Abstract
Groundwater contaminant plumes frequently display a curvilinear anisotropy, which conventional kriging and geostatistical simulation approaches fail to reproduce properly. In many applications, physically relevant coordinate transformations are used to modify the relationships between data points and simplify the specification of nonlinear anisotropy. In this paper, we present a kriging approach that uses a coordinate transformation to improve the interpolation of contaminant concentrations. The proposed alternative flow coordinates (AFC) consist in the hydraulic head and one (2D) or two (3D) streamline-based coordinates. In 2D, the mapping obtained using AFC is similar to that yielded by the natural coordinates of flow (i.e. hydraulic head and stream function). AFC can be generalized to 3D flow and to the presence of wells, which is not the case with the natural coordinates. The performance of the approach is investigated using a simple 3D synthetic case. Kriged concentration maps obtained with the AFC reproduce the curvilinear features found in the reference plume. Performance statistics suggest AFC improves plume delineation compared to conventional kriging on a Cartesian grid. However, the performance of the AFC transformation is limited by the fact that, while it accounts for advection, it does not consider the effects of dispersion on the shape of the plume. This aspect and further testing on complex cases is the subject of ongoing research.
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References
Almendral A, Abrahamsen P, Hauge R (2008) Multidimensional scaling and anisotropic covariance functions. In: Proceedings of the eight international geostatistics congress, Santiago, Chile
Bailly J, Monestiez P, Lagacherie P (2006) Modelling spatial variability along drainage networks with geostatistics. Math Geol 38(5):515–539
Barabas N, Goovaerts P, Adriaens P (2001) Geostatistical assessment and validation of uncertainty for three-dimensional dioxin data from sediments in an estuarine river. Environ Sci Technol 35(16):3294–3301
Boisvert JB, Deutsch CV (2011) Programs for kriging and sequential Gaussian simulation with locally varying anisotropy using non-Euclidean distances. Comput Geosci 37:495–510
Boisvert JB, Manchuk JG, Deutsch CV (2009) Kriging in the presence of locally varying anisotropy using non-Euclidean distances. Math Geosci 41:585–601
Christakos G, Hristopulos DT, Bogaert P (2000) On the physical geometry concept at the basis of space/time geostatistical hydrology. Adv Water Resour 23:799–810
Cirpka OA, Frind EO, Helmig R (1999) Numerical methods for reactive transport on rectangular and streamline-oriented grids. Adv Water Resour 22(7):711–728
Cirpka OA, Frind EO, Helmig R (1999) Streamline-oriented grid generation for transport modelling in two-dimensional domains including wells. Adv Water Resour 22(7):697–710
Curriero FC (1996) The use of non-Euclidean distances in geostatistics. PhD thesis, Kansas State University
Curriero FC (2006) On the use of non-Euclidean distance measures in geostatistics. Math Geol 38(8):907–926
Dagbert M, David M, Crozel D, Desbarats A (1984) Computing variograms in folded strata-controlled deposits. In: Vergy G, David M, Journel A, Marchal A (eds) Geostatistics for natural resources characterization. Reidel, Dordrecht, pp 71–89
Deutsch C, Wang L (1996) Hierarchical object-based stochastic modeling of fluvial reservoirs. Math Geol 28(7):857–879
Deutsch CV (2002) Geostatistical reservoir modeling. Oxford University Press, London
de Fouquet C, Bernard-Michel C (2006) Modèles géostatistiques de concentrations ou de débits le long des cours d’eau. C R Géosci 338:307–318
Frind EO (1982) The principal direction technique: a new approach to groundwater contaminant transport modeling. In: Finite elements in water resources. Proceedings of the 4th international conference, Hanover, Germany, June 1982
Kohavi R, Provost F (1998) Glossary of terms. Mach Learn 30:271–274
Kubat M, Holte RC, Matwin S (1998) Machine learning for the detection of oil spills in satellite radar images. Mach Learn 30:195–215
Legleiter CJ, Kyriakidis PC (2006) Forward and inverse transformation between Cartesian and channel-fitted coordinate systems for meandering rivers. Math Geol 38(8):927–958
Loland A, Host G (2003) Spatial covariance modelling in a complex coastal domain by multidimensional scaling. Environmetrics 14:307–321
Matanga GB (1993) Stream functions in three-dimensional groundwater flow. Water Resour Res 29(9):3125–3133
Monestiez P, Sampson P, Guttorp P (1993) Modeling of heterogeneous spatial correlation structure by spatial deformation. Cah Géostat 3:35–46
Narasimhan TN (2008) Laplace equation and Faraday’s lines of force. Water Resour Res 44:W09412
Rathbun SL (1998) Spatial modelling in irregularly shaped regions: kriging estuaries. Environmetrics 9(2):109–129
Rivest M, Marcotte D, Pasquier P (2012) Sparse data integration for the interpolation of concentration measurements using kriging in natural coordinates. J Hydrol 416–417:72–82. doi:10.1016/j.jhydrol.2011.11.043
Sampson PD, Guttorp P (1992) Nonparametric-estimation of nonstationary spatial covariance structure. J Am Stat Assoc 87(417):108–119
te Stroet CBM, Snepvangers JJJC (2005) Mapping curvilinear structures with local anisotropy kriging. Math Geol 36:635–649
Thompson JF, Warsi ZUA, Mastin CW (1985) Numerical grid generation: foundations and applications. North-Holland, Amsterdam
Ver Hoef JM, Peterson E, Theobald D (2006) Spatial statistical models that use flow and stream distance. Environ Ecol Stat 13:449–464
Xu W (1996) Conditional curvilinear stochastic simulation using pixel-based algorithms. Math Geol 28:937–949
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Financial support for this research was provided by a scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC).
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Rivest, M., Marcotte, D., Pasquier, P. (2012). Interpolation of Concentration Measurements by Kriging Using Flow Coordinates. In: Abrahamsen, P., Hauge, R., Kolbjørnsen, O. (eds) Geostatistics Oslo 2012. Quantitative Geology and Geostatistics, vol 17. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4153-9_42
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DOI: https://doi.org/10.1007/978-94-007-4153-9_42
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