Introduction

As a non-stationary and strongly time-varying signal, seismic data carries the information characterizing underground geological structures and lithology1,2,3,4,5. Time-frequency analysis (TFA) can provide valuable subsurface information and has become an important tool in seismic data processing and interpretation6. The short-time Fourier transform (STFT)7 is a classic TFA technique that controls time-frequency resolution by the width of the analyzing window. Higher time resolution requires narrow time window and similarly in the frequency domain. The continuous wavelet transform (CWT)8 employs the wavelet basis function and a scale parameter to adjust the window length in order to achieve the purpose of variable time-frequency resolution. Unfortunately, both of them cannot simultaneously obtain high resolution both in time and frequency domains due to the Heisenberg’s uncertainty principle9. The Wigner-Ville distribution (WVD)10,11 does not suffer from such deficiency. However, the problem of cross terms still limits its application in seismic data analysis.

In order to solve the above-mentioned problems, a variety of advanced TFA methods have been developed. Reassignment method (RM)12 improved the energy-smeared time-frequency representation (TFR) by relocating the data to new coordinate based on local estimates of instantaneous frequency (IF) and group delay of the signal. Regretfully, such operations lead to the phase information of the signal to be lost, which means the RM cannot reconstruct the original signal. Synchrosqueezing transform (SST)13, firstly introduced by Daubechies and Maes14 in the field of audio processing, condensed the time-frequency spectrum along the frequency direction to enhance the readability of a wavelet-based TFR by the aid of frequency reassignment technique15,16. Unlike the RM, the SST retains the inverse ability, which makes it have wider range of applications. Inspired by the ‘synchrosqueezing’ idea, Thakur and Wu17 sharpened the blurred TFR provided by the STFT and developed the STFT-based SST (FSST), and the detailed theory can be found in18. Huang et al.19 proposed the synchrosqueezing S-transform (SSST) by synchrosqueezing the spectrum of S-transform20,21,22,23. It is worth noting that the above methods cannot address the strongly time-varying signals well because of the hypothesis that the modes making up the signal require weak frequency modulation. Therefore, Oberlin et al.24 introduced the second-order SST to mitigate the limitation of standard FSST by calculating a second-order local estimate of the IF that aims to enhance modes localization, which is named second-order FSST (FSST2). Behera et al.25 presented the theoretical analysis of the FSST2 and explained the hypotheses made on the modes making up the multicomponent signal. Pham and Meignen26 put forward the high-order FSST (FSSTN) that employed a more complex signal model with high-order approximations both for the amplitude and phase to compute more accurate estimate of the IF of the modes constituting the signal. Yu et al.27 developed the synchroextracting transform (SET) in which it only retains the time-frequency information from STFT result that are most relevant to time-varying characteristics of the signal and removes most of the smeared time-frequency energy28. Yu29 proposed the demodulated synchrosqueezing transform by introducing a demodulated operator and obtained the TFR with much more concentrated energy for frequency-modulated signals. Yu et al.30 executed multiple squeezing SST operations to concentrate the time-frequency energy of the SST for a much sharper TFR, which is termed as multisynchrosqueezing transform (MSST). Such a method addresses a wide variety of multicomponent signals comprised of strongly time-varying modes. Mahdavi et al.31 introduced the local maximum synchrosqueezing transform (LMSST) as an efficient method for detection of low-frequency shadow zones in seismic data, which is an indication of presence of natural gas and absorption of high frequency energies within the gas column of the gas reservoir. Fang et al.32 proposed a second-order horizontal multi-synchrosqueezing transform (SHMSST) for hydrocarbon reservoir identification, in which a second-order local group delay deduced from Gaussian-modulated linear chirp model is incorporated into the multi-synchrosqueezing framework. Shirazi et al.33 employed multi-synchrosqueezing transform (MSST) to detect gas reservoirs. It is post-processing mechanism for the STFT that performs the reassignment procedure repeatedly by applying consecutive SST operations. Barabadi et al.34 investigated the efficacy of synchroextracting transform (SET) for amplitude versus offset (AVO) analysis in time-frequency domain. Mehrabi et al.35 integrated CNN and RNN architectures and proposed a hybrid CRNN deep learning model to predict porosity values at new locations where only well-log data in complex carbonate reservoir, which achieves the accurate porosity estimation and valuable insight into reservoir characterization. Moosavi et al.36 combined the support vector machine and a fuzzy system using membership functions to estimate porosity in one of the oil fields in southern Iran. It helps in evaluating the capacity of rock to contain fluid. Jahanjooy et al.37 employed a fuzzy model-based seismic inversion algorithm which considers prior information and fuzzy clustering constraints to obtain more realistic and reliable AI model, which can help to identify and delineate the subsurface channels by providing more contrast and detail of the channel geometry. Jalali et al.38 developed an attribute that uses mixed components of STFT (MC-STFT) to detect gas reservoirs at high resolution. This method takes advantage of extracting three frequency components obtained by STFT, and the results indicate the good performance of MC-STFT in high-resolution gas reservoir detection. Mahzad and Bagheri39 proposed a U-Net network model to predict and reconstruct complex missing geological events including faults and unconformities by leveraging the inherent patterns in seismic data.

