Vol. 12 No.4 (419-427)
ACTA SEISMOLOGICA SINICA
Jul., 1999
Research on percolation model and criticality of seismicity* SHAN-MING KE (;~-j~-~I~) HAO-DING GU (}~, "~ ~B) WEN-JIE ZHAI (,~ 5~,,d~,,) Seismological Bureau of Liaoning Province, Shenyang 110031, China
Abstract Making use of modem nonlinear physics theory and earthquake focus theory, combined with seismicity characteristics, the percolation model of earthquake activity is given in this paper. We take the seismogenic process of a large earthquake as a phase transition process of percolation and apply the renormalization method to phase transition of percolation. The critical property of the system, which is like percolation probability exponential and correlative length exponential, etc, can be calculated under the fixed point as which in the renormalization transformation infinite correlative length in percolation phase transition is taken. The percolation phase transition process of two large earthquakes, which are Haicheng and Tangshan event occurred in 1975 and 1976 respectively, has been discussed by means of seismicity data before and after two shocks. Key words: percolation model
seismicity
percolation phase transition
renormalization method
critical exponential
Introduction Earthquake prediction is a main objective in traditional seismicity study, which is to attempt to find some criterion of precursory earthquake activity by comparing different patterns of seismicity in hopes of predicting occurrence of a major shock. The result is often disappointing. Preearthquake activities can differ greatly for impending earthquake, although all the rupture processes of large earthquakes are essentially the same as far as we know so far. As already discussed in the result of research on seismicity evolution (Gu, Sun, 1992), the self-enlargement behavior of seismicity fluctuation in seismic source system under impending earthquake state means that the system behavior is indeterminate and unpredictable. It follows that we must apply new theory and method to studying complex seismicity phenomena and move round the difficulty of unpredictable behavior. The percolation model is a math one (Broadbent, Hammersley, 1957) to describe random extending and flowing of fluid in disorder medium, for example, the fluid flowing in multiholed medium. Assuming that there is a 2-dimension square lattice and its lattice site or bond is able to be occupied randomly, there appears an infinite percolation group on the points array when the occupied probability attains to some critical value (i.e. percolation threshold). The percolation group consists of occupied sites (bonds) to join each other and is called site percolation. From phase transition theory in physics, it must be pointed out that to appear percolation group means occurrence of percolation phase transition. Percolation group is possessed of statistical self-similar structure and is a typical fractal. Percolation phase transition corresponds to second order phase transition. Percolation probability is regarded as the order parameter in percolation phase transi"Received December 24, 1998; revised May 12, 1999; accepted May 12, 1999.
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tion. While occupied probability is less than percolation threshold percolation probability identically equals zero, and occupied sites (bonds) are independent each other and disordered. While occupied probability is greater than percolation threshold there appears percolation group on the points array, and occupied sites (bonds) have been not independent and is incident to form ordered space structure. Long-range incident grows out of nothing on the points array. The evolution theory of seismicity (Gu, Sun, 1992) has proved that seismicity is disordered in general case, thus earthquake source system related to these events is also a disordered structure. Now we want to know if the fluctuation of seismicity in the disordered structure can be described by percolation model. In fact, rupture process of earthquake source system and percolation phase transition are quite similar. It is well known that seismogenic process of a large earthquake is just the process from preseismic activity to aflershock, and the process always undergoes a sudden change like the occurrence of major earthquake. Before a large earthquake the occurrence of moderate and small shocks in seismogenic region is possessed of randomness, and their space-time structure is disordered, that is, small size fault activity related to moderate and small shocks is disordered, too. Large size fault activity can occur and generate large earthquake while incident of small size fault activity increases gradually and attains to long-range incident. It is the criticality that the system represented by earthquake source or seismicity undergoes. Based on the above idea we attempt to apply percolation model to studying seismicity characteristics under the criticality before large earthquake. Obviously, the study will carry earthquake prediction to a new stage.
