Afterslip, slow earthquakes and aftershocks: Modeling using the rate & state friction law Agnès Helmstetter (LGIT Grenoble) and Bruce Shaw (LDE0 Columbia Univ) Days after Nias earthquake Cumulative number of aftershocks Ex: 2005 m=8.7 Nias earthquake [Hsu et al, Science 2006]
Main questions: relation between coseismic and postseismic slip? relation between afterslip and aftershocks? can we use afterslip to constrain the rheology of the crust (stable/unstable)? mechanisms for aftershock triggering?
Outline intro: observations of afterslip modelling afterslip with rate & state friction laws modelling aftershocks triggered by afterslip
Postseismic deformation observed after most «large» earthquakes duration : sec to decades postseismic displacement 10% of coseismic slip but with huge fluctuations between earthquakes (0-100%!) physical processes: friction: afterslip on mainshock fault diffuse deformation: poro-elasticity or viscous deformation superposition of # mechanisms, ex: Denali [Freed et al 2006]
Spatial distribution of afterslip and aftershocks 2005 m=8.7 Nias 2003 m=8 Tokachi [Hsu et al, Science 2006] [Miyazaki et al, GRL 2004]
Temporal distribution of afterslip and aftershocks [Langbein et al., 2006] 2004 m=6.0 Parkfield earthquake before mainshok after mainshok 1 hour [Peng and Vidale, 2006]
Observations afterslip occurs mostly around coseismic slip overlap between coseismic and afterslip, and aftershock areas displacement log(1+t/t*), with t* hrs to days similar time-dependence for afterslip and aftershocks ( Omori ) but with # characteristic times ( c «t* )
Rate-and-state friction law V 1 V 2 >V 1 µ=τ/σ V friction coefficient µ A D c friction law [Dieterich, 1979]! " = µ = µ # 0 + A ln V $ V * state variable θ age of contacts B slip % & + Bln # ' $ ' * % & B<A: stable µ with V velocity-hardening B>A unstable µ with V velocity weakening dθ/dt = 1 - Vθ/D c parameters values in the lab : - A B 0.01, depend on T,σ, gouge thickness, strain - D c 1-100 µm, depends on roughness and gouge thickness
Rate-and-state friction law From stick-slip to stable sliding fault slip rate V tectonic loading V L displacement postseismic slip stick slip m k nucleation µ(v,θ) fault loading point time transition controlled by stiffness k/k c (or rupture length) and ratio B/A, with k c = (B-A)σ n /D c [Rice and Ruina, 1983] stable sliding: k>k c or B<A stick-slip: k<k c and B>A
Rate-and-state friction law and afterslip slip speed for a slider-block with a fixed loading point relaxation or nucleation of a slip instability after a stress step initial condition: slip rate V 0 and stress τ 0 inertia and tectonic loading negligible tectonic deformation «slip rate «cosesimic slip rate V 0 m k fixed loading point (locked part of the fault) µ 0 (V,θ)
First model: steady-state approximation Scholz [1989], Marone et al [1991] and many others assume - slip-strengthening friction (stable) A>B - steady state θ=constant R&S equations becomes: µ = µ 0 +(A-B) log(v/v 0 ) = µ 0 -kδ/σ n V= θ/d c V= V 0 /(1+t/t*) V 0 δ m k t* = σ n (A-B)/kV 0 µ 0 (V,θ) good fit to afterslip data, and its distribution with depth: mostly above and bellow the seismogenic zone, where A>B constant θ constant V: OK when V varies between m/s to mm/yr? overlap between coseismic rupture, aftershocks and afterslip areas?