In this paper, a second-order signal model is considered in the framework of the STFT-based MSST where the second-order Taylor expansion of the amplitude and phase of the signal is utilized, which is named STFT-based second-order MSST (FMSST2). The FMSST2 not only achieves a highly energy-concentrated TFR, but also allows for signal reconstruction with a high accuracy. The principal contributions of this study are as follows.

  1. (1)

    The study proposed FMSST2 by using a second-order signal model to characterize seismic signal and the FMSST2 is applied to seismic time-frequency analysis.

  2. (2)

    The FMSST2 not only produce a more energy-concentrated TFR, but also recover the original signal well.

  3. (3)

    Synthetic examples and field data are employed to validate the effectiveness of the FMSST2 method in seismic data analysis.

The paper is structured as follows. The principles of the STFT, MSST and FMSST2 methods are detailed in the section “Theory”. The Synthetic examples and field data that are used to verify the effectiveness of the FMSST2 method in characterizing non-stationary signals are provided in the sections “Synthetic examples” and “Field examples”, respectively. The discussion is drawn in the section “Conclusion”.

Theory

Short-time fourier transform

The STFT of a signal f is written as

$${V_f}\left( {t,\eta } \right)=\int_{{ - \infty }}^{\infty } {f\left( \tau \right)g\left( {\tau - t} \right)} {e^{ - i\eta \left( {\tau - t} \right)}}d\tau$$
(1)

where g denotes the window function.

The original signal can be recovered by its STFT using Eq. (2).

$$f\left( t \right)=\frac{1}{{2\pi g\left( 0 \right)}}\int_{{ - \infty }}^{\infty } {{V_f}\left( {t,\eta } \right)} d\eta$$
(2)

where g is non-zero and continuous at 0.

Multisynchrosqueezing transform

The key idea of MSST is to concentrate the energy of a TFR in a stepwise manner by applying multiple SST operations iteratively30.

Considering a signal

$$f\left( \tau \right)=A\left( t \right){e^{i\left( {\varphi \left( t \right)+{\varphi ^\prime }\left( t \right)\left( {\tau - t} \right)} \right)}}$$
(3)

Combining Eq. (1) and Eq. (3), and we have

$$\begin{gathered} {V_f}\left( {t,\eta } \right)=\int_{{ - \infty }}^{\infty } {g\left( {\tau - t} \right)A\left( t \right){e^{i\left( {\varphi \left( t \right)+{\varphi ^\prime }\left( t \right)\left( {\tau - t} \right)} \right)}}} {e^{ - i\eta \left( {\tau - t} \right)}}d\tau \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} =A\left( t \right){e^{i\varphi \left( t \right)}}\mathop g\limits^{ \wedge } \left( {\tau - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ \end{gathered}$$
(4)

where \(\mathop g\limits^{ \wedge } \left( \bullet \right)\) denotes the Fourier transform of the window function g.

The IF is calculated by using the following Eq. 

$$\begin{gathered} {\omega _f}\left( {t,\eta } \right)=\Re \left\{ {\frac{{{\partial _t}V\left( {t,\eta } \right)}}{{iV\left( {t,\eta } \right)}}} \right\} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} =\Re \left\{ {\frac{{{\partial _t}\left( {A\left( t \right){e^{i\varphi \left( t \right)}}\mathop g\limits^{ \wedge } \left( {\tau - {\varphi ^\prime }\left( t \right)} \right)} \right)}}{{i\left( {A\left( t \right){e^{i\varphi \left( t \right)}}\mathop g\limits^{ \wedge } \left( {\tau - {\varphi ^\prime }\left( t \right)} \right)} \right)}}} \right\} \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ={\varphi ^\prime }\left( t \right) \hfill \\ \end{gathered}$$
(5)

where \(\Re \left\{ \bullet \right\}\) stands for the real part of a complex number, \({\partial _\tau }\) is the partial derivative with respect to t.

