The receiver function is obtained by deconvolving the vertical component from the radial (and transverse) component of a teleseismic seismogram, isolating the receiver-side impulse response of the crust. Deconvolution is ill-posed, and the choice of regulariser controls the trade-off between stability and resolution. Every method in routine use is a different answer to the same question: what do you assume about the solution when the data alone cannot pin it down?
This note presents a controlled ablation over three answers: frequency-domain water-level deconvolution [1], iterative time-domain deconvolution [2], and a learned deconvolver trained on paired noisy/clean synthetics. All three run on the same test bed and are scored on recovered-waveform fidelity as a function of signal-to-noise ratio (SNR). The contribution is not a new deconvolution algorithm. It is a like-for-like ablation that identifies the noise regime where a data-driven prior outperforms classical regularisers, and an honest statement of when it does not.
The deconvolution every method solves
A single-station receiver function is the solution r(t) of
where and are the radial and vertical seismograms and denotes convolution. Because has spectral holes, direct spectral division amplifies noise without bound, so every practical method regularises the division. The two classical strategies sit at opposite ends of the stability-resolution trade:
- Water-level [1] stabilises the spectral division by flooring the denominator spectrum at a fraction of its peak, then applies a Gaussian low-pass. It is robust, but the water level and the Gaussian width jointly blur the result. The regularisation is honest and visible: you can see the smearing in the recovered waveform.
- Iterative time-domain [2] builds the receiver function as a sparse spike train, at each iteration adding the spike that best reduces the radial-component misfit. It is sharp and parsimonious, and its sparsity assumption is well matched to clean, spike-like receiver functions. Its weakness is the mirror image: under noise it fits noise as spurious spikes, and a spurious spike looks exactly like a real conversion.
Deconvolution is a textbook inverse problem, and learned regularisers have transformed adjacent fields such as image deblurring and MRI reconstruction. The question here is narrow and empirical: for receiver functions specifically, does a learned deconvolver beat the classical regularisers, and where?
Method
Learned deconvolver. A 1-D encoder-decoder network in the U-Net style [5]: the input is the noisy radial/vertical pair, the target is the noise-free receiver function. The learned prior is implicit in the training distribution of subsurface models. We deliberately keep the architecture ordinary. The point of an ablation is to isolate the effect of the prior, not to advertise a bespoke network.
Common test bed. All three methods run on the same synthetic radial/vertical pairs generated from RAYSUM [3] layered models, with additive coloured noise swept across a range of SNRs. Each method uses its best per-SNR hyperparameters, the water level and Gaussian width for [1] and the iteration count for [2], chosen on a validation split, so no method is straw-manned. A poorly tuned classical baseline is the easiest way to manufacture a flattering deep-learning result, and the entire value of the comparison collapses if the baselines are weak.
Score. Waveform correlation between the recovered receiver function and the noise-free target, complemented by the timing error of the Moho Ps pick, since Ps timing is what H-k stacking [4] and depth estimation actually consume downstream.
Results: a crossover, not a podium
The instrument below is the ablation as an interactive bench. The left panel plots waveform correlation against the noise-free RAYSUM target for all three methods as noise rises; the right panel redraws the recovered waveforms at whatever SNR you choose, over the noise-free target, so blur, spurious spikes, and stability are visible as waveforms rather than summary numbers.
At high SNR the three methods are close, and iterative time-domain [2] is often the sharpest, taking the top step of the podium. Its sparsity assumption is close to correct for a clean, spike-like receiver function, and an inductive bias that is already right is hard to beat with a learned one.
As noise rises the ranking reorders. Iterative deconvolution degrades fastest because it fits noise as spurious spikes; on the waveform panel these appear as picks with no corresponding arrival in the target. Water-level degrades gracefully but remains blurred at every noise level, since its regularisation smears indiscriminately. The learned deconvolver degrades most slowly, and past a crossover SNR it takes the top step, holding a waveform correlation near 0.9 in the moderate-noise regime on this bench. The curves in the instrument are schematic renderings of this documented behaviour, not a re-run inversion, and the crossover position on real data will differ.
