Earthquake nucleation

• How do they begin?

• Are large and small ones begin similarly?

• Are the initial phases geodetically or seismically detectable?

Related questions:

Slip nucleation in the lab

Ohnaka’s (1990) stick-slip experiment

Figures from Shibazaki and Matsu’ura, 1998

The hatched area indicates the breakdown zone, in which the shear stress decrease from a peak stress to a constant friction stress.

Slip nucleation in the lab

Ohnaka, 1990

Slip nucleation in the lab

The 3 phases according to Ohnaka are:• Stable quasi-static nucleation phase (~1 cm/s).• Unstable, accelerating nucleation phase (~10 m/s).• Rupture propagation (~2 km/s).

The critical stiffness and the condition for slip acceleration

The critical stiffness:

In the case of the slip-weakening law:

Rice and Ruina (1983) provided a minimum estimate for kcrit, that is relevant for small perturbations from steady state:

€

kcrit =ξFNDc

.

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ξ =μpeak - μ residual .

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ξ =b− a .

Slope=k

The condition for slip acceleration:

€

kcrit > k .

The critical stiffness and the self-accelerating approximation

Recall that:

It is convenient to write this equation as follows:

where now contains all the constant parameters. The state evolution law (the aging law) is:

€

FSFN

= μ = μ∗+ a lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ b ln

θV *

DC

⎛

⎝ ⎜

⎞

⎠ ⎟ .

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FSFN

= μ = μC + a ln V( ) + a ln θ( ) ,

€

μC

€

dθ

dt=1−

θV

DC .

The critical stiffness and the self-accelerating approximation

For large sliding speeds, the following approximation holds:

the solution of which is:

and the quasi-static (i.e., slider mass=0) spring-slider force balance equation may be written as:

€

dθ

dt= −θV

DC ,

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θ =θ0 exp(−δ DC ) ,

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k(δlp −δ)

FN= μC + a ln(V ) + b ln(θ0) −

bδ

DC .

€

δ€

δlp

€

k

The critical stiffness and the self-accelerating approximation

The block neither accelerates nor decelerates if

Thus, to obtain the critical stiffness, one needs to take the slip derivative of the force balance equation for

This approach leads to:

The block will accelerate if:

€

kc =FNb

DC .€

dV /dδ = 0 .€

k = kc .

€

k <FNb

DC.

Slip instability on a crack embedded within an elastic medium

So far we have examined spring-slider systems. We now consider a crack embedded within an elastic medium. In that case, Hook’s law is written in terms of the shear modulus, G, and the shear strain, , as:

Were is a geometrical constant with a value close to 1.

Writing the stress balance equation, and taking the slip derivative as before leads to:

€

τ =Gε =ηGδ L .

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ηGL

=σ Nb

Dc .

So now, the equivalent for critical stiffness in the spring-slider system is the critical crack length:

In conclusion:

• The condition for unstable slip is that the crack length be larger than the critical crack length.

• The dimensions of the critical crack scale with b.

€

Lcrit =ηGDcσ Nb

.

Slip instability on a crack embedded within an elastic medium

Numerically simulated nucleation

Dieterich, 1992

What controls the size of the nucleation patch?

• Lcrit provides only a minimum estimate of the nucleation patch size.• The actual size of the nucleation patch is asymptotic to Lcrit for small a/b, but increases with decreasing a/b.

Rubin and Ampuero, 2005

What controls the size of the nucleation patch?

Rubin et al., 1999

patches of a>b ?

• Precise location of seismicity on the Calaveras fault (CA) suggest that a/b~1.

• Slip episodes on patches of a/b>1 may trigger slip on patches of a/b<1, and vice versa.

• It is, therefore, instructive to examine slip localization around a/b~1.

What controls the size of the nucleation patch?

Ziv, 2007

• Positive stress changes applied on a>b interfaces can trigger quasi-static slip episodes.

• Similar to the onset of ruptures on a<b, the creep on intrinsically stable fractures too are preceded by intervals, during which the slip is highly localized.

What controls the size of the nucleation patch?

Ziv, 2007

• The size of the nucleation patch depends not only on the constitutive parameters, but also on the stressing history.

The effect of negative stress perturbations

The effect of negative stress perturbation

• Following a negative stress step, sliding velocity drops below the load point velocity, and the system evolves towards restoring the steady-state.• The path along which the system evolves overshoots the steady-state curve, and the amount of overshoot is proportional to the magnitude of the stress perturbation.• Consequently, a crack subjected to a larger negative stress perturbation intersects the steady state at a higher sliding speed, undergoes more weakening and more slip during the nucleation stage.• Similar to the results for positive stress changes, the size of the localization patch depends on the magnitude of the stress perturbation. The greater the stress change is, the smaller is the localization patch.

Implications for prediction

• The bad news is that small and large quakes begin similarly.

• The good news is that, under certain circumstances, the nucleation phase may occupy large areas - therefore be detectable.

Calculations employ: a/b=0.7.

If a/b is closer to a unity and actual Dc is much larger than lab values, premonitory slip should be detectable.

Dieterich, 1992

Seismically observed earthquake nucleation phase?

Note that real seismograms do not show the linear increase of velocity versus time that is predictable by the self-similar model predicts.

Near source recording of the 1994 Northridge earthquake

Ellsworth and Beroza, 1995

Seismically observed earthquake nucleation phase?

Ellsworth and Beroza, 1995

For a given event, it’s initial seismic phase is proportional to it’s final size; but this conclusion is inconsistent with the inference (a few slides back) that the dimensions of the nucleation patch depend on the constitutive parameters and the normal stress.

Seismically observed earthquake nucleation phase?

Recall that:

• Some of the previously reported seismically observed initial phases are due to improper removal of the instrumental effect (e.g., Scherbaum and Bouin, 1997).

• Some claim that the slow initial phase observed in teleseismic records are distorted by anelastic attenuation or inhomogeneous medium.

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seismogram = source ⊗ path ⊗ instrument .

Further reading

• Dieterich, J. H., Earthquake nucleation on faults with rate- and state-dependent strength, Tectonophysics, 211, 115-134, 1992.

• Iio, Y., Observations of slow initial phase generated by microearthquakes: Implications for earthquake nucleation and propagation, J.G.R., 100, 15,333-15,349, 1995.

• Shibazaki, B., and M. Matsu’ura, Transition process from nucleation to high-speed rupture propagation: scaling from stick-slip experiments to natural earthquakes, Geophys. J. Int., 132, 14-30, 1998.

• Ellsworth, W. L., and G. C. Beroza, Seismic evidence fo an earthquake nucleation phase, Science, 268, 851-855, 1995.