# Earthquake Dynamic Triggering and Ground Motion Scaling J. Gomberg, K. Felzer, E. Brodsky.

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Earthquake Dynamic Triggering and Ground Motion Scaling J. Gomberg, K. Felzer, E. Brodsky

We seek to better understand what deformations trigger earthquakes, using observations of both the triggering deformations and triggered earthquakes.

The most commonly observed triggered earthquakes are “aftershocks”. Coyote Lake, California earthquake

Aftershocks occur at all distances,

& occasionally are obvious at remote distances.

We measure linear aftershock densities. number of aftershocks per unit distance,  r, at distance r

Measuring densities from earthquake catalogs.

Effectively, at each r we count the number of aftershocks within  r

Empirically, measured linear aftershock densities are fit by number of aftershocks at distance r M=magnitude  ~constant!

Measured Linear Aftershock Densities, from Southern California Aftershocks within 5 minutes of numerous mainshocks are stacked. From Felzer & Brodsky (2005).

Modeled linear aftershock densities. number of aftershocks at distance r

Modeled linear aftershock densities. number of aftershocks at distance r number of potential nucleation sites per unit distance

Modeled linear aftershock densities. number of aftershocks at distance r number of potential nucleation sites per unit distance probability of nucleation

distribution of nucleation sites per unit volume F(r) = A r (d-3) ‘d’ = dimensionality Number of potential nucleation sites

Sum (integrate) within a volume surrounding the triggering fault, defined by surface S and width  r

cylinder sphere rectangle Surface S is comprised of simple shapes,

D The integration is simple, resulting in an analytic model. D ~ rupture dimension of the triggering fault.

Recall the measured aftershock densities:  ~ constant at all distances! This model illuminates constraints on triggering deformations...

Measured aftershock densities: Modeled aftershock densities.

Measured aftershock densities: in the near field (r< { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/10/2751400/slides/slide_20.jpg", "name": "Measured aftershock densities: in the near field (r<

Measured aftershock densities: in the near field (r<>D)  (r) P(r) r (d-1)

Measured: in the near field  (r) P(r) D 2 r (d-3) Modeled: in the far field  (r) P(r) r (d-1) The probability of nucleation MUST scale in the near field as P(r) constant in the far field as P(r) D 2 r -2

Also, the aftershock density decay rate constrains the nucleation (fault system) dimensionality; d=3-  The probability of nucleation MUST scale in the near field as P(r) constant in the far field as P(r) D 2 r -2

Consistent Probabilities: or

Uncertainties & Resolution Aftershock densities decay as r -   In the most precise cases,  is constant within about +0.1 km -1 at far field distances and +0.3 km -1 in the near field.

Uncertainties & Resolution Our model implies these equalities or If  does not vary with r at all, the equalities require m=n=2. However, the observations permit some variability in  and thus n~2.

Uncertainties & Resolution How much can the scaling of P(r) differ from or ? If P(r) is consistent, ratios of the terms on each side of the equalities should be ~constant, or have slopes that differ by less than the uncertainties in .

Uncertainties & Resolution How much can the scaling of P(r) differ from or ? If P(r) is consistent, ratios of the terms on each side of the equalities should be ~constant, or have slopes that differ by less than the uncertainties in .

Uncertainties & Resolution Deviations from a constant slope can be derived from the derivatives of these curves. How much can the scaling of P(r) differ from or ?

Uncertainties & Resolution Deviations from a constant slope can be derived from the derivatives of these curves. decay rate change (km -1 ) How much can the scaling of P(r) differ from or ?

Uncertainties & Resolution In the far field, the observed decay rate (or slope) is constant within about +0.1 km -1. decay rate change (km -1 ) How much can the scaling of P(r) differ from or ?

Uncertainties & Resolution In the near field, the observed decay rate (or slope) is constant within about +0.3 km -1. decay rate change (km -1 ) How much can the scaling of P(r) differ from or ?

Uncertainties & Resolution Permissible scalings of P(r): or with ~1.8 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/10/2751400/slides/slide_33.jpg", "name": "Uncertainties & Resolution Permissible scalings of P(r): or with ~1.8

Uncertainties & Resolution Permissible scalings of P(r): or ~1.8 { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/10/2751400/slides/slide_34.jpg", "name": "Uncertainties & Resolution Permissible scalings of P(r): or ~1.8

We hypothesize that the probability of nucleation is proportional to the dynamic deformation amplitude. This is consistent with a large rupture being comprised of subevents, & laboratory observations and theoretical models of dynamic loading and failure.

We test various measures of dynamic deformation amplitude. Consistent deformations must scale as or

We test various measures of dynamic deformation amplitude. Strain Rate (acceleration) Strain (velocity) Displacement

Dynamic deformation amplitude = peak value. Strain Rate (acceleration) Strain (velocity) Displacement

Dynamic deformation amplitude = peak value x rupture duration (proportional to D). Strain Rate (acceleration) Strain (velocity) Displacement

Dynamic deformation amplitude = average value x duration = cumulative amplitude. Strain Rate (acceleration) Strain (velocity) Displacement

Published Peak Acceleration & Velocity “Attenuation” Models D scaling ranges from m~0.6 to m~2 with a mean of ~1 for peak accelerations, and m is greater by ~0.5 for peak velocities. This suggests nucleation depends only on peak strain rates (accelerations) times the rupture duration, or perhaps peak strains (velocities) alone.

Published Peak Acceleration & Velocity “Attenuation” Models r scalings are difficult to compare; often b~0 but not always, r represents different distances, the definition of R varies.

Our Deformation and Aftershock Density Scaling Observations The Japanese HiNet seemed ideal for measuring both peak ground motions & aftershock densities. We measure them for 22 M3.0 - 6.1 earthquakes.

Our Deformation and Aftershock Density Scaling Observations Small earthquakes are abundant but have hypocentral depths that make surficial ground motion measurements at far field distances.

Our Deformation and Aftershock Density Scaling Observations We can measure peak ground motion scaling with D and the far field distance decay rate.

Our Deformation and Aftershock Density Scaling Observations Southern California also seemed ideal; but even for 2 recent ~M5 earthquakes all ground motion recordings are in the far field. However, they constrain the scaling of peak motions with distance.

Our Deformation and Aftershock Density Scaling Observations Aftershock densities become uncertain at distances comparable to location errors.

Our Deformation and Aftershock Density Scaling Observations Constraining near field deformations requires large and/or very shallow earthquakes & good luck! We examine peak velocities for 16 M4.4 to M7.9 earthquakes with near field recordings.

Our Deformation and Aftershock Density Scaling Observations Scaling the peak velocity or the distance by rupture dimension D removes all size dependence.

Our Deformation and Aftershock Density Scaling Observations These can be fit by the scaling required for triggering deformations;i.e., D 2 /(  D+r) 2 or D 2 /(  D 2 +r 2 ).

Consistent deformations must scale as or Results Summary

Peak Strains Alone are Consistent or

Results Summary Peak Strain Rate x Rupture Durations are Consistent or

Nucleation Site (Fault Network) Dimensionality: ~1.7 and ~2.2 Results Summary

The probability of triggering an earthquake at a particular location and distance r scales with the size of the triggering earthquake.

The probability of triggering an earthquake anywhere at distance r is scale-independent.

More rigorously quantify scaling measurements. Examine other dynamic deformation measures. Collect & analyze additional near-field observations. Relate inferences to physical models of nucleation. What Next?

Questions? Thank You!

Published Peak Acceleration & Velocity “Attenuation” Models Most relations are generally consistent but very difficult to compare with one another or our model.

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