برآورد پارامتر هرست وابسته به زمان در بزرگی زمین‌لرزه‌های کالیفرنیا

نوع مقاله : Articles

نویسندگان

1 گروه آمار دانشگاه الزهرا، تهران، ایران

2 پژوهشکده زلزله‌شناسی، پژوهشگاه زلزله‌شناسی و مهندسی زلزله، تهران، ایران

چکیده

یکی از مهم‌ترین ویژگی­ها در سری­های زمانی زمین­لرزه، تعیین میزان همبستگی­هایی است که بین زمین­لرزه­ها در یک منطقه خاص وجود دارد. این مقاله به برآورد همبستگی­ها در سری­های زمانی مستخرج از داده­های زمین­لرزه کالیفرنیا، که طی سال­های 2002 تا 2016 رخ­داده­اند، می­پردازد. داده‌ها از 1 جمع آوری شده­اند. میزان همبستگی­ها یا حافظه دور­برد سری زمانی، توسط پارامتر هرست وابسته به زمان، ، و به­صورت موضعی، اندازه­گیری می­شود. پارامتر هرست را به کمک  الگوریتم میانگین متحرک روند­زدا یا به‌اختصار ، برآورد می­کنیم. دلیل استفاده از این روش برآورد، توجه به ساختار مقیاس پایایی زمین­لرزه­ها و همچنین نوسانات پارامتر هرست نسبت به زمان است. جهت ارزیابی دقت این روش، ابتدا پارامتر هرست را در داده­های شبیه­سازی شده برآورد می­کنیم و میانگین مربعات خطا  را، به­عنوان ملاکی از دقت روش، به دست می­آوریم. سپس  را جهت تعیین میزان تغییرات و همبستگی بین زمین­لرزه­های متوالی، در داده­های زمین­لرزه کالیفرنیا، محاسبه می­کنیم. در مواجهه با داده­های زمین­لرزه، مشاهده می‌کنیم که پارامتر هرست، دارای تغییرات قابل‌توجهی نسبت به زمان است درصورتی‌که در داده­های شبیه­سازی شده، این میزان تغییرات دیده نمی­شود. از مزایای روش ، این است که در این روش به هیچ­گونه فرض توزیعی از متغیرهای تصادفی، نیاز نداریم. به­علاوه، این روش، برتری قابل‌توجهی نسبت به روش­های موجک و طیف توان با مرتبه بالا دارد که بر اساس تبدیل لژاندر یا فوریه از گشتاورهای مرتبه  محاسبه می­شوند. با به­کارگیری این روش برآورد در داده­های زمین­لرزه کالیفرنیا، ملاحظه می­شود که مقدار  بین 4/0 و 6/0 نوسان می­کند. در مواردی که مقدار پارامتر از 5/0 بیشتر است، نشان­دهنده‌ی وجود همبستگی دوربرد به مقدار کم است و مقادیر پارامتر هرست کمتر از 5/0 بیانگر عدم همبستگی بین مشاهدات متوالی است.

کلیدواژه‌ها


عنوان مقاله [English]

Time-Dependent Scaling Patterns in Sarpol-e Zahab Earthquakes

نویسندگان [English]

  • Yasaman Maleki 1
  • Mostafa AllamehZadeh 2
1 Faculty of Mathematical Sciences, Alzahra University, Tehran, Iran
2 Seismological Research Center, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran, Iran
چکیده [English]

