Project C2 • Seismic imaging by full waveform inversion

Principal investigators

Project summary

Seismic imaging aims to reconstruct subsurface material parameters (e.g., the pressure wave velocity $v_p$, see figure) from measurements of seismic waves. Mathematically, this translates into a nonlinear inverse problem for parameter identification which is modelled by proper wave equations. A state-of-the-art technique that renders high-resolution images by fitting the full content of measured and computed seismograms is full waveform inversion (FWI). It comes along with several mathematical challenges and demands:

• Due to illposedness, the underlying inverse problem problem needs to be regularized for which mostly gradient-type schemes are employed. At each iteration step, time-dependent PDEs need to be solved. In combination with the slow convergence of gradient-type schemes, the practicality of FWI is restricted by the numerical effort needed, for instance, by explicit time integrators or by the storage requirements for the full wave propagation in space and time.
• The parameter-to-state map, which maps a set of material parameters to a specific wave field and reflects the physical model behind, is highly nonlinear. Therefore, the numerical success of FWI generally relies on good initial guesses for the iterative inversion scheme. Further, phase-dependent mismatches in the seismograms tend to introduce local minima in the corresponding optimization process and can thus additionally spoil the reconstructions (cycle-skipping).

• Multiparameter reconstructions suffer from unbalanced sensitivity dependencies of the wave field data. These enter as cross-talk or other coupling phenomena within the reconstructions of FWI and lead to poor results or severe artifacts for some of the material parameters. To overcome this barrier, preconditioned Newton optimization or other higher-order information need to be incorporated into first order inversion schemes.
• A solid convergence analysis justifying the numerics of FWI for realistic physical models is hard to achieve. Although the wave equations under consideration can be cast into a well-established abstract evolution framework as elaborated during the first funding period, the complex nature of proper function spaces (e.g. $L^\infty$ for the material parameters) and the lack of convenient structural assumptions on the parameter-to-state map (e.g. the tangential cone condition) hinder standard theory to be applicable.

In this project, we develop theoretical foundations and customized numerical solutions addressing the issues from above. Our analytical achievements include:

• Novel theory on nonlinear inverse problems in the context of abstract evolution equations, in particular proofs for the well-posedness of the physical wave models mentioned above under certain initial and boundary conditions, see [KR19, KR16].
• Mathematical verification for the existence of first- and second-order Fréchet derivatives of the parameter-to-state map between appropriate spaces, see [KR19, KR16].
• A new formulation of the viscoacoustic and viscoelastic wave equations which allows the adjoint equations to be evaluated with the same numerical solver as the direct ones, see [Zel18].
• An all-at-once version of time-domain FWI in the viscoelastic regime, see [Rie21].
• A weak$^\ast$-convergence analysis for parameter reconstructions in $L^\infty$ applicable to the abstract evolution equations framework, see [PR21].
• A verification of the tangential cone condition for the semi-discrete FWI setting in the acoustic, elastic and viscoelastic regime, see [ER21, EGR23, EGR22].

Concerning the computational aspects of FWI, we construct implicit hierarchical space-time discretizations leading to efficient parallel schemes. Up to now, we achieved the following in this field:

• Parallel implementation of upwind discontinuous Galerkin discretizations in space using the framework described in [DFW19] for the linear viscoacoustic and viscoelastic equation as considered in [Zel18]. To treat the evolution in time, we employ time stepping schemes as well as space-time discretizations in cooperation with project A3. All the solvers are realized using the M++ library. Finite difference parallel implementation of viscoacoustic and viscoelastic equations described in [Zel18].

  

• Parallel implementation of First Order System Least-Squares (FOSLS) type space-time discretizations for acoustic waves including a variant of a discontinuous Petrov-Galerkin method in cooperation with A3, see [Ern17, EW19].
• Implementation of CG-REGINN, see [BFE21].
• Implementation and testing of a second-order preconditioned truncated Newton method which is capable of mitigating cross-talk and parameter-trade-off in multiparameter viscoelastic FWI in the near surface, see [GPR21].
• Implementation of an all-at-once formulation with the potential to mitigate cycle-skipping effects.
• Realization of numerical experiments using real field data which we acquired during several geophysical expeditions, see [GPB20, GPR22, Ath20].

