Bayesian eikonal tomography using Gaussian processes
DOI:
https://doi.org/10.26443/seismica.v2i2.388Abstract
Eikonal tomography has become a popular methodology for deriving phase velocity maps from surface wave phase delay measurements. Its high efficiency makes it popular for handling datasets deriving from large-N arrays, in particular in the ambient-noise tomography setting. However, the results of eikonal tomography are crucially dependent on the way in which phase delay measurements are predicted from data, a point which has not been thoroughly investigated. In this work, I provide a rigorous formulation for eikonal tomography using Gaussian processes (GPs) to smooth observed phase delay measurements, including uncertainties. GPs allow the posterior phase delay gradient to be analytically derived. From the phase delay gradient, an excellent approximate solution for phase velocities can be obtained using the saddlepoint method. The result is a fully Bayesian result for phase velocities of surface waves, incorporating the nonlinear wavefront bending inherent in eikonal tomography, with no sampling required. The results of this analysis imply that the uncertainties reported for eikonal tomography are often underestimated.References
Al-Naffouri, T., Moinuddin, M., Ajeeb, N., Hassibi, B., & Moustakas, A. (2016). On the Distribution of Indefinite Quadratic Forms in Gaussian Random Variables. IEEE Transactions on Communications, 64(1), 153–165.
Bezanson, J., Edelman, A., Karpinski, S., & Shah, V. (2017). Julia: A Fresh Approach to Numerical Computing. SIAM Review, 59(1), 65–98.
Bodin, T., & Maupin, V. (2008). Resolution Potential of Surface Wave Phase Velocity Measurements at Small Arrays. Geophysical Journal International, 172(2), 698–706.
Brocher, T. (2005). Empirical Relations between Elastic Wavespeeds and Density in the Earth's Crust. Bulletin of the Seismological Society of America, 95(6), 2081–2092.
Butler, R. (2007). Saddlepoint Approximations with Applications. Cambridge University Press.
Butler, R., & Paolella, M. (2008). Uniform Saddlepoint Approximations for Ratios of Quadratic Forms. Bernoulli, 14(1), 140–154.
Chevrot, S., & Lehujeur, M. (2022). Eikonal Surface Wave Tomography with Smoothing Splines— Application to Southern California. Geophysical Journal International, 229(3), 1927–1941.
Daniels, H. (1954). Saddlepoint Approximations in Statistics. The Annals of Mathematical Statistics, 25(4), 631–650.
de Ridder, S., & Biondi, B. (2015). Near-surface Scholte Wave Velocities at Ekofisk from Short Noise Recordings by Seismic Noise Gradiometry. Geophysical Research Letters, 42(17), 7031–7038.
de Ridder, S., & Maddison, J. (2018). Full Wave Field Inversion of Ambient Seismic Noise. Geophysical Journal International.
Dierckx, P. (1993). Curve and surface fitting with splines. Oxford University Press.
Friederich, W., & Wielandt, E. (1995). Interpretation of Seismic Surface Waves in Regional Networks: Joint Estimation of Wavefield Geometry and Local Phase Velocity. Method and Numerical Tests. Geophysical Journal International, 120(3), 731–744.
Friederich, W., Wielandt, E., & Stange, S. (1994). Non-Plane Geometries of Seismic Surface Wavefields and Their Implications For Regional Surface-Wave Tomography. Geophysical Journal International, 119(3), 931–948.
Herrmann, R. (2013). Computer Programs in Seismology: An Evolving Tool for Instruction and Research. Seismological Research Letters, 84(6), 1081–1088.
Hines, T., & Hetland, E. (2018). Revealing Transient Strain in Geodetic Data with Gaussian Process Regression. Geophysical Journal International, 212(3), 2116–2130.
Imhof, J. (1961). Computing the Distribution of Quadratic Forms in Normal Variables. Biometrika, 48(3/4), 419–426.
Kadane, J. (2011). Principles of Uncertainty. Chapman and Hall/CRC.
Kendrick, D. (2002). Stochastic Control for Economic Models. Publisher is required!.
Langston, C. (2007). Spatial Gradient Analysis for Linear Seismic Arrays. Bulletin of the Seismological Society of America, 97(1), 265–280.
Langston, C. (2007). Wave Gradiometry in Two Dimensions. Bulletin of the Seismological Society of America, 97(2), 401–416.
Lehujeur, M., & Chevrot, S. (2020). On the Validity of the Eikonal Equation for Surface-Wave Phase-Velocity Tomography. Geophysical Journal International, 223(2), 908–914.
