Fast wavelet-based stochastic simulation using training images.

*(English)*Zbl 1396.65175Summary: Spatial uncertainty modelling is a complex and challenging job for orebody modelling in mining, reservoir characterization in petroleum, and contamination modelling in air and water. Stochastic simulation algorithms are popular methods for such modelling. In this paper, discrete wavelet transformation (DWT)-based multiple point simulation algorithm for continuous variable is proposed that handles multi-scale spatial characteristics in datasets and training images. The DWT of a training image provides multi-scale high-frequency wavelet images and one low-frequency scaling image at the coarsest scale. The simulation of the proposed approach is performed on the frequency (wavelet) domain where the scaling image and wavelet images across the scale are simulated jointly. The inverse DWT reconstructs simulated realizations of an attribute of interest in the space domain. An automatic scale-selection algorithm using dominant mode difference is applied for the selection of the optimal scale of wavelet
decomposition. The proposed algorithm reduces the computational time required for simulating large domain as compared to spatial domain multi-point simulation algorithm. The algorithm is tested with an exhaustive dataset using conditional and unconditional simulation in two- and three-dimensional fluvial reservoir and mining blasted rock data. The realizations generated by the proposed algorithm perform well and reproduce the statistics of the training image. The study conducted comparing the spatial domain filtersim multiple-point simulation algorithm suggests that the proposed algorithm generates equally good realizations at lower computational cost.

##### MSC:

65T60 | Numerical methods for wavelets |

86A32 | Geostatistics |

86-08 | Computational methods for problems pertaining to geophysics |

##### Keywords:

discrete wavelet transformation; multi-scale analysis; template matching; \(K\)-means clustering; conditional simulation; reservoir characterization
PDF
BibTeX
XML
Cite

\textit{S. Chatterjee} et al., Comput. Geosci. 20, No. 3, 399--420 (2016; Zbl 1396.65175)

Full Text:
DOI

##### References:

[1] | Adams, MD; Kossentini, F, Reversible integer-to-integer wavelet transform for image compression: performance evaluation and analysis, IEEE Trans. Image Process, 8, 1010-1024, (2000) · Zbl 0962.94023 |

[2] | Arpat, G.: Sequential simulation with patterns, PhD thesis, Stanford University (2005) |

[3] | Arpat, G; Caers, J, Conditional simulation with patterns, Math. Geol., 39, 177-203, (2007) |

[4] | Ashikhmin, M.: Synthesizing natural, textures. In: The proceedings of 2001 ACM symposium on interactive 3D graphics, pp. 217-226. Research Triangle Park, North Carolina (2001) |

[5] | Bosch, EH; Gonzalez, AP; Vivas, JG; Easley, GR, Directional wavelets and a wavelet variogram for two-dimensional data, Math. Geosci., 41, 611-641, (2009) · Zbl 1174.42041 |

[6] | Boucher, A, Sub-pixel mapping of coarse satellite remote sensing images with stochastic simulation from training images, Math. Geosci., 41, 265-290, (2009) · Zbl 1162.94305 |

[7] | Can, F; Ismail, SA; Engin, D, Efficiency and effectiveness of query processing in cluster-based retrieval, Inf. Syst., 29, 697-717, (2004) |

[8] | Chatterjee, S; Dimitrakopoulos, R, Multi-scale stochastic simulation with wavelet-based approach, Comput. Geosci., 45, 177-189, (2012) |

[9] | Chatterjee, S; Dimitrakopoulos, R; Mustafa, H, Dimensional reduction of pattern-based simulation using wavelet analysis, Math. Geosci., 44, 343-374, (2012) |

[10] | Daubechies, I.: Ten lectures on wavelets. SIAM, Philadelphia (1992) · Zbl 0776.42018 |

[11] | Demirel, H; Anbarjafari, G, Image resolution enhancement by using discrete and stationary wavelet decomposition, IEEE Trans. Image Process., 20, 1458-1460, (2011) · Zbl 1372.94067 |

[12] | Dimitrakopoulos, R; Mustapha, H; Gloaguen, E, High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena, Math. Geosci., 42, 65-99, (2010) · Zbl 1184.86012 |

