各向异性地层中可控源电磁法一维全参数反演及三维有限体积正演算法研究
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摘要
可控源音频大地电磁法（CSAMT）是在大地电磁法（MT）和音频大地电磁法（AMT）的基础上发展起来的一种地面人工源频率域测深方法，针对MT法场源的随机性和信号微弱，以致观测十分困难这一状况，CSAMT采用可以控制的人工场源，CSAMT广泛的应用于地质普查、勘探石油、天然气、地热、金属矿产、水文、工程、环境保护等各种复杂地电构造的勘探，是一种重要的地球物理勘探方法。
一维水平层状各向同性地层中的CSAMT正反演是非常重要的，因为一维模型简单，存在解析解，作为一个基本的地电模型，有助于理解电磁场的基本物理特性，它也是实际数据的初步解释和分析以及多维反演的初始模型选取的基础。实际地层在一定尺度上总是存在电的各向异性，例如，砂岩页岩的薄交互层是一种典型的各向异性构造。对各向异性地层测量数据进行解释，如果忽略各向异性，将很难得到有效的反演结果。因此，研究各向异性地层中的CSAMT的反演理论是一个非常重要的课题。本文研究各向异性地层中CSAMT的一维正反演理论，讨论各向异性地层对CSAMT一维正反演的影响特征和规律，为实际资料的正确解释提供理论依据和指导。与其他地球物理问题的反演一样，各向异性地层中的一维CSAMT反演，涉及正演计算，模型空间的选择，Fréchet导数的计算，以及反演过程中模型的迭代修正来得到有效的反演结果。
在第二章，利用二维Fourier变换与矩阵分解技术将层状各向异性地层中Maxwell方程转化成两个独立的关于TM波和TE波的传输线方程;借助传输线理论与叠加原理，根据CSAMT只使用电偶极源的特点，改进传输线算法，只引入电流源传输线Green函数，来求解TM波和TE波，建立各向异性地层中频率-波数域电流源电场和磁场并矢Green函数的新算法和新的表达式，相比传统的传输线理论，提高了CSAMT数值模拟的效率。在此基础上,利用传输线Green函数的基本解以及边界条件,推导出广义反射系数与振幅递推公式,得到各个地层中传输线Green函数的解析解;然后利用Fourier逆变换与Bessel公式将CSAMT响应表示为Sommerfeld形式的积分,针对层状模型的特点，本文提出了三次样条插值结合Lommel积分公式的技术，快速计算其数值解，相比地球物理电磁法中常用的数字滤波算法，精度相当，但计算速度更快。
在第三章，选择能够定量反演地下模型参数的参数化模型空间，通过水平和垂直导电率以及水平层界面构成的模型矢量来描述电阻率的空间分布。利用并矢Green函数，结合摄动理论，推导得到了Fréchet导数的解析表达式，表示为Sommerfeld形式的积分，采用第二章提出的三次样条插值结合Lommel积分公式的技术可以有效的求解。进一步的，建立了全参数广义逆反演算法和全参数正则化迭代反演算法来同时反演所有的模型参数，包括横向、纵向电导率和层界面深度。广义逆反演算法采用奇异值分解技术结合阻尼最小二乘法，可以有效的提高反演结果的稳定性和抗噪能力。对于正则化反演，由于CSAMT响应是非线性的，引入偏差原理并结合Cholesky分解自适应选择正则化因子，保证反演过程的稳定性。数值计算结果表明，CSAMT响应对地层纵向电导率的灵敏度要远小于相比对其他模型参数（横向电导率和层界面）的灵敏度。尽管初始模型存在较大误差，反演结果依然能够给出地层模型的主要特征。选择合适的初始层界面可以有效的改善反演结果；全参数的反演相比固定层界面的反演，反演结果更好；即使初始模型的层数与实际地层不符，依然能够给出有效的反演结果。
海洋可控源电磁法（MCSEM）是一种用于海底油气勘探的频率域电磁方法。MCSEM具有提供海底地层电阻率空间分布的能力，在油气层识别和海上油气储层定量评价以及降低海上钻探风险方面发挥着重要作用，已发展成为海上油气勘探的一种重要方法。由于海底地形构造复杂以及地层横向电阻率分布不均匀，在海洋电磁勘探的设计以及海洋电磁资料处理和解释过程中，均需要进行大量的数值模拟，一维和二维的数值模拟技术已经较为成熟或者正在趋于成熟，三维数值模拟成为当前MCSEM的一个研究热点，目前模拟三维MCSEM的数值方法主要有有限元法、有限差分法、有限体积法和积分方程法。基于Yee氏交错网格的有限体积法，对Maxwell方程在各个单元上的积分进行离散处理，能有效降低方程的微分阶数，同时也减少了地层电导率不连续对离散结果的影响，在电磁场数值模拟中得到了较广泛应用。本文采用有限体积算法模型各向异性地层中MCSEM的三维响应，分析、总结各向异性对海洋CSEM三维电磁响应的影响特征。
在第四章，为了有效模拟各向异性地层中海洋可控源电磁法的三维响应，我们建立了一套基于电场矢势与标势分解的耦合势有限体积法。利用电场的矢势和标势分解，将电场分解为无散场和无旋场之和，Maxwell方程转换为关于矢势与标势的混合Helmholtz方程，有效的克服低感应数问题。在此基础上,借助Yee氏交错网格和有限体积法推导出旋度和散度的离散公式以及非均质单元中等效电导率的计算公式，建立混合Helmholtz方程的离散方程。为保证大范围电磁场分布的稳定精确的求解以及多发射源的快速正演，采用直接法求解器PARDISO求解离散方程。此外，为了在不明显降低计算效率的情况下尽可能提高近场的计算精度，采用差异场方法处理各向异性地层3D模拟过程中的源奇异性问题。
