The extraction of instantaneous frequency from ...

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The extraction of instantaneous frequency from seismic data based on the GMWs Ping Wang* and Jinghuai Gao, Xi’an Jiaotong University Summary In this paper, a robust method for the extraction of instantaneous attributes is proposed in wavelet domain. A new class of analytic wavelets, the Generalized Morse Wavelets (GMWs), which have some desirable properties, are applied during the procedure of the proposed method. Compared to the conventional method based on Hilbert transform (HT), the new method is proved to yield higher precision and better anti-noise performance. Experimental results on synthetic signals and real seismic data show the validity of the method.

wavelets, the Generalized Morse Wavelets (GMWs) are applied. Comparisons between the proposed method and HT method are made by calculating IF of the signals. The results on test signals showed higher precision and antinoise performance of the proposed method. Examples from field data demonstrate the excellent performance of the method. The traditional IF estimation method

x ( t ) , the corresponding analytic

For a real-valued signal signal, denoted as

z (t )

can be defined as follows

z ( t ) = x ( t ) + ih ( t )

Introduction Instantaneous frequency (IF), defined as the derivative of the phase of a signal, is an important concept in signal processing that occur in the context of the representation and analysis of non-stationary signal. The IF has been widely used in many different fields, such as seismic, radar, sonar, communications and biomedical applications. As one of the best known seismic attributes, the IF has been found successful applications in subsurface structure analysis (Chopra and Marfurt, 2005) and parameter inversion such as quality (Q) factor (Yang and Gao, 2010).

where the imaginary part

Since Tanner et al. (1979) introduces complex seismic trace analysis in geophysical data processing, a multiuse of applications was found during the development of seismic attributes, and instantaneous attributes was calculated directly from the complex seismic trace. The classical definition of IF, that IF is the derivative of the instantaneous phase of an analytic signal, is proposed by Ville (1948). The analytic signal is composed by a real part and an imaginary part. The real part is the real-valued signal itself while the imaginary part is the Hilbert transform of the real one (Gabor, 1946). However, this method is sensitive to noise and may lead to unphysical values such as negative frequencies (White, 1991), thus it brings difficulty for seismic attributes analysis, especially in a noisy environment.

where f

Although the concept of the instantaneous frequency has been introduced for decades, improving the estimation precision, investigating the physical interpretation, and extending its applications are still active areas of research. In this paper, we present a wavelet-based method using wavelet transform (WT) for the calculation of the analytic counterpart of a real-valued signal and its instantaneous frequency. During the procedure, a new class of analytic

© 2012 SEG SEG Las Vegas 2012 Annual Meeting

h (t )

(1)

is the Hilbert transform of

x ( t ) . Then, the instantaneous attributes are defined as

w (t ) = f ' ( t ) = =

h (t ) d [arctan[ ]] dt x(t )

x (t ) h ' (t ) - x ' (t ) h (t )

(2)

x 2 (t ) + h 2 ( t )

( t ) and w ( t ) represent the instantaneous phase (IP) and instantaneous frequency (IF) of x ( t ) , respectively. Note that w ( t ) is an angular frequency. In a linear algebra notion (Fomel, 2007), where

w

w = D -1n represents

the

vector

of

(3) instantaneous

w ( t ) , n represents the numerator in equation 2, and D is a diagonal operator made from the frequencies

denominator of equation 2. A recipe for avoiding division by zero is adding a small constant e to the denominator (Matheney and Nowack, 1995). Consequently, equation 3 transforms to

winst = ( D + e I ) n -1

(4)

where I stands for the identity operator. Stabilization by e does not, however, prevent instantaneous frequency from being susceptible by noise. Meanwhile, the IF

DOI http://dx.doi.org/10.1190/segam2012-0902.1 Page 1

The extraction of instantaneous frequency from seismic data based on the GMWs estimated from this method may contain physically unreasonable negative values.

where


Continuous wavelet transform

1

ò a

+¥

-¥

where t Î R and s > 0 , conjugate of y

æ t -t x ( t )y * ç è a

y * (t )

ö ÷ dt ø

( t ) . The wavelet function y ( t ) is zero-

mean which is assumed to satisfy the admissibility condition (Holschneider, 1998): ¥

Ky º ò

yˆ (w )

where yˆ (w ) =

ò

¥

2

w

-¥

(6)

y ( t ) e- iwt dt is the Fourier transform

-¥

of the wavelet. The wavelet function is said to be analytic if yˆ (w ) = 0 for w < 0 .