Rate-and-state friction law and afterslip k=0.8 k c velocity weakening regime B>A : transition between postseismic relaxation, slow earthquake, and aftershock as τ 0 velocity strengthening regime B<A : transition between postseismic relaxation and slow earthquake as τ 0, no aftershock power-law relaxation V~1/t p, with p 1
(in-)stability after a stress step initial acceleration dv/dt>0 if stress is large enough : µ>µ a if state variable θ decreases rapidly with time dθ/dt < dθ a /dt and dθ a /dt<0 no acceleration if dθ a /dt>0
(in-)stability after a stress step if V after the stress step, the system can evolve toward: slip-instability («aftershock») or transcient slip event («slow earthquake») condition for instability: V and dθ/dt ("weakening" of friction interface) µ>µ l and k c /k<1 dθ /dt < dθ l /dt <0 instability possible only if k< k c and B>A if V but dθ/dt («healing»), then latter V will : «slow EQ»
(in-)stability after a stress step (1) (2) (3) behavior controlled by both friction parameters k/k c, B/A and stress (1) and (2) postseismic relaxation (4) (5) (3) and (4) transition from postseismic relaxation to slow EQ as µ 0 (5) transition from postseismic relaxation to slow EQ and aftershock as µ 0
(in-)stability after a stress step behavior as a function of distance from steady-state and B/A for k=0.8k c no steady state regime θ ("healing") or θ ("weakening") steady state approx only valid for B<A and steady-state θ θ k«k c
slip rate history - 1D model Simulations with parameters top: B=1.5A and k=0.8k c bottom: B=0.5A and k=2.5 k c µ 0 >µ l µ l >µ 0 >µ a µ 0 <µ a # power-law afterslip regimes, with # slope exponents: B/A or 1 # characteristic times t* µ 0 >µ lss µ 0 =µ lss µ 0 <µ ss
Slip history - 1D model Simulations with: top: B=1.5A and k=0.8k c bottom: B=0.5A and k=2.5 k c µ 0 >µ l µ l >µ 0 >µ a µ 0 <µ a afterslip D c for both B<A and B>A Slip with stress µ 0 >µ lss µ 0 =µ lss µ 0 <µ ss
Slip rate history Superstition Hills EQ [Wennerberg and Sharp, 1997] surface displacement for 6 points along the fault model: (1) R&S friction law in the steady-state regime : µ=µ 0 +(A -B)logV bad fit because in the data V ~1/(1+t/t*) p with p<1 or p>1 (2) initial form of the law [D., 1979] with θ=const. good fit, but : - variations of A and B? - p<1 A<0!? R&S with 2 state variables?
Slip history - 1D model and afterslip data fit (fit of) afterslip data Wennerberg and Sharp [1997] for Superstition Hills 6 points along the fault invert for A, B, k, D c, V 0 and µ o, using broad range of initial values data can be fitted with: V=V 0 /(1+t/t * ) p inversion not constrained! «data» fit B>A fit B<A
Slip history - 1D model and afterslip data for most points, we can t distinguish between B > or < A we don t need A<0 or more complex friction law limits of 1D model?
Conclusions: aferslip and slow EQs R&S friction law can be used to model afterslip data or slow EQs including deviations from log slip history (p< or >1) in both slip-weakening B>A and slip strengthening regimes B<A we don t need along strike variations or temporal variations in B or A to explain overlap between coseismic and afterslip areas we don t need A<0 or more complex friction laws but requires relatively large D c afterslip R&S friction law produces triggered slow EQs, both for B> or <A R&S friction can t be used to invert for the model parameters 6 model parameters, but 3 are enough to fit the data except if good spatial resolution of co- and postseismic slip and using homogeneous or smooth friction parameters?
Afterslip and aftershocks similar time dependence of afterslip rate and aftershock rate ( Omori ) afterslip due to aftershocks, or aftershocks triggered by afterslip? coseismic slip induces: stress increase (on and) around the rupture afterslip stress release in sliping areas reloading on locked parts of the faults aftershocks triggered by afterslip (in unstable areas) [Dieterich 1994, Schaff et al 1998, Perfettini and Avouac 2004, Wennerberg and Sharp 1997, Hsu et al 2006, Savage 2007, ] we use the R&S model of Dieterich [1994] to model the effect of stress changes on seismicity rate, instead of assuming seismicity rate ~ stress rate
Relation between stress changes and seismicity in the R&S model Dieterich [2004] model is equivalent to R: seismicity rate R 0 = R(t=0) N= t R dt 0 r: ref seismicity rate for τ =τ r τ: coulomb stress change (=0 at t=0) t a : nucleation time = Aτ n /τ r short-times regime for T«t a R~R 0 exp(τ/aτ n ) (tides [Cochran et al 2005]) long-times regime for T»t a N~τ (tectonic loading, ) for a stress step τ(0)=δτ : R(t) Omori law with p=1
