Adaptive mixed-norm seismic inversion for non-Gaussian errors
Zhiyong Li 1 3 Guangmin Hu 1 Jiashu Zhang 2
+ Author Affiliations
- Author Affiliations
1 School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China.
2 Sichuan Key Lab of Signal and Information Processing, Southwest Jiaotong University, Chengdu, Sichuan 610031, China.
3 Corresponding author. Email: 359081137@qq.com
Exploration Geophysics 48(4) 413-421 https://doi.org/10.1071/EG16004
Submitted: 13 January 2016 Accepted: 28 April 2016 Published: 8 June 2016
Abstract
The discrepancies between geophysical measurements and forward-modelled data are commonly modelled as Gaussian errors, thereby necessitating the use of least-squares optimisation methods. However, given the many inevitable difficulties and ambiguities in data acquisition, processing, and interpretation, subsurface-property estimation from remote geophysical measurements is subject to non-Gaussian errors. We propose to minimise the misfit with a robust error measure, which is based on a generalised Gaussian distribution. A suboptimal solution is proposed through a mixed-norm functional combination of the l1 norm and l2 norm. A mixed-norm parameter is introduced to determine the relative importance between the l1 norm and l2 norm functional, which is a function of the kurtosis of the errors. The novelty of the proposed mixed-norm algorithm is that no knowledge of the seismic errors is required. The relative contributions of the l1 norm and l2 norm are adjusted based on the partially inverted elastic properties. Both synthetic and field data demonstrate the effectiveness of the proposed algorithm.
Key words: adaptive inversion, mixed norm, non-Gaussian errors, simultaneous inversion.
References
Avseth, P., Mukerji, T., Jørstad, A., Mavko, G., and Veggeland, T., 2001, Seismic reservoir mapping from 3-D AVO in a North Sea turbidite system: Geophysics, 66, 1157–1176| Seismic reservoir mapping from 3-D AVO in a North Sea turbidite system:Crossref | GoogleScholarGoogle Scholar |
Azevedo, L., Nunes, R., Correia, P., Soares, A., Guerreiro, L., and Neto, G. S., 2014, Multidimensional scaling for the evaluation of a geostatistical seismic elastic inversion methodology: Geophysics, 79, M1–M10
| Multidimensional scaling for the evaluation of a geostatistical seismic elastic inversion methodology:Crossref | GoogleScholarGoogle Scholar |
Bachrach, R., 2006, Joint estimation of porosity and saturation using stochastic rock-physics modeling: Geophysics, 71, O53–O63
| Joint estimation of porosity and saturation using stochastic rock-physics modeling:Crossref | GoogleScholarGoogle Scholar |
Bosch, M., 2004, The optimization approach to lithological tomography: combining seismic data and petrophysics for porosity prediction: Geophysics, 69, 1272–1282
| The optimization approach to lithological tomography: combining seismic data and petrophysics for porosity prediction:Crossref | GoogleScholarGoogle Scholar |
Bosch, M., Cara, L., Rodrigues, J., Navarro, A., and Díaz, M., 2007, A Monte Carlo approach to the joint estimation of reservoir and elastic parameters from seismic amplitudes: Geophysics, 72, O29–O39
| A Monte Carlo approach to the joint estimation of reservoir and elastic parameters from seismic amplitudes:Crossref | GoogleScholarGoogle Scholar |
Bosch, M., Carvajal, C., Rodrigues, J., Torres, A., Aldana, M., and Sierra, J., 2009, Petrophysical seismic inversion conditioned to well-log data: methods and application to a gas reservoir: Geophysics, 74, O1–O15
| Petrophysical seismic inversion conditioned to well-log data: methods and application to a gas reservoir:Crossref | GoogleScholarGoogle Scholar |
Bosch, M., Mukerji, T., and Gonzalez, E. F., 2010, Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: a review: Geophysics, 75, 75A165–75A176
| Seismic inversion for reservoir properties combining statistical rock physics and geostatistics: a review:Crossref | GoogleScholarGoogle Scholar |
Bube, K. P., and Langan, R. T., 1997, Hybrid ℓ 1/ℓ 2 minimization with applications to tomography: Geophysics, 62, 1183–1195
| Hybrid ℓ 1/ℓ 2 minimization with applications to tomography:Crossref | GoogleScholarGoogle Scholar |
Byrd, R. H., Lu, P., Nocedal, J., and Zhu, C., 1995, A limited memory algorithm for bound constrained optimization: SIAM Journal on Scientific Computing, 16, 1190–1208
| A limited memory algorithm for bound constrained optimization:Crossref | GoogleScholarGoogle Scholar |
Chen, K., and Sacchi, M. D., 2015, Robust reduced-rank filtering for erratic seismic noise attenuation: Geophysics, 80, V1–V11
| Robust reduced-rank filtering for erratic seismic noise attenuation:Crossref | GoogleScholarGoogle Scholar |
DeCarlo, L. T., 1997, On the meaning and use of kurtosis: Psychological Methods, 2, 292–307
| On the meaning and use of kurtosis:Crossref | GoogleScholarGoogle Scholar |
Downton, J. E., 2005, Seismic parameter estimation from AVO inversion: Ph.D. thesis, University of Calgary.
