ABSTRACT
Traveltime tomography, or traveltime inversion, has been one of the primary seismological tools for decades and has been used to understand the earth’s properties and dynamic processes. An accurate, preferably flexible, eikonal solver to compute the traveltime field is at the heart of the inversion process. However, most conventional eikonal solvers suffer from first-order convergence errors and difficulties dealing with irregular computational grids. Physics-informed neural networks (PINNs) have been introduced to tackle these problems and have successfully addressed these challenges. Nevertheless, these approaches still suffer from slow convergence and unstable training dynamics due to the multiterm nature of the loss function. To improve this, we develop a new formulation for the isotropic eikonal equation, which imposes boundary conditions as hard constraints. We apply the theory of functional connections to the traveltime tomography problem, which allows for using a single loss term to train the PINN model. We also analyze the effect of different traveltime factorizations on the overall inversion performance. The additive factorization yields a better result than the previously used multiplicative factorization. Our framework’s efficiency, stability, and flexibility in tackling various cases, such as topography-dependent and 3D models, are tested through rigorous numerical tests, thus providing an efficient and stable PINN-based traveltime tomography. Compared with existing PINN-based inversion, our framework introduces more stability during the inversion and offers significant convergence speedups.
REFERENCES
- 1977, Determination of the three-dimensional seismic structure of the lithosphere: Journal of Geophysical Research, 82,
277–296 , doi:10.1029/JB082i002p00277 .JGREA2 0148-0227 , - 2002, Traveltime computation with the linearized eikonal equation for anisotropic media: Geophysical Prospecting, 50,
373–382 , doi:10.1046/j.1365-2478.2002.00322.x .GPPRAR 0016-8025 , - 2018, Automatic differentiation in machine learning: A survey: Journal of Machine Learning Research, 18,
1–43 . , - 2004, High-resolution seismic traveltime tomography incorporating static corrections applied to a till-covered bedrock environment: Geophysics, 69,
1082–1090 , doi:10.1190/1.1778250 .GPYSA7 0016-8033 , - 1987, Applications of seismic travel-time tomography: Geophysical Journal International, 90,
285–303 , doi:10.1111/j.1365-246X.1987.tb00728.x .GJINEA 0956-540X , - 2000, Seismic ray method: Cambridge University Press. ,
- 2022, Eikonal tomography with physics-informed neural networks: Rayleigh wave phase velocity in the northeastern margin of the Tibetan Plateau: Geophysical Research Letters, 49,
e2022GL099053 , doi:10.1029/2022GL099053 .GPRLAJ 0094-8276 , - 2023, Physics-informed neural networks for elliptical-anisotropy eikonal tomography: Application to data from the northeastern Tibetan Plateau: Journal of Geophysical Research: Solid Earth, 128,
e2023JB027378 , doi:10.1029/2023JB027378 . , - 2004, Deep seismic imaging of the eastern Nankai trough, Japan, from multifold ocean bottom seismometer data by combined travel time tomography and prestack depth migration: Journal of Geophysical Research: Solid Earth, 109,
B02111 , doi:10.1029/2003JB002689 . , - 1977, Large-scale heterogeneities in the lower mantle: Journal of Geophysical Research, 82,
239–255 , doi:10.1029/JB082i002p00239 .JGREA2 0148-0227 , - 2022, Seismic traveltime tomography of Southern California using Poisson-Voronoi cells and 20 years of data: Journal of Geophysical Research: Solid Earth, 127,
e2021JB023307 , doi:10.1029/2021JB023307 . , - 2016, A new algorithm for three-dimensional joint inversion of body wave and surface wave data and its application to the Southern California plate boundary region: Journal of Geophysical Research: Solid Earth, 121,
3557–3569 , doi:10.1002/2015JB012702 . , - 2010, Full seismic waveform modelling and inversion: Springer Science & Business Media. ,
