基于非均匀Fourier变换的地震数据重建方法研究
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摘要
不规则采样地震数据会对地震数据的多道处理造成严重影响,将非均匀Fourier变换和贝叶斯参数反演方法相结合,对不规则空间带限地震数据进行反演重建.对每一个频率依据最小视速度确定出重建数据的带宽,然后从不规则地震数据中估计出重建数据的空间Fourier系数.将不规则地震数据重建视为信息重建的地球物理反演问题,运用贝叶斯参数反演理论来估计Fourier系数.在反演求解时,使用共轭梯度算法,以保证求解的稳定性,加快解的收敛速度.理论模型和实际资料处理验证了本方法的有效性和实用性.
The uneven seismic data impose serious impact on multi-trace processing algorithms, leading to suboptimal processing and imaging results. Reconstruction of uneven seismic data can be considered as a geophysical inverse problem. In this paper, a new reconstruction method based on non-uniform Fourier transform and Bayesian parameter estimation theory is proposed. The main idea of the method is calculating the band width depending on the minimum apparent velocity for each temporal frequency and estimating the Fourier spectrum from the uneven seismic data. Regarding to this inverse problem, it is solved by the conjugate gradient method for the stable solution and quick convergence. Examples on synthetic as well as on field data show that the proposed method is efficient and applicable.
The uneven seismic data impose serious impact on multi-trace processing algorithms, leading to suboptimal processing and imaging results. Reconstruction of uneven seismic data can be considered as a geophysical inverse problem. In this paper, a new reconstruction method based on non-uniform Fourier transform and Bayesian parameter estimation theory is proposed. The main idea of the method is calculating the band width depending on the minimum apparent velocity for each temporal frequency and estimating the Fourier spectrum from the uneven seismic data. Regarding to this inverse problem, it is solved by the conjugate gradient method for the stable solution and quick convergence. Examples on synthetic as well as on field data show that the proposed method is efficient and applicable.
引文
[1] Nurul Kabir M , Verschuur D J . Restoration of missing offsets by parabolic Radon transform [J]. Geophysical Prospecting, 1995 ,43 (3) : 347-368．
[2] 王维红,裴江云,张剑锋.加权抛物Radon变换叠前地震数据重建[J].地球物理学报,2007,50(3) :851-859． Wang W H, Pei J Y, Zhang J F. Prestack seismic data reconstruction using weighted parabolic Radon transform[J].Chinese J . Geophys. (In Chinese), 2007, 50 (3) :851-859．
[3] 张红梅,刘洪.基于稀疏离散τ-p变换的叠后地震道内插[J].石油地球物理勘探,2006,41(3) :281-285． Zhang H M, LIU H. Interpolation of poststack seismic traces based on sparse discrete τ-p transform[J]. OGP,2006, 41 (3) : 281-285．
[4] Duijndam W, Schonewille M, Hindriks K. Reconstruction of seismic signals, irregularly sampled along on spatial coordinate [J]. Geophysics, 1999, 64(2) :524-538．
[5] 刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究[J].地球物理学报,2004,47(2) :299-305． Liu X W, Liu H, Liu B. A study on algorithm for reconstruction of de-alias uneven seismic data[J]. Chinese J. Geophys. (in Chinese) , 2004 ,47(2) :299-305．
[6] 刘喜武,刘洪,刘彬.非均匀地震数据重建方法及其应用[J].石油物探,2004,43(5) :423-426． Liu X W, Liu H, Liu B. Reconstruction of uneven seismic data and its allocation[J]. GPP,2004,43(5) :423-426．
