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Earthquake triggering Properties of aftershocks and foreshocks and implications for earthquake forecasting Agnès Helmstetter , ISTerre , CNRS, University Grenoble 1. Earthquake triggering. When? Where? What size? Scaling with mainshock size? How ?…. Outline Aftershocks

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Earthquake triggering

Properties of aftershocks and foreshocks

and implications for earthquake forecasting

AgnèsHelmstetter, ISTerre, CNRS, University Grenoble 1

Earthquake triggering

Earthquake triggering

When? Where? What size?

Scaling with mainshock size?




when? where? scaling with mainshock size?

why? : static, dynamic, or postseismic stress change?

model? : ETAS or rate & state


Earthquakes that trigger by chance a larger event

… or part of the nucleation process?

Distribution in time, space and magnitude and comparison with ETAS

Foreshocks and aftershocks of landers california


1 day

Foreshocks and aftershocks of Landers, California






Foreshocks a

few hrs before





Temporal decay of aftershocks

Japan M=9.1

Sumatra M=9.0



Temporal decay of aftershocks




  • stacks for California and rate following the Sumatra and Tohoku M=9 EQs

  • aftershock rate ~1/tp with p≈0.9 (Omori’s law)

  • duration ≈ yrs indep of M

Scaling with mainshock magnitude

Japan M=9.1

Sumatra M=9.0

Scaling with mainshock magnitude




  • Aftershock rate N(M)~10M ~ rupture area

  • Magnitude distribution P(M)~10-M (GR law)

  • Small and large EQs have the same influence on EQ triggering !

Aftershocks where


- aftershocks

1 day after

-- background

1 day before

number of aftershocks


distance from mainshock hypocenter (km)

  • relocated catalog for Southern California

  • [Shearer et al., 2004]

  • average distance

  • ≈ rupture length L

  • <d> ≈ 0.01x10m/2 km

  • max distance ‘+’

    dmax ≈ 7 L

    ≈ 0.07x10m/2 km

Aftershocks : Where?

mainshocks 7<m<7.5


mainshock 2<m<2.5

Aftershocks : What size?

  • aftershocks magnitude distribution = GR law

  • aftershock size does not depend on the mainshock magnitude !

Dynamic triggering

Seismicity remotely triggered by M7.3 Landers EQ [Hill et al 1993]

Long Valley



Dynamic triggering


filtered 5-30 Hz

  • Mostly in geothermal or volcanic areas

  • Dynamic stress change ≈ 1 bar >> static

  • During seismic wave propagation

  • but also in the following days

  • Transient deformation at Long-Valley :

  • change in fluid pressure ?

Summary of observations about aftershocks
Summary of observations about aftershocks

  • aftershock rate decays as N~1/t, for t between a few sec and several yrs, independently of M

  • + short-term remote dynamic triggering by seismic waves

  • number of aftershocks increases as N ~10M~ L2, for 0<M<9.

  • small EQs collectively as important as larger ones for triggering

  • the size of a triggered EQ is not constrained by M

  • typical triggering distance ≈ L ≈ 0.01x10m/2 km,

    • max distance for t<1day ≈ 7L

T riggered seismicity not only aftershocks
Triggered seismicity : not only aftershocks!

  • Other evidences of triggered seismicity, natural and human-induced

  • rainfall (pore pressure changes due to diffusing rain water) [Hainzl et al 2006]

  • CO2 degassing [Chiodini et al 2004; Cappa et al 2009]

  • slow slip events [Segall et al 2006; Lohman & McGuire 2007, Ozawa et al 2007]

  • tides (hydrothermal, volcanic areas or shallow thrust EQs, ∆≈10 kPa, ∆R=10%)[Tolstoy et al 2002 ; Cochran et al 2004]

  • migration of underground water or magma [Hainzl & Fisher 2002]

  • nuclear explosions [Parsons & Velasco 2009]

  • mining (stress concentrations due to the excavation)[McGarr et al., 1975]

  • dams (filling of water reservoirs)[Simpson et al 1988, Gupta 2002]

  • fluid injections or extraction (geothermal power plants, hydraulic fracturing, for oil and gas production, injection of wastewater, extraction of groundwater) [McGarr et al., 2002; Gonzales et al 2012; Ellsworth 2013]

  • … any process that modifies the stress or the pore pressure

What triggers aftershocks





seismicity rate

after a mainshock


What triggers aftershocks?

