Use of convolutional neural networks with encoder-decoder structure for predicting the inverse operator in hydraulic tomography

In this manuscript, we discuss the capabilities of a deep learning algorithm implemented with the Conventional Neural Network concept to characterize the hydraulic properties of aquifers. The algorithm called CNN-HT is designed to predict the inverse operator of hydraulic tomography using a synthetic training dataset in which the hydraulic head data associated with pumping tests are linked to hydraulic transmissivity field. This approach relies on an adaptation of the SegNet network that was initially developed to process image segmentation. The SegNet is composed of encoders and decoders networks. In the encoder, sequential operations with multiple filters, as convolution, batch normalization, max-pooling are performed to identify feature maps of the input data. In the decoder, the up-sampling, convolution, batch normalization and regression operations are used to prepare the output by recovering the loss of spatial resolution that occurred in the encoder process. In this adaptation, we used the least-square iterative formulation at the initial iteration with Jacobian matrix to resize the hydraulic head data to match the size of the output (transmissivity field). This protocol was applied to the hydraulic head data computed numerically by solving the groundwater flow equation for a given transmissivity field, generated geostatistically with Gaussian and spherical variograms. A part of this data was used for training the network and the other part to test its performance. The test step confirmed the effectiveness of this tool in reconstructing the main heterogeneities of the hydraulic properties, and its effectiveness is related to the nature and quantity of the training data. Moreover, the CNN-HT method provided inversion results of the same quality than those obtained with the Gauss-Newton algorithm using the finite difference or adjoint state method in the computation of the Jacobian matrix. However, the computational time is longer in CNN-HT but this time can be less or of the same order as that of Gauss-Newton using finite difference method.