TIME FREQUENCY ANALYSIS OF GROUND MOTION TIME HISTORY OF MICROEARTHQUAKES

Abstract

The modernisation of seismic instrumentation has resulted in better quality data with higher resolution inspiring development of newer computational methodologies for enhancing signal characteristics while processing ground motion time histories for more realistic seismological interpretations. Time-frequency representation and spectral feature extraction from a digitally recorded ground motion time history of an earthquake is one of the cornerstones in seismological signal processing and interpretation. Conventional techniques viz., Frequency domain and Time domain analyses have been developed in the past for frequency decomposition of the seismological data but have failed to capture the time varying frequency contents in time series like seismograms which are complicated convolution of source radiation effects, path of propagation, impedance contrast, characteristics of instrument and ambient noise at recording station. Seismograms reveal non-stationary characteristics having broad range of frequency content. The time variant characteristics of the ground motion time history imply study of the localised temporal distribution of these frequencies and their pattern variations with time. The techniques of Time-Frequency Analyses (TFA) have been used for characterising the frequency content with time in the recorded ground motion time histories which makes use of two dimensional information i.e., time and frequency of the time series simultaneously and elucidates the true structure of the time series (seismogram) when the frequency content of the time series varies with time. The TFA techniques namely Short Term Fourier Transform (STFT), Gabor Transform (GT), Wavelet Transform (WT), Wigner-Ville Distribution (WVD), Choi- William Distribution (CWD) and Cone Shape Distribution (CSD) have been used. The selection of suitable technique for a specific field like seismology is a challenging task. The main objective of the research work was to look into the time frequency distribution in the seismograms recorded from an active seismic region like Garhwal Himalayas and study the frequency variation with time due to varying size and locations of the microearthquakes. The study area for the present study has been selected based on its seismic activity and a set of good instrumentation deployed to record it. The Garhwal Himalaya region is one of the most seismically active regions for which the seismic activity is being monitored for last few decades through deployment of a seismological network which was deployed due to the presence of Tehri dam which is one of the highest dams in Himalayas. Garhwal Himalaya occupies the central sector of Western Himalaya from latitude 290 to 310 and longitude 780 to 800 and has experienced many damaging earthquakes in the past. The region falls under Seismic Zone IV of seismic zoning map of the India (IS: 1893, 2016) and forms the western seismic gap of the Himalaya. ii The data recorded during the period from 2009 to 2011 was considered for the present research work where the array has recorded 1570 events. Out of these 1570 only 63 events (233 seismograms) were selected which were having magnitudes range varying from 1.0 to 3.0 and the hypocentral distances varying from 30 km to 150 km. To study the time variant characteristics of the seismograms using Time Frequency Analysis techniques, the selected data of recorded time history has been further subdivided into different magnitude ranges (Mi, i=1 to 4) and stipulated linear epicentral distance ranges (Dj, j= 1 to 8). TFA was then applied on the data set compiled from Garhwal Himalaya and the comparison carried out between linear and quadratic TFA techniques revealed that while quadratic TFA has good time-frequency resolution, linear TFA better suits to the analyses and synthetics due to its reconstructive characteristics. The Stacking of seismograms have been carried out through spectrograms in time frequency domain which has not only increased the signal to noise ratio but also has supressed the effect of local cultural noise and thus enhances the time-frequency features in the spectrograms of stacked seismograms. The stacking of noisy vertical seismograms has been carried out in Time-Frequency (T-F) domain using discrete Gabor Transform (GT) technique and the Invert average time history of the stacked vertical spectrograms has been obtained using discrete Gabor Expansion (GE) technique. Spectrogram manifests the temporal information of various frequencies in a time history as well as energy variation pattern of frequencies over time and reveals three variable, time, frequency and amplitude and can be used to quantify the maximum energy burst with their incident time and frequency contents both simultaneously. The validation of the stacking is carried out by comparing the frequency ranges present in stacked seismogram with the frequency range in single real seismogram for same magnitude distance range. To study the time variant behaviour of incoming waves both, linear and quadratic Time Frequency Analyses (TFA) techniques, have been applied on the stacked seismograms. After carrying out initial processing on synthetic data Hanning window and Morlet wavelet (fc-0.75 Hz) have been selected for STFT and WT, respectively. The Window length (WL) and timefrequency sampling information (N) in terms of sample has been kept equal to 512 samples for all TFA techniques. The time-scale/frequency spectrums were generated with Continuous Wavelet Transform (CWT) and shown by time-frequency coordinates instead of time-scale. The exercise is carried out for 63 events having 28 pairs of magnitude distance combinations. The section describes the temporal variation of individual frequencies, simulation of these frequencies and finally simulation of frequencies in the range of 1 to 10Hz for