GEOPHYSICS. Seismic anelastic attenuation estimation using prestack seismic gathers. Manuscript ID GEO R1

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1 Seismic anelastic attenuation estimation using prestack seismic gathers Journal: Geophysics Manuscript ID GEO-0-0.R Manuscript Type: Technical Paper Keywords: Q, prestack, attenuation, spectral analysis Area of Expertise: Reservoir Geophysics

2 Page of , VOL. XX, NO. X, (XX-XX, 0); P. XX-XX, FIGS. Seismic anelastic attenuation estimation using prestack seismic gathers Naihao Liu,, Bo Zhang, *, Jinghuai Gao, and Shengjun Li National Engineering Laboratory for Offshore Oil Exploration and Institute of Wave and Information, School of Electronic and Information Engineering, Xi an Jiaotong University, Xi an, Shaanxi, 00, P. R. China, Department of Geological Sciences, University of Alabama, Tuscaloosa, AL,, United States, PetroChina Research Institute of Petroleum Exploration & Development North West Lanzhou, Lanzhou, Gansu, P. R. China. Address: lnhfly@.com (N. Liu), bzhang@ua.edu (B. Zhang), jhgao@mail.xjtu.edu.cn (J. Gao), and li_sj@petrochina.com.cn (S. Li) Corresponding author: Bo Zhang Department of Geological Sciences, The University of Alabama, Tuscaloosa, AL, United States bzhang@ua.edu

3 Page of Liu et al. Abstract The seismic quality factor Q quantifies the effects of anelastic attenuation of seismic waves in the subsurface and can be used in assisting reservoir characterization and as an indicator of hydrocarbons. Usually, Q is estimated using vertical seismic profiles (VSP) and poststack seismic data. In this paper, we propose a workflow to estimate the Q using prestack time migration gathers. Our research is based on the assumption that the source wavelet can be simulated as a constant-phase wavelet. Our workflow begins with compensating the migration stretch of the prestack seismic gathers in the time-frequency domain. We then simulate the log spectral ratio values at zero-offset traces using the log spectral values at non-zero offset traces. We finally determine Q at zero-offset seismic traces using the classical log spectral ratio method (LSRM). We find that there is a linear relationship between a newly defined parameter, M, and the log spectral ratio between the target zone and referred zone. To demonstrate the validity and effectiveness of the proposed method, we first apply it to noise-free and noisy synthetic data examples and then to real seismic data acquired over Sichuan basin, China. Both synthetic and real seismic applications demonstrate the effectiveness of the proposed method in highlighting high anelastic-attenuation zones. List of Keywords: Quality factor, S-transform, Log spectral ratio, Stretch, Prestack seismic data

4 Page of Seismic anelastic attenuation estimation Introduction The seismic wave usually suffers anelastic attenuation and dispersion when it propagates through media in the earth (Ricker, ). The quality factor Q is used to qualify the anelastic attenuation of seismic energy in the subsurface. The quality factor Q is sensitive to parameters such as lithology, porosity, and fluid type in the pores (Johnston et al., ; Wang et al., 00). Thus, Q can be used as a direct indicator of hydrocarbons (Toksöz et al., ). Researchers have successfully improved the time resolution of seismic data using proper inverse Q filtering (Wang, 00). Wang (00) studied the influences of inverse Q filtering, which corrects the wavelet distortion in terms of shape and timing, compensates for energy loss without boosting ambient noise, and produces desirable seismic images with high resolution and high SNR (signal to noise ratio). Yuan et al. (0) proposed a Sparse Bayesian learning-based time-variant deconvolution method to improve the resolution of the seismic data through the Q-filter. Most of the quality factor-estimation methods are based on the quantitative comparison of frequency features. Quan and Harris () proposed the centroid frequency shift (CFS) method. Quan and Harris s method is based on the assumption that the amplitude spectrum of the source wavelet has a Gaussian, boxcar, or triangular shape. Liu et al. () proposed a frequency centroid down-shift method and applied it to cross-hole radar attenuation tomography. Li et al. (0) modified the CFS method and developed a dominant and central frequency shift (DCFS) method to estimate the Q factor using poststack seismic data. Zhang and Ulrych (00) developed the peak frequency shift (PFS) method for Q estimation where the source wavelet is supposed to be Ricker-like. However, Zhang and Ulrych s method is sensitive to the noise type and level contained in the seismic data. Li et al. (00) first calculated the scalogram using the CWT and then estimated the Q factor directly from the scalogram according to the scale shift of the seismic data. Hu et al. (0) derived a conversion formula to calculate the peak frequency from the centroid frequency and developed an improved peak frequency shift (IPFS) method to estimate the Q factor. Wang et al. (0) used a two parameters wavelet (Wang, 0) to simulate the source wavelet and

