Seismic dip estimation based on the two-dimensional Hilbert transform and its application in random noise attenuation a

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1 Seismi dip estimation based on the two-dimensional Hilbert transform and its appliation in random noise attenuation a a Published in Applied Geophysis, 1, (Marh 015) Cai Liu, Changle Chen, Dian Wang, Yang Liu, Shiyu Wang, and Liang Zhang ABSTRACT In seismi data proessing, random noise seriously affets the seismi data quality and subsequently the interpretation. This study aims to inrease the signal-tonoise ratio by suppressing random noise and improve the auray of seismi data interpretation without losing useful information. Hene, we propose a strutureoriented polynomial fitting filter. At the ore of struture-oriented filtering is the haraterization of the strutural trend and the realization of nonstationary filtering. First, we analyze the relation of the frequeny response between twodimensional (D) derivatives and the D Hilbert transform (Riesz transform). Then, we derive the noniterative seismi loal dip operator using the D Hilbert transform to obtain the strutural trend. Seond, we selet polynomial fitting as the nonstationary filtering method and expand the appliation range of the nonstationary polynomial fitting. Finally, we apply variableamplitude polynomial fitting along the diretion of the dip to improve the adaptive strutureoriented filtering. Model and field seismi data show that the proposed method suppresses the seismi noise while proteting strutural information. INTRODUCTION Random noise, whih refers to any unwanted features in data, ommonly ontaminates seismi data. Random noise soures in seismi exploration are roughly divided into three ategories. First, there are external disturbanes suh as wind and human ativities. Seond, there is eletroni instrument noise. Third, there is the irregular interferene owing to seismi explosions. Random noise attenuation is a signifiant step in seismi data proessing. In partiular, the extent of noise suppression in poststak data diretly affets the auray of subsequent proessing and interpretation. Presently, several different random noise attenuation methods are available. Liu College of Geo-exploration Siene and Tehnology, Jilin University, Changhun 13006, China. Qian An Oil Fatory, Jilin Oilfield, CNPC, Songyuan , China. Corresponding Author: Dian Wang ( dianwang@jlu.edu.n)

2 Liu et al. Dip estimation using Riesz transform et al. (006) presented a D multilevel median filter for random noise attenuation, whereas Liu et al. (009b) used a 1D time-varying window median filter. Bekara and van der Baan (009) used the empirial mode deomposition (EMD) method and proposed a filtering tehnique for random noise attenuation in seismi data. Liu et al. (009a) proposed a high-order seislet transform for random noise attenuation. Li et al. (01) applied morphologial omponent analysis to suppress random noise and Liu et al. (01) proposed a novel method of random noise attenuation based on loal frequeny-domain singular value deomposition (SVD). Marashini and Turton (013) assessed the effet of nonloal means random noise attenuator on ohereny. Li et al. (013) used time-frequeny peak filtering to suppress strong noise in seismi data. Liu and Chen (013) used f-x regularized nonstationary autoregression to suppress random noise in 3D seismi data. The abovementioned random noise attenuation methods are limited by their lak of protetion of strutural information. For example, improper filtering may blur small faults, whih may also make the displaement of larger fault ontinuous and onsequently make layers appear ontinuous instead of faulted. Obviously, this hinders fault interpretation, and makes denoising and proteting strutural information important. Fehmers and Hoker (003) applied strutureoriented filtering to fast strutural interpretation. Hoeber et al. (006) applied nonlinear filters, suh as median, trimmed mean, and adaptive Gaussian, over planar surfaes parallel to the strutural dip. Fomel and Guitton (006) suggested the method of plane-wave onstrution by using model reparameterization. Liu et al. (010) applied nonlinear struture-enhaning filtering by using plane-wave predition to preserve strutural information. Liu et al. (011b) proposed a poststak random noise attenuation method by using weighted median filter based on loal orrelation and tried to balane the protetion of fault information and noise attenuation. Struture-oriented filtering inludes struture predition and filtering. Seismi dip is at the ore of struture predition; for, we an use seismi dip to determine strutural trends and ahieve struture protetion. Ottolini (1983) used loal slant stak to formulate a loal seismi dip estimation method. Fomel (00) proposed a seismi dip estimation method based on the plane-wave destrution (PWD) filter. Shleiher et al. (009) ompared different methods of loal dip omputations. The seletion of filtering methods in struture-oriented filters is ritial and polynomial fitting has been suessfully applied to seismi data denoising. Lu and Lu (009) used edge-preserving polynomial fitting to suppress random seismi noise. This method ahieves better results when the trajetories of seismi events are linear or the amplitudes along the trajetories are not onstant. Liu et al. (011a) proposed a novel seismi noise attenuation method by using nonstationary polynomial fitting (Fomel, 009) and shaping regularization (Fomel, 007) for onstraining the smoothness of the polynomial oeffiients. In this paper, we disuss the two-dimensional (D) Hilbert transform and use it to derive the formula for the dip in the plane wave, onstrut a stable algorithm for estimating the dip, and improve the omputational effiieny of Fomel s method (Fomel, 00) without minimizing the preision of the dip estimation. Finally, we use syntheti model and field seismi data to demonstrate the appliability of the

