地震勘探中有关数学近似问题的研究
|
摘要
地震勘探是利用地下介质弹性和密度的差异,通过观测和分析大地对人工激发地震波的响应,推断地下岩层的性质和形态的地球物理勘探方法。
本文通过地震勘探领域动校正、Zoeppritz方程、转换波抽道集这三方面问题的分析,研究地震勘探领域目前所应用的各种近似公式与精确解的相对误差,为合理利用近似公式提供依据和参考。本文首先简单介绍了地震勘探的基础知识,然后根据物理-数学模型求解精确解,分析近似公式,讨论近似公式和精确解的相对误差,最后对结果进行总结。具体结论如下:
1)通过对动校正相对误差分析,发现偏深比越大,相对误差越大,并且高阶近似的相对误差小于低阶近似的相对误差;偏深比在0~2范围内,一阶近似相对误差低于25%,二阶、三阶近似相对误差低于10%。从精度提高程度和计算量的角度考虑,二阶近似更适合用于普遍计算。
2) Zoeppritz方程的精确解表达式极其复杂;当入射角达到临界角后,反射系数变为复数,其辐角为反射系数相角,能量发生突变;针对Ostrander、Goodway模型,相角与入射角有一定的变化关系,随着入射角的增大,相角的幅度也逐渐增大,且类似于线性关系;相同入射角,对的同一切片,反射P波速度与反射S波速度比越大,低误差范围越大;相同反射纵横波速度比,对的同一切片,入射角越小,低误差范围越大;相同入射角,相同反射纵横波速度比,越趋近于0,低误差范围越大。
3)转换点坐标的精确解并不是很复杂,以现在的计算机能力可以较快求解。但是,相对已有的近似公式,它的计算量还是比较大。当大批量的转换波资料生产处理时,需要的时间相对较长。在已有的近似公式中,一阶近似的相对误差最大(14%以内),Thomsen近似公式效果最好(1.1%以内)。当然,迭代公式的精度也很高,但是由于迭代次数的增加会增加计算量,反而增加计算机运行时间。
Seismic exploration is a geophysical method which is applied to infer the elasticity and density differences underground, through bombing with man-made sources and observing, analyzing the seismic wave responses.
In this thesis, normal move out(NMO), Zoeppritz equations and converted wave sorting are implied to discuss the relative error of the current approximated formula and exact solution applied in seismic exploration so that these approximate formula could be utilized reasonably. Three sections are diagramed in this thesis: 1) introduction of some basic knowledge about the seismic exploration; 2) numerical simulation and analyzing, i.e. according to physics-mathematical models, solving the exact solutions, analyzing approximated formula, and discussing the relative errors between approximated formula and exact solutions; 3) summarization. At the basis of the upper study, the following conclusions can be deduced.
1) After analyzing the relative errors of NMO. It was founded that the bigger the deflection deep ration is, the bigger the relative error is, and the relative error of the high-order approximation is less than the one of low-order approximation. When the deflection deep ratio ranges from zero to two, the relative error of first-order approximation is less than 25% and the ones of second-order and third-order approximation are both less than 10%. From the point of increasing the level of precision and computation, the second-order approximation is favorable for general computing.
2) The exact solution of the Zoeppritz equations is extremely complicated. When the incident angle reaches the critical angle, the reflection coefficient becomes plural, the argument becomes the reflectance phase angle and the energy mutates suddenly. In Ostrander, Goodway model, the changes of the phase angle has some relationship with the changes of incident angle. Along with the increasing of the incidence angle, the magnitude of the phase angle also increases, which is relatively a linear relationship. On the condition of same incident angle and same , the bigger P-wave to S-wave velocity ratio is, the bigger the low-order error range is. On the condition of same ratio of reflected wave velocities and same , the smaller the incident angle is, the larger the low-order range is. On the condition of same incident angle and same ratio of reflected wave velocities, the more is close to 0, the larger the low-order error range.
3) Sorting conversion point is not very complicated, and computer could quickly solve up now. However, the process of calculation is relatively complex comparing with the approximated formula. It takes long time when calculating the instant mass data. Among these current approximated formulas, the relative error of first-order approximate is the biggest (less then 14%) and Thomsen approximated formula is the best (less then 1.1%). In addition, the precision of iterative formula is also in a high level, but increasing the number of iteration can also increase the calculation amount and computer running time.
