Deep Learning Inversion of Electrical Resistivity Data

Bin Liu; Qian Guo; Shucai Li; Benchao Liu; Yuxiao Ren; Yonghao Pang; Xu Guo; Lanbo Liu; Peng Jiang

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Deep Learning Inversion of Electrical Resistivity Data

Bin Liu, Qian Guo, Shucai Li, Benchao Liu, Yuxiao Ren, Yonghao Pang, Xu Guo, Lanbo Liu, Peng Jiang

arXiv:1904.05265v2Apr 10, 2019cs.CVcs.AIphysics.geo-ph

TL;DR

ERS inversion is difficult because vertically varying apparent-resistivity patterns can create ambiguity for CNNs. The paper introduces ERSInvNet, which directly maps apparent resistivity data to models while adding tier information, depth weighting, and smooth constraints. Across comprehensive experiments, the combined design achieves the best reported performance, with fast inference and improved accuracy.

Problem
Vertically varying apparent-resistivity patterns can make similar CNN effective-area inputs correspond to different anomalous bodies and model values.
Method
ERSInvNet directly maps apparent resistivity data to resistivity models using a U-Net-based CNN with a tier feature map, depth weighting, and smooth constraints.
Results
The combined tier feature map, smooth constraints, and depth weighting achieve the best overall performance, while inversion curves show relative error within 0.4%.
Takeaways & Limitations
Tier information reduces false anomalies, depth weighting improves deep-region accuracy, and their combination with smooth constraints yields the strongest reported inversion results.

Abstract

from arXiv · show

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.

I. INTRODUCTION

ERS inversion seeks accurate subsurface models from observed data, but CNNs face ambiguity because apparent-resistivity patterns vary vertically. ERSInvNet addresses this with tier information, loss-function constraints, and direct end-to-end mapping.

ERS inversion reconstructs detailed subsurface properties from observed data for environmental, engineering, hydrological, and mining applications.
Apparent resistivity patterns correspond spatially to subsurface anomalies, motivating direct CNN mapping from apparent resistivity data to resistivity models.
Vertically varying apparent-resistivity patterns can make CNNs produce ambiguous outputs because similar effective-area patterns may correspond to different model values.
ERSInvNet uses a U-Net-based CNN with a tier feature map, depth weighting, and smooth constraints to address ambiguity, deep anomalies, and false anomalies.
The proposed network is trained end-to-end without data processing and is reported to provide fast inference and high inversion accuracy.

II. BACKGROUNDS

ERS measurements infer subsurface resistivity from surface electrical responses, while CNNs provide nonlinear mapping through local, weight-sharing operations and receptive fields. These properties motivate careful architectural design for resistivity inversion.

ERS measures potential differences generated by injected current and infers model parameters from observed data through an inverse mapping.
CNNs exploit local, spatially invariant patterns with weight-sharing kernels, unlike apparent-resistivity data whose patterns vary with vertical position.
Weight sharing applies the same convolutional kernel and bias at different positions, while convolutional layers transform inputs through nonlinear weighted operations.
A receptive field is the input neighborhood used to produce an output, and network depth and operations such as pooling affect its size.
Deep CNNs are commonly given excessive layers to ensure nonlinear expressiveness, although whether the nonlinearity is sufficient is difficult to determine.

III. METHODOLOGY

The methodology learns a direct mapping from apparent resistivity data to resistivity models with CNNs. It identifies effective-area ambiguity and supplements the input with vertical-position information.

The method learns the mapping function F from apparent resistivity data d directly to resistivity model m using deep neural networks.
Deep CNNs may emphasize central effective-area patterns even when their receptive field covers the full input.
Similar effective-area patterns at different vertical positions can correspond to different anomalous bodies and model values, creating training ambiguity.
The method supplements apparent-resistivity inputs with vertical tier information because tier positions correlate with anomalous structures at corresponding depths.
The tier feature map is concatenated with the data to provide information related to vertical location.

B. Networks

ERSInvNet is built on U-Net and augments the input with a tier feature map. Its shortcuts combine shallow positional and shape information with deeper features for prediction.

B. Networks: ERSInvNet uses U-Net shortcuts to concatenate shallow feature maps with deep feature maps.
B. Networks: U-Net shortcuts combine low-level position and shape knowledge with high-level features related to final output values.
B. Networks: The network concatenates a tier feature map with apparent-resistivity data to reduce ambiguity caused by vertically varying patterns.
B. Networks: Each tier-feature-map element equals its vertical tier number, and the map has the same spatial dimensions as the input data.

