Detekcja czasu nadejscia fali P sygnalu sejsmicznego metoda najmniejszych kwadratow z wykorzystaniem fali elementarnej jako sygnalu 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)xb(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 where m is the length of filter f(s), n is the length of input wavelet b(t), and the input wavelet b(t) to calculate the filter coefficients came from the seismic signal. In this paper we consider three wavelets shown in Fig. 4. So, if we find a filter, we may then use it to process the seismic signal. Signals with P-phase taken as a reference input wavelet b(t) were processed first. The results are illustrated in Fig. 5. The first distinct spike is the searched time of the P-phase arrival. However, the problem occurs when we want to process signal by filter which is calculated from other reference input wavelet b(t) then the P-phase from the processed signal. The results are illustrated in Fig. 6. Instead of one spike we have got a group of spikes, but the filtration is possible. We have to notice that reference wavelet b(t) was not transformed to fit the P-phase of the processed signal, but they have similar shapes. But the transformation may be rather difficult and will not be presented here. Another problem is that the spike is immersed in noise. As we can find, the noise (Figs. 5 and 6) and the worst effect of filtration illustrated in Fig. 6 are results of ill-conditioning of R. This is critical in inverse matrix operation. We may improve the condition of R by modifying it. The simplest way to achieve this is by adding a small multiply of the identity (Kasina, 1992). This is called the Tikhonov regularization (Neumaier, 1998). The formula The signals processed with regularization procedure are presented in Fig. 7. We can see very good attenuation of noise and much clearer spike (wider, but only one) in the case the input wavelet b(t) to calculate filter coefficients was not taken from the processed signal. The method described in this paper is a first stage of the author's work in the field of automatic pickers of seismic signals. Future work will concentrate on modifying this method, comparison with the most popular STA/LTA algorithm and as well looking for more accurate and reliable detection methods.