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An automatic-differentiation framework for time-lapse electrical resistivity tomography inversion of hydrologic dynamics

Pu Yang, Zhengyang Fang, Yuxin Liu, Xuan Su, Deshan Feng, Hang Chen

arXiv:2608.14661v1cs.LG

TL;DR

Long TL-ERT monitoring sequences are computationally demanding, and changing inversion components can require new gradient derivations or separate implementations. AD-TLERT addresses this with a GPU-accelerated differentiable framework, achieving an approximately 51-fold speedup and reducing average water-content error by approximately 31% for the tested synthetic model.

  • Problem

    Long TL-ERT monitoring sequences are computationally demanding, while changing inversion components may require new gradient derivations or separate implementations.

  • Method

    AD-TLERT is a GPU-accelerated differentiable framework that combines upper-level automatic differentiation with implicit differentiation of the sparse 2.5-D finite-element problem.

  • Results

    AD-TLERT achieved an approximately 51-fold speedup and reduced average water-content error by approximately 31% for the tested synthetic model.

  • Takeaways & Limitations

    The framework supports reuse of the PDE derivative implementation across data-misfit and other inversion choices without re-deriving complete ERT sensitivity.

  • Takeaways & Limitations

    The present GPU implementation cannot represent off-profile heterogeneity, fully three-dimensional topography, or three-dimensional flow paths.

Abstract

from arXiv · show

Time-lapse electrical resistivity tomography (TL-ERT) provides spatially distributed information on subsurface hydrologic changes. However, inversion of long monitoring sequences is computationally demanding. Modifying the data misfit, regularization, model parameterization, or petrophysical transformation may also require new gradient derivations and separate implementations. Here, we present AD-TLERT, a unified, GPU-accelerated framework for time-lapse ERT inversion based on automatic differentiation. The framework integrates model parameterization, differentiable petrophysical transformations, forward modeling, data misfit, regularization and auxiliary constraints into a single computational chain. Alternative inversion formulations can therefore reuse the same PDE derivative implementation without re-deriving the complete ERT sensitivity for each case. Comparisons with pyGIMLi showed close agreement in the forward responses, gradients, and recovered resistivity models. Under the tested configuration, AD-TLERT achieved an approximately 51-fold speedup. Synthetic experiments showed that inversion choices affect the amplitude, geometry, and temporal behavior of recovered anomalies. By propagating gradients through the embedded petrophysical relationship, AD-TLERT enabled direct water-content inversion and yielded more accurate estimates than post-inversion conversion for the tested model. A field application further demonstrated how ERT, temperature, and soil-moisture observations can be combined to image snowmelt-driven hillslope wetting. AD-TLERT provides an efficient and flexible framework for time-lapse ERT inversion and hydrologic interpretation.

1. Introduction

Long-sequence TL-ERT inversion is computationally demanding and sensitive to formulation choices, while extending implementations to new objectives, petrophysical mappings, and constraints often requires problem-specific derivative work. AD-TLERT addresses these challenges with a GPU-accelerated, differentiable framework that unifies inversion components and supports direct hydrologic-variable inversion.

  • Computational and formulation challenges: Long monitoring sequences require repeated forward and sensitivity calculations across many surveys, sources, and wavenumbers, increasing computational cost.This cost can limit inversion of dense monitoring data and systematic testing of alternative formulations.
  • Computational and formulation challenges: Recovered anomalies depend on data misfit, regularization, optimization, and model parameterization choices, with no universally optimal inversion setting.The appropriate formulation depends on data quality, survey coverage, expected anomaly geometry, and prior information.
  • Computational and formulation challenges: Changing one inversion component may require a new gradient derivation or separate implementation, making controlled comparison difficult.This limitation extends to new spatiotemporal objectives, differentiable petrophysical mappings, and GPU sparse solvers.
  • AD-TLERT framework: Alternative misfits, constraints, optimization strategies, and petrophysical parameterizations can be evaluated using the same forward and gradient implementation.The study benchmarks the framework against pyGIMLi and compares representative inversion choices under a common synthetic setting.
  • AD-TLERT framework: AD-TLERT combines differentiable model parameterization, forward modeling, misfit, regularization, and auxiliary constraints within one computational chain.The framework uses implicit differentiation for sparse finite-element problems and GPU-based sparse linear algebra for repeated solves.
  • Hydrologic interpretation: Embedding a differentiable conductivity–saturation relationship enables direct water-content inversion alongside conventional post-inversion conversion.The framework is presented as an efficient and extensible basis for testing inversion assumptions and interpreting hydrologic observations.

