Non-stationary stochastic modulation function definition based on process energy release

Abstract

Many real physical events are characterized by a random nature, as for earthquakes, sea waves, wind pressure, sea waves and so on. A common way to define these events is to think them as a realization of a stochastic process, and the simplest way to model them is to adopt stationary processes characterized mainly by their frequency contents. However, sometime it is necessary to consider also their evolutionary nature in the time, in order to properly take into account the non-stationary nature, as in case of seismic records. In these circumstances, a common mathematical approach is to assume the process as a non-stationary separable one, and then a key problem is to define appropriately a modulation function able to represent the time variation of the physical event. This paper purposes a method based on the energy release velocity definition of the modulation function with the aim to represent time intensity evolution of real non-stationary phenomena. With this scope, the envelope function is assumed related to the manner in which the input energy is built-up over time. Using the Iwan and Hou mathematical formulation, the procedure defines modulation function parameters in a consistent way with excitation input energy evolution. Finally, a closed formulation is obtained to set up the modulation function coherently with the energy release velocity of a single record, or of the mean value of a class of records. An example based on a real seismic event shows the effectiveness of the proposed method.