Aftershock rate decay induced by postseismic creep: A case study from the MW7.9 Wenchuan earthquake sequence
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摘要: 基于Dieterich地震活动性理论，本文推导出计算余震发生率和余震累积次数的一般表达式，其中主震后发震断层内部的剪切应力随时间的演化过程遵从Jeffreys-Lomnitz蠕变模型，且与修正Omori定律直接相关。修正Omori定律中的p值与震后断层的短时应力加卸载过程正相关。采用Rubin和Ampuero 给出的震后断层自维持蠕滑模型本文得出计算余震发生率的近似表达式，并对2008年汶川地震序列进行拟合。结果表明，p值的大小直接对应了速率-状态摩擦定律中摩擦参量b/a，而修正Omori定律中的c值则与速率-状态摩擦定律中的临界滑移Dc相关。对于汶川余震序列而言，拟合结果显示b/a约为1.13，Dc约为2—3 cm。Rubin-Ampuero震后自维持蠕滑描述了震后孕震层内部短暂的速率变化特征，是孕震断层演化过程不可缺少的环节。Abstract: Based on the Dieterich seismicity theory, this study derived a general formula used to calculate decay rate and cumulative frequency of aftershocks following a mainshock, of which the shear stress variation in the fault plane follows the Jeffreys-Lomnitz creep law and is directly related to the modified Omori law. The p value in the modified Omori law is positively correlated with post earthquake loading and unloading process of the transient stress. Combined with the postseismic slip model proposed by Rubin and Ampuero, this study obtained approximate formulas to describe the variation of seismicity and to fit aftershock sequence following the 2008 MW7.9 Wenchuan earthquake. The results show that the p value is directly associated with the ratio of b/a, where a and b are the frictional parameters used in rate- and state-dependent friction, and the c value used in the modified Omori law is associated with Dc, a critical creep distance described by rate- and state-dependent friction. For the MW7.9 Wenchuan earthquake sequence, the fitting results indicate the ratio of b/a and Dc are about 1.13 and 2 to 3 cm, respectively. It needs to be emphasized that postseismic creep proposed by Rubin and Ampuero is an inherent stage during earthquake cycle, which describes characteristics of the transient velocity change in seismogenic zone after the mainshock.
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Keywords:
- aftershock rate /
- postseismic creep /
- stress /
- stressing rate
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图 1
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Figure 1.
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图 2 汶川地震ML≥4.0余震的空间分布
Figure 2. The spatial distribution of the ML≥4.0 aftershocks of Wenchuan earthquake
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图 3 余震累积次数和不同模型最佳拟合曲线的对比
（a） MOL模型与Dieterich模型；（b） 断层蠕滑模型与Dieterich模型
Figure 3. Comparison between the temporal distribution of the cumulative number of triggered events corresponding to real data and the best fits obtained by the different models
（a） MOL model and Dieterich model；（b） Postseismic slip model and Dieterich model
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图 4 由拟合参数给出的余震发生率的衰减曲线
Figure 4. Decay curves of the aftershock rate versus time for different models with the best fitted parameters
表 1 MOL模型参数拟合值
Table 1 Fitted parameters in the MOL model
K p c/d RMS $ {r_{\rm d}^2}$ 