5 Page of Liu et al. estimated the quality factor Q using the peak frequency shift method. The spectral ratio (SR) is one of the most popular methods (Hauge, ; Reine et al., 00) for the Q-factor estimation. The SR-based methods assume that there is a linear relationship between the frequency and the log spectral ratio between the target zone and referred zone (Tonn, ). Using the log spectral ratio method, Dasgupta and Clark () estimated the Q factor from CMP gathers and established the variation of apparent Q versus offset (QVO) to obtain the Q factor estimation at zero offset. Behura and Tsvankin (00) developed a technique for estimating the interval attenuation coefficient from reflection seismic data by combining the spectral ratio method with velocity independent layer stripping. Reine et al. (00) compared the sensitivity of Q factor estimation to time-frequency analysis methods. Their time-frequency analysis methods included the Gabor transform (GT), short-time Fourier transform (STFT), continuous wavelet transform (CWT), and S-transform (ST). They concluded that the time-frequency spectrum computed using the CWT and ST can produce more robust Q estimation. Zhou et al. (00) first obtained the time-frequency spectrum using CWT and then estimated the Q factor by comparing the amplitude of the peak frequencies at the source and target seismic record. Reine et al. (0) developed a robust method for measuring attenuation from prestack surface seismic gathers using the log spectral ratio method and τ p transform. To reduce the presence of spectral interference, Reine et al. introduced a variable-window time-frequency transform and an inversion scheme operating simultaneously in the frequency and traveltime-difference coordinates. Hao et al. (0) assumed the source wavelet to be a Gaussian wavelet and used the log spectral ratio method to estimate Q. They employed a two-parameter generalized S-transform (GST) to calculate the time-frequency spectrum of the seismic traces. In this paper, we propose a novel workflow to estimate Q using prestack seismic gathers. We first derive equations to compensate for the normal move-out (NMO) stretch (Buchholtz, ; Dunkin and Levin, ) in the time-frequency domain. We then estimate Q using the log spectral ratio method. We use the ST to obtain the time-frequency spectrum of the seismic traces. To demonstrate the effectiveness of the proposed method, we first apply it to noise-free and noisy synthetic gathers, then to prestack field data acquired in southwest China to

6 Page of Seismic anelastic attenuation estimation estimate Q and employ Q as a direct indicator of gas. Stretch compensation of prestack seismic gathers in the time-frequency domain In this paper, we use the ST to calculate the time-frequency spectrum of seismic traces. The ST of s( t ) is defined as (Stockwell et al., ) where i ( τ ) = ( ) ( τ ) π ft ST, f s t g t, f e dt, () t f f g( t, f ) = e is a Gaussian window. π Note that the migration stretch would distort the time-frequency features of prestack seismic gathers. We need to compensate the time-frequency distortion of prestack seismic gathers caused by the stretch before Q estimation. The relationship between the stretched, ss ( t ), and original, s ( ) 0 t, seismic traces can be expressed as (Dunkin and Levin, ) t ss( t) = s 0, () β where β is the stretch factor (Dunkin and Levin, ) defined in Appendix A. Equation indicates that the stretched wavelet s ( t ) becomes wider in the time domain and narrower in s the frequency domain compared with those of the original seismic trace. Then, the time-frequency relationship of the stretched and original seismic traces is expressed as where ( τ ) s ( τ, ) 0, τ STs( τ, f) = ST0, β f. () β ST f is the time-frequency spectrum of the original seismic trace s ( t ) and 0 ST f is the time-frequency spectrum of the stretched seismic trace s ( t ). Then, we can compensate the NMO stretch effect in prestack seismic gathers according to equation. Appendix A shows in detail the derivation of equation. Q estimation using prestack seismic gathers (one-layer case) In real applications, we cannot directly achieve the parameters of the source wavelet for Q estimation. We then assume the source wavelet is a constant-phase wavelet in this paper. A s

