This electronic supplement presents the flowchart of the methodology (Table S1) and the results of the methodology obtained using a Brownian passage time (BPT) model (Figs. S1–S4; Matthews et al., 2002; Field and Jordan, 2015), which are only marginally different from the ones obtained with the Poisson model.
In the Method section of the main article, we base all our probability calculations on Poisson’s model. However, other options are possible and we show here an example using a specific renewal model, the BPT model (Matthews et al., 2002; Field and Jordan, 2015).
The probability density function (PDF) of a renewal model f(t) describes the probability to have the first earthquake at time t since the last event. The PDF of the BPT model is
(S1)
in which τ is the earthquake mean recurrence time and is defined by the ratio between the mean and the standard deviation of the recurrence time σ.
Let F(t) represent the cumulative distribution function (CDF) of f(t) and T the time since the last earthquake. The probability of having no earthquakes in a time interval ΔT knowing that no earthquakes happened until T can be rewritten as the probability to have no earthquake until T+ΔT, knowing there has been no earthquakes up to T:
(S2)
Note that compared to the Poisson model, which just needed τ and ΔT to be set, the BPT model needs additionally two more variables: and T. In our calculation, we fix the of all events equal to the of Parkfield’s Mw 6 (). The PDF of the time of the last event for renewal models is given by (Field and Jordan, 2015). The probability of our first approach for renewal models is then:
(S3)
It defines the probability to have no earthquakes in the time period T and T+ΔT, knowing that there has been no earthquake until T but not knowing T. This probability takes into account all possible T given its initial BPT assumptions, avoiding us to sample it.
We now explore the range of possible magnitude and frequency for the largest possible earthquake needed to close the budget.
We use again the Monte Carlo Markov Chain (MCMC) method to sample these probabilities. Each sample is taken from a uniform PDF between 6.4 and 9 for magnitudes, and between 10−6 and 104 yrs−1 for frequencies. We calculate for each sample k the value proportional to the probability as described in the Method section of the main article. For each estimate, we run the sampler for 900,000 steps, rejecting the first 500 steps to ensure accurate sampling of the PDF. The space of magnitude–frequency–time of the last event is then binned into cells of size 0.01 for the moment magnitude axis and logarithmically binned into cells of size 0.01 concerning the frequency axis. The number of samples in each cell divided by the total number of samples approximates the probability as defined by the MCMC method. The results are shown in Figure S1.
The probabilities computed using the BPT model for low-magnitude and high-frequency events are smaller than Poisson model. As already mentioned in the P Hist: Second Approach section of the main article, the marginal cumulative probabilities do not give us any useful information. This is due to the fact that the maximum magnitude and lowest frequency are unbounded a priori. The magnitude for when 95% of the maximum noncumulative probability is reached can instead give us an indication on the magnitude and frequency needed to close the moment budget (Fig. S2). We obtain almost the exact same values as for the Poisson model: an Mw 7.42 with a recurrence time of 2000 yrs is sufficient to close the moment budget at a 95% confidence level. However, the normalized probabilities are lower for the BPT model between Mw 6.4 and 7 than Poisson’s model. Note that the 95% value does not actually represent a probability due to the normalization of the noncumulative probabilities. Higher magnitudes and recurrence times than those estimations are as plausible but would not be necessary.
The hazard probability estimation P Hazard for renewal models is given by Field and Jordan (2015)
(S4)
in which p(T) is the probability distribution for the time since the last event T. It defines the probability to have an earthquake in the time period T and T+ΔT, knowing that there has been no earthquake until T but not knowing T.
Figures S3 shows the hazard estimation for BPT model derived from the probabilities of the magnitude–frequency of the largest earthquake using our second approach described above for a period of time of 30 yrs. for the Poisson model is also indicated in the figure.
The of the BPT model is equivalent to the Poisson’s model for ; they are both almost equal to 1. They then differentiate the BPT model having higher probabilities for . The BPT model joins back to Poisson’s model for larger M Test. This highlights the fact that for the BPT model is significantly higher for events over M Test that have frequencies around ΔT.
Table S1. Flowchart of methodology.
Figure S1. Maximum-magnitude earthquake probability assuming the case where large earthquakes should have a recurrence time lower than the largest earthquake currently observed, using the BPT model. We use here the model MW, do not account for Båth’s law, and suppose that the ratio between postseismic and coseismic moment release is equal to 200%. The black curve represents the magnitude–frequency distribution of Parkfield’s area using the Advanced National Seismic System (ANSS) catalog between 1984 and 2015. The gray curve is the modified magnitude–frequency distribution where the Mw 6 and its aftershocks are fixed to occur every 24.5 yrs. The stars incarnate the historical data (Toppozada et al., 2002) and the black line represents an Mw 7 earthquake with a recurrence time between 140 and 250. The black circle defines the location of an Mw 6.7 event occurring every 140 yrs, our favored scenario.
Figure S2. (a,b) The marginal noncumulative probability normalized by their maximum value for the magnitude and frequency, respectively, for our second approach. They give an indication about the magnitude and frequency of the largest event needed to close the moment budget, using the BPT (gray line) and Poisson model (black dotted line).
Figure S3. Probability to have earthquakes over of a certain magnitude in a period of time of t = 30 yrs considering the probability distribution of a maximum-magnitude earthquake to exist and close the moment budget for the BPT model. We test our favored scenario using model MW, with the Båth’s law not accounted for and with a postseismic moment release equivalent to 200% of the coseismic one. The white dashed line corresponds to for the BPT model and the black dashed line corresponds to for the Poisson model.
Figure S4. Probability to have earthquakes over a certain magnitude in a period of time of t years considering the probability distribution of a maximum-magnitude earthquake to exist and close the moment budget for approach 1. We test our favored scenario using model MW, with the Båth’s law not accounted for and with a postseismic moment release equivalent to 200% of the coseismic one. The white dashed line corresponds to for approach 1, and the black dashed line is for approach 2. (a,b) The results for t = 30 yrs and (c,d) for t = 200 yrs. (a) and (c) suppose a maximum-magnitude possible of Mw 7.5, and (b) and (d) suppose that the maximum-magnitude possible is an Mw 9.
Field, E. H., and T. H. Jordan (2015). Time-dependent renewal-model probabilities when date of last earthquake is unknown, Bull. Seismol. Soc. Am. 105, no. 1, 459–463, doi: 10.1785/0120140096.
Matthews, M. V, W. L. Ellsworth, and P. A. Reasenberg (2002). A Brownian model for recurrent earthquakes, Bull. Seismol. Soc. Am. 92, no. 6, 2233–2250.
Toppozada, T. R., D. M. Branum, M. S. Reichle, and C. L. Hallstrom (2002). San Andreas fault zone, California: M ≥5.5 earthquake history, Bull. Seismol. Soc. Am. 92, no. 7, 2555–2601.
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