基于克里金方法的地球物理三维空间数据网格化
摘要
地质统计学是数学地质领域中一门发展迅速且有着广泛应用前景的新兴学科。地质统计学是以区域化变量理论为基础，以变差函数理论为基本工具，用来研究那些展布于空间并呈现出一定的随机性和结构性的自然现象的一门科学。其核心克里金插值法，是一种无偏的、最小方差的储量估计方法。
随着地质统计学理论体系的不断完善和地质统计学应用水平的不断提高，使得地质统计学拓广应用到影响人类社会发展的各个方面。在许多地球物理方法中，对于数据预处理常常需要对三维数据进行插值，使得不完整或不规则分布的数据网格化。因为现在的许多算法都是针对均匀分布数据的，由于数据不完整或者数据缺失，会造成算法不能运行或运行结果不可预测。而数据缺失不仅仅意味着丢失一些信息，更重要的是它会在各种处理过程中产生许多不必要的噪声，对分辨率和信噪比都有影响。
本文实现了克里金插值法对三维空间地球物理数据体的网格化，同时研究了变差函数的求取以及直线、球状变差函数模型的自动拟合。模型试验中，我们首先针对均匀分布数据进行加密网格化处理，选取地震勘探正演速度体模型数据进行均匀抽稀为原始数据的八分之一，然后进行插值还原，得到数据的估计方差0.6×10-3，误差率约为1.4%。其次验证克里金插值方法对于离散不规则数据的插值效果，我们假设理论位场模型为两个斜磁化球体的ΔT场叠加，把理论模型数据随即抽取一半的数据点，插值还原之后原数据的平均值为-0.558485，插值之后的平均值为-0.560489，插值结果估计方差为1.7×10-3，误差率为6.3%。实际数据处理中，本文选取华北地区地壳S波地震层析成像数据，分别进行了缺失层弥补和随机缺失数据网格化的工作，提供地下结构模型，供后续工作开展。
经过多个例子的比较与实践，普通克里格法对已知数据的分布形式不作要求，且插值精度较高，适用于地球物理不规则分布数据的网格化。
Geostatistics is an emerging disciplines which is in the field of mathematicalgeology, it is developing quickly, and has a widely applied prospects. Geostatistics isbased on the theory of regionalized variable, the variogram theory is the basic tool,which is used to study the randomness and structural phenomena that spread in space.Its core is Kriging interpolation, is a reserve estimates which is unbiased and hasminimum variance.
With the continuous improvement of geostatistical theory and application, it hasapplied in all aspects of human society. Lots of geophysical methods for datapre-processing often need the three-dimensional data interpolation, to make data grid.Because many algorithms always suit for distribution data, because of theincompleted data or missing data will cause the algorithm can not run or the resultsare unpredictable. Missing data does not just mean loss of some information, but moreimportantly, it will generate a lot of unnecessary noises in a variety of processing, thathave an impact on the resolution and signal to noise ratio.
The thesis achieved the Kriging interpolation method on the three-dimensionalgeophysical data grid, strike the variogram, as well as automatic fitting the variogramfunction model of straight line and the spherical. For the model test, first we takeencryption grid processing on uniformly distributed data, which we select the seismicvelocity, uniform pumping dilute eighth of the original data, and then interpolation,the estimation variance is0.6×10-3, the error rate is about1.4%. Second we takeexperience on irregular data, we assume that the theory of potential field model ΔTand superposition of two oblique magnetized sphere, ramdomized simplified secondof the original data, The average value of-0.558485before, after interpolation, theaverage value is-0.560489, the estimated variance is1.7×10-3and the error rate is6.3%. For actual data processing, the thesis selected North China crustal S-waveseismic tomography data, we made up the missing layer and random missing data grid,it has made an underground structure model for the follow-up work.
Comparison of multiple examples and practice, we knew that the ordinary kriging is not required for the form of the distribution of known data, and has a highinterpolation accuracy, so it is suitable for geophysical irregularly distributed datagrid.