7 Page of Liu et al. constant-phase wavelet in the frequency domain (Gao et al., 0) is expressed as ( π f σ 0 ) + iϕ 0 δ0 B( f ) = e, () where σ 0 and δ 0 denote the modulated frequency (i.e. dominant frequency) and energy decay factor of the constant-phase wavelet, respectively. The constant phase ϕ 0 may vary case by case which is determined by the seismic well tie (Peterson et al., ; Wu et al., 0). Considering the propagation of a wave in a half-space with a Q factor for traveltime t, we obtain the amplitude spectrum of the received seismic signal as We obtain the log spectral ratio π ft iπ ft Q B( f, t) = B( f ) e e ( π f σ0 ) + iϕ π ft 0 δ0 iπ ft Q = e e e ( j ) Γ of the offset at time (, ) (, ) ST t f M ST t f Q. t and t ( j ) Γ ( j) = ln +ln(pg)= +ln(pg), (a) ( j ) m = π, p ( j) M t f + 0 () (b) π π p= +, (c) δ f π m= ( π f σ 0), (d) δ 0 where ST stands for the time-frequency result of seismic traces using the S-transform, t and t are the two-way traveltimes to the top and bottom of the target layer, respectively. Obviously, t is the two-way traveltime of the target layer. The terms P and G are simplified here to refer to the ratios between the two reflections for the energy partitioning and geometric spreading terms, respectively (Reine et al., 0). Appendix B shows the derivation of equation in detail. Note that we first calculate the log spectral ratio the j th offset seismic trace. We then obtain ( j ) Γ of (0 ) Γ at the zero-offset trace using a quadratic

8 Page of Seismic anelastic attenuation estimation fitting of Γ at non-zero-offset traces (Dasgupta and Clark, ). Finally, we estimate the Q factor at the zero-offset trace by linearly fitting equation. where N i= (0) M and the log spectral ratio Q estimation using prestack seismic gathers (multi-layer case) In a multi-layer medium, the amplitude attenuation equation is written as Q i and x t= ti = t + V 0 obtain the log spectral ratio as N N π fti iπ f ti Q i= i= i (, ) ( ), (0) Γ using B f t = B f e e () t i are the Q factor and the two-way traveltime within the layer, is the total two-way traveltime at offset ( j) i ST ( t ( ) ( ), f ) j N M N ( j) ST ( t( ), f ) Q N N ( j ) x, j 0,,, = L K. We then Γ = ln +ln(pg) = +ln(pg), (a) ( j) m M N = π tn f + p 0, (b) π π p= +, (c) δ f π m= ( π f σ 0). (d) δ 0 Appendix B shows the derivation of equation in detail. Similarly, we first obtain zero-offset trace through fitting using (0) M N and (0) Γ N. ( j ) N (0) Γ N at the Γ of non-zero-offset traces and then estimate the Q factor Synthetic data examples To demonstrate the validity and effectiveness of the proposed method, we apply it to noise-free and noisy synthetic gathers. Figures a and b show the common-source gathers obtained from a three-layer subsurface model before and after NMO correction, respectively. The maximum traveltime is s and the time sample interval is ms. We have geophones

9 Page of Liu et al. in the synthetic model and the offset increment is 0 m. We choose a Ricker wavelet with a dominant frequency of 0 Hz as the source wavelet (indicated by the solid blue line in Figure a). The Q values of the three layers are 00, 0, and 0, respectively. The velocities in the first layer linearly increase from 000 m/s to 00 m/s. The velocities in the second and third layers are 00 m/s and 000 m/s, respectively. We have muted segmental traces stretched more than 0% in Figure b, shown in Figure c. The red arrow in Figure c indicates the th trace of the synthetic model. Note that the severe stretch at the far offset of the NMO corrected synthetic gather shown in Figure b. Figures a and b show the source Ricker wavelet (solid) and simulated constant-phase wavelet (dashed) in the time and frequency domain, respectively. Note the negligible subtle difference between the source Ricker wavelet and simulated constant-phase wavelet both in the time and frequency domain. Figure shows the computed β for the three layers as a function of offset. The blue, red, and black lines in Figure indicate β values for the first, second, and third layer, respectively. Note that the β values increase with increasing offset. The β value equals at the zero-offset trace which means there is no stretch at all for this trace. Figures a, b, and c show the time-frequency spectrum for the th synthetic trace before NMO correction, after NMO correction, and NMO stretch compensation, respectively. Note that the NMO correction shifts the frequency components toward the lower frequency side and our compensation successfully corrects this time-frequency distortion. Figures a, b, and c show the spectral ratio Γ varying with offset and frequency for the first, second, and third layer using the NMO corrected synthetic gather, respectively. Figures a, b, and c show the spectral ratio Γ varying with offset and frequency for the first, second, and third layer using the stretch-compensated NMO corrected synthetic gather, respectively. Note that the spectral ratios in Figure are distorted and incorrect while Figure shows a smoothing log spectral ratio. Figure shows the theoretical log spectral ratio of the three layers at the zero-offset trace. The blue, red, black lines are theoretical ( 0 ) Γ (0) Γ for the first, second, and third layer, respectively. Note that the curves in Figure have the shape of a parabola. Figures a, b, and c show the variation of the parameter M as a function of offset and frequency. The smooth variations of the parameter M in Figure look like the

