Shear Noise Attenuation and PZ Matching for OBN Data with a New Scheme of Complex Wavelet Transform Can Peng, Rongxin Huang and Biniam Asmerom CGGVeritas Summary In processing of ocean-bottom-node (OBN) data, vertical particle velocity (Vz) data, recorded by geophones, and pressure (P) data, recorded by hydrophones, can be used conjunctively for up- and down-going wavefield separation. However, shear noise attenuation needs to occur in the Vz data before it can be matched to the P data. The noise attenuation and the matching can be achieved in one step by local attribute matching in the dualtree complex wavelet transform (DTCWT) domain. Shear noise on Vz data can be characterized as low frequency and composed of a wide range of local dips. Conventional DTCWT does recursive band analysis only in the LL-band (low-f and low-k band). Therefore, shear noise with k in the higher half-band cannot be wellresolved in the frequency domain due to the poor frequency resolution in the LH band. As a result, the attenuation of high-k shear noise can be inadequate in conventional DTCWT domain. This paper proposes a new scheme to do 2D complex wavelet transformations that can provide adaptive angular-resolving capability for each analysis stage and thus solve the problem of poor high-k shear noise suppression with conventional 2D DTCWT.
Introduction In ocean-bottom-node (OBN) and ocean-bottom-cable (OBC) data processing, pressure (P) data, recorded by hydrophones, and vertical particle velocity (Vz) data, recorded by geophones, are used together for up-going and down-going wave field separation. Up-going waves in P and Vz data have the same polarity, whereas downgoing waves in P data and Vz data have opposite polarity, as shown in Equation 1 (Hoffe, et al. 2000). Therefore, summation of P and Vz can separate the up- and down-going wave fields. However, real Vz data usually contain shear wave noise that does not exist in the P component (Paffenholz and Hays 2006). Thus, the shear wave noise in the Vz component needs to be attenuated (Gordon Poole, et al. 2012) before the Vz and P wavelets can be matched and summed. 1 Shear wave noise attenuation in Vz and matching between P and Vz can be achieved in one step with local attribute matching in the dual-tree complex wavelet transform (DTCWT) domain (Yu, et al. 2011). Yu et al. (2011) apply 2D DTCWT to both P and Vz components to obtain two sets of complex coefficients for each component (Selesnick and Kingsbury 2005). The amplitude of each complex coefficient of Vz is matched to that of P while preserving the phase. By doing this, the wavelet of Vz is matched to that of P, and the shear noise in Vz is also suppressed. At the same time the polarity of Vz traces will be preserved, hence after the processing Vz and P can be directly summed to separate up and down going waves. Conventional 2D DTCWT is implemented in a row-column separable way (Selesnick and Kingsbury 2005), i.e. analysing filters are applied in rows and columns separately in analysis stages; each analysis stage is equivalent to dividing the input F-K domain into four sub-bands: HH band (high-f high-k band), HL band, LH band and LL band. The next analysis stage is recursively performed only within the LL band. Therefore, in conventional 2D DTCWT, the division of the input F-K domain is similar to Figure 1(a). This division pattern has an intrinsic problem: high-k components have poor frequency resolution in the low frequencies. In real field seismic data, shear wave noise in Vz usually exists in a large k range, but with the conventional 2D DTCWT band division, high-k shear noise cannot be adequately isolated in the frequency domain. This inhibits the attenuation of high-k shear wave noise. 2D wavelet transform can also be viewed as decomposing an input on 2D wavelet bases with different orientations. Another drawback of the conventional 2D DTCWT is its limited angle resolving capability. In conventional 2D DTCWT, every stage has three wavelet bands (LH band, HH band and HL band), and each band accommodates two 2D wavelet bases that are conjugate to each other. Thus, in every stage the input is decomposed on 2D wavelet bases with exactly six orientations (Selesnick and Kingsbury 2005), (Yu, et al. 2011). The limitation of six orientations restricts the angular resolving capability of the conventional 2D DTCWT. For high-f or high-k bands (in early analysis stages), 2D wavelet bases with more than six orientations can be constructed. Method To overcome the drawback of conventional 2D DTCWT, we propose a new scheme for 2D complex wavelet transform that provides better angular resolution. We call it High Angular Resolution Complex Wavelet Transform (HARCWT). We keep the same wavelet bases in HARCWT as we do in conventional 2D DTCWT, and the 2D analysis is also achieved in a recursive, row-column separable way; however, the analysis stages in HARCWT are not performed only in the LL band. Instead, we do wavelet analysis in frequency sub-bands for all
k and vice versa. Figure 1 (c) shows the band division of HARCWT. From the band division, we note that all k components have equal resolution in frequency and vice versa. An important advantage of HARCWT is that it has better angular resolution than conventional 2D DTCWT. As mentioned above, in conventional 2D DTCWT, every stage accommodates exactly six orientations of the 2D wavelet bases, regardless of the real angular resolution it can achieve. The six 2D wavelet bases for stage i can be expressed as following:,,,,,,,,,,,,,,, where and are the two wavelet functions for the i-th stage, which are Hilbert transforms to each other; and are the two scaling functions for the i-th stage, also Hilbert transforms to each other (Tay and Kingsbury 1993). Figure 1 (b) shows the six wavelet bases of the first stage of a 4-stage conventional 2D DTCWT for a 128-by-128 sample input. In HARCWT, different stages accommodate different numbers of 2D wavelet orientations. The i-th stage accommodates 2(2n+3-2i) orientations, where n is the total number of stages of the transform; therefore, early stages have more angular resolution. This is reasonable - the early-stage 2D wavelet bases are sharp since they have either higher f or higher k than later stages. The expressions of the 2(2n+3-2i) wavelet functions for stage i of an n-stage HARCWT are given as:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Figure 1 (d) shows the eighteen wavelet bases of the first stage of a 4-stage HARCWT for a 128-by-128 sample input. The wavelet type used in the transform is the same as that in Tay and Kingsbury (1993).