Dubrule, O., 2003, Geostatistics for seismic data integration in earth models, 2003 Distinguished Instructor Short Course (No. 6): SEG Books.
Eidsvik, J., Avseth, P., Omre, H., Mukerji, T., and Mavko, G., 2004, Stochastic reservoir characterization using prestack seismic data: Geophysics, 69, 978–993
| Stochastic reservoir characterization using prestack seismic data:Crossref | GoogleScholarGoogle Scholar |
Grana, D., and Della Rossa, E., 2010, Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion: Geophysics, 75, O21–O37
| Probabilistic petrophysical-properties estimation integrating statistical rock physics with seismic inversion:Crossref | GoogleScholarGoogle Scholar |
Grana, D., Mukerji, T., Dvorkin, J., and Mavko, G., 2012, Stochastic inversion of facies from seismic data based on sequential simulations and probability perturbation method: Geophysics, 77, M53–M72
| Stochastic inversion of facies from seismic data based on sequential simulations and probability perturbation method:Crossref | GoogleScholarGoogle Scholar |
Guitton, A., and Symes, W. W., 2003, Robust inversion of seismic data using the Huber norm: Geophysics, 68, 1310–1319
| Robust inversion of seismic data using the Huber norm:Crossref | GoogleScholarGoogle Scholar |
Gunning, J., and Glinsky, M. E., 2007, Detection of reservoir quality using Bayesian seismic inversion: Geophysics, 72, R37–R49
| Detection of reservoir quality using Bayesian seismic inversion:Crossref | GoogleScholarGoogle Scholar |
Hampson, D. P., Russell, B. H., and Bankhead, B., 2005, Simultaneous inversion of pre-stack seismic data: SEG Technical Program Expanded Abstracts 2005, 1633–1637.
Hong, M.-C., Stathaki, T., and Katsaggelos, A. K., 2002, Iterative regularized least-mean mixed-norm image restoration: Optical Engineering, 41, 2515–2524
| Iterative regularized least-mean mixed-norm image restoration:Crossref | GoogleScholarGoogle Scholar |
Huber, P. J., 1981, Robust statistics: John Wiley.
Jenkinson, A. F., 1955, The frequency distribution of the annual maximum (or minimum) values of meteorological elements: Quarterly Journal of the Royal Meteorological Society, 81, 158–171
| The frequency distribution of the annual maximum (or minimum) values of meteorological elements:Crossref | GoogleScholarGoogle Scholar |
Ji, J., 2012, Robust inversion using biweight norm and its application to seismic inversion: Exploration Geophysics, 43, 70–76
| Robust inversion using biweight norm and its application to seismic inversion:Crossref | GoogleScholarGoogle Scholar |
Kjønsberg, H., Hauge, R., Kolbjørnsen, O., and Buland, A., 2010, Bayesian Monte Carlo method for seismic predrill prospect assessment: Geophysics, 75, O9–O19
| Bayesian Monte Carlo method for seismic predrill prospect assessment:Crossref | GoogleScholarGoogle Scholar |
Larsen, A. L., Ulvmoen, M., Omre, H., and Buland, A., 2006, Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model: Geophysics, 71, R69–R78
| Bayesian lithology/fluid prediction and simulation on the basis of a Markov-chain prior model:Crossref | GoogleScholarGoogle Scholar |
Li, Y., Zhang, J., and Claerbout, J., 2010, Geophysical applications of a novel and robust L1 solver: SEG Technical Program, Expanded Abstracts, 3519–3523.