In this paper, the dynamics of the seismic activity and fractal structures in magnitude time series of Sarpol-e Zahab earthquakes are investigated. In this case, the dynamics of the seismic activity is analyzed through the evolution of the scaling parameter so-called Hurst exponent. By estimating the Hurst parameter, we can investigate how the consecutive earthquakes are related. It has been observed that more than one scaling exponent is needed to account for the scaling properties of earthquake time series. Therefore, the influence of different time-scales on the dynamics of earthquakes is measured by decomposing the seismic time series into simple oscillations associated with distinct time-scales. To this end, the Empirical Mode Decomposition (EMD) method was used to estimate the locally long-term persistence signature derived from the Hurst exponent. As a result, the time dependent Hurst exponent, H(t), was estimated and all values of H>0.5 was obtained, indicating that a long-term memory exists in earthquake time series. The main contribution of this paper is estimating H(t) locally for different time-scales and investigating the long-memory behavior that exists in the non-stationary multifractal time-series. The time-dependent scaling properties of earthquake time series are associated with the relative weights of the amplitudes at characteristic frequencies.
The superiority of the method is the simplicity and the accuracy in estimating the Hurst exponent of earthquakes in each time, without any assumption on the probability distribution of the time series. We investigated the negative and positive autocorrelations that exist between consecutive seismic activities by estimating the time-dependent Hurst parameter, H * (t). To this end, the empirical mode decomposition and the Hilbert-Huang transform are applied. Using this method, the seismic activities are studied locally, and the autocorrelation between consecutive earthquakes are estimated in each time. We have investigated the superiority of the estimator by simulation. Furthermore, the method is applied in estimating H * (t) of earthquakes occurred in Sarpol-e Zahab during November 12, 2017 to January 20, 2018. By estimating the Hurst parameter locally, and considering the values of H * (t) that are greater than 0.5, we identify that the long memory behavior exist in consecutive earthquakes during 25/11/2017 and 13/1/2018. It is also seen that, in spite of small values of H * (t) for some times, which shows the stochastic behavior in earthquakes, the local Hurst parameters are tending to be greater than 0.5, and after this patterns, a peak in magnitude is seen. This paper shows that the combination of the EMD and its associated Hilbert spectral analysis offers a powerful tool to uncover the time-dependent scaling patterns of consecutive seismic activities data. In this paper, we investigate the long-range correlations and trends between consecutive earthquakes by means of the scaling parameter so-called locally Hurst parameter, H(t), and examine its variations in time, to find a specific pattern that exists between foreshocks, main shock and the aftershocks. The long-range correlations are usually detected by calculating a constant Hurst parameter. However, the multi-fractal structure of earthquakes caused that more than one scaling exponent is needed to account for the scaling properties of such processes. Thus, in this paper, we consider the time-dependent Hurst exponent, to realize scale variations in trend and correlations between consecutive seismic activities, for all times. We apply the Hilbert-Huang transform to estimate H(t) for the time series extracted from seismic activities occurred in Iran.
The superiority of the method is discovering some specific hidden patterns that exist between consecutive earthquakes by studying the trend and variations of H(t). Estimating H(t) only as a measure of dependency may lead to misleading results, but using this method, the trend and variations of the parameter is studying to discover hidden dependencies between consecutive earthquakes. Recognizing such dependency patterns can help us in prediction of mainshocks.

کلیدواژه‌ها [English]

