Abstract
The problem of sparsely collected seismic data is one of the main issues in reflection seismology, because most advanced data processing techniques require a dense and regular seismic data grid. We present a geostatistical seismic data interpolation technique based on sequential stochastic simulations with local structural anisotropies. This technique, contrary to conventional existing data-driven seismic interpolation approaches based on sparsity, prediction filters, or rank-reduction, predicts the value of seismic amplitudes at non-sampled locations by exploiting the statistics of the recorded amplitudes, which are used as experimental data for the geostatistical interpolation in the original data domain. Local mean and variance are computed on-the-fly to define intervals of the global conditional distribution function, from where amplitude values are stochastically simulated. The parameters to define subsets of experimental data from which mean and variance are calculated are given by local variogram models, which in turn are obtained from a local dip and azimuth estimation in the t-x-y domain. The geostatistical seismic data interpolation technique is applied to synthetic and real 2D and 3D datasets in both post- and pre-stack domains. Besides being computationally cheaper than other methods, because the interpolation is carried out directly in the original data domain, the proposed technique provides a local quantitative analysis of the reliability of the interpolated seismic samples, which can be exploited in following processing steps.
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Aravkin, A.Y., Kumar, R., Mansour, B., Recht, B., Herrmann, F.J.: Fast methods for denoising matrix completion formulations, with application to robust seismic data interpolation. SIAM J. Sci. Comput. 36, S237–S266 (2014)
Azevedo, L., Soares, A.: Geostatistical methods for reservoir geophysics. Advances in oil and gas exploration & production. Springer International Publishing (2017)
Bahorich, M., Farmer, S.: 3-D seismic coherency for faults and stratigraphic features: The coherence cube. The Leading Edge 14(10), 1053–1058 (1995). https://doi.org/10.1190/1.1437077
Biondi, B., Palacharla, G.: 3-D prestack migration of common-azimuth data. Geophysics 61, 1822–1832 (1996)
Caeiro, M.H., Demyanov, V., Soares, A.: Optimized history matching with direct sequential image transforming for non-stationary reservoirs. Math. Geosci. https://doi.org/10.1007/s11004-015-9591-0 (2015)
Claerbout, J.F.: Earth Soundings Analysis: Processing Versus Inversion. Blackwell Scientific Publications, Cambridge (1992)
Dalley, R.M., Gevers, E.E.A., Stampli, G.M., Davies, D.J., Gastaldi, C.N., Ruijtenberg, P.R., Vermeer, G.J.D.: Dip and azimuth displays for 3D seismic interpretation. First Break 7, 86–95 (1989)
Darche, G: Spatial interpolation using a fast parabolic transform. In: 60th Annual International Meeting, SEG, Expanded Abstracts, pp. 1647–1750 (1990)
Deutsch, C., Journel, A.G.: GSLIB. Geostatistical Software Library and Users’ Guide. Oxford University Press, Oxford (1998)
Doyen, P.M.: Seismic Reservoir Characterization. EAGE, Madrid (2007)
Dubrule, O.: Geostatistics for Seismic Data Integration in Earth Models. SEG/EAGE Distinguished Instructor Short Course Number 6, Tulsa (2003)
Fomel, S.: Applications of plane-wave destruction filters. Geophysics 67, 1946–1960 (2002)
Fomel, S.: Seismic reflection data interpolation with differential offset and shot continuation. Geophysics 68, 733–744 (2003). https://doi.org/10.1190/1.1567243
Gardner, G.H.F., Canning, A.: Effects of irregular sampling on 3-D prestack migration. In: 64th Annual International Meeting, SEG, Expanded Abstracts, pp 1553–1556 (1994)
