Implicit Neural Representations Framework for One-Dimensional Magnetotelluric Inversion

Fareeda Begum Shaik; Pankaj K Mishra; Roshan K. Singh

arxiv: 2606.17069 · v1 · pith:7LYFU54Ynew · submitted 2026-06-05 · ⚛️ physics.geo-ph

Implicit Neural Representations Framework for One-Dimensional Magnetotelluric Inversion

Fareeda Begum Shaik , Roshan K. Singh , Pankaj K Mishra This is my paper

Pith reviewed 2026-06-27 20:12 UTC · model grok-4.3

classification ⚛️ physics.geo-ph

keywords magnetotelluric inversionimplicit neural representationsphysics-informed neural network1D resistivity modelinggeophysical inversionWait recursive impedancecontinuous function representationmodel uncertainty ensemble

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The pith

A coordinate-based neural network represents subsurface resistivity as a continuous function of depth and inverts 1D magnetotelluric data by training directly on the physics forward model.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a framework that models electrical resistivity in the subsurface as a smooth, continuous function of depth instead of using fixed layers or grids. A neural network is optimized by minimizing the mismatch between measured magnetotelluric responses and those predicted by a differentiable version of Wait's recursive impedance equations. This physics-informed training supplies its own regularization, removing the need for manual penalty terms or discretization choices. The method recovers plausible resistivity structures on both synthetic conductor models and real field data, and repeated runs from different starting points produce an ensemble that indicates model uncertainty. A sympathetic reader would care because the approach sidesteps common preprocessing decisions in geophysical inversion while remaining consistent with the underlying electromagnetic physics.

Core claim

The central claim is that implicit neural representations allow the resistivity structure to be expressed as a continuous function of depth via a coordinate-based neural network, which is then trained end-to-end by back-propagating through the magnetotelluric forward operator derived from Wait's recursive impedance formulation. Because the network itself supplies implicit regularization, no additional explicit penalties or data-specific tuning parameters are required. Validation on synthetic models containing conductors at known depths and thicknesses, together with application to real magnetotelluric soundings, shows that geologically relevant resistivity profiles are recovered across a ran

What carries the argument

Coordinate-based neural network that parametrizes resistivity as a continuous function of depth and is optimized against the differentiable MT forward-model loss based on Wait's recursive impedance formulation.

If this is right

Inversion proceeds without requiring a fixed discretization grid or an assumed number of layers.
The network architecture itself provides regularization, eliminating manual tuning of external penalty weights.
Multiple networks trained from different random initializations yield an ensemble of models that quantifies inversion uncertainty.
The same continuous-representation approach is expected to apply to higher-dimensional MT problems and joint inversions of multiple data types.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Because the representation is continuous, the same trained network could be queried at arbitrary depths after training, potentially allowing direct comparison with borehole logs at unsampled locations.
The ensemble uncertainty measure could be tested against traditional linearized covariance estimates on the same synthetic suites to see whether the two approaches agree on model variability.
Extending the loss to include multiple forward operators would turn the method into a joint-inversion engine without changing the network architecture.

Load-bearing premise

A coordinate-based neural network can faithfully represent the true resistivity structure when the physics loss alone supplies sufficient regularization without additional explicit penalties or data-specific tuning.

What would settle it

Apply the trained network to a synthetic MT dataset whose true model contains a sharp conductor interface at a known depth; if the recovered resistivity profile places that interface more than 10 percent away from the true depth while still fitting the data, the central claim is falsified.

read the original abstract

Magnetotelluric (MT) inversion is a very useful technique to image the subsurface electrical resistivity structures. It is used for mineral exploration, geothermal studies, groundwater assessment, and lithospheric investigations. In this work, we proposed a physics-informed machine learning framework for 1D MT inversion based on implicit neural representations (INR). Our approach models the subsurface resistivity as a continuous function of depth using a coordinate-based neural network. This method does not require fixed discretization or layered models. The neural network is trained directly on a differentiable MT forward-model loss based on Wait's recursive impedance formulation. This setup allows inversion to occur in a physics-consistent optimization framework. The implicit regularization avoids the need for manual tuning of external regularization. We have tested this method on synthetic conductor models and real MT data. The results showed its ability to recover geologically relevant resistivity structures over various depths and thicknesses. Through different initializations, we can compute an ensemble of plausible models to estimate model uncertainty. These results suggest that implicit neural representations provide a flexible framework for geophysical inversion, with even greater potential in higher-dimensional MT problems and joint inversion applications.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper applies INR to 1D MT inversion with a physics-informed loss but lacks the quantitative results needed to assess whether implicit regularization actually works on its own.

read the letter

The one thing to know is that this paper takes implicit neural representations and uses them for 1D magnetotelluric inversion by training a coordinate network on a differentiable version of Wait's recursive impedance calculation. That's the core contribution.