Aftershocks triggered by

Static stress changes? postseismic? dynamic?

Coseismic, permanent afterslip, fluids seismic waves







Mechanisms of aftershock triggering

Static stress change

permanent change⇒easy to explain long-time triggering

fast decay with distance ~ 1/r3⇒how to explain distant aftershocks?

Dynamic stress change

short duration⇒how to explain long time triggering?

slower decay with distance ~ 1/r⇒better explains distant aftershocks

Postseismic relaxation

afterslip, fluid flow, viscoelastic relaxation

slow decay with time, ~ seismicity rate ⇒ easy to explain Omori law

but smaller amplitude than coseismic stress change

Mechanisms of aftershocktriggering

Modelling triggered seismicity

Statistical model : ETAS

seismicity rate = background+ triggered seismicity[Kagan, 1981, Ogata 1988…]

R(t,r) = µ(r) + ∑ti<tϕ(t-ti, |r-ri|, mi)

Physical model : coulomb stress change calculations + rate & state model

A ≈ 0.01 parameter of R&S friction law, increase of friction with V

σ: normal stress ; τ: coulomb stress change ;τr’tectonic stressing rate

r : background seismicity rate for τ’=τr’ ; N : cumulated number∫R(t)dt

[Dieterich 1994]






ETAS model

Input : proba that an EQ (t,r,m) triggers another EQ(t’,r’,m’)

Results : multiple interaction between EQs



« direct » Omorilaw

Rd(t) ~1/tp






ETAS : aftershocks and foreshocks (t)

  • Assumptions:

  • Results

Aftershocks +aft. of aft. + …

« global » Omorilaw

Rg(t) » Rd(t)

Rg(t) ≈ 1/tpgwthpg<p


Inverse Omori law

R(t) ~1/tpf


Foreshocks mainshocks aftershocks


inverse Omori law

N(t)~1/(t+c)pfwith pf≤ p

Aftershocks, Omori law


seismicity rate

background rate



“Foreshocks”, “mainshocks”, “aftershocks”

average over many sequences

---- a typical sequence

ETAS model : main results

  • Aftershocks

  • “Global” Omori law with a pglobal≤ pdirect

  • Bath’s law : largest aftershock average magnitude = M-1.2

  • Diffusion of aftershocks

  • Foreshocks

    • Inverse Omori law with pforeshocks≤pdirect

    • Rate of foreshocks independent of mainshock magnitude (if any EQ is a mainshock)

    • Deviation from GR law bforeshocks≤b

    • Migration toward mainshock



T» ta



Rate-and-state : periodic stress changes

  • stress : τ(t) = cos(2πt/T) + τ’r t

  • T« ta or T»ta

  • ta : nucleation time ≈ yrs


short-times regime

for T«ta


tides, seismic waves

long-times regime

for T»ta


tectonic loading


Rate and state triggering by a stress step
Rate-and-state : triggering by a stress step

  • Reproduces Omori law withp=1for a positive stress change

  • Requires a very large ∆ : c=10-4 ta=100 days Aσ=1 MPa⇒∆=15 MPa !



Heterogeneity of eq source and aftershocks
Heterogeneity of EQ source and aftershocks

  • Planar fault with uniform stress drop

    slip∆ EQ rate

  • Real faults : heterogeneous slip and rough faults

  • >hetergoneousstress change in the rupture zone

  • > most aftershocks on or very close to the rupture zone



EQ rate

[Marsan, 2006; Helmstetter & Shaw, 2006]

Slip and shear stress heterogeneity aftershocks

mean stress τ0

Slip and shear stress heterogeneity, aftershocks

Modified « k2 » slip model: U(k) ~ 1/(k+1/L)2.3 [Herrero & Bernard, 1994]

aftershock map

synthetic catalog

R&S model

shear stress

stress drop τ0 =3 MPa


R s model stress heterogeneity and aftershock decay with time



R&S model, stress heterogeneity, and aftershock decay with time

Aftershock rate

heterogeneous ∆




  • triggering at short-timet«ta : Omori law with p<1

  • quiescence at long time (t≈ta≈yrs)

  • [Marsan, 2006; Helmstetter and Shaw, 2006]

Modified k 2 slip model off fault stress change
Modified k2 slip model, off-fault stress change

  • fast attenuation of high frequency τ perturbations with distance



coseismic shear

stress change (MPa)