3 Liu et al. 3 Dip estimation using Riesz transform proposed method. THEORY The extration of strutural information and the seletion of effetive filtering methods are ritial to struture-oriented filters. Beause of the time-spae relation in seismi data, strutural information must satisfy kinematis and kinetis equations. The dip of seismi events reveals strutural features. This study is the first to disuss a alulation method for the loal seismi dip. Noniterative loal dip alulation Following the loal plane-wave equation (Fomel, 00) P (x, t) x we define the loal dip of seismi data σ(x, t) = P (x, t) + σ(x, t) t = 0, (1) P (x, t) P (x, t) /, () x t where P (x, t) is the seismi wave field and σ(x, t) is the loal seismi dip as a funtion of time t and distane x. However, in atual omputations, beause the loal dip is used to determine the diretion of a seismi event, we ignore the dimensions and sampling interval; thus, σ only depends on the sampling data and the loal dimensionless dip is defined as σ = ( P (x, t) P (x, t) / ) x x t t = P x / P y, (3) where P / x and P / y are the partial derivatives of the seismi wave field in the x and y diretion, respetively, and x and t are the respetive sampling intervals in the x and y diretion. Using equation 3, we ompute the loal dip by using the speifi values of the spae- and time-diretional derivatives. Hene, we first disuss the derivative operator. The ideal differentiator frequeny response is F IDD (ω) = iω, π ω π. (4) The ideal differentiator frequeny response is multiplied by a frequeny-dependent linear funtion in the frequeny domain. The diret alulation of the derivative of the signal in the time domain enhanes the high-frequeny random noise and

4 Liu et al. 4 Dip estimation using Riesz transform redues the dip auray. Thus, we analyze the frequeny response of the derivative operator and the frequeny response of the Hilbert transform. We derive the Hilbert transform (Appendix A) and the approximate partial derivative by using the finite impulse response (FIR) filter (Pei and Wang, 001). We use a D Hilbert transform to approximate the partial derivatives of the wave field, whih redues the side effet of strong high-frequeny random noise owing to the derivative algorithm. The redefined noniterative loal dip of the seismi data is σ = ( P x / P y ) = F F T 1 [ P (x)] F F T 1 [ P (y)] = F F T 1 1 [ x P (x)] F F T 1 1 [ y P (y)] F F T 1 [H HT (x)] F F T 1 [H HT (y)] H HT x H HT y, (5) where P (x) is the frequeny response funtion of the partial derivative in the x diretion and P (y) is the frequeny response funtion of the partial derivative in the y diretion. The dimensions are ignored in the derivation and does not depend on the time and spae sampling intervals; thus, we take x = y. H HT (x) is the frequeny response funtion of the Hilbert transform in the x diretion and H HT (y) is the frequeny response funtion of the Hilbert transform in the y diretion. H HT x and H HT y are the omponents of the D Hilbert transform in the x and y diretion, respetively. Using equation 5, we alulate the loal seismi dip attribute by using the D Hilbert transform instead of the derivative operation. Beause division is required in equation 5 and the denominator might beome zero, we add the nonzero onstant ε in the denominator σ H HT x H HT y + ε. (6) Fomel (007) proposed the shaping regularization for imposing regularization onstraints in estimation problems and defined the loal seismi attributes. In this paper, we use the same method to onstrain the division and smooth the loal dip by using the Gaussian smooth operator as the regularization operator. To show the validity of the proposed dip alulation method, we onstrut a syntheti seismi model and add white Gaussian random noise, as shown in Figure 1a. The omponents of the D Hilbert transform in the x and y diretion are shown in Figures 1b and 1, respetively. We obtain the dip of the seismi data by using the ratio of the two omponents and alulate the smoothing onstraints, as shown in Figure 1d. We see that the alulation results an aurately reflet the dip value of the original data at different loations, suh as the tilted layers at the top the underlying strata with the sinusoidal flutuations, and the fault loation. Using the D Hilbert transform and shaping regularization, we obtain the smooth loal dip attribute. Another effetive alulation method of the time-varying and spae-variant seismi loal dip is based on the plan-wave destrution (PWD) filter proposed by Fomel (00). The PWD filter realizes the plane-wave propagation aross different traes, while the total energy of the propagating wave stays invariant, by using an all-pass digital filter in the time domain and a Taylor expansion of the all-pass filter frequeny.