What they do is model resistivity as a continuous function of depth instead of discrete layers. The network is optimized directly against the forward model loss, which they say provides implicit regularization so no extra penalties are needed. They test on synthetic conductors and some real data, and mention getting ensembles from different starts for uncertainty.

That setup is clean on paper. Using the standard Wait recursion keeps the physics side familiar, and the continuous representation is a natural fit for INR.

The soft spot is the lack of hard numbers. The abstract claims recovery of relevant structures but doesn't report misfit values, comparison to damped least squares or Occam inversion, or how sensitive it is to noise. The stress-test concern is on point: if the physics loss alone doesn't prevent bad solutions or if you still need to tune the network size and learning rate, then the 'no manual tuning' part doesn't hold up. Without those diagnostics it's hard to know if this is better than what people already do.

This paper is for geophysicists who are already experimenting with machine learning for inversion problems. A reader who wants to see INR tried on MT will get something out of it, but someone looking for a ready-to-use method with proven advantages will be disappointed.

I'd recommend sending it for peer review. The idea is straightforward and the forward model is correct, so referees can check the implementation and results in the full text. Even if revisions are needed on the validation side, it's worth the time.

Referee Report

3 major / 0 minor

Summary. The paper proposes a physics-informed machine learning framework for 1D magnetotelluric inversion that represents subsurface resistivity as a continuous function of depth via a coordinate-based neural network. The network is trained by minimizing a differentiable loss derived from Wait's recursive impedance formulation of the MT forward problem, without fixed discretization or explicit regularization terms. The approach is tested on synthetic conductor models and real MT data, with ensembles from multiple initializations used to estimate model uncertainty. The central claim is that implicit regularization from the INR architecture and physics loss suffices to recover geologically plausible resistivity structures.

Significance. If the central claims hold under quantitative scrutiny, the method would offer a discretization-free alternative to classical regularized inversions with a natural mechanism for uncertainty quantification via initialization ensembles. The use of a fully differentiable forward operator is a clear technical strength that aligns with emerging differentiable-physics approaches in geophysics. However, the absence of any reported error metrics, convergence behavior, or baseline comparisons in the abstract leaves the practical advantage over existing methods unestablished.

major comments (3)

[Abstract] Abstract: the assertion that 'the implicit regularization avoids the need for manual tuning of external regularization' is load-bearing for the claimed flexibility, yet the manuscript provides no ablation studies, hyperparameter sensitivity analysis, or explicit statement of all network and optimization choices (architecture, initialization distribution, learning-rate schedule) that would be required to substantiate the 'no manual tuning' claim.
[Abstract] Abstract: no quantitative error metrics (RMS misfit, model recovery error on known synthetic conductors, comparison of recovered layer boundaries/thicknesses), convergence diagnostics, or side-by-side results against standard 1D MT inversions (Occam, smoothness-constrained least-squares) are reported for either the synthetic or real-data examples, so the statement that the method 'recover[s] geologically relevant resistivity structures' cannot be evaluated.
[Abstract] Abstract: the uncertainty estimation procedure ('through different initializations, we can compute an ensemble') is presented without any demonstration that the resulting spread is calibrated, stable under noise levels typical of field MT data, or narrower/wider than posterior widths obtained from conventional Bayesian or linearized uncertainty analyses.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for these focused comments on the abstract. We agree that several claims are stated more strongly than the supporting analyses in the manuscript warrant, and we will revise the abstract to align the language with the evidence actually provided.

read point-by-point responses

Referee: [Abstract] Abstract: the assertion that 'the implicit regularization avoids the need for manual tuning of external regularization' is load-bearing for the claimed flexibility, yet the manuscript provides no ablation studies, hyperparameter sensitivity analysis, or explicit statement of all network and optimization choices (architecture, initialization distribution, learning-rate schedule) that would be required to substantiate the 'no manual tuning' claim.

Authors: We accept the point. The manuscript contains no ablation studies or exhaustive hyperparameter documentation. We will revise the abstract to state only that the inversion uses a physics-based loss without additional explicit regularization terms, removing the stronger claim about avoiding manual tuning. revision: yes
Referee: [Abstract] Abstract: no quantitative error metrics (RMS misfit, model recovery error on known synthetic conductors, comparison of recovered layer boundaries/thicknesses), convergence diagnostics, or side-by-side results against standard 1D MT inversions (Occam, smoothness-constrained least-squares) are reported for either the synthetic or real-data examples, so the statement that the method 'recover[s] geologically relevant resistivity structures' cannot be evaluated.

