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The Neuro-Physical Inverter: A Modular Framework for Magnetotelluric Inversion Coupling Ensemble Conditioning with Residual Learning

Jae Deok Kim, Sai Ravela, Rob. L. Evans

arXiv:2610.03225v2cs.LG

TL;DR

Higher-dimensional geophysical inversion motivates methods that remain tractable as dimensionality grows, but ensemble conditioning and learned inversion have distinct limitations. NPI couples ensemble conditioning with constrained residual correction; synthetic and Gabbs Valley results show reduced ensemble-mean error or mid-band mean error while retaining comparable ensemble spread.

  • Problem

    Higher-dimensional inversion motivates tractable alternatives, while ensemble updates are limited by the prior span and can destabilize under mismatched real-data structure.

    Source: §I · p. 1

  • Method

    NPI combines EnsCGP conditioning of a prior ensemble with constrained residual correction, demonstrated in a 1D MT setting.

    Source: §V · p. 13

  • Results

    Synthetic experiments show reduced ensemble-mean error without destabilizing the ensemble, and 81 Gabbs Valley stations show reduced mid-band across-station mean error with comparable ensemble spread.

    Source: §V · p. 13

  • Takeaways & Limitations

    The propagated ensemble spread is an operational measure of how tightly resistivity is constrained under the adopted workflow and prior, not a calibrated Bayesian interval.

    Source: E. Interpretation of the Factor of… · p. 12, G. Limitations and Outlook · p. 13

    The demonstrated rotation-invariant 1D responses discard directional information in the full impedance tensor, so results are effective 1D summaries of resistivity structure.

    Source: G. Limitations and Outlook · p. 13

Abstract

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We present the Neuro-Physical Inverter (NPI), a modular, uncertainty-aware framework for geophysical inversion that couples ensemble-based conditioning with constrained residual learning, demonstrated in the 1D magnetotelluric (MT) setting as a controlled testbed. The framework operates in two stages. An Ensemble-Conditional Gaussian Process (EnsCGP) conditions a prior ensemble of resistivity models on the observed response, producing a physically admissible reference ensemble. A residual-learning neural network then predicts targeted corrections to this reference, trained on synthetic data and fine-tuned per station for field application through a physics-coupled objective. Because an ensemble is conditioned, refined, and propagated through both stages, every estimate carries an associated ensemble spread. Synthetic experiments show that NPI systematically reduces ensemble-mean error without destabilizing the ensemble. Applied to broadband MT data from the Gabbs Valley geothermal region (Nevada, USA), NPI reduces the across-station mean misfit over the mid-period band while retaining comparable ensemble spread. The propagated ensemble yields a factor of uncertainty that serves as an operational measure of constraint within the assumed model class. Both stages are dimension-agnostic in formulation, and the design principles established here are intended to scale to higher-dimensional parameterizations.

I. INTRODUCTION

MT inversion is non-unique and noise-sensitive, while ensemble conditioning and learned inversion have complementary limitations. NPI combines a physically admissible conditioned ensemble with constrained residual learning in a 1D MT demonstration designed to inform higher-dimensional extensions.

I. INTRODUCTION

  • Ensemble conditioning cannot recover structured misfit outside the prior span, while learned inversion can overfit synthetic data and degrade under unfamiliar noise.NPI couples these complementary approaches so a learned corrector can address residual nonlinearity within a physically admissible reference framework.Source: §I · pp. 1–2
  • NPI combines single-pass ensemble conditioning with constrained residual learning in a modular framework intended to extend to higher-dimensional inversion.The 1D MT setting is used as a controlled testbed, not a formulation constraint.Source: §I · p. 2

A. Overview of the Neuro-Physical Inverter (NPI)

  • EnsCGP conditions prior resistivity models on observed MT data to produce a physically admissible ensemble with broadly consistent forward responses.The update is linear and confined to the prior ensemble span, so structured misfit can persist when that span is insufficient or the forward operator is strongly nonlinear.Source: §I · p. 2
  • The residual network predicts depth-dependent corrections from EnsCGP model-consistent responses rather than mapping observations directly to resistivity.Its corrections are constrained to an existing physically admissible reference and target structured misfit.Source: §I · p. 2, A. Overview of the Neuro-Physical… · p. 2

