3D Refraction Travel Time Accuracy Study in High Contrasted Media

Paulo H. B. Alves, Luiz A. Santos, Felipe V. Capuzzo, Marco Cetale

Abstract


Among all the existing methods to solve the eikonal equation, three methods are chosen to verify accuracy, symmetry, reciprocity and error propagation along large offsets of refracted waves in seismic near surface exploration context. Performance is extremely highlighted nowadays and accuracy is being neglected, then an eikonal solver poorly explored in geoscience is used. A classical solver, the Fast Iterative and the modified Fast Sweeping Method are applied in three modeling schemes: a simple two layers model, a large four layers and a complex benchmark model. The three methods compute the first arrival of refracted waves in high contrast media and the results are compared to the analytical solution. A circular geometry is considered in all experiments to explore the method applicability using full azimuth angles. On the first scheme, the errors in travel time are computed among the three methods using different model sample spacing and we discuss accuracy, symmetry and reciprocity of first arrivals. On the second scheme, three circular receivers are placed in different offsets to check errors along refracted wave propagation. Finally, the third scheme, four shots are strategically positioned over the SEG/EAGE Overthrust model in order to compare the full acoustic wavefield with the eikonal solvers and then check the similarities. Although the focus is on methods accuracy, the algorithm run time is also considered and the comparison shows that the modified Fast Sweeping Method is the most accurate. The most computational efficient eikonal solver is the Fast Iterative Method, but its geoscience applicability needs to be cautious, because of its inaccurate results.

Keywords


eikonal solvers; refracted waves; numerical - analytical comparison; accurate first arrivals.

Full Text:

PDF

References


Ahmed, S., S. Bak, J. McLaughlin, and D. Renzi, 2011, A third order accurate fast marching method for the eikonal equation in two dimensions: SIAM Journal on Scientific Computing, 33, 2402–2420, doi: https://doi.org/10.1137/10080258X.

Alkhalifah, T., and S. Fomel, 2001, Implementing the fast marching eikonal solver: spherical versus cartesian coordinates: Geophysical Prospecting, 49, 165–178, doi: https://doi.org/10.1046/j.1365-2478.2001.00245.x.

Bak, S., J. McLaughlin, and D. Renzi, 2010, Some improvements for the fast sweeping method: SIAM Journal on Scientific Computing, 32, 2853–2874, doi: https://doi.org/10.1137/090749645.

Balkaya, Ç., Z. Akç??, and G. Göktürkler, 2010, A comparison of two travel-time tomography schemes for crosshole radar data: Eikonal-equation-based inversion versus ray-based inversion: Journal of Environmental & Engineering Geophysics, 15, 203–218, doi: https://doi.org/10.2113/JEEG15.4.203.

Bording, R. P., 2004, Finite difference modeling - nearly optimal sponge boundary conditions: Presented at the 2004 SEG Annual Meeting, SEG Technical Program Expanded Abstracts. Denver, Colorado. doi: https://doi.org/10.1190/1.1845189.

Bulhões, F. C., M. A. C. Santos, L. A. Santos, and V. T. X. de Almeida, 2021, Regularization effects on 2D seismic refraction tomography: case study on shallow marine environment: Presented at the 17th International Congress of the Brazilian Geophysical Society, SBGf. Rio de Janeiro, Brazil.

Cai, W., P. Zhu, and G. Li, 2023, Improved fast iterative method for higher calculation accuracy of traveltimes: Computers & Geosciences, 174, 105331, doi: https://doi.org/10.1016/j.cageo.2023.105331.

Capozzoli, A., C. Curcio, A. Liseno, and S. Savarese, 2013, A comparison of fast marching, fast sweeping and fast iterative methods for the solution of the eikonal equation: 2013 21st Telecommunications Forum Telfor (TELFOR), IEEE, Belgrade, Serbia, 685–688. doi: https://doi.org/10.1109/TELFOR.2013.6716321.

Cerjan, C., D. Kosloff, R. Kosloff, and M. Reshef, 1985, A nonreflecting boundary condition for discrete acoustic and elastic wave equations: Geophysics, 50, 705–708, doi: https://doi.org/10.1190/1.1441945.

Cerveny`, V., 2001, Seismic ray theory: Cambridge University Press, Cambridge, doi: https://doi.org/10.1017/CBO9780511529399.

Costa, F., F. Capuzzo, A. de Souza Jr, R. Moreira, J. Lopez, and M. Cetale, 2020, Understanding refracted wave paths for Brazilian pre-salt target-oriented imaging, in SEG Technical Program Expanded Abstracts 2020. Virtual event: Society of Exploration Geophysicists, 2375–2380. doi: https://doi.org/10.1190/segam2020- 3426868.1.