10 Page of Seismic anelastic attenuation estimation spectral ratio Γ in Figure. Moreover, the relationship between the variation of the parameter M and the spectral ratio Γ looks like linear, which demonstrates the validity of equations (a) and (a). Figures a, b, and c compare the relationship between (0 ) Γ and (0) M for the first, second, and third layers at the zero-offset trace, respectively. The green lines in Figure a, b, and c are the theoretical (0) Γ and and black lines in Figures a, b, and c are the computed (0) M pairs. The dashed red, blue, ( 0 ) Γ and (0) M pairs using the stretch-compensated synthetic gather, NMO-corrected gather, and poststack data, respectively. Note the perfect matching between the dashed red and solid green lines. The solid green line in Figure 0 denotes the theoretical Q factors. The dashed blue, red, and black lines in Figure 0 are Q values calculated using the NMO-corrected synthetic gathers, stretch-compensated prestack synthetic gather, and poststack data, respectively. Due to the non-linear relationship between (0 ) Γ and M (0), Q factors calculated using the NMO-corrected synthetic gather highly deviate from the theoretical Q values. Q factors computed using the stack trace are greater than the theoretical Q factors. Our proposed method successfully obtains accurate Q estimations. To further test the robustness of our method to noise, we add Gaussian white noise to the synthetic gathers shown in Figure. The SNR is 0 db. Figures a and b show the noisy synthetic gathers before and after NMO correction, respectively. Note that the synthetic gathers are degraded by the heavy Gaussian white noise, especially the third reflection. Figure shows the estimated Q factors using the noisy synthetic gathers. The solid green line in Figure denotes the theoretical Q factors. The dashed blue, red, and black lines in Figure are Q values calculated using the NMO-corrected synthetic gathers, stretch-compensated prestack synthetic gather, and poststack data, respectively. Note that the estimated Q factors using the proposed workflow are still superior to those computed using the other means. Field data The synthetic tests demonstrate the superiority of our method. We apply it to prestack field data to estimate Q and then employ the Q as a direct indicator of gas. The seismic survey is located in the central paleo-uplift at the middle of Sichuan basin. The target zone is located

11 Page 0 of Liu et al. at the Lower Permian Maokou formation and Qixia formation. The lithology of the reservoir is limestone. The fourth member of Maokou (MK) formation and the first member of Qixia (QX) formation are the main gas-producing layers. However, the gas-bearing zones have a very quick lateral change within the formations. The prestack field data shown in this paper is a D line containing CMP gathers. The time sampling interval is ms and each CMP contains offset traces. The minimum and maximum offsets are m and m, respectively. Figure shows two representative prestack CMP gathers. The corresponding CMP numbers are and 0. Figure shows the poststack seismic section with stacked traces. The solid blue lines and dashed red lines in Figures a and b indicate the estimated seismic wavelets (Peterson et al., ; Wu et al., 0) and simulated constant-phase wavelets in the time domain and frequency domain, respectively. The good match between the estimated seismic wavelet and simulated constant-phase wavelet demonstrates the validity and effectiveness of the proposed workflow. Three green curves in Figures and denote the three target horizons. H, H, and H are the top of the MK, the top of QX, the bottom of QX, respectively. Figure is the corresponding estimated Q. The vertical black curves in Figure and vertical red curves in Figure indicate the three borehole locations on our selected seismic section. Well C has high gas production for the MK layer and relatively low production for the QX layer. Well B is regarded as a dry well for both layers. Well A has high production for the QX layer. Note that we have low Q values (high attenuation) (red) for the MK of well C, high Q values (low attenuation) (yellow) for the MK and QX of well B, and low Q values for the QX of well A. This phenomenon indicates that we have very good matching between the low Q values (high attenuation) and high gas production zone. Using the estimated Q profile in Figure, we apply the inverse Q filtering (Wang, 00; Yang et al., 0; Zhang and Gao, 0) to the original poststack seismic data in Figure. Figure shows the seismic profile after the inverse Q filtering. Obviously, the vertical resolution of seismic data is enhanced as marked by the black arrows in Figure. The enhanced resolution of seismic data will allow for more detailed interpretation of seismic structures in the future.