Modified k 2 slip model off fault aftershocks
Modified k2 slip model, off-fault aftershocks

  • seismicity rate and stress change as a function of d/L

  • quiescence for d >0.1L



standard deviation

average stress change

R s and aftershock time decay
R&S and aftershock time decay

  • stacked A.S. for 82 M.S. with 3<M<5 z<50 km in Japan

  • triggering following Omori law decay for 10 s <t<1 yr with pincreasing slightly with time

[Peng et al 2007]


Fit by rate-state model with a Gaussian stress pdf

<∆τ> =0

std(∆τ)/An = 11

ta= 0.9 yrs



Modeling aftershock rate with r s model and heterogeneous static stress change
Modeling aftershock rate with R&S model and heterogeneous static stress change

Sequence pτ* (MPa)ta (yrs)

Morgan Hill M=6.2, 1984 0.68 6.2 78.

Parkfield M=6.0, 2004 0.88 11. 10.

Stack, 3<M<5, Japan* 0.89 12. 1.1

San Simeon M=6.5 2003 0.93 18. 348.

Landers M=7.3, 1992 1.08** 52.

Northridge M=6.7, 1994 1.09** 94.

Hector Mine M=7.1, 1999 1.16** 80.

Superstition-Hills, M=6.6,1987 1.30 ** **

*[Peng et al., 2007]

**we can’t estimate τ* becausep>1

R(t) static stress change




R&S : triggering by afterslip

Mainshock ⇒ coseismic stress change

⇒ afterslip

⇒ postseismicreloading

⇒ aftershocks?

AfterslipPostseismic Aftershock rate stress change



R&S : triggering by static stress changeafterslip

We assume stressing rate due to afterslipdτ/dt~ τ’0/(1+t/t*)qwithq=1.3

seismicity rate

stressing rate

  • Apparent Omoriexponent p(t) decreasesfrom 1.3 to 1

R s model and omori s law
R&S model and Omori’s law static stress change

Deviations from Omori law with p=1 can be explained by :

Coseismictriggering with heterogeneous stress step

  • short-time triggering p≤1, p↘ with t and with stress heterogeneity

  • long-time quiescence

    Postseismic triggering by afterslip

  • Omori law decay with p< or >1





log R

log R




log t

log t

EQ triggering and EQ forecasting static stress change

  • seismicity rate increases a lot (≈104) after a large EQ

  • … but the proba of another large EQ is still very low !

  • limited use for EQ forecasting ?

  • Methods : statistical (ETAS, STEP, kernel smoothing …) or physical models (R&S + Coulomb stress change)

  • ETAS generally provides the best forecasts [Woessner et al 2011; Segou et al 2013]

    • Very simple to use (requires only t,x,y,z,m)

    • Bad modeling of early A.S. spatial distribution

    • … but can be corrected (kernel smoothing of early A.S.) [Helmstetter et al 2006]

  • Coulomb-stress change with R&S

  • Good fit in the far-field, but bad near the rupture (∆ is not accurate)

    • … but can be corrected by assuming a pdf of ∆[Hainzl et al 2009]

    • Usually include only M>6 M.S. (with known slip)

And before the mainshock

And before the mainshock?

Seismicity rate before mainshock

Example : seismicity rate before each M>7 mainshock in California and stack for all M>5 (for R<20 km)

Seismicity rate before mainshock

Seismicity rate before a mainshock

  • Stacks for California and ETAS for California and stack for all M>5 (for R<20 km)mainshock with 2<M<7.5

  • Mainshock : any EQ not preceded by a larger EQ for T=100 days and r<10 km

  • Foreshocks : EQs within 100 days before and 10 km

  • Power-law ↗ of seismicity : inverse Omori law

  • Number of foreshocks ↗ with M because of mainshock selection rules

    • California ETAS

Seismicity rate before a mainshock



Magnitude pdf of foreshocks

  • Stacks for California and ETAS for California and stack for all M>5 (for R<20 km)mainshock with 2<M<7.5

  • For small mainshocks : roll-off Mforeshock<Mmainshock

  • For large mainshocks : increase in the rate of large EQs

  • ETAS theory : P(m)= GR(m,b) + GR(m,b-α) [Helmstetter et al 2003]

    • California ETAS

Magnitude pdf of foreshocks

Spatial distribution of foreshocks m

Spatial distribution of foreshocks (M)

  • small d : similar pdf(d) for all M, but ! location error ↗ with M

  • large d : increase in pdf(d) for all M due to selection rule MF.S. < MM.S.