随着地质统计学理论体系的不断完善和地质统计学应用水平的不断提高，使得地质统计学拓广应用到影响人类社会发展的各个方面。在许多地球物理方法中，对于数据预处理常常需要对三维数据进行插值，使得不完整或不规则分布的数据网格化。因为现在的许多算法都是针对均匀分布数据的，由于数据不完整或者数据缺失，会造成算法不能运行或运行结果不可预测。而数据缺失不仅仅意味着丢失一些信息，更重要的是它会在各种处理过程中产生许多不必要的噪声，对分辨率和信噪比都有影响。
本文实现了克里金插值法对三维空间地球物理数据体的网格化，同时研究了变差函数的求取以及直线、球状变差函数模型的自动拟合。模型试验中，我们首先针对均匀分布数据进行加密网格化处理，选取地震勘探正演速度体模型数据进行均匀抽稀为原始数据的八分之一，然后进行插值还原，得到数据的估计方差0.6×10-3，误差率约为1.4%。其次验证克里金插值方法对于离散不规则数据的插值效果，我们假设理论位场模型为两个斜磁化球体的ΔT场叠加，把理论模型数据随即抽取一半的数据点，插值还原之后原数据的平均值为-0.558485，插值之后的平均值为-0.560489，插值结果估计方差为1.7×10-3，误差率为6.3%。实际数据处理中，本文选取华北地区地壳S波地震层析成像数据，分别进行了缺失层弥补和随机缺失数据网格化的工作，提供地下结构模型，供后续工作开展。
经过多个例子的比较与实践，普通克里格法对已知数据的分布形式不作要求，且插值精度较高，适用于地球物理不规则分布数据的网格化。
Geostatistics is an emerging disciplines which is in the field of mathematicalgeology, it is developing quickly, and has a widely applied prospects. Geostatistics isbased on the theory of regionalized variable, the variogram theory is the basic tool,which is used to study the randomness and structural phenomena that spread in space.Its core is Kriging interpolation, is a reserve estimates which is unbiased and hasminimum variance.
With the continuous improvement of geostatistical theory and application, it hasapplied in all aspects of human society. Lots of geophysical methods for datapre-processing often need the three-dimensional data interpolation, to make data grid.Because many algorithms always suit for distribution data, because of theincompleted data or missing data will cause the algorithm can not run or the resultsare unpredictable. Missing data does not just mean loss of some information, but moreimportantly, it will generate a lot of unnecessary noises in a variety of processing, thathave an impact on the resolution and signal to noise ratio.
The thesis achieved the Kriging interpolation method on the three-dimensionalgeophysical data grid, strike the variogram, as well as automatic fitting the variogramfunction model of straight line and the spherical. For the model test, first we takeencryption grid processing on uniformly distributed data, which we select the seismicvelocity, uniform pumping dilute eighth of the original data, and then interpolation,the estimation variance is0.6×10-3, the error rate is about1.4%. Second we takeexperience on irregular data, we assume that the theory of potential field model ΔTand superposition of two oblique magnetized sphere, ramdomized simplified secondof the original data, The average value of-0.558485before, after interpolation, theaverage value is-0.560489, the estimated variance is1.7×10-3and the error rate is6.3%. For actual data processing, the thesis selected North China crustal S-waveseismic tomography data, we made up the missing layer and random missing data grid,it has made an underground structure model for the follow-up work.
Comparison of multiple examples and practice, we knew that the ordinary kriging is not required for the form of the distribution of known data, and has a highinterpolation accuracy, so it is suitable for geophysical irregularly distributed datagrid.
引文
[1] Briggs I. Machaine contouring using minimum curvature[J]. Geophysics,1974,39:39-48.
[2] ClaytonV. Deutsch. GSLIB:Geostatistical Software Library and User’s Guide[M]. NewYork:Oxford University Press,1998
[3] Forsey D R, Bartels R H. Hierarchical B-Spline Refinement[J]. Computer Graphisics,1988,22(4):205-212.
[4] Hansen R. Interpretive gridding by anisotropic kriging[J]. Geophysics,1993,58,1491-1497.
[5] Lee S, Wolberg G, Shin S Y. Scattered Data International with Multilevel B-Splines[R]. IEEETransaction on Visualization and Computer Graphics,1997,3(3):228-244.
[6] Mark Pilkington,3D magnetic data-space inversion with sparseness constraints[J].Geophysics,2009,74(1):7－15
[7] Ordell L. A scattered equivalent-source method for interpolation and gridding ofpotential-field data in three dimension[J]. Geophysics,1992,57:629-636.
