consideration is valid to times and magnitudes, so that the time and magnitude of the occurrence of a strong earthquake before the occurrence of another one is independent and not correlated with it, which is confirmed by a search with the data used before,
Acknowledgements We thank Dr. Servet Martinez and his collaborators who called our attention to this field of application for fracture statistics, we also thank Dr. Mario Pardo from the Geophysical Department of our University for his collaboration, and one of the authors (G.D.) thanks the National Fund of Scientific anditsTechnological Development from Chile for Grant No. 0545/85. Thanks are also due to R. Toledo for his help with the manuscript.
References Anagnos, T. and Kiremidjian, A.S., 1984. Stochastic time-predictable model for earthquake occurrences. Bull. Seismol. Soc. Am., 74: 2593—2611. Brillinger, D.R., 1982. Seismic risk assessment: some statistical aspects. Earthq. Predict. Res., 1: 183—195. Bullen, K.E., 1955. On the size of strained region prior to an
extreme earthquake. Bull. Seismol. Soc. Am., 45: 1. Freudenthal, A.M., 1968. Statistical approach to brittle fracture. In: H. Liebowitz (Editor), Fracture, An Advanced Treatise. Academic Press, New York, NY, pp. 591—619. Gutenberg, and Richter, C.F., 1956.Bull. Earthquake intensityB.energy and acceleration. Seismol. magnitude Soc. Am., 46: 105—145. Kanamon, J. and Anderson, D.L., 1975. Theoretical basis of some empirical relations in seismology. Bull. Seismol. Soc. Am., 65: 1073—1095. Kagan, Y. and Knopoff, L.,seismicity 1976. Statistical search for nonrandom features of the of strong earthquakes. Phys. Earth Planet. Inter., 12: 291—318. Kagan, Y. and Knopoff, L., 1977. Earthquake risk prediction as a stochastic process. Phys. Earth Planet. Inter., 14: 97-108. Kittl, P., 1984. Analysis of the Weibull distribution function. J.
Appl. Mech., 51: 221—222.
Kittl, P. and Aldunate, R., 1983. Compression fracture statistics of compacted cement cylinders. J. Mater. Sci., 18: 2947—2950. Kitti, P., Lillo, A. and Leon, M., 1984. Resistencia probabilistica a la compresiOn dean grarnto sano. IV Congreso Ingenierla de Minas. Universidad de Atacama, Chile, pp. 1—10. Lomnitz, C., 1974. Global Tectonics and Earthquake Risk. Elsevier, Amsterdam. Patwardhan, A.S., Kullcarni, R.B. and Tocher, D., 1980. A semi-Markov model for characterizing recurrence of great earthquakes. Bull. Seismol. Soc. Am., 70: 323-347. Weibull, W., 1939. A statistical theory of the strength of materials. Ing. Vetenskaps Akad. Handl., 151: 1—45.
Physics of the Earth and Planetary Interiors, 49 (1987) 222—224 Elsevier Science Publishers B.V., Amsterdam — Printed in The Netherlands
Earthquake risk and fracture statistics P. Kitti and G. DIaz Departamento de Ciencia e Ingenieria de Materiales (IDIEM), Facultad de Ciencias Fisicas y Matemtiticas, Universidad de Chile, Casilla 1420, Santiago (Chile) (Received May 27, 1986; revision accepted February 16, 1987)
Kittl, P. and Diaz, G., 1987. Earthquake risk and fracture statistics. Phys. Earth Planet. Inter., 49: 222—224. On the basis of fracture statistics, which establishes a. relation between the cumulative probability of fracture in a brittle body and the stress field to which it is subjected, it is possible to describe some seismological statistics. The cumulative probability of time intervals between two consecutive strong earthquakes and the cumulative probability of the strong earthquakes magnitude, appears as Weibullian. Comparison with Chilean set data gives evidence to the theory.