对高阻油气藏的各向异性的数值模拟表明，MCSEM沿测线方向的电场，随着油气藏的纵向电阻率的增大而增大，而与油气藏的横向电阻率大小无关。对油气藏上方的覆盖层的各向异性的数值模拟表明，沿测线方向的电场，随着覆盖层横向电阻率的增大而增大，同时随着覆盖层纵向电阻率的增大而增大。因此，在进行海洋CSEM三维数据解释时，要特别注意海底各向异性的影响。
Controlled source audio magnetotelluric (CSAMT) is a ground source frequencydomain sounding artificial methods developed on the basis of magnetotelluric (MT)and audio magnetotelluric (AMT). Because the execution ground source is randomand the signal is weak, observation for MT is very difficult. CSAMT workshop usecontrolled sources. CSAMT is an important geophysical exploration methods, whichwidely used in a variety of geological survey, exploration of oil, natural gas,geothermal, metallic minerals, hydrology, engineering, environmental protection andother complex exploration ground structure.
Modeling and inversion of CSAMT in1D isotropic horizontally layeredformations is very important, because1D model is simple and its analytical solutionexists, it helps to understand the basic physical properties of the electromagnetic fieldas a basic geoelectric model, it is also the basis of preliminary interpretation andanalysis of data, selected initial model of the multi-dimensional inversion. Electricalanisotropy is always present on some scale, for example, alternating layers of thinsand shale is a typical anisotropic configuration. Data interpretation without consideranisotropy for the anisotropic formations will be difficult to get effective inversionresults. Therefore, the study of anisotropic formation CSAMT inversion theory is avery important issue.
In this thesis, we study the modeling and inversion of CSAMT in1D anisotropichorizontally layered formations, discuss the impact of anisotropy for the CSAMTresponse, and provide a theoretical basis and practical guidance for the interpretationof the data. Like other geophysical inverse problems, inversion of CSAMT in1Danisotropic horizontally layered formations includes forward calculation, choosemodel space, Fréchet derivatives,and iterative correction to get effectively inversionresults.