[13] | Ding, L; Goshtasby, A; Satter, M, Volume image registration by template matching, Image Vis. Comput., 19, 821-832, (2001) |

[14] | Dumic, E; Grgic, S; Grgic, R, The use of wavelets in image interpolation: possibilities and limitations, Radio Eng, 16, 101-109, (2007) |

[15] | Foufoula-Georgiou, E., Kumar, P.: Wavelets in geophysics. Academic, San Diego (1994) · Zbl 0824.00007 |

[16] | Gardet, C., Ravalec, M.: Multiscale multiple point simulation based on texture synthesis. In: Proceedings of 14th European conference on the mathematics of oil recovery, pp. 1524-1535. Catania, Italy (2014) |

[17] | Gloaguen, E; Dimitrakopoulos, R, Two dimensional conditional simulation based on the wavelet decomposition of training images, Math. Geosci., 41, 679-701, (2009) · Zbl 1174.94003 |

[18] | Goovaerts, P.: Geostatistics for natural resources evaluation (Applied Geostatistics Series). Oxford University Press, New York (1998) |

[19] | Goshtasby, A; Gage, SH; Bartholic, JF, A two-stage cross-correlation approach to template matching, IEEE Trans. Pattern Anal. Mach. Intell., 6, 374-378, (1984) |

[20] | Guardiano, F; Srivastava, RM; Soares, A (ed.), Multivariate geostatistics: beyond bivariate moments, 133-144, (1993), Dordrecht |

[21] | Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning, Data mining, inference, and prediction (Springer Series in Statistics). Springer, New York (2011) · Zbl 0973.62007 |

[22] | Henrion, V; Caumon, V; Cherpeau, N, ODSIM: an object-distance simulation method for conditioning complex natural structures, Math. Geosci., 42, 911-924, (2011) |

[23] | Honarkhah, M; Caers, J, Stochastic simulation of patterns using distance-based pattern modelling, Math. Geosci., 42, 487-517, (2010) · Zbl 1194.86038 |

[24] | Journel, AG; Baafy, E (ed.); Shofield, N (ed.), Deterministic geostatistics: a new visit, 213-224, (1997), Dordrecht |

[25] | Journel, A.: Roadblocks to the evaluation of ore reserved - the simulation overpass and putting more geology into numerical models of deposit. In: Dimitrakopoulos, R. (ed.) Orebody modeling and strategic mine planning, AusIIMM, Melbourn, 2nd Edition, Spectrum Series 14, pp 29-32 (2007) |

[26] | Kim, H.Y., Araújo, S.A.: Grayscale template-matching invariant to rotation, scale, translation, brightness and contrast, PSIVT’07. In: Proceedings of the 2nd Pacific Rim conference on advances in image and video technology, pp. 100-113. Berlin, Heidelberg (2007) |

[27] | Kuglin, C., Hines, D.: The phase correlation image alignment method. In: Proceedings of the IEEE International Conference on Cybernetics and Society, pp. 163-165. San Francisco (1975) · Zbl 0979.62046 |

[28] | Kumar, P, A wavelet based methodology for scale-space anisotropic analysis, Geophys. Res. Lett., 22, 2777-2780, (1995) |

[29] | Lark, RM, Spatial analysis of categorical soil variables with the wavelet transformation, Eur. J. Soil Sci., 56, 779-792, (2005) |

[30] | Le Coz, M; Genthon, P; Adler, PM, Multiple-point statistics for modeling facies heterogeneities in a porous medium: the komadugu-yobe alluvium, lake chad basin, Math. Geosci., 43, 861-878, (2011) |

[31] | Li, B-L; Loehle, C, Wavelet analysis of multiscale permeabilities in the subsurface, Geophysical Research Letters, 22, 3123-3126, (1995) |

[32] | Macías, JAR; Expósito, AG, Efficient computation of the running discrete Haar transform, IEEE Trans. Power Delivery, 21, 504-505, (2006) |

[33] | MacQueen, J.B.: Some methods for classification and analysis of multivariate observations. In: Proceedings of 5th Berkeley symposium on mathematical statistics and probability, vol. 1, pp 281-297. University of California Press, Berkeley (1967) |

[34] | Mallat, S, A theory for multi-resolution signal decomposition: the wavelet representation, IEEE Pattern. Anal. Mach. Intell., 11, 674-693, (1989) · Zbl 0709.94650 |