2. Methodology

AD-TLERT organizes time-lapse ERT inversion as a differentiable computational graph, combining forward modeling, objective terms, constraints, and model transformations. Implicit differentiation reuses the same PDE derivative across inversion formulations and parameterizations.

  • AD-TLERT comprises model definition, GPU-accelerated finite-element forward modeling, misfit evaluation, implicit automatic differentiation, and adaptive model updating.
  • Alternative objective functions, optimization algorithms, and spatial or temporal regularization strategies can be combined without re-deriving complete ERT sensitivities.
  • The framework supports direct resistivity inversion and a petrophysical route that maps hydrologic or rock-physics parameters into resistivity.
  • The finite-element solve is differentiated implicitly, avoiding differentiation of sparse factorization while propagating gradients through the discrete governing equation.The resulting derivative rules support both forward- and reverse-mode calculations.
  • The same implicit ERT derivative can be reused for resistivity, saturation, water content, or another differentiable model parameterization.
  • Differentiable petrophysical transformations propagate gradients through the forward model and regularization, producing petrophysically constrained water-content estimates rather than direct ERT measurements.

3. Synthetic examples and verification

Synthetic experiments constructed hydrologically driven resistivity sequences and tested AD-TLERT against pyGIMLi. The framework closely matched reference responses, gradients, and models while substantially reducing inversion time.

  • 3.1. Synthetic model: Synthetic resistivity dynamics showed rapid shallow changes, delayed intermediate responses, and a nearly stable deep background.Fresh bedrock remained nearly unchanged, consistent with lower porosity and more compact texture.
  • 3.2. Numerical verification and computational performance: Time-lapse inversions used a three-survey moving window to couple neighboring surveys through local temporal constraints.
  • 3.2. Numerical verification and computational performance: 2.61 × 10−4% relative error separated AD-TLERT and pyGIMLi apparent resistivity responses on the tested terrain geometry.
  • 3.2. Numerical verification and computational performance: Both workflows recovered the same resistivity structure, with a 1.63% relative difference between recovered models.
  • 3.2. Numerical verification and computational performance: 51.0-fold speedup was achieved: AD-TLERT required 8.66 min, compared with 441.95 min for the pyGIMLi-based workflow.

4. Effects of inversion choices

AD-TLERT permits inversion choices to be changed within one differentiable workflow. Synthetic tests show that misfit, regularization, and optimization choices mainly alter recovered amplitudes and temporal behavior, while broad structure is more stable.

  • AD-TLERT allows data misfit, regularization, and optimization components to be modified within the same differentiable workflow.
  • Data misfit: All tested misfit formulations recover the main synthetic structure, including the upper-slope resistive region and downslope low-resistivity zone.
  • Data misfit: L2 gives RMSE values of 0.724, 0.755, 0.744, and 0.724 at days 1, 190, 294, and 365, respectively.In the controlled clean synthetic case, L2 provides the strongest baseline for absolute resistivity recovery.
  • Data misfit: Robust objectives recover similar spatial patterns but produce higher resistivity in shallow and intermediate regions and higher RMSE in this clean case.Their reduced sensitivity to large residuals makes the updates more conservative.
  • Data misfit: Data-difference L2 follows the main resistivity decrease with a smoother temporal response while prioritizing changes relative to the reference state.It is most relevant when interpretation focuses on temporal changes rather than absolute resistivity.
  • Spatial regularization: Spatial regularization affects both recovered sections and temporal evolution: first-order smoothness is stable, whereas model-difference smoothness can limit larger changes.Structural priors may under-recover localized anomaly amplitude, while Huber regularization provides a flexible option for sharper wetting fronts.
  • Optimization: Optimizer selection affects convergence and temporal stability more strongly than large-scale structure; Adam produces stronger short-term oscillations after wetting.Gauss–Newton CGLS is the most computationally efficient method in the comparison, while L-BFGS-B is a useful alternative.

5. Differentiable petrophysical parameterization

The study embeds the petrophysical relationship inside AD-TLERT so water content can be updated directly while ERT responses are computed through the corresponding resistivity model. In the synthetic test, direct inversion is more accurate than post-inversion conversion.

