Mathematical model of seismic signal, as a flow of physically non realizable single seismic waves
© V.S. Mostovoy, S.V., Mostovyi, 2016
The new conception of seismic data analysis is proposed. It is based on preliminary studying of seismic background. Its characteristics
are a base for using mathematical models of non realizable seismic signals. The specific mathematical model of the seismic signal is
proposed as well. The peculiarity of the model is that it allows you to simulate the flow of seismic waves of different classes each of
them appears in the stream with specific time delay. This process takes place against the micro-seismic background noise. It is
natural to model the flow of signals by the physically realizable signal. It means those signals which do not have a trace in prehistory.
But this representation of the signal is unacceptable for two reasons. The first one is related to the smoothness of the signal at the time
of its appearance on the seismic record. The second one is related to the fact that the fade of the signal in the noise does not allow us
to determine the time of its appearance on the record accurately. The latter circumstance does not leave us the possibility to simulate the
time of the signal occurrence by using the determined value. Therefore, the time of occurrence of the signal is simulated by random variable
with variance depending on the level of micro-seismic background. We introduce the notion of generalized seismic signal as a function of time
and of the vector of parameters, which determine its shape, the energy, the place in flow of the other signals, spectral characteristics, and in
general behaviour in the entire history of its existence. Any widely spread seismic signal models used in practice are a particular case of this one.
Or in a more rigorous approach to the definition the different particular cases of the signals classes are transformed into the different hyper-planes
into space of parameters.
Key words: seismic signal, stochastic flow, a posterior probability, seismic background noise, mathematical model.