5 Liu et al. 5 Dip estimation using Riesz transform a b d Figure 1: Loal seismi dip based on the D Hilbert transform. Syntheti seismi data (a), time omponent of the D Hilbert transform (b), spae omponent of the D Hilbert transform (), and loal seismi dip (d).

6 Liu et al. 6 Dip estimation using Riesz transform We obtain the relation of the PWD and spae-time-varying loal seismi dip by using the Gauss-Newton algorithm to solve the nonlinear problem of loal seismi dip. This method an be essentially understood as solving an impliit finite-differene sheme for the loal planewave equation. The disadvantage of the PWD-based alulation method is its slow omputation speed, whih is espeially worse at higher order onditions. The omputational ost of the proposed method is proportional to N x N t, where N x N t is the data size, whereas the omputational effiieny of the PWDbased dip estimation method is proportional to N iter N x N t, where N iter is the number of iterations. Hene, to ahieve similar auray, the dip estimation method based on the D Hilbert transform requires a smaller number of iterations than the PWD-based method. The dip of seismi events ontrols the trend of the onstruted seismi model; thus, next, we need to apply filtering along the trend. The seleted filtering method must simultaneously suppress the seismi noise and protet strutural information. Nonstationary polynomial fitting Traditional stationary regression is used to estimate the oeffiients a i, i = 1,,..., N by minimizing the predition error between a master signal s(x) (where x represents the oordinates of a multidimensional spae) and a olletion of slave signals L i (x), i = 1,,..., N (Fomel, 009) E(x) = s(x) N a i L i (x). (7) When x is 1D and N =, L 1 (x) = 1 and L (x) = x, the problem of minimizing E(x) amounts to fitting a straight line a 1 + a 1 x to the master signal. Nonstationary regression is similar to equation 7 but allows the oeffiients a i (x) to vary with x, and the error (Fomel, 009) E(x) = s(x) i=1 N a i (x)l i (x) (8) is minimized to solve for the multinomial oeffiients a i (x). The minimization beomes an ill-posed problem beause a i (x) rely on the independent variables x. To solve the ill-posed problem, we onstrain the oeffiients a i (x). Tikhonov s regularization (Tikhonov, 1963) is a lassial regularization method that amounts to the minimization of the following funtional (Fomel, 009) F (a) = E(x) + ε i=1 N i=1 D[a i (x)], (9) where D is the regularization operator and ε is a salar regularization parameter. When D is a linear operator, the least-squares estimation redues to linear inversion