Goovaerts, P.: Geostatistics for Natural Resources Evaluation. Oxford University Press, New York (1997)
Guo, J., Zhou, X., Yang, H.J.: Efficient f-k domain seismic trace interpolation for spatially aliased data. In: 66th Annual International Meeting, SEG, Expanded Abstracts, pp 1457–1460 (1996)
Gülünay, N.: Seismic trace interpolation in the Fourier transform domain. Geophysics 68, 355–369 (2003). https://doi.org/10.1190/1.1543221
Gülünay, N., Chambers, R.E.: Unaliased f-k domain trace interpolation (UFKI). In: 66th Annual International Meeting, SEG, Expanded Abstracts, pp 1461–1464 (1996)
Hennenfent, G., Herrmann, F.J.: Simply denoise: wavefield reconstruction via jittered undersampling. Geophysics 73, V19–V28 (2008)
Herrmann, F.J., Hennenfent, G.: Non-parametric seismic data recovery with curvelet frames. Geophys. J. Int. 173, 233–248 (2008)
Horta, A, Caeiro, M., Nunes, R., Soares, A.A.: Simulation of continuous variables at meander structures: application to contaminated sediments of a lagoon. In: Atkinson P, Lloyd C (eds.) geoENV VII—Geostatistics for Environmental Applications. Quantitative Geology and Geostatistics, pp 161–172. Springer, The Netherlands (2009)
Jia, Y., Ma, J.: What can machine learning do for seismic data processing: an interpolation application? Geophysics 82(3), V163–V177 (2017). https://doi.org/10.1190/GEO2016-0300.1
Kreimer, N., Stanton, A., Sacchi, M.D.: Tensor completion based on nuclear norm minimization for 5D seismic data reconstruction. Geophysics 78(6), 273–V284 (2013). https://doi.org/10.1190/geo2013-0022.1
Kumar, R, Mansour, H., Herrmann, F.J., Aravkin, A.Y.: Reconstruction of seismic wavefields via low-rank matrix factorization in the hierarchical-separable matrix representation. In: SEG Technical Program Expanded Abstracts, pp 3628–3633 (2013)
Kumar, R., Da Silva, C., Akalin, O., Aravkin, A.Y., Mansour, H., Recht, B., Herrmann, F. J.: Efficient matrix completion for seismic data reconstruction. Geophysics 80(5), V97–V113 (2015)
Liang, J., Ma, J., Zhang, X.: Seismic data restoration via data-driven tight frame. Geophysics 79(3), V65–V74 (2014)
Liu, B., Sacchi, M.D.: Minimum weighted norm interpolation of seismic records. Geophysics 69, 1560–1568 (2004). https://doi.org/10.1190/1.1836829
Liu, Y., Fomel, S.: Seismic data interpolation beyond aliasing using regularized nonstationary autoregression. Geophysics 76, V69–V77 (2011)
Luis J.J., Almeida J.: Stochastic characterization of fluvial sand channels. In: E. Baafi, et al. (eds.) Geostatistics Wollogong 96, pp 465–477, Kluwer Academic Publishers (1997)
Ma, J.: Three-dimensional irregular seismic data reconstruction via low-rank matrix completion. Geophysics 78(5), V181–V192 (2013)
Marfurt, K.J.: Robust estimates of 3D reflector dip and azimuth. Geophysics, 71(4). https://doi.org/10.1190/1.2213049 (2006)
Naghizadeh, M., Sacchi, M.D.: f-x adaptive seismic trace interpolation. Geophysics 74(1), V9–V16 (2009). https://doi.org/10.1190/1.3008547
Naghizadeh, M., Sacchi, M.D.: Beyond alias hierarchical scale curvelet interpolation of regularly and irregularly sampled seismic data. Geophysics 75, WB189–WB202 (2010)
Naghizadeh, M, Sacchi, M.D.: Hierarchical scale curvelet interpolation of aliased seismic data. In: 80th Annual International Meeting, SEG, Expanded Abstracts, pp 3656–3661 (2010)
Naghizadeh, M., Innanen, K.A.: Seismic data interpolation using a fast generalized Fourier transform. Geophysics 76(1), V1–V10 (2011)
Nguyen, T., Winnett, R.: Seismic interpolation by optimally matched Fourier components. In: SEG Technical Program Expanded Abstracts, pp. 3085–3089 (2011)
