基于散射矩阵分解的反射系数二阶近似
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摘要
常规AVO反演利用精确Zoeppritz方程的一阶近似式,更加适用于弱介质变化的反射界面、小角度或小炮检距反射问题,其假设条件导致计算误差,不利于准确提取密度参数,不能很好地预测复杂储层或中深部储层,且无法充分利用近临界角数据;基于反射/透射系数精确表达式进行高阶近似过于繁琐,利用上、下行波特征向量矩阵及矩阵对称性对反射/透射系数进行泰勒二阶展开,其过程物理意义不够明确,且得到的二阶近似式亦不够完善。为此,直接从P-SV平面波入射/散射矩阵出发,给出了一种求取散射矩阵高阶近似的方法,即利用扰动思想将散射矩阵分解为背景矩阵与一阶、二阶扰动矩阵,求取纵波反射系数背景项与一阶、二阶扰动项,推得纵波反射系数的二阶近似公式。模型对比分析表明,所推公式在中高角度乃至近临界角入射情况下具有较高的精度,对密度参数的敏感性更高,为充分利用大炮检距地震数据准确地反演物性参数提供了基础。
The first-order approximation of the exact Zoeppritz equation is widely used in conventional AVO inversions,which is more suitable for the interface with weak medium variations and the problem of small angle or short offset.The assumptions will lead to calculation errors,which are not conducive to accurate inversion of density and prediction of complex reservoirs and medium-deep reservoirs.And seismic data near the critical angle cannot be fully used.The high-order approximation method based on the exact equation of reflection/transmission coefficients is too complicated.The physical meaning of the process of second-order Taylor expansion of the reflection/transmission coefficients based on the eigenvector matrices of up-and-down waves and matrix symmetry is not clear enough.And the second-order approximation is also not perfect enough.In this paper,a high-order approximation of the scattering matrix based on the incident/scattering matrix of P-SV plane wave is presented.The scattering matrix can be decomposed into the background matrix,the first-and second-order perturbation matrices,and the background term based on the theory of perturbations.The background term,first-and second-order perturbation terms can be derived respectively and the secondorder approximation of P wave reflection coefficient can be further obtained.The model comparison analysis shows that the second-order approximation obtained in this paper has higher accuracy in the case of medium-high angle and even nearcritical angle incidence and more sensitive to the density variations.It provides a basis for making full use of large-angle or long-offset seismic data and the accurate inversion of physical parameters.
The first-order approximation of the exact Zoeppritz equation is widely used in conventional AVO inversions,which is more suitable for the interface with weak medium variations and the problem of small angle or short offset.The assumptions will lead to calculation errors,which are not conducive to accurate inversion of density and prediction of complex reservoirs and medium-deep reservoirs.And seismic data near the critical angle cannot be fully used.The high-order approximation method based on the exact equation of reflection/transmission coefficients is too complicated.The physical meaning of the process of second-order Taylor expansion of the reflection/transmission coefficients based on the eigenvector matrices of up-and-down waves and matrix symmetry is not clear enough.And the second-order approximation is also not perfect enough.In this paper,a high-order approximation of the scattering matrix based on the incident/scattering matrix of P-SV plane wave is presented.The scattering matrix can be decomposed into the background matrix,the first-and second-order perturbation matrices,and the background term based on the theory of perturbations.The background term,first-and second-order perturbation terms can be derived respectively and the secondorder approximation of P wave reflection coefficient can be further obtained.The model comparison analysis shows that the second-order approximation obtained in this paper has higher accuracy in the case of medium-high angle and even nearcritical angle incidence and more sensitive to the density variations.It provides a basis for making full use of large-angle or long-offset seismic data and the accurate inversion of physical parameters.
引文
[1] Zoeppritz K.On the reflection and penetration of seismic waves through unstable layers[J].Goettinger Nachr,1919,66-84.
[2] Muskat M,Merest M W.Reflection and transmission coefficients for plane waves in elastic media[J].Geophysics,1940,5(2):115-124.
[3] Koefoed O.On the effect of Poisson’s ratio of rock strata on the reflection coefficients of plane waves[J].Geophysical Prospecting,1995,43(4):381-387.