I. INTRODUCTION

  • NPI propagates an ensemble through both stages, reporting workflow-derived model variability rather than a full posterior.The two-stage decomposition has the network learn the discrepancy between the EnsCGP baseline and target.Source: §I · p. 2

B. Ensemble-Conditional Gaussian Process (EnsCGP)

EnsCGP performs a stabilized, single-pass Gaussian conditioning update of a prior model ensemble using its forward responses and observations. The resulting ensemble feeds a constrained residual network that refines the reference while preserving ensemble propagation.

1) Formulation:

  • The stabilized EnsCGP update conditions prior models toward observations using empirical model–data covariances and a perturbed observation ensemble.The update is a locally linear Kalman-analysis realization of an approximate conditional Gaussian law.Source: B. Ensemble-Conditional Gaussian… · p. 2, 1) Formulation · pp. 2–3
  • The linear correction remains within the affine subspace spanned by the prior, so it cannot introduce structure absent from that ensemble.Model corrections follow covariance directions linking model and data discrepancies; discrepancies orthogonal to those directions do not drive the update.Source: 1) Formulation · p. 3

2) Stabilization:

  • Reduced-rank covariance, optional observation-covariance inflation, and data-space localization stabilize conditioning and limit spurious correlations or poorly matched samples.Localization addresses finite-ensemble spurious correlations, while inflation addresses under-dispersion relative to model–data discrepancy.Source: 2) Stabilization · p. 3

3) Distinction from iterative ensemble inversion:

  • EnsCGP is a single conditioning step, unlike EKI optimization or ES-MDA's repeated assimilation of the same observation for different purposes.The dependency chain uses prior ensemble statistics rather than covariances recomputed from successive updates.Source: 3) Distinction from iterative ensemble… · p. 3
  • EnsCGP reaches the correct mean and spread together in one step under the stated linear Gaussian comparison, whereas continued EKI iteration collapses ensemble spread.ES-MDA is equivalent to a single Kalman update in mean and covariance under its stated inflation condition.Source: 3) Distinction from iterative ensemble… · p. 3

C. Residual-Learning Network (ResNet-1D)

  • ResNet-1D learns a lower-variance, depth-localized residual from the EnsCGP baseline and adds its predicted correction to the reference estimate.The network is deterministic and applied to individual ensemble members using each member's forward-modeled response, yielding a refined ensemble; its architecture is a baseline rather than an optimized design.Source: C. Residual-Learning Network (ResNet-1D) · pp. 3–4

D. Learning in Simulation

Learning in simulation trains the residual network on physically admissible examples generated through forward modeling and ensemble conditioning. The prior ensemble defines which structures EnsCGP can reach and how uncertainty spread varies with depth.

D. Learning in Simulation

  • Synthetic training pairs expose the network to systematic deviations of EnsCGP first guesses from known structures under realistic noise and prior variability.Inputs and targets are generated through forward modeling and ensemble conditioning, anchoring the task to MT physics rather than training a general inverse mapping.Source: D. Learning in Simulation · p. 4

1) Prior Ensemble Construction:

  • The prior ensemble may come from geological models, independent inversions, borehole-constrained profiles, or broad inflated priors; it defines EnsCGP’s reachable affine subspace.Updated members remain within the prior span, so the prior also determines how propagated variability is distributed across depth.Source: 1) Prior Ensemble Construction · p. 4
  • In this study, the prior is a GMRF derived from a regional 3D resistivity model, with covariance inflated to explore broader structures.The chosen source reflects regional-model availability, not a requirement that NPI presuppose a 3D inversion.Source: 1) Prior Ensemble Construction · p. 4
  • Ensemble spread’s depth structure is inherited from the prior, as shown by comparing the GMRF prior with a homogeneous half-space prior.The workflow does not impose that depth structure itself.Source: 1) Prior Ensemble Construction · p. 4

2) Forward Modeling and Frequency Grid:

Synthetic responses are forward-modeled on a fixed frequency grid, and the network learns residual corrections to EnsCGP estimates. Field application adapts the pretrained network station by station using observed-data consistency while limiting departures from the EnsCGP reference.

