The representation theorem and the law governing the summation of random variables implies that, given that the slip amplitudes generated during an earthquake are distributed according to the Lévy law, the recorded ground motions will be approximately distributed according to the Lévy law. According to this formulation, the tails of the probability density functions (PDFs) of the slip and ground motion metrics are both attenuated according to a power law characterized by a single exponent, the Lévy index α.
The Lévy index of the PDF of the peak ground acceleration (PGA) should be independent of the position of the seismometers, whether they are located at the ground surface or at the bottoms of the boreholes. This hypothesis is tested with ground motions recorded during the 2003 Tokachi-oki, Japan, earthquake by seismometers located at the ground surface (K-NET and KiK-net) and in the boreholes (KiK-net). For several subsets of seismometers, we compute the PDF of the PGA and the parameters of the Lévy law that best fit the PDF. For PGAs recorded at the ground surface and in the boreholes, we found that the tails of PDFs of PGA decrease with power law behaviors controlled by α∼1.
The analysis of the PDFs of the random variables associated with the slip spatial distribution of two source models suggests that α∼1.45, though the Cauchy law (α=1) provides a reasonable fit of the same PDFs.