Intelligent Computational Paradigms in Earthquake Engineering. Sailing the Gulf of 3D Digital Content Creation. Meeting National Needs Through Enhanced.

An arbitrary square integrable real-valued function (or, equivalently, the associated Hardy function) can be conveniently analyzed into a suitable family of square integrable wavelets of constant shape, (i.e. Obtained by shifts and dilations from any one of them.) The resulting integral transform is isometric and self-reciprocal if the wavelets satisfy an “admissibility condition” given here.

Explicit expressions are obtained in the case of a particular analyzing family that plays a role analogous to that of coherent states (Gabor wavelets) in the usual $L_2 $ -theory. They are written in terms of a modified $ Gamma $-function that is introduced and studied.

From the point of view of group theory, this paper is concerned with square integrable coefficients of an irreducible representation of the nonunimodular $ax + b$-group.

• 191 Downloads • Abstract Reliable analysis of low-energy earthquakes (microseismic) depends on how accurately one can detect and pick the arrival times, which are strongly influenced by the noise content. The study of microseismic events becomes even more challenging when the sensors are located on the surface because of the poor signal-to-noise ratio (SNR). Consequently, efficient and robust techniques for denoising microseismic data are necessary.

In this study, we propose a method based on an infinite impulse response (IIR) Wiener filter. The proposed method uses statistics based on signal observations (noisy data) and the underlying noise, both recorded by various sensors. The method presented here precludes the need for statistics or prior knowledge of the signal of interest. The second-order statistics of the noise and the noisy data are extracted from the recorded data only. As an advantage, in deriving the filter’s impulse response, no underlying structure of noise is assumed.

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Therefore, our method works for various types of noise, e.g., uncorrelated, spatially correlated, temporally correlated, Gaussian and non-Gaussian noise. Hence, the proposed method can be suitable as well for microseismic data recorded in diverse seismic noise environments. The criteria used to optimize the filter impulse response is the minimization of the mean square error.

The proposed method is tested on synthetic and field data sets and found to be effective in denoising microseismic data with very low SNR ( (-~12 ) dB).