In chapter two, Using the two-dimensional Fourier transform and matrix decomposition techniques, Maxwell equations decompose into two transmission lineequation of TM and TE waves. Appling the transmission line theory and superpositionprinciple, according to the characteristics of CSAMT only using electric dipolesource, we improve the transmission line algorithm by only introducing the currentsource transmission line Green function to solve TM and TE waves. Then weestablish a new algorithm for electric and magnetic current source dyadic Green'sfunctions in frequency-wavenumber domain. Compared to conventionaltransmission line theory, the new algorithm can improve the efficiency of forward ofCSAMT. On this basis, using the fundamental solution of transmission lines Greenfunction and boundary conditions, and appling the generalized reflection coefficientand amplitude recursive formula, we obtain the analytical solutions of transmissionline Green's function for each stratum; then using the inverse Fourier transform andBessel formula, CSAMT response can be expressed as the Sommerfeld integral form.According to the characteristics of the layered model, this paper presents a cubicspline interpolation technique combined Lommel integral formula to calculate theSommerfeld integral. Compared to digital filtering algorithm commonly used ingeo-electromagnetic community, our method is faster with the same accuracy.
In chapter three, we select the parametric model space which can obtain thequantitatively inversion model parameter. Aiming at the characterisation of the1-Danistropic media that is piecewise constant, we first introduce a model vectorcomposed of not only the horizontal and vertical conductivities but also the depths ofhorizontal interfaces. We give the expressions of the spatial distribution ofconductivity in the anistropic media described by the model vector and theperturbation in the conductivity determined by change in the model vector. Then thedyadic Green’s functions given by Michalski and Mosig are applied to model theCSAMT responses in the media and to compute the Fréchet derivatives of theCSAMT response with respect to model vector. In order to enhance the computationalefficiency, we derive the analytic expressions of the Fréchet derivatives in the form ofSommerfeld integral and apply semi-analytical algorithm by the combination of cubicspline interpolation and Lommel integral formula to determine the synthetic CSAMT data and all components of Fréchet derivatives. Furthermore, we establish anadaptively regularized inversion of CSAMT data to simultaneously reconstruct allmodel parameters. Finally, we present numerical results to validate the algorithm.Numerical results showed that CSAMT data is much less sensitive to the verticalconductivity than to other model parameters. The inversion of CSAMT data in1-Danisotropic media recovers the main features of the model despite large initial errorsexist. Choice of initial interfaces is very important to improve inversion results.Whole parameter inversion improves the inversion results much better than theinversion with fixed interfaces but with wrong values. The inversion is also effectivefor the case that the layer number of the initial model being different from that of thetrue model.
Marine controlled source electromagnetic (MCSEM) is the frequency domainelectromagnetic method used for subsea oil and gas exploration. Since its ability toprovide spatial distribution of resistivity of subsea formation, MCSEM is becomingan important tool for both offshore hydrocarbon exploration and gas-hydrateidentification.The seabed may have complex structure including bathymetryvariations, near-surface variations of resistivity, anisotropy, and targets close toresistive basements. Survey design and data processing and interpretation of MCSEMneed a lot of numerical simulations.3D forward method for MCSEM becomes aresearch topic since1D and2D forward technology for MCSEM is becoming mature.The method for3D forward of MCSEM include finite element, finite difference, finitevolume and integral equation method. Based on Yee 's staggered grid, finite volumemethod for Maxwell's equations on each unit's integral discrete treatment, it caneffectively reduce the order of the differential equation, and also can reduce theimpact on the formation of discontinuous discrete conductivity results. In this thesis,we develop a finite volume method to simulate the response of MCSEM in3Danisotropic formation.
In order to effectively simulate three-dimensional marine controlled sourceelectromagnetic (CSEM) response in anisotropic formation, a coupled potential finitevolume method is established. As Electric field decompose by vector potential and scalar potential, the Maxwell’s equation is reformulated into Helmholtz equations interms of coupled scalar-vector potentials with Coulomb gauge. As the results, itovercome the low induction numbers problems. Then Yee’s staggered girds, finitevolume averaging and interpolation technique are used to discrete the Helmholtzequations. After that, a large, sparse and complex linear system with a blockdiagonally dominant structure is obtained. A direct solver PARDISO is applied tostalely and accurately solve the system of large-scale models. In order to improve theaccuracy of the near field without significantly reduce the computational efficiency, amethod using difference fields is proposed to reduce the source singularity effect ofanisotropic formation. The anisotropic modeling examples show that marine CSEMresponse is predominantly sensitive to reservoir vertical resistivity and not to reservoirhorizontal resistivity, provided that the reservoir are thin and high-resistivity; but themarine CSEM response is sensitive to both horizontal and vertical resistivity of theoverburden on top of the reservoir.