Due to the better orientation resolving capability of HARCWT, high-k shear noise can be better isolated and then be better suppressed in the HARCWT domain than in conventional 2D DTCWT domain. Examples Now we test our method of PZ matching for a real-data common node gather. The input pressure record and the Vz component are shown in Figures 2(a) and 2(b). The Vz component of this gather has shear noise with high k in the deep section. Figure 2(c) shows the result of matching the Vz component to the input P using the conventional 2D DTCWT approach. We observe obvious residual high-k shear noise. The matching result in the HARCWT domain is shown in Figure 2(d). When comparing the deep region in Figures 2(c) and 2(d), we see the clear attenuation of the high-k shear noise by HARCWT. Figure 2(e) shows the difference between the matching result with conventional 2D DTCWT and that with HARCWT. Compared with conventional 2D DTCWT, HARCWT has more angular resolution capability, especially for high-k or high-f segments. The energy of high-k shear noise in Vz data can be better separated in certain f-k bands in HARCWT than in conventional 2D DTCWT.As a result the contrast between the noise contaminated f- k band of the Vz data and the noise free f-k band of P data is larger in HARCWT than in conventional 2D DTCWT. Better focusing and larger energy contrasts between the complex wavelet coefficients in P data and Vz data lead to more effective noise energy suppression and better signal preservation with the HARCWT method. Conclusions We propose to do PZ matching and shear noise attenuation in HARCWT domain. Compared to conventional 2D DTCWT, HARCWT has better frequency partitions for all k; hence, it provides more angular resolution than conventional 2D DTCWT. We demonstrated its improvement in shear noise attenuation using a real field data example. Acknowledgements The authors would like to thank CGGVeritas for permission to publish this work. We also owe many thanks to Qing Xu and Shuo Ji for inspirational discussions. We thank Apache for data show rights and Gordon Poole for providing datasets. References Poole, G., Casasanta, L., and Grion, S., 2012, Sparse τ-p Z-noise attenuation for ocean-bottom data: SEG Technical Program Expanded Abstracts, 31, 1-5. Hoffe, B. H., L. R. Lines, and P. W. Cary, 2000, Applications of obc recording: The Leading Edge, 19, 382 382. Ivan W. Selesnick, R. G. B., and N. G. Kingsbury, 2005, The dual-tree complex wavelet transform: IEEE Signal Process-ing Magazine, 22, 123 151. J. Paffenholz, P. Docherty, R. S., and D. Hays, 2006, Shear wave noise on obs vz data - part ii elastic modelling of scat-terers in the seabed: EAGE 68th Conference and Exhibition, B047. Tay, D. B. H., and N. G. Kingsbury, 1993, Flexible design of multidimensional perfect reconstruction fir 2-band filters using transformation of variables: IEEE Transaction on Image Processing, 2, 466 480.
Yu, Z., C. Kumar, and I. Ahmed, 2011, Ocean bottom seismic noise attenuation using local attribute matching filter: SEG Technical Program Expanded Abstracts, 30, 3586 3590. 0 k knyquist f fnyquist 0 (a) k knyquist (b) f fnyquist (c) (d) Figure 1: (a) the F-K division of the conventional 2D DTCW, the shadowed bands are generated in the first stage; (b) the wavelet bases for the first analysis stage of a 4-stage 2D DTCWT, which are concentrated along both directions and uneasy to distinguish the orientation;(c) the F-K division of HARCWT, the shadowed bands are generated in the first stage; (d) the wavelet bases for the first analysis stage of a 4-stage HARCWT, which have better angle resolution compared with those in (b). The input size is 128 128 a b c d e Figure 2: (a) the input P gather; (b) the input Vz gather; (c) Vz gather matched with conventional 2D DTCWT; (d) Vz gather matched with HARCWT; (e) The difference between (c) and (d)