Keywords

1 Introduction

Earthquake ground motions are inherently non-stationary in nature. For better understanding about the characterization of local site effects subject to earthquake, reliable earthquake signal processing is essential. A time history is a most widely used method for explaining the ground motion. The strong motion duration of an earthquake is the time interval during which most of the energy is contained. Because seismic waves are scattered during propagation, it shows a time evolving frequency composition. The records of these seismic waves exhibit non-stationary characteristics. Seismic signal shows non-stationary characteristics due to attenuation and absorption of seismic energy [1].

Damage of structures depends on earthquake’s time duration, their amplitude and frequency content of waves. Fourier-based analysis neglects the time duration information of dominant frequency of strong ground motion signal. Fourier transform can identify the frequencies which are present in signal but it does not reveal, where changes occur in the frequency contents [2].

To overcome this type of problem, joint time–frequency representations have been introduced which is a representation of both time and frequency. For the proper interpretation of seismic data in terms of varying frequency content, there is a need of time–frequency representation techniques jointly [2,3,4]. A proper analysis of ground motions is very necessary, if building design is to be constructed in seismically active areas, predicting earthquakes, quantification of damage from the recorded motions. To overcome this type of problem, joint time–frequency representations have been introduced which is a representation of both time and frequency. For the proper interpretation of seismic data in terms of varying frequency content, there is a need of time–frequency representation techniques jointly [2].

2 Time–Frequency Distributions: Fundamental Ideas

Time–frequency representation unfolds temporal information and maps a time series into 2D quantity of time and frequency with effective characterization of the time–frequency image. It describes how the spectral content of the signal changes with time. Through the view of mathematical point, this joint distribution will provide fractional energy of signal’s total energy at frequency (ω) and time (t) [5].

If the ground motion signal is represented by x(t), then the energy density of signal is represented by \(\left| {x\left( t \right)} \right|^{2}\).

The total energy Ex is

$$Ex = \int {\left| {x\left( t \right)} \right|^{2} {\text{d}}t}$$

The frequency domain representation x(f) for the signal x(t) is

$$x\left( f \right) = \int\limits_{ - \infty }^{\infty } {x\left( t \right){\text{e}}^{ - j2\pi ft} {\text{d}}t}$$

x(t) and x(f) are uniquely related.

The energy or intensity per unit frequency at frequency f is \(\left| {x\left( f \right)} \right|^{2}\).

The total energy

$$E_{x} = \int {\left| {x\left( f \right)} \right|^{2} {\text{d}}f}$$

This should be equal the total energy of the signal calculated directly from the time waveform

$$E_{x} = \int {\left| {x\left( t \right)} \right|^{2} {\text{d}}t } = \int {\left| {x\left( f \right)} \right|^{2} {\text{d}}f}$$

Jointly distribution of energy of x(t) over both the time and frequency variables amounts to looking for an energy distribution \(P_{x} \left( {t,f} \right)\) such that

$$E_{x} = \mathop \int \limits_{ - \infty }^{\infty } \mathop \int \limits_{ - \infty }^{\infty } P_{x} \left( {t,f} \right){\text{d}}t{\text{d}}f$$

For analysis of non-stationary signal, TFD is an appropriate tool. Various time–frequency representation method has been developed in last two decades for analysis of non-stationary signals. Some early form of JTF representation is STFT and CWT. The S transform is also one of JTF representation method [6]. It has some unique advantage as it gives frequency-dependent resolution and also maintain a direct relationship with Fourier spectrum [7].

3 Method

3.1 Stockwell Transform

It is an extension of the ideas of the short-time Fourier transform and is based on a moving and scalable localizing Gaussian window [7] so before to define the Stockwell transform, a general idea about STFT and CWT has been described here.

3.2 Short-Time Fourier Transform

Fourier transform, when made the function of time gives this STFT, i.e. this technique involves the division of signal into narrow time slots using the window template such that the segmented signal is considered to be stationary [8]. The FT of the segmented signal is then taken to get the frequency spectrum of that particular segment. So it can be defined as Fourier transform of the product of the signal and shifted version of window function [9, 10].

The mathematical relation that gives the STFT of a signal p(t) is

$$S\left( {\tau ,\omega } \right) = \mathop \int \limits_{ - \infty }^{ + \infty } p\left( t \right)w\left( {t - \tau } \right){\text{e}}^{ - j\omega t} {\text{d}}t$$

where

\(p\left( t \right)\) is the signal to be transformed.

w(t) is the analysis window.

\(S\left( {\tau ,\omega } \right)\) is the STFT of the signal.

\(\tau\) is the centre position of the window.

The width of the window length is fixed, which leads to the disadvantage of fixed time–frequency resolution. Because of fixed window length, there is a difference in number of cycles in that window along frequencies and that makes it inconvenient of having good time resolution comparatively frequency resolution at higher frequency [11].

