Where does crosstalk come from?

We cannot dig under a region to see its rocks. We send waves into the ground, record the echo at the surface, and let a computer reconstruct maps of the rock properties that would explain it.

In practice we want several maps at once: for example, the speed of compressional waves and the speed of shear waves. Here we call them simply field 1 and field 2.

The data never tell the whole story, so the honest answer is not a single map but a collection of possible maps, all compatible with what was measured. Statisticians call that collection the posterior distribution. Looking at it, a pattern shows up: where one possible map has field 1 higher at some point, field 2 tends to be higher (or lower) at the same point. The fields are linked.

Geophysicists call this link crosstalk, or interparameter trade-off: part of what appears in one map may really be a trace of the other (Operto et al., 2013). It matters, because decisions such as where to drill or how to interpret a rock are made by reading these maps.

Links can also be put in on purpose

Before any data arrive, the inversion already holds a belief about what the fields look like: the prior. In the testbed of this study the prior builds each field from a few smooth patterns, and a fraction \(s\) of those patterns is shared by both fields. With \(s = 0\) the fields are independent; with \(s = 1\) they move together. Draw a few pairs and watch the cloud of points tilt as \(s\) grows.

Try it: draw two fields from the prior

lowhigh

Shared fraction s =

0.5

Draw another pair

The link you see in an inversion result could therefore have been put there by the prior, by the measurement, or by both. The rest of this page is about telling them apart.

The link between the fields in the posterior can come from three places. In the final map all three are mixed, and nothing shows who did what.

Switching each suspect off

To learn whether a suspect matters, switch it off and see what changes. Each switch is an experiment one can actually run:

• Operator off: observe each field separately, with the same noise. The information about each field stays; only the mixing goes.
• Prior off: keep each field's own statistics but make the two independent. Every field still looks the same on its own; only the link between them goes.

Two switches give four combinations. In each, we measure how linked the fields are in the posterior, with a dependence measure \(D\):

	operator off	operator on
prior off	\(D(\varnothing)\): what is left with no source	\(D(O)\): the physics alone
prior on	\(D(P)\): the prior alone	\(D(OP)\): the real situation, what the scientist sees

A fair split: the Shapley value

The total change, \(D(OP) - D(\varnothing)\), still has to be split between the suspects. We use the Shapley value (Shapley, 1953), a rule from game theory for sharing credit in a group effort: each player's credit is what it adds on joining the group, averaged over every order in which the players could arrive. With two players,

\[ \varphi_O = \tfrac12\big[(D(O) - D(\varnothing)) + (D(OP) - D(P))\big], \qquad \varphi_P = \tfrac12\big[(D(P) - D(\varnothing)) + (D(OP) - D(O))\big], \]

and the shares always add up to the total: \(\varphi_O + \varphi_P = D(OP) - D(\varnothing)\). The Shapley value already apportions variance among inputs in sensitivity analysis (Owen, 2014), also when the inputs are dependent (Herin et al., 2022). Here the players are not inputs drawn from a distribution but switches on the problem itself.

A share can be negative: a suspect can remove dependence. That is why the study always reports a dependence measure with a sign.

To test the method we need a problem where the right answer is known. The testbed is artificial and fully controlled, and its posterior has a closed form.

The testbed

• Fields: two maps of 16 × 16 points, 512 numbers in all, drawn from the Gaussian prior of the first section with shared fraction \(s\).
• Measurement: each field is blurred and sampled every few points (the stride; a larger stride means fewer data). The two fields' data are mixed with a coupling \(\kappa\): at \(\kappa = 0\) each datum sees one field, as \(\kappa\) grows each datum sees more of both.
• Noise: Gaussian, with a variance you choose. Small noise and a small stride mean informative data.

With a Gaussian prior of covariance \(C_p\) and a linear measurement \(G\) with noise variance \(\sigma^2\), the posterior is Gaussian too, with covariance

\[ C = C_p - C_p G^\top \big(G C_p G^\top + \sigma^2 I\big)^{-1} G C_p . \]

From \(C\) we read the dependence between the fields. The main measure is the mean signed pointwise correlation: the correlation \(\rho(x)\) between field 1 and field 2 at each point \(x\), averaged over the 256 points. Two others are available: the same average without the sign, and the mutual information between the whole fields.