Oporeza, V., Sacchi, M.D.: Simultaneous seismic data denoising and reconstruction via multichannel singular spectrum analysis. Geophysics 76, V25–V32 (2011)
Porsani, M.J.: Seismic trace interpolation using half-step prediction filters. Geophysics 64, 1461–1467 (1999)
Ronen, J.: Wave-equation trace interpolation. Geophysics 52, 973–984 (1987)
Sacchi, M.D., Ulrych, T.J.: Estimation of the discrete Fourier Transform, a linear inversion approach. Geophysics 61, 1128–1136 (1996)
Sacchi, M.D., Ulrych, T.J., Walker, C.J.: Interpolation and extrapolation using a high-resolution discrete Fourier transform. IEEE Trans. Signal Process. 46, 31–38 (1998)
Soares, A.: Geostatistical estimation of orebody geometry: morphological kriging. Math. Geol. 22(7), 787–802 (1990)
Soares, A.: Direct sequential simulation and co-simulation. Math. Geol. 33(8), 911–926 (2001)
Spitz, S.: Seismic trace interpolation in the F-X domain. Geophysics 56, 785–794 (1991). https://doi.org/10.1190/1.1443096
Stolt, R.H.: Seismic data mapping and reconstruction. Geophysics 67, 890–908 (2002). https://doi.org/10.1190/1.1484532
Stroet, C., Judith, J.: Mapping curvilinear structures with local anisotropy kriging. Math. Geol. 37(6) (2005)
Trad, D., Ulrych, T.J., Sacchi, M.D.: Accurate interpolation with high-resolution time-variant Radon transforms. Geophysics 67(2), 644–656 (2002)
Trad, D.: Interpolation and multiple attenuation with migration operators. Geophysics 68, 2043–2054 (2003). https://doi.org/10.1190/1.1635058
Trad, D.: Five-dimensional interpolation: recovering from acquisition constraints. Geophysics 74(6), V123–V132 (2009). https://doi.org/10.1190/1.3245216
Trickett, S.R.: F-xy Eigenimage noise suppression. Geophysics 68, 751–759 (2003)
Trickett, S., Burroughs, L., Milton, A., Walton, L., Dack, R.: Rank-reduction-based trace interpolation. In: SEG Technical Program Expanded Abstracts, pp 3829–3833 (2010)
Turquais, P., Asgedom, E.G., Söllner, W.: Structured dictionary learning for interpolation of aliased seismic data. In: 87th Annual Meeting, SEG, pp 4257–4261 (2017)
Van Dedem, E.J., Verschuur, D. J.: 3-D surface-related multiple elimination and interpolation. In: 68th Annual International Meeting, SEG, pp 1321–1324 (1998)
Wang, Y.: Seismic trace interpolation in the f-x-y domain. Geophysics 67, 1232–1239 (2002)
Wang, J., Ng, M, Perz, M.: Fast high resolution Radon transforms by greedy least-squares methods. In: SEG Expanded Abstracts, vol. 28, pp 3128–3132 (2009)
Xu, W.: Conditional curvilinear stochastic simulation using pixel-based algorithms. Math. Geol. 28(7), 937–949 (1996)
Yang, S., Ma, J., Osher, S.: Seismic data reconstruction via matrix completion. In: UCLA CAM, pp 12–14 (2012)
Acknowledgements
The authors gratefully acknowledge the support of the CERENA (strategic project FCT-UID/ECI/04028/2013) and Parallel Geoscience for data availability and the use of SPW 4.0. L. M. Pinheiro for proposing the challenge. A special acknowledgement goes to Gary F. Margrave, Mostafa Naghizadeh, and the CREWES consortium for making available an exhaustive Matlab software library used to test the f -x interpolation technique.
Funding
The authors thank the EU Youth Guarantee program, the Italian Ministry of Education and the Autonomous Region Friuli-Venezia Giulia for financial support.
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Turco, F., Azevedo, L. & Herold, D. Geostatistical interpolation of non-stationary seismic data. Comput Geosci 23, 665–682 (2019). https://doi.org/10.1007/s10596-019-9812-6
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DOI: https://doi.org/10.1007/s10596-019-9812-6