12 Page of Seismic anelastic attenuation estimation Conclusion The stretch generated by the NMO would distort the time-frequency features of the seismic trace. As a result, the relationship between the log spectral ratio (LSR) and frequency is no longer linear. In this paper, we proposed a novel workflow to estimate the quality factor Q using the log spectral ratio method. Our method is based on the assumption that the source wavelet is characterized a constant-phase wavelet. To avoid the influence of the stretch effect on the Q estimation, we compensate the stretch in the time-frequency domain using the stretch compensated factor. Noise-free and noisy synthetic data examples illustrate the validity and effectiveness of the proposed method. The estimated quality factor Q in the real application accurately highlights the gas zone of the reservoir, where the extracted low Q values match high gas production zone well. Moreover, we improve the vertical resolution of the original seismic data after applying the inverse Q filtering technique using the estimated Q section. Note that the source wavelet is often temporal and spatial varying in real applications. Moreover, seismic wavelets are time-varying and complex during seismic waves propagation in the subsurface. The simulated constant-phase wavelet matches the seismic wavelet well both in the time and frequency domain. Nevertheless, the simulated constant-phase wavelet cannot describe the time-varying characteristic of the real seismic wavelet. As a result, we will introduce a time-varying wavelet to estimate the Q factor using prestack seismic gathers in the future. Acknowledgments We thank the National Natural Science Foundation of China (Nos. 00 and 0) and the National Science and Technology Major Project (Nos. 0ZX and 0ZX00) for their financial support. We also want to thank Dr. Xiaokai Wang, Guowei Zhang, and Ph.D. candidate Hao Wu for their fruitful discussions about this work.

13 Page of Liu et al. APPENDIX A Stretch compensation of prestack seismic gathers in the time-frequency domain Assuming a layer-cake model, we obtain the hyperbolic traveltime equation as t x = +, (A-) t0 v( t0 ) where t 0 is two-way traveltime at zero offset, t is two-way traveltime at offset x, v is the NMO velocity (the root-mean square velocity for flat layered media). And, we have 0 t v t0 t = x ( ). Then, the stretch-compensated factor β is calculated as (Dunkin and Levin, ) β x v ( t ) t + t0 t 0 0 = x= = t t0 x v'( t0 ) x v '( t0 ) v ( t0) t0 v ( t0 ) t0 Then, the relationship between the original and stretched seismic traces is expressed as where ( τ ) STs ( τ, ) 0, t β ( τ, ) = 0 ( τ, ) iπ ft STs f s g t f e dt = 0 β, = 0 = ST0 f t iπ fβ β t τ t t s g f e β d β β β τ t t f / iπβ f β t f β β t s e e β d β π β τ, β, β. (A-) (A-) ST f is the time-frequency spectrum of the original seismic trace s ( t ) and 0 f is the time-frequency spectrum of the stretched seismic trace. Using equation A-, we compensate the NMO stretch in prestack seismic gathers and obtain the compensated time-frequency spectrum.

14 Page of Seismic anelastic attenuation estimation APPENDIX B Q estimation using prestack seismic gathers. Q estimation using prestack seismic gathers for multi-layer case The ST in the frequency domain is defined as (Stockwell et al., ) π α, f iπατ, ( ) = ( + ) ST τ f S α f e e d α (B-) where S ( f ) is the Fourier transform of the analyzed seismic signal s( t ). To derive the equation to estimate the Q factor using the ST, we first rewrite equations and in equations B- and B-. ( π f σ 0 ) + iϕ 0 δ0 B( f ) = e, (B-) where σ 0, δ 0, and ϕ 0 denote the dominant frequency, delay factor, and phase of the constant-phase wavelet, respectively. In a multi-layer medium, the amplitude attenuation equation is rewritten as where N i= Q i and x t= ti = t + V 0 N N π fti iπ f ti Q i= i= i (, ) ( ), B f t = B f e e (B-) t i are the Q factor and the two-way traveltime within the layer, is the total two-way traveltime at offset ( j ) x, j 0,,, = L K. Using the constant-phase wavelet and the amplitude spectrum of the received seismic signal in equations B- and B-, the ST of the wavelet B( f, t ) can be expressed as