C. Loss Function

The loss function combines a depth-weighted value-fitting term with smoothness regularization to improve deep-anomaly inversion and reduce undesirable model variations.

C. Loss Function: The data value term uses mean square error to penalize differences between predicted and groundtruth resistivity values.The formulation applies the error at each model position.
C. Loss Function: Depth weighting assigns greater emphasis to deeper regions, addressing the difficulty of accurately recovering deep anomalies.The weighting follows the idea of compensating for the natural decay of the static field.
C. Loss Function: The depth-weighting parameters λ and β represent grid/electrode-location effects and control the distribution of depth weights, respectively.Their configuration is specified later in the paper.
C. Loss Function: The value term is modified by multiplying each squared prediction error by the depth-weighting function.This makes the loss place more emphasis on depth-dependent errors.
C. Loss Function: Smooth constraints are introduced because ERS inversion is ill-posed and typically produces nonunique, unstable solutions.The supplied formulation uses smoothness to reduce sudden changes between adjacent model grids.

2) Smooth Constraints:

The smoothness term regularizes predicted resistivity models by penalizing abrupt changes between neighboring grid cells, and it is incorporated into the final differentiable loss.

2) Smooth Constraints:: The smooth term penalizes absolute differences between horizontally and vertically adjacent predicted resistivity values.This reduces sudden changes between neighboring grids.
2) Smooth Constraints:: The smooth term acts as regularization and is also known as total variation loss.It is used to constrain the spatial variation of the inversion result.
2) Smooth Constraints:: The final loss combines the depth-weighted value term with the smoothness term scaled by α and normalized by Z.The operations and losses remain differentiable for end-to-end training.
2) Smooth Constraints:: ERSInvNet is the resulting end-to-end network formed from the differentiable operations and loss components.Its design adapts a U-Net backbone with a tier feature map and depth weighting for vertically varying apparent resistivity data.

A. Data Set Preparation

The study prepares synthetic ERS model–data pairs using selected electrode configurations and evaluates inversion with weighted statistical metrics alongside qualitative inspection.

A. Data Set Preparation: The ERSInv data set contains 36 214 synthetic resistivity-model and apparent-resistivity-data pairs, each with spatial dimension [64×304].The models are designed with reference to real two-dimensional ERS scenarios.
A. Data Set Preparation: Wenner and Wenner–Schlumberger arrays are used together because they provide good vertical and appropriate horizontal resolution.The apparent resistivity observations are generated by forward modeling on resistivity models.
A. Data Set Preparation: Input and output values are normalized to [0, 1], and the two electrode-configuration data types are combined with the tier feature map as three input channels.The simulated electrical fields are generated using finite-element methods.
A. Data Set Preparation: Training uses SGD with batch size 5, learning rate 0.1, momentum 0.9, weight decay 1e-4, and 500 optimization epochs.Validation is performed once after each training epoch.
A. Data Set Preparation: Performance is assessed visually and quantitatively using weighted mean square error and weighted correlation coefficient.The weighting gives larger importance to regions far from true anomalies because false distant anomalies are not preferred.

V. RESULTS AND DISCUSSION

Experiments show that ERSInvNet can reconstruct anomalous bodies accurately, while its tier feature map reduces ambiguity and false anomalies. Training losses decrease without apparent overfitting, though deep anomalies remain difficult to predict.

Experiment 1: ERSInvNet accurately localized anomalous bodies and predicted model values, with relative resistivity-profile error within 0.4% except near anomaly boundaries.Smooth constraints restrict resistivity-value mutations near boundaries, producing localized deviations there.
Experiment 1: Deep anomalies were harder to recover because their surface responses lacked obvious patterns in apparent resistivity data.The deepest 2000 Ω·m body was reconstructed at 1400 Ω·m, whereas shallower bodies were closer to model values.
Experiment 1: Both training and validation losses decreased gradually, falling from 0.008 to 0.001 by epoch 500 without indicating overfitting.The reported decline appeared likely to continue beyond 500 epochs.
Experiment 2: Adding the tier feature map improved anomaly morphology and boundaries while suppressing false anomalies near multiple true bodies.Examples include removal of false anomalies around a low-resistivity body and elimination of three high-resistivity false anomalies.