[4] Bortfeld R.Approximation to the reflection and transmission coefficients of the plane longitudinal and transverse waves[J].Goephysical Prospecting,1961,9(4):485-502.
[5] Richards P G,Frasier C W.Scattering of elastic waves from depth-dependent inhomogeneities[J].Geophy-sics,1976,41(3):441-458.
[6] Aki K,Richards P G.Quantitative Seismology:Theory and Methods[M].W H Freeman and Co Cambridge,1980.
[7] Ostrander W J.Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence[J].Geophysics,1984,49(10):1637-1648.
[8] Shuey R T.A simplification of the Zoeppritz equa-tions[J].Geophysics,1985,50(4):609-614.
[9] Smith G C,Gidlow P M.Weighted stacking for rock property estimation and detection of gas[J].Goephy-sical Prospecting,1987,35(9):993-1014.
[10] Hilterman F.Is AVO the seismic signature of lithology? A case history of Ship Shoal-South addition[J].The Leading Edge,1990,9(6):15-22.
[11] 郑晓东.Zoeppritz方程的近似及其应用[J].石油地球物理勘探,1991,26(2):129-144.ZHENG Xiaodong.Approximation and application of Zoeppritz equations[J].Oil Geophysical Prospecting,1991,26(2):129-144.
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[13] Fatti J L,Smith G C,Vail P J,et al.Detection of gas in sandstone reservoirs using AVO analysis:A 3-D seismic case history using the Geostack technique[J].Geophysics,1994,59(9):1362-1376.
[14] Goodway B,Chen T,Downton J.Improved AVO fluid detection and lithology discrimination using Lamé petrophysical parameters: “λρ,μρ,λμ fluid stack”,from P and S inversions[C].SEG Technical Program Expanded Abstracts,1997,16:183-186.
[15] Xu Y,Bancroft J C.AVO case study:extraction of Lame’s parameters from vertical component seismic data[R].CREWES Project,Canada,1998.
[16] Gray D,Goodway B,Chen T.Bridging the gap: Using AVO to detect changes in fundamental elastic constants[C].SEG Technical Program Expanded Abstracts,1999,18:852-855.
[17] Russell B H,Gray D,Hampson D P.Linearized AVO and poroelasticity[J].Geophysics,2011,76(3):C19-C29.
[18] 宗兆云,印兴耀,张峰,等.杨氏模量和泊松比反射系数近似方程及叠前地震反演[J].地球物理学报,2013,55(11):3786-3794.ZONG Zhaoyun,YIN Xingyao,ZHANG Feng,et al.Reflection coefficient equation and pre-stack seismic inversion with Young’s modulus and Poisson ratio[J].Chinese Journal of Geophysics,2012,55(11):3786-3794.
[19] 邓炜,印兴耀,宗兆云,等.基于逆算子估计的高阶AVO非线性反演[J].石油地球物理勘探,2016,51(5):955-964.DENG Wei,YIN Xingyao,ZONG Zhaoyun,et al.High-order nonlinear AVO inversion based on estimated inverse operator[J].Oil Geophysical Prospecting,2016,51(5):955-964.
[20] 黄捍东,王彦超,郭飞,等.基于佐普里兹方程的高精度叠前反演方法[J].石油地球物理勘探,2013,48(5):740-749.HUANG Handong,WANG Yanchao,GUO Fei,et al.High precision prestack inversion algorithm based on Zoeppritz equations[J].Oil Geophysical Prospecting,2013,48(5):740-749.
[21] Xu Y,Bancroft J C.Joint AVO analysis of PP and PS data[R].Crewes Research Reports,1997,1-44.
[22] Wang Y H.Approximations to Zoeppritz equations and their use in AVO analysis[J].Geophysics,1999,64(6):1920-1927.
[23] Ramos C B.Useful approximations for converted-wave AVO[J].Geophysics,2001,66 (6):1721-1734.
[24] Ursin B,Stovas A.Second-order approximations of the reflection and transmission coefficients between two visco-elastic isotropic media[J].Journal of Seismic Exploration,2001,9(3):223-233.
[25] Charles P.Ursenbach.Optimal Zoeppritz approximations[R].CREWES Research Report,2002,14,1-10.
