Abstract
An ensemble-based method for seismic inversion to estimate elastic attributes is considered, namely the iterative ensemble Kalman smoother. The main focus of this work is the challenge associated with ensemble-based inversion of seismic waveform data. The amount of seismic data is large and, depending on ensemble size, it cannot be processed in a single batch. Instead a solution strategy of partitioning the data recordings in time windows and processing these sequentially is suggested. This work demonstrates how this partitioning can be done adaptively, with a focus on reliable and efficient estimation. The adaptivity relies on an analysis of the update direction used in the iterative procedure, and an interpretation of contributions from prior and likelihood to this update. The idea is that these must balance; if the prior dominates, the estimation process is inefficient while the estimation is likely to overfit and diverge if data dominates. Two approaches to meet this balance are formulated and evaluated. One is based on an interpretation of eigenvalue distributions and how this enters and affects weighting of prior and likelihood contributions. The other is based on balancing the norm magnitude of prior and likelihood vector components in the update. Only the latter is found to sufficiently regularize the data window. Although no guarantees for avoiding ensemble divergence are provided in the paper, the results of the adaptive procedure indicate that robust estimation performance can be achieved for ensemble-based inversion of seismic waveform data.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Asch, M., Bocquet, M., Nodet, M.: Data Assimilation: Methods, Algorithms, and Applications. Society for Industrial and Applied Mathematics, Philadelphia (2016). https://doi.org/10.1137/1.9781611974546
Bishop, C.H., Etherton, B.J., Majumdar, S.J.: Adaptive sampling with the ensemble transform kalman filter. part i: Theoretical aspects. Mon. Weather Rev. 129(3), 420–436 (2001). https://doi.org/10.1175/1520-0493(2001)129<0420:ASWTET>2.0.CO;2
Bocquet, M., Sakov, P.: An iterative ensemble Kalman smoother. Q. J. Roy. Meteorol. Soc. 140(682), 1521–1535 (2014). https://doi.org/10.1002/qj.2236
Cardinali, C., Pezzulli, S., Andersson, E.: Influence-matrix diagnostic of a data assimilation system. Q. J. Roy. Meteorol. Soc. 130(603), 2767–2786 (2004). https://doi.org/10.1256/qj.03.205
Carrassi, A., Bocquet, M., Bertino, L., Evensen, G.: Data assimilation in the geosciences: an overview of methods, issues, and perspectives. WIREs Climate Change 9(5), e535 (2018). https://doi.org/10.1002/wcc.535
Emerick, A.A.: Deterministic ensemble smoother with multiple data assimilation as an alternative for history-matching seismic data. Computational Geosciences, https://doi.org/10.1007/s10596-018-9745-5(2018)
Emerick, A.A., Reynolds, A.C.: Ensemble smoother with multiple data assimilation. Comput. Geosci. 55, 3–15 (2013). https://doi.org/10.1016/j.cageo.2012.03.011
Evensen, G.: Data Assimilation. Springer, https://doi.org/10.1007/978-3-642-03711-5 (2009)
Evensen, G.: Analysis of iterative ensemble smoothers for solving inverse problems. Comput. Geosci. 22(3), 885–908 (2018). https://doi.org/10.1007/s10596-018-9731-y
Fillion, A., Bocquet, M., Gratton, S.: Quasi-static ensemble variational data assimilation: a theoretical and numerical study with the iterative ensemble kalman smoother. Nonlinear Process. Geophys. 25(2), 315–334 (2018). https://doi.org/10.5194/npg-25-315-2018
Fowler, A., van Leeuwen, P.J.: Measures of observation impact in non-gaussian data assimilation. Tellus A: Dynamic Meteorology and Oceanography 64(1), 17192 (2012). https://doi.org/10.3402/tellusa.v64i0.17192
Gebraad, L., Boehm, C., Fichtner, A.: Bayesian elastic full-waveform inversion using hamiltonian monte carlo. J. Geophys. Res. Solid Earth, 125(3). https://doi.org/10.1029/2019JB018428 (2020)
