L^p-Order Time–Frequency Spectra and Its Application in Edge Detection

Abstract

Generalized Hilbert transform (GHT) has been widely and successfully used in signal processing and interpretation for many years. However, the order parameter in GHT can only be chosen in a certain range, and the GHT result is not stable with respect to the chosen order parameter, which usually leads to poor-stability and low-resolution outputs. To improve the stability of GHT and broaden its order parameter chosen range, and finally achieve high-efficiency and high-resolution results, we extend GHT by introducing the sign function, the two frequency integration parameters, and propose the L^p-order time–frequency spectra (LTFS). In LTFS, the sign function aims to broaden the chosen range of the order parameter into any positive numbers, and then leads to high-stability and high-resolution LTFS results; the two frequency integration parameters aim to optimize the computation efficiency and output resolution of LTFS. Synthetic one-dimensional (1D) channel data testing, synthetic two-dimensional (2D) seismic data, and actual three-dimensional (3D) seismic data examples demonstrate favorable capability of LTFS for edge detection.

1. Introduction

The generalized Hilbert transform (GHT) [1] improves the noise robustness and resolution of the Hilbert transform (HT) [2], by introducing the order parameter (a real number) to the time–frequency spectra, which are obtained by time–frequency analysis methods such as the short-time Fourier transform (STFT) [3], wavelet transform [4,5,6], S transform [7,8], generalized S transform [9], EMD [10,11,12], VMD [13], modified short-time Fourier transform [14,15,16], synchrosqueezing transform [17,18,19,20], and so on.

Due to the good performance of noise robustness and resolution, GHT has been widely used in signal processing and interpretation. However, GHT results depend on the time–frequency spectra and the order parameter; and for a chosen order parameter, the output resolution of GHT is unique, which limits its application. In addition, the order parameter in GHT can only be chosen in a certain range; and the GHT result is not stable with respect to the chosen order parameter, which usually leads to poor-stability and low-resolution output.

In order to improve the stability of GHT and broaden its order parameter chosen range, thus obtain high-stability and high-resolution results; we first extend GHT by introducing the sign function, and then propose the high-stability and high-resolution GHT (SGHT). In SGHT, due to the sign function, the order parameter can be chosen as any positive numbers. Thus, we can obtain high-stability and high-resolution GHT results. In addition, the frequency integral interval in GHT and SGHT is fixed, which limits the corresponding computation efficiency and output resolution. In order to improve the computation efficiency and output resolution of GHT and SGHT, and finally obtain high-efficiency and high-resolution results, we extend SGHT by introducing the two frequency integration parameters, and then propose the L^p-order time–frequency spectra (LTFS). In LTFS, we can flexibly choose the order parameter and the two frequency integration parameters, and finally optimize the computation efficiency and output resolution of LTFS for more general application.

The remainder of this paper is organized as follows. First, we introduce the principle of GHT and LTFS. Then, we use synthetic two-dimensional (2D) seismic data to test the capability of LTFS for edge detection, with the comparison of HT and GHT. Finally, we apply LTFS to actual three-dimensional (3D) seismic data, with comparison of HT, GHT, curvature [21,22], and coherence [22,23].

2. Generalized Hilbert Transform (GHT)

GHT extends HT (Appendix A) by introducing the order parameter in the time–frequency domain to enhance the noise robustness and resolution of HT, which make GHT easier for new applications than HT.

2.1. Theory of GHT

If complex numbers $S(t,f)$ denote the time–frequency spectra of the inputted real signal $x(t)$, which are obtained by the short-time Fourier transform (STFT) [3], wavelet transform [4,5,6], S transform [7,8], generalized S transform [9], EMD [10,11,12], VMD [13], modified short-time Fourier transform [14,15,16], synchrosqueezing transform [17,18,19,20], and so on; then, $S(t,f)$ can be modified as

$$R(t,f)=\left\{\begin{array}{cc}\begin{array}{l}2S(t,f)\\ S(t,f)\end{array}& \begin{array}{l}f>0\\ f=0\end{array}\\ 0& f<0\end{array}\right.$$

(1)

and further considering two real numbers $S_1(t,f)$ and $S_2(t,f)$ as the real and imaginary parts of $R(t,f)$, respectively, then, we have

$$\left\{\begin{array}{l}{S}_1(t,f)=\mathrm{Re}\left[R(t,f)\right]\\ S_2(t,f)=\mathrm{Im}\left[R(t,f)\right]\end{array}\right.$$

(2)

where $\mathrm{Re}\left[ \right]$ and $\mathrm{Im}\left[ \right]$ takes the real and imaginary parts of a complex number, respectively.

