Statistical Methods and Modeling of Seismogenesis. Eleftheria Papadimitriou

are represented in Figure 1.4, taken from Lasocki and Papadimitriou (2006). We can see that the kernel estimate better fits the observed data, though the difference between these two estimates in the larger magnitude range is pretty tiny. However, due to the significant impact of the magnitude CDF on the parameters of hazard [1.17] and [1.18], the difference between the mean return period estimates is dramatic. For instance, for magnitude 6.5, the return period obtained from the exponential distribution model is close to 100 years. The kernel estimate of this return period is less than 10 years. Lasocki and Papadimitriou (2006) compared the “exponential” and “kernel” estimates of the mean return period for different magnitudes, with the estimates drawn from the actual observations done in the preceding 94 years (Figure 1.5). The differences between the “kernel” estimates and the assessments from actual observations were insignificant compared to the huge deviation of the “exponential” estimate. This led to the conclusion that the “kernel” estimate was much better than the “exponential” one.

      Also, many other studies indicated big differences between the “exponential” and “kernel” estimates of hazard parameters. The mentioned Monte Carlo analyses by Kijko et al. (2001), and the actual data studies by Lasocki and Papadimitriou (2006), suggest that in the case when these estimates differ, the “kernel” estimate is more accurate. All of this favors the kernel estimation of magnitude distribution functions for the seismic hazard assessment.

      The PSHA, which uses the kernel estimation of magnitude distribution as an alternative to the parametric model [1.19] and [1.20], has been implemented on the IS-EPOS Platform (tcs.ah-epos.eu, Orlecka-Sikora et al. 2020). The kernel estimation of magnitude distribution is also applied in the SHAPE software package for time-dependent seismic hazard analysis (Leptokaropoulos and Lasocki 2020). SHAPE is open-source, downloadable from https://git.plgrid.pl/projects/EA/repos/seraapplications/browse/SHAPE_Package.

Figure 1.4. Comparison of the estimates obtained from the kernel estimation method (solid lines) and from the exponential model of magnitude distribution (dashed lines) for the data from the Northern Aegean Sea. Left – the CDF estimates juxtaposed with the cumulative histogram of magnitude data. Right – the mean return period estimates. Reprinted from Lasocki and Papadimitriou (2006, Figures 5b and 6b)

Figure 1.5. “Kernel” (circles) and “exponential” (squares) estimates of the mean return periods juxtaposed with the mean return period estimates drawn from the actual 94 year-long observations of seismicity in the Northern Aegean Sea (diamonds). Reprinted from Lasocki and Papadimitriou (2006, Figure 7b)

1.5. Interval estimation of magnitude CDF and related hazard parameters

      Orlecka-Sikora (2004, 2008) presented a method for assessing the confidence intervals of the CDF, which had been estimated by the kernel estimation. The method is based on the bias-corrected and accelerating method by Efron (1987), the smoothed bootstrap and the second-order bootstrap samples, and is called the iterative bias-corrected and accelerating method (IBCa).

To simplify the notation let: {M_i}, i = 1,.., n be an n-element sample of already randomized magnitudes [1.21], be the kernel estimate of CDF of magnitude [1.30] or [1.32], h be the optimal bandwidth obtained from [1.10], {ω_i}, i = 1,.. n be the weights [1.7] and α define the confidence interval of length 1 − 2α. The IBCa method begins by generating the jackknife and the smoothed bootstrap samples of the first and second orders.

      The j-th jackknife sample, j ≤ n, is the n − 1 element sample {M₁, M₂,.., M_j−1, M_j+1, .., M_n} that is the initial sample from which the j-th element has been removed. Hence, we can have, at most, n jackknife samples.

      The first-order smoothed bootstrap samples are obtained in the same way as previously (equation [1.25]). The k-th smoothed bootstrap sample, ℳ_(k), is composed of:

      [1.33]

where is the standard bootstrap sample obtained by n times uniform selection, with replacement from the original data {M_i}, i = 1,.., n, and εi are the standard normal random numbers. We can have any number of the first-order bootstrap samples.

The second-order smoothed bootstrap samples are based on the first-order bootstrap sample, say ℳ_(k), and the CDF kernel estimate from ℳ_(k), say built on the bandwidth from [1.10], h_(k), and weights from [1.7]. A j-th second-order smoothed bootstrap sample, ℳ_(k,j), is composed of:

      [1.34]

      where is the standard bootstrap sample from ℳ_(k).

      The IBCa method can be presented in the following steps:

       – Step 1. Generate n jackknife samples, estimate kernel CDF-s from the jackknife samples, and evaluate the accelerating constant:

      [1.35]

      where stands for the arithmetic mean of

       –