In: IEEE International Symposium Time-Frequency and Time-Scale Analysis, France, pp. 345-348, 1996
SEISMIC SIGNAL DETECTION WITH TIME-FREQUENCY MODELS Carlos Rivero-Moreno
Boris Escalante-Ramírez
Electrical Engineering Department Graduate Division, Faculty of Engineering / DEPFI National University of Mexico / UNAM Apdo. Postal 70-256, México, D.F., 04510, MEXICO
[email protected] [email protected]
ABSTRACT We start this work by reviewing the pattern of a seismic signal. Then, we apply a time-frequency analysis to the signal by means of the Smoothed Pseudo Wigner-Ville Distribution (SPWVD) in order to obtain instantaneous frequency (IF) information. Based on the time-frequency behavior, we estimate a pattern to characterize the seismic signal. At the same time, we analyze the energy signal envelope, which is the derivative of the filtered cumulative energy. With the energy behavior, we estimate the different transitions along the seismic signal. The main objective of this work is the characterization of seismic signals, and their detection above a specific level of energy, so that, we can eventually build a seismic alarm.
to obtain later the instantaneous frequency (IF) and its derivative as a function of time. With these results, we verify whether the IF falls within the frequency range that characterizes a seismic signal generated in Guerrero’s coast. At the same time, we obtain the instantaneous energy envelope by deriving the filtered cumulative energy. This envelope is used to detect the transitions along the seismic signal. With the cumulative energy, we activate the alarm signal when it exceeds some threshold.
2. BACKGROUND The seismic signals are sensed in three orthornormal components designated North (N), East (E), and Vertical (V).
1. INTRODUCTION This work presents a methodology to characterize seismic signals acquired by seismometers. When the seismic signal is characterized only by its energy level, or its amplitude in the worst case, and activates an alarm signal when these levels exceed some threshold, the probability of false alarm becomes high, since non-seismic signals (perturbations) are often sensed by the seismometers. For instance, the vibration generated by a cow's kick near the sensor, as has already occurred once, could produce a signal that exceeds the threshold and activates the alarm signal. It is therefore necessary to find more reliable methods to discriminate seismic signals from other signals. We can characterize a seismic signal by its amplitude, but it is necessary to define more parameters that provide a better description of the signal’s behavior in time. In order to obtain further information of the signal we propose to obtain a frequency-based parameter. The frequency characterization will be obtained through a timefrequency representation. We use the smoothed pseudo Wigner-Ville Distribution (SPWVD) in order
Figure 1. Earthquake’s oscilogram.
These three components or axes are represented in fig. 1. The oscilogram corresponds to an earthquake took from a seismic base in Guerrero, Mexico. The plot shows the two main types of seismic waves: p-wave, or primary wave, which appears inside the earth before the earthquake is felt on the earth's surface. The s-wave, or secondary wave, represents the beginning of the earthquake, as it is followed by the Surface waves.
346
TFTS’ 96
3. METHODOLOGY 3.1. Time-Frequency characterization An important characterization of a signal is obtained through its frequency representation. The fundamental transform between the time domain (signal x(t)) and the frequency domain (spectrum X(f)) is the Fourier transform. The latter provides an overall description of the signal in terms of its frequency content. For signals whose frequency content change in time, the Fourier transform does not represent a good model. Moreover, it is important to note that the proposed method must be able to work in real-time. Therefore, we have to model the frequency content as function of time. For this reason, it is necessary to find a method which can be implemented by windowing the signal x(t). Besides, this method must provide a good time-frequency resolution and show the frequencies with high energy content. A time-frequency energy representation which satisfies a large number of mathematical properties is the Wigner-Ville distribution (WVD) [2] defined by
We show in fig. 2 the North-South component. The signal’s frequency band estimation along the time axis is difficult to do from the SPWVD since distribution differences exist not only between earthquakes but also between directional components of a single event. For this reason, we propose to use the IF as the loci of the energy concentration in frequency along the time. Furthermore, this is a WVD’s property [2]. Computing the IF is done by
∫ f fWx (t , f )df f x (t ) = ∫ f Wx (t , f )df We show the IF in the same plot together with the SPWVD (fig. 2). An expected result appears when the IF matches the SPWVD’s high energy level, or crosses near to it.
τ τ − j 2πfτ Wx ( t , f ) = ∫ x ( t + ) x ∗ ( t − ) e dτ 2 2 τ
Because the algorithm must work in real-time, and since the signal is digitized, windowing has to be applied in the discrete-time case [4]. When we apply windowing in the time domain and compute the WVD, we are actually obtaining the pseudo WignerVille distribution (PWVD) [4]. The PWVD is, in fact, a "short-time WVD" using a running analysis window [2]. This means that the signal is smoothed in frequency. On the other hand, due to the discrete nature of the data, in order to avoid aliasing in the WVD, the sampling frequency must be at least twice the Nyquist rate [4]. Due to the interference terms, it is better to use the smoothed operation in the time domain too. This means to filter the WVD, which is similar to multiplying it by a frequency window. By doing so, we obtain more resolution in both time and frequency. The SPWVD [2] is defined by
SPWVDx(g,H) (t, f ) = ∫ ∫ g(t − t’)H( f − f ’)Wx(t’, f ’)dt’df ’ t’ f ’
For each directional component, the SPWVD of the seismic signal is computed. The distribution in the three components present similar characteristics.
