Advances in Extraction of Signal From Ground Motion Time Histories Using Time-Frequency Analysis

Introduction

A simple 1-D frequency transformation (viz., Fourier Transform) is not sufficient to reveal basic characteristics of ground motion time histories (seismogram) whose frequency content varies with time. Major research efforts have been dedicated in the past decades to seismological data processing to interpret the physical processes using various outcomes of data processing techniques. The extraction of time of the incoming seismic phases use automatic phase pickers which make use of time domain as well as frequency domain analyses. Most of the event detectors and phase pickers are based on the STA (Short Term Average) and LTA (Long Term Average) algorithm (Sharma et al 1990, Sharma, 1992a and b). However, due to non-stationary nature of seismograms, techniques like Time-Frequency Analysis (TFA) which includes time information for the frequencies present in signals are preferred. TFA techniques maps a one-dimensional signal into two-dimensional as a function of time and frequency i.e. Time- Frequency (T-F) domain and show how the spectral content of the signal changes with time. Time- Frequency domain representation a is better frame work then time domain or frequency domain to employ for feature extraction analysis such as the concentration of energy in certain frequency band or at certain time and existence of frequency components and respective localised time in different phases of the waves. In this concern, different types of TFA approaches have been developed for analysis and processing the non-stationary signals like seismograms. The Time-Frequency distributions consist of two categories which can be classified as linear time-frequency representation and quadratic time- frequency distributions (Cohen, 1995, Hlawatsch and Boundreaux-Bartels, 1992; Devi and Sharma, 2016). The linear time-frequency distributions satisfy the basic property of linearity principle/superposition principle which is the desired characteristic in any application involving multicomponent signals, whereas, quadratic time-frequency distribution violate this principle because of quadratic structure of time-frequency representation. The approaches which comes under the linear time-frequency distribution are Short-Time Fourier Transform (STFT), Gabor Transform (GT) and Wavelet Transform (WT). On the other hand Wigner-Ville Distribution (WVD), Choi–Williams Distribution (CWD) and Cone Shaped Distribution (CSD) approaches belongs to quadratic time-frequency distribution and generally known as the classes of Cohen’s Distribution. Later two CWD and CSD are also called reduced interference distribution which is an extension of the WVD (Jeong and Williams, 1992).

The widely used Short-Time Fourier Transform (STFT) technique has been well documented by Nawad and Quatieri (1988) and Allen (1977). Huerta-Lopez et al., (2000) and Upergui-Botero et. al., (2012) applied STFT for characterisation of seismic/earthquake waves which is essential for understanding the wave propagation phenomena. The characteristics of waves include the variation of ground motion intensity and spectral content with time. Gabor Transform is a special case of STFT with application of Gaussian window function and the discrete Gabor expansion is the inverse of the discrete Gabor transform. Hence, the pair of the discrete Gabor transform, and discrete Gabor expansion provides a feasible bridge for converting an arbitrary signal from 1D the time domain into 2D the time-frequency domain, or vice versa (Qian et al., 1993, 1999 and 2003; Devi and Sharma, 2016b). The Gabor Transform and expansion has been implemented for the spectral enhancement and removal of culture noise from the doppler ultrasound signals and seismic signals (Zhang et al., 2005; Kumar et. al., 2015; Devi & Sharma, 2016b). STFT and GT both deals with uncertainty principle which precludes to simultaneously obtaining fine time-frequency resolution. The uncertainty principle is given by , where is time resolution and is the frequency resolution and are not arbitrary parameters. In order to achieve good resolution depending upon the signal characteristics a trade off between is generally considered.