- 2009, Fast sweeping method for the factored eikonal equation: Journal of Computational Physics, 228,
6440–6455 , doi:10.1016/j.jcp.2009.05.029 .JCTPAH 0021-9991 , - 2010, Understanding the difficulty of training deep feedforward neural networks:
Proceedings of the 13th International Conference on Artificial Intelligence and Statistics ,249–256 . , - 2012, Waveform relocated earthquake catalog for Southern California (1981 to June 2011): Bulletin of the Seismological Society of America, 102,
2239–2244 , doi:10.1785/0120120010 .BSSAAP 0037-1106 , - 1989, Multilayer feedforward networks are universal approximators: Neural Networks, 2,
359–366 , doi:10.1016/0893-6080(89)90020-8 .NNETEB 0893-6080 , - 1977, Three-dimensional seismic ray tracing: Journal of Geophysics, 43,
95–113 .JGEOD4 0340-062X , - 2014, Adam: A method for stochastic optimization: arXiv preprint, doi:
10.48550/arXiv.1412.6980 . , - 2002, User’s guide to HYPOINVERSE-2000, a Fortran program to solve for earthquake locations and magnitudes: Open File Report 02-171: US Geological Survey, 1–123. ,
- 2006, An adjoint state method for three-dimensional transmission traveltime tomography using first-arrivals: Communications in Mathematical Sciences, 4,
249–266 , doi:10.4310/CMS.2006.v4.n1.a10 .1539-6746 , - 2011, Helmholtz surface wave tomography for isotropic and azimuthally anisotropic structure: Geophysical Journal International, 186,
1104–1120 , doi:10.1111/j.1365-246X.2011.05070.x .GJINEA 0956-540X , - 2009, Eikonal tomography: Surface wave tomography by phase front tracking across a regional broad-band seismic array: Geophysical Journal International, 177,
1091–1110 , doi:10.1111/j.1365-246X.2009.04105.x .GJINEA 0956-540X , - 2007, Ambient noise Rayleigh wave tomography of New Zealand: Geophysical Journal International, 170,
649–666 , doi:10.1111/j.1365-246X.2007.03414.x .GJINEA 0956-540X , - 1993, Static corrections — A review, Part 1: The Leading Edge, 12,
43–49 , doi:10.1190/1.1436912 . , - 2020, Self-adaptive physics-informed neural networks using a soft attention mechanism: arXiv preprint, doi:
10.48550/arXiv.2009.04544 . , - 2012, Shear wave tomography of China using joint inversion of body and surface wave constraints: Journal of Geophysical Research: Solid Earth, 117,
B01311 , doi:10.1029/2011JB008349 . , - 2006, A review of the adjoint-state method for computing the gradient of a functional with geophysical applications: Geophysical Journal International, 167,
495–503 , doi:10.1111/j.1365-246X.2006.02978.x .GJINEA 0956-540X , - 1992, Finite-difference solution of the eikonal equation along expanding wavefronts: Geophysics, 57,
478–487 , doi:10.1190/1.1443263 .GPYSA7 0016-8033 , - 2019, On the spectral bias of neural networks:
International Conference on Machine Learning ,5301–5310 . , - 2019, Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations: Journal of Computational Physics, 378,
686–707 , doi:10.1016/j.jcp.2018.10.045 .JCTPAH 0021-9991 , - 2012, Seismic structure of the Southeast Australian lithosphere from surface and body wave tomography: Tectonophysics, 572,
111–122 , doi:10.1016/j.tecto.2011.11.016 .TCTOAM 0040-1951 , - 2008, Seismic ray tracing and wavefront tracking in laterally heterogeneous media: Advances in Geophysics, 49,
203–273 , doi:10.1016/S0065-2687(07)49003-3 .ADGOAR 0065-2687 , - 2004, Wave front evolution in strongly heterogeneous layered media using the fast marching method: Geophysical Journal International, 156,
631–647 , doi:10.1111/j.1365-246X.2004.02153.x .GJINEA 0956-540X , - 2020, Extreme theory of functional connections: A physics-informed neural network method for solving parametric differential equations: arXiv preprint, doi:
10.48550/arXiv.2005.10632 . , - 1996, A fast marching level set method for monotonically advancing fronts: Proceedings of the National Academy of Sciences, 93,
1591–1595 , doi:10.1073/pnas.93.4.1591 . , - 2012, LLNL-G3DV3: Global P wave tomography model for improved regional and teleseismic travel time prediction: Journal of Geophysical Research: Solid Earth, 117,
