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A rate-state model for aftershocks triggered by dislocation on a rectangular fault: a review and new insights
Issued date
January 2007
Issue/vol(year)
6/49 (2006)
Language
English
Pages
1259-1273
Abstract
We compute the static displacement, stress, strain and the Coulomb failure stress produced in
an elastic medium by a finite size rectangular fault after its dislocation with uniform stress drop
but a non uniform dislocation on the source. The time-dependent rate of triggered earthquakes
is estimated by a rate-state model applied to a uniformly distributed population of faults whose
equilibrium is perturbated by a stress change caused only by the first dislocation. The rate of
triggered events in our simulations is exponentially proportional to the shear stress change, but
the time at which the maximum rate begins to decrease is variable from fractions of hour for
positive stress changes of the order of some MPa, up to more than a year for smaller stress
changes. As a consequence, the final number of triggered events is proportional to the shear
stress change. The model predicts that the total number of events triggered on a plane containing
the fault is proportional to the 2/3 power of the seismic moment. Indeed, the total
number of aftershocks produced on the fault plane scales in magnitude, M, as 10M. Including
the negative contribution of the stress drop inside the source, we observe that the number of
events inhibited on the fault is, at long term, nearly identical to the number of those induced
outside, representing a sort of conservative natural rule. Considering its behavior in time, our
model does not completely match the popular Omori law; in fact it has been shown that the
seismicity induced closely to the fault edges is intense but of short duration, while that expected
at large distances (up to some tens times the fault dimensions) exhibits a much slower decay.
an elastic medium by a finite size rectangular fault after its dislocation with uniform stress drop
but a non uniform dislocation on the source. The time-dependent rate of triggered earthquakes
is estimated by a rate-state model applied to a uniformly distributed population of faults whose
equilibrium is perturbated by a stress change caused only by the first dislocation. The rate of
triggered events in our simulations is exponentially proportional to the shear stress change, but
the time at which the maximum rate begins to decrease is variable from fractions of hour for
positive stress changes of the order of some MPa, up to more than a year for smaller stress
changes. As a consequence, the final number of triggered events is proportional to the shear
stress change. The model predicts that the total number of events triggered on a plane containing
the fault is proportional to the 2/3 power of the seismic moment. Indeed, the total
number of aftershocks produced on the fault plane scales in magnitude, M, as 10M. Including
the negative contribution of the stress drop inside the source, we observe that the number of
events inhibited on the fault is, at long term, nearly identical to the number of those induced
outside, representing a sort of conservative natural rule. Considering its behavior in time, our
model does not completely match the popular Omori law; in fact it has been shown that the
seismicity induced closely to the fault edges is intense but of short duration, while that expected
at large distances (up to some tens times the fault dimensions) exhibits a much slower decay.
References
Belardinelli, M., A. Bizzarri, and M. Cocco (2003), Earthquake triggering by static and dynamic
stress change, J. Geophys. Res., 108 (B3), 2135, doi:10.1029/2002JB001779.
Boatwright, J., and M. Cocco (1996), Frictional constraints on crustal faulting, J. Geophys.
Res., 101 (10), 13,895–13,910.
Bonafede, M., and A. Neri (2000), Effects induced by an earthquake on its fault plane: a boundary
element study, Geophys. J. Int., 141 (1), 43–56, doi:10.1046/j.1365-246X.2000.00074.x.
Chinnery, M. (1961), The deformation of the ground around surface faults, Bull. Seism. Soc.
Am., 51, 355–372, doi:10.1046/j.1365-246X.2000.00074.x.
Cocco, M., and J. R. Rice (2002), Pore pressure and poroelasticity effects in coulomb stress analysis
of earthquake interactions, J. Geophys. Res., 107(B2), 2030, doi:10.1029/2000JB000138.
Console, R., M. Murru, and A. Lombardi (2003), Refining earthquake clustering models, J.
Geophys. Res., 108 (B10), 2468, doi:10.1029/2002JB002130.
Dieterich, J. (1986), A model for the nucleation of earthquake slip, Geophy. Monogr. Ser., 37,
36–49.
Dieterich, J. (1992), Earthquake nucleation on faults with rate and state dependent strenght,
Tectonophysics, 211, 115–134.
Dieterich, J. (1994), A constitutive law for rate of earthquake production and its application
to earthquake clustering, J. Geophys. Res., 99 (18), 2601–2618.