Spatial distribution of foreshocks time

timebefore M.S. (day).

Spatial distribution of foreshocks (time)

  • apparent migration towardmainshock.

Foreshocks asesimic loading

  • Swarms sometimes detected before California and stack for all M>5 (for R<20 km)mainshocks (not explained by ETAS) ex : M=9 Tohoku [Marsan et al, 2013]

  • «Repeating» EQs (triggered by aseimic slip?) and low-frequency noise

  • ex : m=7.6 Izmit[Bouchon et al 2011] or M=9 Tohoku [Kato et al 2012]

  • Slow slip event

    • Ex : M=8.1 Iquique [Ruiz et al, 2014]

  • Accelerating foreshock sequences followed by enhanced aftershock rate

  • Stack of M>6.5 mainshocks worldwide [Marsan et al 2014]

  • Foreshock / aftershock ratio istoo large

  • Stack for 2.5<M<5.5 mainshocks in California [Shearer 2012]

  • Foreshocks do not promote the mainshock (∆<0)

  • Landers M=7.3 and other EQs in California M4.7-6.4 [Dodge et al 1995,1996]

  • Accelerating slip predicted by R&S friction law and lab friction experiments … but very small slip (≈ Dc) and difficult to detect [Dieterich 1992]

Foreshocks = asesimic loading ?

Asesimic loading before mainshocks

  • but in most cases nothing special occurs before California and stack for all M>5 (for R<20 km)mainshocks

  • and most slow EQs, repeating EQs or swarms are not followed by mainshocks !

  • need to consider whole seismicity (not only before mainshocks) to check that these patterns are really unusual !

Asesimic loading before mainshocks?

Swarms before mainshocks

  • fitting seismicity with ETAS with variable background µ( California and stack for all M>5 (for R<20 km)t,r) to detect deviations = transient [Marsan et al, 2013]

  • Transient before

  • Tohoku, Jan-Feb/2011

  • ≈30 days, 40 km

  • ●all EQs

  • ●transient

  • but several other swarms detected not related to large EQs …

Swarms before mainshocks

Repeating eqs before mainshocks

  • accelerating repeating California and stack for all M>5 (for R<20 km)EQs with very similar waveforms during the last 44 mn before M=7.6 1999 Izmit EQ [Bouchon et al 2011]

  • 18 events with 0.3<M<2.7, distant by <20 m

Repeating EQs before mainshocks

Normalized waveforms, chronological order

Waveforms of the 1st and 2nd ev.

Top : filter <3 Hz

Repeating eqs before mainshocks1

Repeating EQs before mainshocks

  • repeating EQs : large correlation -> same exact location?

Slow slip events before mainshocks

Intense foreshock activity and a SSE before M=8.1 Iquique California and stack for all M>5 (for R<20 km)[Ruiz et al 2014]

Slow slip events before mainshocks



SSE with slip≈1m following the largest M6.7 foreshock 15 days before mainshock (or unusually large afterslip?)

Foreshock activity related to enhanced aftershocks

  • Stacked seismicity rate with M>4 before and after M>6.5 mainshocks in the worldwide ANSS catalog [Marsan et al 2014]

  • Population A :

  • Significant precursory

  • acceleration

  • Population B :

  • No significant precursory

  • acceleration

  • This pattern cannot be explained by ETAS, incompleteness, or # in M.S. M

  • episodic creep that preceded the M.S. and lasted during the A.S. sequence?

Foreshock activity related to enhanced aftershocks

Foreshocks did not trigger each other and did not trigger the mainshock

M7.3 mainshock mainshocks in the worldwide ANSS catalog

Stress change due to the Landers foreshocks did not trigger the mainshock (∆<0)… but results depend on relocation method

M7.3 mainshock

Foreshocks did not trigger each other and did not trigger the mainshock?





[Marsan 2014]

SHLK catalog

M3.6 foreshock

In SHLK catalog

[Dodge et al 1995]

Conclusion mainshocks in the worldwide ANSS catalog

  • earthquake triggering explains most properties of EQ catalogs

  • triggering mechanism : static? dynamic? postseismic?