  • Hilbert Spectral Analysis
  • Hurst Parameter
  • Time-Dependent Scaling
  • Estimating
  1. Mart-Montoya, L.A., Aranda-Camacho, N.M., Quimbay, C.J. (2014) Long-range correlations and trends in Colombian seismic time series. Physica A: Statistical Mechanics and its Applications. 421(C), 124-133.
  2. Shadkhoo1, Sh., Ghanbarnejad, F., Jafari, Gh.R., Tabar, M.R.R. (2009) Scaling behavior of earthquakes’ inter-events time series. Cent. Eur. J.Phys. 7(3), 620-623.
  3. Masci, F. and Thomas, J.N. (2013) Review Article: On the relation between the seismic activity and the Hurst exponent of the geomagnetic field at the time of the 2000 Izu swarm. Nat. Hazards Earth Syst. Sci., 13, 2189-2194.
  4. Shao, Y.H., Gu, G.F., Jiang, Z.Q., Zhou, W.X., Sornette, D. (2012) Comparing the performance of FA, DFA and DMA using different synthetic long-range correlated time series. Scientific Reports 2(835).
  5. Huang, Y., Johanse, A., Lee, M.W., Saleur, H., Sornett, D. (2000) Artifactual log-periodicity in finite size data Relevance for earthquake aftershocks. J. Geophys. Res. Cond-mat/9911421. 105(10), 25451-25471.
  6. Hurst, H.E. (1951) Long term storage capacity of reservoirs. Trans. Am. Soc. Civil Eng., 116, 770-808.
  7. Zheng, Z., Yamasaki, K., Tenenbaum, J., Podobnik, B., Tamura, Y., Stanley, H.E. (2012) Scaling of seismic memory with earthquake size. Phys. Rev. E., 86. 011107.
  8. Ortiz, J.P., Aguilera, R.C., Balankin, A.S., Ortiz, M.P., Rodriguez, J.C.T., Mosqueda, M.A.A., Cruz, M.A.M., Yu, W. (2016) Seismic activity seen through evolution of the Hurst exponent model in 3D. Fractals, 24(4), 1650045.
  9. Peng, C.K., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Simons, M., Stanley, H.E. (1993) Finite-size effects on long-range correlations: Implications for analyzing DNA sequences. Phys. Rev. E., 47(5), 3730.
  10. Buldyrev, S.V., Goldberger, A.L., Havlin, S., Mantegna, R.N., Matsa, M.E., Peng, C.K., Simons, M., Stanley, H.E. (1993) Long-range correlation properties of coding and noncoding DNA sequences: GenBank analysis. Phys. Rev. E., 51(5), 5084-91.
  11. Taqqu, M.S., Teverovsky, V., Willinger, W. (1995) Estimators for long-range dependence: an empirical study. Fractals, 3(4), 785-798.
  12. Ivanova, K., Ausloos, M. (1999) Application of the detrended fluctuation analysis (DFA) method for describing cloud breaking. Physica A, 274(1), 349-354.
  13. Stanley, H.E., Afanasyev, V., Amaral, L.A.N., Buldyrev, S.V., Goldberger, A.L., Havlin, S., Leschhorn, H., Maass, P., Mantegna, R.N., Peng, C.K., Prince, P.A., Salinger, M.A., Stanley, M.H.R., Viswanathan, G.M. (1996) Fluctuations in the dynamics of complex systems: from DNA and physiology to econophysics, Physica A., 224, 302-321.
  14. Carbone, A., Castelli, G., Stanley, H.E., (2004) Time-dependent Hurst exponent in financial time series. Physica A, 344, 267-271.
  15. Alessio, E., Carbone, A., Castelli, G., Frappietro, V. (2002) Second-order moving average and scaling of stochastic time series, Eur. Phys. J. B., 27, 197-200.
  16. Yue, J., Dong, K., Shang, P., (2010) Time-Dependent Hurst Exponent in Traffic Time Series. 2010 IEEE International Conference on Information Theory and Information Security.
  17. Carbone, A., Castelli, G., Stanley, H.E. (2004) Analysis of clusters formed by the moving average of a long-range correlated time series. Phys. Rev. E., 69(2), 026105.
  18. Carbone, A., Castelli, G. in: Schimanskey-Geier, L., Abbot, D., Neimann, A., Van, C., den Broeck (Eds.) (2003) Noise in Complex Systems and Stochastic Dynamics. Proc. SPIE 5114406.
  19. Carbone, A. (2007) Algorithm to estimate the Hurst exponent of high-dimensional fractals. Phys. Rev, E76, 056703.
  20. Carbone, A., Detrending Moving Average Algorithm: a brief review (2009), Science and Technology for Humanity (TIC-STH), IEEE Toronto International Conference.
  21. Peng, C.K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L. (1994) Mosaic organization of DNA nucleotides. Phys. Rev. E., 49(2), 1685-9.
  22. Chen, Z., Ivanov, P.CH., Hu, K., Stanley, H.E. (2002) Effect of nonstationarities on detrended fluctuation analysis. Phys. Rev. E., 65, 041107.
  23. Hu, K., Chen, Z., Ivanov, P.CH., Carpena, P., Stanley, H.E. (2001) Effect on trends on detrended fluctuation analysis. Phys. Rev. E., 64, 011114.
  24. Smith, S.W. (2003) Digital filters, in Digital Signal Processing: A Practical Guide for Engineers and Scientist. Elsevier Science, Burlington, MA, 261-343.