Thus, the GHT result of $x(t)$ can be expressed as

$$g(t)=g_3(t)+i\cdot g_4(t)$$

(3)

and

$$\left\{\begin{array}{l}{g}_1(t)={\int }_0^{f_N}{\left[S_1(t,f)\right]}^{p}df\\ g_2(t)={\int }_0^{f_N}{\left[S_2(t,f)\right]}^{p}df\\ g_3(t)={\left[g_1(t)\right]}^{p^{-1}}\\ g_4(t)={\left[g_2(t)\right]}^{p^{-1}}\end{array}\right.$$

(4)

where $f_N$ is the Nyquist frequency [24,25,26]; $p$ is the order parameter and $p>0$.

Therefore, the instantaneous amplitudes, phases, and frequencies of the GHT result $g(t)$ can be obtained by

$$\left\{\begin{array}{l}\tan \left[{\theta }_1(t)\right]=g_3{}^{-1}(t)\cdot g_4(t)\\ f_1(t)=\frac{1}{2\pi }\frac{\partial {\theta }_1(t)}{\partial t}\\ A_1(t)=\sqrt{g_3{}^2(t)+g_4{}^2(t)}\end{array}\right.$$

(5)

and $A_1(t)$, ${\theta }_1(t)$, and $f_1(t)$ are referred to as the instantaneous amplitudes, phases, and frequencies of $x(t)$ obtained by GHT, respectively.

2.2. Analysis of GHT

According to Equations (1)–(4), the GHT result $g(t)$ depends on the time–frequency spectra $S(t,f)$ and the order parameter $p$. For a chosen order parameter, the output resolution of GHT is unique, which limits its application. As $S(t,f)$ is related to the time–frequency analysis methods [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20], here we only analyze the impact of the chosen order parameter $p$ on the GHT result $g(t)$.

If $c$ denotes the collection of positive numbers, then it can be expressed as

$$c=\left\{\begin{array}{c}\begin{array}{c}\left(2k_1-1\right)\cdot {\left(2k_2\right)}^{-1},\begin{array}{l}2k_1\cdot {\left(2k_2-1\right)}^{-1}\hfill \end{array}\end{array},\begin{array}{l}\left(2k_1-1\right)\cdot {\left(2k_2-1\right)}^{-1}\hfill \end{array}\end{array}\right\}$$

(6)

where { } takes the number collection; $k_1$ and $k_2$ are positive integer numbers; if $A$ and $B$ both are real numbers, and

$$B=A^c=\left\{\begin{array}{l}\sqrt[(2k_2)]{A^{(2k_1-1)}}\\ \sqrt[(2k_2-1)]{A^{(2k_1)}}\\ \sqrt[(2k_2-1)]{A^{(2k_1-1)}}\end{array}\right.$$

(7)

then, the following four conclusions are drawn:

(i) For $A<0$, then $B=\sqrt[(2k_2)]{A^{(2k_1-1)}}$ does not exist. Therefore, when $S_n(t,f)<0$ or $g_n(t)<0$ ($n=\left\{1,2\right\}$), $p$ cannot be chosen as $\left(2k_1-1\right)\cdot {\left(2k_2\right)}^{-1}$.

(ii) For $A<0$, then $B=\sqrt[(2k_2-1)]{A^{(2k_1)}}>0$. Therefore, when $S_n(t,f)<0$ or $g_n(t)<0$ and $p$ is chosen as $2k_1\cdot {\left(2k_2-1\right)}^{-1}$, the signs of $S_n(t,f)$ and $g_n(t)$ are changed and GHT obtains instability result (as shown in Figure 1d).

