Detekcja czasu nadejścia fali P sygnału sejsmicznego metoda̧ najmniejszych kwadratów z wykorzystaniem fali elementarnej jako sygnału wzorcowego

An approach known from exploration geology is used to identify the P phase arrival of seismic events. This approach was derived by Robinson and Treitel (1980). They used it to increase the number of details available on a time series by the so-called shaping filtration or spiking filtration. The author of this paper uses this method to determine P-phase arrival time in a seismic signal from coal mine. The accuracy of detection of the P-phase determines the epicenter localization accuracy in some method of epicenter localization. The aim of the investigation is to evaluate the efficiency of this method for this problem. The method uses spiking (shaping) filtration, because our goal is to construct a filter which converts the P-phase (wave) into a single spike. The spike location determines the P-phase arrival. The filter is constructed using the least-square method. To do this, we have to make an assumption regarding the shape of P-phase we are looking for. This is the major difficulty, because we need a priori information about this shape. The procedure of time-domain least-squares filtering is presented in Fig. 1. Given an input wavelet b_t, which represents the shape of P-wave, and a desired output d_t, the problem is to find a filter whose output c_t = f_t × b_t deviates from the desired output in a least-squares sense. If we denote as error e_t the difference between the desired output d_t and the actual output c_t, the least-squares criterion requires minimization of the energy of error signal min J = min ∑e_t² As a result of minimization (see formulae (1) to (11)) we obtain normal equations.