Determination of zero crossing frequency and likelihood function for maximum displacement in real time earthquake signal
P.K. DUTTA1, O.P. MISHRA2, M. K. NASKAR1
1. Research Fellow, Advanced Digital Embedded System Lab, Electronics and Telecomm. Dept., Jadavpur University, Kolkata, India and Faculty,Electronics & Automation, D.G. Shipping, Kolkata, India
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2. Scientist,National Centre of Seismology, Ministry of Earth Sciences, New Delhi, India
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3. Advanced Digital Embedded System Lab, Jadavpur University,Electronics and Tele- Comm. Dept., Kolkata, IndiaEmail:
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Abstract:
In the proposed work, we analyze high-frequency directivity effect based on accelerometric data to observe whether rupture process of this event had a strong directivity effect. Directivity analysis of ground motion in the direction of slip propagation for fault rupture is close to that of the shear wave. In the proposed study, special attention has been made to the non-stationarity in frequency contents of ground motions for studying the statistical properties and attenuation laws of non-stationarity in frequency contents due to their nonparametric nature. In contrast to displacements inferred through integration of seismic data alone for the characterization of the non-stationary in frequency contents of ground motions has been proposed to study the short time amplitude analysis of the waveform signals. We estimate the zero crossing rates for waveforms for maximizing the correlation between two events that occur on the periodogram output. New criteria in the selection and synthesis of ground motion using a zero crossing rate for better warning scenarios. Our results show that Kalman filter is better in linearizing the system process and measurements and can be used to derive non-stationary characteristics of the future envelope of the response spectra for early detection of earthquake aftershock analysis. Based on the accelerometric data available for the Sikkim Earthquake of 2011 for seismic signature of the earthquake as triggering function for the slip as forecast errors were correlated with elements of the information set. Our results prove that directivity effect of a seismic rupture can be found from acceleration traces.
KeyWords—earthquake ground motion, non-stationarity, frequency content, filter, Sikkim Earthquake, zero-crossing rate
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P.K. Dutta, O.P. Mishra and M.K. Naskar
I. Introduction
Real time strong motion data is often acquired from dense seismic network recording for giving a rapid earthquake response. The goal of the strong motion data processing algorithm involves giving suitable response for a certain ground shaking through correlation of peak ground acceleration amplitude with seismic intensity [1,2] using signal processing analysis. The modeling, statistical properties and attenuation laws of non-stationarity in frequency contents of peak ground acceleration have been the open issues in earthquake engineering. Due to time bound constraints in the real world, acceleration data and signal amplification are often nonlinear and non- stationary which makes the processing of such signals a difficult task constraining analysis for amplitude variation for signals. In order to document the characteristic feature of signal amplification from ground motion acceleration data which causes the proposed study has taken recorded peak ground acceleration data of magnitude 6.9 earthquake in the Himalayan states Nepal and Sikkim (India) that hit on 2nd,October, 2011 from IMD acquired from 10 stations recorded during the earthquake event. Directivity of the rupture process is a parameter of the seismic source that plays an important role in the generation of stronger ground motions which is distributed in an elongated pattern centered along the axis of the fault. In the direction of directivity, stations that see the rupture coming, the duration of the apparent source-time function [3] is shorter than the real duration of the process on the fault. As a consequence at equal epicentral distances and for the same site conditions, the ground motions will be higher in the direction of directivity. The proposed work outlines tools for non-stationary and nonlinear acceleration time series analysis to have good potential for application in a wider geophysical context. It has been found that long period pulses which may or may not be visible in the acceleration traces, but they are readily distinguishable in the velocity traces. These velocity pulses appear to be stronger when the rupture propagates towards the site (forward directivity). It can be proved that rupture directivity can be proved using acceleration data traces