一维水平层状各向同性地层中的CSAMT正反演是非常重要的，因为一维模型简单，存在解析解，作为一个基本的地电模型，有助于理解电磁场的基本物理特性，它也是实际数据的初步解释和分析以及多维反演的初始模型选取的基础。实际地层在一定尺度上总是存在电的各向异性，例如，砂岩页岩的薄交互层是一种典型的各向异性构造。对各向异性地层测量数据进行解释，如果忽略各向异性，将很难得到有效的反演结果。因此，研究各向异性地层中的CSAMT的反演理论是一个非常重要的课题。本文研究各向异性地层中CSAMT的一维正反演理论，讨论各向异性地层对CSAMT一维正反演的影响特征和规律，为实际资料的正确解释提供理论依据和指导。与其他地球物理问题的反演一样，各向异性地层中的一维CSAMT反演，涉及正演计算，模型空间的选择，Fréchet导数的计算，以及反演过程中模型的迭代修正来得到有效的反演结果。
在第二章，利用二维Fourier变换与矩阵分解技术将层状各向异性地层中Maxwell方程转化成两个独立的关于TM波和TE波的传输线方程;借助传输线理论与叠加原理，根据CSAMT只使用电偶极源的特点，改进传输线算法，只引入电流源传输线Green函数，来求解TM波和TE波，建立各向异性地层中频率-波数域电流源电场和磁场并矢Green函数的新算法和新的表达式，相比传统的传输线理论，提高了CSAMT数值模拟的效率。在此基础上,利用传输线Green函数的基本解以及边界条件,推导出广义反射系数与振幅递推公式,得到各个地层中传输线Green函数的解析解;然后利用Fourier逆变换与Bessel公式将CSAMT响应表示为Sommerfeld形式的积分,针对层状模型的特点，本文提出了三次样条插值结合Lommel积分公式的技术，快速计算其数值解，相比地球物理电磁法中常用的数字滤波算法，精度相当，但计算速度更快。
在第三章，选择能够定量反演地下模型参数的参数化模型空间，通过水平和垂直导电率以及水平层界面构成的模型矢量来描述电阻率的空间分布。利用并矢Green函数，结合摄动理论，推导得到了Fréchet导数的解析表达式，表示为Sommerfeld形式的积分，采用第二章提出的三次样条插值结合Lommel积分公式的技术可以有效的求解。进一步的，建立了全参数广义逆反演算法和全参数正则化迭代反演算法来同时反演所有的模型参数，包括横向、纵向电导率和层界面深度。广义逆反演算法采用奇异值分解技术结合阻尼最小二乘法，可以有效的提高反演结果的稳定性和抗噪能力。对于正则化反演，由于CSAMT响应是非线性的，引入偏差原理并结合Cholesky分解自适应选择正则化因子，保证反演过程的稳定性。数值计算结果表明，CSAMT响应对地层纵向电导率的灵敏度要远小于相比对其他模型参数（横向电导率和层界面）的灵敏度。尽管初始模型存在较大误差，反演结果依然能够给出地层模型的主要特征。选择合适的初始层界面可以有效的改善反演结果；全参数的反演相比固定层界面的反演，反演结果更好；即使初始模型的层数与实际地层不符，依然能够给出有效的反演结果。
海洋可控源电磁法（MCSEM）是一种用于海底油气勘探的频率域电磁方法。MCSEM具有提供海底地层电阻率空间分布的能力，在油气层识别和海上油气储层定量评价以及降低海上钻探风险方面发挥着重要作用，已发展成为海上油气勘探的一种重要方法。由于海底地形构造复杂以及地层横向电阻率分布不均匀，在海洋电磁勘探的设计以及海洋电磁资料处理和解释过程中，均需要进行大量的数值模拟，一维和二维的数值模拟技术已经较为成熟或者正在趋于成熟，三维数值模拟成为当前MCSEM的一个研究热点，目前模拟三维MCSEM的数值方法主要有有限元法、有限差分法、有限体积法和积分方程法。基于Yee氏交错网格的有限体积法，对Maxwell方程在各个单元上的积分进行离散处理，能有效降低方程的微分阶数，同时也减少了地层电导率不连续对离散结果的影响，在电磁场数值模拟中得到了较广泛应用。本文采用有限体积算法模型各向异性地层中MCSEM的三维响应，分析、总结各向异性对海洋CSEM三维电磁响应的影响特征。
在第四章，为了有效模拟各向异性地层中海洋可控源电磁法的三维响应，我们建立了一套基于电场矢势与标势分解的耦合势有限体积法。利用电场的矢势和标势分解，将电场分解为无散场和无旋场之和，Maxwell方程转换为关于矢势与标势的混合Helmholtz方程，有效的克服低感应数问题。在此基础上,借助Yee氏交错网格和有限体积法推导出旋度和散度的离散公式以及非均质单元中等效电导率的计算公式，建立混合Helmholtz方程的离散方程。为保证大范围电磁场分布的稳定精确的求解以及多发射源的快速正演，采用直接法求解器PARDISO求解离散方程。此外，为了在不明显降低计算效率的情况下尽可能提高近场的计算精度，采用差异场方法处理各向异性地层3D模拟过程中的源奇异性问题。
对高阻油气藏的各向异性的数值模拟表明，MCSEM沿测线方向的电场，随着油气藏的纵向电阻率的增大而增大，而与油气藏的横向电阻率大小无关。对油气藏上方的覆盖层的各向异性的数值模拟表明，沿测线方向的电场，随着覆盖层横向电阻率的增大而增大，同时随着覆盖层纵向电阻率的增大而增大。因此，在进行海洋CSEM三维数据解释时，要特别注意海底各向异性的影响。
Controlled source audio magnetotelluric (CSAMT) is a ground source frequencydomain sounding artificial methods developed on the basis of magnetotelluric (MT)and audio magnetotelluric (AMT). Because the execution ground source is randomand the signal is weak, observation for MT is very difficult. CSAMT workshop usecontrolled sources. CSAMT is an important geophysical exploration methods, whichwidely used in a variety of geological survey, exploration of oil, natural gas,geothermal, metallic minerals, hydrology, engineering, environmental protection andother complex exploration ground structure.
