Generalized stable inverse Q filtering

Journal of Applied Geophysics 169 (2019) 214–225

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Journal of Applied Geophysics journal homepage: www.elsevier.com/locate/jappgeo

Generalized stable inverse Q filtering Yan Zhao a,b,⁎, Ningbo Mao a,b, Jing Xu c a b c

Key Laboratory of Exploration Technologies for Oil and Gas Resources (Yangtze University), Ministry of Education, Wuhan, China Hubei Cooperative Innovation Center of Unconventional Oil and Gas, Yangtze University, Wuhan, China Research Institute of Petroleum Exploration and Development, Dagang Oilfield Company, CNPC, Tianjin, China

a r t i c l e

i n f o

Article history: Received 29 December 2018 Received in revised form 8 July 2019 Accepted 8 July 2019 Available online 11 July 2019 Keywords: Stable inverse Q filtering Generalized Compensation bandwidth Resolution Signal-to-noise ratio

a b s t r a c t The conventional stable inverse Q filtering method overcomes the instability of the inverse Q filtering and can effectively perform amplitude compensation and phase correction. However, its shortcoming is that it cannot solve the problem of excessive compensation for shallow seismic records and insufficient compensation for deep seismic records. The essence is that the compensation bandwidth and intensity of the amplitude compensation operator cannot be adjusted. We propose a generalized stable inverse Q filtering method. By introducing a parameter into the stable amplitude compensation operator based on stabilization factor, the method can flexibly and effectively handle the compensation intensity and frequency bandwidth to achieve high resolution and high signal-to-noise ratio seismic data. The test results of theoretical synthetic seismic data and real seismic data show that this method has unique value in frequency band selection of amplitude compensation according to the quality of seismic data, which can improve the resolution and get high signal-to-noise ratio seismic data. © 2019 Published by Elsevier B.V.

1. Introduction Inverse Q filtering is often used to perform amplitude compensation and phase correction for seismic data, and the predecessors have done a lot of research (Robinson, 1979, 1982; Kjartansson, 1979; Hale, 1981; Hargreaves and Calvert, 1991; Wang, 2002, 2003, 2006, 2008; James and Knight, 2003; Van der Baan, 2012 Braga and Moraes, 2013; Wang and Chen, 2014; Zhang et al., 2014, 2015, 2017a, 2017b). The stability and noise immunity of the amplitude compensation operator are two important problems of the inverse Q filtering algorithm. For stability issue, Wang (2006) proposed a stabilized inverse Q filtering method through adding a constant stabilization factor in amplitude compensation term. Liu et al. (2013) introduced an iterative method into inverse Q filtering and proposed a stable iterative inverse Q filtering. Zhang et al. (2013) introduced the regularization method for solving the ill-posed problem into the inverse Q filtering, and proposed an inverse Q filtering based on regularization method for the wavefield continuation. Zhang et al. (2017b) proposed a balanced filtering method, which based on phase stable inverse Q filtering, however, the method is sensitive to noise. For issue of noise immunity, Chen et al. (2014) combined a time-varying bandpass filter with stable inverse Q filtering to propose a band-limited stable inverse Q filtering method. However, the ⁎ Corresponding author at: 111 Daxue Road, Caidian, Wuhan, Hubei province 430100, China. E-mail address: [email protected] (Y. Zhao).

https://doi.org/10.1016/j.jappgeo.2019.07.007 0926-9851/© 2019 Published by Elsevier B.V.

shortcoming of this method is that the high cutoff frequency of the bandpass filter is a function of the Q value, and there is no direct relationship with the signal-to-noise ratio (SNR) of seismic data. Zhao et al. (2014) proposed an inverse Q filtering method for suppressing noise with the help of time-frequency domain SNR. Zhang et al. (2015) proposed a self-adaptive inverse Q filtering method. The method considered the dynamic range of seismic data and the time-varying gain limit is used to adapt to the cutoff angle frequency of the effective frequency band. It can effectively control the noise beyond the effective bandwidth of the seismic data. Wei and Chen (2017) improved the performance of inverse Q filtering by introducing the noise suppression coefficients. Zhao and Mao (2018) proposed an amplitude compensation operator based on varying stabilization factor, can effectively avoid the amplification of noise energy in shallow seismic records by inverse Q filtering. Wei et al. (2018) developed an improved stable inverse Q filtering algorithm by shaping regularization scheme and achieved noise suppression by controlling the frequency bandwidth. In this paper, we developed a generalized stable inverse Q filtering algorithm. By introducing parameters into the amplitude compensation term, it is possible to handle the bandwidth of the amplitude compensation, and perform amplitude compensation and phase correction for the seismic data more flexibly and effectively. The structure of this paper is as follows: Firstly, the basic principles of inverse Q filtering is reviewed. Secondly, a generalized stable inverse Q filtering algorithm is introduced and the characteristics of the amplitude compensation term are analyzed. Finally, the noisy synthetic and

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(c) Fig. 1. Operator Λλ(τ, ω) with different p values when λ is a constant stabilization factor. (a) p = 1. (b) p = 0.5. (c) p = 1.8.

real seismic records are used to further demonstrate its flexibility and superiority. It shows that the proposed method has unique value in frequency band selection and can better improve the SNR and resolution of seismic data.

