We present a new algorithm for 3-D forward modelling and spectral inversion of resistivity and time-domain full-decay induced polarization (IP) data. To our knowledge, all algorithms available for handling 3-D spectral inversion of full-decay IP data use a time-domain approximation to Poisson's equation in the forward response. To avoid this approximation, we compute the response in the frequency domain solving the full version of Poisson's equation for a range of frequencies (10^–8–10⁴ Hz) and then transform the response into the time domain, where we account for the transmitted current waveform. Solving Poisson's equation in 3-D is computationally expensive and in order to balance accuracy, time, and memory usage we introduce the following: (1) We use two separate meshes for the forward response and the model update, respectively. The forward mesh is an unstructured tetrahedral mesh allowing for local refinements whereas the model (inversion) mesh is a node-based structured mesh, where roughness constraints are easily implemented. By decoupling the two meshes, they can be tuned for optimizing the forward accuracy and the inversion resolution, independently. (2) A singularity removal method known from resistivity modelling has been adapted to the complex IP case and is applied to minimize the numerical errors caused by the fast changing potential close to the source electrodes. The method includes splitting the potential field into a primary part (response of a homogenous background) and a secondary part (from the anomalies). Two different forward meshes are then used to compute the forward response: a dense mesh for the primary potential field (only computed once for each frequency) and a coarser mesh for the secondary potential field (computed in each iteration step of the inversion). With this method, the singularity is minimized and the memory usages is decreased significantly at the same time. (3) Finally, we are sparsing (downsampling) the Jacobian matrix based on a threshold value of the normalized sensitivity. The Jacobian computation is performed by time-transforming the frequency-domain Jacobian obtained through the adjoint method. The Jacobian downsampling is carried out before the time-transform in the frequency domain, thus avoiding the time-transformation of the Jacobian elements with negligible sensitivity. We invert resistivity data and all IP time-gates simultaneously and use the Gauss–Newton model update to minimize the L2 misfit function. We invert the resistivity data and all IP time-gates simultaneously and use the Gauss–Newton model update to minimize the L2 misfit function. We demonstrate the performance of our inversion approach with a synthetic data example with 3-D anomalies and a field example, where lithology logs verify the results. The data sets contain 1256 quadrupole measurements with 33 IP time-gates each. The inversions results show good data fits and model retrieval. The inversion takes approximately one hour per iteration using four CPUs. With this speed and accuracy, we believe this modelling and inversion approach will be a strong tool for 3-D spectral inversion of resistivity and full-decay IP field data for both surface and borehole applications.

中文翻译：

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电阻率和全衰减感应极化数据的3-D时域频谱反演-泊松方程的完整解和电流波形的建模

我们提出了一种用于电阻率和时域全衰减感应极化（IP）数据的3-D正向建模和频谱反演的新算法。据我们所知，所有可用于处理全衰减IP数据的3-D频谱反演的算法都在正向响应中使用时域近似泊松方程。为了避免这种近似，我们在频域中计算响应，以求解一系列频率范围（10 ^–8 –10 ⁴赫兹（Hz），然后将响应转换到时域，其中我们考虑了所传输的电流波形。在3-D中求解泊松方程在计算上很昂贵，并且为了平衡准确性，时间和内存使用，我们引入以下内容：（1）我们分别使用两个单独的网格进行正向响应和模型更新。前向网格是允许局部细化的非结构化四面体网格，而模型（反演）网格是基于节点的结构化网格，可以轻松实现粗糙度约束。通过解耦两个网格，可以分别调整它们以优化前向精度和反演分辨率。（2）电阻率建模中已知的奇异点消除方法已适用于复杂的IP情况，并用于最小化源电极附近电位快速变化所引起的数值误差。该方法包括将势场分成主要部分（对同质背景的响应）和次要部分（来自异常）。然后使用两个不同的前向网格来计算前向响应：用于主电势场的密集网格（每个频率仅计算一次）和用于第二电势场的粗网格（在反演的每个迭代步骤中计算）。使用此方法，可以最大程度地减少奇点并同时显着减少内存使用。（3）最后，我们基于归一化灵敏度的阈值来稀疏（下采样）Jacobian矩阵。雅可比计算是通过对通过伴随方法获得的频域雅可比矩阵进行时间转换来执行的。雅可比下采样是在频域中的时间变换之前进行的，因此避免了雅可比元素的时间变换，其灵敏度可以忽略不计。我们同时反转电阻率数据和所有IP时间门，并使用Gauss-Newton模型更新来最小化L2失配功能。我们同时反转电阻率数据和所有IP时间门，并使用Gauss-Newton模型更新来最小化L2失配功能。我们通过具有3-D异常的综合数据示例和现场示例来演示反演方法的性能，岩性记录验证结果的位置。数据集包含1256个四极测量，每个测量具有33个IP时间门。反演结果表明良好的数据拟合和模型检索。使用四个CPU，每次迭代大约需要花费一小时的时间。凭借这种速度和准确性，我们相信这种建模和反演方法将是用于地面和钻孔应用的电阻率和全衰减IP场数据的3-D光谱反演的强大工具。