2) Forward Modeling and Frequency Grid:

  • A fixed reference frequency grid keeps network inputs consistent across training, testing, and field stations, regardless of acquisition frequencies.Inputs use forward-modeled EnsCGP ensemble-mean responses, and Gaussian noise is added at the complex-impedance level before apparent resistivity and phase are computed.Source: 2) Forward Modeling and Frequency Grid · p. 4

3) Construction of Training Pairs:

  • The network takes EnsCGP ensemble-mean responses as input and predicts residuals between true and EnsCGP ensemble-mean log-resistivity profiles.This targets systematic corrections relative to the reference rather than an unconstrained inverse mapping.Source: 3) Construction of Training Pairs · p. 4

4) Training Procedure:

  • Training uses standardized inputs and targets, Huber loss, AdamW, validation-based learning-rate scheduling, early stopping, and K-fold cross-validation.Huber loss limits sensitivity to large residuals from high-contrast structures or unfavorable noise realizations.Source: 4) Training Procedure · p. 4

5) Synthetic Testing:

  • Synthetic testing applies the fixed trained network to EnsCGP-conditioned responses and evaluates refinements against known profiles.The workflow uses separate training and testing subsets; Figure 1 depicts the inputs and residual targets used for training.Source: 5) Synthetic Testing · p. 4, Figure 1 · p. 5

E. Adaptation to Real MT Data

  • For real observations, station-wise fine-tuning uses a physics-coupled objective that improves forward-data fit while regularizing departures from the EnsCGP reference.After fine-tuning, the network is applied to each ensemble member to yield a refined ensemble.Source: E. Adaptation to Real MT Data · p. 5

1) Physics-coupled Data Misfit Loss:

The real-data refinement objective combines explicit forward-data fit with smoothness and reference-model penalties. A target normalized RMS misfit controls how far refinement departs from the EnsCGP reference and how much structure is permitted.

1) Physics-coupled Data Misfit Loss:

  • Data misfit is measured by explicitly forward-modeling the predicted resistivity and comparing its response with observations.The weights are inverse standard deviations, rather than a learned surrogate for the forward operator.Source: 1) Physics-coupled Data Misfit Loss · p. 5

2) Smoothness Regularization:

  • Smoothness regularization penalizes first- and second-order derivatives, with α balancing curvature against first-derivative regularization.The stated penalties constrain the structure of the resistivity profile.Source: 2) Smoothness Regularization · p. 5

3) Reference-model Regularization:

  • The reference-model penalty constrains departures from the EnsCGP ensemble mean, requiring data-fit reductions to justify corrections.This implements minimum-change refinement around the reference.Source: 3) Reference-model Regularization · p. 5

4) Selection of the Target Misfit Level:

  • A lower target misfit permits more fine-scale structure but risks fitting noise; a higher target keeps estimates simpler and nearer the reference.The default target nRMS is 1 under correctly specified Gaussian noise, and structure appearing only under a tighter target is not uniquely required by the data.Source: 4) Selection of the Target Misfit Level · p. 6

5) Optimization and Ensemble Propagation:

On held-out synthetic tests, residual refinement reduces ensemble-mean resistivity error and data misfit. It centers truth within the ensemble but produces wider-than-needed spread, while EnsCGP spread is closer to its error; ensemble propagation retains uncertainty context through refinement.