[26] Tarantola A.A strategy for nonlinear elastic inversion of seismic reflection data[J].Geophysics,1986,51(10):1893-1903.
[27] Behural J,Kabir N,Crlder R,et al.Density extraction from P-wave AVO inversion:Tusealoosa Trend example[C].SEG Technical Program Expanded Abstracts,2007,26:1466-1470.
[28] 郭强,张宏兵,曹呈浩,等.Zoeppritz方程叠前多参数反演及密度敏感性分析[J].石油地球物理勘探,2017,52(6):1218-1225.GUO Qiang,ZHANG Hongbing,CAO Chenghao,et al.Density-sensitivity analysis about prestack multi-parameter inversion based on the exact Zoeppritz equation[J].Oil Geophysical Prospecting,2017,52(6):1218-1225.
[29] 印兴耀,王慧欣,曹丹平,等.利用三参数AVO近似方程的深层叠前地震反演[J].石油地球物理勘探,2018,53(1):129-135.YIN Xingyao,WANG Huixin,CAO Danping,et al.Three term AVO approximation of Kf-fm-ρ and prestack seismic inversion for deep reservoirs[J].Oil Geophysical Prospecting,2018,53(1):129-135.
[30] Lavaud B,Kabir N,Chavent G.Pushing AVO inversion beyond linearized approximation[J].Journal of Seismic Exploration,1999,8(11):279-302.
[31] 石玉梅,姚逢昌,孙虎生,等.地震密度反演及地层孔隙度估计[J].地球物理学报,2010,53(1):197-204.SHI Yumei,YAO Fengchang,SUN Husheng,et al.Density inversion and porosity estimation using seismic data[J].Chinese Journal of Geophysics,2010,53(1):197-204.
[2] Muskat M,Merest M W.Reflection and transmission coefficients for plane waves in elastic media[J].Geophysics,1940,5(2):115-124.
[3] Koefoed O.On the effect of Poisson’s ratio of rock strata on the reflection coefficients of plane waves[J].Geophysical Prospecting,1995,43(4):381-387.
[4] Bortfeld R.Approximation to the reflection and transmission coefficients of the plane longitudinal and transverse waves[J].Goephysical Prospecting,1961,9(4):485-502.
[5] Richards P G,Frasier C W.Scattering of elastic waves from depth-dependent inhomogeneities[J].Geophy-sics,1976,41(3):441-458.
[6] Aki K,Richards P G.Quantitative Seismology:Theory and Methods[M].W H Freeman and Co Cambridge,1980.
[7] Ostrander W J.Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence[J].Geophysics,1984,49(10):1637-1648.
[8] Shuey R T.A simplification of the Zoeppritz equa-tions[J].Geophysics,1985,50(4):609-614.
[9] Smith G C,Gidlow P M.Weighted stacking for rock property estimation and detection of gas[J].Goephy-sical Prospecting,1987,35(9):993-1014.
[10] Hilterman F.Is AVO the seismic signature of lithology? A case history of Ship Shoal-South addition[J].The Leading Edge,1990,9(6):15-22.
[11] 郑晓东.Zoeppritz方程的近似及其应用[J].石油地球物理勘探,1991,26(2):129-144.ZHENG Xiaodong.Approximation and application of Zoeppritz equations[J].Oil Geophysical Prospecting,1991,26(2):129-144.
[12] 杨绍国,周熙襄.Zoeppritz方程的级数表达式及近似[J].石油地球物理勘探,1994,29(4):399-412.YANG Shaoguo,ZHOU Xixiang.The series expressions and approximations of the Zoeppritz equations[J].Oil Geophysical Prospecting,1994,29(4):399-412.
[13] Fatti J L,Smith G C,Vail P J,et al.Detection of gas in sandstone reservoirs using AVO analysis:A 3-D seismic case history using the Geostack technique[J].Geophysics,1994,59(9):1362-1376.
[14] Goodway B,Chen T,Downton J.Improved AVO fluid detection and lithology discrimination using Lamé petrophysical parameters: “λρ,μρ,λμ fluid stack”,from P and S inversions[C].SEG Technical Program Expanded Abstracts,1997,16:183-186.
