Bayesian-Deep-Learning Estimation of Earthquake Location from Single-Station Observations

1 Introduction

• Prior neural networks applied to earthquake data did not quantify uncertainty in their outputs.Uncertainty can help identify cases for expert analysis in earthquake signal detection and arrival-time measurement.Source: §1 · pp. 2–3
• The paper approaches single-station earthquake location as regression using two Bayesian neural networks.One estimates epicentral distance and P travel time jointly with epistemic and aleatory uncertainties; the other estimates back-azimuth and its epistemic uncertainty.Source: §1 · p. 3
• The predicted distance and back-azimuth are used to estimate epicenter, origin time, and depth with confidence intervals.The model is built and evaluated using a global dataset.Source: §1 · pp. 3–4
• The approach is intended for rapid source characterization with limited observations and for locating sparsely recorded earthquakes.The introduction also identifies earthquake early-warning systems as a possible application.Source: §1 · p. 4

Dataset

• For training and testing, the study selected about 150,000 STEAD earthquake waveforms recorded less than 110 km from epicenters, with signal-to-noise ratios of at least 25 dB and correctly oriented north–south/east–west components.Source: Dataset · p. 4

Network design

• A multi-task temporal-convolutional network estimates epicentral distance and P travel time, while a separate multi-input convolutional network estimates back-azimuth.The distance–travel-time network receives three-component waveforms and a binary P-to-S arrival window to highlight informative waveform segments.Source: Dataset · pp. 4–5

Back-azimuth estimation

• The back-azimuth network represents angles as (cos θ, sin θ) on a unit circle and combines a 1.5-second waveform window with covariance and eigenvector features.The authors do not estimate aleatory uncertainty for back-azimuth because stable estimation proved technically difficult; they estimate epistemic uncertainty using Monte Carlo dropout.Source: Dataset · pp. 5–6

Regression results

• The network estimated P travel time with a 0.66-second standard deviation, epicentral distance with 0.23-km mean error and 5.42-km standard deviation, and back-azimuth with R^2 = 0.87 and ~1-degree mean error.The back-azimuth result was considered good because only 1.5 seconds of waveforms were used.Source: Dataset · p. 7

Error and uncertainty patterns

• Distance and P travel-time errors increase slightly with station–event distance, deeper events show distance-estimation bias and higher uncertainties, and larger-magnitude events have higher uncertainties.Aleatory uncertainties reflect distance and travel-time errors better than epistemic uncertainties; the relationship between SNR and regression errors is visible, but its relationship with estimated uncertainties is less clear.Source: Dataset · pp. 7–8

Event location

• The authors combine distance and back-azimuth predictions to locate test events and derive origin time and rough depth from P travel time, assuming straight-line propagation at 5.6 km/s.Example error ellipses show a larger uncertainty contribution from back-azimuth than from origin time; the authors describe location errors as reasonable given catalog-location uncertainties.Source: Dataset · pp. 8–9

• Depth estimates are within an acceptable range despite not training the network end-to-end for depth, because inconsistent catalog labels make learning difficult; estimates combine predicted P-travel distance and epicentral distance.Source: Dataset · pp. 9–10
• Even when location errors are large, often due to back-azimuth error, predictions can point toward the source; large uncertainties can flag less reliable estimates, while origin time remains close to ground truth and may support event association.Source: Dataset · p. 10
• Predicted locations recover regional seismicity patterns, including the San Andreas creeping-section lineation and Geysers cluster, with sparse outliers.These are averaged event-based estimates without removing high-uncertainty estimates or weighting results.Source: Dataset · p. 10
• 7.3 km, 0.4 seconds, and 6.7 km are the mean errors for epicenter, origin time, and depth, respectively, across the test set.The error distributions are presented in Figure 12.Source: Dataset · p. 11
• Larger events tend to have larger errors, plausibly due to sparser training data, while small events perform strongly; depth errors are larger for very shallow and >20 km events, and predicted uncertainty correlates with error.Source: Dataset · p. 11

3 Discussion and Conclusions

• The network learns a general mapping from raw three-component seismograms, and tests indicate it can perform well when arrival picks are unavailable.The approach may also work with a portion of the waveform, such as P-wave-only input, through augmentation.Source: §3 · pp. 12, 14
• Distance and back-azimuth predictions can be made without a local velocity model, and the model was trained and tested using a global dataset.The authors state that distance and back-azimuth predictions could be extended to regional and teleseismic distances by training on waveforms from those events.Source: §3 · p. 12
• The current approximate depth-estimation procedure does not extend to regional or teleseismic events because of more complex wave propagation and Earth sphericity.The authors state that distance and back-azimuth predictions could be extended by training on waveforms from those events.Source: §3 · p. 12
• A multi-task network for distance and travel time, paired with separate networks for back-azimuth, improves overall performance.The approach uses multi-task learning for closely related tasks and separate networks for the other task.Source: §3 · p. 12
• Larger, high-quality labeled training sets could support more flexible networks and potentially reduce epistemic uncertainty, but building such datasets is challenging.The authors identify waveform and metadata quality, including pick accuracy, as relevant to dataset construction.Source: §3 · p. 12
• The method does not account for stations whose orientations differ from geographic orientations and is not applicable to borehole stations.This excludes high-signal-to-noise data provided by borehole instruments.Source: §3 · p. 13
• The largest uncertainty in location results comes from back-azimuth uncertainty, which may relate to station installation or orientation error.The authors note that learning orientation angle with neural networks is technically challenging and suggest more advanced continuous-orientation methods might improve results.Source: §3 · p. 13
• The neural-network approach shows superior performance to traditional single-station location methods and can provide rapid location estimates directly from single instruments.The authors note potential use for rapid public reporting or earthquake early warning.Source: §3 · p. 13

Bayesian deep learning

• Aleatory uncertainty captures uncertainty in information the data cannot explain and can be heteroscedastic or homoscedastic.The section also distinguishes epistemic and ontological uncertainty as other types contributing to model predictions.Source: Bayesian deep learning · p. 15
• Monte Carlo dropout approximates a posterior over models by sampling thinned networks during training and testing, providing a computationally efficient approximation.At test time, dropout approximates averaging predictions from thinned networks using the un-thinned network's weights.Source: Bayesian deep learning · p. 16
• The model estimates an output and data-dependent aleatory variance together, using a customized loss that learns the variance without uncertainty labels.The variance represents heteroscedastic uncertainty and can account for differing noise levels across inputs.Source: Bayesian deep learning · pp. 16–17
• Predictive uncertainty combines epistemic and aleatory uncertainty, with its estimate based on Monte Carlo dropout samples.T denotes the number of Monte Carlo dropout samples.Source: Bayesian deep learning · p. 17

Temporal Convolution Networks

• Earthquake signals have temporal dependencies arising from elastic-wave propagation, motivating neural networks that model sequences.The section notes that P-waves arrive before S-waves, which arrive before surface waves.Source: Bayesian deep learning · p. 17
• The distance and P-travel-time network uses causal, dilated convolutions and residual blocks to expand its receptive field and support deeper networks.Causal convolutions use information from the past, while dilation increases the receptive field; each residual block contains two dilated causal-convolution layers.Source: Bayesian deep learning · pp. 18–19

Authors’ contributions

• S.M.M. designed the project and networks, implemented the software, performed training and tests, and wrote the manuscript.Source: Authors’ contributions · p. 25
• G.C.B. led the project and helped with the manuscript.Source: Authors’ contributions · p. 25
• All authors contributed ideas to the project.Source: Authors’ contributions · p. 25