5) Optimization and Ensemble Propagation:

A. Synthetic Results

2) Performance on Synthetic Test Examples:

  • Across three representative examples, NPI reduces EnsCGP amplitude bias while preserving dominant large-scale structure and closely tracking the end-to-end estimate in forward responses.Figure 3 compares EnsCGP, NPI, and RTO-TKO; RTO-TKO’s ensemble mean is broadly comparable across these examples.Source: 2) Performance on Synthetic Test… · pp. 6–7, Figure 3 · p. 7
  • Adding the residual stage to EnsCGP reduces RMS resistivity error from 0.131 to 0.111 and nRMS data misfit from 1.355 to 0.962 across test examples.Source: 2) Performance on Synthetic Test… · p. 7
  • NPI’s truth percentile falls between 49th and 55th at every depth, while its spread-to-error ratio is 1.40 and coverage exceeds both targets.Coverage reaches 96.8% against 90% and 98.6% against 95%; EnsCGP’s ratio is 0.97 and its coverage is 88.6% against 90% and 94.1% against 95%.Source: 2) Performance on Synthetic Test… · p. 7

B. Real-Data Results

In the Gabbs Valley field application, NPI improves station-level fits while retaining ensemble information, and its per-station application is faster than RTO-TKO. The results are based on rotationally invariant 1D responses and are not intended to resolve 3D structure.

2) Study Area and MT Dataset:

  • The workflow uses 81 USGS broadband MT stations across southeastern Gabbs Valley, with rotationally invariant impedances supplying 1D apparent-resistivity and phase responses.These responses discard anisotropy and higher-dimensional structure to remain internally consistent with 1D assumptions; Figure 4 locates the stations alongside faults, geothermal systems, and temperature-binned wells.Source: 2) Study Area and MT Dataset · p. 8, 1) Overview · p. 8, Figure 4 · p. 9

3) Application of NPI to Real MT Data:

  • At GV129, NPI brings the forward response into close agreement with observations where EnsCGP retains substantial structured misfit.The residual corrections remain anchored to the EnsCGP estimate and can address structure outside the prior span.Source: 3) Application of NPI to Real MT Data · p. 8
  • At GV129 and GV161, RTO-TKO's mean uncertainty factor is 1.8–3.0 times NPI's; at GV137 its deepest interval spans over five orders of magnitude versus less than a factor of two for NPI.Above roughly 10 km, mean resistivity estimates agree closely at all three stations; below that depth, they diverge at GV129 and GV137.Source: 3) Application of NPI to Real MT Data · p. 8
  • For 500 ensemble members at GV129, EnsCGP conditioning plus NPI fine-tuning and propagation takes 13.0 s, versus 640.1 s for RTO-TKO.The reported comparison uses an AMD Ryzen 5 PRO 4650U workstation with no GPU; NPI's synthetic pretraining cost is incurred upfront and amortized across stations using the same parameterization.Source: 3) Application of NPI to Real MT Data · pp. 8–9
  • NPI ensemble-mean resistivity estimates are compared with regional 3D model slices for context, while the factor of uncertainty U summarizes spread and inherits its depth structure from the prior.The regional comparison is a consistency check, not a 3D resolution claim; larger U indicates greater uncertainty within the assumed 1D model class.Source: 1) Overview · p. 8, 3) Application of NPI to Real MT Data · p. 9

4) Data Misfit Diagnostics:

Across the Gabbs Valley stations, NPI reduces mid-period data misfit while retaining comparable or slightly greater ensemble spread. The section also frames EnsCGP's prior-dependent limits and the residual network's role, with synthetic comparisons and scope caveats for end-to-end learning.

4) Data Misfit Diagnostics:

  • For both stages, misfit is lowest over mid-periods, where signal-to-noise is highest, and rises toward band edges as data quality degrades.Figure 7's panels (a) and (c) show apparent-resistivity and phase error; panels (b) and (d) show relative 95% confidence-interval half-width.Source: 4) Data Misfit Diagnostics · pp. 9–10, Figure 7 · p. 10
  • NPI lowers mid-band mean apparent-resistivity error from EnsCGP's roughly 25–30% to below ∼10%, and phase error from 11–16% to a few percent.Across the band, relative ensemble spread is comparable to or modestly larger than EnsCGP's; the figure reports across-station error and spread over all 81 stations.Source: Figure 7 · p. 10, 4) Data Misfit Diagnostics · p. 10