The FSST uses a frequency reassignment operator to squeeze the TFR result of the STFT, which is defined as

$${{\rm T}_f}\left( {t,\omega } \right)=\int {{V_f}\left( {t,\eta } \right)\delta \left( {\omega - {\omega _f}\left( {t,\eta } \right)} \right)} d\eta$$
(6)

Then, the MSST method is formulated as

$$\begin{gathered} {T_f}^{{\left[ 2 \right]}}\left( {t,\omega } \right)=\int_{R} {{{\rm T}_f}\left( {t,\omega } \right)\delta \left( {\omega - {\omega _f}\left( {t,\eta } \right)} \right)} d\eta \hfill \\ {T_f}^{{\left[ 3 \right]}}\left( {t,\omega } \right)=\int_{R} {{T_f}^{{\left[ 2 \right]}}\left( {t,\omega } \right)\delta \left( {\omega - {\omega _f}\left( {t,\eta } \right)} \right)} d\eta \hfill \\ \vdots \hfill \\ {T_f}^{{\left[ N \right]}}\left( {t,\omega } \right)=\int_{R} {{T_f}^{{\left[ {N - 1} \right]}}\left( {t,\omega } \right)\delta \left( {\omega - {\omega _f}\left( {t,\eta } \right)} \right)} d\eta \hfill \\ \end{gathered}$$
(7)

where \({{\rm T}_f}\left( {t,\omega } \right)\) in Eq. (6) is denoted by \({T_f}^{{\left[ 1 \right]}}\left( {t,\omega } \right)\), and N is the iteration number.

And, the \({T_f}^{{\left[ 1 \right]}}\left( {t,\omega } \right)\) is substituted into \({T_f}^{{\left[ 2 \right]}}\left( {t,\omega } \right)\), we get

$${T_f}^{{\left[ 2 \right]}}\left( {t,\omega } \right)=\int_{R} {{V_f}\left( {t,\eta } \right)\delta \left( {\omega - {\omega _f}\left( {t,{\omega _f}\left( {t,\eta } \right)} \right)} \right)} d\eta$$
(8)

Similarly,

$${T_f}^{{\left[ 3 \right]}}\left( {t,\omega } \right)=\int_{R} {{V_f}\left( {t,\eta } \right)\delta \left( {\omega - {\omega _f}\left( {t,{\omega _f}\left( {t,{\omega _f}\left( {t,\eta } \right)} \right)} \right)} \right)} d\eta$$
(9)

After N iterations, the MSST can be expressed as

$${T_f}^{{\left[ N \right]}}\left( {t,\omega } \right)=\int_{R} {{V_f}\left( {t,\eta } \right)\delta \left( {\omega - {\omega _f}\left( {t,{\omega _f}\left( {t,{\omega _f}\left( {t, \cdots ,{\omega _f}\left( {t,\eta } \right)} \right)} \right)} \right)} \right)} d\eta$$
(10)

Finally, the mode is reconstructed by

$${f_k}\left( t \right) \approx \frac{1}{{2\pi g\left( 0 \right)}}\int_{{\left| {\omega - {\varphi _k}\left( t \right)} \right|<d}} {{T_f}^{{\left[ N \right]}}\left( {t,\omega } \right)} d\omega$$
(11)

where \({\varphi _k}\left( t \right)\)is an estimate of IF, d denotes a compensation factor.