(iii) For $A<0$, then $B=\sqrt[(2k_2-1)]{A^{(2k_1-1)}}<0$. Therefore, when $S_n(t,f)<0$ and $g_n(t)<0$, $p$ can be chosen as $\left(2k_1-1\right)\cdot {\left(2k_2-1\right)}^{-1}$; and correspondingly, GHT preserves the signs of $S_n(t,f)$ and $g_n(t)$.

(iv) For $A\ge 0$, then $B=A^c\ge 0$. Therefore, when $S_n(t,f)\ge 0$ and $g_n(t)\ge 0$, $p$ can be chosen as any number; and correspondingly, GHT preserves the signs of $S_n(t,f)$ and $g_n(t)$.

Above all, the order parameter $p$ can be only chosen as $\left(2k_1-1\right)\cdot {\left(2k_2-1\right)}^{-1}$ and $2k_1\cdot {\left(2k_2-1\right)}^{-1}$; the GHT result with $p=2k_1\cdot {\left(2k_2-1\right)}^{-1}$ usually has a serious instability problem, which limits its further application.

2.3. Synthetic 1D Channel Data Testing

Here, we used one set of synthetic 1D channel data (as shown in Figure 1a) to analyze the GHT, with comparison of HT. The time–sample interval of the synthetic 1D channel data is one millisecond, thus, the Nyquist frequency f_N is 500 Hz.

Figure 1b shows the instantaneous amplitudes obtained by HT, which indicate the edges of the synthetic channel; however, the HT result shows lower resolution than we desired. Figure 1c shows the instantaneous amplitudes obtained by GHT with p = 1.0. Figure 1d shows the instantaneous amplitudes obtained by GHT with p = 0.8 (ratio of an even number 4 and an odd number 5). Figure 1e shows the instantaneous amplitudes obtained by GHT p = 0.6 (ratio of two odd numbers 3 and 5).

We can see that when p = 1.0 and 0.6, GHT results effectively detected the edges of the synthetic channel and have a higher resolution than the HT result; whereas, GHT obtains a lower resolution or stability result with p = 0.8, due to the corresponding sign changing. Comparison of Figure 1c–e shows that GHT results are unstable with changing order parameter p.

3. L^p-Order Time–Frequency Spectra (LTFS)

LTFS aims to extend GHT by introducing the sign function, and two frequency integration parameters to produce high-stability and high-resolution results with high efficiency.

3.1. Theory of High-Stability GHT (SGHT)

In order to improve the stability of GHT and broaden its order parameter chosen range, and finally achieve high-stability and high-resolution GHT results, we can extend Equation (4) by introducing the sign function and propose the SGHT given by

$$s(t)=s_3(t)+i\cdot s_4(t)$$

(8)

and

$$\left\{\begin{array}{l}{s}_1(t)={\int }_0^{f_N}\mathrm{Sign}\left[S_1(t,f)\right]\cdot {\left|S_1(t,f)\right|}^{p}df\\ s_2(t)={\int }_0^{f_N}\mathrm{Sign}\left[S_2(t,f)\right]\cdot {\left|S_2(t,f)\right|}^{p}df\\ s_3(t)=\mathrm{Sign}\left[s_1(t)\right]\cdot {\left|s_1(t)\right|}^{p^{-1}}\\ s_4(t)=\mathrm{Sign}\left[s_1(t)\right]\cdot {\left|s_1(t)\right|}^{p^{-1}}\end{array}\right.$$

(9)

where || takes the absolute value of a real number; $\mathrm{Sign}\left[ \right]$ denotes the sign function.

Thus, instantaneous amplitudes, phases, and frequencies of $s(t)$ can be obtained by

$$\left\{\begin{array}{l}\tan \left[{\theta }_2(t)\right]=s_3{}^{-1}(t)\cdot s_4(t)\\ f_2(t)=\frac{1}{2\pi }\frac{\partial {\theta }_2(t)}{\partial t}\\ A_2(t)=\sqrt{s_3{}^2(t)+s_4{}^2(t)}\end{array}\right.$$

(10)

and $A_2(t)$, ${\theta }_2(t)$, and $f_2(t)$ are referred to as the instantaneous amplitudes, phases, and frequencies of $x(t)$ obtained by SGHT, respectively.

