基于地震二阶矩方法的中小地震破裂过程研究

探究地震破裂过程是揭示发震断层的震源物理特征和评估地震灾害的重要手段。大地震虽然能够提供丰富的震源信息，但其在特定断层发生频次低、复发周期长，可供研究的实际案例有限。相比之下，依据震级-频次关系，中小地震数量众多，适合作为系统性分析的研究对象。通过解析大量中小地震破裂过程，我们能够深入理解断层的活动规律，进而为概率性地震危险性分析和断层强度非均匀性评估提供有力支撑。解析中小地震破裂过程常用方法包括谱分解法、频率域谱比 法、震源时间函数伸缩法以及地震二阶矩方法等。其中，地震二阶矩方法具有合理的模型复杂度、广泛的震级适用范围及丰富的解析参数优势。前人在开展地震二阶矩反演研究时是利用时间域反卷积得到的视震源时间函数的持续时长，但时间域反卷积对波形数据相关性和信噪比的要求较高，这导致在处理区域台网中震级较小的地震时存在困难。针对这一问题，本文发展了抗噪性和稳定性更高的频率域地震二阶矩方法。同时，为了克服传统网格叠加算法在计算视震源时间函数和视震源谱时出现的假高频问题，本文创新性地提出了半解析数值正演方法。在此基础上，我们使用理论模型测试验证了频率域地震二阶矩方法的有效性。我们基于新发展的时间域和频率域地震二阶矩方法，利用布设在加拿大西部Fox Creek水力压裂实验场和滇西地区维西-乔后-巍山断裂附近的短周期密集台阵，解析了加拿大Fox Creek诱发地震和2021年云南M_S6.4漾濞地震序列的破裂过程，进而得到了相应区域目标地震的破裂长度、宽度、速度、时长、方向性及应力降等震源信息。加拿大Fox Creek诱发地震研究结果显示，目标地震破裂方向与该区域水力压裂井的注入方向基本平行，这表明水力压裂活动对地震破裂过程有一定影响。此外，同一簇中位置非常相近的目标地震其水平破裂方向和垂直破裂方向基本一致， 但也存在相反的情况，这反映了该区域的断层强度和应力水平存在一定的非均匀性。对于云南漾濞地震序列，我们观察到大部分目标地震的水平破裂方向与该震区乔后-巍山主断层走向以及微小地震活动分布趋势相符，进一步验证了地震二阶矩方法和研究结果的可靠性。此外，部分余震的破裂方向存在与主震不一致以及指向共轭断层的情况，这揭示出断层系统在强度、应力状态以及几何结构上的复杂性。

Investigating the seismic rupture processes is vital for unveiling the physical characteristics of earthquake sources and assessing seismic hazards. While major earthquakes offer valuable insights, their infrequent occurrence on specific faults and lengthy recurrence intervals limit available case studies. Conversely, small-to-moderate earthquakes, owing to their more frequent incidence based on the magnitude-frequency relationship, serve as suitable subjects for systematic analysis. By resolving the rupture processes of numerous small-to-moderate earthquakes, we gain profound insights into fault activity patterns, providing strong support for probabilistic seismic hazard analysis and evaluating the non-uniformity of fault strength. Several methods are commonly employed to resolve the rupture processes of small-to-moderate earthquakes, including spectral decomposition, frequency-domain spectral ratio method, source time function stretching method, and seismic second moments method. Among these, the seismic second moments method offers advantages regarding reasonable model complexity, applicability across a wide range of magnitudes, and rich analytical parameters. In previous studies, researchers utilized time-domain deconvolution to estimate the duration of the apparent source time function in seismic second moments inversion. However, this method imposes stringent requirements on waveform data correlation and signal-to-noise ratio, posing challenges in analyzing small-magnitude earthquakes within regional seismic networks. To address these challenges, this study develops the frequency-domain seismic second moments method with enhanced noise resistance and stability. Additionally, this study innovatively proposes a semi-analytical numerical simulation method to address the false high-frequency signals encountered in traditional grid superposition algorithms when calculating the apparent source time function and apparent source spectra. Building upon this approach, we conduct theoretical model testing to validate the effectiveness of the frequency-domain seismic second moments method. Based on the newly developed time-domain and frequency-domain second seismic moments method, we resolve the rupture processes of induced earthquakes in Fox Creek, western Canada, and the 2021 Yunnan M_S6.4 Yangbi earthquake sequence. This analysis utilizes short-period dense arrays deployed at the Fox Creek hydraulic fracturing experimental site in western Canada and near the Weixi-Qiaohou-Weishan fault on the western boundary of the Sichuan-Yunnan block. Consequently, we obtain essential source parameters such as rupture length, width, velocity, duration, directivity, and