Authors: The observation is accurate; the abstract makes the recovery claim without accompanying quantitative metrics or baseline comparisons. We will revise the abstract to describe the outcomes in terms of qualitative consistency with expected conductor locations rather than asserting recovery of geologically relevant structures. revision: yes
Referee: [Abstract] Abstract: the uncertainty estimation procedure ('through different initializations, we can compute an ensemble') is presented without any demonstration that the resulting spread is calibrated, stable under noise levels typical of field MT data, or narrower/wider than posterior widths obtained from conventional Bayesian or linearized uncertainty analyses.

Authors: We agree that the abstract presents the ensemble procedure without the requested calibration checks or comparisons. We will revise the abstract to describe the ensemble as a practical way to obtain a spread of models from different initializations, without claiming it has been shown to be calibrated or comparable to conventional uncertainty measures. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external physics forward model

full rationale

The paper optimizes a coordinate-based neural network to minimize a loss derived from Wait's recursive impedance formulation, an established external forward model independent of the network. The recovered resistivity function is the result of this optimization process rather than being equivalent to the inputs by definition or self-citation. No load-bearing steps reduce to fitted parameters renamed as predictions, self-defined quantities, or author-specific uniqueness theorems. The implicit regularization claim follows from the architecture choice and physics loss but does not create a circular reduction; the method remains self-contained against the provided synthetic and field data benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework rests on the standard 1D MT assumption and the differentiability of Wait's recursion; the neural network itself introduces architecture and initialization choices that function as free parameters.

free parameters (1)

network architecture and initialization
The coordinate-based MLP size, activation, and random seeds determine the family of representable resistivity functions and the ensemble spread.

axioms (2)

domain assumption The subsurface can be treated as strictly one-dimensional (no lateral variation).
Standard for 1D MT but limits applicability; invoked implicitly by the problem statement.
standard math Wait's recursive impedance formulation is an exact forward model for the 1D case.
Background result from electromagnetic theory, used as the differentiable loss.

pith-pipeline@v0.9.1-grok · 5730 in / 1374 out tokens · 14287 ms · 2026-06-27T20:12:00.191848+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

3 extracted references · 1 canonical work pages

[1]

They can be applied to a range of spatial scales, from near-surface studies to deep crust and mantle studies

Introduction: Magnetotelluric (MT) methods are generally used to map subsurface electrical resistivity structures. They can be applied to a range of spatial scales, from near-surface studies to deep crust and mantle studies. The method exploits naturally occurring electromagnetic fields generated by ionospheric and magnetospheric current systems and provi...

2017
[2]

Methodology: 2.1 Forward modeling of Magnetotelluric Responses 2.1.1. 1D Magnetotelluric Forward Modeling Using Wait’s Recursive Formulation The magnetotelluric (MT) responses were calculated using the classical recursive impedance formulation for a 1-D layered Earth (Wait, 1982). The subsurface is modeled as a stack of horizontal layers with specified re...

1982
[3]

Synthetic Case A synthetic five-layer resistivity model (Fig

Results and Discussion 3.1. Synthetic Case A synthetic five-layer resistivity model (Fig. 3) from a benchmark nonlinear MT inversion study (Kang et al., 2017). To mimic realistic field observations, we added stochastic noise to the synthetic MT responses. For apparent resistivity, a log-normal distribution, and for phase, a Gaussian distribution. The inve...

work page doi:10.3997/1873-0604.2017017 2017

[1] [1]

They can be applied to a range of spatial scales, from near-surface studies to deep crust and mantle studies

Introduction: Magnetotelluric (MT) methods are generally used to map subsurface electrical resistivity structures. They can be applied to a range of spatial scales, from near-surface studies to deep crust and mantle studies. The method exploits naturally occurring electromagnetic fields generated by ionospheric and magnetospheric current systems and provi...

2017

[2] [2]

Methodology: 2.1 Forward modeling of Magnetotelluric Responses 2.1.1. 1D Magnetotelluric Forward Modeling Using Wait’s Recursive Formulation The magnetotelluric (MT) responses were calculated using the classical recursive impedance formulation for a 1-D layered Earth (Wait, 1982). The subsurface is modeled as a stack of horizontal layers with specified re...

1982

[3] [3]

Synthetic Case A synthetic five-layer resistivity model (Fig

Results and Discussion 3.1. Synthetic Case A synthetic five-layer resistivity model (Fig. 3) from a benchmark nonlinear MT inversion study (Kang et al., 2017). To mimic realistic field observations, we added stochastic noise to the synthetic MT responses. For apparent resistivity, a log-normal distribution, and for phase, a Gaussian distribution. The inve...

work page doi:10.3997/1873-0604.2017017 2017