Time-lapse electrical resistivity tomography (ERT) is a popular geophysical method to estimate three-dimensional (3D) permeability fields from electrical potential difference measurements. Traditional inversion and data assimilation methods are used to ingest this ERT data into hydrogeophysical models to estimate permeability. Due to ill-posedness and the curse of dimensionality, existing inversion strategies provide poor estimates and low resolution of the 3D permeability field. Recent advances in deep learning provide us with powerful algorithms to overcome this challenge. This paper presents a deep learning (DL) framework to estimate the 3D subsurface permeability from time-lapse ERT data. To test the feasibility of the proposed framework, we train DL-enabled inverse models on simulation data. Each measurement in both synthetic and field data is standardized by removing the mean and scaling the time-series to unit variance. This pre-processing step is necessary to bring simulation data closer to field observations. Subsurface process models based on hydrogeophysics are used to generate this synthetic data. Training performed on limited simulation data resulted in the DL model over-fitting. An advanced data augmentation based on mixup is implemented to generate additional training samples to overcome this issue. This mixup technique creates weakly labeled (low-fidelity) samples from strongly labeled (high-fidelity) data. The weakly labeled training data is then used to develop DL-enabled inverse models and reduce over-fitting. As both time-lapse ERT (1133048 features/realization) and 3D permeability (585453 features/realization) data samples are from a high-dimensional space, principal component analysis (PCA) is employed to reduce dimensionality. Encoded ERT and encoded permeability are generated using the trained PCA estimators. A deep neural network is then trained to map the encoded ERT to encoded permeability. This mixup training and unsupervised learning allowed us to build a fast and reasonably accurate DL-based inverse model under limited simulation data. Results show that proposed weak supervised learning can capture salient spatial features in the 3D permeability field. Quantitatively, the average mean squared error (in terms of the natural log) on the strongly labeled training, validation, and test datasets is less than 0.5. The $R^2$-score (global metric) is greater than 0.75, and the percent error in each cell (local metric) is less than 10%. Finally, an added benefit in terms of computational cost is that the proposed DL-based inverse model is at least $\mathcal{O}\left(10^4\right)$ times faster than running a forward model once it is trained. Data generation, DL model training, and hyperparameter tuning to identify optimal neural network architectures utilized high-performance computing resources while the DL inference is performed on a standard laptop. Approximately, $\mathcal{O}\left(10^5\right)$ processor hours are used for generating data and DL tuning and training. We acknowledge that the data generation and DL model development are expensive. But once a DL model is trained, it can be re-used for inversion rapidly for the given system, with set physics and domain. Note that traditional inversion may require multiple forward model simulations (e.g., in the order of 10 to 1000), which are very expensive. This computational savings $\left(\approx \mathcal{O}\left(10^5\right)-\mathcal{O}\left(10^7\right)\right)$ makes the proposed DL-based inverse model attractive for subsurface imaging and real-time ERT monitoring applications due to fast and yet reasonably accurate estimations of permeability field.

延时电阻率断层扫描 (ERT) 是一种流行的地球物理方法，用于从电势差测量值估计三维 (3D) 渗透率场。传统的反演和数据同化方法用于将 ERT 数据引入水文地球物理模型以估计渗透率。由于不适定性和维数灾难，现有的反演策略对 3D 渗透率场的估计较差且分辨率较低。深度学习的最新进展为我们提供了强大的算法来克服这一挑战。本文提出了一种深度学习 (DL) 框架，用于根据延时 ERT 数据估计 3D 地下渗透率。为了测试所提出框架的可行性，我们在模拟数据上训练了支持深度学习的逆模型。通过去除平均值并将时间序列缩放为单位方差，对合成数据和现场数据中的每次测量进行标准化。这个预处理步骤对于使模拟数据更接近现场观察是必要的。基于水文地球物理学的地下过程模型用于生成这种合成数据。在有限的模拟数据上进行的训练导致 DL 模型过拟合。实现了基于 mixup 的高级数据增强，以生成额外的训练样本来克服这个问题。这种混合技术从强标记（高保真）数据中创建弱标记（低保真）样本。然后使用弱标记的训练数据来开发支持 DL 的逆模型并减少过度拟合。由于延时 ERT（1133048 个特征/实现）和 3D 渗透率（585453 个特征/实现）数据样本均来自高维空间，因此采用主成分分析 (PCA) 来降低维度。使用经过训练的 PCA 估计器生成编码的 ERT 和编码的渗透率。然后训练深度神经网络以将编码的 ERT 映射到编码的渗透率。这种混合训练和无监督学习使我们能够在有限的模拟数据下构建快速且合理准确的基于 DL 的逆模型。结果表明，提出的弱监督学习可以捕捉 3D 渗透率领域中的显着空间特征。在数量上，强标记的训练、验证和测试数据集的平均均方误差（以自然对数表示）小于 0.5。这 使用经过训练的 PCA 估计器生成编码的 ERT 和编码的渗透率。然后训练深度神经网络以将编码的 ERT 映射到编码的渗透率。这种混合训练和无监督学习使我们能够在有限的模拟数据下构建快速且合理准确的基于 DL 的逆模型。结果表明，提出的弱监督学习可以捕捉 3D 渗透率领域中的显着空间特征。在数量上，强标记的训练、验证和测试数据集的平均均方误差（以自然对数表示）小于 0.5。这 使用经过训练的 PCA 估计器生成编码的 ERT 和编码的渗透率。然后训练深度神经网络以将编码的 ERT 映射到编码的渗透率。这种混合训练和无监督学习使我们能够在有限的模拟数据下构建快速且合理准确的基于 DL 的逆模型。结果表明，提出的弱监督学习可以捕捉 3D 渗透率领域中的显着空间特征。在数量上，强标记的训练、验证和测试数据集的平均均方误差（以自然对数表示）小于 0.5。这 这种混合训练和无监督学习使我们能够在有限的模拟数据下构建快速且合理准确的基于 DL 的逆模型。结果表明，提出的弱监督学习可以捕捉 3D 渗透率领域中的显着空间特征。在数量上，强标记的训练、验证和测试数据集的平均均方误差（以自然对数表示）小于 0.5。这 这种混合训练和无监督学习使我们能够在有限的模拟数据下构建快速且合理准确的基于 DL 的逆模型。结果表明，提出的弱监督学习可以捕捉 3D 渗透率领域中的显着空间特征。在数量上，强标记的训练、验证和测试数据集的平均均方误差（以自然对数表示）小于 0.5。这$R^2$-score（全局度量）大于 0.75，每个单元格（局部度量）的百分比误差小于 10%。最后，在计算成本方面的另一个好处是，所提出的基于 DL 的逆模型至少$\mathcal{○}\left(10^4\right)$训练后比运行正向模型快几倍。数据生成、深度学习模型训练和超参数调整以识别最佳神经网络架构利用高性能计算资源，而深度学习推理是在标准笔记本电脑上执行的。大约，$\mathcal{○}\left(10^5\right)$处理器小时用于生成数据和 DL 调整和训练。我们承认数据生成和 DL 模型开发成本很高。但是，一旦训练了 DL 模型，它就可以快速地重新用于给定系统的反演，具有设定的物理场和域。请注意，传统的反演可能需要多个正向模型模拟（例如，大约 10 到 1000 次），这非常昂贵。这种计算节省$\left(\approx \mathcal{○}\left(10^5\right)-\mathcal{○}\left(10^7\right)\right)$由于对渗透率场的快速且合理准确的估计，所提出的基于 DL 的逆模型对地下成像和实时 ERT 监测应用具有吸引力。