stress drop for the target earthquakes in the respective regions. The study of induced earthquakes in Fox Creek, Canada, shows that the rupture directivities of target earthquakes generally align with the injection orientation of hydraulic fracturing wells in the region. This indicates that hydraulic fracturing activities have a certain influence on the earthquake rupture processes. Additionally, the horizontal and vertical rupture directivities are generally consistent for target earthquakes within the same cluster that are very close to each other. However, there are instances where they differ, indicating heterogeneity in the region's fault strength and stress levels. Regarding the Yangbi earthquake sequence in Yunnan, it is observed that the horizontal rupture directivities of most target earthquakes align with the orientation of the Qiaohou-Weishan main fault and the distribution trend of microseismic activity in the seismic zone, further validating the reliability of the seismic second moments method and the research findings. Furthermore, some aftershocks exhibit rupture directivities inconsistent with the main shock and pointing toward conjugate faults, revealing the complexity of the fault system in terms of strength, stress state, and geometric structure.

[1] SCHOLZC, COWIEPA. Determination of total strain from faulting using slip measurements [J]. Nature, 1990, 346(6287): 837-839.
[2] SCHOLZ C H. Earthquakes and friction laws[J]. Nature, 1998, 391(6662): 37-42.
[3] KIKUCHI M, KANAMORI H. Inversion of complex body waves[J]. Bulletin of the Seismological Society of America, 1982, 72(2): 491-506.
[4] OLSON A H, APSEL R J. Finite faults and inverse theory with applications to the 1979 Imperial Valley earthquake[J]. Bulletin of the Seismological Society of America, 1982, 72(6A): 1969-2001.
[5] HARTZELL S H, HEATON T H. Inversion of strong ground motion and teleseismic waveform data for the fault rupture history of the 1979 Imperial Valley, California, earthquake[J]. Bulletin of the Seismological Society of America, 1983, 73(6A): 1553-1583.
[6] CHEN Y T, ZHOU J Y, NI J C. Inversion of near-source-broadband accelerograms for the earthquake source-time function[J]. Tectonophysics, 1991, 197(1): 89-98.
[7] 许力生,陈运泰.震源时间函数与震源破裂过程[J].地震地磁观测与研究,2002,23(6):1-8.
[8] GUTENBERG B, RICHTER C F. Seismicity of the earth and associated phenomena[M]. Princeton University Press, 1949.
[9] HAO J, JI C, YAO Z. Slip history of the 2016 𝑀W 7.0 Kumamoto earthquake: Intraplate rupture in complex tectonic environment[J]. Geophysical Research Letters, 2017, 44(2): 743-750.
[10] YE L, LAY T, KANAMORI H, et al. Rupture characteristics of major and great (𝑀W ≥ 7.0) megathrust earthquakes from 1990 to 2015: 1. Source parameter scaling relationships[J]. Journal of Geophysical Research: Solid Earth, 2016, 121(2): 826-844.
[11] YUE H, SHEN Z K, ZHAO Z, et al. Rupture process of the 2021 𝑀7.4 Maduo earthquake and implication for deformation mode of the Songpan-Ganzi terrane in Tibetan Plateau[J]. Proceedings of the National Academy of Sciences, 2022, 119(23): e2116445119.
[12] 朱音杰,罗艳,赵里,等. 利用区域宽频地震数据反演2021年5月云南漾濞𝑀S6.4地震震源破裂过程[J]. 地球物理学报,2022,65(3): 1021-1031.
[13] HARTZELL S H. Earthquake aftershocks as Green’s functions[J]. Geophysical Research Letters, 1978, 5(1): 1-4.
[14] MUELLER C S. Source pulse enhancement by deconvolution of an empirical Green’s function [J]. Geophysical Research Letters, 1985, 12(1): 33-36.
[15] SHEARERPM, PRIETOGA, HAUKSSONE. Comprehensive analysis of earthquake source spectra in southern California[J]. Journal of Geophysical Research: Solid Earth, 2006, 111: B06303.
