# An elastic full-waveform inversion based on wave-mode separation

Yingming Qu^{1}

^{4}Jinli Li

^{2}

^{3}Zhenchun Li

^{1}Jianping Huang

^{1}

^{1} School of Geosciences, China University of Petroleum, Qingdao 266580, China.

^{2} Institute of Geophysical and Geochemical Exploration, Chinese Academy of Geological Sciences, Langfang 065000, China.

^{3} The National Center for Geological Exploration Technology, Langfang 065000, China.

^{4} Corresponding author. Email: qu_geophysics@yahoo.com

*Exploration Geophysics* - https://doi.org/10.1071/EG16158

Submitted: 19 December 2016 Accepted: 18 July 2017 Published online: 15 September 2017

## Abstract

Multi-parameter elastic full-waveform inversion (EFWI) attempts to find high resolution model parameters that are able to match observed data exactly by minimising residuals between the observed and predicted data. However, the coupling of V* _{p}* and V

*, and the cross-talk artefacts between P- and S-wave modes increase non-uniqueness and ill-conditionedness. We propose a new EFWI method based on P- and S-wave mode separation to mitigate these problems. In this method, we derive the gradient formulas with respect to various wave modes using a P- and S-wave mode separated first-order velocity-stress wave equation, and use a step search method in subspace to calculate the corresponding step lengths. The algorithm, called wave-mode separation EFWI (SEFWI), appears to be helpful to weaken non-uniqueness and ill-conditionedness of conventional EFWI by decoupling multiple parameters. Numerical examples conducted with a synthetic dataset modelled on a simple model with anomalies reveal that SEFWI can reduce the cross-talk artefacts between P- and S-wave modes. Synthetic tests on the Marmousi2 model demonstrate that SEFWI yields better inversion results than conventional EFWI. Although the computational cost of SEFWI per iteration is 1.81 times as much as that of EFWI, the total computational cost is almost at the same level, because of its faster convergence rate.*

_{s}**Key words:** elastic full-waveform inversion, encoding multi-source, multi-scale decomposition, P- and S-wave mode separation, step search method in subspace, time-domain.

## References

Aki, K., and Richards, P., 2002,*Quantitative seismology*(2nd edition): University Science Books.

Boonyasiriwat, C., Valasek, P., Routh, P., Cao, W., Schuster, G., and Macy, B., 2009, An efficient multiscale method for time-domain waveform tomography:

*Geophysics*,

**74**, WCC59–WCC68

| An efficient multiscale method for time-domain waveform tomography:CrossRef |

Brossier, R., Operto, S., and Virieux, J., 2009, Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion:

*Geophysics*,

**74**, WCC105–WCC118

| Seismic imaging of complex onshore structures by 2D elastic frequency-domain full-waveform inversion:CrossRef |

Brossier, R., Operto, S., and Virieux, J., 2010, Which data residual norm for robust elastic frequency-domain full waveform inversion:

*Geophysics*,

**75**, R37–R46

| Which data residual norm for robust elastic frequency-domain full waveform inversion:CrossRef |

Bunks, C., Saleck, F., Zaleski, S., and Chavent, G., 1995, Multiscale seismic waveform inversion:

*Geophysics*,

**60**, 1457–1473

| Multiscale seismic waveform inversion:CrossRef |

Cheong, S., Shin, C., Pyun, S., Min, D. J., and Suh, S., 2004, Efficient calculation of steepest descent direction for source-independent waveform inversion using normalized wavefield by convolution: SEG Technical Program, Expanded Abstracts, 1842–1845.

Choi, Y., and Alkhalifah, T., 2011, Source-independent time-domain waveform inversion using convolved wavefields: Application to the encoded multisource waveform inversion:

*Geophysics*,

**76**, R125–R134

| Source-independent time-domain waveform inversion using convolved wavefields: Application to the encoded multisource waveform inversion:CrossRef |

Choi, Y., Shin, C., Min, D. J., and Ha, T., 2005, Efficient calculation of the steepest descent direction for source-independent seismic waveform inversion: An amplitude approach:

*Journal of Computational Physics*,

**208**, 455–468

| Efficient calculation of the steepest descent direction for source-independent seismic waveform inversion: An amplitude approach:CrossRef |

Chung, W., Shin, C., and Pyun, S., 2010, 2D elastic waveform inversion in the Laplace domain:

*Bulletin of the Seismological Society of America*,

**100**, 3239–3249

| 2D elastic waveform inversion in the Laplace domain:CrossRef |

Drossaert, H. F., and Giannopoulos, A., 2007, A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves:

