Non-linear filter, using the wavelet transform, applied to seismological records

As any process in Nature, seismic records are affected by noise that the analyst would want to eliminate. One of the most common techniques used to minimise this noise effect is the application of linear filters, which reduce the bandwidth of the signal. This method is based on the Fourier Transform, and therefore any perturbation on the coefficients affects the entire record. We have developed a non-linear filter based on the multiresolution analysis of the Discrete Time Wavelet Transform (DTWT). The main idea is to use the time-frequency localisation properties of the wavelet decomposition. Each coefficient is associated to a window on the time-frequency plane, so any perturbation would only affect the time and frequency range of the correspondent window. The procedure we propose has three stages: periodic noise elimination, spikes reduction and, finally, the non-linear filtering. The non-linear filter acts by thresholding the wavelet coefficients. The thresholding estimator will depend on the signalnoise ratio (SNR) in each of the frequency bands associated to the wavelet decomposition. We have compared the proposed method to the coherent structures method (Mallat, 1998) and to two 4th order linear filter banks (Butterworth and Elliptic filters), applying all of them to a synthetic database, and a real earthquake database recorded by the Short Period ROA Network. The proposed method improves the SNR in the 87% of the tested events, being the relative rms error less than three, and the maximum amplitude relative error less than 10% in the 90% of the synthetic database.