基于显式垂直比例因子的显式分形插值地震数据重建
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摘要
在前人工作的基础上对分形插值方法作了详细的探讨,给出了分形插值函数的显式表达方式.在量纲分析的基础上给出了垂直比例因子的局部显式表达式,旨在提高地震道插值重建的精度及突出局部信息,并从单道地震图的角度分析其在地震道插值重建中的应用效果.研究了垂直比例因子的变化对分形插值精度的影响.数值实验表明,随着垂直比例因子的增大,分形垂直的误差逐渐增大,二者之间呈显出指数增长的趋势.该法克服了随机分形插值方法必须进行多步迭代的弱点,提高了计算效率.通过对理论地震道插值重建的分析,说明了本文分形插值方法的高精度和高效率.本文提出的显式分形插值方法既能够突出地震道数据的局部信息,又较好地保持了地震道数据的总体变化趋势.
In order to improve the accuracy of the reconstructed seismic data,this paper makes a detailed study into the fractal interpolation method on the basis of the former work.The explicit expression of the fractal interpolation function is applied and the locally explicit expression for the vertical scaling factors is put forward on the basis of dimension analysis.At the same time the interpolating accuracy is analyzed from the single trace seismographs.The influence of the vertical scaling factors on the precision of the fractal interpolation has been investigated.The numerical experiments demonstrate that the interpolating residual is in proportion to the exponent function with increasing vertical scaling factors.This explicit fractal interpolation method avoids the iteration that is inevitable in the traditional interpolation method,and then it improves the computational efficiency.By analyzing the theoretical seismograms and the reconstructed seismograms and the differences between them,the numerical results demonstrate that the fractal interpolation method put forward here has high accuracy.The method not only makes the local information obvious but also preserves the overall characteristics well of the seismic data.
In order to improve the accuracy of the reconstructed seismic data,this paper makes a detailed study into the fractal interpolation method on the basis of the former work.The explicit expression of the fractal interpolation function is applied and the locally explicit expression for the vertical scaling factors is put forward on the basis of dimension analysis.At the same time the interpolating accuracy is analyzed from the single trace seismographs.The influence of the vertical scaling factors on the precision of the fractal interpolation has been investigated.The numerical experiments demonstrate that the interpolating residual is in proportion to the exponent function with increasing vertical scaling factors.This explicit fractal interpolation method avoids the iteration that is inevitable in the traditional interpolation method,and then it improves the computational efficiency.By analyzing the theoretical seismograms and the reconstructed seismograms and the differences between them,the numerical results demonstrate that the fractal interpolation method put forward here has high accuracy.The method not only makes the local information obvious but also preserves the overall characteristics well of the seismic data.
引文
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[2]Barsley M.F.Fractal functions and interpolation[J].Con-structive Approximation,1986(2):303~329.
[3]胡瑞安,胡纪阳,徐树公.分形的计算机图象及其应用[M].北京:中国铁道出版社,1995,6l~85.HU Rui-an,HU Ji-yang,XU Shu-gong.Computer Graphicsand Applications of Fractal[M].Beijing:China Railway Pub-lishing House,1995,6l~85.
[4]宋万寿,杨晋吉.一种地表造型方法[J].小型微型计算机系统,1996(3):32~36.Song W S,Yang J J.Modeling method for the earth’s surface[J].Mini-Micro Systems,1996(3):32~36.
[5]Xie H P.Fractals in rock mechanics[M].Netherlands:A.A.Balkema Publishers,1993,70~78.
[6]Larner K,Gibson B,Rothman D.Trace interpolation and thedesign of seismic surveys[J].Geophysics,1981,46:407~415.
[7]Kabir M M N,Verschuur D J.Restoration of missing offsetsby parabolic Radon Transformation[J].Geophys.Prosp.1995,43:347~368.
[8]Ronen,J.Wave equation trace interpolation[J].Geophysics,1987,52:973~984.
[9]Spitz S.Seismic trace interpolation in F-X domain[J].Geo-physics,1991,56:785~794.
[10]Porsani M J.Seismic trace interpolation using half-step pre-diction filters[J].Geophysics,1999,64:1461~1467.
[11]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(Sup-pl.):150~153.Meng X H,Hu C S,Liu H Y,et al.The application of non-uniform fast fourier transform to interpolation of geophysicaldata[J].Journal of Changchun University of Science andTechnology(in Chinese),2000,30(Suppl.):150~153.
[12]张红梅,刘洪.基于稀疏离散τ-p变换的非均匀地震道重建[J].石油物探,2006,45(2):141~145.Zhang H M,Liu H.Sparseness discrete-τp transform in ir-regular seismic trace reconstruction[J].Geophysical Prospec-ting for Petroleum,2006,45(2):141~145.
[13]Craig M,Wittenbrink.IFS Fractal Interpolation for 2D and3D Visualization[C].Proceedings of the 6th IEEE Visualiza-tion Conference(VISUALIZATION'95),1995:77~84.
[14]孙博文,葛江华,张彤.分形插值函数及其应用[J].机电与控制学报,1996,19(4):540~543.Sun B W,Ge J H,Zhang T.Fractal interpolated function andits application[J].Electric Machines and Control,1996,19(4):540~543.
[15]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.
[16]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.
[17]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 ofcombination of fractal interpolation and linear interpolation[J].Science of Surveying and Mapping,2005,30(2):76~77,80.
[18]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.
[19]沙震,阮火军编著.分形与拟合[M].杭州:浙江大学出版社,2005:123~130.Sha Z,Ruan H J.Fractals and fitting[M].Hangzhou:Zhengjiang University Press,2005:123~130.
[20]龙晶凡.分形插值的条件[J].北京师范大学学报(自然科学版),2001,37(3):289~291.Long J F.The condition of fractal interpolation[J].Journalof Beijing Normal University(Natural Science),2001,37(3):289~291.
[21]冯志刚,周其生.关于一类分形插值稳定性问题[J].安庆师范学院学报(自然科学版),1998,4(4):18~21.Feng Z G,Zhou Q S.Stability of a kind of fractal interpola-tion[J].Journal of Anqing Teachers College(Natural Sci-ence),1998,4(4):18~21.
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