*Geophysics*,

**72**, T9–T17

| A nonsplit complex frequency-shifted PML based on recursive integration for FDTD modeling of elastic waves:CrossRef |

Gélis, C., Virieux, J., and Grandjean, G., 2007, Two-dimensional elastic full waveform inversion using Born and Rytov formulations in the frequency domain:

*Geophysical Journal International*,

**168**, 605–633

| Two-dimensional elastic full waveform inversion using Born and Rytov formulations in the frequency domain:CrossRef |

Guitton, A., Ayeni, G., and Diaz, E., 2012, Constrained full-waveform inversion by model reparameterization:

*Geophysics*,

**77**, R117–R127

| Constrained full-waveform inversion by model reparameterization:CrossRef |

Hu, W., 2014, FWI without low frequency data-beat tone inversion: SEG Technical Program, Expanded Abstracts, 1116–1120.

Kamei, R., and Pratt, R. G., 2013, Inversion strategies for visco-acoustic waveform inversion:

*Geophysical Journal International*,

**194**, 859–884

| Inversion strategies for visco-acoustic waveform inversion:CrossRef |

Kang, S. G., Bae, H. S., and Shin, C., 2012, Laplace–Fourier-domain waveform inversion for fluid–solid media:

*Pure and Applied Geophysics*,

**169**, 2165–2179

| Laplace–Fourier-domain waveform inversion for fluid–solid media:CrossRef |

Kennett, B. L. N., Sambridge, M. S., and Williamson, P. R., 1988, Subspace methods for large inverse problems with multiple parameter classes:

*Geophysical Journal International*,

**94**, 237–247

| Subspace methods for large inverse problems with multiple parameter classes:CrossRef |

Köhn, D., De Nil, D., Kurzmann, A., Przebindowska, A., and Bohlen, T., 2012, On the influence of model parametrization in elastic full waveform tomography:

*Geophysical Journal International*,

**191**, 325–345

| On the influence of model parametrization in elastic full waveform tomography:CrossRef |

Krebs, J. R., Anderson, J. E., Hinkley, D., Neelamani, R., Lee, S., Baumstein, A., and Lacasse, M. D., 2009, Fast full-wavefield seismic inversion using encoded sources:

*Geophysics*,

**74**, WCC177–WCC188

| Fast full-wavefield seismic inversion using encoded sources:CrossRef |

Lee, K. H., and Kim, H. J., 2003, Source-independent full-waveform inversion of seismic data:

*Geophysics*,

**68**, 2010–2015

| Source-independent full-waveform inversion of seismic data:CrossRef |

Li, Z., Zhang, H., Liu, Q., and Han, W., 2007, Numeric simulation of elastic wavefield separation by staggering grid high-order finite-difference algorithm:

*Oil Geophysical Prospecting*,

**42**, 510–515

| 1:CAS:528:DC%2BD1cXitFKjtrg%3D&md5=a3e7c91ad12241b5a78245d472a9ba49CAS |

| 1:CAS:528:DC%2BD1cXitFKjtrg%3D&md5=a3e7c91ad12241b5a78245d472a9ba49CAS |

Liu, Y., Symes, W. W., and Li, Z., 2013, Multisource least-squares extended reverse-time migration with preconditioning guided gradient method: SEG Technical Program, Expanded Abstracts, 3709–3715.

Liu, Y., Yang, J., Chi, B., and Dong, L., 2015, An improved scattering-integral approach for frequency-domain full waveform inversion:

*Geophysical Journal International*,

**202**, 1827–1842

| An improved scattering-integral approach for frequency-domain full waveform inversion:CrossRef |

Martin, G., Wiley, R., and Marfurt, K., 2006, Marmousi2: an elastic upgrade for Marmousi:

*The Leading Edge*,

**25**, 156–166

| Marmousi2: an elastic upgrade for Marmousi:CrossRef |

Mora, P., 1987, Nonlinear two-dimensional elastic inversion of multioffset seismic data:

*Geophysics*,

**52**, 1211–1228

| Nonlinear two-dimensional elastic inversion of multioffset seismic data:CrossRef |

Park, E., Ha, W., Chung, W., Shin, C., and Min, D. J., 2013, 2D Laplace-domain waveform inversion of field data using a power objective function:

*Pure and Applied Geophysics*,

**170**, 2075–2085

| 2D Laplace-domain waveform inversion of field data using a power objective function:CrossRef |

Plessix, R. E., and Solano, C. A. P., 2015, Modified surface boundary conditions for elastic waveform inversion of low-frequency wide-angle active land seismic data:

*Geophysical Journal International*,

**201**, 1324–1334

| Modified surface boundary conditions for elastic waveform inversion of low-frequency wide-angle active land seismic data:CrossRef |

