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Seismicity Acceleration Model and Its Application to Several Earthquake Vol.12 No.1 (35~45) ACTA SEISMOLOGICA SINICA Jan., 1999 Seismicity acceleration model and its applica- tion to several earthquake regions in China WEN-ZHENG YANG (~ ~i~) LI MA (~ ~) Center for Analysis and Prediction, China Seismological Bureau, Beijing i 00036, China Abstract With the theory of subcritical crack growth, we can deduce the fundamental equation of regional seismicity accel- eration model. Applying this model to intraptate earthquake regions, we select three earthquake subplates: North China Subplate, Chuan-Dian Block and Xinjiang Subplate, and divide the three subplates into seven researched regions by the difference of seismicity and tectonic conditions. With the modified equation given by Somette and Sammis (1995), we analysis the seismicity of each region. To those strong earthquakes already occurred in these region, the model can give close fitting of magnitude and occurrence time, and the result in this article indicates that the seismicity acceleration model can also be used for describing the seismicity of intraplate. In the article, we give the magnitude and occurrence time of possible strong earthquakes in Shanxi, Ordos, Bole-Tuokexun, Ayinke- Wuqia earthquake regions. In the same subplate or block, the earthquake periods for each earthquake region are similar in time interval. The constant a in model can be used to describe the intensity of regional seismicity, and for the Chinese Mainland, a is 0.4 generally. To the seismicity in Taiwan and other regions with complex tectonic conditions, the model does not fit well at present. Key words: seismicity acceleration model subcritical crack growth China earthquake region fit Introduction Before the occurrence of a strong regional earthquake, the seismicity in the region is com- monly changed from weak to strong, this phenomenon bespeaks the process of energy release in the region. After the occurrence of strong regional earthquake and its aftershock, the seismicity in the region recovers silence, as time goes by, changed from weak to strong again, later, another strong earthquake would occur. If we call the interval time between two strong earthquakes in a region to be an earthquake period, and the seismicity in this region would be changed from weak to strong in the period, then it would be beneficial for us to acknowledge which stage is this seis- micity being and to set up a model to predict the following strong earthquake. In one earthquake period, we define remain time is the end time of an earthquake (when the next strong earthquake occurs) minus the time of the earthquake event in the period (Figure 1). Pa- l~emaintime(tr__~ pazachos (1973), Jones and Monlar (1979) found the number of earthquake events in a time unit is in Figurel The definition of earthquake period direct proportion to its remain time to the minus and remain time * Received May 28, 1998; revised September 1, 1998; accepted September 16, 1998. 36 ACTA SEISMOLOGICASINICA Vol. 12 some power. Varnes (1989) pointed out the following relationship exists between the energy of foreshock series and time d.(2 /dt = C /(tf - t)" (1) where t is the currence time of one foreshock, tf is the time of mainshock occurrence, C and n are constants, .62 is a measurable quantity which is used for describing strain. Equation (1) can also be written as X~ = A + B(t~ - t) ~ (2) where A and B are constants, m=l-n, .Ocan be gotten through the following equation lgD = cM + d (3) When M is magnitude, c=1.5, .(2 representes seismic moment, d is constant; when M is Benioff strain release or square root of moment, c=0.75, d is constant; when M is the number of earth- quake events, c=0, d=0 (Bufe, et al, 1994). Varnes and Bufe found Benioff strain release to be especially useful to analysis practical catalogue, and Robinson also use Benioff strain release to analysis the seismicity of New Zealand. Equation (3) can be written as Y_,Mo = A + B(t r - t)" (4) where Y_,Mo is the observed accumulated seismic moment, to some power a range from 0 to 1 (usually to be 0.5), at time t. A, B and m are constants to be found, along with the time of ultimate failure (tf). With equation (4), Bufe and Varnes (1993) analyzed two earthquake periods in San Francis- co Bay region of the United State (1906 San Francisco Ms=8.3 earthquake and 1989 Loma Prieta Ms=7.1 earthquake as the end earthquake event of tx, o period respectively). Using the earthquake events in each period except the last event to fit the last event with equation (4), they got accurate tf and Mf for each of the two strong earthquake in the region. Sornette and Sammis (1995) modi- fied equation (4) by leading into the conception of normalized group, which make equation (4) be with discrete hierarchical structure, and get the following form ~M°=A+ B(tf-t)'II+Cc°s(2~ lg(tf-t)lg2 +tp )1 (5) Where C, 2 and (p are constants to be determined. Generally, equation (5) provides a better fit to the observed seismicity data than equation (4). Robinson (1997) used equation (5) to analysis the seismicity of New Zealand systematically, and divided several hazardous regions in New Zealand for the coming years. 