Modeling and inversion of CSAMT in1D isotropic horizontally layeredformations is very important, because1D model is simple and its analytical solutionexists, it helps to understand the basic physical properties of the electromagnetic fieldas a basic geoelectric model, it is also the basis of preliminary interpretation andanalysis of data, selected initial model of the multi-dimensional inversion. Electricalanisotropy is always present on some scale, for example, alternating layers of thinsand shale is a typical anisotropic configuration. Data interpretation without consideranisotropy for the anisotropic formations will be difficult to get effective inversionresults. Therefore, the study of anisotropic formation CSAMT inversion theory is avery important issue.
In this thesis, we study the modeling and inversion of CSAMT in1D anisotropichorizontally layered formations, discuss the impact of anisotropy for the CSAMTresponse, and provide a theoretical basis and practical guidance for the interpretationof the data. Like other geophysical inverse problems, inversion of CSAMT in1Danisotropic horizontally layered formations includes forward calculation, choosemodel space, Fréchet derivatives,and iterative correction to get effectively inversionresults.
In chapter two, Using the two-dimensional Fourier transform and matrix decomposition techniques, Maxwell equations decompose into two transmission lineequation of TM and TE waves. Appling the transmission line theory and superpositionprinciple, according to the characteristics of CSAMT only using electric dipolesource, we improve the transmission line algorithm by only introducing the currentsource transmission line Green function to solve TM and TE waves. Then weestablish a new algorithm for electric and magnetic current source dyadic Green'sfunctions in frequency-wavenumber domain. Compared to conventionaltransmission line theory, the new algorithm can improve the efficiency of forward ofCSAMT. On this basis, using the fundamental solution of transmission lines Greenfunction and boundary conditions, and appling the generalized reflection coefficientand amplitude recursive formula, we obtain the analytical solutions of transmissionline Green's function for each stratum; then using the inverse Fourier transform andBessel formula, CSAMT response can be expressed as the Sommerfeld integral form.According to the characteristics of the layered model, this paper presents a cubicspline interpolation technique combined Lommel integral formula to calculate theSommerfeld integral. Compared to digital filtering algorithm commonly used ingeo-electromagnetic community, our method is faster with the same accuracy.
