显式分形插值在有限元叠前逆时偏移成像中的应用
|
摘要
以分形理论为基础,借助于压缩映射原理和不动点理论及实变函数理论,给出了分形插值函数的显式表达形式,同时给出了垂直比例因子的局部显式表达式.利用该显式分形插值方法对稀疏采样的地震记录进行了插值加密处理,并对原始记录和分形插值重建记录进行了叠前逆时偏移处理.数值重建实验表明,显式分形插值重建的剖面与原始地震剖面非常类似,单道地震记录则表明显式分形插值重建的地震道是原始地震道的良好近似,缺失道的振幅和相位都得到了很好的估计.叠前偏移结果则表明,分形插值叠前偏移剖面是原始记录偏移剖面的良好近似,二者的分辨率几乎一样.
Based on the fractal theories,contractive mapping principles as well as the fixed point theory,by means of affine transform,this paper develops a novel Explicit Fractal Interpolation Function(EFIF)and puts forward the locally explicit vertical scaling factor.Interpolation reconstruction has been made to the sparse sampling data by the explicit fractal interpolation method and the pre-stack reverse-time migration has been made using the original and reconstructed seismic data.The numerical experiments demonstrate that the seismic section and seismograms reconstructed by the explicit fractal interpolation method resemble the original ones very well.The waveform of the missing traces could be estimated very well and also the amplitudes of the interpolated traces are good approximations of the original ones.The imaging sections obtained by using the fractal interpolated data resemble the original ones very well.The numerical experiments demonstrate that even with the sparse sampling we can still obtain the high accurate images of the earth′s interior structure by means of the explicit fractal interpolation method.
Based on the fractal theories,contractive mapping principles as well as the fixed point theory,by means of affine transform,this paper develops a novel Explicit Fractal Interpolation Function(EFIF)and puts forward the locally explicit vertical scaling factor.Interpolation reconstruction has been made to the sparse sampling data by the explicit fractal interpolation method and the pre-stack reverse-time migration has been made using the original and reconstructed seismic data.The numerical experiments demonstrate that the seismic section and seismograms reconstructed by the explicit fractal interpolation method resemble the original ones very well.The waveform of the missing traces could be estimated very well and also the amplitudes of the interpolated traces are good approximations of the original ones.The imaging sections obtained by using the fractal interpolated data resemble the original ones very well.The numerical experiments demonstrate that even with the sparse sampling we can still obtain the high accurate images of the earth′s interior structure by means of the explicit fractal interpolation method.
引文
[1]Larner K.,Gibson B,Rothman,D.Trace interpolation andthe design of seismic surveys[J].Geophysics,1981,46:407~415.
[2]Kabir M.M.N.,and Verschuur D.J.,Restoration of missingoffsets by parabolic Radon Transformation[J].Geophys.Prosp.1995,43:347~368.
[3]Ronen J.Wave equation trace interpolation[J].Geophysics,1987,52:973~984.
[4]Spitz S.Seismic trace interpolation in F-X domain[J].Geo-physics,1991,56:785~794.
[5]Porsani M J.Seismic trace interpolation using half-step predic-tion filters[J].Geophysics,1999,64:1461~1467.
[6]孟小红,胡朝顺,刘海英,等.利用非均匀快速傅立叶变换进行地球物理数据插值[J].长春科技大学学报,2000,30(Suppl):150~153.Meng X H,Hu C S,Liu H Y,et al.The application of nonu-niform fast fourier transform to interpolation of geophysicaldata[J].Journal of Changchun University of Science andTechnology(in Chinese),2000,30(Suppl.):150~153.
[7]Craig M,Wittenbrink.IFS Fractal Interpolation for 2D and3D Visualization[J].Proceedings of the 6th IEEE Visualiza-tion Conference(VISUALIZATION'95),1995:77~84.
[8]孙博文,葛江华,张彤.分形插值函数及其应用[J].哈尔滨电工学院学报,1996,19(4):540~543.Sun B W,Ge J H,Zhang T.Fractal Interpolated Functionand Its Application[J].Electric Machines and Control,1996,19(4):540~543.
[9]杨松林.分形插值方法及其应用[J].铁道师院学报,2000,17(3):9~14.Yang S L.Fractal Interpolation and Its Applications[J].Jour-nal of Suzhou Railway Teachers College,2000,17(3):9~14.
[10]徐群芳.分段分形插值对平面曲线拟合的研究[J].雁北师范学院学报,2003,19(5):5~7.Xu Q F.Study on Subsection Fractal Interpolation for PlaneCurve Fitting[J].Journal of Yanbei Normal University,2003,19(5):5~7.
[11]范玉红,栾元重,王永,等.分形插值与传统插值方法相结合的研究[J].测绘科学,2005,30(2):76~77,80.Fan Y H,Luan Y Z,Wang Y,et al.The method study of com-bination of fractal interpolation and linear interpolation[J].Sci-ence of Surveying and Mapping,2005,30(2):76~77,80.
[12]杨杰.分形插值及其分形维数研究[J].武汉工业学院学报,2006,25(1):9~11.Yang J.Study of fractal interpolation and fractal dimension[J].Journal of Wuhan Polytechnic University,2006,25(1):9~11.