Dieterich, J. (1995), Earthquake simulations with time-dependent nucleation and long-range
interactions, Nonlinear Processes in Geophysics, 2, 109–120.
Enescu, B., and K. Ito (2002), Spatial analysis of the frequency-magnitude distribution and
decay rate of aftershock activity of the 2000 Western Tottori earthquake, Earth Planet Space,
54, 847–859.
Felzer, K. R., T. W. Becker, R. E. Abercrombie, G. Ekstrm, and J. R. Rice (2001), Triggering
of the 1999 MW 7.1 Hector Mine earthquake by aftershocks of the 1992 MW 7.3 Landers
earthquake, J. Geophys. Res., 107 (B9), 2190, doi:10.1029/2001JB000911.
Gomberg, J., N. Beeler, M. Blanpied, and P. Bodin (1998), Earthquake triggering by transient
and static deformations, J. Geophys. Res., 103 (12), 24,411–24,426.
Gomberg, J., N. Beeler, and M. Blanpied (2000), On rate-state and Coulomb failure models,
J. Geophys. Res., 105 (14), 7857–7872.
Gomberg, J., P. Reasenberg, M. Cocco, and M. Belardinelli (2005), A frictional population
model of seismicity rate change, J. Geophys. Res., 110 (B05S03), doi:10.1029/2004JB003404.
Gross, S., and C. Kisslinger (1997), Estimating tectonic stress rate and state with Landers
aftershocks, J. Geophys. Res., 102 (B4), 7603–7612.
Harris, R. (1998), Introduction to special section: Stress triggers, stress shadows, and implications
for seismic hazard, J. Geophys. Res., 103 (12), 24,347–24,358.
Helmstetter, A. (2003), Is earthquake triggering driven by small earthquakes?, Phys. Res. Lett.,
91 (058501).
Helmstetter, A., Y. Kagan, and D. Jackson (2005), Importance of small earthquakes
for stress transfers and earthquakes triggering, J. Geophys. Res., 110 (B05S08), doi:
10.1029/2004JB003286.
Iwasaki, T., and R. Sato (1979), Strain field in a semi-infinite medium due to an inclined
rectangular fault, J. Phys. Earth, 27, 117–123.
Kagan, Y. Y., and D. D. Jackson (1991), Long-term earthquake clustering, Geophys. J. Int.,
104, 117–133.
Kanamori, H., and D. L. Anderson (1975), Theoretical basis for some empirical realtions in
seismology, Bull. Seism. Soc. Am., 65, 1073–1095.
Keilis-Borok, V. (1959), On estimation of the displacement in an earthquake source and of
source dimensions, Annali di Geofisica, 12, 2,205–2,214.
Kilb, D., J. Gomberg, and P. Bodin (2002), Aftershock triggering by complete coulomb stress
changes, J. Geophys. Res., 107, doi:10.1029/2001JB0002002.
King, G., and M. Cocco (2001), Fault interaction by elastic stress. changes: New clues from
earthquake sequences, Adv. Geophys., 44, 1–39.
Knopoff, L. (1957), Energy release in earthquake, Tech. Rep. 90, Institute of Gephysics, University
of California, U.S.A.
Kostrov, B., and S. Das (1998), Principles of Earthquake Source Mechanics, Cambridge University
Press.
Marsan, D. (2003), Triggering of seismicity at short timescales following Californian earthquakes,
J. Geophys. Res., 108 (B5), 2266, doi:10.1029/2002JB001946.
Mendoza, C., and S. Hartzell (1988), Aftershock patterns and main shock faulting, Bull. Seism.
Soc. Am., 78, 1438–1449.
Nostro, C., L. Chiaraluce, M. Cocco, D. Baumont, and O. Scotti (2005), Coulomb stress changes
caused by repeated normal faulting earthquakes during the 1997 Umbria-Marche (central
Italy) seismic sequence, J. Geophys. Res., 110, B05S20, doi:10.1029/2004JB003386.
Ogata, Y. (1988), Statistical models for earthquake occurrence and residual analysis for point
processes, J. Amer. Statist. Assoc., 83, 9–27.
Ogata, Y. (1998), Space-time point-process models for earthquake occurrences, Ann. Inst.
Statist. Math., 50, 379–402.
Okada, Y. (1985), Surface deformation due to shear and tensile faults in an half-space, Bull.