本文通过地震勘探领域动校正、Zoeppritz方程、转换波抽道集这三方面问题的分析,研究地震勘探领域目前所应用的各种近似公式与精确解的相对误差,为合理利用近似公式提供依据和参考。本文首先简单介绍了地震勘探的基础知识,然后根据物理-数学模型求解精确解,分析近似公式,讨论近似公式和精确解的相对误差,最后对结果进行总结。具体结论如下:
1)通过对动校正相对误差分析,发现偏深比越大,相对误差越大,并且高阶近似的相对误差小于低阶近似的相对误差;偏深比在0~2范围内,一阶近似相对误差低于25%,二阶、三阶近似相对误差低于10%。从精度提高程度和计算量的角度考虑,二阶近似更适合用于普遍计算。
2) Zoeppritz方程的精确解表达式极其复杂;当入射角达到临界角后,反射系数变为复数,其辐角为反射系数相角,能量发生突变;针对Ostrander、Goodway模型,相角与入射角有一定的变化关系,随着入射角的增大,相角的幅度也逐渐增大,且类似于线性关系;相同入射角,对的同一切片,反射P波速度与反射S波速度比越大,低误差范围越大;相同反射纵横波速度比,对的同一切片,入射角越小,低误差范围越大;相同入射角,相同反射纵横波速度比,越趋近于0,低误差范围越大。
3)转换点坐标的精确解并不是很复杂,以现在的计算机能力可以较快求解。但是,相对已有的近似公式,它的计算量还是比较大。当大批量的转换波资料生产处理时,需要的时间相对较长。在已有的近似公式中,一阶近似的相对误差最大(14%以内),Thomsen近似公式效果最好(1.1%以内)。当然,迭代公式的精度也很高,但是由于迭代次数的增加会增加计算量,反而增加计算机运行时间。
Seismic exploration is a geophysical method which is applied to infer the elasticity and density differences underground, through bombing with man-made sources and observing, analyzing the seismic wave responses.
In this thesis, normal move out(NMO), Zoeppritz equations and converted wave sorting are implied to discuss the relative error of the current approximated formula and exact solution applied in seismic exploration so that these approximate formula could be utilized reasonably. Three sections are diagramed in this thesis: 1) introduction of some basic knowledge about the seismic exploration; 2) numerical simulation and analyzing, i.e. according to physics-mathematical models, solving the exact solutions, analyzing approximated formula, and discussing the relative errors between approximated formula and exact solutions; 3) summarization. At the basis of the upper study, the following conclusions can be deduced.
1) After analyzing the relative errors of NMO. It was founded that the bigger the deflection deep ration is, the bigger the relative error is, and the relative error of the high-order approximation is less than the one of low-order approximation. When the deflection deep ratio ranges from zero to two, the relative error of first-order approximation is less than 25% and the ones of second-order and third-order approximation are both less than 10%. From the point of increasing the level of precision and computation, the second-order approximation is favorable for general computing.
2) The exact solution of the Zoeppritz equations is extremely complicated. When the incident angle reaches the critical angle, the reflection coefficient becomes plural, the argument becomes the reflectance phase angle and the energy mutates suddenly. In Ostrander, Goodway model, the changes of the phase angle has some relationship with the changes of incident angle. Along with the increasing of the incidence angle, the magnitude of the phase angle also increases, which is relatively a linear relationship. On the condition of same incident angle and same , the bigger P-wave to S-wave velocity ratio is, the bigger the low-order error range is. On the condition of same ratio of reflected wave velocities and same , the smaller the incident angle is, the larger the low-order range is. On the condition of same incident angle and same ratio of reflected wave velocities, the more is close to 0, the larger the low-order error range.
3) Sorting conversion point is not very complicated, and computer could quickly solve up now. However, the process of calculation is relatively complex comparing with the approximated formula. It takes long time when calculating the instant mass data. Among these current approximated formulas, the relative error of first-order approximate is the biggest (less then 14%) and Thomsen approximated formula is the best (less then 1.1%). In addition, the precision of iterative formula is also in a high level, but increasing the number of iteration can also increase the calculation amount and computer running time.
引文
[1]陆孟基.地震勘探原理.上册[M].东营:石油大学出版社,1993.1(2004.1重印)
[2] Tariq A. Velocity analysis using nonhyperbolic moveout in transversely isotropic media. Geophysics, 1997, 62(6):1839~1854
[3] Knott C G. On the reflection and refraction of elastic wave with seismological application. Phil Mag 5th ser,1899, 48:64~97
[4] Hilterman F. Seismic lithology. SEG-continuing education,1983
[5] Shuey R T. A simplification of the Zoeppritz equations. Geophysics, 1985,50:609~614
[6] Smith G C and Gidlow P M. Weighted stacking for rock property estimation and detection of gas. Geophysical Prospecting, 1987, 35:993~1014
[7]郑晓东. Zoeppritz方程的近似及其应用.石油地球物理勘探, 1991,26(2):129~144
[8] Fatti J L et al.Detection of gas in sandstone reservoirs using A V O analysis. Geophysics, 1994, 59:1362~1376
[9] Xu Y and Bancroft J C.A V O case study:Extraction of Lame's parameters from vertical component seismic data. Presented at Geotriad'98, Canada. 1998
[10] Goodway B Chen T and Downton J.Improved A V O fluid detection and lithology discrim ination using Lame petrophysical parameters,“λ”,“μ”and“λfluid stack”, from P and S inversions. CSEG Expand Abstracts, 1997, 148~151
[11]华东师范大学数学系.数学分析.下册(第三版).北京:高等教育出版社,2001(2002重印)
[12]渥.伊尔马滋. (刘怀山等译).地震资料分析——地震资料处理、反演和解释.北京:石油工业出版社, 2006
[13]杨绍国,周熙襄.Zoeppritz方程的级数表达式及近似.石油地球物理勘探,1994,29(4):406~ 407
[14] Aki K,Richards P G.Quantitative seismology theory and methods [M].USA,1980
[15] Ostrander W J. Plane wave reflection coefficients for gas sands at nonnormal angles of incidence.Geophysics,1984,49:1637~1648
[16] Yong Xu. AVO developments applied to Blackfoot 3C-2D broadband line. M.SC. Thesis of University of Calgary,1999
[17]张永刚,王赟,王妙月.目前多分量地震勘探中的几个关键问题[J].地球物理学报. 2004, 47(1):151~154.