Try it: switch each source on and off

Informative data Weak data Tug of war A confirmatory point

ρ = −1ρ = +1

Operator coupling κ =

0.5

Prior shared fraction s =

0.75

Noise variance 0.001 0.01 0.1 1 Stride 2 (64 data per field) 4 (16 data per field) 8 (4 data per field)

Dependence measure

signed correlation unsigned mutual information

What the exact case shows

How much of the prior survives depends on the data. Try Informative data and then Weak data. The prior is the same and links the fields strongly, and the operator does not mix them. Yet almost none of the prior's link survives when the data are informative, and nearly all of it when they say little. The data erase the prior belief to the extent that they can speak for themselves; in large samples, prior dependence is in general not retained by the posterior (Hagar & Stevens, 2023).

Operator and prior can pull in opposite directions. In this testbed the measurement tends to create negative dependence (one field up, the other down) and the prior positive dependence. On the study's grid of settings the two shares had opposite signs in of the settings where both were present. Try Tug of war: with a prior that ties the fields almost perfectly, the final map still shows a slightly negative link.

When the prior is a neural network there is no formula for the posterior any more. All we can do is draw possible maps with a program called a sampler, and estimate the dependence from those draws.

A sampler that is slightly off can create or remove dependence by itself, like a thermometer that reads two degrees high. And more draws do not help: they only make the wrong value more precise. The demo below uses a posterior where the exact answer is known. The bias slider makes the sampler shrink the link between the fields by a fraction \(b\).

Try it: estimate the dependence from samples

Samples per estimate 16 64 256 Sampler bias b =

0

Draw again

Posterior of one confirmatory point (κ = 0.25, s = 0.75, noise variance 0.1, stride 4), sampled exactly; the biased sampler draws from the same posterior with the cross-field covariance multiplied by 1 − b.

How the real sampler was checked

The study's sampler is a twisted sequential Monte Carlo method (Wu et al., 2023). Before any result, it had to reproduce exact answers on problems like the one above, in 512 dimensions, within margins fixed in advance. It failed twice, was diagnosed and corrected, and passed on a third, fresh seed, with its worst case at 0.92 of the allowed margin.

It was then compared with a sampler built differently, DDSMC (Ekström Kelvinius et al., 2025), and with an ablation of its own twist, on a network trained in 128 dimensions: all 45 comparisons across five independently trained networks agreed, judged against the gap expected between two exact samplers.

Training networks and drawing thousands of maps is expensive. There is a shortcut: replace the network by a Gaussian surrogate that keeps only the mean and the correlations of its training examples. For the surrogate the attribution is exact and free, by the formulas of the previous sections.

Previous work suggests that diffusion networks in their generalizing regime denoise much like that Gaussian surrogate, at least over a band of noise levels and, for large networks, transiently during training (Li et al., 2024). Hence the study's hypothesis:

Rules written before the data

To avoid adjusting the criterion after seeing the result, the test was pre-registered: the pass rules, the margins and even the analysis program were written and stored before the experiments ran. When a rule had to change, the change was recorded as a dated amendment, always before the data it would judge existed. Two requirements had to hold together, at eight settings (operator coupling \(\kappa \in \{0.25, 0.75\}\), prior dependence \(s \in \{0.25, 0.75\}\), informative or weak data), with four independently trained networks per setting:

• Ranking: across the eight settings, the surrogate must order each share the way the network does. Agreement is measured with Kendall's \(\tau\), where 1 is perfect agreement; the thresholds were computed in advance from how much \(\tau\) an exactly correct surrogate would lose to estimation noise and training variance.
• Magnitude: the predicted total dependence may not be off by more than 4.5% at any of the eight settings.

Is the network copying its examples?

A network that memorizes its training set turns the posterior into a lookup over training examples (Siahkoohi & Sabeddu, 2026), and the surrogate would not apply. Every network was therefore tested for copying with an exact test that counts near-duplicates of training examples against those of held-out examples (Majhi & Bagchi, 2026). of the networks trained for the test passed: none of their samples is a near-duplicate of a training example.

The surrogate orders the suspects almost exactly as the network does. It does not get the size right when the prior links the fields only weakly.

Magnitude error at the eight confirmatory settings

• weak prior dependence (s = 0.25)
• strong prior dependence (s = 0.75)

Each bar is one setting: operator coupling κ and whether the data are informative or weak. The rule required every bar below the dashed line.

At the four settings with weak prior dependence the surrogate overstates the total dependence by ; at the four with strong dependence the error is , inside the tolerance. Where it fails, the network leaves less dependence in the posterior than the surrogate predicts.

A negative result is still a result. The pre-registration anticipated it: if the hypothesis failed, the map of where it fails would be the contribution. And the test had controls that rule out the easy explanations: the networks were not copying, and the samplers agreed on the settings that were used.

Looking inside the networks themselves, before any data enter: a trained network carries less dependence between the fields than its own training examples had.

