Journal of Seismology 7: 413–429, 2003. © 2003 Kluwer Academic Publishers. Printed in the Netherlands. 413 Non-linear filter, using the wavelet transform, applied to seismological records A. Pazos 1 , M.J. Gonz´ alez 2 & G. Alguacil 3,4 1 Real Instituto y Observatorio de la Armada, Cecilio Pujaz´ on s/n, 1100 San Fernando, C´ adiz, Spain, e-mail: pazos@roa.es; 2 Dpto. de Matem´ aticas, Universidad de C´ adiz, 11510 Pto. Real, C´ adiz, Spain, e-mail: ma- jose.gonzalez@uca.es; 3 Instituto Andaluz de Geof´ ısica, Observatorio de la Cartuja, Universidad de Granada, 18.071 Granada, Spain, e-mail: alguacil@iag.ugr.es; 4 Dpto. de F´ ısica Te´ orica y del Cosmos, Facultad de Ciencias, Universidad de Granada, 18.071 Granada Received 24 June 2001; accepted in revised form 16 May 2002 Key words: de-noising, seismic signals, filter, noise, non-linear, signal-noise ratio, wavelet Abstract 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 signal- noise 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 4 th 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. Introduction Any digital or analogue seismic record is always af- fected by noise due mainly to either natural sources (i.e., wind, water currents, tides) or artificial sources (i.e., traffic, industry), even when the instrument noise is negligible. Besides that, ‘spikes’, caused by radio transmissions can also be found in the telemetric nets. It is very difficult to build a mathematical model to represent the noise, since it varies from station to sta- tion depending on where it is located, or on the time of the day the record is registered. The short period network of the Royal Naval Ob- servatory (ROA) in San Fernando (Figure 1) is located in the South of Spain. A detection algorithm, based on the well-known STA/LTA algorithm (McEvilly and Majer, 1982), operates over the nine stations of the ROA network. It provides a detection of about 94% of the events identified by the analyst, in spite of the poor signal-noise ratio (SNR) of most of the local events (strong winds from West or East are usually present in this area). Many phase-picking methods use a filter bank (Evans and Allen, 1983; Gledhill, 1985; Moltshan et al., 1964) or non-sinusoidal transforms (Goforth and Herrin, 1981; Andrew et al., 1882) in order to de- compose the signal into several frequency bands and to choose the one with the best SNR.