Читать книгу Statistical Methods and Modeling of Seismogenesis - Eleftheria Papadimitriou - Страница 14

The kernel estimation of magnitude distribution follows the general kernel estimation methods presented in Silverman (1986), with some adaptations to the specific features of magnitude (Kijko et al. 2001; Orlecka-Sikora and Lasocki 2005; Lasocki and Orlecka-Sikora 2008).

As already mentioned, magnitude datasets contain many repetitions. The kernel estimation of distribution functions is applicable for continuous random variables; hence, firstly, we should randomize magnitudes according to [1.21].

The estimation is based on the sample being representative of a population. The definition of the catalog completeness level, Mc, implicates the use of the magnitudes M ≥ Mc only.

The magnitude PDF is a steeply, exponentially like, the decreasing function. The larger the magnitudes are, the more sparse they are in data samples. In the seismic hazard studies, we are mainly interested in larger magnitudes. To ensure a better estimation of the distribution functions in the sparse data range, we use the estimators with the adaptive kernel [1.5] and [1.6]. For the left-hand side limited distribution of magnitude they take the form of:

[1.29]

[1.30]

The magnitude PDF has the global maximum at the catalog completeness level, Mc, and is zero for M < Mc. For this reason the data sample is mirrored symmetrically around Mc, and the estimation is carried out using the estimator [1.12] in the way described in section 1.1.

When the existence of a strict, single value upper bound to the magnitude range, Mmax, is assumed, the estimators of magnitude distribution functions are:

[1.31]

[1.32]

Kijko et al. (2001) compared the performances of the kernel estimation of magnitude distribution functions and the estimation based on the exponential distribution model [1.20]. For this purpose, they estimated the distribution functions, using Monte Carlo samples drawn from two distributions, mimicking real instances of magnitude distribution. The considered starting distributions were the exponential distribution [1.20] and the bi-component distribution, comprised of a dominant exponential component and a secondary normal component. For the samples drawn from the exponential distribution, the kernel estimates were only insignificantly worse than the estimates obtained with the use of the model [1.20]. For the samples drawn from the bi-component distribution, the kernel estimates fitted the starting distribution well, whereas the estimates based on the exponential model [1.20] deviated strongly from the starting distribution. Kijko et al. (2001) used these results as an argument advocating for more frequent use of the kernel estimation of magnitude distribution, particularly in seismic hazard studies.