[7] 孟小红,郭良辉,张致付,等.基于非均匀快速傅里叶变换的最小二乘反演地震数据重建[J].地球物理学报,2008,(1) :235-241． Meng X H, Guo L H, Zhang Z F, et al. Reconstruction of seismic data with least squares inversion based on non-uniform fast Fourier transform[J]. Chinese J. Geophys. (in Chinese), 2008,51(1) :235-241．
[8] 孟小红,刘国峰,周建军.大间距地震数据重建方法研究[J].地球物理学进展,2006． 21(3) :687-691． Meng X H, Liu G F, Zhou J J. The study of reconstruction of large gap seismic data [J]. Progress in Geophysics (in Chinese), 2006, 21(3) :687-691．
[9] 王立明,刘保华,王揆洋.等间距地震道插值的傅里叶重建法[J].中国石油大学学报(自然科学版),2008,32(6) :51-56． Wang L M, Liu B H, Wang K Y. Interpolating equal group interval seismic traces by Fourier reconstruction[J]. Journal of China University of Petroleum, 2008, 32(6) :51-56．
[10] 李信富,李小凡.分形插值地震数据重建方法研究[J].地球物理学报,2008,51(4) :1196-1201． Li X F, Li X F. Seismic data reconstruction with fractal interpolation [J]. Chinese J. Geophys. (in Chinese), 2008, 51 (4) : 1196-1201．
[11] 李信富,李小凡.基于显式垂直比例因子的显式分形插值地震数据重建[J].地球物理学进展.2008,23(4) :1085-1091． Li X F, Li X F. Seismic data reconstruction by explicit fractal interpolation based on explicit vertical scaling factors [J]. Progress in Geophysics (in Chinese), 2008, 23(4) :1085-1091．
[12] 李冰,刘洪,李幼铭.三维地震数据离散光滑插值的共轭梯度法[J].地球物理学报,2002,45(5) :691-699． Li B, Liu H, Li Y M. 3-D seismic data discrete smooth interpolation using conjugate gradient [J]. Chinese J. Geophys. (in Chinese), 2002,45(5) :691-699．
[13] Naghizadeh M, Sacchi D. Sampling considerations for band-limited fourier reconstruction of aliased seismic data[J]. 71~(st) EAGE Conference & Exhibition,2009．
[14] Schonewille M,Klaedtke A, Vigner A. Anti-Aliasing, Anti-Leakage Fourier Transform [J]. 71st EAGE Conference & Exhibition,2009．
[15] Fomel S ,Liu Y. Four methods of data regularization[J]. 71~(st) EAGE Conference & Exhibition,2009．
[16] Claerbout J, Nichols D. Interpolation beyond aliasing by (tau-x) domain PEFs [J]. Expanded Abstracts of 53th Annual Internat EAGE Mtg, 1991, 1-10．
[17] Spitz S. Seismic trace interpolation in the f-x domain [J]. Geophysics, 1991, 56(6) :785-794．
[18] Soubaras R. Spatial interpolation of aliased seismic data [J]. Expanded Abstracts of 67th Annual Internat SEG Mtg, 1997, 1167-1190．
[19] 国九英,周兴元,俞寿朋.F-X域等道距道内插[J].石油地球物理勘探,1996,31(1) :28-34． Guo J Y, Zhou X Y, Yu SP. Iso2inlerval trace interpolation in F-X domain [J]. OGP, 1996, 31(1) :28-34．
[20] 国九英,周兴元.F-K域等道距道内插[J].石油地球物理勘探,1996,31(2) :211-218． Guo J Y, Zhou X Y. Iso2interval trace interpolation in F-K domain [J]. OGP, 1996, 31(2) :211-218．
[21] Ji J. Interpolation using prediction error filter simulation (PEFs) [J]. Expanded Abstracts of 73th Annual Internat SEG Mtg ,1993,1170-1173．
[22] Fomel S. Applications of plane-wave destruction filters [J]. Geophysics, 2002, 67(6) : 1946-1960．
[23] Gulunay N. Seismic trace interpolation in the Fourier transform domain [J]. Geophysics, 2003, 68(1) : 355-369．
[24] Naghizadeh M, Sacchi D. f-x adaptive seismic-trace interpolation[J]. Geophysics,2009,74(1) :9-16．
[25] Ronen J. Wave equation trace interpolation [J]. Geophysics, 1987, 52(7) :973-984．