Finally, we have come closer towards the goal to incorporate dispersion, attenuation and anisotropy in a 3D setting, thus reducing the gap between mathematical idealization and practical needs by extending the model step by step to a realistic physical scenario, see [PGB19, PSS18]. Our vision is to realize highly reliable reconstruction algorithms complying with industrial standards. To this end, we combine up-to-date techniques for inverse problems with massively parallel explicit and implicit direct solvers for the coarse level of adaptive hierarchy space-time discretizations which allow for locally very exact resolutions and provide efficient coarse approximation for preconditioning. This is done in collaboration with A3.

To find out more about FWI, checkout our playground software for educational purposes to experiment with acoustic waves in inhomogeneous media, see PyFWI. Within the program, the user can observe a simple FWI-algorithm at work.

Publications

1. M. Eller, R. Griesmaier, and A. Rieder. Tangential cone condition for the full waveform forward operator in the viscoelastic regime: the non-local case. SIAM J. Appl. Math., 84(2):412–432, April 2024. URL https://doi.org/10.1137/23M1551845. [preprint] [bibtex]

2. L. Gao, Y. Pan, A. Rieder, T. Bohlen, and W. Mao. Multiparameter 2-D viscoelastic full-waveform inversion of Rayleigh waves: a field experiment at Krauthausen test site. Geophys. J. Int., 234(1):297–312, July 2023. URL https://doi.org/10.1093/gji/ggad072. [preprint] [bibtex]

3. L. Pieronek and A. Rieder. On the iterative regularization of non-linear illposed problems in $L^{\infty}$. Numer. Math., 154:209–247, June 2023. URL https://doi.org/10.1007/s00211-022-01359-7. [preprint] [files] [bibtex]

4. L. Gao, Y. Pan, A. Rieder, and T. Bohlen. Multiparameter viscoelastic full-waveform inversion of shallow-seismic surface waves with a pre-conditioned truncated Newton method. Geophys. J. Int., 227(3):2044–2057, December 2021. URL https://doi.org/10.1093/gji/ggab311. [preprint] [bibtex]

5. T. Bohlen, M. R. Fernandez, J. Ernesti, C. Rheinbay, A. Rieder, and C. Wieners. Visco-acoustic full waveform inversion: From a DG forward solver to a Newton-CG inverse solver. Comput. Math. Appl., 100:126–140, October 2021. URL https://doi.org/10.1016/j.camwa.2021.09.001. [preprint] [bibtex]

6. M. Eller and A. Rieder. Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime. Inverse Problems, 37(8):085011, 17, July 2021. URL https://doi.org/10.1088/1361-6420/ac11c5. [preprint] [bibtex]

7. A. Rieder. An all-at-once approach to full waveform seismic inversion in the viscoelastic regime. Math. Methods Appl. Sci., 44(8):6377–6388, May 2021. URL https://doi.org/10.1002/mma.7190. [preprint] [bibtex]

8. Y. Pan and L. Gao. Random objective waveform inversion of surface waves. Geophysics, 85(4):1JA–Z18, July 2020. URL https://doi.org/10.1190/GEO2019-0613.1. [bibtex]

9. L. Gao, Y. Pan, and T. Bohlen. 2D multiparameter viscoelastic shallow-seismic full-waveform inversion: reconstruction tests and first field-data application. Geophys. J. Int., 222(1):560–571, July 2020. URL https://doi.org/10.1093/gji/ggaa198. [preprint] [bibtex]

10. A. Kirsch and A. Rieder. Inverse problems for abstract evolution equations II: Higher order differentiability for viscoelasticity. SIAM J. Appl. Math., 79(6):2639–2662, December 2019. URL https://doi.org/10.1137/19M1269403. See corrigendum. [preprint] [bibtex]

11. Y. Pan, L. Gao, and T. Bohlen. High-resolution characterization of near-surface structures by surface-wave inversions: From dispersion curve to full waveform. Surv. Geophys., 40(2):167–195, March 2019. URL https://doi.org/10.1007/s10712-019-09508-0. [preprint] [bibtex]