1 Percolation model of seismieity Percolation model has been extensively applied to describing critical phenomena of system. In seismology seismologist has studied the problems such as linking of rupture by means of percolation model (Bebbington, et al, 1989; Robertson, et al, 1995). The latest research result shows that extremely simple case of seismic rupture process corresponds to the bond percolation model (Wu, 1998). Therefore, we can take moderate and small earthquake activity which are neighboring in space as bond linking group. By the research of distribution and structure of the group near by the percolation threshold, we shall discuss critical characteristics of the system during percolation phase transition. We regard geologic environment in which a large earthquake is forming as a whole generalized seismic source system (Gu, Sun, 1992), and simulate it to 2-dimension array which is composed of basic seismogenic blocks with rupture intensity o); We can relate moderate and small shocks with the fault activity of basic seismogenic blocks during seismogenic process. According to the results of research of rock rupture, the rupture probability of basic seismogenic blocks is assumed to satisfy second Weibull distribution p(cr r < ao-)= Pa = 1-exp[- (ax) 2]
(1)
where x=o-/cr0, o-is applying stress, o-0 is a the reference intensity of basic seismogenic blocks, a is the size parameter. Therefore, the probability of o)
When rupture probability p~ attains to a critical value, i.e. percolation threshold Pc, there appears a infinite percolation group in the system, which consists of basic seismogenic blocks linked
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up by ruptures, and percolation phase transition has occurred this moment. The order parameter during phase transition is an important physical quantity, and describes ordered state of the system. The order parameter is percolation probability for percolation model. When rupture probability is p~, the probability for any basic seismogenic block on the array to belong to percolation group is referred to as percolation probability, which is written as p~o (Pl). When pl is less than percolation threshold Pc, percolation probability is identically vanishing because there exists no percolation group in the system. When p~ is greater than percolation threshold, there appears percolation group in the system and percolation probability p~ (P0 would increase quickly with the increase ofpl. It means that infinite group annexes other finite groups, and the whole array has been occupied by an infinite group. Percolation probability near by percolation threshold can be written as follows (Stanley, 1985): Poo(P,) - I P , - Pcl p
(3)
where fl is referred to as percolation probability exponential. Another important physical quantity in the percolation phase transition is correlation length ~. In our system, it refers to an average distance between any two basic seismogenic blocks in the group. When rupture probability of the system does not attain to Pc, incident length of the system is finite. The largest rupture scale just is incident length. At critical point of phase transition, incident length of the system also tends to infinite, i.e. so called long-range incident, because there exists percolation group. The rupture far exceeding basic seismogenic block will lead to the occurrence of macroscopic sudden change in the system. Near by percolation threshold, correlation length is expressed as - [ p ~ - p c ] -v
(4)
where v is correlation length exponential. We have discussed several critical exponentials of the system which attains to percolation threshold. They not only describe characteristics of the system in criticality, but are related to the fractal number of percolation group. The relation can be expressed as (Takayasu, 1986).
D = d(5) where D is the fractal number of percolation group, d is space dimension. At critical point of phase transition, fractal number of the system approximates to one of percolation group because there exists percolation group. With the increase of rupture probability, percolation group annexes other finite groups, and whole system is occupied by an infinite group. Therefore, fractal number of the system gradually approximates to space dimension.
2 Research on percolation phase transition of seismicity by renormalization method It will be firstly faced with the difficulty of infinite degree of freedom for traditional math method to describe a physical system in criticality, because there are infinite particles or large and small cracks, e t c in this system. Secondly, we can not describe the fluctuation of phase state related with different scales only by a few critical parameters. It is the renormalization method that can overcome above difficulty and provide a technique simplified phase state fluctuation. Applying the method, firstly the system would satisfy nonscale. For a generalized seismic focal system to
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relate large and small shock events with large and small cracks, Gu, et al have discussed selfenlargement behavior of seismicity fluctuation of the system at critical point and proved that system behavior is independent of rupture scale (Gu, Sun, 1992; Gu, Chert, 1997). In reality, focal system contains various large and small rupture scale, thus renormalization method is allowed to be used. The purpose of renormalization is to simplify describing fluctuation behavior of phase state of the system and is able to obtain quantitative expression of the system in criticality. Correlation length of the system at the critical point of occurrence of second order phase transition tends to infinite. Infinite correlation length is consistent with whole macroscopic rupture in earth medium. The system possesses invariance (i.e. scale invariance) under scale transformation and must