We have developed a memory and operation-count efficient 2.5D inversion algorithm of electrical resistivity (ER) data that can handle fine discretization domains imposed by other geophysical (e.g, ground penetrating radar or seismic) data. Due to numerical stability criteria and available computational memory, joint inversion of different types of geophysical data can impose different grid discretization constraints on the model parameters. Our algorithm enables the ER data sensitivities to be directly joined with other geophysical data without the need of interpolating or coarsening the discretization. We have used the adjoint method directly in the discretized Maxwell’s steady state equation to compute the data sensitivity to the conductivity. In doing so, we make no finite-difference approximation on the Jacobian of the data and avoid the need to store large and dense matrices. Rather, we exploit matrix-vector multiplication of sparse matrices and find successful convergence using gradient descent for our inversion routine without having to resort to the Hessian of the objective function. By assuming a 2.5D subsurface, we are able to linearly reduce memory requirements when compared to a 3D gradient descent inversion, and by a power of two when compared to storing a 2D Hessian. Moreover, our method linearly outperforms operation counts when compared with 3D Gauss-Newton conjugate-gradient schemes, which scales cubically in our favor with respect to the thickness of the 3D domain. We physically appraise the domain of the recovered conductivity using a cutoff of the electric current density present in our survey. We evaluate two case studies to assess the validity of our algorithm. First, on a 2.5D synthetic example, and then on field data acquired in a controlled alluvial aquifer, where we were able to match the recovered conductivity to borehole observations.

中文翻译：

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使用离散伴随方法有效地反演2.5D电阻率数据

我们已经开发了一种电阻率（ER）数据的高效存储和可操作计数的2.5D反演算法，该算法可处理其他地球物理（例如，探地雷达或地震）数据所施加的精细离散域。由于数值稳定性标准和可用的计算内存，不同类型的地球物理数据的联合反演会对模型参数施加不同的网格离散约束。我们的算法使ER数据敏感度可以直接与其他地球物理数据结合，而无需内插或粗化离散化。我们已经在离散的麦克斯韦稳态方程中直接使用了伴随方法来计算数据对电导率的敏感性。在这样做，我们没有对数据的雅可比行列进行有限差分近似，并且避免了存储大而密集的矩阵的需要。相反，我们利用稀疏矩阵的矩阵向量乘法，并在不求助于目标函数的Hessian的情况下，对我们的反演例程使用梯度下降法找到了成功的收敛性。通过假设2.5D地下，与3D梯度下降反演相比，我们能够线性地减少内存需求，而与存储2D粗麻布相比，我们可以线性地减少2的幂。此外，与3D Gauss-Newton共轭梯度方案相比，我们的方法在线性上胜过运算次数，3D Gauss-Newton共轭梯度方案在3D域的厚度方面对我们有利。我们使用调查中存在的电流密度的临界值来物理评估恢复的电导率的范围。我们评估了两个案例研究，以评估我们算法的有效性。首先，在一个2.5D合成实例上，然后在受控冲积含水层中采集的现场数据，在这里我们能够将恢复的电导率与钻孔观测值进行匹配。