The article advances a 3-D adaptive regularization nonlinear inversion of the logging while drilling (LWD) ultradeep multicomponent resistivity (LWD-UDMCR) by the Gauss-Newton (GN) method. We manage to reconstruct the pixel-based horizontal and vertical conductivities simultaneously in a goal domain outside an arbitrary dipping borehole. The piecewise constant functions are used to describe the spatial distribution of the block-based and the pixel-based conductivity. The background formation is assumed as the horizontally layered transversely isotropic (TI) media, and the background electromagnetic (EM) fields are determined analytically by the transmission line method (TLM). We then use the 3-D finite volume method (FVM) of secondary field coupled potentials and parallel direct sparse solver (PARDISO) to simulate the tool responses and Fréchet derivatives simultaneously. Through the projection operator and OpenMP parallel technique, we further enhance the computational efficiency of the pixel-based explicit Fréchet derivatives and set up a complete normalization linearized response. After that, the large normal equation from the quadratic objective function is solved by the preconditioned conjugate gradient (PCG) to determine the gradient of the objective function. By properly controlling the maximum component of the gradient per iteration step, we acquire an adaptive regularization factor so that the stabilization of the inversion solution is assured as well as the realization of the best fit of the input data with the modeling logs. Finally, numerical tests validate the algorithm and antinoise ability.

中文翻译：

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基于二次场耦合势有限体积法和显式 Frechet 导数的各向异性地层随钻测井超深电阻率三维自适应正则化非线性反演


本文提出了一种采用高斯-牛顿 (GN) 方法对随钻测井 (LWD) 超深多分量电阻率 (LWD-UDMCR) 进行 3 维自适应正则化非线性反演。我们设法在任意倾斜钻孔外的目标域中同时重建基于像素的水平和垂直电导率。分段常数函数用于描述基于块和基于像素的电导率的空间分布。假设背景形成为水平分层的横观各向同性（TI）介质，并且通过传输线法（TLM）分析确定背景电磁（EM）场。然后，我们使用二次场耦合势的 3-D 有限体积法 (FVM) 和并行直接稀疏求解器 (PARDISO) 同时模拟工具响应和 Fréchet 导数。通过投影算子和OpenMP并行技术，我们进一步提高了基于像素的显式Fréchet导数的计算效率，并建立了完整的归一化线性响应。之后，通过预条件共轭梯度（PCG）求解二次目标函数的大正规方程，以确定目标函数的梯度。通过适当控制每个迭代步骤梯度的最大分量，我们获得自适应正则化因子，从而保证反演解的稳定性以及实现输入数据与建模日志的最佳拟合。最后通过数值测试验证了算法和抗噪声能力。