Li, Z. Y., Song, B. B., Zhang, J. S., and Hu, G. M., 2015, Joint elastic and petrophysical inversion using prestack seismic and well log data: Exploration Geophysics, ,
Mardia, K. V., 1970, Measures of multivariate skewness and kurtosis with applications: Biometrika, 57, 519–530
| Measures of multivariate skewness and kurtosis with applications:Crossref | GoogleScholarGoogle Scholar |
Mukerji, T., Jørstad, A., Avseth, P., Mavko, G., and Granli, J. R., 2001, Mapping lithofacies and pore-fluid probabilities in a North Sea reservoir: seismic inversions and statistical rock physics: Geophysics, 66, 988–1001
| Mapping lithofacies and pore-fluid probabilities in a North Sea reservoir: seismic inversions and statistical rock physics:Crossref | GoogleScholarGoogle Scholar |
Ostrander, W., 1984, Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence: Geophysics, 49, 1637–1648
| Plane-wave reflection coefficients for gas sands at nonnormal angles of incidence:Crossref | GoogleScholarGoogle Scholar |
Pun, W. H., and Jeffs, B. D., 1995, Adaptive image restoration using a generalized Gaussian model for unknown noise: IEEE Transactions on Image Processing, 4, 1451–1456
| Adaptive image restoration using a generalized Gaussian model for unknown noise:Crossref | GoogleScholarGoogle Scholar | 1:STN:280:DC%2BD1c7jt1Cmuw%3D%3D&md5=df733f4a71e724e19cfca7555793b450CAS | 18291976PubMed |
Ramirez, A., and Bosch, M., 2013, Seismic and petrophysical inversion using the optimization approach: Revista de la Facultad de Ingeniería Universidad Central de Venezuela, 28, 53–64
Saltzer, R., Finn, C., and Burtz, O., 2005, Predicting V shale and porosity using cascaded seismic and rock physics inversion: The Leading Edge, 24, 732–736
| Predicting V shale and porosity using cascaded seismic and rock physics inversion:Crossref | GoogleScholarGoogle Scholar |
Sengupta, M., and Bachrach, R., 2007, Uncertainty in seismic-based pay volume estimation: analysis using rock physics and Bayesian statistics: The Leading Edge, 26, 184–189
| Uncertainty in seismic-based pay volume estimation: analysis using rock physics and Bayesian statistics:Crossref | GoogleScholarGoogle Scholar |
Tarantola, A., 1987 Elsevier., , ,
Taylor, H. L., Banks, S. C., and McCoy, J. F., 1979, Deconvolution with the ℓ1 norm: Geophysics, 44, 39–52
| Deconvolution with the ℓ1 norm:Crossref | GoogleScholarGoogle Scholar |
Ulvmoen, M., Omre, H., and Buland, A., 2010, Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 2—Real case study: Geophysics, 75, B73–B82
| Improved resolution in Bayesian lithology/fluid inversion from prestack seismic data and well observations: Part 2—Real case study:Crossref | GoogleScholarGoogle Scholar |
Wang, Y., 2016, Seismic inversion: theory and applications: John Wiley.
Zhang, J. S., Lv, S. F., Liu, Y., and Hu, G. M., 2013, AVO inversion based on generalized extreme value distribution with adaptive parameter estimation: Journal of Applied Geophysics, 98, 11–20
| AVO inversion based on generalized extreme value distribution with adaptive parameter estimation:Crossref | GoogleScholarGoogle Scholar |