Seism. Soc. Am., 75 (4), 1135–1154.
Okada, Y. (1992), Internal deformation due to shear and tensile faults in an half-space, Bull.
Seism. Soc. Am., 82 (2), 10181040.
Ruina, A. (1983), Slip instability and state variable friction laws, J. Geophys. Res., 88 (B12),
10,359–10,370.
Stein, R. (1999), The role of stress transfer in earthquake occurrence, Nature, 402, 605–609.
Stein, R., A. Barka, and J. Dieterich (1997), Progressive failure on the North Anatolian fault
since 1939 by earthquake stress triggering, Geophy. J. Int., 128, 594–604.
Toda, S., and R. Stein (2000), Did stress triggering cause the large off-fault aftershocks of the 25
March 1998 Mw=8.1 Antarctic plate earthquake?, Geophys. Res. Lett., 27 (15), 2301–2304.
Toda, S., and R. Stein (2002), Response of the San Andreas fault to the 1983 Coalinga-Nunez
earthquakes: an application of interaction-based probabilities for Parkfield, J. Geophys. Res.,
107 (B6), 2126, doi:10.1029/2001JB000172.
Toda, S., and R. Stein (2003), Toggling of seismicity by the 1997 Kagoshima earthquake couplet:
a demonstration of time-dependent stress transfer, J. Geophys. Res., 108 (B12), 2567, doi:
10.1029/2003JB002527.
Toda, S., R. Stein, P. Reasenberg, J. Dieterich, and A. Yoshida (1998), Stress transferred by the
1995 Mw=6.9 Kobe, Japan, shock: Effect on aftershocks and future earthquake probabilities,
J. Geophys. Res., 103 (B10), 24,54324,566.
Udias, A. (1999), Principles of Seismology, Cambridge University Press.
Utsu, T. (1969), Aftershocks and earthquake statistics (i) – investigation of aftershocks and
other earthquake sequences based on a new classification of earthquake sequences, J. of
Faculty of Science, Hokkaido Univ., Series VII (Geophysics), 3.
Utsu, T., Y. Ogata, and R. Matsuura (1995), The centenary of the Omori formula for a decay
law of aftershock activity, J. Phys. Earth, 43, 1–33.
Wiemer, S. (2000), Introducing probabilistic aftershock hazard mapping, Geophys. Res. Lett.,
27 (20), 3405–3408.
Wiemer, S., and K. Katsumata (1999), Spatial variability of seismicity parameters in aftershock
zones, J. Geophys. Res., 104 (B6), 13,135–13,152.
Wiemer, S., and M. Wyss (2002), Spatial and temporal variability of the b-value in seismogenic
volumes, Adv. Geophysics, 45, 259–302.
Wiemer, S., M. Gerstenberger, and E. Hauksson (2002), Properties of the Aftershock Sequence
of the 1999 Mw 7.1 Hector Mine Earthquake: implications for aftershock hazard, Bull. Seism.
Soc. Am., 92 (4), 1127–1240.
Yamanaka, Y., and K. Shimazaki (1990), Scaling relationship between the number of aftershocksand
the size of the mainshock, J. Phys. Earth., 38, 305–324.
Ziv, A. (2003), Foreshocks, aftershocks, and remote triggering in quasi-static fault models, J.
of Geophys. Res. (Solid Earth), 108, 14,1–14,13.
Ziv, A., and A. M. Rubin (2003), Implications of rate-and-state friction for properties of aftershock
sequence: Quasi-static inherently discrete simulations, J. of Geophys. Res., 108, 2051,
doi:10.1029/2001JB001219.
stress change, J. Geophys. Res., 108 (B3), 2135, doi:10.1029/2002JB001779.
Boatwright, J., and M. Cocco (1996), Frictional constraints on crustal faulting, J. Geophys.
Res., 101 (10), 13,895–13,910.
Bonafede, M., and A. Neri (2000), Effects induced by an earthquake on its fault plane: a boundary
element study, Geophys. J. Int., 141 (1), 43–56, doi:10.1046/j.1365-246X.2000.00074.x.
Chinnery, M. (1961), The deformation of the ground around surface faults, Bull. Seism. Soc.
Am., 51, 355–372, doi:10.1046/j.1365-246X.2000.00074.x.
