Over several decades, much research has been done to develop 3D electromagnetic inversion algorithms. Due to the computational complexity and the memory requirements for 3D time-domain electromagnetic (TEM) inversion algorithms, many real-world surveys are inverted within one dimension. To speed up calculations and manage memory for 3D inversions of TEM data, we have developed an approach using three uncoupled meshes: an inversion mesh, a forward-model mesh, and a mesh for Jacobian calculations. The inversion mesh is a coarse regular and structured mesh, such that constraints are easily enforced between the model parameters. Forward responses are calculated on a dense unstructured mesh to obtain accurate electromagnetic fields, whereas the Jacobian is calculated on a coarse unstructured mesh. We found that using a coarse mesh for the Jacobian is sufficient for the inversion to converge and, equally important, that it provides a significant speed boost in the overall inversion process, compared to calculating it on the forward-modeling mesh. The unstructured meshes are made of tetrahedral elements, and the electromagnetic fields are calculated using the finite-element method. The inversion optimization uses a standard Gauss-Newton formulation. For further speed up and memory optimizing of the inversion, we use domain decomposition for calculating the responses for each transmitter separately and parallelize the problem over domains using OpenMP. Compared to a 1D solution, the accuracy for the Jacobian is 1%–5% for the dense mesh and 2%–7% for the coarse mesh, but the calculation time is approximately five times faster for the coarse mesh. We also examined the algorithm on a small ground-based TEM data set acquired in an area where a 3D earth distorts the electromagnetic fields to such a degree that a 1D inversion is not feasible.

中文翻译：

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使用有限元和三重网格公式对时域电磁数据进行3D反演

几十年来，为开发3D电磁反演算法已进行了大量研究。由于3D时域电磁（TEM）反演算法的计算复杂性和存储要求，许多现实世界的调查都在一维范围内被反演。为了加快计算速度并管理TEM数据的3D反演的内存，我们开发了一种使用三个非耦合网格的方法：反演网格，正向模型网格和用于Jacobian计算的网格。反演网格是粗糙的规则网格和结构化网格，因此可以轻松地在模型参数之间实施约束。在密集的非结构化网格上计算正向响应以获得准确的电磁场，而在粗糙的非结构化网格上计算雅可比行列。我们发现，对于反演收敛，使用粗糙网格作为雅可比行列就足够了，同等重要的是，与在正向建模网格上进行计算相比，它在整个反演过程中提供了显着的速度提升。非结构化网格由四面体元素组成，并且使用有限元方法计算电磁场。反演优化使用标准的高斯-牛顿公式。为了进一步提高反演的速度和优化内存，我们使用域分解来分别计算每个发送器的响应，并使用OpenMP对问题进行并行化处理。与一维解决方案相比，致密网格的Jacobian精度为1％–5％，粗网格的精度为2％–7％，但是粗网格的计算时间大约要快五倍。我们还研究了在3D地球使电磁场失真到无法进行1D反演的程度的区域中获取的小型地面TEM数据集上的算法。