7 Liu et al. 7 Dip estimation using Riesz transform (Fomel, 009) where a = [a 1 (x)a (x) a N (x)] T, a = A 1 d, (10) d = [L 1 (x)s(x)l (x)s(x) L N (x)s(x)] T, and the elements of matrix A are A ij (x) = L i (x)l j (x) + ε δ ij D T D. Figure : Least-squares linear fitting ompared with nonstationary polynomial fitting. Next, we use a simple signal to simulate the variation of the amplitude of a nonstationary event with random noise (dashed line in Figure ). In Figure, the dot dashed line denotes the results of the least-squares linear fitting and the solid line denotes the results of the nonstationary polynomial fitting. We ompare the leastsquares linear fitting and nonstationary polynomial fitting results, and we find that the nonstationary polynomial fitting models the urve variations more aurately for events with variable amplitude, partiularly for 40 < x < 60. SYNTHETIC DATA TESTS We onstrut a new struture-oriented filtering method based on the D Hilbert transform with nonstationary polynomial fitting and apply it to syntheti data (Figure 1a). The loal seismi dip (Figure 1d) ontrols the trend of the event, and we apply nonstationary polynomial fitting along the diretion of the dip for fast strutural interpretation using struture-oriented filtering. We ahieve ontinuous model protetion in the diretion of dip, and noise attenuation and fault protetion beause of the use of nonstationary polynomial fitting. Nine sampling points are used in the struture-oriented filtering and five sampling points in the nonstationary polynomial fitting. The filtering results are shown in Figure 3a and the differene profile is shown in Figure 3b. Figure 3a shows that the upper tilted layer, the lower sinusoidal layer, and the fault information are preserved, while the noise is learly suppressed. Random noise onstitutes most of the differene profile without any tilted layer and fault

8 Liu et al. 8 Dip estimation using Riesz transform a b d e Figure 3: Analysis of results using different struture-oriented filtering. Nonstationary polynomial fitting (a), differene profile of nonstationary polynomial fitting (b), loal PWD-based dip (), median filter (d), and differene profile of median filter (e).

9 Liu et al. 9 Dip estimation using Riesz transform information left beause the loal seismi dip annot reflet the trend of the layers and owing to the attenuation of the limited effetive information. To ompare the proposed method with the PWD-based loal dip estimation method (Figure 3) with similar dip auray (Figure 1d), we hoose the median filter and show the results in Figure 3d and the differene profile in Figure 3e. We ompare the two profiles after the appliation of the median filter. We find more useful strutural information than the method we proposed. That means the method we proposed has better effet. FIELD DATA TESTS a b Figure 4: Comparison of proessing results. filtering (), Differene profile (d). d Field data (a), Loal dip (b), After For field data proessing, we hose the D profile of 3D poststak data (Liu and Chen, 013). The shallow strutures are simple planar layers and the deep strutures are omplex urved layers. First, we use the proposed method, whih is based on the D Hilbert transform, to ompute the orresponding loal seismi dip attribute (Figure 4b). From Figure 4b, we see that the dip hanges smoothly and steadily in

10 Liu et al. 10 Dip estimation using Riesz transform the midshallow layer orresponding to the ontinuous event in the profile, whereas the variation of the dip in the deep layer is relatively larger, whih haraterizes the bending event in the mid-deep layer. The trend of the loal seismi events an be determined by using the dip attribute; thus, we selet the filtering window, whih is determined by the dip, and use nonstationary polynomial fitting for filter proessing. The window size of the struture-oriented data onsists of 11 sampling points and the window size of the nonstationary polynomial fitting omprises seven sampling points. Figure 4 shows the denoising results. We see that the random noise in the raw profile is suppressed, the whole setion is learer, and the ontinuity of the plane event (0.1s-0.3s) in the shallow layer and the urved event (below 0.3s) in the deep layer has improved. The differene profile (Figure 4d) shows that the removed noise is mainly irrelevant random noise and the information is well preserved. CONCLUSIONS We propose a seismi dip estimate method based on the D Hilbert transform. We ompute the stable dip by using the noniterative approximation relation within the middle frequeny band, and improve the omputational effiieny relative to the iterative dip algorithm based on the PWD filter. We ombine the proposed method with nonstationary polynomial fitting to suppress the seismi random noise using the omputed loal seismi dip. We predit the seismi struture trend using the struture-oriented window based on the seismi dip, while balaning the random noise attenuation and signal preservation via filtering with the nonstationary polynomial fitting. The proposed method suppresses the seismi noise and strongly depend on the of dip trend predition. The auray of omputed dip is diretly affeted by filtering. The method is not appliable at strong noise onditions. We use syntheti model and field data proessing, to demonstrate the appliability of the proposed method. APPENDIX A: HILBERT TRANSFORM DERIVATION FOR APPROXIMATING THE PARTIAL DERIVATIVE Derivation of the FIR transfer funtion for the frequeny response of digital differentiators First, to haraterize the FIR for signal differentiators, we transform the Leibniz series arsinx to power series (Lehmer, 1985) 1 x arsinx 1 x = x [1 + (m)!! (m + 1)!! xm ]. (A-1)