15 Page of where (, ) = ( α+, ) π α / f i πατ ST t f B f t e e d = Liu et al. N N N π( α+ f) σ 0 π( f + α) ti + iϕ0 iπ( f + α) πα ti i ti δ Q π α / f 0 i= i= i i= e e e e e d = e e e e e e α N N N ( π f σ 0 ) π fti ( πα ) + ( π f σ 0 )πα παti + iϕ 0 iπ f ti δ Q 0 i= i= i δ Q 0 i= i π α / f α π π N π π t i + α + ( π f σ 0) α δ 0 δ f Q 0 i= i N N ( π f σ 0 ) π fti + iϕ 0 iπ f ti δ Q 0 i= i= i = e e e e dα = N N ( π f σ 0 ) π fti + iϕ 0 iπ f ti δ Q 0 i= i= i p + q α α e e e e d = e = ( π f σ 0 ) + iϕ 0 δ0 π π p= +, δ f e α N N π fti q q iπ f ti pα Q i= i= i p p e e e dα N N ( π f σ 0 ) π fti q + iϕ 0 iπ f ti δ Q 0 i= i= i p π e e e e p 0 q N π = N ti m i= i, π m f. Q and = ( π σ ) 0 δ We then calculate the log spectral ratio (LSR) at offset 0 x ( j ) dα (B-)

16 Page of ST t ( j) Γ N = ln ST t Seismic anelastic attenuation estimation ( ( ), f N ) ( ( N ), f) q( N ) p N N ( π f σ 0 ) π fti q( N ) + iϕ 0 iπ f ti δ Q 0 i i i p = = π e e e e p = ln + ln( PG) N N ( π f σ 0 ) π fti π ft q( N ) + iϕ 0 iπ f t N i π δ Q 0 i i i Q p = = N e e e e e p ln e = π ft q N ( N ) Q p N e N N πt i πt i N t i π i= i + ln( PG) m m π ftn i= Qi i= Qi = ln( PG) Q p p N N N π ft N π t i π t i π mt N = ln( PG) Q N p i= Qi p i= Qi p Q N π = p Q p e N ti t N tn π m + + π f + ln( PG) + i= Qi QN QN p N π ti t N tn πtn m = f ln( PG), p i= Qi QN QN QN p (B-) where t i is the two-way traveltime in the i th layer and t( N ) t i. Note that we write Q N instead of Q in equation B- for convenience. The terms P and G are simplified here ( j ) N to refer to the ratios between the two reflections for the energy partitioning and geometric spreading terms, respectively (Reine et al., 0). We easily conclude Q Q = N i= because the Q factor usually ranges from tens to hundreds. Therefore, the contribution of the first term in equation B- is negligible. We then obtain a linear relationship ( ( N ), ) ( ( N ), f) ST t ( j) ( j) f πtn m M N Γ N = ln + ln( PG) f ln( PG) ln( PG), ( j) + + = + ( j) ST t QN p QN (B-)

17 Page of where ( j) m M N = πtn f + p Liu et al. and t is the two-way traveltime within the N th N layer.. Q estimation using prestack seismic gathers for one-layer case When considering the one-layer case (i.e. N = in equations B-, B-, and B-), the equation B- turns into where ( π f σ 0 ) q i π ft + ϕ 0 δ0 iπ ft Q p π ST( t, f) = e e e e, (B-) p π π π p= +, = t δ f 0 π m f 0. δ q m and = ( π σ ) Q (, ) (, ) ST t f j ln ln( PG) ST t f ( ) Γ = + π t ( t + t ) πt m = + f ln( PG), + + p Q Q p 0 (B-) where t and t are the two-way traveltime at the top and bottom of our target layer. At last, we obtain a linear relationship for one-layer case where (, ) (, ) ST t f ( j) ( j) πt m M Γ = ln + ln( PG) f ln( PG) ln( PG). ( j ) + + = + ( j) ST t f Q p Q π m. p ( j) M = t f + (B-)