Pratt, R., 1990, Frequency-domain elastic modeling by finite differences: a tool for crosshole seismic imaging:

*Geophysics*,

**55**, 626–632

| Frequency-domain elastic modeling by finite differences: a tool for crosshole seismic imaging:CrossRef |

Pratt, R., 1999, Seismic waveform inversion in the frequency domain, part I: theory and verification in a physic scale model:

*Geophysics*,

**64**, 888–901

| Seismic waveform inversion in the frequency domain, part I: theory and verification in a physic scale model:CrossRef |

Pratt, R., and Shipp, R., 1999, Seismic waveform inversion in the frequency domain, part 2: fault delineation in sediments using crosshole data:

*Geophysics*,

**64**, 902–914

| Seismic waveform inversion in the frequency domain, part 2: fault delineation in sediments using crosshole data:CrossRef |

Pratt, R., Shin, C., and Hicks, G. J., 1998, Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion:

*Geophysical Journal International*,

**133**, 341–362

| Gauss-Newton and full Newton methods in frequency-space seismic waveform inversion:CrossRef |

Prieux, V., Brossier, R., Operto, S., and Virieux, J., 2013

*a*, Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation:

*Geophysical Journal International*,

**194**, 1640–1664

| Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 1: imaging compressional wave speed, density and attenuation:CrossRef |

Prieux, V., Brossier, R., Operto, S., and Virieux, J., 2013

*b*, Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 2: imaging compressive-wave and shear-wave velocities:

*Geophysical Journal International*,

**194**, 1665–1681

| Multiparameter full waveform inversion of multicomponent ocean-bottom-cable data from the Valhall field. Part 2: imaging compressive-wave and shear-wave velocities:CrossRef |

Qu, Y., Huang, J., Li, Z., Li, Q., Zhao, L., and Li, X., 2015

*a*, Elastic wave modeling and pre-stack reverse time migration of irregular free-surface based on layered mapping method:

*Chinese Journal of Geophysics*,

**58**, 544–560

| Elastic wave modeling and pre-stack reverse time migration of irregular free-surface based on layered mapping method:CrossRef |

Qu, Y., Li, Z., Huang, J., Li, Q., and Li, J., 2015

*b*, Multiple dual-variable grid encoding full time inversion based on an optimized encoding function: 2015 Workshop: Depth Model Building: Full-Waveform Inversion, 18–19 June 2015, Beijing, China, 125–129.

Qu, Y., Li, Z., Huang, J., and Li, J., 2016, Multi-scale full waveform inversion for areas with irregular surface topography in an auxiliary coordinate system:

*Exploration Geophysics*, ,

| Multi-scale full waveform inversion for areas with irregular surface topography in an auxiliary coordinate system:CrossRef |

Qu, Y., Li, Z., Huang, J., and Li, J., 2017

*a*, Viscoacoustic anisotropic full waveform inversion:

*Journal of Applied Geophysics*,

**136**, 484–497

| Viscoacoustic anisotropic full waveform inversion:CrossRef |

Qu, Y, Li, Z, Huang, J, Li, J, and Guan, Z, 2017

*b*, Elastic full-waveform inversion for surface topography:

*Geophysics*,

**82**, R269–R285

| Elastic full-waveform inversion for surface topography:CrossRef |

Ren, Z., and Liu, Y., 2016, A hierarchical elastic full-waveform inversion scheme based on wavefield separation and the multistep-length approach:

*Geophysics*,

**81**, R99–R123

| A hierarchical elastic full-waveform inversion scheme based on wavefield separation and the multistep-length approach:CrossRef |

Sambridge, M. S., Tarantola, A., and Kennett, B. L. N., 1991, An alternative strategy for nonlinear inversion of seismic waveforms:

*Geophysical Prospecting*,

**39**, 723–736

| An alternative strategy for nonlinear inversion of seismic waveforms:CrossRef |

Scales, J. A., 1987, Tomographic inversion via the conjugate gradient method:

*Geophysics*,

**52**, 179–185

| Tomographic inversion via the conjugate gradient method:CrossRef |

Shin, C., and Cha, Y. H., 2008, Waveform inversion in the Laplace domain:

*Geophysical Journal International*,

**173**, 922–931

| Waveform inversion in the Laplace domain:CrossRef |

Shin, C., and Cha, Y. H., 2009, Waveform inversion in the Laplace–Fourier domain:

*Geophysical Journal International*,

**177**, 1067–1079

| Waveform inversion in the Laplace–Fourier domain:CrossRef |

Sirgue, L., and Pratt, R. G., 2004, Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies:

*Geophysics*,

**69**, 231–248

| Efficient waveform inversion and imaging: a strategy for selecting temporal frequencies:CrossRef |

Solano, C. A., Stopin, A., and Plessix, R. E., 2013, Synthetic study of elastic effects on acoustic full waveform inversion: 75th EAGE Conference and Exhibition incorporating SPE EUROPEC 2013, Extended Abstracts, Th P10 10.