Aftershocks triggered by afterslip numerical solution of R-τ relation assuming reloading due to afterslip is of the form dτ/dt ~ V (elastic stress transfer) ~ τ 0 /(1+t/t * ) p with p=0.8 t * t * dτ/dt τ R ~ stress rate for t»t * when p<1 short time cut-off of Omori law for EQ rate is always larger than for stress rate
Aftershocks triggered by afterslip numerical solution of R-τ relation assuming reloading due to afterslip is of the form dτ/dt ~ V (elastic stress transfer) ~ τ 0 /(1+t/t * ) p with p=1.3 t * stress step afterslip Δτ= 2MPa t * dτ/dt τ τ r short times: R 0 intermediate times: R ~ dτ/dt ~ 1/t p with p>1 large times: R = aftershock rate for a stress step of the same anplitude
Conclusions: afershocks triggered by afterslip R&S friction law can be used to model aftershock rate aftershock rate decreases as a modified Omori law with p< or >1 apparent exponent changes with time because afterslip is comparable to coseismic slip, the number of aftershocks triggered by afterslip must be significant but t * for afterslip days» c of aftershocks sec short times aftershocks can t be explained by afterslip many models reproduce Omori law, not very helpful complex relation between stress and seismicity history, so be careful when using seismicity to identify possible triggering processes
The end For details see: Helmstetter, A., and B. E. Shaw, Afterslip and aftershocks in the rate-and-state friction law, submitted to J. Geophys. Res. (March 2007) Draft availble at: http://www.arxiv.org/abs/physics/0703249).
Analytical approximations afterslip p=1 aftershock nucleation [Dieterich 1992, 1994] slow EQ afterslip p=1 afterslip, p=b/a
Rate-and-state friction law and EQs Slip speed for a slider-block with a constant loading rate log slip speed V l postseismic slip <0 or >0 τ change triggered EQ slow EQ EQ delayed EQ k m µ(v,θ) nucleation V l B>A B A B<A t a = Aσ/τ time
Rate-and-state friction law From stick-slip to stable sliding A-B stable unstable A-B (A-B)σ z (km) z (km) [Scholz, 1998]
Spatial distribution of afterslip and aftershocks 2002 m=7.8 Denali [Freed et al, JGR 2006]
Temporal distribution of afterslip and aftershocks 2005 m=8.7 Nias earthquake [Hsu et al, Science 2006] Days after Nias earthquake Cumulative number of aftershocks log variation of afterslip and number of M>0 aftershocks with time ~log(1+t/t*) with same characteristic time t* 1 day (!?) but looking at M>5 events gives t* 10-3 days
Temporal distribution of afterslip and aftershocks 1999 m=7.6 ChiChi earthquake [Perfettini and Avouac 2004]
Time decay of afterslip Afterslip measured using creep-meters Marone et al., [JGR 1991]; Marone [AREPS, 1998] log variation of afterslip with time no surface afterslip if coseismic rupture reaches the surface (Landers)
Rate-and-state friction law friction µ depends on slip rate V and state variable θ derived from friction experiment [Dieterich, 1979] and theoretical models applied to earthquakes, landslides, material sciences, aseismic slip models the transition between stable and unstable slip in the crust, afterslip, slow earthquakes, and earthquake triggering due to stress changes
First model of afterslip Model of Scholz [1990], Marone et al., [JGR 1991]; Marone [AREPS, 1998] Shallow afterslip due to slip deficit during the EQ
(in-)stability after a stress step behavior as a function of distance from steady-state and B/A for k=2.5 k c no steady state regime θ ("healing") or θ ("weakening") steady state approx only valid for B<A and steady-state healing θ weakening θ k«k c
Friction as a function of slip rate - 1D model (1) (2) (3) (4) (5) (6) Simulations with parameters top: B=1.5A and k=0.8k c bottom: B=0.5A and k=2.5 k c history of µ(v) can t be used µ 0 >µ l µ l >µ 0 >µ a µ 0 <µ a to measure A or B-A, because slope depends on model parameters µ 0 >µ lss µ 0 =µ lss µ 0 <µ ss
Friction as a function of slip rate 2005 M=8.7 Nias EQ [Hsu et al., 2006] GPS displacement afterslip map slip rate and stress Model: [Perfettini 2004] µ = µ0 +A logv/v0 A is <0!!??
Friction as a function of slip rate 2003 M=8.0 Tokachi EQ [Miyazaki et al., 2004] model with R&S friction A, B, C : steady-state friction µ=µ 0 +(A -B)logV (A-B)σ 0.6 MPa? D : Velocity weakening B>A?