[26] Chemingui N, Biondi B. Handling the irregular geometry in wide azimuth surveys [J]. Expanded Abstracts of 61th Annual Internat SEG Mtg, 1991, 32-35．
[27] Zwartjes P, Gisolf A. Fourier reconstruction with sparse inversion [J]. Geophysical Prospecting, 2007, 55(2) :199-221．
[28] Zwartjes P, Sacchi D. Fourier reconstruction of nonuniformly sampled, aliased seismic data [J]. Geophysics, 2007, 72(1) :21-32．
[29] Thorson J R, Claerbout J F. Velocity-stack and slant-stack stochastic inversion [J]. Geophysics, 1985, 50(12) :2727-2741．
[30] Zwartjes P, Duijndam A J. Optimizing reconstruction for sparse spatial sampling[J]. Expanded Abstracts of 70th Annual Internat SEG Mtg,2000,2162-2165．
[31] 李庆阳,关治,白峰彬.数值计算原理[M].北京:清华大学出版社,218-224． Li Q Y, Guan Z, Bai F B. The Theory of Numerical Calculation [M]. Peiking: Tinghua University Press, 218-224．
[32] Feichtinger H,Grochenig K,Strohmer T. Efficient Numerical Methods in Non-uniform Sampling Theory[J]. Numer. Math. , 1995, 69:423-440．
[2] 王维红,裴江云,张剑锋.加权抛物Radon变换叠前地震数据重建[J].地球物理学报,2007,50(3) :851-859． Wang W H, Pei J Y, Zhang J F. Prestack seismic data reconstruction using weighted parabolic Radon transform[J].Chinese J . Geophys. (In Chinese), 2007, 50 (3) :851-859．
[3] 张红梅,刘洪.基于稀疏离散τ-p变换的叠后地震道内插[J].石油地球物理勘探,2006,41(3) :281-285． Zhang H M, LIU H. Interpolation of poststack seismic traces based on sparse discrete τ-p transform[J]. OGP,2006, 41 (3) : 281-285．
[4] Duijndam W, Schonewille M, Hindriks K. Reconstruction of seismic signals, irregularly sampled along on spatial coordinate [J]. Geophysics, 1999, 64(2) :524-538．
[5] 刘喜武,刘洪,刘彬.反假频非均匀地震数据重建方法研究[J].地球物理学报,2004,47(2) :299-305． Liu X W, Liu H, Liu B. A study on algorithm for reconstruction of de-alias uneven seismic data[J]. Chinese J. Geophys. (in Chinese) , 2004 ,47(2) :299-305．
[6] 刘喜武,刘洪,刘彬.非均匀地震数据重建方法及其应用[J].石油物探,2004,43(5) :423-426． Liu X W, Liu H, Liu B. Reconstruction of uneven seismic data and its allocation[J]. GPP,2004,43(5) :423-426．
[7] 孟小红,郭良辉,张致付,等.基于非均匀快速傅里叶变换的最小二乘反演地震数据重建[J].地球物理学报,2008,(1) :235-241． Meng X H, Guo L H, Zhang Z F, et al. Reconstruction of seismic data with least squares inversion based on non-uniform fast Fourier transform[J]. Chinese J. Geophys. (in Chinese), 2008,51(1) :235-241．
[8] 孟小红,刘国峰,周建军.大间距地震数据重建方法研究[J].地球物理学进展,2006． 21(3) :687-691． Meng X H, Liu G F, Zhou J J. The study of reconstruction of large gap seismic data [J]. Progress in Geophysics (in Chinese), 2006, 21(3) :687-691．
[9] 王立明,刘保华,王揆洋.等间距地震道插值的傅里叶重建法[J].中国石油大学学报(自然科学版),2008,32(6) :51-56． Wang L M, Liu B H, Wang K Y. Interpolating equal group interval seismic traces by Fourier reconstruction[J]. Journal of China University of Petroleum, 2008, 32(6) :51-56．
[10] 李信富,李小凡.分形插值地震数据重建方法研究[J].地球物理学报,2008,51(4) :1196-1201． Li X F, Li X F. Seismic data reconstruction with fractal interpolation [J]. Chinese J. Geophys. (in Chinese), 2008, 51 (4) : 1196-1201．
