利用传输线理论、Sommerfeld积分快速计算以及最小平方拟合技术研究建立多分量感应测井数据的一种新的快速参数化迭代反演算法,同时重构水平层状横向同性地层的纵、横向电阻率以及水平层界面深度.首先,通过Fourier变换与传输线理论给出频率波数域中电磁场并矢Green函数在各个地层中的解析解,并利用三次样条插值和贝塞尔函数递推公式建立Sommerfeld积分的半解析算法,快速计算多分量感应的测井响应.然后在此基础上,利用摄动理论建立磁场并矢Green函数与模型向量间变化关系的摄动方程,并将摄动方程中各个积分转化为Sommerfeld积分,实现正演模拟的同时用半解析算法快速确定多分量感应测井响应的Fréchet导数.最后,利用归一化处理和奇异值分解技术,同时反演所有地层的纵、横向电阻率和层界面深度,实现输入数据和反演模型的模拟数据优化拟合.理论模型的数值结果验证了该反演算法的有效性及抗噪性.%We advance a new fast parameterized inversion algorithm to simultaneously reconstruct both horizontal and vertical conductivities, and the horizontal interface depth of each layer from the multicomponent induction logging (MCIL) data in horizontal layered transversely isotropic (TI) formations by transmission line method (TLM) , fast computation of Sommerfeld integral and least square fitting technique. We give the analytic solution of tensor Green's function in the frequency-wavenumber domain in the TI formations based on Fourier transformation and transmission line method (TLM). The novel semi-analytic algorithm is set up to compute Sommerfeld integrals by combination of the cubic spline interpolation with the recursive formulae of Bessel function. Thus we can efficiently compute the responses of MCIL responses in the formations. Then, we apply the perturbation principle to derive perturbation equations about the relation between changes in MCIL response and perturbations in model parameters. The perturbation equations are transformed into the forms of Sommerfeld integrals. As a result , we can efficiently compute MCIL responses and their Fréchet derivatives with respect to all model parameter vectors at the same time. Finally, we iteratively reconstruct all the model parameters to realize the best fit of the input logging data with the modeling data by the normalization of Fréchet derivative and singular-value-decomposition (SVD). Theoretical inversion results validate the inversion method and its anti-noise performance.
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