In the absence of strong motion records at the future construction sites, different theoretical and semi-empirical approaches are used to estimate the initial seismic vibrations of the soil. If there are records of weak earthquakes on the site and the parameters of the fault that generates the calculated earthquake are known, then the empirical Green’s function can be used. Initially, the empirical Green’s function method in the formulation of Irikura was applied for main shock record modelling using its aftershocks under the following conditions: the magnitude of the weak event is only 1–2 units smaller than the magnitude of the main shock; the focus of the weak event is localized in the focal region of a strong event, hearth, and it should be the same for both events. However, short-termed local instrumental seismological investigation, especially on seafloor, results usually with weak microearthquakes recordings. The magnitude of the observed micro-earthquakes is much lower than of the modeling event (more than 2). To test whether the method of the empirical Green’s function can be applied under these conditions, the accelerograms of the main shock of the earthquake in L'Aquila (6.04.09) with a magnitude Mw = 6.3 were modelled. The microearthquake with ML = 3,3 (21.05.2011) and unknown origin mechanism located in mainshock’s epicentral zone was used as the empirical Green’s function. It was concluded that the empirical Green’s function is to be preprocessed. The complex Fourier spectrum smoothing by moving average was suggested. After the smoothing the inverses Fourier transform results with new Green’s function. Thus, not only the amplitude spectrum is smoothed out, but also the phase spectrum. After such preliminary processing, the spectra of the calculated accelerograms and recorded correspond to each other much better. The modelling demonstrate good results within frequency range 0,1–10 Hz, considered usually for engineering seismological studies.
A New Proof for Lyapunov-Type Inequality on the Fractional Boundary Value Problem