STFT-based multisynchrosqueezing transform using a second-order signal model

Now, considering a second-order signal model

$${f_2}\left( \tau \right)=A\left( t \right){e^{i\left( {\varphi \left( t \right)+{\varphi ^\prime }\left( t \right)\left( {\tau - t} \right)+0.5{\varphi ^{\prime \prime }}\left( t \right){{\left( {\tau - t} \right)}^2}} \right)}}$$
(12)

Moreover, the window function is selected as Gaussian function

$$g\left( t \right)={e^{ - \frac{{{t^2}}}{{2{\sigma ^2}}}}}$$
(13)

Then, we substitute Eq. (12) and Eq. (13) into Eq. (1), the STFT of signal (12) can be derived as

$$\begin{gathered} {V_{2,f}}\left( {t,\eta } \right)=\int_{{ - \infty }}^{\infty } {{e^{ - \frac{{{{\left( {\tau - t} \right)}^2}}}{{2{\sigma ^2}}}}}A\left( t \right){e^{i\left( {\varphi \left( t \right)+{\varphi ^\prime }\left( t \right)\left( {\tau - t} \right)+0.5{\varphi ^{\prime \prime }}\left( t \right){{\left( {\tau - t} \right)}^2}} \right)}}} {e^{ - i\eta \left( {\tau - t} \right)}}d\tau \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} =A\left( t \right){e^{i\varphi \left( t \right)}}\sqrt {\frac{{2{\sigma ^2}\pi }}{{1 - i{\sigma ^2}{\varphi ^{''}}\left( t \right)}}} {e^{ - \frac{{{\sigma ^2}{{\left( {\eta - {\varphi ^\prime }\left( t \right)} \right)}^2}}}{{2\left( {1 - i{\sigma ^2}{\varphi ^{\prime \prime }}\left( t \right)} \right)}}}} \hfill \\ \end{gathered}$$
(14)

According to Eq. (5), we get

$${\omega _{2,f}}\left( {t,\eta } \right)={\varphi ^\prime }\left( t \right)+\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right)$$
(15)

According to Eq. (15), we can further obtain \(\omega _{{2,f}}^{{\left[ 2 \right]}}\left( {t,\eta } \right)\)and \(\omega _{{2,f}}^{{\left[ 3 \right]}}\left( {t,\eta } \right)\)

$$\begin{gathered} \omega _{{2,f}}^{{\left[ 2 \right]}}\left( {t,\eta } \right)={\varphi ^\prime }\left( t \right)+\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{''}}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{''}}\left( t \right)} \right]}^2}}}}}\left( {{\omega _{2,f}}\left( {t,\eta } \right) - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} ={\varphi ^\prime }\left( t \right)+{\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^2}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ \end{gathered}$$
(16)
$$\omega _{{2,f}}^{{\left[ 3 \right]}}\left( {t,\eta } \right)={\varphi ^\prime }\left( t \right)+{\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^3}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right)$$
(17)

Furthermore, we can derive that

$$\begin{gathered} \left| {\omega _{{2,f}}^{{\left[ 2 \right]}}\left( {t,\eta } \right) - {\varphi ^\prime }\left( t \right)} \right|={\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^2}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} <\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} =\left| {{\omega _{2,f}}\left( {t,\eta } \right) - {\varphi ^\prime }\left( t \right)} \right| \hfill \\ \end{gathered}$$
(18)
$$\begin{gathered} \left| {\omega _{{2,f}}^{{\left[ 3 \right]}}\left( {t,\eta } \right) - {\varphi ^\prime }\left( t \right)} \right|={\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^3}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} <{\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^2}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right) \hfill \\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} =\left| {\omega _{{2,f}}^{{\left[ 2 \right]}}\left( {t,\eta } \right) - {\varphi ^\prime }\left( t \right)} \right| \hfill \\ \end{gathered}$$
(19)

Equations (18) and (19) illustrate that the IF of the FMSST2 (N = 3) is much closer to the true IF \({\varphi}^{\prime}\left( t \right)\), which means the FMSST2 (N = 3) can provide the more concentrated result than the original FMSST2.

By analogy, the IF estimate \(\omega _{{2,f}}^{{\left[ N \right]}}\left( {t,\eta } \right)\) after N iterations can be represented as

$$\omega _{{2,f}}^{{\left[ N \right]}}\left( {t,\eta } \right)={\varphi ^\prime }\left( t \right)+{\left( {\frac{{{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}{{{1 \mathord{\left/ {\vphantom {1 {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}} \right. \kern-0pt} {{\sigma ^4}+{{\left[ {{\varphi ^{\prime \prime }}\left( t \right)} \right]}^2}}}}}} \right)^N}\left( {\eta - {\varphi ^\prime }\left( t \right)} \right)$$
(20)

Based on Eqs. (18) and (19), it can be observed that the obtained IF estimate will be closer and closer the true IF of the signal as the number of iterations increases, that is, the \(\omega _{{2,f}}^{{\left[ N \right]}}\left( {t,\eta } \right)\)is the desired IF estimate.