A. EnsCGP as a Representational Object

  • EnsCGP's linear update cannot recover structures outside the prior span, and its Bayesian reading is qualified by that restriction and the joint-Gaussian approximation.The update is single-pass rather than iterative; the paper describes the factor of uncertainty as constraint within the assumed model class, not a posterior.Source: A. EnsCGP as a Representational Object · p. 10

B. Role of the Residual Network in a Modular Framework

  • On the synthetic test set, the end-to-end network has slightly lower RMS resistivity error than NPI, while their nRMS data misfits are nearly identical.The accuracy comparison depends on a known true model, unlike real stations where only nRMS data misfit is available.Source: B. Role of the Residual Network in a… · p. 11
  • With capacity matched, NPI's errors change only a few percent while the end-to-end network's roughly double; under distribution shift, NPI's advantage over the end-to-end network widens.The end-to-end network's Monte Carlo dropout ensemble is markedly overconfident relative to NPI in both settings.Source: B. Role of the Residual Network in a… · p. 11
  • The ensemble originates in EnsCGP and is propagated through the residual network, which refines ensemble members rather than generating an ensemble itself.This ordering is the basis of the paper's uncertainty-aware design.Source: B. Role of the Residual Network in a… · p. 11
  • Table I does not test unlabeled per-station adaptation, and the end-to-end network is not applied to real Gabbs Valley stations; validating such an adaptation scheme is outside the comparison's scope.The distribution-shift ablation is a step toward portability, but does not include per-station fine-tuning or real MT data.Source: B. Role of the Residual Network in a… · p. 11
  • The residual network targets structured misfit the conditioning stage cannot capture, learning corrections to the EnsCGP reference while remaining tied to MT physics through forward modeling.It learns low-variance residuals from the noise-mitigated EnsCGP mean response and is regularized toward the reference; the specific network is a working baseline, not the contribution itself.Source: B. Role of the Residual Network in a… · pp. 10–11

C. Bias Removal and Generalization Across Settings

The residual-learning target is designed to transfer across geological settings because EnsCGP absorbs prior- and location-dependent structure, though that transfer is not quantified here. Ensemble spread depends on the prior and is a workflow-derived constraint measure, not a calibrated Bayesian interval.

C. Bias Removal and Generalization Across Settings

  • EnsCGP removes prior-dependent bias from the residual target, which is designed to promote transfer across settings, but cross-region generalization is not quantified.Training and test profiles here come from a common prior.Source: C. Bias Removal and Generalization… · p. 11

D. Prior-Dependence of Ensemble Spread

  • The depth profile of U(z) is prior-dependent: different priors can yield nearly identical NPI point estimates but markedly different ensemble spreads.The prior bounds the spread available to conditioning, while the data determine how much remains after conditioning.Source: D. Prior-Dependence of Ensemble Spread · pp. 11–12
  • U(z) measures constraint within the assumed model class and prior; informative regional priors yield tighter spread where they are informative.Broad or homogeneous priors yield spread reflecting data resolution alone.Source: D. Prior-Dependence of Ensemble Spread · p. 12

E. Interpretation of the Factor of Uncertainty

  • The refined ensemble is not generally a conditional law, so U(z) is workflow-derived spread rather than a calibrated Bayesian interval.Per-station fine-tuning uses the same observation used in conditioning, so the Bayesian interpretation of the conditioning stage does not extend to the correction.Source: E. Interpretation of the Factor of… · p. 12
  • Field-station U(z) has no equivalent calibration guarantee because true resistivity is unavailable, but it can flag robust or weakly constrained features.Empirical coverage is testable on synthetic data with known truth, but does not establish calibration against the true Bayesian posterior.Source: E. Interpretation of the Factor of… · p. 12

F. Toward Higher-Dimensional Applications

NPI is demonstrated in 1D, but its conditioning and residual-learning principles do not assume a 1D parameterization. The conclusions present it as a scalable, uncertainty-aware framework, while noting that higher-dimensional forward costs and retraining remain practical challenges.