[15] Xu Y,Bancroft J C.AVO case study:extraction of Lame’s parameters from vertical component seismic data[R].CREWES Project,Canada,1998.
[16] Gray D,Goodway B,Chen T.Bridging the gap: Using AVO to detect changes in fundamental elastic constants[C].SEG Technical Program Expanded Abstracts,1999,18:852-855.
[17] Russell B H,Gray D,Hampson D P.Linearized AVO and poroelasticity[J].Geophysics,2011,76(3):C19-C29.
[18] 宗兆云,印兴耀,张峰,等.杨氏模量和泊松比反射系数近似方程及叠前地震反演[J].地球物理学报,2013,55(11):3786-3794.ZONG Zhaoyun,YIN Xingyao,ZHANG Feng,et al.Reflection coefficient equation and pre-stack seismic inversion with Young’s modulus and Poisson ratio[J].Chinese Journal of Geophysics,2012,55(11):3786-3794.
[19] 邓炜,印兴耀,宗兆云,等.基于逆算子估计的高阶AVO非线性反演[J].石油地球物理勘探,2016,51(5):955-964.DENG Wei,YIN Xingyao,ZONG Zhaoyun,et al.High-order nonlinear AVO inversion based on estimated inverse operator[J].Oil Geophysical Prospecting,2016,51(5):955-964.
[20] 黄捍东,王彦超,郭飞,等.基于佐普里兹方程的高精度叠前反演方法[J].石油地球物理勘探,2013,48(5):740-749.HUANG Handong,WANG Yanchao,GUO Fei,et al.High precision prestack inversion algorithm based on Zoeppritz equations[J].Oil Geophysical Prospecting,2013,48(5):740-749.
[21] Xu Y,Bancroft J C.Joint AVO analysis of PP and PS data[R].Crewes Research Reports,1997,1-44.
[22] Wang Y H.Approximations to Zoeppritz equations and their use in AVO analysis[J].Geophysics,1999,64(6):1920-1927.
[23] Ramos C B.Useful approximations for converted-wave AVO[J].Geophysics,2001,66 (6):1721-1734.
[24] Ursin B,Stovas A.Second-order approximations of the reflection and transmission coefficients between two visco-elastic isotropic media[J].Journal of Seismic Exploration,2001,9(3):223-233.
[25] Charles P.Ursenbach.Optimal Zoeppritz approximations[R].CREWES Research Report,2002,14,1-10.
[26] Tarantola A.A strategy for nonlinear elastic inversion of seismic reflection data[J].Geophysics,1986,51(10):1893-1903.
[27] Behural J,Kabir N,Crlder R,et al.Density extraction from P-wave AVO inversion:Tusealoosa Trend example[C].SEG Technical Program Expanded Abstracts,2007,26:1466-1470.
[28] 郭强,张宏兵,曹呈浩,等.Zoeppritz方程叠前多参数反演及密度敏感性分析[J].石油地球物理勘探,2017,52(6):1218-1225.GUO Qiang,ZHANG Hongbing,CAO Chenghao,et al.Density-sensitivity analysis about prestack multi-parameter inversion based on the exact Zoeppritz equation[J].Oil Geophysical Prospecting,2017,52(6):1218-1225.
[29] 印兴耀,王慧欣,曹丹平,等.利用三参数AVO近似方程的深层叠前地震反演[J].石油地球物理勘探,2018,53(1):129-135.YIN Xingyao,WANG Huixin,CAO Danping,et al.Three term AVO approximation of Kf-fm-ρ and prestack seismic inversion for deep reservoirs[J].Oil Geophysical Prospecting,2018,53(1):129-135.
[30] Lavaud B,Kabir N,Chavent G.Pushing AVO inversion beyond linearized approximation[J].Journal of Seismic Exploration,1999,8(11):279-302.
[31] 石玉梅,姚逢昌,孙虎生,等.地震密度反演及地层孔隙度估计[J].地球物理学报,2010,53(1):197-204.SHI Yumei,YAO Fengchang,SUN Husheng,et al.Density inversion and porosity estimation using seismic data[J].Chinese Journal of Geophysics,2010,53(1):197-204.