[16] CALDERONI G, ROVELLI A, BEN-ZION Y, et al. Along-strike rupture directivity of earthquakes of the 2009 L’Aquila, central Italy, seismic sequence[J]. Geophysical Journal International, 2015, 203(1): 399-415.
[17] SHEARERPM, ABERCROMBIERE, TRUGMANDT, et al. Comparing EGF methods for estimating corner frequency and stress drop from P wave spectra[J]. Journal of Geophysical Research: Solid Earth, 2019, 124(4): 3966-3986.
[18] PRIETO G A, FROMEN T B, YU C, et al. Earthquake rupture below the brittle-ductile transition in continental lithospheric mantle[J]. Science Advances, 2017, 3(3): e1602642.
[19] MCGUIRE J J, ZHAO L, JORDANTH. Teleseismic inversion for the second degree moments of earthquake space-time distributions[J]. Geophysical Journal International, 2001, 145(3): 661-678.
[20] MCGUIRE J J. Estimating finite source properties of small earthquake ruptures[J]. Bulletin of the Seismological Society of America, 2004, 94(2): 377-393.
[21] MENG H, MCGUIRE J J, BEN-ZION Y. Semiautomated estimates of directivity and related source properties of small to moderate Southern California earthquakes using second seismic moments[J]. Journal of Geophysical Research: Solid Earth, 2020, 125(4): e2019JB018566.
[22] WARREN L M, SHEARER P M. Mapping lateral variations in upper mantle attenuation by stacking P and PP spectra[J]. Journal of Geophysical Research: Solid Earth, 2002, 107(B12): ESE 6-1-ESE 6-11.
[23] AKI K. Scaling law of seismic spectrum[J]. Journal of Geophysical Research, 1967, 72(4): 1217-1231.
[24] PRIETO G A, SHEARER P M, VERNON F L, et al. Earthquake source scaling and selfsimilarity estimation from stacking P and S spectra[J]. Journal of Geophysical Research: Solid Earth, 2004, 109: B08310.
[25] BRUNEJN. Tectonic stress and the spectra of seismic shear waves from earthquakes[J]. Journal of Geophysical Research, 1970, 75(26): 4997-5009.
[26] KANE D L, SHEARER P M, GOERTZ-ALLMANN B P, et al. Rupture directivity of small earthquakes at Parkfield[J]. Journal of Geophysical Research: Solid Earth, 2013, 118(1): 212-221.
[27] TRUGMAN D T, SHEARER P M. Application of an improved spectral decomposition method to examine earthquake source scaling in Southern California[J]. Journal of Geophysical Research: Solid Earth, 2017, 122(4): 2890-2910.
[28] 赵翠萍,陈章立,华卫,等. 中国大陆主要地震活动区中小地震震源参数研究[J]. 地球物理学报,2011, 54(6): 1478-1489.
[29] 左可桢,陈继锋,蒲举,等. 2016-01-21青海门源𝑀S6.4地震震前应力降变化特征研究[J].大地测量与地球动力学,2018,38(6): 629-633.
[30] 周少辉,蒋海昆,曲均浩,等. 应力降研究进展综述[J]. 中国地震,2018,34(4): 591-605.
[31] 盛敏汉,储日升,王清东,等. 2019年12月26日湖北应城𝑀4.9地震震源参数及余震活动性研究[J]. 中国地震,2021,37(2): 463-471.
[32] MAYEDA K, WALTER W R. Moment, energy, stress drop, and source spectra of western United States earthquakes from regional coda envelopes[J]. Journal of Geophysical Research: Solid Earth, 1996, 101(B5): 11195-11208.
[33] SATO T, HIRASAWA T. Body wave spectra from propagating shear cracks[J]. Journal of Physics of the Earth, 1973, 21(4): 415-431.
[34] 梁尚鸿, 李幼铭,束沛镒,等. 利用区域地震台网P、S振幅比资料测定小震震源参数[J].地球物理学报,1984,27(3): 249-257.
[35] ABERCROMBIE R E. Investigating uncertainties in empirical Green’s function analysis of earthquake source parameters[J]. Journal of Geophysical Research: Solid Earth, 2015, 120(6): 4263-4277.
[36] ABERCROMBIE R E, POLI P, BANNISTER S. Earthquake directivity, orientation, and stress drop within the subducting plate at the Hikurangi margin, New Zealand[J]. Journal of Geophysical Research: Solid Earth, 2017, 122(12): 10176-10188.