Lin, F.C., & Ritzwoller, M. (2011). Helmholtz Surface Wave Tomography for Isotropic and Azimuthally Anisotropic Structure: Helmholtz Surface Wave Tomography. Geophysical Journal International, 186(3), 1104–1120.
Lin, F.C., Ritzwoller, M., & Snieder, R. (2009). Eikonal Tomography: Surface Wave Tomography by Phase Front Tracking across a Regional Broad-Band Seismic Array. Geophysical Journal International, 177(3), 1091–1110.
Lindgren, F., Rue, H., & Lindstrom, J. (2011). An Explicit Link between Gaussian Fields and Gaussian Markov Random Fields: The Stochastic Partial Differential Equation Approach: Link between Gaussian Fields and Gaussian Markov Random Fields. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 73(4), 423–498.
Mandic, V., Tsai, V., Pavlis, G., Prestegard, T., Bowden, D., Meyers, P., & Caton, R. (2018). A 3D Broadband Seismometer Array Experiment at the Homestake Mine. Seismological Research Letters, 89(6), 2420–2429.
McHutchon, A. (2014). Nonlinear Modelling and Control Using Gaussian Processes. (Doctoral dissertation, Cambridge).
Muir, J. (2023). GP-Eikonal. Journal is required!.
Muir, J., & Zhan, Z. (2021). Wavefield-Based Evaluation of DAS Instrument Response and Array Design. Geophysical Journal International, 229(1), 21–34.
Padonou, E., & Roustant, O. (2015). Polar Gaussian Processes for Predicting on Circular Domains. Journal is required!, 24.
Pollitz, F. (2008). Observations and Interpretation of Fundamental Mode Rayleigh Wavefields Recorded by the Transportable Array (USArray). Journal of Geophysical Research: Solid Earth, 113(B10).
Rasmussen, C., & Williams, C. (2006). Gaussian Processes for Machine Learning. MIT Press.
Rasmussen, C. (2003). Gaussian Processes to Speed up Hybrid Monte Carlo for Expensive Bayesian Integrals. Bayesian Statistics, 7, 651–659.
Solak, E., Murray-smith, R., Leithead, W., Leith, D., & Rasmussen, C. (2002). Derivative Observations in Gaussian Process Models of Dynamic Systems. In Advances in Neural Information Processing Systems. MIT Press.
Titsias, M. (2009). Variational Learning of Inducing Variables in Sparse Gaussian Processes. In Proceedings of the Twelth International Conference on Artificial Intelligence and Statistics (pp. 567–574). PMLR.
Treister, E., & Haber, E. (2016). A Fast Marching Algorithm for the Factored Eikonal Equation. Journal of Computational Physics, 324, 210–225.
Tromp, J., & Dahlen, F. (1993). Variational Principles for Surface Wave Propagation on a Laterally Heterogeneous Earth - III. Potential Representation. Geophysical Journal International, 112, 195–209.
Valentine, A., & Sambridge, M. (2020). Gaussian Process ModelstextemdashII. Lessons for Discrete Inversion. Geophysical Journal International, 220(3), 1648–1656.
Valentine, A., & Sambridge, M. (2020). Gaussian Process ModelstextemdashI. A Framework for Probabilistic Continuous Inverse Theory. Geophysical Journal International, 220(3), 1632–1647.
White, M., Fang, H., Catchings, R., Goldman, M., Steidl, J., & Ben-Zion, Y. (2021). Detailed Traveltime Tomography and Seismic Catalogue around the 2019 Mw 7.1 Ridgecrest, California, Earthquake Using Dense Rapid-Response Seismic Data. Geophysical Journal International, 227(1), 204–227.
Wielandt, E. (1993). Propagation and Structural Interpretation of Non-Plane Waves. Geophysical Journal International, 113(1), 45–53.
Wilson, A., & Nickisch, H. (2015). Kernel Interpolation for Scalable Structured Gaussian Processes (KISS-GP). Proceedings of the 32 nd International Conference on Machine Learning, 1775–1784.
Yang, Y., Atterholt, J., Shen, Z., Muir, J., Williams, E., & Zhan, Z. (2022). Sub-Kilometer Correlation Between Near-Surface Structure and Ground Motion Measured With Distributed Acoustic Sensing. Geophysical Research Letters, 49, e2021GL096503.
Zhan, Z. (2020). Distributed Acoustic Sensing Turns Fiber-Optic Cables into Sensitive Seismic Antennas. Seismological Research Letters, 91(1), 1–15.
Additional Files
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Jack Muir
This work is licensed under a Creative Commons Attribution 4.0 International License.
Funding data
-
European Commission
Grant numbers 101027079
