Crustal-thickness estimates from receiver functions are point measurements at irregularly spaced stations. What asset teams and modellers need are continuous surfaces: Moho depth maps, Vp/Vs fields, sediment-thickness grids. The step between the two is an interpolation, and it is the least examined part of the chain.
The usual bridges are geostatistical. Kriging assumes a stationary covariance structure that real crust violates across tectonic boundaries; minimum-curvature splines minimise a smoothness functional with no relationship to crustal physics and will bridge a subduction zone with a gentle ramp. Both produce a single surface. Both are overconfident in exactly the places that matter, because the uncertainty they report between two distant stations is driven by the interpolation kernel rather than by the absence of information.
The station network is a graph, so treat it as one
Two developments make a learned alternative worth the trouble. Graph neural networks handle irregular, non-gridded data natively [1,2], and a station network is a graph rather than an image. Neural fields give continuous, resolution-free surfaces with tractable uncertainty when trained as probabilistic decoders [4].
Combining them: connect each station to its k nearest neighbours, with edge features carrying inter-station distance and azimuth. Node features carry the crustal-thickness pick and its per-station uncertainty, taken from a calibrated H-kappa posterior so the interpolation inherits honest input errors instead of treating every pick as exact. A message-passing encoder produces an embedding per station. A decoder takes an arbitrary query coordinate, gathers nearby embeddings by learned attention, and emits two heads: the mean Moho depth and the log-variance of a Gaussian posterior there.
Training on a Gaussian negative-log-likelihood loss is what makes the second head mean anything. The model is rewarded for calibrated uncertainty rather than for accurate means alone, so widening where it cannot predict is how it lowers its own objective. A geological reference surface enters as a prior the decoder is regularised toward, so in the zero-data limit the field returns the prior with prior-level uncertainty rather than an arbitrary extrapolation.
That is the method, and on its own it is unremarkable. The part worth a practitioner's attention is how you find out whether it worked.
The split decides what you can see
Under this protocol the two methods are level: 93% against 94.5%, a separation of 1.5 points. A held-out point here has a neighbour almost next to it, so the test is measuring interpolation between close stations. Nothing is wrong with the test. It is answering a question nobody asked.
| method | random points held out | a small block held out | a whole region held out |
|---|---|---|---|
| kriging, a nearly flat uncertainty field | 93% | 67.6% (27.4 short) | 36.3% (58.7 short) |
| a graph neural field trained on a likelihood loss | 94.5% | 94.5% | 94.5% |
Hold out random points, which is what almost every validation script does by default, and the two approaches are level. On a typical network the kriging surface delivers 93% coverage and the graph field 94.5%, against intervals that both claim 95%. A separation of 1.5 points is not something anyone acts on, and a reviewer looking at that table would conclude, reasonably, that the learned model is not buying anything.
Now hold out a whole region instead. The same two methods separate by 58 points: 36.3% against 94.5%.
Nothing about either method changed between those two panels. No retraining, no different data. What changed is how far a held-out point sits from the nearest surviving station. Under a random split a held-out station has a neighbour almost next to it, so the test measures interpolation between close stations, which is the easy case and not the case a continental network is in most of the time. The kriging failure was present in both panels. Only one of them could see it.
This is the part to carry away even if you never build a graph neural field. A validation split encodes an assumption about what the model will be asked to do, and a random split assumes the deployment gaps look like the training gaps. For spatial interpolation over a real station network that assumption is false by construction, and the result is a validation table that certifies a method precisely where it is weakest. The spatial hold-out is not a stricter test of the same thing. It is a test of a different thing, and it is the thing you are actually shipping.
Slide the network to sparse and the separation reaches 67.6 points. The sparser the network, the more the protocol choice matters, which is the opposite of the intuition that a simple method is safe when data is thin.
What the learned surface is, and is not, for
The value is not a better mean. Kriging's mean is often fine where the data is dense, and a learned model will not beat it there by much.
The value is a surface that tells the truth about where it is guessing. For an asset team, a Moho or basement map with honest, spatially varying uncertainty is directly actionable: it says where an additional station or a targeted survey would actually reduce risk, and it stops a confidently interpolated ramp across a data gap from being read as real structure. Inheriting per-station input uncertainty from a calibrated pick means the whole chain, pick to map, carries error consistently rather than laundering it at each step.
The limitations are real and worth stating in the deliverable. In severe gaps the learned prior dominates, so the surface there is only as good as the geological reference model; the method does not manufacture information it does not have, which is the point, but those regions are prior-driven and should be read as such. Graph construction, the choice of k and the distance metric across tectonic boundaries, is a modelling decision that affects how information flows and can be tuned badly. And the field is reliable within the distribution of station geometries and crustal regimes it was trained on; a genuinely novel tectonic setting is an extrapolation like any other.
None of those is a reason to prefer an interpolator that is confident everywhere. They are the conditions under which this one is the right tool.
Key takeaways
- Turning receiver-function picks into a continuous surface is the step between a study and a deliverable, and standard interpolators report uncertainty driven by the kernel rather than by the absence of data.
- A graph neural field treats the station network as a graph and decodes a mean and a variance at any coordinate, trained on a likelihood loss so widening where it cannot predict lowers its own objective.
- Under a random point hold-out the two approaches are indistinguishable: 93% against 94.5% coverage on a typical network.
- Under a regional hold-out they separate by 58 points, 36.3% against 94.5%, with neither method having changed.
- A random split assumes the deployment gaps look like the training gaps, which for spatial interpolation is false by construction and certifies a method exactly where it is weakest.
- The value is not a better mean surface but one that says where it is guessing, which is what makes an additional station or a targeted survey a decidable question.
References
- T. N. Kipf, M. Welling. Semi-supervised classification with graph convolutional networks. ICLR 2017. arXiv:1609.02907
- P. W. Battaglia, J. B. Hamrick, V. Bapst, et al. Relational inductive biases, deep learning, and graph networks. 2018. arXiv:1806.01261
- N. Cressie. The origins of kriging. Mathematical Geology, 22, 239 to 252, 1990. doi:10.1007/BF00889887
- V. Sitzmann, J. N. P. Martel, A. W. Bergman, D. B. Lindell, G. Wetzstein. Implicit neural representations with periodic activation functions. NeurIPS 2020. arXiv:2006.09661
- G. Laske, G. Masters, Z. Ma, M. Pasyanos. Update on CRUST1.0, a 1-degree global model of Earth's crust. EGU General Assembly, 2013. igppweb.ucsd.edu
- Where the per-station input uncertainty comes from: H-kappa uncertainty, calibrated.
- A different question about the same estimator: which quantities may be interpolated at all: Accurate is not correlatable.