According to Equations (8) and (9), as

$$\left\{\begin{array}{l}\mathrm{Sign}\left[\mathrm{Sign}\left[S_n(t,f)\right]\cdot {\left|S_n(t,f)\right|}^p\right]=\mathrm{Sign}\left[S_n(t,f)\right]\\ \mathrm{Sign}\left[\mathrm{Sign}\left[s_n(t)\right]\cdot {\left|s_n(t)\right|}^p\right]=\mathrm{Sign}\left[s_n(t)\right]\end{array}\right.$$

(11)

thus, the order parameter $p$ in SGHT can be chosen as any positive real number, which means that

$$p=c=\left\{\begin{array}{c}\begin{array}{c}\left(2k_1-1\right)\cdot {\left(2k_2\right)}^{-1},\begin{array}{l}2k_1\cdot {\left(2k_2-1\right)}^{-1}\hfill \end{array}\end{array},\begin{array}{l}\left(2k_1-1\right)\cdot {\left(2k_2-1\right)}^{-1}\hfill \end{array}\end{array}\right\}$$

(12)

3.2. Theory of LTFS

Considering the amplitude spectra dynamic range [16,27,28], the actual signal belongs to the frequency band limited signals; and it means that amplitude spectra of actual signal ranges only in a certain frequency range, not from zero to the Nyquist frequency. Thus, in order to improve the computation efficiency and output resolution of SGHT, and finally obtain high-efficiency and high-resolution results; we extend SGHT by introducing the two frequency integration parameters, and then propose the LTFS given by

$$h_{f_1,f_2}(t)=h_3(t)+i\cdot h_4(t)$$

(13)

and

$$\left\{\begin{array}{l}{h}_1(t)={\int }_{f_1}^{f_2}\mathrm{Sign}\left[S_1(t,f)\right]\cdot {\left|S_1(t,f)\right|}^{p}df\\ h_2(t)={\int }_{f_1}^{f_2}\mathrm{Sign}\left[S_2(t,f)\right]\cdot {\left|S_2(t,f)\right|}^{p}df\\ h_3(t)=\mathrm{Sign}\left[s_1(t)\right]\cdot {\left|s_1(t)\right|}^{p^{-1}}\\ h_4(t)=\mathrm{Sign}\left[s_1(t)\right]\cdot {\left|s_1(t)\right|}^{p^{-1}}\end{array}\right.$$

(14)

where $f_1$ and $f_2$ both are the frequency integration parameters, and $0\le f_1\le f_2\le f_N$; for actual seismic data, usually $f_1=0$ Hz and $f_2\le 120$ Hz.

Correspondingly, instantaneous amplitudes, phases, and frequencies of $h_{f_1,f_2}(t)$ can be obtained by

$$\left\{\begin{array}{l}\tan \left[{\theta }_3(t)\right]=h_3{}^{-1}(t)\cdot h_4(t)\\ f_3(t)=\frac{1}{2\pi }\frac{\partial {\theta }_3(t)}{\partial t}\\ A_3(t)=\sqrt{h_3{}^2(t)+h_4{}^2(t)}\end{array}\right.$$

(15)

and $A_3(t)$, ${\theta }_3(t)$, and $f_3(t)$ are referred to as the instantaneous amplitudes, phases, and frequencies of $x(t)$ obtained by LTFS, respectively.

3.3. Analysis of LTFS

According to Equations (13) and (14), the output of LTFS depends on the time–frequency spectra, the sign function, the order parameter, and the two frequency integration parameters. A comparison of SGHT and LTFS shows that LTFS is a general representation of SGHT, and SGHT is the special case of LTFS. Thus, we can flexibly choose the order parameter and the two frequency integration parameters, to optimize the output resolution and computation efficiency of LTFS, respectively. Finally, high-stability and high-resolution results can be obtained with high efficiency.