x (t )

through

its wavelet coefficients C (a,t ) can be achieved using x y

the reconstruction formula:

c (t ) =

1 Ky

æ t - t ö dadt C x (a,t )y ç ÷ 2 -¥ y è a ø a

+¥

ò ò 0

+¥

(7)

[ a1 , a2 ] allows for the isolation

and identification of the processes with characteristic scale comprises between

a1

and

For a real-valued signal be derived that complex,

x (t )

a2 .

x ( t ) , when y ( t )

is real, it can

x ( t ) = c ( t ) . However, while y ( t ) is the real part of

c (t )

is

(Gao, 1999), i.e.:

x ( t ) = Re éë c ( t ) ùû

(8)

x2 (t ) + y 2 (t )

In a linear algebra notion, equation 4 transforms to

w 'inst = ( D '+ e I ) n ' -1

where

w 'inst

(11)

represents the vector of instantaneous

w ( t ) obtained by the proposed method, n ' the numerator in equation 10, and D ' is a

represents diagonal operator made from the denominator. The choice of wavelet

The result of the wavelet analysis of a signal is highly correlated to the choice of the wavelet. Particular attention, therefore, should be paid to wavelet determination. A commonly used complex-valued wavelet is the Morlet wavelet, which was first introduced by Goupillaud, Grossman and Morlet (1984), and given by

y s ( t ) = as e

- (1/ 2 ) t 2

[eis t - e

- (1/ 2 )s 2

]

(12)

yˆs (w ) = as e -(1/ 2 )(w -s ) [1 - e -ws ]

(13)

where s is the carrier wave frequency. The second term in (12) and (13) is a correction necessary to enforce zero mean, while as normalizes the wavelet amplitude. For

sufficiently large s , e.g. s > 5.33 , the values of the second term are so small that can be neglected. Meanwhile, the values of yˆs (w ) for w £ 0 are so small that y s can be considered as an analytic wavelet. The generalized Morse wavelets (GMWs) are a twoparameter family of wavelets, defined, in the frequency domain, by (Olhede and Walden, 2002)

yˆ b ,g (w ) = U (w ) a b ,g w b e -w

g

(14)

æ eg ö a b ,g º 2 ç ÷ èb ø

b /g

(15)

is a normalizing constant. To be a valid wavelet, one must have b > 0 and g > 0 .

(9)

In spite of its usefulness, the Morlet wavelet suffers from some limitations. On top of that, the Morlet wavelet is not,

obtained, which can be expressed as


(10)

( t ) , the signal x ( t ) is

By using CWT with analytic wavelet function y

c ( t ) = x ( t ) + iy ( t )

x (t ) y ' (t ) - x ' (t ) y (t )

where U (w ) is the Heaviside step function and

The calculation of IF

analytic counterpart of the real-valued

y (t ) d [arctan[ ]] dt x (t )

2

The integration over the whole scale interval allows for the reconstruction of the signal. The integration over a restricted scale interval

Then, the

frequencies

dw < ¥

The synthesis of the continuous time signal

=

(5)

is the complex

c (t ) .

is the imaginary part of

w (t ) =

The continuous wavelet transform (CWT) of a signal x ( t ) Î L2 ( ¡ ) is defined as

Cyx ( a,t ) =

y (t )

instantaneous frequency is computed as follows:



The extraction of instantaneous frequency from seismic data based on the GMWs however, exactly analytic -- it is only approximately analytic for sufficiently large s . The advantage of using precisely, as opposed to approximately, analytic wavelets such as the generalized Morse wavelets was demonstrated by S.C. Olhede, et al. (2003), who showed that even small amounts of leakage to negative frequencies can result in spurious variation of the transform phase. If we narrow the time window of both wavelets in order to increase time resolution, the Morlet wavelet exhibits leakage to negative frequencies while the Morse wavelets remain analytic even for highly time-localized parameter settings. Furthermore, the Morlet wavelet depend on just one parameter, implying that it is not very versatile, because the correction term cannot be neglected for s < 5.33 , the parameter choices are very restricted. By varying two parameters, the GMWs can be given a broad range of characteristics while remaining exactly analytic. Although it is true that the Morlet Wavelets has optimal joint timefrequency concentration in the Heisenberg sense, it is also true that there are other criteria available. “The whole set of generalized Morse wavelets are optimally localized in that they maximize the eigenvalues of a joint timefrequency localization operator (……) and indeed this is the way the generalized Morse wavelets were initially constructed." (Lilly and Olhede, 2009)

(Figure 1a) is a synthetic chirp function with linearly varying frequency. The second signal (Figure 1b) is a piece of a synthetic seismic trace obtained by convolving a 50Hz Ricker wavelet with synthetic reflectivity. The last test example (Figure 1c) is a real trace extracted from a seismic image.