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21 Page 0 of Liu et al. List of figure captions Figure. The synthetic gathers. Synthetic gathers (a) before and (b) after NMO correction, (c) NMO correction gather after muting the severely stretched signals at far offset. The Q factors of the three layers are 00, 0, and 0, respectively. The red arrow in Figure c indicates the th trace. Figure. The source Ricker wavelet and simulated constant-phase wavelet. The Ricker wavelet (solid) and simulated constant-phase wavelet (dashed) in the (a) time and (b) frequency domain, respectively. The dominant frequencies of the Ricker wavelet and constant-phase wavelet are both 0 Hz. Figure. The stretch-compensated factor β values for the first (blue), second (red), and third (black) layers. Figure. Time-frequency representations of the th Time-frequency spectra for the th correction, and (c) NMO stretch compensation. trace using the S-transform. trace (a) before NMO correction, (b) after NMO Figure. The log spectral ratio Γ varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the NMO-corrected synthetic gather, respectively. Figure. The log spectral ratio Γ varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the stretch-compensated NMO corrected synthetic gather. Figure. The theoretical log spectral ratio are the theoretical ( 0 ) Γ at zero offset. The blue, red, and black lines (0) Γ for the first, second, and third layers, respectively. Figure. Variation of the parameter M as a function of frequency and offset for the (a) first, (b) second, and (c) third layer. Figure. The relationship between layer. (0) Γ and (0) M for the (a) first, (b) second, (c) third Figure 0. The estimated Q values. The solid green line denotes the theoretical Q factors. The dashed blue, red, and black lines are the estimated Q factor calculated using the

22 Page of Seismic anelastic attenuation estimation NMO-corrected synthetic gathers, stretch-compensated synthetic gathers, and poststack trace, respectively. Figure. The noisy synthetic gathers (a) before and (b) after NMO correction. The SNR equals 0 db. Figure. The estimated Q values. The solid green line denotes the theoretical Q factors. The dashed blue, red, and black lines are the estimated Q factors calculated using the NMO-corrected synthetic gathers, stretch compensated synthetic gathers, and post-stack trace, respectively. Figure. Selected two prestack CMP gathers from field data. The CMP numbers are and 0. Figure. The poststack seismic section. Three green curves denote three horizons, labeled H, H, and H. Three vertical black lines indicate the three borehole locations on our selected seismic section. Figure. Estimated seismic wavelets (solid blue lines) and simulated constant-phase wavelets (dashed red lines) in the (a) time and (b) frequency domain. Figure. Estimated Q factors by the proposed method. The white, pink, and red curves denote the three horizons H, H, and H, respectively. Three vertical red lines indicate the three borehole locations on our selected seismic section. Figure. The seismic profile after the inverse Q filtering using the estimated Q profile in Figure.

23 Page of Figure. The synthetic gathers. Synthetic gathers (a) before and (b) after NMO correction, (c) NMO correction gather after muting the severely stretched signals at far offset. The Q factors of the three layers are 00, 0, and 0, respectively. The red arrow in Figure c indicates the th trace. x0mm (00 x 00 DPI)

24 Page of Figure. The synthetic gathers. Synthetic gathers (a) before and (b) after NMO correction, (c) NMO correction gather after muting the severely stretched signals at far offset. The Q factors of the three layers are 00, 0, and 0, respectively. The red arrow in Figure c indicates the th trace. x0mm (00 x 00 DPI)

25 Page of Figure. The synthetic gathers. Synthetic gathers (a) before and (b) after NMO correction, (c) NMO correction gather after muting the severely stretched signals at far offset. The Q factors of the three layers are 00, 0, and 0, respectively. The red arrow in Figure c indicates the th trace. x0mm (00 x 00 DPI)

26 Page of Figure. The source Ricker wavelet and simulated constant-phase wavelet. The Ricker wavelet (solid) and simulated constant-phase wavelet (dashed) in the (a) time and (b) frequency domain, respectively. The dominant frequencies of the Ricker wavelet and constant-phase wavelet are both 0 Hz. x0mm (00 x 00 DPI)

27 Page of Figure. The source Ricker wavelet and simulated constant-phase wavelet. The Ricker wavelet (solid) and simulated constant-phase wavelet (dashed) in the (a) time and (b) frequency domain, respectively. The dominant frequencies of the Ricker wavelet and constant-phase wavelet are both 0 Hz. x0mm (00 x 00 DPI)

28 Page of Figure. The stretch-compensated factor values for the first (blue), second (red), and third (black) layers. x0mm (00 x 00 DPI)