Therefore, the expression of the FMSST2 can be rewritten as

$${T_{2,f}}^{{\left[ N \right]}}\left( {t,\omega } \right)=\int_{R} {{V_{2,f}}\left( {t,\eta } \right)\delta \left( {\omega - \omega _{{2,f}}^{{\left[ N \right]}}\left( {t,\eta } \right)} \right)} d\eta$$
(21)

The mode is finally reconstructed by replacing \({T_f}^{{\left[ N \right]}}\left( {t,\omega } \right)\) by \({T_{2,f}}^{{\left[ N \right]}}\left( {t,\omega } \right)\) in Eq. (11).

$${f_k}\left( t \right) \approx \frac{1}{{2\pi g\left( 0 \right)}}\int_{{\left| {\omega - {\varphi _k}\left( t \right)} \right|<d}} {{T_{2,f}}^{{\left[ N \right]}}\left( {t,\omega } \right)} d\omega$$
(22)

Synthetic examples

In this section, we utilize two numerical signals to compare the performance of the STFT, FSST, MSST and FMSST2 methods in time-frequency energy concentration.

Synthetic signal 1

The first numerical example is a signal consisting of two components with a sampling frequency of 100 Hz and time of 4 s (See Fig. 1(a)), which was previously used by Yu et al.40.

$$\begin{gathered} {s_1}\left( t \right)=\left( {1 - 0.6{e^{ - 0.3t}}} \right) \times \sin \left( {2\pi \left( {38t+3\sin \left( {1.5t} \right)} \right)} \right) \hfill \\ {s_2}\left( t \right)={e^{ - 0.4t}} \times \sin \left( {2\pi \left( {15t+2\sin \left( {4.5t} \right)} \right)} \right) \hfill \\ s\left( t \right)={s_1}\left( t \right)+{s_2}\left( t \right) \hfill \\ \end{gathered}$$
(23)

Figure 2 shows the TFR results of synthetic signal 1 after applying the STFT, FSST, MSST and FMSST2 methods, and the partially enlarged results (See magenta rectangles) are depicted in Fig. 3. It can be found that the STFT has the lowest time-frequency resolution, and the energy of TFR smears around IF trajectories (See Fig. 2(a) and Fig. 3(a)). In comparison, the TFR results produced by the FSST and MSST show more concentrated energy than that from the STFT (See Fig. 2(b) and Fig. 2(c)). From zoomed view, we can observe that the FSST result seems to smear heavily (See Fig. 3(b)), which means that the MSST is more suitable for characterizing the strongly time-varying signal than the FSST (See Fig. 3(c)). The synthetic signal 1 processed by the FMSST2 method is displayed in Fig. 2(d), it is obvious that the FMSST2 method achieves a highly energy-concentrated TFR (See Figs. 2(d) and 3(d)), which is more helpful in characterizing non-stationary signals.

In order to further evaluate the FMSST2 method to address noisy signal, we generate a noisy synthetic signal 1 (Fig. 1(b)) with a signal-to-noise ratio (SNR) of 4 dB by adding Gaussian noise to the original synthetic signal 1 (Fig. 1(a)). Figure 4 plots the corresponding TFRs of the STFT, FSST, MSST and FMSST2 methods in the presence of noise, and the partially enlarged results (See magenta rectangles) are in Fig. 5. As can be clearly seen, in the STFT result, the noise has a great influence on the time-varying feature (See Fig. 4(a)), so that the IF trajectories are not well described. In contrast, in the results of the other three methods, the FSST, MSST and FMSST2, the time-frequency energy is greatly enhanced. However, the FMSST2 methods gives the sparsest result among all the methods (See Fig. 4(d) and Fig. 5(d)).

Furthermore, we also show the reconstructed signal (See Fig. 6) by the FMSST2 method for synthetic signal 1. It can be seen that the reconstructed signal is highly consistent with the original signal 1, which validates the superior reconstruction performance of the FMSST2 method.