different magnitude-distance pairs. iii The spectrograms reveal large number of energy bursts in the 5 to 9Hz range of frequency. These bursts may be attributed to refractions and reflections of direct waves from the heterogeneities inside the earth. The largest energy concentrated burst in the whole seismogram has appeared at about 9 sec to 11 sec with respect to p-wave arrival and are reported with lower frequency and high amplitude in the seismograms. To explore the frequency pattern variation with time of the stacked seismogram, two different frequency components have been selected for demonstration. Both the frequencies are selected to represent the maxima in frequency and time domains respectively, viz., one frequency component is corresponding to the highest frequency amplitude called as dominant frequency (fd) in frequency domain/Fourier Transform and second corresponds to the maximum amplitude of the seismogram in time domain. The frequency amplitude coefficients corresponding to dominant frequency and frequency corresponding to the maximum amplitude in seismograms have been extracted from the GT spectrogram of TFA. The present TFA has been carried out using GT due its inversion capability to reproduce the spectrogram back into time domain. The time domain representation in terms of sine and cosine components is extra advantage for using it in pattern recognition and its feasibility for restitution of original time histories. The frequencies between 1 to 10Hz have been selected from the stacked average vertical seismograms for each of the magnitude-distance pairs to synthesize a seismogram. Simulation has been done as the amplitude modulator functions are developed for the individual frequencies of the target seismogram by decomposing into localised basic window function using Discrete Gabor Transform and Discrete Gabor Expansion techniques. The identified Gabor amplitude coefficients (A’s) corresponding to individual frequencies (1 to 10 Hz) are extracted from the original GT spectrogram. A time series (x(t)fj, j = 1, to 10 Hz for f frequencies) of each frequency component can be generated using these identified Gabor amplitude coefficients envelope as x(t)fj = Aifj cos(2π(fj)t + φfj), where, Aifj and φfj are the Gabor amplitude coefficients and phase difference corresponding to jth frequency component for a time series, respectively. Aifj acts as amplitude modification function and cos(2π(fj)t + φfj) as stationary process for a particular jth frequency component and their multiplication results in the target time series. Here, i represents the number of coefficients used for different delay times in the time series. The target seismogram time history has been formed by stacking all the time series of frequencies 1 to 10Hz component in time-frequency domain having a signal to noise ratio equal to 6.27E-04 and FFT with standard deviation equal to 7.5904E-06. Since, the time variation of identified Gabor amplitude coefficients envelope pattern for particular frequency resembles with Fourier iv spectrum of Gaussian-modulated signal, a Gaussian-modulated sinusoidal pattern has been used to generalise the concept of simulation of synthetic time series of 1 to 10 Hz frequencies components. The Gaussian-modulated sinusoidal generates a time series for the pattern as 𝑦𝑖 (𝑡) = 𝐴𝑖 𝑒𝑥𝑝(−𝑘 (𝑖Δ𝑡 − 𝑑)2) cos(2𝜋 𝑓𝑐 (𝑖Δ𝑡 − 𝑑)) and 𝑘 = 5𝜋2𝑏2𝑓𝑐 2 𝑞 ln(10) 𝑓𝑜𝑟 𝑖 = 0, 1, 2, … … . . , 𝑁 − 1 where, Ai is amplitude at delay d, b is the normalized bandwidth, q is the attenuation, d is the delay in sec., fc is the central frequency (Hz), Δ𝑡 is time sampling interval (sec) and N is the samples. Factor 𝑒𝑥𝑝(−𝑘 (𝑖Δ𝑡 − 𝑑)2) serves as a damping factor which limits the energy of the wave to a small interval centred on Δ𝑡 = 𝑑. Thus, factor A𝑒𝑥𝑝 (−𝑘 𝑡2) represents the envelope of the Gaussian-modulated sine pattern. First the time history model for particular frequency component is computed by giving appropriate initial input parameters values of Ai at different delay time values. The Square Deviation between the target and estimated individual time series is summed up over time and is optimised by changing the initial input parameters. The synthetic model seismogram time history has been formed by stacking all the time series of frequencies 1 to 10Hz component in time-frequency domain having a signal to noise ratio equal to 8.32E-04 and FFT with standard deviation equal to 8.613E-06. The coefficient of determination (R2) between target seismogram and synthetic model seismogram has been found to be 0.922 which reveals a good matching with the target seismogram of these frequencies. The same exercise has been carried out for the rest of the data set and the results shows that the present methodology can be used for seismogram synthesis in future. The following are the main conclusions drawn from the present research work. 1. A new approach has been developed to synthesize seismograms using TFA techniques. 2. The comparison of TFA techniques to analyse and synthesize seismograms reveal that STFT and GT techniques are better performers due to their inherent advantages of reconstructive properties. 3. The analysis of non-stationary seismograms from Garhwal Himalaya shows that the selected window length/ size and time-frequency sampling information (N) should be kept same to avoid some of the spectral leakage. 4. The time variation of specific frequencies in non-stationary seismograms of microearthquakes can be extracted faithfully using linear TFA techniques. The Gabor amplitude coefficients of temporal variation of frequencies obtained from spectrograms can be used for synthesising a seismogram. v 5. Gaussian-modulated sinusoidal functions can be used to model the non-stationarity of the individual time series of a specific frequency for a magnitude-distance pair which can further be extended to synthesize a non-stationary seismogram for a set of frequencies.