[35] | Mallat, S.: A wavelet tour of signal processing. Academic, San Diego (1998) · Zbl 0937.94001 |

[36] | Mao, S., Journel, A.G.: Generation of a reference petrophysical and seismic 3D data set: the Stanford V reservoir. In: Stanford center for reservoir forecasting annual meeting. http://ekofisk.stanford.edu/SCRF.html (1999), (2009). Accessed 23 February 2013 · Zbl 1119.86313 |

[37] | Mariethoz, G; Renard, P, Reconstruction of incomplete data sets or images using direct sampling, Math. Geosci., 42, 245-268, (2010) · Zbl 1184.86015 |

[38] | Mariethoz, G; Renard, P, Special issue on 20 years of multiple-point statistics: part 2, Math. Geosci., 46, 517-518, (2014) · Zbl 1323.00065 |

[39] | Meyer, Y., Ryan, R.D.: Wavelets: algorithms and applications. Society for industrial and applied mathematics, Philadelphia (1993) |

[40] | Mustafa, H., Chatterjee, S., Dimitrakopoulos, R., Graf, T.: Wavelet-based pattern simulation for geologic heterogeneity recognition: implications in subsurface flow and transport simulations. Adv. Water Resour. (2012). doi:10.1016/j.advwatres.2012.11.018 |

[41] | Mustapha, H; Dimitrakopoulos, R, High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns, Math. Geosci., 42, 457-485, (2010) · Zbl 1194.86040 |

[42] | Portilla, J; Simoncelli, EP, A parametric texture model based on joint statistics of complex wavelet coefficients, Int. J. Comput. Vis., 40, 49-71, (2000) · Zbl 1012.68698 |

[43] | Quddus, A; Gabbouj, M, Wavelet-based corner detection technique using optimal scale, Pattern Recogn. Lett., 23, 215-220, (2002) · Zbl 0998.68652 |

[44] | Strebelle, S, Conditional simulation of complex geological structures using multiple-point statistics, Math. Geol., 34, 1-21, (2002) · Zbl 1036.86013 |

[45] | Strebelle, S; Zhang, T; Leuangthong, O (ed.); Deutsch, C V (ed.), Non-stationary multiple-point geostatistical models, 235-244, (2004), Dordrecht |

[46] | Strebelle, S; Cavelius, C, Solving speed and memory issues in multiple-point statistics simulation program SNESIM, Math. Geosci., 46, 171-186, (2014) · Zbl 1322.65037 |

[47] | Toftaker, H; Tjelmeland, H, Construction of binary multi-grid Markov random field prior models from training images, Math. Geosci., 45, 383-409, (2013) · Zbl 1321.86034 |

[48] | Tibshirani, R; Walther, G; Hastie, T, Estimating the number of clusters in a data set via the gap statistic, J. R. Statist. Soc. B, 63, 411-423, (2001) · Zbl 0979.62046 |

[49] | Tran, T., Mueller, U.A., Bloom, L.M.: Multi-scale conditional simulation of two-dimensional random processes using Haar wavelets. In: Proceedings of GAA symposium, perth, pp. 56-78 (2002) |

[50] | Vannucci, M; Corradi, F, Covariance structure of wavelet coefficients: theory and models in a Bayesian perspective, J. R. Statis. Soc. B, 61, 971-986, (1999) · Zbl 0940.62023 |

[51] | Walnut, D.: An introduction to wavelets analysis. Birkhauser, Boston (1998) |

[52] | Wei, L., Levoy, M.: Fast texture synthesis using tree-structured vector quantization. In: Proceedings of SIGGRAPH 2000 (2000) · Zbl 0940.62023 |

[53] | Wu, J; Zhang, T; Journel, A, Fast FILTERSIM simulation with score-based distance, Math. Geosci., 40, 773-788, (2008) · Zbl 1174.86311 |

[54] | Zhang, T, MPS-driven digital rock modeling and upscaling, Math. Geosci., (2015) |

[55] | Zhang, T; Switzer, P; Journel, A, Filter-based classification of training image patterns for spatial simulation, Math. Geol., 38, 63-80, (2006) · Zbl 1119.86313 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.