Cocco, M., and J. R. Rice (2002), Pore pressure and poroelasticity effects in coulomb stress analysis
of earthquake interactions, J. Geophys. Res., 107(B2), 2030, doi:10.1029/2000JB000138.
Console, R., M. Murru, and A. Lombardi (2003), Refining earthquake clustering models, J.
Geophys. Res., 108 (B10), 2468, doi:10.1029/2002JB002130.
Dieterich, J. (1986), A model for the nucleation of earthquake slip, Geophy. Monogr. Ser., 37,
36–49.
Dieterich, J. (1992), Earthquake nucleation on faults with rate and state dependent strenght,
Tectonophysics, 211, 115–134.
Dieterich, J. (1994), A constitutive law for rate of earthquake production and its application
to earthquake clustering, J. Geophys. Res., 99 (18), 2601–2618.
Dieterich, J. (1995), Earthquake simulations with time-dependent nucleation and long-range
interactions, Nonlinear Processes in Geophysics, 2, 109–120.
Enescu, B., and K. Ito (2002), Spatial analysis of the frequency-magnitude distribution and
decay rate of aftershock activity of the 2000 Western Tottori earthquake, Earth Planet Space,
54, 847–859.
Felzer, K. R., T. W. Becker, R. E. Abercrombie, G. Ekstrm, and J. R. Rice (2001), Triggering
of the 1999 MW 7.1 Hector Mine earthquake by aftershocks of the 1992 MW 7.3 Landers
earthquake, J. Geophys. Res., 107 (B9), 2190, doi:10.1029/2001JB000911.
Gomberg, J., N. Beeler, M. Blanpied, and P. Bodin (1998), Earthquake triggering by transient
and static deformations, J. Geophys. Res., 103 (12), 24,411–24,426.
Gomberg, J., N. Beeler, and M. Blanpied (2000), On rate-state and Coulomb failure models,
J. Geophys. Res., 105 (14), 7857–7872.
Gomberg, J., P. Reasenberg, M. Cocco, and M. Belardinelli (2005), A frictional population
model of seismicity rate change, J. Geophys. Res., 110 (B05S03), doi:10.1029/2004JB003404.
Gross, S., and C. Kisslinger (1997), Estimating tectonic stress rate and state with Landers
aftershocks, J. Geophys. Res., 102 (B4), 7603–7612.
Harris, R. (1998), Introduction to special section: Stress triggers, stress shadows, and implications
for seismic hazard, J. Geophys. Res., 103 (12), 24,347–24,358.
Helmstetter, A. (2003), Is earthquake triggering driven by small earthquakes?, Phys. Res. Lett.,
91 (058501).
Helmstetter, A., Y. Kagan, and D. Jackson (2005), Importance of small earthquakes
for stress transfers and earthquakes triggering, J. Geophys. Res., 110 (B05S08), doi:
10.1029/2004JB003286.
Iwasaki, T., and R. Sato (1979), Strain field in a semi-infinite medium due to an inclined
rectangular fault, J. Phys. Earth, 27, 117–123.
Kagan, Y. Y., and D. D. Jackson (1991), Long-term earthquake clustering, Geophys. J. Int.,
104, 117–133.
Kanamori, H., and D. L. Anderson (1975), Theoretical basis for some empirical realtions in
seismology, Bull. Seism. Soc. Am., 65, 1073–1095.
Keilis-Borok, V. (1959), On estimation of the displacement in an earthquake source and of
source dimensions, Annali di Geofisica, 12, 2,205–2,214.
Kilb, D., J. Gomberg, and P. Bodin (2002), Aftershock triggering by complete coulomb stress
changes, J. Geophys. Res., 107, doi:10.1029/2001JB0002002.
King, G., and M. Cocco (2001), Fault interaction by elastic stress. changes: New clues from
earthquake sequences, Adv. Geophys., 44, 1–39.
Knopoff, L. (1957), Energy release in earthquake, Tech. Rep. 90, Institute of Gephysics, University
of California, U.S.A.
Kostrov, B., and S. Das (1998), Principles of Earthquake Source Mechanics, Cambridge University
Press.
Marsan, D. (2003), Triggering of seismicity at short timescales following Californian earthquakes,
J. Geophys. Res., 108 (B5), 2266, doi:10.1029/2002JB001946.
Mendoza, C., and S. Hartzell (1988), Aftershock patterns and main shock faulting, Bull. Seism.