  • but some discrepancies : swarms, heterogeneity, excess of foreshocks …

  • need to model accurately «normal» seismicity to detect deviations

  • deviations from normal seismicity ⇒ aseismic loading?

  • detection of “aseismic loading” : from EQ catalogs? Geodesy?

  • aseismic loading = precursor (part of nucleation)?

  • or aseismic loading = potential triggering factor (like foreshocks)?

  • implication for EQ forecasting :

  • ↗ in seismicity rate ⇒ ↗ in the proba of a future large event?

  • Or can we do better?

Tutorial : statistical analyses of EQ catalogs to reveal nucleation and triggering patterns

  • distribution of aftershocks and foreshocks in time, space and magnitude

  • transient increase in catalog incompleteness after a large EQ, implication for the temporal decay of aftershocks

  • how to identify foreshocks, mainshock and aftershocks?

  • comparison of foreshocks and aftershocks properties in ETAS model or in the R&S model

  • can we estimate ETAS model parameters (p, c, α, µ, b …) from stacked aftershock sequences?

  • how dependent are the results on : parameter choices (windows in time, space, magnitude …), location errors, catalog incompleteness …?

Tutorial nucleation and triggering patterns

  • download and unzip


  • Archive with EQ catalogs, matlab codes, ETAS program

  • You alsoneedmatlab and a fortran compiler to use the ETAS simulator

Tutorial : earthquake catalogs nucleation and triggering patterns

  • ANSS catalog for California

    • M≥1 ; 31≦ lat ≦ 43°N ; -127 ≦ lon ≦ -110°

  • Relocated SHLK catalog for California

    • M≥0 ; 31.4 ≦ lat ≦ 37°N ; -121.5≦ lon ≦ -114°

  • Worldwide ANSS catalog

    • M≥4

  • ETAS catalog :

    • GR law : b=1, M0=0, md=2

    • Aftershock : productivity K(m)~10αmwithα=1

    • Omori law : p=1.1, c=0.001 day

    • Aftershock spatial distribution : Φ(r,M)~1/(r+d010M/2)1+µ

    • with d0=0.01 km and µ=1

    • Uniform background, R=1000 km, Zmax=50 km, 2 M≥2 EQs / day

Tutorial : codes nucleation and triggering patterns

  • demo.m :

  • plots of earthquakes in space and time to illustrate clustering

  • aftershock rate following a large EQ and fit by Omori's law using MLE

  • transient changes in completeness magnitude mc after large Eqs

  • estimation of mc for different time and space windows (by fitting the magpdf by the product of a GR law and an erf function)

  • stack_aft.m

  • stack of aftershocks sequences for different classes of mainshock magnitude

  • simple selection rules (time, space and magnitude windows, following [Helmstetter et al 2005]

  • aftershock rate as a function of time, distance and magnitude including correction for time-dependent completeness

  • scaling of aftershock productivity with mainshock magnitude

  • comparison of California or worlwide seismicity and an ETAS catalog

Tutorial : codes nucleation and triggering patterns

  • stack_for.m :

  • stack of foreshock sequences for different classes of mainshock magnitude

  • foreshock rate as a function of time, distance and magnitude

  • comparison of California or worldwide seismicity and an ETAS catalog

  • aft_RS

  • aftershock rate due to a static stress change using the rate-and-state model, as a function of time and space [Dieterich 1994]

Tutorial : codes nucleation and triggering patterns

  • Toolbox :

  • omori_synt_cat : generates a EQ times following Omori’s law

  • Omori_fit : fit aftershock time decay by Omori’ law using Max. Likelihood

  • get_pm_erfGR : estimation of mc and b by fitting a magnitude distribution by the product of a GR law and an erf function

  • get_for, get_aft : selection of F.S., M.S. and A.S. using windows in t, r, and m. Computes rates of EQs in t, r and m.

  • get_mc : compute completeness magnitude for each EQ due to increase in etection threshold following large EQs[Helmstetter et al 2006]

R s triggering by a stress step
R&S : triggering by a stress step nucleation and triggering patterns



R(r) for t<ta

Stress change for a dislocation of length L: τ(r)~(1-(L/r)3)-1/2 -1

  • Very few events for r>2L

  • «diffusion» of aftershocks with time

  • Shape of R(r) depends on time, very # from τ(r)

  • Difficult to guess triggering mechanisms from the decrease of R(r)




R(r) for t>ta