U ðz þ Δz; ωÞ ¼ U ðz; ωÞ expðikΔzÞ; where i ¼

2. Theory of inverse Q filtering

2

ð1Þ

where U(z, ω) is the wavefield, z is distance, ω is radial frequency. k is

ð2Þ

pffiffiffiffiffiffiffiffi −1. The distance increment Δz can be replaced by

Δz ¼ vðω0 ÞΔτ;

Inverse Q filtering is based on 1D wave equation in frequency domain (Wang, 2002) ∂ U ðz; ωÞ 2 þ k U ðz; ωÞ ¼ 0; ∂z2

wave number. One of the solution of Eq. (1) given by

ð3Þ

where v(ω0) is the phase velocity at the dominant frequency ω0. Δτ is the traveltime increment. The earth Q effect is introduced in the definition of the wavenumber k: k¼

  ω i 1− ; vðωÞ 2Q

ð4Þ

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Amplitude

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Amplitude

(b)

Amplitude

(a)

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(c)

(d)

Fig. 2. Comparison of the amplitude compensation curves in Fig. 1 at different time samples. (a) t = 0.3 s. (b) t = 0.5 s. (c) t = 0.7 s. (b) t = 0.9 s.

where Q is the frequency independent quality factor, the phase velocity v(ω) defined by  γ ω vðωÞ ¼ vðω0 Þ  ; ω0 γ¼

  2 1 1 tan−1 ≈ ; π 2Q πQ

ð5Þ

ð6Þ

(Kjartansson, 1979). Then, the continuation equation of inverse Q filtering can be obtained by substituting Eqs. (3)–(6) into Eq. (2), which is  −γ    −γ  ω ωΔτ ω exp i U ðτ þ Δτ; ωÞ ¼ U ðτ; ωÞ exp ωΔτ ; ω0 2Q ω0

ð7Þ

U(τ + Δτ, ω) and U(τ, ω) are the wavefield at τ + Δτ and τ, respectively. 3. Generalized stable inverse Q filtering In the conventional stable inverse Q filtering algorithms, a stabilization factor is added to the denominator and the numerator of the amplitude compensation term, and the high-cutoff frequency of the amplitude compensation is gradually attenuated (Wang, 2006). In this paper, we propose a generalized stable inverse Q filtering, its theoretical expression can be written as U ðτ þ Δτ; ωÞ ¼ U ðτ; ωÞΛλ ðΔτ; ωÞΘðΔτ; ωÞ;

ð8Þ

where Λλ ðΔτ; ωÞ ¼

α p ðΔτ; ωÞ þ λ ; α 1þp ðΔτ; ωÞ þ λ

  −γ  ω ωΔτ ; α ðΔτ; ωÞ ¼ exp − ω0 2Q

ð9Þ

ð10Þ

  −γ  ω ΘðΔτ; ωÞ ¼ exp i ωΔτ : ω0

ð11Þ

Λλ(τ, ω) is the amplitude compensation term, Θ(τ, ω) is the phase compensation term. p is real numbers greater than 0. λ can be the constant stabilization factor, λ = exp (−0.23Glim + 1.63), in Wang (2006) or the varying stabilization factor, λ = mωnQ(t)/t, in Zhao and Mao (2018). Assume that the gain limit of the amplitude compensation function in Eq. (9) is C, which means the maximum value of Eq. (9). When λ is a constant stabilization factor, the relationship between the parameters p, λ and the gain limit C is   1þp   p 1 1þp 1þp ¼p C C ; −ð1 þ pÞ λ p p

ð12Þ

the detailed derivation process can be found in Appendix A. Compared with the traditional stable inverse Q filtering methods, the difference of this method is the introduction of the parameter p. When p = 1, the method degenerates into the conventional method. When p ≠ 1, the method will show some different characteristics. To show the superiority of the method, the amplitude compensation law as a function of p value are shown in Figs. 1 to 4. Figs. 1 and 2 show the operator Λλ(τ, ω) with different p values when λ is a constant stabilization factor, where Q = 25, Glim = 15. Fig. 1(a), (b) and (c) show the three dimentional stereogram of the amplitude compensation law when p = 1, 0.5, 1.8, respectively. We can see that when the p value is less than 1, the maximum value of the operator is increased, and the compensation band is also widened. Conversely, when the p value is greater than 1, the maximum value of the operator is reduced, and the compensation band is also narrow. Fig. 2 shows the amplitude compensation curves of the operators in Fig. 1 at different traveltime, wherein the