although very less work has been done in this regard [4]. In order to analyze directivity characteristics from time series data, stochastic analysis needs to be conducted that can exploit powerful recursive methods of estimation analysis of data generated for non-stationary and nonlinear time series data as in earthquake monitoring. It has been found seismic waves observed through earthquake accelerogram record manifest clearly non-stationary characteristics, as well as wide frequency content. Based on the acceleration-time series records, earthquake magnitude and the on-site ground-motion intensity could be estimated and secondary estimates for warning can be issued. Several researchers like [5] found characteristic measurement of the predominant period (tP, max) in the few seconds after the P-wave arrival onset, or that proposed by [6] based on the peak displacement amplitude. Understanding the structure of the earth’s crust and mechanical nucleation models prevalent in geo-analysis and initiation ofseismogenesis [7-8], there is a requirement to initiate analysis of slip velocity locally as there exist gradients due to heterogeneity that complicates any interpretation of the seismic sequence in terms of a precursory process. We present a less subjective and more real time oriented measurement using integration of accelerometer data that avoids problematic baseline corrections suggested by [9-10] as function form fitting developed in time domain averaging in short term vs long term averaging (STA) of strong motion data giving improved broadband record of ground displacements, spanning the broadest possible spectrum of static deformation. The proposed approach is suitable for dense networks and real-time processing required by early warning systems and rapid earthquake response. In current seismological practice, strong-motion displacements are obtained from double integration of accelerometer data. The first characteristic measures variations with time of the intensity of the ground motion (acceleration, velocity or displacement). At the onset of the earthquake rupture, with the arrival of the first seismic wave, energy of the earthquake builds up rapidly to a maximum value [11] for a certain time and then decreases slowly until it vanishes. The second characteristic involves identifying mean square error variations with time of the frequency content having a tendency to shift to lower frequencies as time increases. The information that is inferred from accelerogram outputs involve peak acceleration;duration of shaking ;strength of shaking due to peak ground acceleration, peak ground velocity and peak ground displacement (PGD) extracted by processing accelerograms[12]. The first step developed in the proposed work involves an optimal modeling technique based on acceleration data and identifying the characteristic functions like short term energy and zero crossing rates. Being specifically designed for real-time monitoring, our method provides distinct advantages for rapid earthquake response, determining the intensity of a large earthquake more quickly or deciding whether the rupture will propagate over part of or the entire fault for shaking and non-shaking behavior. An analysis of past records shows that the strength of the oscillations exhibited by the number of zero crossings and extremes in a given interval and the non-stationary random nature revealed by the time-dependent variance function are important characteristics. Using this information, a stationary random process modulated by a deterministic function has been developed to find their mean square error has been compared. In the proposed work, peak motion behavior for accelerogram data is found for Sikkim Earthquake and our results show that larger distance for hypocenter presents a relatively slower rate of decay measured using zero crossing rate. Being specifically designed for real-time monitoring, the proposed method provides distinct advantages for rapid earthquake response in determining the magnitude of a large earthquake more quickly or deciding whether the peak in amplitude is going to occur at a future defined instant. Due to the lack of quantification of systematic differences in ground motion time-histories (acceleration, velocity and displacement), and linear and nonlinear structural responses introduced by various processing techniques, it was therefore necessary to apply the auto regressive model to limit segments of data, as large variations in mean square error is observed as the earthquake progressed. We have used kalman filter to find smoothed estimation of the parameters of the acceleration time series model using filtering, smoothing and prediction in which the parameters being estimated by mean square error is checked for optimal balance among the parameters for a recursion based model that varies with time [13].