be possessed of self-similarity. We can obtain a variety of critical exponentials at critical point of the system by scale invariance. We select the basic seismogenic block containing four squares as an operation unit (Figure 1). The mechanism of stress transfer between blocks in the unit is introduced in computing and assuming that the unit was not regarded as whole rupture until all blocks had ruptured. In order to measure resulting rupture of blocks from stress transfer, we can use conditional probability P~b to express. It means that the probability of occurrence of rupture for the block while stress (a-b)cr is transferred to unruptured block with stress bo-. It can be written as follows (Smalley, Turcotte, 1985). Figurel Illustration of the two dimenp(bo" < o'/<_ act) (6) sional source models with four basic seismogenic blocks P"'b= p(cr;>b~r) per unit It is obtained by calculation: Pu,b
--
P~ - Pb 1 -Ph
-
(7)
-
Therefore, after renormalization we have a new array structure and the relation between new array probability P l and P t, i.e. renormalization group transformation p; = p~ + 4 p ? ( p , - p ~ ) + 6 p ~ ( p z - p l ) Z + 1 2 p Z ( p z - p ~ ) C o 4 - p 2 ) + 4 p , ( p 4 / 3 - p l ) 3 + 12P, (P4/3-
Pl)2(P4- P4/3) + 12Pl C°4/3- P l ) ( P 2 - P4/3)z+
(8)
24p,(P4/3 - P,)(Pz - P4/3)(Pa - Pz)= R(Pl) After renormalization group transformation, correlation length at critical point still is infinite, so P~=Pc is a fixed point of renormalization group transformation and equation (8) can be expressed as
pc=R(pc)
(9)
From equation (9) three fixed points solved are such as Pc = 0, 0.170 7, 1, where pc=0, 1 are stable fixed points; pc=0.170 7 is unsteady one which is a critical point of phase transition. We make a Taylor expansion near by critical point:
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P; =Pc + 2 ( P l - P c ) + ° ( P l - P c ) 2
423 (10)
Thus from (4) we have v=
lng
=0.808 4
(11)
ln2 Where g=~/~:'=2, it is relative variation of correlation length after renormalization.
~,dPl )p,=p, It is same to get fl = 0.428 9. The fractal number of percolation group at the critical point can be given from equation (5), that is D = d-(fl/v)
= 1.470
(12)
Different critical exponentials in percolation threshold relating to percolation process of seismicity have been obtained by renormalization method as stated above. We can apply seismicity data before and after a large earthquake to discussing critical conditions of occurrence of large shock by critical exponential.
3 Research of earthquake example Having expounded the percolation model of seismicity, a practical applying study has been performed by seismicity data before and after Haicheng earthquake, in 1975 and Tangshan shock, 1976. The criticality of seismicity in focal system before a large earthquake is quantitatively discussed. For Haicheng earthquake, selected study region is from 122°E to 124°E, from 40°N to 42°N. The study region is divided into small areas with 0.2°×0.2 ° as basic seismogenic unit. Seismogenic region and basic unit depend on seismic focal scale of practical earthquake. Time interval of seismicity studied is from 1970 to 1995. Selected lower limit of magnitude is ME=3.0. The occurrence (or fault activity) probability p of basic unit in seismogenic region is 0.75 F computed at first. The probability represents 0. 50Ithat the basic units in which shocks have oc~" curred makes up percentage of total seismo0. 0 0 L ~ - " - ~ I J I genic units. Figure 2 gives the p - t curve from 1970 1975 1980 1985 1990 1995 1970 to 1995. As seen in the figure thatp value ¥e~ has increased to near by percolation threshold Figure 2 The p-t diagram in the percolation phase transition process for pc= 0.170 7 in February, 1975. Thus Haicheng Haichengearthquake earthquake occurred. Then p value increased rapidly with aftershock activity, it attained to 0.46 in December, 1995. Figure 3 shows a picture of percolation phase transition near by percolation threshold. As seen in Figure 3 that the infinite group, that is, percolation group which is composed of ruptured basic seismogenic units is formed from up to down. Percolation probability exponential, correlation length exponential and fractal number, etc, can be obtained from equations (3), (4) and (5). The results are shown in Table 1.
0.z5|~- J
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Table 1 The calculatin~ result near b~,percolation threshold for Haichen~;earthquake a-mo-d 1975-02-03 1975-02-03 1975-02-04 1975-02-04 1975-02-04 122"E
p 017 0.18 0.19 0.20 0.21
fl 0.331 0.436 0.498 0.485 0.513
123"
5 2 0 8 1
124" :42*N
~ 31.70 37.43 37.33 53.71 53.18
v 0.478 7 0.774 3 0.917 1 1.128 1227
D 1.307 1 437 1 457 1.569 1.582
0. 75[-
O. 25 F 0, 0 0 |
J
1972
197,1
19t78
1976 Year
41"
t
1980
Figure 4 The p-t diagram in the percolation phase transition process for Tangshan earthquake
VA
117"E
118"
1.19" 40. 6*N
N
40*
~
Figure 3 The percolation group in the percolation phase transition process for Haicheng earthquake (p = 0.20)
39.6 °
Concerning Tangshan earthquake in 1976, selected study region is from l17°E to 119°E, from 38.6°N to 40.6°N. Time range is from June of 1972 to 1980. The division of basic seismogenic unit and lower limit of magnitude 38. 6* are same as Haicheng earthquake. Figure 4 Figure 5 The percolation group in the percolation phase transition process for Tangshan and Figure 5 show p-t curve and percolation earthquake (p = 0.22) phase transition for Tangshan earthquake respectively. As seen in Figure 4 that p value was approximately equal to percolation threshold in February, 1976. After Tangshan earthquake in July 1976, p value quickly increased and attained to 0.54 in 1980. As seen from Figure 5 that the percolation group which consists of ruptured basic units is already formed from left to right. Different critical exponentials near by percolation threshold are shown in Table 2.