29 Page of Figure. Time-frequency representations of the th trace using the S-transform. Time-frequency spectra for the th trace (a) before NMO correction, (b) after NMO correction, and (c) NMO stretch compensation. x0mm (00 x 00 DPI)

30 Page of Figure. Time-frequency representations of the th trace using the S-transform. Time-frequency spectra for the th trace (a) before NMO correction, (b) after NMO correction, and (c) NMO stretch compensation. x0mm (00 x 00 DPI)

31 Page 0 of Figure. Time-frequency representations of the th trace using the S-transform. Time-frequency spectra for the th trace (a) before NMO correction, (b) after NMO correction, and (c) NMO stretch compensation. x0mm (00 x 00 DPI)

32 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the NMO-corrected synthetic gather, respectively. x0mm (00 x 00 DPI)

33 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the NMO-corrected synthetic gather, respectively. x0mm (00 x 00 DPI)

34 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the NMO-corrected synthetic gather, respectively. x0mm (00 x 00 DPI)

35 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the stretch-compensated NMO corrected synthetic gather. x0mm (00 x 00 DPI)

36 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the stretch-compensated NMO corrected synthetic gather. x0mm (00 x 00 DPI)

37 Page of Figure. The log spectral ratio varying with offset and frequency for the (a) first, (b) second, and (c) third layer using the stretch-compensated NMO corrected synthetic gather. x0mm (00 x 00 DPI)

38 Page of Figure. The theoretical log spectral ratio at zero offset. The blue, red, and black lines are the theoretical for the first, second, and third layers, respectively. x0mm (00 x 00 DPI)

39 Page of Figure. Variation of the parameter as a function of frequency and offset for the (a) first, (b) second, and (c) third layer. x0mm (00 x 00 DPI)

40 Page of Figure. Variation of the parameter as a function of frequency and offset for the (a) first, (b) second, and (c) third layer. x0mm (00 x 00 DPI)

41 Page 0 of Figure. Variation of the parameter as a function of frequency and offset for the (a) first, (b) second, and (c) third layer. x0mm (00 x 00 DPI)

42 Page of Figure. The relationship between and for the (a) first, (b) second, (c) third layer. x0mm (00 x 00 DPI)

43 Page of Figure. The relationship between and for the (a) first, (b) second, (c) third layer. x0mm (00 x 00 DPI)

44 Page of Figure. The relationship between and for the (a) first, (b) second, (c) third layer. x0mm (00 x 00 DPI)

45 Page of Figure 0. The estimated Q values. The solid green line denotes the theoretical Q factors. The dashed blue, red, and black lines are the estimated Q factor calculated using the NMO-corrected synthetic gathers, stretch-compensated synthetic gathers, and poststack trace, respectively. x0mm (00 x 00 DPI)

46 Page of Figure. The noisy synthetic gathers (a) before and (b) after NMO correction. The SNR equals 0 db. x0mm (00 x 00 DPI)

47 Page of Figure. The noisy synthetic gathers (a) before and (b) after NMO correction. The SNR equals 0 db. x0mm (00 x 00 DPI)

48 Page of Figure. The estimated Q values. The solid green line denotes the theoretical Q factors. The dashed blue, red, and black lines are the estimated Q factors calculated using the NMO-corrected synthetic gathers, stretch compensated synthetic gathers, and post-stack trace, respectively. x0mm (00 x 00 DPI)

49 Page of Figure. Selected two prestack CMP gathers from field data. The CMP numbers are and 0. x0mm (00 x 00 DPI)

50 Page of Figure. The poststack seismic section. Three green curves denote three horizons, labeled H, H, and H. Three vertical black lines indicate the three borehole locations on our selected seismic section. x0mm (00 x 00 DPI)

51 Page 0 of Figure. Estimated seismic wavelets (solid blue lines) and simulated constant-phase wavelets (dashed red lines) in the (a) time and (b) frequency domain. x0mm (00 x 00 DPI)

52 Page of Figure. Estimated seismic wavelets (solid blue lines) and simulated constant-phase wavelets (dashed red lines) in the (a) time and (b) frequency domain. x0mm (00 x 00 DPI)

53 Page of Figure. Estimated Q factors by the proposed method. The white, pink, and red curves denote the three horizons H, H, and H, respectively. Three vertical red lines indicate the three borehole locations on our selected seismic section. x0mm (00 x 00 DPI)

54 Page of Figure. The seismic profile after the inverse Q filtering using the estimated Q profile in Figure. x0mm (00 x 00 DPI)