According to the Weibull theory (Weibull, 1939; Freudenthal, 1968; Kitti, 1984), the cumulative failure probability of a brittle body in a nonhomogeneous stress field, is given by F( a)
=
1
—
exp
(1 — -~-
f4) [a ( ,~)}
dV
}
(1)
where J’~is the volume unit, V the body volume, a is the maximum tractional stress undergone by the body of fracture, a(r) is the stress field, r is the vector position and 4) is usually the function
q(a)
=
J
~
= ~
—
=
where i is the average position vector. The volume V is the mass subjected to stress, and supposing that a uniform increase of stress produces a practically uniform movement, then it is possible to put V = At (5) where t is the time measured between earthquakes, and A is a constant. Formula (4) may be transformed into the following expression
(o_ai~~
ao
10
(2)
/
[V 1— exp~—~~~4)[a(i)]}
0031-9201/87/$03.50
in
~(t)
=in t+1n{~~-4)[a(~)]}
(6)
a
In formula (2) aL is the stress below which no fracture occurs at all, a0 and m are constants of the body. If the integral of formula (1) is evaluated where there is a substantial increase of stress with respect to the initial state when the average stress 0, and with the integral average theorem is obtained, with aL = o F(a)
and with some transformations 1 V
(3)
© 1987 Elsevier Science Publishers B.V.
which allows us to make a Weibull diagram by plotting in E(t) versus in t, which gives a straight line. To make such a Weibull plot, we used a set of 110 strong Chilean earthquakes that occurred between the latitudes of 180 S and 410 S from 1906 to 1975 (M. Pardo, personal communication, 1975; Regional Catalogue of Earthquakes) (Fig. 1). The slope of the straight line passing through the points is practically 1, which represents a strong support
223
99
0/,
eqs. 7 and 8 we get
-
900/. 80!.
-
70/. 60°!.
C=~/j7j~•10fl/2 (9) 2M where C is an additive constant that may be a=C.lOa/
-
50°!. 40•!. 30°!. 950/,
--
200/.
-
Jo.,.
-
,j/0
ignored here. In formula (4), because a = a(M) after eq. 9, E(a) is only a function of the experimental probability F(M) of a certain earthquake of magnitude M, and hence ~(a) ~(M) and formula (4) is transformed into 1 V E(M)=lnlF(M) =~j~4)(a)
-
0
o I °!.
-
______________________
1/365
I 1/36
I 1/12
1/4
I 1/2
I 1
I .1.1 2 4 6
Time between Earthquakes 1 year
Fig. 1. Weibull plot of the cumulative probability F(t) of time (1906—1975). intervals between two consecutives strong earthquakes, in Chile
j
=
74)(C’. iOa/2M) m VC 10~u,,’2M 10 2ML (10) a~ a/ where C. 10~~/2ML = GL and V is supposed to be equal for all the earthquakes. If we plot in ~(M) vs. ln(10~/2M 10a/2ML) for 64 earthquakes of magnitude M ~ 6 that occurred in Chile between the latitudes of 180 S and 410 S from 1906 to 1975 (Pardo, 1975), we get the .graph shown in Fig. 2.
(
—
—
—
)
—
for this theory. The supposition V = At is in accordance with (Lomnitz, 1974) the tectonics of the Chilean zone, and fracture statistics come from the propagation of faults that occur during seismological movements (Kanamori and Anderson, 1975). Probabilistic models of strong earthquakes are developed in different ways (Kagan and Knopoff, 1976, 1977; Patwardhan et al., 1980; Brillinger, 1982), and in some cases (Anagnos and Kiremidjian, 1984) the Weibull distribution is used. Regarding the distribution of earthquakes of magnitudes M, the relationship between the energy E dispersed in an earthquake and the magnitude M thereof, is according to Gutenberg and Richter (1956), Kanamori and Anderson (1975) as follows iogE=aM+/3 (7) where a 1.5 and fi 11.8. For a finite volume V the seismic energy liberated, according to Bullen (1955) is2kV (8) E = ~ia where ~ is the rigidity and k is a coefficient. From —
—
~
95
0/• 0/
/
.
~ ~ 70°!. 60 ~/. 50 °/~
~oY. 30 °/. 20
1.
~ ~o
I
&
o s ./. 2 •/.
—
I •/.
—
a I 60
I I 65 10 75 Earthquake Megnltude hi
I 80
I 85
Fig. 2. Weibull plot of the cumulative probability F(M) of the strong earthquakes with magnitude minor or equal to M, in Chile (1906-1975).
