地震数据重建方法原理及运用
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摘要
在数值分析领域中,许多方法都是借助于插值公式导出的。几乎所有的经典数值微分、数值求积和常微分方程的数值积分公式,都可以从插值公式中推导出来。地震勘探领域常见的插值重建方法按照数学原理大致可以分为三类:适用于均匀采样的预测误差滤波插值方法,基于算子的插值方法(如DMO和偏移算子)和在最小二乘意义下估算特定数学变换系数进行插值的方法。这里从数学原理上系统地阐述了这几种插值方法的基本原理,并利用数值试验和实际数据进行了检验。
Interpolation formulas are the starting points in the derivations of many methods in several areas of numerical analysis.Almost all the classical methods of numerical differentiation,numerical quadrature and numerical integration of ordinary differential equations are directly derivable from interpolation formulas.For the seismic data,the most commonly used methods based on a variety of principles have been proposed and in common they fall into the following three categories: Prediction-error filter based techniques frequently are used for uniformly sampled aliased data;To use operators based on the seismic experiment,such as DMO or migration operators;A third approach is based on estimating the coefficients of a particular transform that describe the data in a least-square sense.This paper systematically first introduces the basic principles for the above-mentioned interpolation methods on the points of mathematics,and then classifies the above-mentioned methods into several categories with respect to the application area of them in seismology and clarify the particular characteristic of each category.Finally,the numerical experiments and real data reconstruction demonstrate the validity and efficiency of the method.
Interpolation formulas are the starting points in the derivations of many methods in several areas of numerical analysis.Almost all the classical methods of numerical differentiation,numerical quadrature and numerical integration of ordinary differential equations are directly derivable from interpolation formulas.For the seismic data,the most commonly used methods based on a variety of principles have been proposed and in common they fall into the following three categories: Prediction-error filter based techniques frequently are used for uniformly sampled aliased data;To use operators based on the seismic experiment,such as DMO or migration operators;A third approach is based on estimating the coefficients of a particular transform that describe the data in a least-square sense.This paper systematically first introduces the basic principles for the above-mentioned interpolation methods on the points of mathematics,and then classifies the above-mentioned methods into several categories with respect to the application area of them in seismology and clarify the particular characteristic of each category.Finally,the numerical experiments and real data reconstruction demonstrate the validity and efficiency of the method.
引文
[1]年静波,汤磊,刘喜武.地质统计分析在地双资料砂泥岩分布预测中的应用[J].石油物探,2004,43(3):278.
[2]SPITZ S.Seismic trace interpolation in the f-x domain[J].Geophysics,1991,56(6):785.
[3]CANNING A,GARDNER G HF.Regularizing3-D da-ta sets with DMO[J].Geophysics,1996,61(4):1103.
[4]RONEN J.Wave equation trace interpolation[J].Geo-physics,1987,52(7):973.
[5]JAKUBOWICZ H.Wavefield reconstruction[C].Inter-nat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1994,64:1557.
[6]JAKUBOWICZ H.Seismic data acquisition[P].US Pa-tent,1997,5:648.
[7]LAMER K,GIBSONB,ROTHMAND.Trace interpola-tion and the design of seismic surveys[J].Geophysics,1981,46(1):407.
[8]LU L.Application of local slant-stack to trace interpo-lation[C].4Internat.Mtg.,Soc.Expl.Geophys.Ex-panded Abstracts,1985,55:560.
[9]KING G A,LEONG T K,FLINCHBAUGH B E.Therole of interpolation in seismic resolution[C].Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1984,54:766.
[10]MONK D,MCBEATHR G,WASONC B.Interpolationof aliased seismic traces[P].US Patent,1993,5:235.
[11]KOSTOV,C.,Finite aperture slant-stack transforms[J]SEP,1989,61:261.
[12]DARCHE G.Spatial interpolation using fast parabolictransform[C].Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1990,60:1647.
[13]KABIR MMN,VERSCHUUR D J.Applications of thegeneralized Radon transform[C].Internat.Mtg.,Eur.Assn.Geosci.Eng.,Extended Abstracts,1992,54:52.
[14]SCHONEWILLE MA,DUIJNDAMA J W.Decomposi-tion of signal and noise for sparsely sampled data usinga mixed-Fourier/Radon transform[C].Internat.Mtg.,Soc.Exp.Geophys.,Expanded Abstract,1996,66:1438.
[15]SACCHI M,ULRYCHT.High-resolution velocity gath-ers and offset space reconstruction[J].Geophysics,1995,60(4):1169.
[16]SPITZ S.Trace interpolation beyond aliasing and with-out picking[C].Internat.Mtg.,Soc.Exp.Geo-phys.,Expanded Abstracts,1989,59:1121.
[17]JI J.Interpolation using prediction-error filter simula-tion(PEFs)[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1993,63:1170.
[18]MANIN M,SPITZ S.Wavefield de-aliasing for acqui-sition configurations leading to coarse sampling[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Ab-stracts,1995,65:930.
[19]PORSANI M J.Seismic trace interpolation using half-step prediction filters[J].Geophysics,1999,64(5):1461.
[20]SOUBARAS R.Spatial interpolation of aliased seismicdata[C].Internat.Mtg.,Soc.Exp.Geophys.,Ex-panded Abstracts,1997,67:1167.
[21]CLAERBOUT J,NICHOLS D.Interpolation beyondaliasing by(t-x)domain PEFs[C].Internat.Mtg.,Eur.Assn.Geosci.Eng.,Expanded Abstracts,1991,53:1.
[22]PIEPRZAK A W,MCCLEAN J W.Trace interpolationof severely aliased events[C].Internat.Mtg.,Soc.Exp.Geophys.,Expanded Abstracts,1988,58:658.