The Renyi entropy is an indicator used to evaluate the energy concentration of a TFR result, and a lower Renyi entropy value means a more concentrated TFR result41,42. Here, Renyi entropy is employed to quantitatively evaluate the performance of each TFA method. For the synthetic signal 1 and its noisy version, the corresponding Renyi entropies are listed in Table 1, respectively. It is obvious that the FMSST2 method provides the most concentrated TFR result among all of the four TFA methods. The Renyi entropy is defined by

$$I= - \frac{1}{2}{\log _2}\left( {\frac{{\iint_{{{R^2}}} {{{\left| {T\left( {t,\omega } \right)} \right|}^3}d\omega dt}}}{{\iint_{{{R^2}}} {\left| {T\left( {t,\omega } \right)} \right|d\omega dt}}}} \right)$$
(24)

where \(T\left( {t,\omega } \right)\) denotes the TFR result.

Fig. 1
figure 1

(a) A synthetic signal comprised of two components, (b) the synthetic signal contaminated with random noise.

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Fig. 2
figure 2

The TFR results of the synthetic signal 1 by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Fig. 3
figure 3

The zoomed TFR results of the synthetic signal 1 in magenta rectangles in Fig. 2 by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Fig. 4
figure 4

The TFR results of the noisy synthetic signal 1 by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Fig. 5
figure 5

The zoomed TFR results of the noisy synthetic signal 1 in magenta rectangles in Fig. 4 by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Fig. 6
figure 6

The original synthetic signal 1 and the reconstructed synthetic signal 1.

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Table 1 Renyi entropies by several TFA methods for synthetic signal 1.
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Synthetic signal 2

The second synthetic signal consists of a 20 Hz Ricker wavelet at 0.1s, a 50 Hz Ricker wavelet with 270°rotated phase at 0.5s, and three 30 Hz Ricker wavelets which are closely spaced at 0.8s, recorded with 2ms sampling interval and 500 time samples (Fig. 7(a)). Results of the STFT, FSST, MSST and FMSST2 methods for synthetic signal 2 are displayed in Fig. 8. The STFT shows the poor energy concentration and a low-resolution time-frequency map (See Fig. 8(a)). The FSST methods exhibits the better energy concentration and higher time-frequency resolution compared to the STFT method, but the energy is still dispersed along the frequency (See Fig. 8(b)). Both the MSST and FMSST2, by contrast, perform better in terms of energy concentration. Nevertheless, the MSST methods cannot well characterize the second Ricker wavelet (See Fig. 8(c)). Compared to the previous methods, the FMSST2 method depicts the highest resolution both in time and frequency directions (See Fig. 8(d)), which makes it better at characterizing seismic signals.

To test the noise robustness of the FMSST2 method, Gaussian noise is added to the original synthetic signal 2, and the noise contaminated signal with a SNR of 4 dB is shown in Fig. 7(b). TFR results of applying the STFT, FSST, MSST and FMSST2 methods are displayed in Fig. 9. As could be observed, the energy is dispersed and the resolution is low both in time and frequency in the result of the STFT (See Fig. 9(a)). Compared to the STFT method, the FSST and MSST methods, illustrated in Fig. 9(b) and Fig. 9(c), provide higher resolution in the time-frequency map. However, the energy divergence can still be found, especially in the result of FSST method. The higher resolution both in time and frequency is achieved for the TFR by the FMSST2 method in Fig. 9(d). Therefore, it is concluded that the FMSST2 method is more robust to noise.

In addition, the Renyi entropies for the STFT, FSST, MSST and FMSST2 methods are presented in Table 2 for the synthetic signal 2 and its noisy signal, respectively. It indicates the better performance of the FMSST2 method compared to other methods.

Fig. 7
figure 7

(a) A synthetic seismic signal composed of several Ricker wavelets, (b) the synthetic seismic signal contaminated with random noise.