224
Because we have limited ourselves to the consideration of magnitudes M ~ 6 a correction may be carried~outin formula (10) and we obtain it in this way 1 E M) = ln 1 [F( M) + FL]
(
—
VCm I 10a/2M
—
10a/2ML
,n
a0
=
where FL is the cumulative probability neglected, and here ML = 5.9 in order to avoid a point at infinity. Since the graph obtained is nearly a straight line, this implies that FL 0. This means that in this seismological zone an earthquake with M < 6 does not imply a fracture of the rock. The slope of the straight line is m = 0.87. In safe granite (K.ittl et al., 1984) m 2, this difference implies that the great mass of rock subjected to stress is very dispersive, In the present work it is worthwhile to emphasize the contributions of fracture statistics to seismological risk, which consists roughly of the graphs in Figs. 1 and 2. These graphs are represented by formulas (6) and (11); it is probable that a careful research of tensions in situ and fracture properties of rock at high pressures can give more information about seismological risk. Finally if we remember the foundations of statistical mechanics (Kittl and Aldunate, 1983; Kittl, 1984) these two formulas, (6) and (11), mean the complete aleatory of the seismological events in this case. In equations this funda.ment is as follows
.P( V1 +
V2) =
.P( V1) .P( V2) .
(12)
which means that the probability of a fracture F not occurring in the volume V1 + V2 is the product of the probabilities of the fracture not occurring in the volumes V1 and V2. This is an application of the theorem of probability of independent events. The same consideration is valid to times and magnitudes, so that the time and magnitude of the occurrence of a strong earthquake before the occurrence of another one is independent and not correlated with it, which is confirmed by a search with the data used before,
Acknowledgements We thank Dr. Servet Martinez and his collaborators who called our attention to this field of application for fracture statistics, we also thank Dr. Mario Pardo from the Geophysical Department of our University for his collaboration, and one of the authors (G.D.) thanks the National Fund of Scientific anditsTechnological Development from Chile for Grant No. 0545/85. Thanks are also due to R. Toledo for his help with the manuscript.
References Anagnos, T. and Kiremidjian, A.S., 1984. Stochastic time-predictable model for earthquake occurrences. Bull. Seismol. Soc. Am., 74: 2593—2611. Brillinger, D.R., 1982. Seismic risk assessment: some statistical aspects. Earthq. Predict. Res., 1: 183—195. Bullen, K.E., 1955. On the size of strained region prior to an
extreme earthquake. Bull. Seismol. Soc. Am., 45: 1. Freudenthal, A.M., 1968. Statistical approach to brittle fracture. In: H. Liebowitz (Editor), Fracture, An Advanced Treatise. Academic Press, New York, NY, pp. 591—619. Gutenberg, and Richter, C.F., 1956.Bull. Earthquake intensityB.energy and acceleration. Seismol. magnitude Soc. Am., 46: 105—145. Kanamon, J. and Anderson, D.L., 1975. Theoretical basis of some empirical relations in seismology. Bull. Seismol. Soc. Am., 65: 1073—1095. Kagan, Y. and Knopoff, L.,seismicity 1976. Statistical search for nonrandom features of the of strong earthquakes. Phys. Earth Planet. Inter., 12: 291—318. Kagan, Y. and Knopoff, L., 1977. Earthquake risk prediction as a stochastic process. Phys. Earth Planet. Inter., 14: 97-108. Kittl, P., 1984. Analysis of the Weibull distribution function. J.
Appl. Mech., 51: 221—222.
Kittl, P. and Aldunate, R., 1983. Compression fracture statistics of compacted cement cylinders. J. Mater. Sci., 18: 2947—2950. Kitti, P., Lillo, A. and Leon, M., 1984. Resistencia probabilistica a la compresiOn dean grarnto sano. IV Congreso Ingenierla de Minas. Universidad de Atacama, Chile, pp. 1—10. Lomnitz, C., 1974. Global Tectonics and Earthquake Risk. Elsevier, Amsterdam. Patwardhan, A.S., Kullcarni, R.B. and Tocher, D., 1980. A semi-Markov model for characterizing recurrence of great earthquakes. Bull. Seismol. Soc. Am., 70: 323-347. Weibull, W., 1939. A statistical theory of the strength of materials. Ing. Vetenskaps Akad. Handl., 151: 1—45.