Inverse Modeling of Moving Average Isotropic Kernels for Non-parametric Three-Dimensional Gaussian Simulation
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Inverse Modeling of Moving Average Isotropic Kernels for Non-parametric Three-Dimensional Gaussian Simulation Oscar Peredo1 · Julián M. Ortiz1,2 · Oy Leuangthong3
Received: 6 October 2014 / Accepted: 3 July 2015 © International Association for Mathematical Geosciences 2015
Abstract Moving average simulation can be summarized as a convolution between a spatial kernel and a white noise random field. The kernel can be calculated once the variogram model is known. An inverse approach to moving average simulation is proposed, where the kernel is determined based on the experimental variogram map in a non-parametric way, thus no explicit variogram modeling is required. The omission of structural modeling in the simulation work-flow may be particularly attractive if spatial inference is challenging and/or practitioners lack confidence in this task. A non-linear inverse problem is formulated in order to solve the problem of discrete kernel weight estimation. The objective function is the squared euclidean distance between experimental variogram values and the convolution of a stationary random field with Dirac covariance and the simulated kernel. The isotropic property of the kernel weights is imposed as a linear constraint in the problem, together with lower and upper bounds for the weight values. Implementation details and examples are presented to demonstrate the performance and potential extensions of this method. Keywords Moving average · Convolution · Gaussian simulation · Variogram · Inverse problems
1 Introduction Geostatistical simulation is useful to quantify uncertainty of a variable in space and assess its performance to various processes. These processes may be related to flow,
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Oscar Peredo [email protected]
1
ALGES Laboratory, Advanced Mining Technology Center, University of Chile, Santiago, Chile
2
Department of Mining Engineering, University of Chile, Santiago, Chile
3
SRK Consulting (Canada) Inc., Toronto, Canada
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Math Geosci
transport, thresholding, etc., and they require computing multiple numerical realizations that discretize the geographical space. The generation of multiple realizations can be cumbersome, since the spatial structure must be inferred and imposed a priori in the numerical models. Many methods exist for generating the simulations once a structural model (a variogram) has been determined (Chilès and Delfiner 2012). However, a constant challenge is to simplify this inference process as much as possible and minimize the degree of uncertainty involved. In the context of parametric variogram modeling, novel attempts to semi-automatize or fully automatize this process can be found in Cressie (1985), Pardo-Igúzquiza (1999), Emery (2010), and Desassis and Renard (2013). In these works the variables to be fitted are parameters from structures defined a priori (spherical, gaussian, exponential and others) minimizing some form of least-squares metric. Even though this approach is fast and accurate, the predictions might be seriously biased if the true variogram is not closely a
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