[23]PIEPRZAK A W,MCCLEAN J W.Interpolation of se-verely aliased events[P].US Patent,1990:4,922.
[24]PAN K,FIELDS J L.Seismic trace interpolation using f-k filtering[P].US Patent,1986,4:594.
[25]GUO J,ZHOU X,YANG H J.Efficient f-k domainseismic trace interpolation for spatially aliased data[C].Internat.Mtg.,Soc.Expl.Geophys.,Expand-ed Abstracts,1996,66:1457.
[26]G L NAY N,CHAMBERS R E.Un-aliasedf-k do-main trace interpolation(UFKI)[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1996,66:1461.
[27]G L NAY N,CHAMBERS R E.Generalizedf-k do-main trace interpolation[C].Internat.Mtg.,Soc.Ex-pl.Geophys.Expanded Abstracts,1997,67:1100.
[28]CHEMINGUI N,BIONDI B.Handling the irregulargeometry in wide-azimuth surveys[C].Internat.Meeting,Soc.Expl.Geophys.,Expanded Abstracts,1999,66:3.
[2]SPITZ S.Seismic trace interpolation in the f-x domain[J].Geophysics,1991,56(6):785.
[3]CANNING A,GARDNER G HF.Regularizing3-D da-ta sets with DMO[J].Geophysics,1996,61(4):1103.
[4]RONEN J.Wave equation trace interpolation[J].Geo-physics,1987,52(7):973.
[5]JAKUBOWICZ H.Wavefield reconstruction[C].Inter-nat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1994,64:1557.
[6]JAKUBOWICZ H.Seismic data acquisition[P].US Pa-tent,1997,5:648.
[7]LAMER K,GIBSONB,ROTHMAND.Trace interpola-tion and the design of seismic surveys[J].Geophysics,1981,46(1):407.
[8]LU L.Application of local slant-stack to trace interpo-lation[C].4Internat.Mtg.,Soc.Expl.Geophys.Ex-panded Abstracts,1985,55:560.
[9]KING G A,LEONG T K,FLINCHBAUGH B E.Therole of interpolation in seismic resolution[C].Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1984,54:766.
[10]MONK D,MCBEATHR G,WASONC B.Interpolationof aliased seismic traces[P].US Patent,1993,5:235.
[11]KOSTOV,C.,Finite aperture slant-stack transforms[J]SEP,1989,61:261.
[12]DARCHE G.Spatial interpolation using fast parabolictransform[C].Internat.Mtg.,Soc.Expl.Geophys.,Expanded Abstracts,1990,60:1647.
[13]KABIR MMN,VERSCHUUR D J.Applications of thegeneralized Radon transform[C].Internat.Mtg.,Eur.Assn.Geosci.Eng.,Extended Abstracts,1992,54:52.
[14]SCHONEWILLE MA,DUIJNDAMA J W.Decomposi-tion of signal and noise for sparsely sampled data usinga mixed-Fourier/Radon transform[C].Internat.Mtg.,Soc.Exp.Geophys.,Expanded Abstract,1996,66:1438.
[15]SACCHI M,ULRYCHT.High-resolution velocity gath-ers and offset space reconstruction[J].Geophysics,1995,60(4):1169.
[16]SPITZ S.Trace interpolation beyond aliasing and with-out picking[C].Internat.Mtg.,Soc.Exp.Geo-phys.,Expanded Abstracts,1989,59:1121.
[17]JI J.Interpolation using prediction-error filter simula-tion(PEFs)[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1993,63:1170.
[18]MANIN M,SPITZ S.Wavefield de-aliasing for acqui-sition configurations leading to coarse sampling[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Ab-stracts,1995,65:930.
[19]PORSANI M J.Seismic trace interpolation using half-step prediction filters[J].Geophysics,1999,64(5):1461.
[20]SOUBARAS R.Spatial interpolation of aliased seismicdata[C].Internat.Mtg.,Soc.Exp.Geophys.,Ex-panded Abstracts,1997,67:1167.
[21]CLAERBOUT J,NICHOLS D.Interpolation beyondaliasing by(t-x)domain PEFs[C].Internat.Mtg.,Eur.Assn.Geosci.Eng.,Expanded Abstracts,1991,53:1.
[22]PIEPRZAK A W,MCCLEAN J W.Trace interpolationof severely aliased events[C].Internat.Mtg.,Soc.Exp.Geophys.,Expanded Abstracts,1988,58:658.
[23]PIEPRZAK A W,MCCLEAN J W.Interpolation of se-verely aliased events[P].US Patent,1990:4,922.
[24]PAN K,FIELDS J L.Seismic trace interpolation using f-k filtering[P].US Patent,1986,4:594.
[25]GUO J,ZHOU X,YANG H J.Efficient f-k domainseismic trace interpolation for spatially aliased data[C].Internat.Mtg.,Soc.Expl.Geophys.,Expand-ed Abstracts,1996,66:1457.
[26]G L NAY N,CHAMBERS R E.Un-aliasedf-k do-main trace interpolation(UFKI)[C].Internat.Mtg.,Soc.Expl.Geophys.Expanded Abstracts,1996,66:1461.
[27]G L NAY N,CHAMBERS R E.Generalizedf-k do-main trace interpolation[C].Internat.Mtg.,Soc.Ex-pl.Geophys.Expanded Abstracts,1997,67:1100.
[28]CHEMINGUI N,BIONDI B.Handling the irregulargeometry in wide-azimuth surveys[C].Internat.Meeting,Soc.Expl.Geophys.,Expanded Abstracts,1999,66:3.
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