12. N. Thiel, T. Hertweck, and T. Bohlen. Comparison of acoustic and elastic full-waveform inversion of 2D towed-streamer data in the presence of salt. Geophys. Prospect., 67(2):349–361, February 2019. URL https://doi.org/10.1111/1365-2478.12728. [bibtex]

13. F. Wittkamp, N. Athanasopoulos, and T. Bohlen. Individual and joint 2D elastic full-waveform inversion of Rayleigh and Love waves. Geophys. J. Int., 216(1):350–364, January 2019. URL https://doi.org/10.1093/gji/ggy432. [bibtex]

14. Y. Pan, S. Schaneng, T. Seinweg, and T. Bohlen. Estimating S-wave velocities from 3D 9-component shallow seismic data using local Rayleigh-wave dispersion curves – A field study. J. Appl. Geophys., 159:532–539, December 2018. URL https://doi.org/10.1016/j.jappgeo.2018.09.037. [bibtex]

15. A. Kirsch and A. Rieder. Inverse problems for abstract evolution equations with applications in electrodynamics and elasticity. Inverse Problems, 32(8):085001, 24, June 2016. URL https://doi.org/10.1088/0266-5611/32/8/085001. [preprint] [bibtex]

16. T. Bohlen and F. Wittkamp. Three-dimensional viscoelastic time-domain finite-difference seismic modelling using the staggered Adams–Bashforth time integrator. Geophys. J. Int., 204(3):1781–1788, March 2016. URL https://doi.org/10.1093/gji/ggv546. [bibtex]

Preprints

17. A. Rieder. On the mathematical foundation of full waveform inversion in viscoelastic vertically transverse isotropic media. CRC 1173 Preprint 2024/26, Karlsruhe Institute of Technology, December 2024. [bibtex]

18. F. Margotti, M. Pauleti, and A. Rieder. Inexact Newton regularizations with uniformly convex stability terms: a unified convergence analysis. CRC 1173 Preprint 2023/12, Karlsruhe Institute of Technology, April 2023. [bibtex]

19. M. Eller, R. Griesmaier, and A. Rieder. Tangential cone condition for the full waveform forward operator in the elastic regime: the non-local case. CRC 1173 Preprint 2022/48, Karlsruhe Institute of Technology, September 2022. Revised version from October 2022. [bibtex]

Theses

20. C. Rheinbay. All-at-once and reduced solvers for visco-acoustic full waveform inversion. PhD thesis, Karlsruhe Institute of Technology (KIT), November 2023. [bibtex]

21. N. Athanasopoulos. Challenges in near-surface seismic full-waveform inversion of field data. PhD thesis, Karlsruhe Institute of Technology (KIT), November 2020. [bibtex]

22. U. C. Zeltmann. The viscoelastic seismic model: existence, uniqueness and differentiability with respect to parameters. PhD thesis, Karlsruhe Institute of Technology (KIT), December 2018. [bibtex]

23. J. Ernesti. Space-time methods for acoustic waves with applications to full waveform inversion. PhD thesis, Karlsruhe Institute of Technology (KIT), December 2017. [bibtex]

Other references

24. J. Ernesti and C. Wieners. A space-time discontinuousa Petrov–Galerkin method for acoustic waves. In U. Langer and O. Steinbach, editors, Space-Time Methods: Applications to Partial Differential Equations, volume 25 of Radon Series on Computational and Applied Mathematics, chapter 3, pages 89–116. De Gruyter, Berlin/Boston, September 2019. [preprint] [bibtex]

25. W. Dörfler, S. Findeisen, C. Wieners, and D. Ziegler. Parallel adaptive discontinuous Galerkin discretizations in space and time for linear elastic and acoustic waves. In U. Langer and O. Steinbach, editors, Space-Time Methods: Applications to Partial Differential Equations, volume 25 of Radon Series on Computational and Applied Mathematics, chapter 2, pages 61–88. De Gruyter, Berlin/Boston, September 2019. [preprint] [bibtex]