Da Silva, S., A. Karsou, R. Moreira, J. Lopez, and M. Cetale, 2022, Klein-Gordon equation and variable density effects on acoustic wave propagation in Brazilian pre-salt fields: 83rd EAGE Annual Conference & Exhibition, European Association of Geoscientists & Engineers, Madrid, 1–5. doi: https://doi.org/10.3997/2214- 4609.202210385.

Dang, F., and N. Emad, 2014, Fast iterative method in solving eikonal equations: a multi-level parallel approach: Procedia Computer Science, 29, 1859–1869, doi: https://doi.org/10.1016/j.procs.2014.05.170.

De Matteis, R., A. Romeo, G. Pasquale, G. Iannaccone, and A. Zollo, 2010, 3D tomographic imaging of the southern apennines (Italy): A statistical approach to estimate the model uncertainty and resolution: Studia Geophysica et Geodaetica, 54, 367–387, doi: https://doi.org/10.1007/s11200-010-0022-x.

Detrixhe, M., F. Gibou, and C. Min, 2013, A parallel fast sweeping method for the eikonal equation: Journal of Computational Physics, 237, 46–55, doi: https://doi.org/10.1016/j.jcp.2012.11.042.

Farber, R., 2016, Parallel programming with openacc: Newnes, doi: https://doi.org/10.1016/B978-0-12-410397- 9.00001-9.

Farra, V., 1993, Ray tracing in complex media: Journal of Applied Geophysics, 30, 55–73, doi: https://doi.org/10.1016/0926-9851(93)90018-T.

Herrmann, M., 2003, A domain decomposition parallelization of the fast marching method: Technical Report: Presented at the Defense Technical Information Center.

Hicks, G. J., 2002, Arbitrary source and receiver positioning in finite-difference schemes using Kaiser windowed sinc functions: Geophysics, 67, 156–165, doi: https://doi.org/10.1190/1.1451454.

Hole, J., and B. Zelt, 1995, 3-D finite-difference reflection traveltimes: Geophysical Journal International, 121, 427–434, doi: https://doi.org/10.1111/j.1365-246X.1995.tb05723.x.

Hong, S., and W.-K. Jeong, 2016, A multi-gpu fast iterative method for eikonal equations us- ing on-the-fly adaptive domain decomposition: Procedia Computer Science, 80, 190–200, doi: https://doi.org/10.1016/j.procs.2016.05.309.

Huang, G., and S. Luo, 2020, Hybrid fast sweeping methods for anisotropic eikonal equation in two-dimensional tilted transversely isotropic media: Journal of Scientific Computing, 84, 1–30, doi: https://doi.org/10.1007/s10915-020-01280-3.

Huang, Y., 2021, Improved fast iterative algorithm for eikonal equation for GPU computing (version 3): arXiv, doi: https://doi.org/10.48550/ARXIV.2106.15869.

Jeong, W.-K., and R. T. Whitaker, 2008, A fast iterative method for eikonal equations: SIAM Journal on Scientific Computing, 30, 2512–2534, doi: https://doi.org/10.1137/060670298.

Kearey, P., M. Brooks, and I. Hill, 2002, An introduction to geophysical exploration, 4: John Wiley & Sons. Koketsu, K., 2000, Finite difference traveltime calculation for head waves travelling along an irregular interface:

Geophysical Journal International, 143, 729–734, doi: https://doi.org/10.1046/j.1365-246X.2000.00269.x. Lecomte, J., E. Campbell, and J. Letouzey, 1994, Building the SEG/EAGE overthrust velocity macro model:

EAGE/SEG Summer Workshop-Construction of 3-D Macro Velocity-Depth Models, European Association of Geoscientists & Engineers, cp–96. doi: https://doi.org/10.3997/2214-4609.201407587.

Linde, N., A. Tryggvason, J. E. Peterson, and S. S. Hubbard, 2008, Joint inversion of crosshole radar and seismic traveltimes acquired at the south oyster bacterial transport site: Geophysics, 73, G29–G37, doi: https://doi.org/10.1190/1.2937467.

Lopez, J., F. Neto, M. Cabrera, S. Cooke, S. Grandi, and D. Roehl, 2020, Refraction seismic for pre-salt reservoir characterization and monitoring: Presented at the SEG International Exposition and Annual Meeting. Virtual event, SEG Technical Program Expanded Abstracts. doi: https://doi.org/10.1190/segam2020-3426667.1.

Luo, S., and J. Qian, 2012, Fast sweeping methods for factored anisotropic eikonal equations: multiplicative and additive factors: Journal of Scientific Computing, 52, 360–382, doi: https://doi.org/10.1007/s10915-011-9550-y.

Noble, M., A. Gesret, and N. Belayouni, 2014, Accurate 3-D finite difference computation of trav- eltimes in strongly heterogeneous media: Geophysical Journal International, 199, 1572–1585, doi: https://doi.org/10.1093/gji/ggu358.