11 Liu et al. 11 Dip estimation using Riesz transform We substitute sin( ω ) for x, and after rearrangement and trunation of the first M terms, we obtain ω 1 sin ω = sin ω [ 1 + M (m)!! (m + 1)!! ( ) ] (A-) m 1 osω + o(( 1 osω ) M+1 ) and after manipulation [ ω = sin ω M ( ) ] m osω (m)!! 1 osω o(( 1 osω ) M+1 ) (m + 1)!! [ M ( ) ] m (m)!! 1 osω = sinω o(( 1 osω ) M+1 ). (m + 1)!! (A-3) We ignore the higher order terms and we obtain the (M + )th-order ausal transfer funtion of the derivative operator as { ˆF DD (z) 1 M z z M (m)!! + [ (1 ] } z 1 ) m (m + 1)!! z (M m). (A-4) 4 Derivation of the FIR transfer funtion for the frequeny response of the Hilbert transform The ideal frequeny response of the Hilbert transform is expressed as H IHT (ω) = i sgn ω = i ω { i, π < ω < 0 ω =. (A-5) i, 0 < ω < π From equations 4 and A-5, we obtain the differene as 1/ ω. For sgn x = x x = xf(x ), x 0 and f(u) = 1 u, u > 0, the Taylor series of f(u) at enter is expressed f(u) = 1 [1 + (m 1)!! (m)!! (A-6) ] ( 1 u ) m, (A-7) where (m 1)!! = (m 1), (m)!! = (m). Consequently, the signum funtion sgnx is expressed sgn x = x ) ] m (m 1)!! [1 + (1 x. (A-8) (m)!!

12 Liu et al. 1 Dip estimation using Riesz transform We substitute sinω for x, based on sgnω=sgn(sinω) for π < ω < π, trunate the series at the first M terms, and obtain the sinusoidal power series of the signum funtion as sgn ω = sinω M ( ) m ] (m 1)!! [1 + 1 sin ω + ((1 sin ω ) M+1 ) (A-9) (m)!! The series in A-9 onverges for 1 < 1 sinω < 1; that is, has to be larger than 1/. On the other hand, the expansion enter in the x-domain is assoiated to the frequeny enter in the ω-domain via the relation = sin ω. Therefore, = sin ω must be less than or equal to 1. Aordingly, is onstrained by 1/ < 1 and the orresponding ω is within the range [π/4, π/]. Clearly, the ideal frequeny response is well approximated within the middle frequeny band. Multiplying A-9 by i and substituting z z 1 for sinω, the transfer funtion for the zero phase FIR of the i Hilbert transform is expressed as { [ H HT (z, ) z M z 1 (m 1)!! ( ) ] z z 1 m } (A-10) (m)!! To obtain the ausal transfer funtion, H HT (z, ) is multiplied by z M 1 and the resultant transfer funtion of the FIR Hilbert transform of the (M +)th-order is Ĥ HT (z, ) 1 z { z M + M [ (m 1)!! z (M m) z + 1 (m)!! ( ) ] 1 z m } (A-11) For M=0, the transfer funtions of equations A-4 and A-11 are approximated as Ĥ HT (z, ) 1 z (A-1) ˆF DD (z) 1 z (A-13) We ompare equations A-1 and A-13, and we onlude that these two transfer funtions in middle frequeny band of the frequeny domain differ by the onstant oeffiient. 1 REFERENCES Bekara, M., and M. van der Baan, 009, Random and oherent noise attenuation by empirial mode deomposition: Geophysis, 74, V89 V98. Fehmers, G. C., and C. F. W. Hoker, 003, Fast strutural interpretation with struture-oriented filtering: Geophysis, 68,

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