In chapter three, we select the parametric model space which can obtain thequantitatively inversion model parameter. Aiming at the characterisation of the1-Danistropic media that is piecewise constant, we first introduce a model vectorcomposed of not only the horizontal and vertical conductivities but also the depths ofhorizontal interfaces. We give the expressions of the spatial distribution ofconductivity in the anistropic media described by the model vector and theperturbation in the conductivity determined by change in the model vector. Then thedyadic Green’s functions given by Michalski and Mosig are applied to model theCSAMT responses in the media and to compute the Fréchet derivatives of theCSAMT response with respect to model vector. In order to enhance the computationalefficiency, we derive the analytic expressions of the Fréchet derivatives in the form ofSommerfeld integral and apply semi-analytical algorithm by the combination of cubicspline interpolation and Lommel integral formula to determine the synthetic CSAMT data and all components of Fréchet derivatives. Furthermore, we establish anadaptively regularized inversion of CSAMT data to simultaneously reconstruct allmodel parameters. Finally, we present numerical results to validate the algorithm.Numerical results showed that CSAMT data is much less sensitive to the verticalconductivity than to other model parameters. The inversion of CSAMT data in1-Danisotropic media recovers the main features of the model despite large initial errorsexist. Choice of initial interfaces is very important to improve inversion results.Whole parameter inversion improves the inversion results much better than theinversion with fixed interfaces but with wrong values. The inversion is also effectivefor the case that the layer number of the initial model being different from that of thetrue model.
Marine controlled source electromagnetic (MCSEM) is the frequency domainelectromagnetic method used for subsea oil and gas exploration. Since its ability toprovide spatial distribution of resistivity of subsea formation, MCSEM is becomingan important tool for both offshore hydrocarbon exploration and gas-hydrateidentification.The seabed may have complex structure including bathymetryvariations, near-surface variations of resistivity, anisotropy, and targets close toresistive basements. Survey design and data processing and interpretation of MCSEMneed a lot of numerical simulations.3D forward method for MCSEM becomes aresearch topic since1D and2D forward technology for MCSEM is becoming mature.The method for3D forward of MCSEM include finite element, finite difference, finitevolume and integral equation method. Based on Yee 's staggered grid, finite volumemethod for Maxwell's equations on each unit's integral discrete treatment, it caneffectively reduce the order of the differential equation, and also can reduce theimpact on the formation of discontinuous discrete conductivity results. In this thesis,we develop a finite volume method to simulate the response of MCSEM in3Danisotropic formation.
In order to effectively simulate three-dimensional marine controlled sourceelectromagnetic (CSEM) response in anisotropic formation, a coupled potential finitevolume method is established. As Electric field decompose by vector potential and scalar potential, the Maxwell’s equation is reformulated into Helmholtz equations interms of coupled scalar-vector potentials with Coulomb gauge. As the results, itovercome the low induction numbers problems. Then Yee’s staggered girds, finitevolume averaging and interpolation technique are used to discrete the Helmholtzequations. After that, a large, sparse and complex linear system with a blockdiagonally dominant structure is obtained. A direct solver PARDISO is applied tostalely and accurately solve the system of large-scale models. In order to improve theaccuracy of the near field without significantly reduce the computational efficiency, amethod using difference fields is proposed to reduce the source singularity effect ofanisotropic formation. The anisotropic modeling examples show that marine CSEMresponse is predominantly sensitive to reservoir vertical resistivity and not to reservoirhorizontal resistivity, provided that the reservoir are thin and high-resistivity; but themarine CSEM response is sensitive to both horizontal and vertical resistivity of theoverburden on top of the reservoir.
引文
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