Figure 1: Test signals. (a) Synthetic chirp signal with linear frequency change. (b) Synthetic seismic trace from convolution of a synthetic reflectivity with a Ricker wavelet. (c) Real seismic trace from a marine survey.

It is well known that HT method is sensitive to noise. On the other hand, when a noisy signal is transformed into time-scale domain by using a proper wavelets, the energy distribution of the useful signal will be confined in a small close subspace V of the time-scale domain; while the energy distribution of the noise will disperse in a larger close subspace V ¢ (even in the whole time-scale space). The two subspace, V and V ¢ , may be separated or partly overlapped. No matter how, it is obvious that in subspace V , the noise is suppressed and the signal-to-noise ratio will be improved.

Figure 2: The IF of test signals from Figure 1, calculated by traditional HT method.

The closer the selected wavelets are to the true pattern of the input data, the more concentrated the energy distribution in the time-scale domain. So, when calculating the instantaneous frequency, we should first choose an analytic wavelet function with its real part similar (or closer) to the seismic wavelet, and then decompose the seismic data into time-scale space. By varying two parameters, the GMWs allow for more flexibility and can be selected so as to best match the data to be analyzed. The GMWs are used to calculate the analytic counterpart of a real-valued signal and its instantaneous frequency in the proposed method. Examples In order to illustrate the performance of the method, three test signals shown in Figure 1 are used. The first signal


Figure 3: The IF of test signals from Figure 1, calculated by the proposed method.



The extraction of instantaneous frequency from seismic data based on the GMWs The comparisons between the traditional method and the proposed approach are given. Figure 2 shows the IF calculated by traditional HT method and Figure 3 shows the IF calculated by the proposed approach using the GMWs with b = 8 and g = 3 . These two methods appear indistinguishable, and the IF calculated by them are nearly identical. Both the two methods can give high precision of the IF estimate when there is no noise. However, when the observed signal is noisy, the IF obtained by the traditional method is inaccurate. Figure 4 shows the noisy signal by adding white Gaussian noise to the test signals shown in Figure 1. Figure 5 and Figure 6 shows the IF calculated by traditional HT and the proposed approach with the same parameters, respectively. It is seen that the IF calculated by the traditional method (Figure5) appears to be noisy and contain physically unreasonable negative values, while the proposed method can give a more accurate estimation of IF in noisy environment, compared to the traditional HT method.

Figure 4: The noisy test signals obtained by adding white Gaussian noise to the test signals shown in Figure 1

Figure 6: The IF of noisy test signals from Figure 4, calculated by the proposed method. To further confirm the effectiveness of the proposed method, we applied it to a stacked real 3D seismic data from CNOOC. Figure 7a shows a time slice of the 3D data, Figure 7b shows the IF obtained by traditional HT method, and Figure 7c shows the IF obtained by our method. It is clear that the abnormal region is more distinct in Figure 3c due to the better anti-noise performance of the proposed method.

Figure 7: Example on real data from CNOOC. (a) A time slice of the data set. IF calculated by (b) traditional HT, and (c) the proposed method. Conclusions In this paper, a wavelet-based method is presented for the instantaneous frequency estimation, and the GMWs were used during the procedure. The IF estimated by the proposed approach is robust to noise. The result on both synthetic signals and filed data show that the proposed method outperforms the traditional HT method. Acknowledgments

Figure 5: The IF of noisy test signals from Figure 4, calculated by traditional HT method.


We thank National Natural Science Foundation of China (40730424) and National Science & Technology Major Project (2011ZX05023-005) for their support.


http://dx.doi.org/10.1190/segam2012-0902.1


EDITED REFERENCES Note: This reference list is a copy-edited version of the reference list submitted by the author. Reference lists for the 2012 SEG Technical Program Expanded Abstracts have been copy edited so that references provided with the online metadata for each paper will achieve a high degree of linking to cited sources that appear on the Web. REFERENCES

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