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(c) Fig. 3. Operator Λλ(τ, ω) with different p values when λ is a varying stabilization factor. (a) p = 1. (b) p = 0.5. (c) p = 1.8.

moments corresponding to Fig. 2(a), (b), (c) and (d) are 0.3 s, 0.5 s, 0.7 s and 0.9 s, respectively. It displays that the parameter p value affects the compensation bandwidth, high cutoff frequency and maximum compensation value. Figs. 3 and 4 show the operator Λλ(τ, ω) with different p values when λ is a varying stabilization factor, where Q = 25. Fig. 3(a), (b) and (c) show the three dimentional stereogram of the

amplitude compensation law when p = 1, 0.5, 1.8, respectively. Fig. 4 shows the amplitude compensation curves of the operators in Fig. 3 at different traveltime, wherein the moments corresponding to Fig. 4(a), (b), (c) and (d) are 0.3 s, 0.5 s, 0.7 s and 0.9 s, respectively. We can see that the effect of the change in p value on the amplitude compensation laws are similar to the case when λ is a constant stabilization factor.

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(b)

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(a)

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(c)

(d)

Fig. 4. Comparison of the amplitude compensation curves in Fig. 3 at different time samples. (a) t = 0.3 s. (b) t = 0.5 s. (c) t = 0.7 s. (b) t = 0.9 s.

From these figures, it can be seen that compared with the amplitude compensation operator in conventional methods, the operator Λλ(τ, ω) has strong flexibility, and the frequency bandwidth and intensity of the amplitude compensation can be effectively handled through parameter p according to the quality of seismic data. 4. Theoretical examples In this section, three examples are used to further illustrate the superiority of our method in this paper. Fig. 5 shows the results of generalized stable inverse Q filtering for noise-free synthetic records, in which λ is a constant stabilization factor. Fig. 5(a) displays the noise-free synthetic seismic data. The source is a Ricker wavelet with 35 Hz dominant frequency, and the time sampling interval is 0.002 s. The results of our method in the case of p = 1 and p = 0.2 are shown in Fig. 5(b) and (c). It can be seen from Fig. 5(b) that when the Q value is small, the amplitude of the deep seismic record is not satisfactory, which is also pointed out by Zhang et al. (2017b). This situation is improved when the p value is decreased, as shown in Fig. 5(c), the energy of the deep and shallow seismic records is substantially the same. Figs. 6 and 7 show the results of generalized stable inverse Q filtering for noise-added synthetic records, in which λ is a varying stabilization factor. The difference is in SNR of seismic records, as shown in Figs. 6(a) and 7(a). For weaker noise data, although the conventional method (p = 1) improves the quality of seismic records, the amplitude energy of the same deep seismic record is not satisfactorily compensated, as shown in Fig. 6(b). The result of our method (p = 0.4) is shown in Fig. 6(c), and we can see that the energy of the deep seismic record is compensated to a large extent, although it also partially amplifies the noise energy. For stronger noise data, the traditional method (p = 1) over-amplifies the noise energy and reduces the quality of the seismic record, as shown in Fig. 7(b).

We can choose to increase the p value in our method (p = 1.6) and appropriately reduce the amplitude and bandwidth of the compensation operator to achieve the purpose of both compensating the effective signal and suppressing the noise. The corresponding results are shown in Fig. 7(c). It can be seen from the above three examples that the method in this paper can flexibly select the parameter p in the operator Λλ(τ, ω) according to the quality of the seismic data, and handle the frequency bandwidth for the amplitude compensation to better improve the SNR and resolution of the seismic record after inverse Q filtering. 5. Real data examples In this part, we test the performance of our method with real seismic data, in which λ is a constant stabilization factor. In order to display the details of the amplitude compensation after inverse Q filtering, the seismic records are displayed with 300 ms automatic gain control (AGC). Fig. 8(a) shows the real seismic records. Without knowledge of the Q value, Q = 80 is set in the example. Fig. 8(b) and (c) show the results of our method with Glim = 3, p = 1 and Glim = 3, p = 0.4, respectively. It can be seen that the resolution of the seismic record in Fig. 8(b) is reduced compared to Fig. 8(a), while the resolution of the seismic record in Fig. 8(c) is improved. Magnified views of the yellow boxes are shown in Fig. 9, we can see that compared to Fig. 9(a), the events in Fig. 9 (b) becomes thicker and the resolution decreases, but the events in Fig. 9(c) becomes finer and more continuous, and the resolution is also effectively improved, especially the seismic image in the black ellipse. Fig. 10 shows the comparison of the normalized logarithmic amplitude spectra of the seismic records in Fig. 8, where blue line, red line and yellow line indicate the amplitude spectra of the seismic records in Fig. 8(a), (b) and (c), respectively. Fig. 10(a) shows the spectra comparison of the overall seismic records, and the time window range is 0–1000 ms. It can be seen that under the parameters selected in this