B10302 , doi:10.1029/2012JB009525 . , - 2021, Spiral: A multiresolution global tomography model of seismic wave speeds and radial anisotropy variations in the crust and mantle: Geophysical Journal International, 227,
1366–1391 , doi:10.1093/gji/ggab277 .GJINEA 0956-540X , - 2021, Eikonet: Solving the eikonal equation with deep neural networks: IEEE Transactions on Geoscience and Remote Sensing, 59,
10685–10696 , doi:10.1109/TGRS.2020.3039165 .IGRSD2 0196-2892 , - 2009, First-arrival traveltime tomography based on the adjoint-state method: Geophysics, 74, no. 6,
WCB1–WCB10 , doi:10.1190/1.3250266 .GPYSA7 0016-8033 , - 2023, A stable neural network-based eikonal tomography using hard-constrained measurements: Authorea. ,
- 2022, Upwind, no more: Flexible traveltime solutions using physics-informed neural networks: IEEE Transactions on Geoscience and Remote Sensing, 60,
1–12 , doi:10.1109/TGRS.2022.3218754 .IGRSD2 0196-2892 , - 2017, Slope tomography based on eikonal solvers and the adjoint-state method: Geophysical Journal International, 209,
1629–1647 , doi:10.1093/gji/ggx111 .GJINEA 0956-540X , - 1983, Earthquake locations and three-dimensional crustal structure in the Coyote Lake area, central California: Journal of Geophysical Research: Solid Earth, 88,
8226–8236 , doi:10.1029/JB088iB10p08226 . , - 2016, A fast marching algorithm for the factored eikonal equation: Journal of Computational Physics, 324,
210–225 , doi:10.1016/j.jcp.2016.08.012 .JCTPAH 0021-9991 , - 1987, A fast algorithm for two-point seismic ray tracing: Bulletin of the Seismological Society of America, 77,
972–986 , doi:10.1785/BSSA0770030972 .BSSAAP 0037-1106 , - 1988, Finite-difference calculation of travel times: Bulletin of the Seismological Society of America, 78,
2062–2076 .BSSAAP 0037-1106 , - 2009, An overview of full-waveform inversion in exploration geophysics: Geophysics, 74, no. 6,
WCC1–WCC26 , doi:10.1190/1.3238367 .GPYSA7 0016-8033 , - 2021a, PINNtomo: Seismic tomography using physics-informed neural networks: arXiv preprint, doi:
10.48550/arXiv.2104.01588 . , - 2021b, PINNeik: Eikonal solution using physics-informed neural networks: Computers & Geosciences, 155,
104833 , doi:10.1016/j.cageo.2021.104833 . , - 2022, Respecting causality is all you need for training physics-informed neural networks: arXiv preprint, doi:
10.48550/arXiv.2203.07404 . , - 2021, Detailed traveltime tomography and seismic catalogue around the 2019 Mw7.1 Ridgecrest, California, earthquake using dense rapid-response seismic data: Geophysical Journal International, 227,
204–227 , doi:10.1093/gji/ggab224 .GJINEA 0956-540X , - 2020, Pykonal: A Python package for solving the eikonal equation in spherical and Cartesian coordinates using the fast marching method: Seismological Research Letters, 91,
2378–2389 , doi:10.1785/0220190318 .SRLEEG 0895-0695 , - 2022, Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems: Computer Methods in Applied Mechanics and Engineering, 393,
114823 , doi:10.1016/j.cma.2022.114823 .CMMECC 0045-7825 , - 2006, 3D seismic refraction traveltime tomography at a groundwater contamination site: Geophysics, 71, no. 5,
H67–H78 , doi:10.1190/1.2258094 .GPYSA7 0016-8033 , - 1998, Three-dimensional seismic refraction tomography: A comparison of two methods applied to data from the faeroe basin: Journal of Geophysical Research: Solid Earth, 103,
7187–7210 , doi:10.1029/97JB03536 . , - 2003, Double-difference tomography: The method and its application to the Hayward fault, California: Bulletin of the Seismological Society of America, 93,
1875–1889 , doi:10.1785/0120020190 .BSSAAP 0037-1106 , - 1998, Nonlinear refraction traveltime tomography: Geophysics, 63,
1726–1737 , doi:10.1190/1.1444468 .GPYSA7 0016-8033 , - 2005, A fast sweeping method for eikonal equations: Mathematics of Computation, 74,
603–627 , doi:10.1090/S0025-5718-04-01678-3 .MCMPAF 0025-5718 ,