Soc. Am., 78, 1438–1449.
Nostro, C., L. Chiaraluce, M. Cocco, D. Baumont, and O. Scotti (2005), Coulomb stress changes
caused by repeated normal faulting earthquakes during the 1997 Umbria-Marche (central
Italy) seismic sequence, J. Geophys. Res., 110, B05S20, doi:10.1029/2004JB003386.
Ogata, Y. (1988), Statistical models for earthquake occurrence and residual analysis for point
processes, J. Amer. Statist. Assoc., 83, 9–27.
Ogata, Y. (1998), Space-time point-process models for earthquake occurrences, Ann. Inst.
Statist. Math., 50, 379–402.
Okada, Y. (1985), Surface deformation due to shear and tensile faults in an half-space, Bull.
Seism. Soc. Am., 75 (4), 1135–1154.
Okada, Y. (1992), Internal deformation due to shear and tensile faults in an half-space, Bull.
Seism. Soc. Am., 82 (2), 10181040.
Ruina, A. (1983), Slip instability and state variable friction laws, J. Geophys. Res., 88 (B12),
10,359–10,370.
Stein, R. (1999), The role of stress transfer in earthquake occurrence, Nature, 402, 605–609.
Stein, R., A. Barka, and J. Dieterich (1997), Progressive failure on the North Anatolian fault
since 1939 by earthquake stress triggering, Geophy. J. Int., 128, 594–604.
Toda, S., and R. Stein (2000), Did stress triggering cause the large off-fault aftershocks of the 25
March 1998 Mw=8.1 Antarctic plate earthquake?, Geophys. Res. Lett., 27 (15), 2301–2304.
Toda, S., and R. Stein (2002), Response of the San Andreas fault to the 1983 Coalinga-Nunez
earthquakes: an application of interaction-based probabilities for Parkfield, J. Geophys. Res.,
107 (B6), 2126, doi:10.1029/2001JB000172.
Toda, S., and R. Stein (2003), Toggling of seismicity by the 1997 Kagoshima earthquake couplet:
a demonstration of time-dependent stress transfer, J. Geophys. Res., 108 (B12), 2567, doi:
10.1029/2003JB002527.
Toda, S., R. Stein, P. Reasenberg, J. Dieterich, and A. Yoshida (1998), Stress transferred by the
1995 Mw=6.9 Kobe, Japan, shock: Effect on aftershocks and future earthquake probabilities,
J. Geophys. Res., 103 (B10), 24,54324,566.
Udias, A. (1999), Principles of Seismology, Cambridge University Press.
Utsu, T. (1969), Aftershocks and earthquake statistics (i) – investigation of aftershocks and
other earthquake sequences based on a new classification of earthquake sequences, J. of
Faculty of Science, Hokkaido Univ., Series VII (Geophysics), 3.
Utsu, T., Y. Ogata, and R. Matsuura (1995), The centenary of the Omori formula for a decay
law of aftershock activity, J. Phys. Earth, 43, 1–33.
Wiemer, S. (2000), Introducing probabilistic aftershock hazard mapping, Geophys. Res. Lett.,
27 (20), 3405–3408.
Wiemer, S., and K. Katsumata (1999), Spatial variability of seismicity parameters in aftershock
zones, J. Geophys. Res., 104 (B6), 13,135–13,152.
Wiemer, S., and M. Wyss (2002), Spatial and temporal variability of the b-value in seismogenic
volumes, Adv. Geophysics, 45, 259–302.
Wiemer, S., M. Gerstenberger, and E. Hauksson (2002), Properties of the Aftershock Sequence
of the 1999 Mw 7.1 Hector Mine Earthquake: implications for aftershock hazard, Bull. Seism.
Soc. Am., 92 (4), 1127–1240.
Yamanaka, Y., and K. Shimazaki (1990), Scaling relationship between the number of aftershocksand
the size of the mainshock, J. Phys. Earth., 38, 305–324.
Ziv, A. (2003), Foreshocks, aftershocks, and remote triggering in quasi-static fault models, J.
of Geophys. Res. (Solid Earth), 108, 14,1–14,13.
Ziv, A., and A. M. Rubin (2003), Implications of rate-and-state friction for properties of aftershock
sequence: Quasi-static inherently discrete simulations, J. of Geophys. Res., 108, 2051,
doi:10.1029/2001JB001219.
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article