[18]许世勇,马在田.快速有效的转换波共转换点叠加技术[J].地球物理学报,2002,45(4): 558
[19]郭向宇,凌云,魏修成.PS转换波共转换点的几种计算方法及实际应用.石油物探,2002, 41(2):142
[20] Thomsen Leon. Converted-wave reflection seismology over inhomogeneous, anisotropic media. Geophysics,1999,64(3):678~690
[21]周竹生,王卫华.一种快速、高精度的共转换点轨迹计算方法[J].石油地球物理勘探,1993, 28(1):37~45
[22]毕丽飞,李振春,王延光等.转换波处理中共转换点的定位.勘探地球物理进展,2005, 28(5):353
[23]崔秀,韩立国,葛勇.共转换点道集的迭代算法及应用.世界地质,1998,17(4):53~54
[24]徐东艳,孟晓刚.MATLAB函数库查询辞典[M].北京:中国铁道出版社,2005.12(2006.11重印)
[2] Tariq A. Velocity analysis using nonhyperbolic moveout in transversely isotropic media. Geophysics, 1997, 62(6):1839~1854
[3] Knott C G. On the reflection and refraction of elastic wave with seismological application. Phil Mag 5th ser,1899, 48:64~97
[4] Hilterman F. Seismic lithology. SEG-continuing education,1983
[5] Shuey R T. A simplification of the Zoeppritz equations. Geophysics, 1985,50:609~614
[6] Smith G C and Gidlow P M. Weighted stacking for rock property estimation and detection of gas. Geophysical Prospecting, 1987, 35:993~1014
[7]郑晓东. Zoeppritz方程的近似及其应用.石油地球物理勘探, 1991,26(2):129~144
[8] Fatti J L et al.Detection of gas in sandstone reservoirs using A V O analysis. Geophysics, 1994, 59:1362~1376
[9] Xu Y and Bancroft J C.A V O case study:Extraction of Lame's parameters from vertical component seismic data. Presented at Geotriad'98, Canada. 1998
[10] Goodway B Chen T and Downton J.Improved A V O fluid detection and lithology discrim ination using Lame petrophysical parameters,“λ”,“μ”and“λfluid stack”, from P and S inversions. CSEG Expand Abstracts, 1997, 148~151
[11]华东师范大学数学系.数学分析.下册(第三版).北京:高等教育出版社,2001(2002重印)
[12]渥.伊尔马滋. (刘怀山等译).地震资料分析——地震资料处理、反演和解释.北京:石油工业出版社, 2006
[13]杨绍国,周熙襄.Zoeppritz方程的级数表达式及近似.石油地球物理勘探,1994,29(4):406~ 407
[14] Aki K,Richards P G.Quantitative seismology theory and methods [M].USA,1980
[15] Ostrander W J. Plane wave reflection coefficients for gas sands at nonnormal angles of incidence.Geophysics,1984,49:1637~1648
[16] Yong Xu. AVO developments applied to Blackfoot 3C-2D broadband line. M.SC. Thesis of University of Calgary,1999
[17]张永刚,王赟,王妙月.目前多分量地震勘探中的几个关键问题[J].地球物理学报. 2004, 47(1):151~154.
[18]许世勇,马在田.快速有效的转换波共转换点叠加技术[J].地球物理学报,2002,45(4): 558
[19]郭向宇,凌云,魏修成.PS转换波共转换点的几种计算方法及实际应用.石油物探,2002, 41(2):142
[20] Thomsen Leon. Converted-wave reflection seismology over inhomogeneous, anisotropic media. Geophysics,1999,64(3):678~690
[21]周竹生,王卫华.一种快速、高精度的共转换点轨迹计算方法[J].石油地球物理勘探,1993, 28(1):37~45
[22]毕丽飞,李振春,王延光等.转换波处理中共转换点的定位.勘探地球物理进展,2005, 28(5):353
[23]崔秀,韩立国,葛勇.共转换点道集的迭代算法及应用.世界地质,1998,17(4):53~54
[24]徐东艳,孟晓刚.MATLAB函数库查询辞典[M].北京:中国铁道出版社,2005.12(2006.11重印)