  • Parameterization: The conversion workflow instead inverts resistivity and converts the recovered models to saturation and water content afterward.Its estimates inherit smoothing and amplitude bias from the resistivity model.
  • Parameterization: Direct AD-TLERT inversion updates water content itself and computes the corresponding resistivity before each ERT forward solve.
  • Synthetic comparison: Direct inversion reduces RMSE to 0.042, 0.044, 0.044, and 0.042, an average reduction of about 31% relative to conversion.
  • Synthetic comparison: Conversion tends to overestimate water content, whereas direct inversion substantially reduces this positive bias and more closely follows the true time series.The direct result remains smoother than the true response and slightly underestimates some short-term wet-period peaks.
  • Interpretation: Including the petrophysical transformation inside inversion improves consistency between the recovered model and the target physical parameter.The recovered water content remains a petrophysically constrained estimate rather than a post-processing product.

6. Field data inversion

The field application uses time-lapse ERT with temperature correction and TMC soil-moisture observations to image snowmelt-driven hillslope wetting. Adding TMC observations as a soft constraint substantially improves agreement with sensor trends.

  • Field workflow: The field workflow corrects resistivity for temperature and uses TMC water-content data as a soft constraint.
  • Resistivity and water-content changes: Later surveys show heterogeneous negative resistivity changes mainly in shallow and intermediate profile regions relative to the 26 March baseline.
  • Hydrologic interpretation: The inferred pattern agrees with an interpretation in which snowmelt-driven wetting and flow vary with hillslope position, bedrock structure, and near-surface storage.Shallow lateral flow dominates the steep upper slope, while lower colluvial areas show more vertical or groundwater-related dynamics.
  • Sensor comparison: The lower recovered amplitudes are expected because TMC probes measure local point water content, whereas ERT provides a weighted and regularized estimate over a broader volume.

7. DISCUSSION

AD-TLERT separates PDE derivatives from upper-level inversion choices, enabling efficient reuse across formulations. Comparisons support numerical consistency and substantial computational gains, while current scope is limited by 2.5-D modeling and fixed petrophysical parameters.

  • Framework contribution: AD-TLERT separates the PDE derivative from upper-level inversion formulation, allowing alternative objectives, transformations, and constraints to reuse the same derivative implementation.
  • Validation: AD-TLERT matches pyGIMLi in forward responses, gradients, and recovered resistivity models, while ResIPy recovers similar large-scale structures and time-lapse changes.
  • Computational performance: AD-TLERT reduces combined forward-and-gradient time by approximately 10.3-fold and lowers peak GPU memory from 961 to 485 MiB against ADERT.The comparison used an ADERT 2.5-D reference model.
  • Framework contribution: The framework provides a reusable platform for testing how inversion assumptions affect time-lapse ERT results and hydrologic interpretation.
  • Limitations: The present implementation is limited to 2.5-D problems and cannot represent off-profile heterogeneity, fully three-dimensional topography, or three-dimensional flow paths.Full 3-D extension would impose substantially greater memory and computational demands.
  • Limitations: Synthetic water-content improvements depend on an adequate prescribed petrophysical relationship and do not remove uncertainty in porosity, pore-water conductivity, or related parameters.Those parameters are treated as known, and their uncertainty is not propagated into recovered models.

8. CONCLUSIONS

AD-TLERT combines automatic differentiation with implicit differentiation and GPU acceleration to make time-lapse ERT inversion flexible, efficient, and extensible. Tests showed close agreement with pyGIMLi, large computational savings, improved direct water-content estimation, and multimodal field constraints.

  • Framework: AD-TLERT combines automatic differentiation of upper-level inversion components with implicit differentiation of the sparse 2.5-D finite-element problem.Data misfits, regularization terms, model transformations, and auxiliary-observation constraints can reuse the same PDE derivative implementation.
  • Validation: Close agreement with pyGIMLi was obtained for apparent resistivities, gradients, and recovered resistivity models.
  • Computational performance: 441.95 to 8.66 min reduced elapsed inversion time, corresponding to an approximately 51-fold speedup under the tested configuration.The comparison used the tested hardware and inversion configuration.
  • Synthetic experiments: Alternative data-misfit, regularization, and optimization choices changed recovered amplitudes, spatial structure, temporal behavior, and computational cost.These alternatives were evaluated within the same computational framework.
  • Petrophysical inversion: Direct water-content inversion reduced average water-content error by approximately 31% for the tested synthetic model compared with post-inversion conversion.Differentiable petrophysical parameterization enabled estimation directly rather than through conversion after resistivity inversion.
  • Field application: Temperature information and soil-moisture observations were incorporated to constrain snowmelt-driven hillslope wetting in a field application.The framework extends ERT toward process-relevant subsurface variables and complementary observations.

Data Availability

The repository provides AD-TLERT source code and synthetic-example materials, while the field observations are available through ESS-DIVE. Supplementary material also documents numerical implementation and benchmark settings.

