Abstract
The magnitude frequency distribution of earthquakes is generally observed to depend on regional differential stress. The typical signature is a decrease of the b-value with increasing differential stress, corresponding to an increased probability of larger events. At the scale of a seismogenic fault, regional stress changes translate into shear and normal stress variations that both influence the nucleation process and the maximum extent of earthquake ruptures. Recent models have shown that normal stress changes cannot explain the observed b-value dependency, suggesting that b-value is primarily controlled by shear stress conditions on faults resulting from regional loading. The relationship between shear stress and b-value is however still not clear. Here, I investigate numerically the shear stress contribution on magnitude statistics by analyzing the b-value evolution with shear stress changes along a planar rate-and-state interface containing hierechical asperities and forced by slow tectonic loading under constant normal stress. The b-value is shown to decrease from 1.2 to 0.7 when shear stress increases by 1 MPa which is in agreement with fracture mechanics predictions. An approximate expression relating shear stress and b value is proposed and validated by the simulations, where the rate of b-value decrease with shear stress is provided by the inverse stress drop of typical earthquakes. The effect of shear stress along a planar hierarchical interface is therefore too strong to explain current observations.
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Availability of data and material
Earthquake catalogs and average stress timeseries used in this study are available at Dublanchet (2020a).
Code availability
The code used to perform simulations is available on request.
References
Abercrombie, R. E., & Rice, J. R. (2005). Can observations of earthquake scaling constrain slip weakening? Geophysical Journal International, 162(2), 406–424.
Aki, K. (1965). Maximum likelihood estimate of b in the formula log n= a-bm and its confidence limits. Bulletin Earthquake Research Institute, Tokyo University, 43, 237–239.
Amitrano, D. (2003). Brittle–ductile transition and associated seismicity: Experimental and numerical studies and relationship with the b value. Journal of Geophysical Research: Solid Earth, 108(B1), 2044–2058.
Aochi, H., & Ide, S. (2009). Complexity in earthquake sequences controlled by multiscale heterogeneity in fault fracture energy. Journal of Geophysical Research: Solid Earth, 114(B3), 305–317.
Bizzarri, A., & Cocco, M. (2003). Slip-weakening behavior during the propagation of dynamic ruptures obeying rate-and state-dependent friction laws. Journal of Geophysical Research: Solid Earth, 108(B8), 2373–2393.
Carlson, J. M., Langer, J. S., & Shaw, B. E. (1994). Dynamics of earthquake faults. Reviews of Modern Physics, 66(2), 657.
Cattania, C. (2019). Complex earthquake sequences on simple faults. Geophysical Research Letters, 46(17–18), 10384–10393.
Chen, Y., & Huang, C. (2006). Time-dependent b value for aftershock sequences. Available in web
Dieterich, J. H. (1979). Modeling of rock friction-1. Experimental results and constitutive equations. Journal of Geophysical Research, 84, 2161–2168.
Dublanchet, P. (2018). Inferring fault slip rates from cumulative seismic moment in a multiple asperity context. Geophysical Journal International, 216(1), 395–413.
Dublanchet, P. (2020a). Catalogs and stress history for "stress dependent b value variations in a heterogeneous rate-and-state fault model". https://doi.org/10.5281/zenodo.3655290
Dublanchet, P. (2020b). Stress-dependent b value variations in a heterogeneous rate-and-state fault model. Geophysical Research Letters, 47(13):e2020GL087434
Dublanchet, P., Bernard, P., & Favreau, P. (2013). Interactions and triggering in a 3-d rate-and-state asperity model. Journal of Geophysical Research: Solid Earth, 118(5), 2225–2245.
El-Isa, Z. H., & Eaton, D. W. (2014). Spatiotemporal variations in the b-value of earthquake magnitude-frequency distributions: Classification and causes. Tectonophysics, 615, 1–11.
Fehlberg, E. (1969). Low-order classical Runge–Kutta formulas with stepsize control and their application to some heat transfer problems
Galis, M., Ampuero, J. P., Mai, P. M., Cappa, F. (2017). Induced seismicity provides insight into why earthquake ruptures stop. Science Advances, 3(12):eaap7528
Gerstenberger, M., Wiemer, S., & Giardini, D. (2001). A systematic test of the hypothesis that the b value varies with depth in California. Geophysical Research Letters, 28(1), 57–60.
Goebel, T. H., Kwiatek, G., Becker, T. W., Brodsky, E. E., & Dresen, G. (2017). What allows seismic events to grow big? Insights from b-value and fault roughness analysis in laboratory stick-slip experiments. Geology, 45(9), 815–818.
Goebel, W., Schorlemmer, T., Becker, T., Dresen, G., & Sammis, C. (2013). Acoustic emissions document stress changes over many seismic cycles in stick-slip experiments. Geophysical Research Letters, 40(10), 2049–2054.
Gulia, L., Rinaldi, A. P., Tormann, T., Vannucci, G., Enescu, B., & Wiemer, S. (2018). The effect of a mainshock on the size distribution of the aftershocks. Geophysical Research Letters, 45(24), 13–277.
Gutenberg, B., & Richter, C. F. (1944). Frequency of earthquakes in California. Bulletin of the Seismological Society of America, 34(4), 185–188.
Heimisson, E. R. (2020). Crack to pulse transition and magnitude statistics during earthquake cycles on a self-similar rough fault. Earth and Planetary Science Letters, 537, 116202.
Hillers, G., Ben-Zion, Y., & Mai, P. (2006). Seismicity on a fault controlled by rate-and state-dependent friction with spatial variations of the critical slip distance. Journal of Geophysical Research: Solid Earth, 111(B1), 403–425.