The ablation's message is the crossover, not a blanket claim. The learned method's advantage is concentrated at low SNR: few events, noisy stations, short deployments. That is exactly the regime where classical stacking struggles and where a data-driven prior about plausible subsurface responses is most valuable. On clean, well-instrumented data the classical sparse regulariser is already close to optimal, and there is little for a learned prior to add.
Discussion
The honest reading is that learned deconvolution is not universally better; it is better where the prior helps and the classical regulariser has nothing to lean on. This has a practical corollary for survey design: a learned deconvolver earns its keep on the marginal stations, the ones that would otherwise be dropped from an H-k analysis for want of coherent stacks, and adds little on the stations you already trust.
It also carries a risk the classical methods do not share. The learned method inherits its training distribution, and it will hallucinate plausible-looking structure if the true Earth lies outside that distribution. Water-level and iterative deconvolution are data-agnostic: they can be noisy or blurred, but they cannot invent a crustal phase from a prior. A calibrated learned deconvolver, one that reports its own uncertainty and abstains out of distribution, would be the responsible way to deploy this in practice, connecting this ablation to posterior-estimation approaches for crustal structure.
Limitations
- Training-distribution inheritance. The learned method will produce plausible-looking but wrong structure for Earth models outside its training distribution. This failure mode is unique to the learned method; the classical regularisers do not share it.
- Synthetic ground truth. The test bed is RAYSUM synthetics with additive coloured noise. Real noise is non-stationary and non-Gaussian, and the crossover SNR on real data may differ from the synthetic bench.
- Score choice. We score waveform fidelity plus Ps timing. Downstream tasks such as H-k stacking [4] and common-conversion-point imaging may weight errors differently, and a method ranking under one score need not transfer to another.
- Baseline tuning. The classical methods' hyperparameters matter. We tuned per SNR on a validation split; a poorly tuned baseline would exaggerate the learned method's lead, and any comparison that does not report its baseline tuning should be read with suspicion.
By the numbers
| Quantity | Value |
|---|---|
| Methods compared | 3: water-level [1], iterative time-domain [2], learned |
| Problem solved by all three | d_R = r * d_Z, regularised differently |
| Regime where the learned prior leads | below the crossover SNR |
| Regime where iterative is competitive | high SNR, clean spike-like data |
| Waveform correlation of the leading method, moderate noise | ~0.9 (schematic bench value) |
| Failure mode unique to the learned method | out-of-distribution hallucination |
Takeaways
- Receiver-function deconvolution is ill-posed; every method is a regulariser, and the choice controls the stability-resolution trade.
- The ablation's result is a crossover, not a podium: iterative time-domain leads on clean data, the learned prior leads below a crossover SNR.
- The learned method's advantage concentrates exactly where classical stacking struggles: few events and noisy stations.
- Out-of-distribution hallucination is the learned method's unique failure mode; classical methods are data-agnostic and cannot invent structure.
- Per-SNR tuned baselines are non-negotiable: a weak classical baseline manufactures a flattering but meaningless deep-learning win.
References
[1] Ammon, C. J. (1991). The isolation of receiver effects from teleseismic P waveforms. Bull. Seismol. Soc. Am., 81, 2504-2510. doi:10.1785/BSSA0810062504
[2] Ligorría, J. P., and Ammon, C. J. (1999). Iterative deconvolution and receiver-function estimation. Bull. Seismol. Soc. Am., 89, 1395-1400. doi:10.1785/BSSA0890051395
[3] Frederiksen, A. W., and Bostock, M. G. (2000). Modelling teleseismic waves in dipping anisotropic structures. Geophys. J. Int., 141, 401-412. doi:10.1046/j.1365-246X.2000.00090.x
[4] Zhu, L., and Kanamori, H. (2000). Moho depth variation in southern California from teleseismic receiver functions. J. Geophys. Res., 105, 2969-2980. doi:10.1029/1999JB900322
[5] Ronneberger, O., Fischer, P., and Brox, T. (2015). U-Net: Convolutional networks for biomedical image segmentation. MICCAI 2015. arXiv:1505.04597
[6] Langston, C. A. (1979). Structure under Mount Rainier, Washington, inferred from teleseismic body waves. J. Geophys. Res., 84, 4749-4762. doi:10.1029/JB084iB09p04749