[8] Webring M. MINC:A gridding program based on minimum cur-vature[M]. U.S. Geol. Surv.Open File Rep.1981,81-1230.
[9]陈勖.地质统计学软件开发与应用[D].吉林大学,2009
[10]高建军,陈小宏,李景叶.三维不规则地震数据重建方法[J].石油地球物理勘探,2011,46(1):40
[11]郭良辉,孟小红,等.位场数据网格化的反插值法[J].地球物理学报,2006,49(4):1183
[12]郭良辉,孟小红,刘国峰,等.地球物理不规则分布数据的空间网格化[J].物探与化探,2005.29(5):439
[13]侯景儒,黄竞先.地质统计学及其在矿产储量计算中的应用[M]，地质出版社，1982.
[14]侯景儒,尹镇南,立维明,等.实用地质统计学[M].北京:地质出版社，1998:31-44
[15]侯景儒.地质统计学发展现状及对若干问题的讨论[J].黄金地质,1996.2(1):1-10
[16]侯景儒.地质统计学及其在中国矿业领域中的应用[J].中国矿业,1993.1(4):34-39
[17]侯景儒.地质统计学及其在中国矿业领域中的应用[J].中国矿业，1993.1(4):34-40
[18]侯景濡.地质统计学在我国的应用及其发展[J].地质与勘探，1991.27（4）：36-38
[19]侯景濡.中国地质统计学（空间信息统计学）发展的回顾与前景[J].地质与勘探,1997.33(1):53-58
[20]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):692.
[21]李庆扬,王能超,亿大义.数值分析[M].清华大学出版,2009:3-44
[22]刘国峰,孟小红,岳延波,等.克里金法在GPS数据内插中的应用[J].物探与化探,2006.30(2):178
[23]孟健康,马小明.Kriging空间分析法及其在城市大气污染中的应用[J].数学的实践与认识，2002，32(2):309-312.
[24]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,2008,51(1):235.
[25]孟小红,候建全,梁宏英,等.离散光滑插值在地球物理位场中的快速实现[J].物探与化探,2002,26(4):302.
[26]任建国.用普通克立格法计算永平铜矿床储量[J].矿产与地质,2001(1):69-72.
[27]孙红泉,康永尚，杜惠芝.实用地质统计学程序集M.地质出版社，1997.
[28]孙洪泉.地质统计学及其应用[M].北京:中国矿业大学出版社,1990.
[29]孙进,张佳荣,侯斌.矿产储量计算经典统计与地质统计学方法的对比分析[J].采矿技术2005(2):81-82.
[30]王银宏,彭润民等,克立格算法的若干思考[J].地质与勘探，2004(2):77-79.
[31]肖斌,赵鹏大,侯景儒.地质统计学新进展[J].地球科学进展,2000.15(3):294-296
[32]肖斌,赵鹏大,侯景儒.等.山东归来庄金矿床金异常分布及其时空演化的地质统计学研究[J].现代地质,1999.13(4):419-424
[33]姚长利,郑元满,张聿文.重磁异常三维物性反演随机子域法方法技术[J].地球物理学报,2007,50(5),1576.
[34]印兴耀刘永社.储层建模中地质统计学整合地震数据的方法及研究进展[J].石油地球物理勘探2002.37(4):423-430
[35]余先川,侯景儒,俞晨.析取克立格法及其应用的研究[J]，地球科学进展.2004(6)547-549.
[36]张魁.基于最小二乘直线拟合的小目标测量[J].电子设计工程,200918(7):176-177
[37]张仁铎.空间变异理论及其应用[M].科学出版社.2005.
[38]赵鹏大,李万亨，矿床勘查与评价[M].地质出版社，1988.10.
[39]朱燕玲,过仲阳,张朔,等.基于Kriging的三维地质体建模方法研究[J].计算机时代,2010.1:26
[2] ClaytonV. Deutsch. GSLIB:Geostatistical Software Library and User’s Guide[M]. NewYork:Oxford University Press,1998
[3] Forsey D R, Bartels R H. Hierarchical B-Spline Refinement[J]. Computer Graphisics,1988,22(4):205-212.
[4] Hansen R. Interpretive gridding by anisotropic kriging[J]. Geophysics,1993,58,1491-1497.