Ide, S., & Aochi, H. (2005). Earthquakes as multiscale dynamic ruptures with heterogeneous fracture surface energy. Journal of Geophysical Research: Solid Earth, 110(B11), 303–312.
Jaeger, J. C., Cook, N. G., & Zimmerman, R. (2009). Fundamentals of rock mechanics. Wiley.
Knopoff, L., Kagan, Y. Y., & Knopoff, R. (1982). b values for foreshocks and aftershocks in real and simulated earthquake sequences. Bulletin of the Seismological Society of America, 72(5), 1663–1676.
Kwiatek, G., Goebel, T., & Dresen, G. (2014). Seismic moment tensor and b value variations over successive seismic cycles in laboratory stick-slip experiments. Geophysical Research Letters, 41(16), 5838–5846.
Lambert, V., Lapusta, N., & Faulkner, D. (2021). Scale dependence of earthquake rupture prestress in models with enhanced weakening: Implications for event statistics and inferences of fault stress. Journal of Geophysical Research: Solid Earth, 126(10), 886–914.
Lawn, B. (1993). Fracture of brittle solids. Cambridge University Press.
Marone, C. (1998). Laboratory-derived friction laws and their application to seismic faulting. Annual Review of Earth and Planetary Sciences, 26(1), 643–696.
Molchan, G., Kronrod, T., & Nekrasova, A. (1999). Immediate foreshocks: Time variation of the b-value. Physics of the earth and planetary interiors, 111(3–4), 229–240.
Passelègue, F. X., Almakari, M., Dublanchet, P., Barras, F., Fortin, J., & Violay, M. (2020). Initial effective stress controls the nature of earthquakes. Nature Communications, 11(1), 1–8.
Rice, J. R. (1992). Fault stress states, pore pressure distributions, and the weakness of the San Andreas fault. In: International geophysics (vol. 51, pp. 475–503). Elsevier, Academic Press.
Rice, J. R. (1993). Spatio-temporal complexity of slip on a fault. Journal of Geophysical Research, 98, 9885–9907.
Rice, J. R., & Ben-Zion, Y. (1996). Slip complexity in earthquake fault models. Proceedings of the National Academy of Sciences, 93(9), 3811–3818.
Romanet, P., Bhat, H. S., Jolivet, R., & Madariaga, R. (2018). Fast and slow slip events emerge due to fault geometrical complexity. Geophysical Research Letters, 45(10), 4809–4819.
Rubin, A., & Ampuero, J. (2005). Earthquake nucleation on (aging) rate and state faults. J Geophys Res, 110, B11312.
Ruina, A. L. (1983). Slip instability and state variable friction laws. Journal of Physical Research, 88, 10359–10370.
Sagy, A., Brodsky, E. E., & Axen, G. J. (2007). Evolution of fault-surface roughness with slip. Geology, 35(3), 283–286.
Scholz, C. (1968). The frequency-magnitude relation of microfracturing in rock and its relation to earthquakes. Bulletin of the seismological society of America, 58(1), 399–415.
Scholz, C. H. (2015). On the stress dependence of the earthquake b value. Geophysical Research Letters, 42(5), 1399–1402.
Scholz, C. H. (2019). The mechanics of earthquakes and faulting. Cambridge University Press.
Schorlemmer, D., Wiemer, S., & Wyss, M. (2005). Variations in earthquake-size distribution across different stress regimes. Nature, 437(7058), 539.
Spada, M., Tormann, T., Wiemer, S., & Enescu, B. (2013). Generic dependence of the frequency-size distribution of earthquakes on depth and its relation to the strength profile of the crust. Geophysical Research Letters, 40(4), 709–714.
Tal, Y., Goebel, T., & Avouac, J. P. (2020). Experimental and modeling study of the effect of fault roughness on dynamic frictional sliding. Earth and Planetary Science Letters, 536, 116133.
Thomas, M. Y., Lapusta, N., Noda, H., & Avouac, J. P. (2014). Quasi-dynamic versus fully dynamic simulations of earthquakes and aseismic slip with and without enhanced coseismic weakening. Journal of Geophysical Research: Solid Earth, 119(3), 1986–2004.
Turcotte, D. L., & Schubert, G. (2002). Geodynamics. Cambridge University Press.
Wiemer, S., & Katsumata, K. (1999). Spatial variability of seismicity parameters in aftershock zones. Journal of Geophysical Research: Solid Earth, 104(B6), 13135–13151.
Wiemer, S., Gerstenberger, M., & Hauksson, E. (2002). Properties of the aftershock sequence of the 1999 m w 7.1 hector mine earthquake: implications for aftershock hazard. Bulletin of the Seismological Society of America, 92(4), 1227–1240.
Zielke, O., Galis, M., & Mai, P. M. (2017). Fault roughness and strength heterogeneity control earthquake size and stress drop. Geophysical Research Letters, 44(2), 777–783.
Ziv, A., & Rubin, A. (2003). Implications of rate-and-state friction for properties of aftershock sequence: Quasi-static inherently discrete simulations. Journal of Geophysical Research: Solid Earth, 108(B1), 2051–2065.
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The author would like to thank Martijn van den Ende, an anonymous reviewer and the editor for their constructive comments that greatly improved the manuscript.
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Dublanchet, P. Shear Stress and b-value Fluctuations in a Hierarchical Rate-and-State Asperity Model. Pure Appl. Geophys. 179, 2423–2435 (2022). https://doi.org/10.1007/s00024-022-03039-3
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DOI: https://doi.org/10.1007/s00024-022-03039-3