Test 1 : Dependence on detection parameters

We construct a synthetic catalog of seismicity with the same features as the Aleutian seismic catalog (our study area in this paper) but uniformly distributed in space. This catalog consists of N=3226 events with magnitude ranging between 2 and 5.9, in an area of 600×600 km² and during T=3648 days (see Figure 5.14).

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Figure 5.14 – (a) Location of the simulated earthquakes. The circles have sizes corresponding to the rupture size of the earthquakes. (b) Cumulative number of earthquakes as a function of time. (c) Magnitudes versus time. The earthquakes produced by the background transient are marked in red.

Earthquake magnitudes follow a Gutenberg-Richter law with b= 1. The ETAS parameters inverted for this catalog are α = 1.525, p = 1.135, c= 0.002 day, γ = 2.450, L₀ = 0.100 km, κ₀ = 0.014. We add to this catalog a background transient (called T1). This transient increases locally the background rate 1473 times, from 1.51×10⁻⁶ earthquakes/day/km² to 2.22×10⁻³ earthquakes/day/km². It starts at t= 3258 day, lasts for 5 days, affects a disk of 50 km radius and generates 71 events, either background or triggered by new earthquakes.

We first apply our method with detection scales τ = 5 days and L = 50 km, i.e, the same scales as the transient (see Figure 5.15).

We find that this transient is effectively detected with a significance level greater than 99%

for these detection parameters. Our discretization in space and time happens to divide this 5-day, 50-km radius transient into neighboring cells instead of just one.

CAS DES ˆILES AL´EOUTIENNES

We also tested if the transient is detected with other detection parameters. We usedL=30, 50 and 100 km, andτ=1, 5, 10 and 30 days. All 12 combinations of these detection parameters gave a >99 % significance for at least one of the three cells affected by the transient.

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Transient Transient

< 95 % > 99 %

Figure 5.15 – Log-likelihood gain Δ for the synthetic catalog with one transient, for cells withτ

= 5 days and L = 50 km. The Δ values marking the 95% and 99% significance levels are shown by dashed lines ; they are obtained by the Monte-Carlo method described in section 2.3. The transient is well detected at the 99% significance level. It is spread over three cells, and therefore shows up three times on this graph.

We then add a second transient T2, less intense than T1. It has a total duration of 2 days, but its maximum activity is focused on a one-day period. It affects a 50 km-radius disk, and corresponds to an increase of the background rate by a factor of 400. It generates 8 new earthquakes. It is located 30 km away from T1 and occurs about 4 days before it. We again check how the detection of T2 depends on detection parameters Land τ. Figure 5.16summarizes our results. The transient can be detected with a significance level greater than 70 % , only when

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Figure 5.16 – Probability of finding transient T2 of typical size L^∗ = 50 km and duration τ^∗ = 2 day for different values of the detection parameters L andτ.

the detection parameters L and τ are similar to the size and duration of the transient. In the 110

5.2 Detection and characterization of transient forcing episodes affecting earthquake activity in the Aleutian Arc system.

next section, we present a simple analytical approximation to explain this result.

5.2.9.2 Test 2 : Dependence of the log-likelihood ∆ℓ on detection parameters We consider a spatio-temporal cell i, for which the background seismicity estimated by the null hypothesis is µ⁽ⁱ⁾₀ and the time-varying background rate is µ⁽ⁱ⁾₁ . Equation (5.11) can be rewritten as

∆ℓ⁽ⁱ⁾ =ℓ⁽ⁱ⁾₁ −ℓ⁽ⁱ⁾₀ =−

µ⁽ⁱ⁾₁ −µ⁽ⁱ⁾₀ τL²+

N

X

j=1

ln µ⁽ⁱ⁾₁ +νj

µ⁽ⁱ⁾₀ +νj

!

(5.13)

where N is the number of earthquakes inside cell i.

We assume that during a transient the background rate increases up to µ⁽ⁱ⁾₁ ≫ µ⁽ⁱ⁾₀ . We also consider that the interaction terms are small, so that νj ≪ µ⁽ⁱ⁾₀ for all earthquakes j.

We further assume that during the transient most earthquakes are background events, so that

µ⁽ⁱ⁾₁ −µ⁽ⁱ⁾₀

τL² ≈N.

We can thus approximate the gain in likelihood as

∆ℓ⁽ⁱ⁾ ≃ −N +Nlnx , (5.14)

where x=µ⁽ⁱ⁾₁ /µ⁽ⁱ⁾₀ .

We denote by L^∗ and τ^∗ the size and duration of the transient, and by Θ^∗ = [L^∗, τ^∗] the corresponding set of scales. We analyze how the gain in likelihood depends on the departure of the detection parameters Θ = [L, τ] from Θ^∗. To do so, we let one parameter vary while the other is kept constant.

If Θ ≪ Θ^∗, i.e., we observe the seismicity over cells much smaller than the scale of the transient, the number of earthquakes N in the cell grows linearly with Θ, while x does not change. Since x ≫ 1, we find that ∆ℓ⁽ⁱ⁾ also grows linearly : ∆ℓ⁽ⁱ⁾ ∼ Θ. For Θ ≫ Θ^∗, the number N increases very slowly with Θ since we only pick extra earthquakes outside the burst of activity caused by the transient, while the estimated gain in background rate x decreases as x∼ Θ¹. This causes ∆ℓ⁽ⁱ⁾ to decrease : ∆ℓ⁽ⁱ⁾ ∼ −Nln Θ.