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example, the amplitude spectra energy of high frequency components is effectively compensated. Fig. 10(b), (c) and (d) show the spectra comparison of the seismic records with the time windows 0–320 ms, 320–640 ms and 640–900 ms, respectively. It can be seen that the high-frequency amplitude energy after the proposed inverse Q filtering method is effectively compensated in the spectra of the three

windowed seismic records, except for the part of high-frequency energy in Fig. 10(d). However, the high frequency component energy is obviously suppressed in the other two windowed seismic records after the conventional stable inverse Q filtering method, except the seismic records with 0–320 ms window. This further embodies the value of this method.

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6. Conclusion In this paper, a generalized stable inverse Q filtering is proposed. The method handles the frequency bandwidth by introducing a parameter into the amplitude compensation operator, so

that it can more flexibly compensate amplitude and correct phase for seismic data. The test results of theoretical seismic records and real seismic records show that the proposed method can improve the SNR and resolution more effectively according to the quality of seismic data.

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Fig. 10. Comparison of the normalized logarithmic amplitude spectra of the seismic records in Fig. 8, where blue line, red line and yellow line indicate the amplitude spectra of the seismic records in Fig. 8(a), (b) and (c), respectively. (a) time window 0–1000 ms. (b) time window 0–320 ms. (c) time window 320–640 ms. (d) time window 640–960 ms.

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Acknowledgement

C is a constant. If Λ'λ(Δτ, ω1) = 0, then

The research is jointly supported by the National Natural Science Foundation of China (41604094) and Open Fund of Key Laboratory of Exploration Technologies for Oil and Gas Resources (Yangtze University), Ministry of Education (K2018-13).

ε ¼ pB1þp ðΔτ; ω1 Þ−ð1 þ pÞBp ðΔτ; ω1 Þ:



Appendix A. Appendix

U ðτ þ Δτ; ωÞ ¼ U ðτ; ωÞΛλ ðΔτ; ωÞΘðΔτ; ωÞ;

ðA  1Þ

ðA  10Þ

Substituting Eq. (A-10) into Eq. (A-9), BðΔτ; ω1 Þ ¼

The theoretical expression of the proposed generalized stable inverse Q filtering can be written as

225

 1þp C: p

ðA  11Þ

Substituting Eqs. (A-4) and (A-11) into Eq. (A-10), and we can obtain   1þp   p 1 1þp 1þp ¼p C C : −ð1 þ pÞ λ p p

ðA  12Þ

where the amplitude compensation function Λλ ðΔτ; ωÞ ¼

α p ðΔτ; ωÞ þ λ ; α 1þp ðΔτ; ωÞ þ λ

ðA  2Þ

  −γ  ω ωΔτ : α ðΔτ; ωÞ ¼ exp − ω0 2Q

ðA  3Þ

when λ in Eq. (A-2) is a constant stabilization factor, suppose ε¼

1 ; λ

ðA  4Þ 

BðΔτ; ωÞ ¼ exp

ω ω0

−γ

 ωΔτ ; 2Q

ðA  5Þ

Eq. (A-2) can be written as 1 1 þ Bp ðΔτ; ωÞ ε Λλ ðΔτ; ωÞ ¼ 1 1 þ ; B1þp ðΔτ; ωÞ ε

ðA  6Þ

change the form, there is Λλ ðΔτ; ωÞ ¼

B1þp ðΔτ; ωÞ þ εBðΔτ; ωÞ B1þp ðΔτ; ωÞ þ ε

:

ðA  7Þ

Taking B(Δτ, ω) as an independent variable, derive Eq. (A-7) and simplify it, then there is h i ε ð1 þ pÞBp ðΔτ; ωÞ þ ε−pB1þp ðΔτ; ωÞ Λλ ðΔτ; ωÞ ¼ : h i2 B1þp ðΔτ; ωÞ þ ε 0

ðA  8Þ

Suppose ω1 is the corresponding angular frequency when the amplitude compensation function Λλ(Δτ, ω) takes the maximum value, so Λλ ðΔτ; ω1 Þ ¼

B1þp ðΔτ; ω1 Þ þ εBðΔτ; ω1 Þ B1þp ðΔτ; ω1 Þ þ ε

¼ C;

ðA  9Þ

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