边界条件和偏移距的变化对分形插值精度的影响
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摘要
地震数据空间道插值是地震资料预处理的重要内容之一。本文以分形理论为基础,借助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式及垂直比例因子的局部显式表达式。正演模拟中采用了吸收边界条件,边界条件的应用使得靠近边界的地震道分形插值重建的误差要高于远离边界的地震道,即随着检波器逐渐靠近边界,分形插值重建的误差逐渐增大。偏移距的变化也会对分形插值重建的精度产生影响,随着偏移距的增大,分形插值重建的误差有增大的趋势,但不明显。总体上来说,分形插值重建的地震道是原始地震道的良好近似,相位和振幅都得到了很好的恢复。
Spatial trace interpolation is one of the important issues in seismic data processing.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 and put forward the locally explicit vertical scaling factor.Absorbing boundary condition has been applied in forward modeling.The residuals of the reconstructed seismograms near the boundary are larger than that far away from the boundary.The changes of the offsets also affect the precision of the fractal interpolation,i.e.,with the increasing offsets the residuals are becoming larger and larger,but the trend is not obvious.All in all,the seismograms reconstructed by 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 a good approximation of the original ones.
Spatial trace interpolation is one of the important issues in seismic data processing.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 and put forward the locally explicit vertical scaling factor.Absorbing boundary condition has been applied in forward modeling.The residuals of the reconstructed seismograms near the boundary are larger than that far away from the boundary.The changes of the offsets also affect the precision of the fractal interpolation,i.e.,with the increasing offsets the residuals are becoming larger and larger,but the trend is not obvious.All in all,the seismograms reconstructed by 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 a good approximation of the original ones.
引文
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[8]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):150-153.Meng X H,Hu C S,Liu H Y,et al.The applicationof nonuniform fast fourier transform to interpolationof geological data[J].Journal of Changchun Universi-ty Science and Technology,2000,(30):150-153.
[9]WangY.Seismic trace interpolation in the f-x-y do-main[J].Geophysical Prospecting,2003,(51):75-87.
[10]孟小红,周建军,郭良辉.非均匀Fourier变换在地震数据重建中的应用[M]∥张中杰,高锐,吕庆田,等.中国大陆地球深部结构与动力学研究.北京:科学出版社,2004.501-513.Meng X H,Zhou J J,Guo L H.Application ofnonuniform FFT in reconstruction of seismic data[M]∥Zhang Z J,Gao R,Lu。Q T,et al.Research ofDeep Structure and Dynamics of the Earth of China.Beijing:Science Press,2004.501-513.
[11]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 re-construction of de-alias uneven seismic data[J].Chi-nese Journal of Geophysics,2004,47(2):299-305.
[12]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularlysampled along one spatial direction[J].Geophysics,1999,(64):524-538.
[13]Kabir M M N,Verschuur D J.Restoration of missingoffsets by parabolic Radon Transformation[J].Geo-physical Prospecting,1995,(43):347-368.
[14]杨绍国,何明星.地震剖面的随机分形插值方法[J].物探化探计算技术,1998,20(3):218-221.Yang S G,He M X.The random fractal interpolationof seismic section[J].Computing Techniques for Geo-physical and Geochemical Exploration,1998,20(3):218-221.
[15]刘刚,胡远来,贾.随机分形插值法在地震数据处理中的应用[J].物探化探计算技术,2002,24(4):304-308.Liu G,Hu Y L,Jia Y.The application of the randomfractal interpolation to seismic data processing[J].Computing Techniques for Geophysical and Geochemi-cal Exploration,2002,24(4):304-308.
[16]沙震,阮火军.分形与拟合[M].杭州:浙江大学出版社,2005.123-128.Sha Z,Ruan H J.Fractal and Fitting[M].Hangzhou:Zhejiang University Press,2005.123-128.
[2]Lu L.Application of local slant-stack to trace interpo-lation[C]∥55thAnnual International Meeting of theSociety of Exploration Geophysics.Geophys.Expand-ed Abstracts,Washington D C:[s.n.],1985.560-562.
[3]RonenJ.Wave equation trace interpolation[J].Geo-physics,1987,(52):973-984.
[4]Ja。gerC,Hertweck P,Hubral P.The unified approachand its applications:wave-equation based trace inter-polation[C]∥72ndAnnual International Meeting of theSociety of Exploration Geophysics.Expanded Ab-stracts,Salt Lake City:[s.n.],2002.2178-2181.
[5]Gu。lu。nay N.F-X domain least-square tau-p and tau-q[C]∥60thAnnual International Meeting of the Societyof Exploration Geophysics.Expanded Abstracts,SanFrancisco:[s.n.],1990.1607-1610.
[6]Gu。lu。nay N,Chambers R E.Un-aliased f-k domaintrace interpolation(UFKI)[C]∥66thAnnual Interna-tional Meeting of the Society of Exploration Geo-physics.Expanded Abstracts,Denver CO:[s.n.],1996.1461-1464.
[7]Gu。lu。nay N,Chambers R E.Generalized f-k domaintrace interpolation[C]∥67thAnnual InternationalMeeting of the Society of Exploration Geophysics.Geophys.Expanded Abstracts,Alberta:[s.n.],1997.1100-1103.
[8]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):150-153.Meng X H,Hu C S,Liu H Y,et al.The applicationof nonuniform fast fourier transform to interpolationof geological data[J].Journal of Changchun Universi-ty Science and Technology,2000,(30):150-153.
[9]WangY.Seismic trace interpolation in the f-x-y do-main[J].Geophysical Prospecting,2003,(51):75-87.
[10]孟小红,周建军,郭良辉.非均匀Fourier变换在地震数据重建中的应用[M]∥张中杰,高锐,吕庆田,等.中国大陆地球深部结构与动力学研究.北京:科学出版社,2004.501-513.Meng X H,Zhou J J,Guo L H.Application ofnonuniform FFT in reconstruction of seismic data[M]∥Zhang Z J,Gao R,Lu。Q T,et al.Research ofDeep Structure and Dynamics of the Earth of China.Beijing:Science Press,2004.501-513.
[11]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 re-construction of de-alias uneven seismic data[J].Chi-nese Journal of Geophysics,2004,47(2):299-305.
[12]Duijndam A J W,Schonewille M A,Hindriks C O H.Reconstruction of band-limited signals,irregularlysampled along one spatial direction[J].Geophysics,1999,(64):524-538.
[13]Kabir M M N,Verschuur D J.Restoration of missingoffsets by parabolic Radon Transformation[J].Geo-physical Prospecting,1995,(43):347-368.
[14]杨绍国,何明星.地震剖面的随机分形插值方法[J].物探化探计算技术,1998,20(3):218-221.Yang S G,He M X.The random fractal interpolationof seismic section[J].Computing Techniques for Geo-physical and Geochemical Exploration,1998,20(3):218-221.
[15]刘刚,胡远来,贾.随机分形插值法在地震数据处理中的应用[J].物探化探计算技术,2002,24(4):304-308.Liu G,Hu Y L,Jia Y.The application of the randomfractal interpolation to seismic data processing[J].Computing Techniques for Geophysical and Geochemi-cal Exploration,2002,24(4):304-308.
[16]沙震,阮火军.分形与拟合[M].杭州:浙江大学出版社,2005.123-128.Sha Z,Ruan H J.Fractal and Fitting[M].Hangzhou:Zhejiang University Press,2005.123-128.
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