This simple argument shows that the likelihood gain is effectively maximized when Θ = Θ^∗: a transient is more likely to be detected if the detection parameters are tuned to its characteristic scales (its significance is maximum). We test this behavior and the existence of the two end-member regimes using the synthetic catalog containing transient T2 (see Table 5.6).

This table confirms the strong correlation between the maximum likelihood gain and the detection parameters when they are close to the transient scales.

CAS DES ˆILES AL´EOUTIENNES

L=30 km

1/24 1/12 1/6 1/2 1 day 2 5 10 30 100 365 days

N 0.21 0.42 0.83 2.5 5 5 6 6 6 7 7

µ¹/µ⁰ ×5052 ×4954 ×4937 ×4843 ×4852 ×2390 ×1176 ×568 ×171 ×57 ×13

∆ℓ 1.2 2.6 4.9 12.8 19.7 16.9 19.2 15.9 11.1 10.2 4.8

P(∆ℓ) 9.9% 14.9% 21.6% 100% 100% 100% 100% 100% 100% 100% 99.1%

L=50 km

1/24 1/12 1/6 1/2 1 day 2 5 10 30 100 365 days

N 0.21 0.42 0.83 2.5 5 6 7 7 7 8 8

µ¹/µ⁰ ×1404 ×1359 ×1344 ×1298 ×1319 ×817 ×381 ×181 ×54 ×18 ×4

∆ℓ 1.2 2.2 4.0 9.5 15 18.5 19.2 15.2 9.8 7.4 2.1

P(∆ℓ) 9.5% 12.7% 18.1% 99.8% 100% 100% 100% 100% 100% 99.6% 87.9%

L=100 km

1/24 1/12 1/6 1/2 1 day 2 5 10 30 100 365 days

N 0.29 0.58 1.17 3.5 7 8 9 9 12 13 23

µ¹/µ⁰ ×483 ×474 ×438 ×434 ×435 ×255 ×112 ×51 ×21 ×6 ×3

∆ℓ 0.78 1.6 2.9 6.7 13.1 14.6 13.5 9.7 10.0 5.2 6.0

P(∆ℓ) 6.8% 10.4% 13.7% 99.3% 100% 100% 100% 99.8% 100% 96.9% 95.8%

Table 5.6 – Mean number of earthquakes N, ratio of the transient over the background rate, and likelihood gain ∆ℓ for varying detection parametersL and τ, for the ransient T2. The characteristic size and duration of the transient are L^∗ = 50 km andτ^∗ = 2 days, but the activity is maximum for 1 day. We consider all the cells affected by the increased background rate during the transient, and average ∆ℓover these cells. We add the probabilityP(∆ℓ) to quantify the probability that a temporal change in background rate is significant, i.e, anomalous compared with the other values of ∆ℓfor this catalog.

5.2.9.3 Test 3 : Aftershock sequences seen as transients

The interaction model (ETAS) we use is based on ensemble-averaged laws. Departure from these laws is expected in reality. For example, the number of aftershocks directly triggered by a mainshock of magnitude m is, according to the model, a Poisson law, identical for all the mainshocks with magnitude m; fluctuations in these numbers are not predicted by the model outside the natural dispersion of Poisson random deviates. A mainshock triggering many more aftershocks than predicted by the model will then lead to the detection of a transient with our approach.

While it could be argued that such a vigorous aftershock sequence is indeed the signature of an actual transient, for example corresponding to strong afterslip, it is difficult with our approach to evaluate its reality since we do not account for possible, natural departures from the ensemble-averaged laws. We therefore test whether a very vigorous aftershock sequence can cause our method to detect a transient. To do so, we focus on the biggest mainshock of the synthetic catalog, of magnitude m = 6. It triggers an aftershock sequence of about 300 aftershocks. We artificially decrease the magnitude of the mainshock from 6 to 3 in steps of 0.5, 112

5.2 Detection and characterization of transient forcing episodes affecting earthquake activity in the Aleutian Arc system.

and apply our method with detection parameters L = 30 km andτ = 1 day.

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Mainshock magnitude

(a) (b)

Figure5.17 – (a) Probability that an aftershock sequence is identified as a transient when artificially decreasing the mainshock magnitude below its true valuem= 6, using detection scalesL= 30 km and τ = 1 day. This probability is larger than 90% when the mainshock magnitude is decreased by more than 1.5, i.e., when this mainshock triggered e^(α×1.5) = 9.85 more events that expected according to the ETAS model. (b) Magnitudes of all earthquakes in the aftershock sequence as a function of time.

Figure5.17shows the probability that the sequence is identified as a transient. For a decrease of 1.5 of magnitude, the probability to find a transient exceeds 90%. In theory, this aftershock sequence is too strong by a factor e^α×1.5 10 for a mainshock of magnitudem = 6−1.5 = 4.5.

Departures from the mean productivity law by a factor of 10 have been observed (Tahir et Grasso,2013) in real catalogs. We thus expect the detection of transients embedded in aftershock sequences. Section 5.2.4.3 describes a possible way to discriminate these from transients not related to highly productive mainshocks.

Chapitre 6