N
Table 2 The calculating result near by percolation threshold for Tangshan earthquake a-mo-d 1976-02-22 1976-07-28 1976-07-28 1976-07-28 1976-07-28 1976-07-28
p 0.17 0.18 0.19 0.20 0.21 022
fl 0.331 5 0.492 2 0.559 1 0~600 6 0.630 4 0.584 7
¢ 33.25 34.72 36.28 54.17 56.89 57.62
v 0,482 3 0.758 3 0.909 7 1 131 1.248 1.347
D 1.313 1.351 1,385 1.469 1495 1.566
Calculation results show that whether Haicheng earthquake or Tangshan shock of which the critical characteristics of pre-earthquake activity near by percolation threshold value are same. Before major shock occurred, both percolation probability exponential and correlation length exponential gradually increase from lower value and jump over their critical exponential, that is, 0.428 9 and 0.808 4. Meanwhile, fractal number of percolation group and correlation length of
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focal system are also gradually augmented with the increase of occurrence probability p.
4 Discussion and conclusions (1) There appeared evident different seismicity before Haicheng earthquake and Tangshan shock, but their transition behavior of pre-earthquake activity is not obviously different. Plentiful foreshocks before Haicheng earthquake mostly occurred in ruptured basic seismogenic uoits and seismic occurrence probability did not obviously increase. On the contrary, though there was no any sensible pre-earthquake for Tangshan shock, rupture probability as a whole focal system had attained to percolation threshold as early as February, 1976. In terms of this significance, from February of 1976 to July in which Tangshan shock occurred the focal system was in criticality all along. Compared with Haicheng earthquake, they showed different self-enlargement behavior of fluctuation at transition point and different unpredictability of system behavior. (2) According to research on different critical exponential of seismicity near by percolation threshold in some area, a quantitative judgement which is to understand if an earthquake will occur in the area for medium and short term can be given. Because percolation probability as order parameter of percolation phase transition is a measure of percolation group size, and correlation length is a measure of incidental extent of ruptured basic units in the system, therefore critical exponential of percolation phase transition possesses obvious physical meaning. They are necessary condition with which the system would generate macroscopic sudden change. Final occurrence of sudden change will depend on influence of various factors on fluctuation. (3) In practical application we can select some study region with 2 ° x 2 ° and divide it into basic units with 0.2°x0.2 °. Time range studying can be selected as about five years. The step width of sliding can be selected as one year when computing seismic occurrence probability in the study region. As long as the occurrence probability attain to percolation threshold and percolation probability exponential and correlation length exponential amount to critical value, it is possible to give a conclusion predicting a large earthquake in the study region for medium and short term. The research result is only primary, of course, new study will be advanced in future research on earthquake prediction. (4) As for the process forming percolation group near by transition point, it easily make us associate with the concept relating to seismicity belt and seismic gap, etc, before large earthquake. In fact, the process in which a group gradually augments and forms percolation group near by percolation threshold can be realized by seismic belts and gap. The theory of percolation would maybe become theory basis of quantitative research on forming seismic belt and gap. We would like to express heartfelt thanks to Professor ZHONG-LIANGWU for his warm support and help. This project is supported by the Scientific and Technological Foundation of Liaoning Province (972016) and Scientific Research Foundation of Seismological Bureau of Liaoning Province (97-Q-02). References Bebbington M, Vere-Jones D, Zheng X. 1989. Percolation theory: a model for rock fracture? Geophys J Int, 100:215-220 Broadbent S R, Hammersley J M. 1957. Percolation processes. Proceeding of the Cambridge Philosophical, 53:629-641 Gu H D, Sun W F. 1992. Self-organization and evolution of seismicity. Acta Geophysica Sinica, 35(1): 25-36 (in Chinese) Gu H D, Chen Y T. 1997. Physics of seismic gap and nonscale of focal system. In: Chen Y T edit-in-chie£ Advance of Chinese Seismology Research. Beijing: Seismological Press, 37-40 (in Chinese)
