EXTENDING EARTHQUAKES’ REACH

THROUGH CASCADING

D. Marsan and O. Lengliné

SCIENCE 319, 1076-1079, 2008

Presented by Marcello GoriGe 277 – Professor Jean-Philippe Avouac

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THE CASCADING LADDER OF AFTERSHOCKS

1. Earthquakes can trigger other earthquakes, regardless of their size.

2. The causality of “mainshock A triggered aftershock B” needs to be modified into “mainshock A triggered C1, which triggered C2, …, which triggered B” Cascading

3. If indirect triggering is important in the overall aftershock budget, then direct triggering must be confined to spatial ranges and times shorter than the size of the total aftershock sequence.

4. The cascading effect is therefore dominated by small shocks.

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NEW VERSATILE MODEL

1. Previous models have the limitations of being either heavily parameter-dependent or model-dependent. (however, some model-dependent stochastic declustering models relaxed the binary-linking (mean-field) approximation.)

2. The new “model-independent stochastic declustering” (MISD) is a rapidly converging algorithm with a small number of hypotheses: linearity and mean-field.

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THE MODEL

The observed (dressed) seismicity rate is defined as the number of earthquakes per unit time and unit area at position and time .

𝜆 (𝑥 , 𝑡 )=𝜆0+∑𝑡 𝑖<𝑡

𝜆𝑖 (𝑥 ,𝑡 )

: uniform background rate density : (bare) contribution of earthquake that occurred at

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THE ASSUMPTIONS

1. The triggering process is linear (i.e. the bare contributions sum up)

2. A mean-field response to the occurrence of an earthquake is assumed, i.e. an event has a unique ‘mother’ mainshock as opposed to several (Harris, 1963; Vere-Jones, 1977; Helmstetter and Sornette, 2002)

3. This response is supposed to depend only on its magnitude, two earthquakes of equal magnitude are modeled similarly

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THE ALGORITHM I

1. From an a priori kernel and , the triggering weights are

2. The updated bare rates are computed as

the background weights are and the normalization coefficient are such that .

𝜆 (|∆ 𝑥|,∆ 𝑡 ,𝑚 )= 1𝑁𝑚𝛿 𝑡𝑆 (|Δ𝑥|,𝛿𝑟 ) ∑

𝑖 , 𝑗∈ 𝐴𝑤 𝑖 , 𝑗

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THE ALGORITHM II

The a posteriori background rate is

𝜆 (|∆ 𝑥|,∆ 𝑡 ,𝑚 )= 1𝑁𝑚𝛿 𝑡𝑆 (|Δ𝑥|,𝛿𝑟 ) ∑

𝑖 , 𝑗∈ 𝐴𝑤 𝑖 , 𝑗

: the set of pair such that , , : discretization parameters : the number of earthquakes such that and is the surface covered by the disk

with radii

𝜆0=1𝑇 𝑆∑

𝑗=1

𝑁

𝑤0 , 𝑗

: the duration of the time series containing earthquakes : the surface analyzed

Convergence is reached when the weights (or the rates) do not substancially change during an iteration

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TESTING THE MODEL

• Seismicity of SoCal from 1 January 1984 to 31 December 2002 was analyzed

• Only the earthquakes in the catalog were considered (because the method uses weight matrix)

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RESULTS I

Omori-Utsu decaying law:

: productivity : time : constant : p-value

Best spacial decaying law:

: distance : bare influence length

Bare Dressed

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RESULTS II• The -value increases with (as observed for dressed aftershock

sequences in Ouillon and Sornette, 2005), but saturates for • The dressed -values are to smaller than the bare ones

The productivity :• Bare, • Dressed,

• The bare influence length is remarkably small, growing as , which is close to the dependence expected for the rupture length of small to intermediate size earthquakes (Scholz, 1991)

• The dressed ones are about times as much

• The duration of the direct (bare) aftershock sequence is invariant of the magnitude and generally short

• The dressed one lasts for longer and shows a dependence on the magnitude as

“This implies that short-lasting triggering mechanisms, acting at the time scales of few days, could be the key process, along with the cascading effect, in controlling earthquake dynamics.”

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RESULTS II

The mean distance between mainshock and aftershock (with time separating the two events) is:• Constant for bare aftershocks• Growing as for dressed aftershocksSimilar conclusions are reached by normalizing this distance by the mainshock’s influence length

• The duration of the direct (bare) aftershock sequence is invariant of the magnitude and generally short

• The dressed one lasts for longer and shows a dependence on the magnitude as

“This implies that short-lasting triggering mechanisms, acting at the time scales of few days, could be the key process, along with the cascading effect, in controlling earthquake dynamics.”

“This shows that cascading triggering drives the expansion of aftershocks zones.”

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RESULTS III

The number of events directly triggered by all the events of a given magnitude decreases with the magnitude “This demonstrates the importance of small shocks in controlling the regional seismicity.”

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Conclusions

“Large earthquakes casually trigger aftershocks during a relatively short time span. However, they condition regional seismicity for a much longer time period and over larger, time-increasing distances, through the local triggering caused by their aftershocks.”

The cascading effect is therefore dominated by small shocks.

The cascading of aftershock triggering is characterized by a scale-invariant anatomy, making earthquake declustering an ill-defined problem.

Interestingly, “if small and large earthquakes are dynamically similar, then the initiation process is scale-invariant and therefore the size of the earthquakes is inherently unpredictable.” (Kanamori and Brodsky, 2004), on the scaling relations of dynamic parameters.

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