Electronic Supplement to
Impact of Earthquake Rupture Extensions on Parameter Estimations of Point-Process Models

by S. Hainzl, A. Christophersen, and B. Enescu

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We performed additional tests concerning the robustness of the simulation results presented in Hainzl et al. (2008). Our analysis indicates that neither the magnitude range, nor observational errors, nor spatial heterogeneities of the background activity affect the parameter estimation significantly. It is, however, found that using an isotropic spatial kernel (Eq.2, Hainzl et al., 2008) where d is scaled with the mainshock magnitude leads to a slightly better estimation than assuming a constant d-value. This agrees in principle with the observation that estimates with variable d(M) lead to higher alpha-values (Zhuang et al., 2004) than those with constant d (Console et al., 2003).

Here we perform additional simulations to test the impact of a physically reasonable scaling of the d-parameter with earthquake magnitude. Furthermore, we test possible correlations between the magnitude of the largest earthquake in the analyzed earthquake catalog and the estimated α-value, and finally, we test the impact of inhomogeneous background activity.

Simulations

We assume a self-similar aftershock pattern, that is, we assume that in Eq.(2), the d-parameter scales according to d = d₀L, where d₀ is a constant related to the aspect ratio between the characteristic width and length of the aftershock cloud. For the magnitude-dependence of the rupture length, we use again the empirical relation of Wells and Coppersmith (1994): L = √10−^3.49+0.91M [km]. In the following, d₀ is set to 0.01.

We analyze Monte Carlo simulations of the ETAS model with pre-set α=1. The other parameters are chosen by fitting the ETAS model to the seismicity preceding the 1992 M7.3 Landers, California, earthquake in the region -119W to -115.5W and 32.5N to 36.5N and the time interval between t₁=1/1/1984 and t₂=6/27/1992. The data set included N=1537 earthquakes with magnitude M ≥ 3. We optimized the parameters μ,K, c, p, q, q by maximum likelihood method and obtained: μ=0.06 [1/day]; K=0.007; c=0.006 days; p=1.08; and q=1.45.

The simulations are done for a 100 km x 100 km box and a time interval of 1 year. Earthquake magnitudes are randomly chosen from a Gutenberg-Richter distribution with b=1, within M_min and M_max which are specified below. To allow a targeted simulations, a mainshock with a magnitude of M_max is set in each simulation to occur at time 0 in the center of the box.

Parameter estimation

For each simulation, the estimation of all model parameters is performed again by the maximum likelihood method. To study the impact of taking the spatial distribution into account, we use the following three models for inversion:

1. the “no-space” model which uses only the time-sequence of earthquake occurrences (i.e. neglect the spatial information)
2. the “d=const” model which assumes that d is constant; that means, we use in this case a slightly wrong model for the spatial kernel.
3. the “d(M)” model which uses the true kernel d = d₀L(M) with the Wells & Coppersmith scaling.

Test 1: Results for the d(M)-model

For the first test, we used a constant magnitude range M_max−M_min=3.5 to ensure similar statistics, but change the size of the mainshock. In this way, we can test the effect of the relative size of the mainshock with regard to total size of the region under consideration. Furthermore, it allows us to analyze the impact of epicenter location errors.

Figure 1 shows the estimated α-parameter for the three different inversion methods. It can be clearly seen that only the “no-space”-model is able to reproduce the input-value of α, whereas both models using the space-informations clearly underestimate the α-value. However, the underestimation is less pronounced for the model using the correct scaling of d.

Real earthquake catalogs will always consist of location errors. Theoretically, it can be expected that for mainshocks with a rupture length in the order of the location error, the apparent aftershock distribution will be almost isotropic in space and thus the bias of inversions using isotropic kernels should become smaller. Therefore we add realistic Gaussian distributed location errors on the true (simulated) epicenters and repeat the estimation. For a standard deviation of 1km, the results are shown in Fig. 1. The bias decreases slightly in the case of the “d=const”-model but does not vanish. For larger location errors, we find that the inversion with the space-dependent models becomes completely unstable and that the uncertainties of the α-estimations become huge. The likely reason is that the power-law kernel cannot fit the almost Gaussian distributed aftershock activity well. Finally, we add magnitude errors with standard deviation of 0.1 and repeat the parameter estimations. We find that this does not influence the results systematically.

Test 2: Dependence on M_max

Now we will test whether the estimation of the α-parameter depends on the magnitude range of our observation. For that, we analyze simulations where the minimum magnitude is set to M_min=3, whereas M_max is varying between 4.5 and 7. The resulting α-estimation based on the “no-space”- model is shown in Fig.2. We find that the true α-parameter can be reconstructed in all cases without any systematic error, however, the uncertainty is largely increased for smaller M_max. The enlarged uncertainty is mainly related to the smaller number of events in the simulated catalogs with small M_max.

Test 3: Inhomogeneous background activity

Background activity is likely to be related to fault structures and thus cannot generally be expected to be uniform in space. This could be another reason for the underestimation of the α-parameter.

To test this, we modeled the probability distribution of the background seismicity by a fractal surface of dimension D (Turcotte, 1997), where we set the ratio between the absolute maximum and minimum to 1000. For the simulation, we neglected the anisotropy of aftershock clouds and used an isotropic kernel in order to analyze the possible effect of inhomogeneous background distributions separately. The “d(M)”-model is in this case the true model for the triggered activity but not for the description of the background activity. The results of the parameter inversion for the “d(M)”-model are shown in Fig.3. The estimations are scattering around the true values and the inhomogeneity seems to have no significant effect on the parameter estimation.

It is important to note that this is not an exhaustive test because of the large variety of possible model setups, in particular, alternative spatial distributions of the background seismicity. Thus our investigation is not conclusive, however, it seems to indicate that the spatial inhomogeneity of the background seismicity does not significantly influence the α-estimation.

References

Console R., M. Murru, and A.M. Lombardi (2003), Refining earthquake clustering models, J. Geophys. Res., 108 (B10), 2468, doi:10.1029/2002JB002130.

Hainzl, S., A. Christophersen, and B. Enescu (2008), Impact of earthquake rupture extensions on parameter estimations of point-process models, Bull. Seismol. Soc. Am.

Turcotte, D. L. (1997). Fractals and Chaos in Geology and Geophysics, Cambridge University Press, 1997.

Zhuang, J., Y. Ogata, and D. Vere-Jones (2004), Analyzing earthquake clustering features by using stochastic reconstruction, J. Geophys. Res.,109 (B5), B05301