Wavelet analysis in approximation of small-amplitude seismic signals

Application of wavelet transform method to seismic signals from coal mines for the purpose of approximating noisy signals and detecting the P phase arrival was shown (Fig. 1). The wavelet transform is a very useful tool in the analysis of nonstationary signals such as a seismic signal. It allows us to resolve features of the signal at various scales. At each scale we can observe an approximation and details (low-frequency, and high-frequency) components of the original signal (Fig. 3). The process of this decomposition has two stages: decomposition of the original signal into approximation and details coefficients and then reconstruction of the approximation and details (Fig. 4). To do so, we use two pairs of filters associated with wavelet and scaling function (see. Eqs. 9 to 13). What is interesting for us, is the that that the details at finest scales contain noise (Fig. 5), which can be suppressed by thresholding the wavelet coefficient using soft or hard thresholding (eq. 14, 15). The noise suppression results seem to be better than those of the traditional low-pass Butterworth filtering (Fig. 7). Because we do "amplitude denosing" instead of frequency denoising, we can suppress also low frequency noise (by thresholding the details coefficients at lower scales). The P phase arrival detection can be done due to observation of the increase of the value of details at the moment of P phase arrival (Fig. 6). We can gain this effect by multiplying details at few scales (Fig. 8 and Fig. 9).