F. Toward Higher-Dimensional Applications

  • The EnsCGP update applies to 2D sections or 3D volumes wherever the forward operator can be evaluated for each ensemble member.Its update depends on empirical cross-covariance between model parameters and forward-modeled responses.Source: F. Toward Higher-Dimensional… · p. 12
  • Higher-dimensional use makes forward-ensemble cost dominant, increasing the importance of localization, reduced-rank conditioning, or surrogate forward operators.A surrogate could reduce repeated exact evaluations at the cost of some accuracy.Source: F. Toward Higher-Dimensional… · p. 12

G. Limitations and Outlook

  • The framework is demonstrated in 1D, and changing to 2D or 3D requires network retraining; comparative validation is also confined to 1D.The conclusions position its dimension-agnostic design for scaling, rather than reporting higher-dimensional validation.Source: G. Limitations and Outlook · p. 13, §V · p. 13

V. CONCLUSIONS

  • The two-stage design can apply to geophysical inverse problems with a defined forward operator and model-response ensembles, using conditioning as a baseline for neural residual correction.The framework is not intended to replace established methods such as Occam inversion or trans-dimensional MCMC.Source: G. Limitations and Outlook · p. 13, §V · p. 13
  • Across 81 broadband Gabbs Valley stations, NPI reduced mid-band mean error while retaining comparable ensemble spread.The framework propagates an ensemble through conditioning and refinement, and reports a depth-dependent diagnostic of constraint.Source: §V · p. 13
  • The depth dependence of ensemble spread is inherited from the prior: the informed GMRF tightens spread at depth, while the homogeneous prior widens it.Robust near-surface features are recovered under both priors.Source: §V · p. 13

APPENDIX A: RESNET-1D ARCHITECTURE

The appendices describe the residual network and synthetic data, then examine prior sensitivity and compare stitched NPI sections with a regional 3D inversion. These comparisons show that point estimates and large-scale near-surface patterns can remain consistent even as spread and fine-scale structure depend on the prior and fit target.

APPENDIX A: RESNET-1D ARCHITECTURE

  • ResNet-1D maps log apparent resistivity and phase through a convolutional network to depth-dependent log-resistivity corrections added to the EnsCGP mean.The network is presented as a working baseline rather than an optimized design.Source: Appendix A · pp. 13–14, Figure 8 · p. 14

APPENDIX B: SYNTHETIC DATASET DISTRIBUTION

  • The synthetic dataset samples 1D resistivity models from the reduced-rank GMRF prior, with clean apparent-resistivity and phase responses reflecting its geologic variability.The three representative test examples are drawn from the same distribution.Source: Figure 9 · p. 14, Appendix B · p. 14

APPENDIX C: PRIOR SENSITIVITY COMPARISON

  • At GV09, both priors reproduce observed responses, but mean apparent-resistivity and phase errors are 5% and 2% for GMRF versus 9% and 3% for the homogeneous prior.Both fits meet the fine-tuning stopping criterion; the GMRF fit is tighter.Source: Appendix C · pp. 14–15, Figure 10 · p. 15
  • Changing the prior at GV09 leaves the NPI point estimate comparatively stable, while ensemble-spread depth structure changes sharply.Mean U is 2.57 above and 2.79 below 10 km for GMRF (ratio 1.09), versus 1.69 and 3.74 for the homogeneous prior (ratio 2.22).Source: Appendix C · pp. 14–15

APPENDIX D: PSEUDO-2D CROSS-SECTIONS AND CONSISTENCY WITH REGIONAL 3D INVERSION

  • The near-surface conductor has relatively low U(z) at most stations, with spread increasing toward the base of the interval as MT sensitivity decreases.Locally elevated near-surface U(z) may reflect site-specific effects, prior misspecification, or multidimensional structure.Source: Appendix D · p. 15
  • NPI pseudo-2D sections reproduce large-scale near-surface conductivity patterns from the regional 3D inversion along both profiles.A shallow conductive layer at roughly 0–2 km appears across stations and coincides with the 3D near-surface conductor, but NPI does not constrain its thickness or internal structure.Source: Appendix D · p. 15, Figure 11 · p. 15