If we compare LTFS results with $R(t,f)$, GHT, and SGHT results, then

$$\left\{\begin{array}{l}{h}_{f_1,f_2}(t)=R(t,f),\begin{array}{l}\mathrm{if}\begin{array}{l}{f}_1=f_2=f\hfill \end{array}\hfill \end{array}\\ h_{f_1,f_2}(t)=g(t),\begin{array}{l}\mathrm{if}\begin{array}{l}\begin{array}{l}p=\left(2k_1-1\right)\cdot {\left(2k_2-1\right)}^{-1},f_1=0,f_2=f_N\hfill \end{array}\hfill \end{array}\hfill \end{array}\\ h_{f_1,f_2}(t)=s(t),\begin{array}{l}\mathrm{if}\begin{array}{l}\begin{array}{l}{f}_1=0,f_2=f_N\hfill \end{array}\hfill \end{array}\hfill \end{array}\end{array}\right.$$

(16)

and if we consider we consider $T_1$, $T_2$, and $T_3$ as the time costs of GHT, SGHT, and LTFS, respectively; then, according to Equations (4), (9), and (14), we have

$$\left\{\begin{array}{l}{T}_2=T_1\\ T_1\cdot T_3{}^{-1}=f_N\cdot {\left(f_2-f_1\right)}^{-1}\end{array}\right.$$

(17)

Above all, SGHT has higher stability than GHT without computation efficiency decreasing, while LTFS has higher computation efficiency than SGHT without stability decreasing. Moreover, LTFS have higher stability and computation efficiency than GHT, and finally obtain high-stability and high-resolution results with high efficiency.

3.4. Synthetic 1D Channel Data Testing

Figure 2a shows the instantaneous amplitudes of the synthetic 1D channel data in Figure 1a, which are obtained by LTFS with p = 1.0, f₁ = 0 Hz, and f₂ = f_N = 500 Hz. Figure 2b shows the instantaneous amplitudes of the synthetic 1D channel data in Figure 1a, which are obtained by LTFS with p = 0.8, f₁ = 0 Hz, and f₂ = f_N = 500 Hz. Figure 2c shows the instantaneous amplitudes of the synthetic 1D channel data in Figure 1a, which are obtained by LTFS with p = 0.6, f₁ = 0 Hz, and f₂ = f_N = 500 Hz. LTFS has successfully detected the abrupt and gradual changes no matter which order parameter is chosen, and the LTFS results become more and more sharp with decreasing order parameter p.

According to Figure 1 and Figure 2, we can see that GHT and LTFS results with p = 1.0 are the same, while GHT and LTFS results with p = 0.6 are the same; in addition, LTFS results with p = 1.0 have much higher resolutions than GHT results with p = 1.0, which benefits from the sign function used in LTFS. Comparison of Figure 2a–c shows that the LTFS resolutions are increased with decreasing order parameter p.

4. Examples

In the three examples shown in this paper, the time–frequency spectra are generated by the modified short-time Fourier transform [14] given by

$$S(t,f)={\int }_{-\infty }^{+\infty }z(\tau +t)w(\tau )\exp (-i2\pi f\tau )d\tau$$

(18)

and

$$w(\tau )=\frac{a{\left|f\right|}^b}{\sqrt{2\pi }}\exp \left[-\frac{a^2{\left|f\right|}^{2b}}{2}{\tau }^2\right]$$

(19)

where $a>0$ and $b\ge 0$.

4.1. Synthetic 2D Seismic Data Example

For a comparative study of the LTFS with HT and GHT in seismic edge detection, one set of synthetic 2D seismic data (as shown in Figure 3a) with three horizontal reflections and one gradient reflection (the gradient is 1 millisecond per trace) is generated by a 20 Hz dominant frequency zero-phase Ricker wavelet. The time–sample interval is one millisecond, thus the Nyquist frequency is 500 Hz and the reflection coefficients of the four reflections are 1.0, −1.0, 1.0, and −1.0, respectively.

Figure 3b shows the instantaneous amplitudes obtained by HT; Figure 3c shows the instantaneous amplitudes obtained by GHT, with p = 0.4 (ratio of an even number 2 and an odd number 5). Figure 3d shows the instantaneous amplitudes obtained by LTFS, with p = 0.4, f₁ = 0 Hz, and f₂ = f_N = 500 Hz. Figure 3e shows the instantaneous amplitudes obtained by GHT with p = 0.2 (ratio of two odd numbers 1 and 5), which is the same as the LTFS result with p = 0.2, f₁ = 0 Hz, and f₂ = f_N = 500 Hz. Figure 3f shows the instantaneous amplitudes obtained by LTFS, with p = 0.2, f₁ = 10 Hz, and f₂ = 100 Hz.

