Abstract
Estimating observation error covariance matrix properly is a key step towards successful seismic history matching. Typically, observation errors of seismic data are spatially correlated; therefore, the observation error covariance matrix is non-diagonal. Estimating such a non-diagonal covariance matrix is the focus of the current study. We decompose the estimation into two steps: (1) estimate observation errors and (2) construct covariance matrix based on the estimated observation errors. Our focus is on step (1), whereas at step (2) we use a procedure similar to that in Aanonsen et al. 2003. In Aanonsen et al. 2003, step (1) is carried out using a local moving average algorithm. By treating seismic data as an image, this algorithm can be interpreted as a discrete convolution between an image and a rectangular window function. Following the perspective of image processing, we consider three types of image denoising methods, namely, local moving average with different window functions (as an extension of the method in Aanonsen et al. 2003), non-local means denoising and wavelet denoising. The performance of these three algorithms is compared using both synthetic and field seismic data. It is found that, in our investigated cases, the wavelet denoising method leads to the best performance in most of the time.
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Aanonsen, S., Aavatsmark, I., Barkve, T., Cominelli, A., Gonard, R., Gosselin, O., Kolasinski, M., Reme, H.: Effect of scale dependent data correlations in an integrated history matching loop combining production data and 4D seismic data. In: SPE Reservoir Simulation Symposium (2003)
Abadpour, A., Bergey, P., Piasecki, R.: 4D seismic history matching with ensemble kalman filter-assimilation on Hausdorff distance to saturation front. In: SPE Reservoir Simulation Symposium. Society of Petroleum Engineers. SPE-163635-MS (2013)
Buades, A., Coll, B., Morel, J. M.: A non-local algorithm for image denoising. In: IEEE Computer Society Conference On Computer vision and Pattern Recognition 2005 (CVPR 2005) (2005)
Candès, E. J., Donoho, D. L.: Ridgelets: a key to higher-dimensional intermittency?. Philos. Trans. R. Soc. Lond. A Math. Phys. Eng. Sci. 357, 2495–2509 (1999)
Candès, E. J., Donoho, D. L.: New tight frames of curvelets and optimal representations of objects with piecewise C 2 singularities. Commun. Pur. Appl. Math. 57, 219–266 (2004)
Chang, S. G., Yu, B., Vetterli, M.: Adaptive wavelet thresholding for image denoising and compression. IEEE Trans. Image Process. 9, 1532–1546 (2000)
Do, M. N., Vetterli, M.: The contourlet transform: an efficient directional multiresolution image representation. IEEE Trans. Image Process. 14, 2091–2106 (2005)
Donoho, D. L., Johnstone, I. M.: Adapting to unknown smoothness via wavelet shrinkage. J. Am. Stat. Assoc. 90, 1200–1224 (1995)
Donoho, D. L., Johnstone, J. M.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81, 425–455 (1994)
Emerick, A. A., Reynolds, A. C.: History matching time-lapse seismic data using the ensemble Kalman filter with multiple data assimilations. Comput. Geosci. 16, 639–659 (2012)
Fahimuddin, A., Aanonsen, S., Skjervheim, J. A.: Ensemble based 4D seismic history matching–integration of different levels and types of seismic data. In: 72Nd EAGE Conference & Exhibition (2010)
Harris, F. J.: On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66, 51–83 (1978)
Hennenfent, G., Herrmann, F. J.: Seismic denoising with nonuniformly sampled curvelets. Computing in Science & Engineering 8, 16–25 (2006)
Ioup, J. W., Ioup, G. E.: Noise removal and compression using a wavelet transform. In: 68Th Annual International Meeting, Society of Exploration Geophysicists, Expanded Abstracts, pp 1076–1079 (1998)
Jansen, M.: Noise reduction by wavelet thresholding, vol. 161. Springer Science & Business Media (2012)
Johnstone, I. M., Silverman, B. W.: Wavelet threshold estimators for data with correlated noise. J. R. Stat. Soc. Ser. B Stat. Methodol. 59, 319–351 (1997)
Katterbauer, K., Hoteit, I., Sun, S.: History matching of electromagnetically heated reservoirs incorporating full-wavefield seismic and electromagnetic imaging. SPE J. 20, 923–941 (2015). SPE-173896-PA
Leeuwenburgh, O., Arts, R.: Distance parameterization for efficient seismic history matching with the ensemble Kalman filter. Comput. Geosci. 18, 535–548 (2014)
Luo, X., Bhakta, T., Jakobsen, M., Nævdal, G.: An ensemble 4D seismic history matching framework with sparse representation based on wavelet multiresolution analysis. SPE J. (2016). in press
Luo, X., Bhakta, T., Jakobsen, M., Nævdal, G.: An ensemble 4D seismic history matching framework with wavelet multiresolution analysis—a 3D benchmark case study. In: 15Th European Conference on the Mathematics of Oil Recovery (ECMOR), Amsterdam (2016)
Miao, X., Cheadle, S.: Noise attenuation with wavelet transforms. In: 1998 SEG Annual Meeting. Society of Exploration Geophysicists (1998)
Nguyen, M., Mars, J.: Filtering surface waves using 2D discrete wavelet transform. In: 1999 SEG Annual Meeting. Society of Exploration Geophysicists (1999)
Oppenheim, A. V., Schafer, R. W., Buck, J. R.: Discrete-time signal processing. Prentice Hall, Englewood Cliffs (1989)
Shan, H., Ma, J., Yang, H.: Comparisons of wavelets, contourlets and curvelets in seismic denoising. J. Appl. Geophys. 69, 103–115 (2009)
Skjervheim, J.A., Evensen, G., Aanonsen, S.I., Ruud, B.O., Johansen, T.A.: Incorporating 4D seismic data in reservoir simulation models using ensemble Kalman filter. SPE J. 12, 282–292 (2007). SPE-95789-PA
Trani, M., Arts, R., Leeuwenburgh, O.: Seismic history matching of fluid fronts using the ensemble Kalman filter. SPE J. 18, 159–171 (2012). SPE-163043-PA
Wang, Z., Bovik, A. C., Sheikh, H. R., Simoncelli, E. P.: Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13, 600–612 (2004)
Zhang, H., Liu, T., Zhang, Y.: Denoising of seismic data via multi-scale ridgelet transform. Earthq. Sci. 22, 493–498 (2009)
Zhang, R., Ulrych, T. J.: Physical wavelet frame denoising. Geophysics 68, 225–231 (2003)
Zhao, Y., Li, G., Reynolds, A.: Characterization of the measurement error in time-lapse seismic data and production data with an EM algorithm. Oil & Gas Science and Technology-Revue de l’IFP 62, 181–193 (2007)
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Luo, X., Bhakta, T. Estimating observation error covariance matrix of seismic data from a perspective of image denoising. Comput Geosci 21, 205–222 (2017). https://doi.org/10.1007/s10596-016-9605-0
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DOI: https://doi.org/10.1007/s10596-016-9605-0