APPENDIX E: SENSITIVITY OF NPI RESULTS TO THE FINE-TUNING TARGET

Fine-tuning target sensitivity is tested at two stations using nRMS targets of 1.00 and 1.50. The relaxed target suppresses finer conductive-zone structure while preserving the large-scale conductor, making fit level the operative control on model complexity.

APPENDIX E: SENSITIVITY OF NPI RESULTS TO THE FINE-TUNING TARGET

  • Relaxing nRMS from 1.00 to 1.50 suppresses finer conductive-zone structure at both stations, while the large-scale conductor persists.The effect acts similarly to stronger regularization through early stopping rather than a loss weight.Source: Appendix E · p. 16, Figure 12 · p. 16
  • Structure appearing only under the tighter target is not uniquely required by the data, consistent with non-uniqueness.The test compares GV06 and GV120 at target nRMS values of 1.00 and 1.50.Source: Appendix E · p. 16
  • The fit target controls how strongly the fixed regularization weights bind and thereby acts as the workflow’s operative model-complexity control.Figure 12 shows ensemble means and 95% intervals alongside forward responses for the two target levels.Source: Appendix E · p. 16, Figure 12 · p. 16

APPENDIX F: PHASE TENSOR ANALYSIS AND DIMENSIONALITY DIAGNOSTICS

The appendices use phase-tensor diagnostics to contextualize the 1D field-data assumption and compare NPI with capacity-matched alternatives. They also describe the computational setup for those comparisons.

Phase tensor analysis

  • The phase-tensor skew angle β is insensitive to galvanic distortion and indicates dimensionality continuously, rather than assigning a strict structural class.Source: Appendix F · p. 16, Figure 13 · p. 16
  • At shorter periods, most stations show compact, weakly skewed phase-tensor ellipses; at longer periods, some show stronger 3D effects in the deeper crust.WALDIM analysis is consistent with the observed depth- and period-dependent transitions.Source: Appendix F · p. 16, Figure 13 · p. 16
  • The dimensionality diagnostics provide qualitative context, are not used in the inversion, and are not quantitatively linked to NPI ensemble spread.They motivate caution at stations and periods where multidimensionality is significant.Source: Appendix F · p. 16

Computational environment

  • EnsCGP and RTO-TKO use parallelism differently: EnsCGP forward models across four workers, while RTO-TKO draws form a serial chain with frequency-parallel Jacobian assembly.NPI propagates ensemble members in a batch, then forward-models their responses in a serial loop.Source: Appendix F · p. 17

Capacity-matched ablation

  • The capacity-matched ablation applies a reduced backbone identically to NPI’s residual correction and the end-to-end mapping, isolating the two-stage decomposition from model capacity.CompactNet1D has 2,257 parameters, a 75-fold reduction from the main architecture.Source: Appendix F · p. 17
  • With capacity matched, NPI’s RMS resistivity error and nRMS data misfit are each roughly 40% lower than the end-to-end network’s in distribution, widening under distribution shift.Source: Table IV · p. 18, Appendix F · p. 18
  • The comparison evaluates 1,000 held-out in-distribution and 1,000 homogeneous-half-space-prior examples without per-station fine-tuning, using matched ensemble or sample sizes.Figure 14 shows representative examples; Table IV reports averages over the full test sets.Source: Appendix F · p. 18, Figure 14 · p. 18, Table IV · p. 18
  • NPI’s coverage remains closer to nominal than the end-to-end network’s in both settings, while the latter reaches only 59–67% coverage against 90%/95% nominal in distribution.The end-to-end network’s coverage degrades further under distribution shift; EnsCGP’s also degrades, but less severely than the end-to-end network’s.Source: Appendix F · p. 18, Table IV · p. 18
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