76.290 796 433 386 3 1.129 714 921 398 86 0.174 700 213 127 629 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 201 251 5 1.129 714 919 607 34 0.174 700 212 134 727 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 599 558 6 1.129 714 921 574 52 0.174 700 214 565 022 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 420 031 1 1.129 714 920 241 93 0.174 700 213 526 721 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 407 508 2 1.129 714 920 303 60 0.174 700 213 369 922 4.723 176 877 126 94 0.999 064 715 754 963 76.290 795 889 337 1 1.129 714 918 858 40 0.174 700 210 045 648 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 203 779 0 1.129 714 919 793 27 0.174 700 211 916 160 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 358 633 6 1.129 714 920 749 10 0.174 700 212 952 576 4.723 176 877 126 94 0.999 064 715 754 963 76.290 796 339 850 3 1.129 714 920 639 84 0.174 700 212 665 134 4.723 176 877 126 93 0.999 064 715 754 963 76.290 796 251 218 9 1.129 714 919 679 88 0.174 700 212 577 751 4.723 176 877 126 94 0.999 064 715 754 963 注：表中K∈[0，500]；p∈[0，5]；c∈[−10, 10] （ Utsu et al，1995 ；Utsu，2002；Hainzl et al，2016） 。
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表 2 Dieterich模型参数拟合值
Table 2 Fitted parameters in the Dieterich model
ta/d exp[Δτco/（aσ）] Δτco/（aσ） RMS $ r_{\rm d}^2$ 
5 882.015 239 530 44 123 970.933 974 009 5.093 319 873 110 68 16.132 130 498 174 8 0.989 089 148 314 182 5 882.014 868 234 99 123 970.963 284 002 5.093 319 975 789 32 16.132 130 498 171 8 0.989 089 148 314 186 5 882.015 309 131 30 123 970.909 997 818 5.093 319 789 117 38 16.132 130 498 174 2 0.989 089 148 314 182 5 882.014 592 899 16 123 970.994 215 420 5.093 320 084 148 10 16.132 130 498 172 7 0.989 089 148 314 184 5 882.014 503 677 40 123 970.987 109 376 5.093 320 059 254 25 16.132 130 498 172 8 0.989 089 148 314 184 5 882.014 976 173 96 123 970.948 098 090 5.093 319 922 590 10 16.132 130 498 169 5 0.989 089 148 314 189 5 882.014 915 449 27 123 970.957 968 035 5.093 319 957 166 45 16.132 130 498 172 2 0.989 089 148 314 185 5 882.015 214 323 03 123 970.935 318 443 5.093 319 877 820 50 16.132 130 498 170 4 0.989 089 148 314 188 5 882.015 150 196 45 123 970.949 854 423 5.093 319 928 742 88 16.132 130 498 170 6 0.989 089 148 314 187 5 882.015 052 767 42 123 970.941 083 807 5.093 319 898 017 70 16.132 130 498 167 5 0.989 089 148 314 192 注：表中ta∈[0，106]，exp[Δτco/（aσ）]∈[0，106] （Dieterich，2007； Kame et al，2013 ）。
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表 3 断层蠕滑模型参数拟合值
Table 3 Fitted parameters in the postseismic slip model
θo/d b/a exp[Δτco/（aσ）] Δτco/（aσ） RMS $ r_{\rm d}^2$ 
0.174 700 212 670 846 1.129 714 920 608 64 60 844.601 650 449 9 4.784 222 052 123 64 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 212 368 259 1.129 714 920 027 86 60 844.601 626 682 0 4.784 222 051 953 99 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 213 831 794 1.129 714 921 162 83 60 844.601 370 971 4 4.784 222 050 128 78 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 212 381 837 1.129 714 920 578 23 60 844.601 713 419 6 4.784 222 052 573 10 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 212 075 125 1.129 714 919 810 53 60 844.601 679 670 0 4.784 222 052 332 20 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 212 385 557 1.129 714 920 392 20 60 844.601 712 262 3 4.784 222 052 564 84 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 212 418 767 1.129 714 919 972 89 60 844.601 611 131 1 4.784 222 051 842 99 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 213 236 443 1.129 714 920 438 69 60 844.601 424 260 9 4.784 222 050 509 15 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 214 115 631 1.129 714 921 173 05 60 844.601 283 946 5 4.784 222 049 507 62 4.723 176 877 126 94 0.999 064 715 754 963 0.174 700 211 246 645 1.129 714 919 459 10 60 844.601 866 461 2 4.784 222 053 665 47 4.723 176 877 126 94 0.999 064 715 754 963 注：表中θo∈[0，1]，b/a∈[0，5]，exp[Δτco/（aσ）]∈[0，106] （Dieterich，2007）。
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