Sun, R., Chow, J., and Chen, K. J., 2001, Phase correction in separating P-and S-waves in elastic data:

*Geophysics*,

**66**, 1515–1518

| Phase correction in separating P-and S-waves in elastic data:CrossRef |

Sun, R., McMechan, G. A., and Chuang, H., 2011, Amplitude balancing in separating P- and S-waves in 2D and 3D elastic seismic data:

*Geophysics*,

**76**, S103–S113

| Amplitude balancing in separating P- and S-waves in 2D and 3D elastic seismic data:CrossRef |

Tarantola, A., 1984, Inversion of seismic reflection data in the acoustic approximation:

*Geophysics*,

**49**, 1259–1266

| Inversion of seismic reflection data in the acoustic approximation:CrossRef |

Tarantola, A., 1986, A strategy for nonlinear elastic inversion of seismic reflection data:

*Geophysics*,

**51**, 1893–1903

| A strategy for nonlinear elastic inversion of seismic reflection data:CrossRef |

Vigh, D., and Starr, E. W., 2008, 3D prestack plane-wave, full-waveform inversion:

*Geophysics*,

**73**, VE135–VE144

| 3D prestack plane-wave, full-waveform inversion:CrossRef |

Vigh, D., Starr, B., Kapoor, J., and Li, H., 2010, 3D full waveform inversion on a Gulf of Mexico WAZ data set: SEG Technical Program, Expanded Abstracts, 957–961.

Virieux, J., and Operto, S., 2009, An overview of full-waveform inversion in exploration geophysics:

*Geophysics*,

**74**, WCC1–WCC26

| An overview of full-waveform inversion in exploration geophysics:CrossRef |

Wang, Y., and Dong, L., 2015, Multi-parameter full waveform inversion for acoustic VTI media using the truncated Newton method:

*Chinese Journal of Geophysics*,

**58**, 2873–2885

| Multi-parameter full waveform inversion for acoustic VTI media using the truncated Newton method:CrossRef |

Wang, B. L., and Goo, J. H., 2010, Fast full inversion of multi-shot seismic data: SEG Technical Program, Expanded Abstracts, 1055–1058.

Wu, R. S., Luo, J., and Wu, B., 2013, Ultra-low-frequency information in seismic data and envelope inversion: SEG Technical Program, Expanded Abstracts, 3078–3042.

Xue, Z., Chen, Y., Fomel, S., and Sun, J., 2014, Imaging incomplete data and simultaneous-source data using least-squares reverse-time migration with shaping regularization: SEG Technical Program, Expanded Abstracts, 3991–3996.

Xue, Z., Chen, Y., Fomel, S., and Sun, J., 2016, Seismic imaging of incomplete data and simultaneous-source data using least-squares reverse-time migration with shaping regularization:

*Geophysics*,

**81**, S11–S20

| Seismic imaging of incomplete data and simultaneous-source data using least-squares reverse-time migration with shaping regularization:CrossRef |

Xue, J, Zhu, H, and Fomel, S, 2017, Full-waveform inversion using seislet regularization:

*Geophysics*,

**82**, A43–A49

| Full-waveform inversion using seislet regularization:CrossRef |

Yang, J., Liu, Y., and Dong, L., 2014, A multi-parameter full waveform inversion strategy for acoustic media with variable density:

*Chinese Journal of Geophysics*,

**57**, 628–643

| A multi-parameter full waveform inversion strategy for acoustic media with variable density:CrossRef |

Zhang, Q., and McMechan, G. A., 2010, 2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media:

*Geophysics*,

**75**, D13–D26

| 2D and 3D elastic wavefield vector decomposition in the wavenumber domain for VTI media:CrossRef |

Zhou, B., and Greenhalgh, S. A., 2003, Crosshole seismic inversion with normalized full-waveform amplitude data:

*Geophysics*,

**68**, 1320–1330

| Crosshole seismic inversion with normalized full-waveform amplitude data:CrossRef |

Zhou, W., Brossier, R., Operto, S., and Virieux, J., 2015, Full waveform inversion of diving & reflected waves for velocity model building with impedance inversion based on scale separation:

*Geophysical Journal International*,

**202**, 1535–1554

| Full waveform inversion of diving & reflected waves for velocity model building with impedance inversion based on scale separation:CrossRef |