2 ANALYSES OF TIME SERIES DATA
Accelerometers are easy to design and maintain due to their less sensitivity and sample data obtained in time series can be extracted by multiplication of scaling factors. Acceleration spectrum is one of the most direct and common functions used to describe the frequency content of strong ground earthquake shaking [14]. A time series is a sequence of random variables y(t)= f{y(t); t =0;±1;±2;:::} representing the potential observations of the process, which have a common finite expected value E(xt)=µ and a set of auto-covariances C(yt;ys)= Ef(yt -µ) (ys- µ)} = γ|t-s| which depend only on the temporal separation τ = |t-s| of the time ‘t’ and ‘s’ and not on their absolute values. We present a new solution to the classical problem of deriving displacements from seismic data suitable for real-time monitoring. Time series analysis is essentially concerned with evaluating the properties of the probability model which generated the observed time series. One way of describing a stochastic process is to specify the joint probability distribution of xt1, ..,X tnfor any set of time tl,…, tnand any value of ‘n’. It is observed that most time series are stochastic as the future values are only partly determined by past values, so that exact predictions are impossible and by the idea that future values have a probability conditioned by knowledge of past values. It is an iterative process, where it first computes the mean- shift value for the current point position, then moves the point to its mean-shift value as the new position, then computes the mean-shift until it fulfills certain conditions. Some studies of filtering and deformation analysis were performed in order to detect failures and outliers, and to Figure 1: Plot for identifying acceleration time series energy spectrum and zero crossing points to see where the maximum transitions in shaking occurred from Gangtok data
increase the reliability of the deformation analysis. The first problem is the estimation of the arrival time of the seismic signal. When an earthquake occurs, its location is estimated from arrival times of the seismic waves at several different observatories. This is non stationary by nature while several notions have been set forth for practical purpose, the zero-crossing rate is used to describe the non-stationary in frequency contents and the instantaneous spectrum is used to describe the non-stationary both in amplitude and frequency contents of spectra.Energy of a seismic signal is another parameter for classifying as the high energy because of its periodicity and the unshaking part has zero crossing rate (ZCR) and high energy. Advantage of using zero crossing rate in non-stationary signals like earthquakes is that one can plot the signal in a time frequency space enabling the energy distribution in the signal to be observed. Based on the rupture analysis, we calculate the arias intensity pattern which says that total energy which is delivered during an earthquake depends on the zero crossing rate variations[15] is said to occur if successive samples have different algebraic signs as in Figure 1.
The rate at which zero crossings occur is a simple measure of the frequency content of a signal [16]. Identifying characteristics of structural seismic response based on amplitude and frequency duration and the non stationary properties in amplitude and frequency contents can also influence the structural seismic response significantly. Maximum amplitudes correspond to a point in the time-frequency plane where several time-frequency characteristics of the signal concentrate where ‘x’is slip associated with the system identified from zero crossing rate ‘ y’as the measured output or the associated acceleration. Since high frequencies imply high zero crossing rates, and low frequencies imply low zero-crossing rates, there is a strong correlation between zero-crossing rate and energy distribution with frequency. Time domain representation of edge signals looking at zero crossing to identify samples time when negative and the next is positive as the time between the successive zero crossings and measure T as the successive crossing in the same direction. Peak detection analysis for complete extraction of the energy ratio of the signal based on zero crossing edge detection end point detection based on short term analysis and zero cross rate to identify the features of the waveform.A reasonable generalization is that thezerocrossing rate is high, the signal is non periodicas when the zero-crossing rate is low then the signal is periodic.Spectral analysis and improved algorithms for time domain representation of edge signals looking at zero crossings to identify the points in time when the sample is negative and the next is positive ; we measure time between the successive zero crossings in the same direction. A reasonable generalization is that if the zero-crossing rate is high, the shaking of the body is non periodic if the zero-crossing rate is low, signal is periodic which is helpful in predicting the model. Efficient warningfor time domain representation of edge signals looking at zero crossings can identify the instant time when the sample is negative and the next is positive to evaluate time between the successive zero crossings and measure ‘T’ as the successive crossing in the same direction as the rupture follows a slip occurs in more or less an independent manner generating high frequency waves [17]. There is a gradually decreasing tendency in the slopes of the cumulate zero-crossing curves. The zero-crossing model fits the actual result perfectly and the precision is satisfactory. The non-stationarity in frequency contents is an important property of earthquake ground motions besides the conventional properties in amplitude, frequency and duration. The study has been differentiated into three parts. The first part involves analysis of elastic deformation based on signal processing analysis involving the energy of the spectrum through analysis of the Arias Intensity and the zero crossing rate, in the second phase as the arrival time is estimated by using the locally stationary autoregressive model.