[5] Lee S, Wolberg G, Shin S Y. Scattered Data International with Multilevel B-Splines[R]. IEEETransaction on Visualization and Computer Graphics,1997,3(3):228-244.
[6] Mark Pilkington,3D magnetic data-space inversion with sparseness constraints[J].Geophysics,2009,74(1):7－15
[7] Ordell L. A scattered equivalent-source method for interpolation and gridding ofpotential-field data in three dimension[J]. Geophysics,1992,57:629-636.
[8] Webring M. MINC:A gridding program based on minimum cur-vature[M]. U.S. Geol. Surv.Open File Rep.1981,81-1230.
[9]陈勖.地质统计学软件开发与应用[D].吉林大学,2009
[10]高建军,陈小宏,李景叶.三维不规则地震数据重建方法[J].石油地球物理勘探,2011,46(1):40
[11]郭良辉,孟小红,等.位场数据网格化的反插值法[J].地球物理学报,2006,49(4):1183
[12]郭良辉,孟小红,刘国峰,等.地球物理不规则分布数据的空间网格化[J].物探与化探,2005.29(5):439
[13]侯景儒,黄竞先.地质统计学及其在矿产储量计算中的应用[M]，地质出版社，1982.
[14]侯景儒,尹镇南,立维明,等.实用地质统计学[M].北京:地质出版社，1998:31-44
[15]侯景儒.地质统计学发展现状及对若干问题的讨论[J].黄金地质,1996.2(1):1-10
[16]侯景儒.地质统计学及其在中国矿业领域中的应用[J].中国矿业,1993.1(4):34-39
[17]侯景儒.地质统计学及其在中国矿业领域中的应用[J].中国矿业，1993.1(4):34-40
[18]侯景濡.地质统计学在我国的应用及其发展[J].地质与勘探，1991.27（4）：36-38
[19]侯景濡.中国地质统计学（空间信息统计学）发展的回顾与前景[J].地质与勘探,1997.33(1):53-58
[20]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):692.
[21]李庆扬,王能超,亿大义.数值分析[M].清华大学出版,2009:3-44
[22]刘国峰,孟小红,岳延波,等.克里金法在GPS数据内插中的应用[J].物探与化探,2006.30(2):178
[23]孟健康,马小明.Kriging空间分析法及其在城市大气污染中的应用[J].数学的实践与认识，2002，32(2):309-312.
[24]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,2008,51(1):235.
[25]孟小红,候建全,梁宏英,等.离散光滑插值在地球物理位场中的快速实现[J].物探与化探,2002,26(4):302.
[26]任建国.用普通克立格法计算永平铜矿床储量[J].矿产与地质,2001(1):69-72.
[27]孙红泉,康永尚，杜惠芝.实用地质统计学程序集M.地质出版社，1997.
[28]孙洪泉.地质统计学及其应用[M].北京:中国矿业大学出版社,1990.
[29]孙进,张佳荣,侯斌.矿产储量计算经典统计与地质统计学方法的对比分析[J].采矿技术2005(2):81-82.
[30]王银宏,彭润民等,克立格算法的若干思考[J].地质与勘探，2004(2):77-79.
[31]肖斌,赵鹏大,侯景儒.地质统计学新进展[J].地球科学进展,2000.15(3):294-296
[32]肖斌,赵鹏大,侯景儒.等.山东归来庄金矿床金异常分布及其时空演化的地质统计学研究[J].现代地质,1999.13(4):419-424
[33]姚长利,郑元满,张聿文.重磁异常三维物性反演随机子域法方法技术[J].地球物理学报,2007,50(5),1576.
[34]印兴耀刘永社.储层建模中地质统计学整合地震数据的方法及研究进展[J].石油地球物理勘探2002.37(4):423-430
[35]余先川,侯景儒,俞晨.析取克立格法及其应用的研究[J]，地球科学进展.2004(6)547-549.
[36]张魁.基于最小二乘直线拟合的小目标测量[J].电子设计工程,200918(7):176-177
[37]张仁铎.空间变异理论及其应用[M].科学出版社.2005.
[38]赵鹏大,李万亨，矿床勘查与评价[M].地质出版社，1988.10.
[39]朱燕玲,过仲阳,张朔,等.基于Kriging的三维地质体建模方法研究[J].计算机时代,2010.1:26