  The inverse problem of electrical resistivity surveys (ERSs) is difficult
because of its nonlinear and ill-posed nature. For this task, traditional
linear inversion methods still face challenges such as suboptimal approximation
and initial model selection. Inspired by the remarkable nonlinear mapping
ability of deep learning approaches, in this article, we propose to build the
mapping from apparent resistivity data (input) to resistivity model (output)
directly by convolutional neural networks (CNNs). However, the vertically
varying characteristic of patterns in the apparent resistivity data may cause
ambiguity when using CNNs with the weight sharing and effective receptive field
properties. To address the potential issue, we supply an additional tier
feature map to CNNs to help those aware of the relationship between input and
output. Based on the prevalent U-Net architecture, we design our network
(ERSInvNet) that can be trained end-to-end and can reach a very fast inference
speed during testing. We further introduce a depth weighting function and a
smooth constraint into loss function to improve inversion accuracy for the deep
region and suppress false anomalies. Six groups of experiments are considered
to demonstrate the feasibility and efficiency of the proposed methods.
According to the comprehensive qualitative analysis and quantitative
comparison, ERSInvNet with tier feature map, smooth constraints, and depth
weighting function together achieve the best performance.