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Robertson M C. Sammls C G, Sahimi M, et al. 1995. Fractal analysis of three-dimensional spatial distributions of earthquake with a percolation interpretation. J Geophys Res, 100:609-620 Stanley H E. 1985. Renormalization and percolation. Advance of Physics Research, S(I): 1-65 (in Chinese) Smatley JR R F, Turcotte D L. 1985. A renormalizationgroup approach to the stick-slip behavior of faults. JGeophysRes, 90:1 894 I 900 Yakayasu H. 1986. Shen B M, Chang Z W trans. 1989. FractalBeijing: SeismologicalPress, 42-63 (in Chinese) Wu Z L 1998. Implicationsof a percolationmodel for earthquake "nucleation".GeophysJ lnt, 133:104-110
Appendix T h e calculation o f critical indices on percolatJgn p h a s e transition process o f seismicity According to the results of research of rock ruptm I, the rupture probability of basic seismogenic blocks is assumed to satisfy second Weibull distribution: p ( ~ < aa)=po=l-exp[ - (ax) 2]
(1)
where x=o'/cr0, ~ is applying stress, a 0 is a reference intensity of basic seismogenic blocks, a is the size parameter. Therefore, the probability of o)
(2)
Thus from equations (1) and (2), we obtain
p~ = 1 - (l - pE )°2
(3)
In order to measure resulting rupture of blocks from stress transfer, we can use conditional probability p,. ~ to express. It means that, the probability of occurrence of rupture for the block while stress (a-b)cr is transferred to unruptured block with stress ba. It can be written as follows:
p(ba bo-)
(4)
We obtain Po from a density integral of probability:
[aa/al, dpl
P. = ~o
dx
dx
Thus
p(ba
' ax:
p(o-; > b o ' ) = l - p(o'! _
po-p
Pt,
Therefore from equation (4), we obtain P° p.,h
-
-
-
1 -
Pb
(5)
p~
We select the basic seismogenic block containing four squares as an operation unit. The mechanism of stress transfer between blocks in the unit is introduced in computing and assuming that the unit was not regarded as whole rupture until all blocks had ruptured. Therefore, after renormalization we have a new array structure and the relation between new array probability Pl and p~, i.e. renormalization group transformation: t p,=p~ +4p?(1-p,)p4, , +6p~(l-p,)
2
2 b2,,
+2p2,,(I-P2,,)P4, z]+4p,(1-P,
)3
x
2 4'3 +3P4/3,1(|--P4/3,1) 2 [P2,4/3 2 ]~IpJ/ {p43/3,1 +3P4/3,1(I--P4/3,1)P4, + 2P2,4/3(1-- P2,4/3~4,2
(6)
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From equations (3) and (5), we obtain Pl = p4 + 4 p ( ( p 4 - Pl ) + 6p~ (P2 - P, )2 + 12p~ (P2 - Pl ) ( P 4 - P 2 ) + 4p, (P4/3 - Pl )3 + 12pt (P4/3 - PI )2 (P4 - P4/3 ) + 12p, (P4/3 - P, )(P2 - P4/3)2 +
(7)
2 4 p , (P4/3 - P, )(P2 - P,/3 )(P4 - ' P 2 ) = R ( P l )
where R (P0 is a function o f p v After renormalization group transformation, correlation length at critical point still is infinite, so PJ=Pc is fixed point of renormalization group transformation and (7) can be expressed as Po=R( Pc )
(8)
From equation (8) three fixed points solved is such as pc=0, 0.170 7, 1, where pc=0, 1 are stable fixed points, pc=0.170 7 is unsteady one which is a critical point of phase transition. We make a Taylor expansion near by critical point: Pl = Pc + 2(Pl - Pc )+ °(Pl -Pc)2
(9)
where 2
~2.357
(dpl)
Therefore from p~(p,)-lp,
- pcl p
We obtain p',: - )./Jp~ f l = lngl ~0.428 9 In2
(10)
From
~-IP,- Pot-" We obtain
v=
lng2
z 0.808 4
(11)
In2 where g~ = p'~/p,~ .~ 1.444, g2 = ~ , ~ ' = 2. They are respectively relative variation of percolation probability and correlation length after renormalization. The fractal number of percolation group at the critical point can be given from above results, that is D = d - ( f l / v ) ~ 1.470
where d = 2, it is space dimension.
(12)