[37] SILVER P G, JORDAN T H. Total-moment spectra of fourteen large earthquakes[J]. Journal of Geophysical Research: Solid Earth, 1983, 88(B4): 3273-3293.
[38] MENG H, FAN W. Immediate foreshocks indicating cascading rupture developments for 527 M 0.9 to 5.4 Ridgecrest earthquakes[J]. Geophysical Research Letters, 2021, 48(19): e2021GL095704.
[39] KANEKO Y, SHEARER P M. Variability of seismic source spectra, estimated stress drop, and radiated energy, derived from cohesive-zone models of symmetrical and asymmetrical circular and elliptical ruptures[J]. Journal of Geophysical Research: Solid Earth, 2015, 120(2): 1053-1079.
[40] MCGUIRE J J, KANEKO Y. Directly estimating earthquake rupture area using second moments to reduce the uncertainty in stress drop[J]. Geophysical Journal International, 2018, 214(3): 2224-2235.
[41] MCGUIRE J J. A MATLAB toolbox for estimating the second moments of earthquake ruptures [J]. Seismological Research Letters, 2017, 88(2A): 371-378.
[42] BERTERO M,BINDI D,BOCCACCI P ,et al. Application of the projected Landweber method to the estimation of the source time function in seismology[J]. Inverse Problems, 1997, 13(2): 465-486.
[43] LANZA V,SPALLAROSSA D,CATTANEOM, et al. Source parameters of small events using constrained deconvolution with empirical Green’s functions[J]. Geophysical Journal International, 1999, 137(3): 651-662.
[44] SHEARER P M. Introduction to seismology[M]. 2th ed. New York: United States of America by Cambridge University Press, 2009.
[45] AKI K, RICHARDS P G. Quantitative seismology[M]. University Science Books, 2002.
[46] MCGUIRE J J. A MATLAB Toolbox for Estimating the Second Moments of Earthquake Ruptures[J]. Seismological Research Letters, 2017, 88(2A): 371-378.
[47] BOYD S P, VANDENBERGHE L. Convex optimization[M]. Cambridge University Press, 2004.
[48] GRANT M C, BOYD S P. Graph implementations for nonsmooth convex programs[C]//Recent advances in learning and control. Springer London, 2008: 95-110.
[49] PRIETO G A, PARKER RL,VERNON III F L. A Fortran 90 library for multitaper spectrum analysis[J]. Computers & Geosciences, 2009, 35(8): 1701-1710.
[50] BOATWRIGHT J. A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy[J]. Bulletin of the Seismological Society of America, 1980, 70(1): 1-27.
[51] ABERCROMBIERE,BANNISTERS,RISTAUJ,etal.Variabilityofearthquakestressdropin a subduction setting, the Hikurangi Margin, New Zealand [J]. Geophysical Journal International, 2017, 208(1): 306-320.
[52] HUANG Y, BEROZA G C, ELLSWORTH W L. Stress drop estimates of potentially induced earthquakes in the Guy-Greenbrier sequence[J]. Journal of Geophysical Research: Solid Earth, 2016, 121(9): 6597-6607.
[53] KANEKO Y, SHEARER P M. Seismic source spectra and estimated stress drop derived from cohesive-zone models of circular subshear rupture[J]. Geophysical Journal International, 2014,197(2): 1002-1015.
[54] SILVER P. Retrieval of source-extent parameters and the interpretation of corner frequency[J]. Bulletin of the Seismological Society of America, 1983, 73(6A): 1499-1511.
[55] ESHELBY J D. The determination of the elastic field of an ellipsoidal inclusion, and related problems[J]. Mathematical and Physical Sciences, 1957, 241(1226): 376-396.
[56] MADARIAGA R. Dynamics of an expanding circular fault[J]. Bulletin of the Seismological Society of America, 1976, 66(3): 639-666.
[57] BAO X, EATON D W. Fault activation by hydraulic fracturing in western Canada[J]. Science, 2016, 354(6318): 1406-1409.
[58] SCHULTZ R, WANG R. Newly emerging cases of hydraulic fracturing induced seismicity in the Duvernay East Shale Basin[J]. Tectonophysics, 2020, 779: 228393.