[13]张美根,王妙月.各向异性弹性波有限元叠前逆时偏移[J].地球物理学报,2001,44(5):711~719.Zhang M G,Wang M Y.Prestack finite element reverse-timemigration for anisotropic elastic waves[J].Chinese Journal ofGeophysics,2001,44(5):711~719.
[14]常旭,刘伊克,桂志先.反射地震零偏移距逆时偏移方法用于隧道超前预报[J].地球物理学报,2006,49(5):1482~1488.Chang X,Liu Y K,Gui Z X.Zero-offset reverse time migra-tion for prediction ahead of tunnel face[J].Chinese J.Geo-phys.(in Chinese),2006,49(5):1482~1488.
[15]Claerbout J F.著,许云译.地震成像理论及方法[M].北京:石油工业出版社,1991.Claerbout J F,Xu Y.Seismic image theory and method[M].Beijing:Petroleum Industry Press,1991.
[16]Barnsley M F.Fractal functions and interpolation[J].Con-structive Approximation,1986,16(2):303~329.
[17]Geoltrain S,Brac J.Can we image complex structures withfirst-arrival traveltime?[J].Geophysics,1993,58(4):564~575.
[2]Kabir M.M.N.,and Verschuur D.J.,Restoration of missingoffsets by parabolic Radon Transformation[J].Geophys.Prosp.1995,43:347~368.
[3]Ronen J.Wave equation trace interpolation[J].Geophysics,1987,52:973~984.
[4]Spitz S.Seismic trace interpolation in F-X domain[J].Geo-physics,1991,56:785~794.
[5]Porsani M J.Seismic trace interpolation using half-step predic-tion filters[J].Geophysics,1999,64:1461~1467.
[6]孟小红,胡朝顺,刘海英,等.利用非均匀快速傅立叶变换进行地球物理数据插值[J].长春科技大学学报,2000,30(Suppl):150~153.Meng X H,Hu C S,Liu H Y,et al.The application of nonu-niform fast fourier transform to interpolation of geophysicaldata[J].Journal of Changchun University of Science andTechnology(in Chinese),2000,30(Suppl.):150~153.
[7]Craig M,Wittenbrink.IFS Fractal Interpolation for 2D and3D Visualization[J].Proceedings of the 6th IEEE Visualiza-tion Conference(VISUALIZATION'95),1995:77~84.
[8]孙博文,葛江华,张彤.分形插值函数及其应用[J].哈尔滨电工学院学报,1996,19(4):540~543.Sun B W,Ge J H,Zhang T.Fractal Interpolated Functionand Its Application[J].Electric Machines and Control,1996,19(4):540~543.
[9]杨松林.分形插值方法及其应用[J].铁道师院学报,2000,17(3):9~14.Yang S L.Fractal Interpolation and Its Applications[J].Jour-nal of Suzhou Railway Teachers College,2000,17(3):9~14.
[10]徐群芳.分段分形插值对平面曲线拟合的研究[J].雁北师范学院学报,2003,19(5):5~7.Xu Q F.Study on Subsection Fractal Interpolation for PlaneCurve Fitting[J].Journal of Yanbei Normal University,2003,19(5):5~7.
[11]范玉红,栾元重,王永,等.分形插值与传统插值方法相结合的研究[J].测绘科学,2005,30(2):76~77,80.Fan Y H,Luan Y Z,Wang Y,et al.The method study of com-bination of fractal interpolation and linear interpolation[J].Sci-ence of Surveying and Mapping,2005,30(2):76~77,80.
[12]杨杰.分形插值及其分形维数研究[J].武汉工业学院学报,2006,25(1):9~11.Yang J.Study of fractal interpolation and fractal dimension[J].Journal of Wuhan Polytechnic University,2006,25(1):9~11.
[13]张美根,王妙月.各向异性弹性波有限元叠前逆时偏移[J].地球物理学报,2001,44(5):711~719.Zhang M G,Wang M Y.Prestack finite element reverse-timemigration for anisotropic elastic waves[J].Chinese Journal ofGeophysics,2001,44(5):711~719.
[14]常旭,刘伊克,桂志先.反射地震零偏移距逆时偏移方法用于隧道超前预报[J].地球物理学报,2006,49(5):1482~1488.Chang X,Liu Y K,Gui Z X.Zero-offset reverse time migra-tion for prediction ahead of tunnel face[J].Chinese J.Geo-phys.(in Chinese),2006,49(5):1482~1488.
[15]Claerbout J F.著,许云译.地震成像理论及方法[M].北京:石油工业出版社,1991.Claerbout J F,Xu Y.Seismic image theory and method[M].Beijing:Petroleum Industry Press,1991.
[16]Barnsley M F.Fractal functions and interpolation[J].Con-structive Approximation,1986,16(2):303~329.
[17]Geoltrain S,Brac J.Can we image complex structures withfirst-arrival traveltime?[J].Geophysics,1993,58(4):564~575.
版权所有：© 2023 中国地质图书馆 中国地质调查局地学文献中心