Measured as mutual information against draws from each network's own Gaussian surrogate, the deficit is with weak prior dependence and with strong dependence.

The first suspicion was under-training: perhaps the network had not yet learned the link. To test it, a follow-up was registered before it ran: sixteen networks trained eight times longer, to 160 000 steps, measuring the deficit along the way.

Dependence lost by the network along training

Mean of four networks per line, with the range of the four shaded. Orange: weak prior dependence; blue: strong. Solid: 4096 training examples; dashed: 1024.

The answer was the opposite of the suspicion. In three of the four settings the deficit grew with training in all four networks. In the fourth it was already high from the first checkpoint and stayed flat. Under-training explains none of them, and no checkpoint copies its training set.

Training time is therefore not a neutral technical detail: it decides how much of the real dependence between the fields reaches the answer.

Limits

Everything here is on one artificial problem with smooth Gaussian fields and a linear measurement (each datum is a weighted sum of the fields). Fields with sharp boundaries, nonlinear physics such as elastic wave propagation, and real seismic data are left to the next chapters of the thesis. Classical tools measure the operator's coupling locally, from the Hessian (Pan et al., 2018); comparing them with the operator share is also future work.

Next: Study II measures, under controlled conditions, how the dependence between fields and the network's tendency to copy its examples depend on the number of examples and on training time.

Generated from the thesis bibliography by scripts/build_page.py. Entries marked † have bibliographic fields not yet checked against the publisher record.

1. Bonnaire, T., Urfin, R., Biroli, G., Mézard, M. (2025). Why diffusion models don't memorize: The role of implicit dynamical regularization in training. Advances in Neural Information Processing Systems (NeurIPS). Cited in: Why the surrogate misses
2. Ekström Kelvinius, F., Zhao, Z., Lindsten, F. (2025). Solving linear-Gaussian Bayesian inverse problems with decoupled diffusion sequential Monte Carlo. Proceedings of the 42nd International Conference on Machine Learning (ICML) 267, pp. 15148–15181. Cited in: Checking the instrument
3. Hagar, L., Stevens, N. T. (2023). Posterior Ramifications of Prior Dependence Structures. arXiv preprint arXiv:2312.06437. arXiv:2312.06437 † Cited in: The exact case
4. Herin, M., Il Idrissi, M., Chabridon, V., Iooss, B. (2022). Proportional marginal effects for global sensitivity analysis. arXiv preprint arXiv:2210.13065. arXiv:2210.13065 † Cited in: Three suspects
5. Li, X., Dai, Y., Qu, Q. (2024). Understanding generalizability of diffusion models requires rethinking the hidden Gaussian structure. Advances in Neural Information Processing Systems 37, pp. 57499–57538. doi:10.52202/079017-1833 Cited in: A shortcut, and its test, Why the surrogate misses
6. Majhi, S., Bagchi, P. (2026). The Null Is the Hard Part: Exact Tests for Memorization in Generative Models. arXiv preprint arXiv:2610.00251. arXiv:2610.00251 † Cited in: A shortcut, and its test
7. Operto, S., Gholami, Y., Prieux, V., Ribodetti, A., Brossier, R., Métivier, L., Virieux, J. (2013). A guided tour of multiparameter full-waveform inversion with multicomponent data: From theory to practice. The Leading Edge 32(9), 1040–1054. doi:10.1190/tle32091040.1 Cited in: Two linked fields
8. Owen, A. B. (2014). Sobol' indices and Shapley value. SIAM/ASA Journal on Uncertainty Quantification 2(1), 245–251. doi:10.1137/130936233 Cited in: Three suspects
9. Pan, W., Geng, Y., Innanen, K. A. (2018). Interparameter trade-off quantification and reduction in isotropic-elastic full-waveform inversion: synthetic experiments and Hussar land data set application. Geophysical Journal International 213(2), 1305–1333. doi:10.1093/gji/ggy037 Cited in: What stays
10. Shapley, L. S. (1953). A value for n-person games. Contributions to the Theory of Games, Volume II pp. 307–317. † Cited in: Three suspects
11. Siahkoohi, A., Sabeddu, D. (2026). On the role of memorization in learned priors for geophysical inverse problems. arXiv preprint arXiv:2603.19629. arXiv:2603.19629 Cited in: A shortcut, and its test
12. Wu, L., Trippe, B. L., Naesseth, C. A., Blei, D. M., Cunningham, J. P. (2023). Practical and Asymptotically Exact Conditional Sampling in Diffusion Models. Advances in Neural Information Processing Systems (NeurIPS). Cited in: Checking the instrument