Podvin, P., and I. Lecomte, 1991, Finite difference computation of traveltimes in very contrasted velocity models: a massively parallel approach and its associated tools: Geophysical Journal International, 105, 271–284, doi: https://doi.org/10.1111/j.1365-246X.1991.tb03461.x.

Qin, F., Y. Luo, K. B. Olsen, W. Cai, and G. T. Schuster, 1992, Finite-difference solution of the eikonal equation along expanding wavefronts: Geophysics, 57, 478–487, doi: https://doi.org/10.1190/1.1443263.

Rawlinson, N., and M. Sambridge, 2004, Wave front evolution in strongly heterogeneous layered media using the fast marching method: Geophysical Journal International, 156, 631–647, doi: https://doi.org/10.1111/j.1365- 246X.2004.02153.x.

Rawlinson, N., and M. Sambridge, 2005, The fast marching method: an effective tool for tomographic imag- ing and tracking multiple phases in complex layered media: Exploration Geophysics, 36, 341–350, doi: https://doi.org/10.1071/EG05341.

Robinson, E. A., and D. Clark, 2017, Basic geophysics: Society of Exploration Geophysicists. Geophysical Monograph Series, 22, 376 pp, doi: https://doi.org/10.1190/1.9781560803461.

Sethian, J. A., 1999, Fast marching methods: SIAM review, 41, 199–235, doi: https://doi.org/10.1137/S0036144598347059.

Shen, Y., and J. Zhang, 2020, Refraction wavefield migration: Geophysics, 85, Q27–Q37, doi: https://doi.org/10.1190/geo2020-0141.1.

Sheriff, R. E., and L. P. Geldart, 1995, Exploration seismology: Cambridge University Press, doi: https://doi.org/10.1017/CBO9781139168359.

Tryggvason, A., and B. Bergman, 2006, A traveltime reciprocity discrepancy in the Podvin & Lecomte time3d finite difference algorithm: Geophysical Journal International, 165, 432–435, doi: https://doi.org/10.1111/j.1365- 246X.2006.02925.x.

Van Trier, J., and W. W. Symes, 1991, Upwind finite-difference calculation of traveltimes: Geophysics, 56, 812–821, doi: https://doi.org/10.1190/1.1443099.

Vidale, J., 1988, Finite-difference calculation of travel times: Bulletin of the Seismological Society of America, 78, 2062–2076, doi: 10.1785/BSSA0780062062.

Waheed, U. B., and T. Alkhalifah, 2017, A fast sweeping algorithm for accurate solution of the tilted transversely isotropic eikonal equation using factorization: Geophysics, 82, WB1–WB8, doi: https://doi.org/10.1190/geo2016-0712.1.

Waheed, U. B., C. E. Yarman, and G. Flagg, 2015, An iterative, fast-sweeping-based eikonal solver for 3D tilted anisotropic media: Geophysics, 80, C49–C58, doi: https://doi.org/10.1190/geo2014-0375.1.

White, M. C., H. Fang, N. Nakata, and Y. Ben-Zion, 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: https://doi.org/10.1785/0220190318.

Yang, J., and F. Stern, 2017, A highly scalable massively parallel fast marching method for the eikonal equation: Journal of Computational Physics, 332, 333–362, doi: https://doi.org/10.1016/j.jcp.2016.12.012.

Yordkayhun, S., A. Tryggvason, B. Norden, C. Juhlin, and B. Bergman, 2009, 3D seismic traveltime tomography imaging of the shallow subsurface at the CO2SINK project site, Ketzin, Germany: Geophysics, 74, G1–G15, doi: https://doi.org/10.1190/1.3026553.

Zhao, H., 2005, A fast sweeping method for eikonal equations: Mathematics of Computation, 74, 603–627, doi: https://doi.org/10.1090/S0025-5718-04-01678-3.

Zhao, H., 2007, Parallel implementations of the fast sweeping method: Institute of Computational Mathematics and Scientific / Engineering Computing, 25, 4, 421–429, doi: https://www.jstor.org/stable/43693378.




DOI: http://dx.doi.org/10.22564/brjg.v41i1.2295

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.







>> Brazilian Journal of Geophysics - BrJG (online version): ISSN 2764-8044
a partir do v.37n.4 (2019) até o presente

Revista Brasileira de Geofísica - RBGf (online version): ISSN 1809-4511
v.15n.1 (1997) até v.37n.3 (2019)

Revista Brasileira de Geofísica - RBGf (printed version): ISSN 0102-261X
v.1n.1 (1982) até v.33n.1 (2015)

 

Brazilian Journal of Geophysics - BrJG
Sociedade Brasileira de Geofísica - SBGf
Av. Rio Branco 156 sala 2509
Rio de Janeiro, RJ, Brazil
Phone/Fax: +55 21 2533-0064
E-mail: editor@sbgf.org.br

 

Creative Commons