1 Theory of seismicity acceleration model We can build the relationship between regional seismicity and time according to existed rock mechanics theory and experiment. If we regard regianal seismicity as energy release of crack due to the tectonic block continues to grow under long time stress, the growth of crack coming to ut- most would make the rock ultimately broke, and lead to a strong earthquake, then the theory of crack growth would do more help to build seismicity-time model. About the theory of crack growth, there are detail literatures on fracture mechanics of crack, but in relationship with practi- No. I YANG, W. Z. et al: SEISMICITY ACCELERATION MODEL AND ITS APPIdCATION 37 cal seismicity, the theory of subcritical crack growth (Atkinson, 1979) may provide a better ex- planation. For crack with arbitrary mode, in a homogeneous and linear elastic medium, the stress com- ponents near the crack top are proportional to r z/2, where r is the distance measured from the crack, the coefficient of the r la term in the stress is the operator of stress intensity (k), which char- acterizes the intensity of the stress field at the end of crack. Classical fracture mechanics postu- lates that the crack in a linear elastic solid will propagate once a critical stress intensity factor Kc has been reached, however, under the circumstance of long period of loading the classical fracture mechanics approach breaks down, especially if high temperatures or reactive environments are presented. By observing in glass, engineering material, rocks and minerals individually, Atkinson (1982, 1984) found that significant rates of crack extension can occur at value k, which may be substantially lower than Kc, this phenomenon is known as subcritical crack growth. For a two-dimensional crack in any mode, the stress intensity factor is given by k = Yfr(~X) I/2 (6) where fir is the far-field stress, Y is a numerical modification factor, X is the crack length for a two- dimensional crack or the radius for a circular crack. Charles (1958) used a different stress dependence of chemical reactives involved in static fa- tigue of grass and arrived at the following expression for subcritical crack growth J( = voA exp(-H / RT)k" (7) Vo and n are constants, n>2, Aexp(-H/RT) is the constant of reaction equilibrium. This equation also fits a large number of experimental data on many different classes of materials, including rocks and minerals (Atkinson, 1982, 1984). Equation (7) is most commonly used to describe ex- perimental data on subcritical crack growth in geological material. Combining (6) and (7) J( = voA exp(-H / RT) x [Yf r (~ X) I/2 ]" (8) During the subcritical crack growth, the 6~ can be treated as a constant, and v=vo, X=Xo, then v o = voA exp(-H / RT) x [Y8 r (it X o )1/2 ],, (9) Combining (8) and (9) k = vo[(XIXo)l/2]" (10) v0 is the crack velocity at time zero and Xo is the crack length at time zero. Integrating to the equation (10), we can get i -12/(2-n) X= X~ 2-')/2 n- 2 Vot .| (11) 2 x:' l Das and Scholz (1981) simply set the quantity in brackets equal to zero, since this is where X goes to infinity at time to failure. X0 2 tf - -- -- vo n- 2 (12) tf is the failure time, which only depends on the initial condition and not on the final condition. 38 ACTA SEISMOLOGICA SINICA Vol. 12 Substituting (12) into (11), then ( t -~,,/(2-,,) -v f In fracture mechanics, the energy release rate G is defined to be the energy flux to the crack tip zone per unit crack length advance (per unit width along crack front) and there is a relationship as following G = Ck 2 (14) where C is constant. Combining (7), (13), (14) and the definition of G, we get E= A+ B(tf -t) m (15) where A is constant, represents the total energy released from time zero to failure; B and m are constant, E is the total cumulative energy released from time zero to t. Equation (15) gives the re- lationship between energy and time of subcritical crack growth, and it shows the cumulative ener- gy is proportional to the power index of the remaining time to failure. Equation (4) and equation (15) looks alike in form, but there exists some differences.
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