One another problem through which STFT suffer is leakage due to window effect. Fourier transform of rectangular window used in this transform is a 'Sinc' function that has narrow main lope width and larger side lopes, which result spectral leakage.

Continuous wavelet transform: The basic and main principles of wavelet are review here briefly to understand its application in seismic signal. The continuous wavelet transform is the cross-correlation function, which is calculated using the signal correlated with wavelets. These wavelets are generated through the original mother wavelet by its scaled and translated versions [12,13,14]. Mother wavelet is termed as the main function, and the modified functions are called wavelets.

Mathematical formulation of the CWT is given by [15]

$$W_{{\text{s}}} \left( {a,\tau } \right) = \mathop \int \limits_{ - \infty }^{ + \infty } p\left( t \right)\frac{1}{{\sqrt {\left| a \right|} }}\psi *\left( {\frac{t - \tau }{a}} \right){\text{d}}t$$

where \(p\left( t \right)\) is the signal.

\(W_{{\text{s}}} \left( {a,\tau } \right)\) are the CWT coefficients.

\(\psi\)(\(t\)) is the mother wavelet, dilated by scale \(a\) and shifted in time \(\tau\).

\(\frac{1}{{\sqrt {\left| a \right|} }}\) is known as the multiplication factor, and it ensures the normalization of energy, means the wavelet which has unit energy at all scales.

Like STFT, the wavelet transforms also suffer from uncertainty principle [16]. Both a good time and frequency resolution cannot be achieved simultaneously.

3.3 Stockwell Transform

The Stockwell transform is a mid way between STFT and CWT. This time–frequency representation method has quite similarity to STFT while because of the use of multiresolution tactics, it makes somewhere closer to wavelet transform [11]. A big advantage of S-transform is that because of its simple concept, it gives a simple understanding of multiresolution approach as have been introduced in wavelets and hardly require any additional knowledge excepting STFT [11]. In Stockwell transform for window function, Gaussian window is used, which can be time shifted by \(\tau\) and is inversely proportional to the linear frequency f [10]. The S-transform may comparable to the CWT as the Gaussian template is comparable to the mother wavelet with a phase shift.

The mathematical expression for the Gaussian taper is given as

$$w\left( t \right) = {\text{e}}^{{\frac{{ - t^{2} }}{{2\sigma^{2} }}}},$$

where \(\sigma\) is taper width and inversely proportional to the frequency \(\sigma = \frac{k}{\left| f \right|}.\) The parameter k can be tuned to obtain better frequency localization at the cost of reduced time localization by controlling the width of Gaussian taper [15] here the parameter k is considered 1. Gaussian taper function is normalized to achieve the Gaussian template.

The S transform can be derived from STFT by replacing the window function w(t) with the Gaussian function shown as

$$w\left( t \right) = \frac{\left| f \right|}{{\sqrt {2\pi } }}{\text{e}}^{{\frac{{ - t^{2} f^{2} }}{2}}}$$

As STFT defined as

$$S(\tau ,f) = \mathop \int \limits_{ - \infty }^{ + \infty } p\left( t \right) w\left( {t - \tau } \right){\text{e}}^{ - j2\pi ft} .$$

Then, the S transform is defined as

$$S_{{\text{T }}} \left( {\tau ,f} \right) = \frac{\left| f \right|}{{\sqrt {2\pi } }}p\left( t \right){\text{e}}^{{\frac{{ - \left( {t - \tau } \right)^{2} f^{2} }}{2}}} {\text{e}}^{ - j2\pi ft}$$

where \(S_{{\text{T }}} \left( {\tau ,f} \right)\) represents the S-transform of signal.

\(p\left( t \right)\) is continuous time signal.

\(\tau\) parameter controls the position of the Gaussian window on t-axis.

The advantage of S-transform is that it may give multiresolution analysis beside keeping the absolute phase of each frequency [17]. The Gaussian window which localization is inversely proportional to the frequency is an improvement over STFT as fixed width window used in this.

S-transform’s phase referenced to time origin which gives useful and accompanying information about the spectra, and this is not obtainable from locally referenced phase information in continuous wavelet transform [10].

4 Result

To demonstrate the performance of ST, one non-stationary synthetic signal and one real-time earthquake signal record are considered.