[11] 李信富,李小凡.基于显式垂直比例因子的显式分形插值地震数据重建[J].地球物理学进展.2008,23(4) :1085-1091． Li X F, Li X F. Seismic data reconstruction by explicit fractal interpolation based on explicit vertical scaling factors [J]. Progress in Geophysics (in Chinese), 2008, 23(4) :1085-1091．
[12] 李冰,刘洪,李幼铭.三维地震数据离散光滑插值的共轭梯度法[J].地球物理学报,2002,45(5) :691-699． Li B, Liu H, Li Y M. 3-D seismic data discrete smooth interpolation using conjugate gradient [J]. Chinese J. Geophys. (in Chinese), 2002,45(5) :691-699．
[13] Naghizadeh M, Sacchi D. Sampling considerations for band-limited fourier reconstruction of aliased seismic data[J]. 71~(st) EAGE Conference & Exhibition,2009．
[14] Schonewille M,Klaedtke A, Vigner A. Anti-Aliasing, Anti-Leakage Fourier Transform [J]. 71st EAGE Conference & Exhibition,2009．
[15] Fomel S ,Liu Y. Four methods of data regularization[J]. 71~(st) EAGE Conference & Exhibition,2009．
[16] Claerbout J, Nichols D. Interpolation beyond aliasing by (tau-x) domain PEFs [J]. Expanded Abstracts of 53th Annual Internat EAGE Mtg, 1991, 1-10．
[17] Spitz S. Seismic trace interpolation in the f-x domain [J]. Geophysics, 1991, 56(6) :785-794．
[18] Soubaras R. Spatial interpolation of aliased seismic data [J]. Expanded Abstracts of 67th Annual Internat SEG Mtg, 1997, 1167-1190．
[19] 国九英,周兴元,俞寿朋.F-X域等道距道内插[J].石油地球物理勘探,1996,31(1) :28-34． Guo J Y, Zhou X Y, Yu SP. Iso2inlerval trace interpolation in F-X domain [J]. OGP, 1996, 31(1) :28-34．
[20] 国九英,周兴元.F-K域等道距道内插[J].石油地球物理勘探,1996,31(2) :211-218． Guo J Y, Zhou X Y. Iso2interval trace interpolation in F-K domain [J]. OGP, 1996, 31(2) :211-218．
[21] Ji J. Interpolation using prediction error filter simulation (PEFs) [J]. Expanded Abstracts of 73th Annual Internat SEG Mtg ,1993,1170-1173．
[22] Fomel S. Applications of plane-wave destruction filters [J]. Geophysics, 2002, 67(6) : 1946-1960．
[23] Gulunay N. Seismic trace interpolation in the Fourier transform domain [J]. Geophysics, 2003, 68(1) : 355-369．
[24] Naghizadeh M, Sacchi D. f-x adaptive seismic-trace interpolation[J]. Geophysics,2009,74(1) :9-16．
[25] Ronen J. Wave equation trace interpolation [J]. Geophysics, 1987, 52(7) :973-984．
[26] Chemingui N, Biondi B. Handling the irregular geometry in wide azimuth surveys [J]. Expanded Abstracts of 61th Annual Internat SEG Mtg, 1991, 32-35．
[27] Zwartjes P, Gisolf A. Fourier reconstruction with sparse inversion [J]. Geophysical Prospecting, 2007, 55(2) :199-221．
[28] Zwartjes P, Sacchi D. Fourier reconstruction of nonuniformly sampled, aliased seismic data [J]. Geophysics, 2007, 72(1) :21-32．
[29] Thorson J R, Claerbout J F. Velocity-stack and slant-stack stochastic inversion [J]. Geophysics, 1985, 50(12) :2727-2741．
[30] Zwartjes P, Duijndam A J. Optimizing reconstruction for sparse spatial sampling[J]. Expanded Abstracts of 70th Annual Internat SEG Mtg,2000,2162-2165．
[31] 李庆阳,关治,白峰彬.数值计算原理[M].北京:清华大学出版社,218-224． Li Q Y, Guan Z, Bai F B. The Theory of Numerical Calculation [M]. Peiking: Tinghua University Press, 218-224．
[32] Feichtinger H,Grochenig K,Strohmer T. Efficient Numerical Methods in Non-uniform Sampling Theory[J]. Numer. Math. , 1995, 69:423-440．
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