[59] 张捷,况文欢,张雄,等. 全球油气开采诱发地震的研究现状与对策[J]. 地球与行星物理论评,2021, 52(3): 239-265.
[60] EATON D W, IGONIN N, POULIN A, et al. Induced seismicity characterization during hydraulic-fracture monitoring with a shallow-wellbore geophone array and broadband sensors[J]. Seismological Research Letters, 2018, 89(5): 1641-1651.
[61] ZHANG F, WANG R, CHEN Y, et al. Spatiotemporal variations and quantifying mechanisms during multistage hydraulic fracturing in Western Canada[J]. Journal of Geophysical Research: Solid Earth, 2022, 127(8): e2022JB024744.
[62] ZHANG H, EATON D W, RODRIGUEZ G, et al. Source-mechanism analysis and stress inversion for hydraulic-fracturing-induced event sequences near Fox Creek, Alberta[J]. Bulletin of the Seismological Society of America, 2019, 109(2): 636-651.
[63] HUI G, CHEN S, CHEN Z, et al. An integrated approach to characterize hydraulic fracturing induced seismicity in shale reservoirs[J]. Journal of Petroleum Science and Engineering, 2021, 196: 107624.
[64] OJO A O, KAO H, VISSER R, et al. Spatiotemporal changes in seismic velocity associated with hydraulic fracturing-induced earthquakes near Fox Creek, Alberta, Canada[J]. Journal of Petroleum Science and Engineering, 2022, 208: 109390.
[65] ZHUW, BEROZAGC. PhaseNet: a deep-neural-network-based seismic arrival-time picking method[J]. Geophysical Journal International, 2019, 216(1): 261-273.
[66] KANAMORIH. The energy release in great earthquakes[J]. Journal of Geophysical Research,1977, 82(20): 2981-2987.
[67] 吕江宁,沈正康,王敏. 川滇地区现代地壳运动速度场和活动块体模型研究[J]. 地震地质, 2003, 25(4): 543-554.
[68] 常祖峰,常昊,臧阳,等. 维西—乔后断裂新活动特征及其与红河断裂的关系[J]. 地质力学学报,2016, 22(3): 517-530.
[69] YANGT, LI B, FANGL, et al. Relocation of the Foreshocks and Aftershocks of the 2021 𝑀S 6.4 Yangbi Earthquake Sequence, Yunnan, China[J]. Journal of Earth Science, 2022, 33(4): 892-900.
[70] 王莹,赵韬,胡景,等. 2021年云南漾濞6.4级地震序列重定位及震源机制解特征分析[J].地震地质,2021,43(4): 847-863.
[71] 万永革, 王昱茹,靳志同. 2021年云南漾濞6.4级地震震源区地壳应力不均匀性研究[J].地震地质,2023,45(4): 1025-1040.
[72] 刘昌伟,常祖峰,王光明,等. 2021年云南漾濞𝑀S6.4地震震区断裂构造特征[J]. 地震研究,2023, 46(3): 323-331.
[73] GONG W, YE L, QIU Y, et al. Rupture directivity of the 2021 𝑀W 6.0 Yangbi, Yunnan earthquake[J]. Journal of Geophysical Research: Solid Earth, 2022, 127(9): e2022JB024321.
[74] WANG W, HE J, WANG X, et al. Rupture process models of the Yangbi and Maduo earthquakes that struck the eastern Tibetan Plateau in May 2021[J]. Science Bulletin, 2022, 67(5): 466-469.
[75] COCHARDA, MADARIAGAR. Dynamic faulting underrate-dependent friction[J]. Pure and Applied Geophysics, 1994, 142: 419-445.
[76] FUKUYAMA E, MADARIAGA R. Integral equation method for plane crack with arbitrary shape in 3D elastic medium[J]. Bulletin of the Seismological Society of America, 1995, 85(2): 614-628.
[77] AOCHI H, FUKUYAMA E, MATSU’URA M. Spontaneous rupture propagation on a non-planar fault in 3-D elastic medium[J]. Pure and Applied Geophysics, 2000, 157: 2003-2027.
[78] ZHANG H, CHEN X. Dynamic rupture on a planar fault in three-dimensional half space—I. Theory[J]. Geophysical Journal International, 2006, 164(3): 633-652.
[79] WEI X, XU J, LIU Y, et al. The slow self-arresting nature of low-frequency earthquakes[J]. Nature Communications, 2021, 12(1): 5464.