For more clear comparison, we extracted the corresponding results in the yellow rectangles in Figure 3. Figure 4a–f shows the zoomed-in views of yellow rectangles in Figure 3a–f, respectively; and we can see that Figure 4a–f clearly detected the edges between the second horizontal layer and the gradient layer when trace numbers are greater than 40, 42, 34, 24, 21, and 21, respectively. In contrast, the GHT result in Figure 4c shows strong artifacts (or instabilities) due to the sign changing during GHT. A comparison of Figure 4b–f indicates that Figure 4e,f obtains higher-resolution results than Figure 4b–d and Figure 4e,f has no obvious differences.

Table 1. Time costs of GHT and LTFS results in Figure 3e,f.

Methods	GHT with p = 0.2 (or LTFS with p = 0.2, f1 = 0 Hz, and f2 = 500 Hz)	LTFS with p = 0.2, f1 = 10 Hz, and f2 = 100 Hz
Time costs (s)	T1 = 4.21	T3 = 0.77
Ratios of time costs	T1/T3 = 5.47

shows the time costs of GHT and LTFS results in Figure 3e,f. Figure 4f has higher computation efficiency than Figure 4e due to the chosen frequency integration parameters f₁ = 10 Hz and f₂ = 100 Hz.

Above all, LTFS have higher resolution and computation efficiency than GHT, and finally obtain high-efficiency and high-resolution results.

4.2. Actual 3D Seismic Data Example

In order to better use the actual 3D seismic data for geologic body detection by LTFS, a four-step procedure was followed. First, HT was applied to the input actual 3D seismic data in the time (or depth) direction to generate its analytical representation. Second, MSTFT was applied to the analytical representation of the input actual 3D seismic data in the time direction to generate the time–frequency spectra. Third, the 3D instantaneous amplitudes obtained by HT, GHT, and LTFS were calculated. Finally, time slices of the instantaneous amplitudes obtained by HT, GHT, and LTFS were extracted for comparison.

Figure 5a,b shows the seismic data of inline 160 and crossline 96 extracted from the input actual 3D seismic data, respectively. The time–sample interval is one millisecond, thus the Nyquist frequency is 500 Hz.

Figure 6a shows a time slice of 1.7 s extracted from the input actual 3D seismic data. Figure 6b shows the same time slice of instantaneous amplitudes obtained by HT. Figure 6c shows the same time slice of instantaneous amplitudes obtained by GHT with p = 6 (an even number). Figure 6d shows the same time slice of the instantaneous amplitudes obtained by LTFS with p = 6, f₁ = 0 Hz, and f₂ = 90 Hz. For comparison, the same time slices of curvature and third-generation coherence (C3) attributes used for characterizing the channels are shown as Figure 6e,f, respectively.

Figure 7a–f shows the time slices in Figure 6a–f, respectively, and with two interpreted channels (shown as the two yellow curves). Comparison of Figure 7b with Figure 7a,c–f indicates that HT shows the main features of the two channels, but lacks the details obtained by GHT, LTFS, most positive curvatures, and third-generation coherences. A comparison of Figure 7d with Figure 7a–c,e,f indicates that Figure 7d is consistent with the two interpreted channels, and exhibits higher resolution and more detail than Figure 7a–c,e,f.

5. Conclusions

In this paper, we propose the LTFS to extend GHT by introducing the sign function, the two frequency integration parameters. The sign function broadens the chosen range of the order parameter into any positive numbers and leads to high-stability and high-resolution results. The two frequency integration parameters not only control the computation efficiency of LTFS, but also control its output resolution. Thus, we can flexibly choose the order parameter and two frequency integration parameters to optimize the computation efficiency and output resolution of LTFS. Synthetic 1D channel data testing, synthetic 2D seismic data, and actual 3D seismic data examples reveal that LTFS has higher resolution than GHT and can be widely used in signal processing and interpretation.