An Efficient Algorithm of Both Fréchet Derivative and Inversion of MCIL Data in a Deviated Well in a Horizontally Layered TI Formation Based on TLM Modeling
An Efficient Algorithm of Both Fréchet Derivative and Inversion of MCIL Data in a Deviated Well in a Horizontally Layered TI Formation Based on TLM Modeling
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DOI：
10.1109/tgrs.2014.2305669
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发表时间：
2014-04
影响因子：
8.2
通讯作者：
Shouwen Yang;Jianxun Wang;Jianmei Zhou;Tianzhu Zhu;Hongnian Wang
中科院分区：
文献类型：
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作者：
Shouwen Yang;Jianxun Wang;Jianmei Zhou;Tianzhu Zhu;Hongnian Wang
本文基于传输线法（TLM）建立了一种有效的水平层状横观各向同性（TI）地层多分量感应测井（MCIL）数据反演的Frechet导数算法，以便从MCIL数据中同时重建包括水平和垂直电导率、水平界面、井斜角和斜井下仪器方位角的模型向量。首先，TI模型中的MCIL响应可以通过TLM获得的谱电磁场和基于三次样条插值的索末菲积分的半解析计算来有效地确定。然后，玻恩近似执行，以获得MCIL Fréchet导数作为两个频谱电磁场在无限域中的乘积的六重积分。利用二维Dirac函数的积分性质，将Fréchet导数的六重积分进一步简化为两重积分：一重是径向谱域的Sommerfeld积分，另一重是垂直空间域的定积分.它们是有效地计算类似的MCIL模拟的半解析方法。因此，我们得到MCIL响应的变化和模型向量的小扰动之间的线性方程。然后，通过Frechet导数的归一化和奇异值分解技术，迭代地修改所有模型参数，以实现反演模型的输入数据与合成数据之间的最佳拟合。最后，我们应用数值结果来研究Frechet导数的特性，并验证反演方法及其抗噪声能力。
In this paper, we set up an efficient Fréchet derivative algorithm of inversion of multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations based on transmission line method (TLM) in order to simultaneously reconstruct the model vector including both horizontal and vertical conductivities, horizontal interfaces, borehole dipping angle, and tool azimuth in deviated well from the MCIL data. First, MCIL responses in the TI model are efficiently determined by both the spectrum EM fields obtained by TLM and the semianalytical computation of Sommerfeld integrals based on the cubic spline interpolation. Then, the Born approximations are executed to derive the MCIL Fréchet derivatives as the sixfold integrals of the products of two spectrum EM fields in infinite domains. By using the integral characters of the 2-D Dirac function, the sixfold integrals of Fréchet derivatives are further simplified into twofold integrals: One is the Sommerfeld integral in the radial spectrum domain, and the other is the definite integral in the vertical spatial domain. They are efficiently computed by the semianalytical approach similar to the MCIL simulation. Therefore, we obtain the linear equations between changes in the MCIL responses and small perturbations in the model vector. After that, we iteratively modify all of the model parameters to realize the best fit between the input data and the synthetic data of the inverted model by the normalization of the Fréchet derivative and singular value decomposition technique. Finally, we apply numerical results to investigate the characteristics of the Fréchet derivatives and to validate the inversion method and its antinoise ability.