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Fig. 8
figure 8

The TFR results of the synthetic signal 2 obtained by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Fig. 9
figure 9

The TFR results of the noisy synthetic signal 2 obtained by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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Table 2 Renyi entropies by several TFA methods for synthetic signal 2.
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Field examples

In this section, the FMSST2 method is applied on a real seismic data containing a gas-filled reservoir (Fig. 10). This data includes 100 seismic traces with 2 ms sampling interval and 150 samples in each trace. First, we extract a single trace 55 from 2D seismic section, plotted in Fig. 11, which is characterized by a strong amplitude around 0.64 s. Figure 12 shows the TFR results of the trace 55 corresponding to the STFT, FSST, MSST and FMSST2 methods. As can be clearly seen, all TFR results have similar energy variation, that is, the energy intensity increases with time, and a strong time-frequency energy appears at around 0.64 s. It should be noted that the STFT method suffers from a poor resolution both in time and frequency directions, thus, it cannot provide the better delineation of strong spectral energy in the time-frequency map (See Fig. 12(a)). Owing to the improved time-frequency resolution, the methods, FSST and MSST, generate the relatively sparse TFR results and show more detailed time-frequency characteristics compared to the STFT method (See Fig. 12(b) and Fig. 12(c)). However, among all methods, the FMSST2 method has the most concentrated energy and the highest time-frequency resolution (See Fig. 12(d)), which can help us extract richer time-frequency information, thus facilitating seismic data processing and interpretation.

Fig. 10
figure 10

2D time migrated seismic section indicating the gas-filled reservoir.

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Fig. 11
figure 11

A seismic trace from 2D time migrated seismic section in Fig. 10.

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Fig. 12
figure 12

The TFR results of a single trace signal shown in Fig. 11 obtained by (a) the STFT method, (b) the FSST method, (c) the MSST method, and (d) the FMSST2 method.

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When seismic waves propagate through a gas-bearing reservoir, the loss of energy for all frequency components of seismic signals will occur. Moreover, the high frequencies are attenuated more seriously than the low frequencies. This energy distribution of frequency content leads to the formation of low-frequency shadow zone, which has been regarded as a potential hydrocarbon indicator43,44,45. Now, we apply the STFT, FSST, MSST and FMSST2 methods to the 2D stacked seismic data, and extract the 15 Hz and 35 Hz frequency slices, respectively. The constant frequency slices obtained by the STFT, FSST, MSST, and FMSST2 at, respectively, 15 Hz and 35 Hz are displayed in Fig. 13. The STFT and FSST methods show the rough spectral characteristics due to lower time-spatial resolution, while the MSST and FMSST2 methods produce much sparse results that can characterize the spectral characteristics in detail. As shown in Fig. 13, a strong spectral energy appears at 15 Hz frequency slice, then the energy reduces with frequency increasing (35 Hz). As it is observed, the spectral energy zone is better delineated by the FMSST2 method with highly energy-concentrated frequency slices and higher time-spatial resolution (See Fig. 13(g) and Fig. 13(h)), which improves identification of the low-frequency shadow well. Illustration of frequency anomalies on seismic section demonstrates that the FMSST2 method could be considered as an alternative for detection of hydrocarbon.

Fig. 13
figure 13

2D frequency slices of applying (a) the STFT method for frequency of 15 Hz, (b) the STFT for frequency of 35 Hz, (c) the FSST method for frequency of 15 Hz, (d) the FSST for frequency of 35 Hz, (e) the MSST method for frequency of 15 Hz, (f) the MSST for frequency of 35 Hz, (g) the FMSST2 method for frequency of 15 Hz, and (h) the FMSST2 for frequency of 35 Hz.

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Conclusion

In this paper, we introduce a high-resolution time-frequency analysis method, FMSST2, for seismic data analysis. It makes full use of an iterative strategy that reassigns the time-frequency coefficients to raise the energy concentration of the TFR based on a second-order signal model. The FMSST2 method can not only achieve a highly energy-concentrated time-frequency representation, but also reconstruct the original signal with a high accuracy. Two numerical examples, including a strongly time-varying signal and a synthetic seismic record, reveal that the FMSST2 is more accurate and provides more sparser time-frequency map compared to traditional methods such as STFT, FSST and MSST. It should be noted that the FMSST2 exhibits several advantages such as robustness against noise, energy concentration and more interpretable time-frequency information. However, one should be aware of computational cost since it involves executing multiple STFT operations using a second-order signal model. Therefore, improving the computational efficiency of the FMSST2 method is a future research direction. The low frequency shadow zone in seismic data is an indication of gas-filled reservoir. The high-resolution TFR makes it possible to accurately describe the boundaries of the low frequency shadow zone. The FMSST2 is applicable to extract accurate frequency slices from seismic data with high precision and the lowest energy dispersion. Application on field data containing a gas-filled reservoir further approves its capability in identification of low frequency shadow zone, which makes this technique more promising in the field of seismic data processing and interpretation.