Figure 2. Instantaneous frequency and smoothed pseudo Wigner-Ville distribution in the North-South direction.
This can also be clearly observed if we apply the reassignment method proposed by P. Flandrin and F. Auger [3] in the SPWVD in order to obtain the reassigned smoothed pseudo Wigner-Ville distribution (RSPWVD) (fig. 3). The matching between the IF and the RSPWVD supports the idea of using the former as an efficient time-frequency characterization of the signal. The reassignment method is not suitable for our purposes due to the real-time constraint.
TFTS’ 96
Figure 3. Reassignment version of the SPWVD shown on figure 2 and its Instantaneous frequency.
Comparing the IF in the three directional components, they appear similar for the same earthquake. We can observe (fig. 4) that, in general, the IF is smooth with few oscillations within a certain frequency band. This behavior (fig. 4) can be tracked in the IF derivative. For strong oscillations, the derivative (change rate) exceeds a fixed threshold set by means of experimental measurements. Similar results were obtained by processing several earthquakes, all of them recorded in stations in Guerrero, Mexico. Comparing the different earthquakes, we set thresholds in frequency between 0.1 and 0.2 of the normalized frequency, such as fig. 4 shows. The thresholds for the IF derivative were set between -0.01 and 0.01. All thresholds are valid for any earthquake taken from a Guerrero’s station.
347
transition is detected and if the frequency behavior indicates seismic signal, then the transitions are included in the set {flat, p, s}. Otherwise, the transition is considered as no-seismic. Two types of energy measurements are considered: the cumulative energy (CE) and the instantaneous energy envelope (FCED), which is obtained by deriving the filtered cumulative energy. The CE is used to measure the Richter magnitude, or to fix an energy level that can be used to activate an alarm. This level represents a large quantity since it is produced by the energy that shakes the earth. This energy is calculated [1] by 2 ∗
E x (t ) = ∫ x (t ) x (t )dt = ∫ x (t ) dt t
t
The FCED is used to detect the transitions mentioned above. We use a FIR filter to smooth the CE which presents abrupt changes in its slope when a p-wave to s-wave transition occurs. The threshold estimated for several earthquakes is 0.015 in the FCED. If the FCED exceeds this threshold, then the signal is in swave. If not, it is in p-wave or flat. The FCED is derived from a total normalized energy EN(t) computed by
E N (t ) = n(t ) + e(t ) + v ( t ) 2
2
2
where 0 ≤ i ( t ) ≤ 1, i = n , e, v 2
and n,e, and v are the directional axes. It is therefore possible to fix a threshold in the FCED in order to detect the transitions of any earthquake.
Figure 4. Instantaneous frequency f(t) and its derivative df(t)/dt for the three directional components.
3.2. Energy analysis The energy analysis consists in the detection from flat (no signal) to p-wave, p-wave to s-wave, s-wave to p-wave, and p-wave to flat. Any important
Figure 5. Cumulative energy, instantaneous energy envelope, and oscilogram of the three axes from the earthquake shown on figure 1.
348
TFTS’ 96
We show in the plot above (fig. 5) the results of the energy analysis. When the CE’s slope changes suddenly, the FCED exceeds the threshold indicating s-wave. This is reflected in the oscilogram.
4. CONCLUSIONS We propose an algorithm for seismic signal detection. It consists of two main parts: frequency analysis for signal detection, and energy analysis for pattern recognition. The summarized algorithm is represented by the block diagram of fig. 6.
Figure 6. Block diagram which represents the complete algorithm.
Both parts are independent when they are computed. The instantaneous frequency estimation indicates whether the received signal is or not a seismic signal, while the energy analysis is used to detect transitions in the signal. If it is a seismic signal, then the transition is identified according to the type of seismic signal {flat, p, s}. The algorithm can be implemented in real-time this being an important issue, since it allows to implement a seismic alarm to work in the coast of Guerrero.
REFERENCES [1] A. Papoulis, "Signal Analysis," McGraw-Hill Book Co., New York, 1977. [2] F. Hlawatsch and G.F. Boudreaux-Bartels, "Linear and Quadratic Time-Frequency Signal Representations," IEEE Signal Proc. Mag., pp. 21-67, 1992.
[3] F. Auger and P. Flandrin, "The why and how of TimeFrequency Reassignment," Proc. IEEE-SP Symp. on Time-Frequency and Time-Scale Analysis, pp. 197 200, Philadelphia, PA, Oct. 25-28, 1994. [4] T.A.C.M. Claasen and W.F.G. Mecklenbräuker, "The Wigner Distribution--A Tool for Time-Frequency Signal Analysis; Part II: Discrete Time Signals," Philips J. Res., Vol. 35, pp. 217-250, 1980.