  • Repository: AD-TLERT source code, hydrological-model files, synthetic resistivity models, and a compact AD-TLERT, ResIPy, and pyGIMLi comparison are provided in the repository.
  • Reproducibility: Complete scripts and configuration files for reproducing the reported experiments were scheduled for release in the same repository upon acceptance.
  • Field data: Field ERT, soil-moisture, and temperature data for the snowmelt application are available from the ESS-DIVE repository.
  • Numerical implementation: The supplementary implementation describes batched finite-element assembly, sparse-factorization reuse, implicit derivatives, batched sensitivities, and GPU CGLS updates.
  • Benchmark documentation: The benchmark settings for synthetic windowed inversions and the pyGIMLi comparison are listed in Table S1.

S2. Objective functions and time-lapse constraints

The framework combines weighted data-misfit terms with spatial and temporal regularization for time-lapse inversion. Baseline and later surveys are treated differently, while structural and petrophysical choices alter the regularized model space.

  • Data transformation and weighting: Synthetic measurements use a uniform relative error of ϵ_i = 0.03 when constructing data weights.
  • Data-misfit functions: Weighted log-L2 and smooth log-L1 objectives are available, with the latter solved by iteratively reweighted least squares.
  • Data transformation and weighting: The first survey is fitted in absolute log-data space, whereas later surveys are fitted relative to that baseline.
  • Spatial and temporal regularization: Spatial smoothness penalizes differences across neighboring mesh cells, using equal and opposite coefficients for each interior edge.
  • Spatial and temporal regularization: A structure-guided operator assigns unit weight within hydros­tratigraphic units and relative weight 0.05 across unit boundaries.
  • Time-lapse constraints: Three-survey windows are inverted independently, overlapping estimates are averaged in natural-log resistivity space, and petrophysical fields use the same differentiable mapping.

S3. Synthetic hydrologic and petrophysical models

The synthetic benchmark uses a heterogeneous hillslope hydrologic model and unit-dependent petrophysical relationships to generate resistivity sequences. AD-TLERT closely matches independent inversion results while providing a faster automatic-differentiation benchmark under matched ADERT settings.

  • Synthetic hydrologic model: The synthetic hillslope is 250 m long, approximately 60 m deep, and forced with water-year 2023 conditions at hourly resolution.
  • Synthetic hydrologic model: The subsurface contains regolith, fractured bedrock, and fresh bedrock, with heterogeneous conductivity fields in the first two units.
  • Petrophysical model: Synthetic resistivity follows the unit-dependent relationship σ(S) = σpS^n + σsS^(n−1), with ρ(S) = σ(S)^−1.
  • ResIPy comparison: ResIPy yields absolute-model ln ρ RMSE values from 0.616 to 0.679, compared with 0.647 to 0.718 for AD-TLERT at four representative times.
  • ResIPy comparison: Time-lapse ∆ln ρ RMSE values are 0.061, 0.042 and 0.008 for AD-TLERT, versus 0.067, 0.043 and 0.009 for ResIPy.
  • ADERT benchmark: AD-TLERT and ADERT show closely matching predictions and gradients, with logarithmic-prediction correlation 0.9970 and gradient cosine similarity 0.9982.
  • ADERT benchmark: 10.3 times faster: AD-TLERT completes the combined forward-and-gradient calculation in 0.2657 s versus 2.7394 s for ADERT.
  • ADERT benchmark: Peak GPU memory decreases from 961 MiB to 485 MiB, while both implementations recover the main synthetic resistivity structures.

S5. Field-data processing and auxiliary observations

The field-data workflow corrects resistivity for temperature and incorporates temperature and soil-moisture observations as auxiliary information. Sensor observations enter the inversion through a separate weighted soft constraint in physical water-content space.

  • Temperature model and correction: Seasonal ground temperature is modeled as a depth- and day-of-year-dependent function fitted at sensor locations and interpolated across the ERT domain.
  • Temperature model and correction: Resistivity is corrected to a reference temperature of Tref = 6.5 °C using ct = 2.0 per cent °C^−1.
  • Auxiliary observations: Six monitoring stations measure soil moisture at depths of 0.1, 0.3, 0.6 and 0.8 m every 15 minutes before hourly averaging.
  • Auxiliary observations: Temperature observations define CT, while soil-moisture observations enter the inversion as a separate soft water-content constraint.
  • Auxiliary observations: The sensor term is assembled in physical water-content space and differentiated through the same parameter mapping as the ERT data term.
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