Gineste, M., Eidsvik, J., Zheng, Y.: Ensemble-based seismic inversion for a stratified medium. Geophysics 85(1), R29–R39 (2020). https://doi.org/10.1190/geo2019-0017.1
Hunt, B.R., Kostelich, E.J., Szunyogh, I.: Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter. Physica D: Nonlinear Phenomena 230(1-2), 112–126 (2007). https://doi.org/10.1016/j.physd.2006.11.008
Kennett, B.: ERZSOL3. http://www.spice-rtn.org/library/software/ERZSOL3.htmlhttp://www.spice-rtn.org/library/software/ERZSOL3.html, accessed on 2017-09-01 (2005)
Kennett, B: Seismic Wave Propagation in Stratified Media. ANU Press (2011)
Leeuwen, P.J.V., Cheng, Y., Reich, S.: Nonlinear Data Assimilation, Frontiers in Applied Dynamical Systems: Reviews and Tutorials, vol 2. Springer International Publishing, https://doi.org/10.1007/978-3-319-18347-3 (2015)
Liu, J., Kalnay, E., Miyoshi, T., Cardinali, C.: Analysis sensitivity calculation in an ensemble kalman filter. Q. J. Roy. Meteorol. Soc. 135(644), 1842–1851 (2009). https://doi.org/10.1002/qj.511
Mannseth, T., Fossum, K.: Assimilating spatially dense data for subsurface applications–balancing information and degrees of freedom. Comput. Geosci. 22(5), 1323–1349 (2018). https://doi.org/10.1007/s10596-018-9755-3
Muir, J.B., Tsai, V.C.: Geometric and level set tomography using ensemble Kalman inversion. Geophys. J. Int. 220(2), 967–980 (2020). https://doi.org/10.1093/gji/ggz472
Raanes, P.N., Stordal, A.S., Evensen, G.: Revising the stochastic iterative ensemble smoother. Nonlinear Process. Geophys. 26(3), 325–338 (2019). https://doi.org/10.5194/npg-26-325-2019
Rafiee, J., Reynolds, A.C.: Theoretical and efficient practical procedures for the generation of inflation factors for ES-MDA. Inverse Problems 33(11), 115003 (2017). https://doi.org/10.1088/1361-6420/aa8cb2
Rodgers, CD: Inverse Methods for Atmospheric Sounding. World Scientific Publishing, https://doi.org/10.1142/3171 (2000)
Sacher, W., Bartello, P.: Sampling errors in ensemble kalman filtering. part i: Theory. Mon. Weather. Rev. 136(8), 3035–3049 (2008). https://doi.org/10.1175/2007MWR2323.1
Sakov, P., Oliver, D.S., Bertino, L.: An iterative enKF for strongly nonlinear systems. Mon. Weather. Rev. 140(6), 1988–2004 (2012). https://doi.org/10.1175/mwr-d-11-00176.1
Sheriff, R.E., Geldart, L.P.: Exploration Seismology, 2nd edn. Cambridge University Press, https://doi.org/10.1017/CBO9781139168359 (1995)
Shirangi, M.G., Emerick, A.A.: An improved tsvd-based levenberg–marquardt algorithm for history matching and comparison with gauss–newton. J. Pet. Sci. Eng. 143, 258–271 (2016). https://doi.org/10.1016/j.petrol.2016.02.026
Thurin, J., Brossier, R., Métivier, L.: Ensemble-based uncertainty estimation in full waveform inversion. Geophys. J. Int. 219(3), 1613–1635 (2019). https://doi.org/10.1093/gji/ggz384
Zupanski, D., Hou, A.Y., Zhang, S.Q., Zupanski, M., Kummerow, C.D., Cheung, S.H.: Applications of information theory in ensemble data assimilation. Q. J. Roy. Meteorol. Soc. 133(627), 1533–1545 (2007). https://doi.org/10.1002/qj.123
Acknowledgements
We thank the Norwegian Research Council and partners of the Uncertainty in Reservoir Evaluation (URE) project and the GAMES consortium at the Norwegian University of Science and Technology (NTNU) for the financial support (grant no. 294404). We further thank BP for data. We also thank the three reviewers for constructive comments that improved the paper.
Funding
Open access funding provided by NTNU Norwegian University of Science and Technology (incl St. Olavs Hospital - Trondheim University Hospital).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Gineste, M., Eidsvik, J. Batch seismic inversion using the iterative ensemble Kalman smoother. Comput Geosci 25, 1105–1121 (2021). https://doi.org/10.1007/s10596-021-10043-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-021-10043-4