  1. A.

    Synthetic Signal

    A synthetic signal quite similar to synthetic example [18] is generated with three sinusoidal components to illustrate the features of ST. The synthetic signal is a sum of three components which consists of sinusoidal waves. The sampling frequency of the signal is 500 Hz, and the signal-to-noise ratio (SNR) is 10 db. The length of signal is 10 s. Here, \(\eta\)(t) is the Gaussian noise. The details of synthetic signal are given by the following equation and are shown in Fig. 1 with its three components.

$$S\left( t \right) = S_{1} \left( t \right) + S_{2} \left( t \right) + S_{3} \left( t \right) + \eta \left( t \right)$$
$$S_{1} \left( t \right) = \left[ {2 + 0.2\sin \left( t \right)} \right] \cdot \sin \left[ {2\pi \left( {3t + 0.6\sin \left( {\text{t}} \right)} \right)} \right]$$
$$S_{2} \left( t \right) = 0.6\left[ {1 + 0.3\sin \left( {2t} \right) \cdot \exp \left( { - \frac{t}{20}} \right) \cdot \cos \left[ {2\pi \left( {5t + 0.6t^{1.8} + 0.3\cos \left( t \right)} \right)} \right]} \right]$$
$$S_{3} \left( t \right) = 0.4{\text{sin}}\left[ {2\pi \left( {9t} \right)} \right]$$
Fig. 1
figure 1

Developed synthetic signal in (a). b–d represents the three components S1(t), S2(t) and S3(t), respectively,

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The frequency of S1(t) is the lowest among all the three components but the amplitude of which is highest. The frequency of S2(t) increases with time, and amplitude becomes smaller when frequency becomes higher. The frequency of S3(t) is constant, and its amplitude is smallest. The specially designed signal components are so that the higher is the amplitude of a component, the lower the frequency of the component.

Figure 1a shows the synthetic signal and its components considered for the analysis. The TF representations of synthetic signal using STFT, CWT, ST are shown in Fig. 2a–c, respectively. Hanning window with window length (196 in sample) used for STFT and Morse wavelet has been used for CWT; the width factor (K) is 1 for ST. All three components of S(t), namely S1(t), S2(t), S3(t) are visible in the result of TF spectra of ST while in the result of CWT only one frequency component S1(t) is clearly visible.

Fig. 2
figure 2

Time–frequency representations of synthetic signal a STFT b CWT c ST

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  1. B.

    Real-Time Earthquake Record

    The real-time earthquake record is of Uttarkashi earthquake occurred in October 1991. This earthquake was recorded at 13 stations of a strong motion network installed by Department of Earthquake Engineering, Indian Institute of Technology, Roorkee in Garhwal Himalaya under the research scheme funded by Department of Science and Technology, Govt. of India.

    The acceleration record of Uttarkashi earthquake recorded at Barkot station at 20 October 1991 at 02:53 IST with a moment magnitude of 6.8. Signal was recorded through a three channel (North–South, Vertical, East–West) accelerograph. The frequency sampling rate of accelerogram’s recording is 50 sps. N-S component of this accelerogram, which duration is 31.74 s used for analysis.

The acceleration time history of the signal and corresponding FT of the signal are shown in Fig. 3a, b. T-F representation of seismic signal using STFT, CWT and ST is shown in Fig. 4a–c, respectively. The parameter used for analysis through different methods is given in Table 1.

Fig. 3
figure 3

a Acceleration record of N-S component of Uttarkashi earthquake recorded at Badkot station, b frequency spectrum of acceleration record

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Fig. 4
figure 4

Time–frequency representations of Uttarkashi station record (a) STFT  (b) CWT  (c) ST

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Table 1 Paramaters used in various T-F Representations
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Figure 4a–c shows at least 3–4 frequency components. These components are difficult to find only frequency domain based analysis. These are marked as f1, f2, f3 and f4. The time localization of f1, f2, f3, f4 is very good in the result of ST comparatively STFT and CWT. In the result of ST, the frequencies are annotated at different time as f1 = 7.5 Hz at 5.6 s, f2 = 2.5 Hz at 6.5 s, f3 = 9 Hz at 7.2 s and f4 = 8.5 Hz at 8.5 s. It may visualized from the results that resolution of time–frequency representation obtained using ST is better as compared to the STFT and CWT representation.

5 Conclusion

The Stockwell transform-based joint time–frequency technique is introduced for seismic signals and is validated for real-time ground motion signal. The time–frequency representations of synthetic signal using STFT, CWT, ST are compared. All three components of S(t), namely S1(t), S2(t), S3(t) are visible in the result of time frequency spectra of Stockwell transform while in the result of CWT only one frequency component S1(t) is clearly visible. The time–frequency representations of real-time signal using STFT, CWT, ST are also compared. These components are difficult to find only frequency domain-based analysis. These are marked as f1, f2, f3 and f4. The time localization of f1, f2, f3, f4 is very good in the result of ST comparatively STFT and CWT. It may visualized from the results that resolution of time–frequency representation obtained using ST is better as compared to the STFT and CWT representation. Stockwell transform is capable for improving the resolution of non-stationary signals as well as clearly identified the spots of concentration in energy. For seismic signals, the S-transform provides good time localization. Frequency localization is not so good; it shows frequency smearing. Frequency smearing problem may be minimized through synchrosqueezing of Stockwell transform.