C. Results of Experiment 3

Experiment 3 separates the effects of smooth constraints and depth weighting. Smooth constraints reduce false anomalies but can weaken boundaries, while depth weighting improves deep-anomaly morphology; together, the full configuration performs best.

Results of Experiment 3: Smooth constraints produced fewer false anomalies but poorer anomaly-boundary accuracy than the configuration without them.Their principal reported contribution was suppressing false anomalies, at the cost of boundary delineation.
Results of Experiment 3: Depth weighting improved anomaly morphology and resistivity values, especially in deep regions, compared with the configuration without it.The comparison identifies depth weighting as beneficial to inversion accuracy.
Results of Experiment 3: The full SD configuration, combining the tier feature map, smooth constraints, and depth weighting, achieved the best overall performance.SD had the highest WR and lowest WMSE across the reported validation and test comparisons.
Results of Experiment 3: Applying smooth constraints alone increased WMSE by 74.7%, while the combined SD configuration restrained the negative effects of the individual components.The text attributes the increase to sacrificed boundary accuracy and suggests false deep anomalies contributed to the depth-weighting-only degradation.
Results of Experiment 3: Validation loss curves indicate robustness to the smoothness factor, with α = 0.2 yielding better performance.The study reports loss curves for different α values rather than a single universal optimum.

D. Results of Experiment 4

Noise affects inversion quality unevenly: deeper, smaller, and high-resistivity anomalies are more vulnerable, while larger shallow low-resistivity bodies are comparatively stable.

Results of Experiment 4: Under 1-dBw noise, one anomaly retained its shape and location but became much smaller; at 3 dBw, its position shifted.Noise intensity therefore altered both recovered size and location in this example.
Results of Experiment 4: Larger, shallower low-resistivity bodies were less affected by noise than deeper or smaller anomalies.Two larger inclined low-resistivity bodies showed little difference across noise levels.
Results of Experiment 4: Noise substantially degraded high-resistivity inversion, causing some bodies to disappear and deeper portions to be missed.Upper boundaries could remain accurate while deeper parts were lost.
Results of Experiment 4: Overall, deeper depth and smaller anomaly volume increased noise sensitivity, while anomalies producing smaller observed-data amplitudes were more susceptible.The authors propose future transfer-learning fine-tuning to improve antinoise ability.

E. Results of Experiment 5

Experiment 5 benchmarks ERSInvNet against iterated linear inversion on unseen synthetic models and applies it to a field survey. The field result identifies low-resistivity zones consistent with drilling evidence.

Results of Experiment 5: ERSInvNet was benchmarked against RES2DINV iterated linear inversion using synthetic anomalies with positions, sizes, and resistivities unseen during training.The comparison was designed to test performance on unprecedented synthetic configurations.
Results of Experiment 5: The field survey used one 180 m 2-D-ERT profile with 3 m electrode spacing and compared ERSInvNet inversion against three boreholes.Boreholes K01, K02, and K03 were located at 82 m, 94 m, and 118 m along the line.
Results of Experiment 5: ERSInvNet identified two low-resistivity areas below 10 Ω·m, including a suspected aquifer and a possible water-bearing karst cave.R1 extended from 21–89 m and 18–30 m depth; R2 extended from 111–124 m and 6–19 m depth.
Results of Experiment 5: The low-resistivity zones were broadly consistent with fragmented dolomite limestone intervals observed in boreholes K01 and K03.The reported correspondence supported an inferred high possibility of water leakage and the recommendation for appropriate grouting.

VI. CONCLUSION

ERSInvNet addresses ambiguity from vertically varying apparent-resistivity patterns with a tier feature map, while smooth constraints and depth weighting improve inversion behavior. Experiments show that combining these components achieves the best results and supports fast, accurate inversion.

ERSInvNet supplements CNN input with a tier feature map to address ambiguity from vertically varying apparent-resistivity patterns.The network uses a CNN-based architecture for resistivity inversion.
Smooth constraints and a depth weighting function are added to the training loss to reduce false anomalies and improve deep-region accuracy.
36 214 synthetic apparent-resistivity/model pairs support training, validation, and testing of ERSInvNet.
Using the tier feature map, smooth constraints, and depth weighting function together achieves the best performance in qualitative and quantitative comparisons.The combined design outperforms the individual component uses across the reported evaluation.
ERSInvNet provides faster testing-time inference and more accurate inversion results than traditional methods in the reported experiments.Field-data experiments further demonstrate promising performance and potential for resistivity inversion.