Computationally efficient generation of gaussian conditional simulations over regular sample grids

The generation over two-dimensional grids of normally distributed random fields conditioned on available data is often required in reservoir modeling and mining investigations. Such fields can be obtained from application of turning band or spectral methods. However, both methods have limitations. First, they are only asymptotically exact in that the ensemble of realizations has the correlation structure required only if enough harmonics are used in the spectral method, or enough lines are generated in the turning bands approach. Moreover, the spectral method requires fine tuning of process parameters. As for the turning bands method, it is essentially restricted to processes with stationary and radially symmetric correlation functions. Another approach, which has the advantage of being general and exact, is to use a Cholesky factorization of the covariance matrix representing grid points correlation. For fields of large size, however, the Cholesky factorization can be computationally prohibitive. In this paper, we show that if the data are stationary and generated over a grid with regular mesh, the structure of the data covariance matrix can be exploited to significantly reduce the overall computational burden of conditional simulations based on matrix factorization techniques. A feature of this approach is its computational simplicity and suitability to parallel implementation.     

General joint conditional simulations using a fast fourier transform method

A procedure for generating joint statistically homogeneous random fields is examined. The method is based on the spectral representation theorem. It handles large fields easily and is both rapid and flexible. Algorithm development and examples are presented. The procedure is adapted further to include the possibility of generating fields that are jointly conditioned on data from two related fields.     

Production of conditional simulations via the LU triangular decomposition of the covariance matrix

This paper reviews the turning band method and fast Fourier transform method of producing a nonconditional simulation of a multinormal random function with a given covariance structure. A review of the two common methods of conditioning the simulation to honor the data shows that they are formally equivalent. Another method for directly pondering a conditional simulation based on the LU triangular decomposition of the covariance matrix is presented. Computational and implementation difficulties are discussed.     

The practice of fast conditional simulations through the LU decomposition of the covariance matrix

The LU-matrix approach to conditional simulations allows fast generation of large numbers of realizations for a given stochastic process. Simplicity, flexibility, and quality are its main advantages. Its implementation for cases where dense grids and/or large numbers of conditioning data cause computational problems is discussed. A case study is presented.     

Bayesian Modelling of Spatial Data Using Markov Random Fields, With Application to Elemental Composition of Forest Soil

Spatial datasets are common in the environmental sciences. In this study we suggest a hierarchical model for a spatial stochastic field. The main focus of this article is to approximate a stochastic field with a Gaussian Markov Random Field (GMRF) to exploit computational advantages of the Markov field, concerning predictions, etc. The variation of the stochastic field is modelled as a linear trend plus microvariation in the form of a GMRF defined on a lattice. To estimate model parameters we adopt a Bayesian perspective, and use Monte Carlo integration with samples from Markov Chain simulations. Our methods does not demand lattice, or near-lattice data, but are developed for a general spatial data-set, leaving the lattice to be specified by the modeller. The model selection problem that comes with the artificial grid is in this article addressed with cross-validation, but we also suggest other alternatives. From the application of the methods to a data set of elemental composition of forest soil, we obtained predictive distributions at arbitrary locations as well as estimates of model parameters.     

Analyzing reproduction of correlations in Monte Carlo simulations: application to mine project valuation

Monte Carlo (MC) simulations are extensively used to assess risk in mining ventures; however, the correlation between the inputs used to build the models is often overlooked. We observed how value-at-risk (VaR) of a mining venture was affected by running MC simulations, using two different input correlation methods: Spearman's rank correlation and copulas using Kendall's tau. The goal was to compare different correlation approaches on risk analysis associated with uncertain parameters of mining ventures and uncover which one would yield the most accurate result. Three case studies were carried out to compare correlation structures. Modelling the input variable correlations was better achieved using copulas since they were able to capture a wider range of correlations that did not make any linearity assumptions. In the case study based on MC simulations, the impact of the input correlation choice on the VaR was rather severe with an approximate 9% difference between the results obtained with Spearman's correlations and the Normal copula correlations.     

Empirical Orthogonal Functions Analysis Applied to the Inverse Problem in Hydrogeology: Evaluation of Uncertainty and Simulation of New Solutions

To fulfil the need to generate more realistic solutions, stochastic inverse simulations in hydrogeology are now constrained on both piezometric head and hydraulic conductivity data. These inverse techniques, often based on geostatistics, allow modifications of an initial solution conditioned only on hydraulic conductivity data to arrive at a final solution that also matches observed heads. By repeating the process as many times as necessary with different initial solutions, one generates an ensemble of final solutions thereby addressing the uncertainty of the inverse problem. This requires a method able to handle the whole ensemble and to work on its relevant characteristics. From this standpoint, the analysis by Empirical Orthogonal Functions (EOF) appears promising. The method builds an orthogonal decomposition of the covariance matrix, calculated over the whole set of solutions, and the areas in space where the first functions have a greater influence corresponding to locations of maximum uncertainty in the solutions. These locations depend both on the hydraulic characteristics of the flow problem and on the spatial distribution of available data. The EOF analysis is used on a synthetic problem that mimics a possible behavior of the Culebra aquifer of the Waste Isolation Pilot Plant (WIPP, New Mexico). The method also allows new solutions to be generated at lower computational cost by a random composition of the functions obtained by the EOF analysis. These new solutions keep the main characteristics of the initial ensemble and because they can be conditioned, they return very good results when they are used to solve the direct problem.     

蒙特卡罗法在势流计算中的应用研究

针对水动力学实际问题多存在复杂几何边界的状况,提出了用不规则游动网格求解偏微分方程的蒙特卡罗法,建立了相应的随机游动模型。选择具有复杂自由面的堰闸流动问题作为算例,验证了新方法的正确性。与有限元法相比,蒙特卡罗法解势流等线性问题时更灵活,可以根据需要,单独计算流动区域内任意一点的未知物理量,且所用计算容量较少。     

Analysis of cation distribution in the octahedral sheet of dioctahedral 2:1 phyllosilicates by using inverse Monte Carlo methods

An inverse Monte Carlo (MC) method was developed to determine the distribution of octahedral cations (Al³⁺, Fe³⁺, and Mg²⁺) in bentonite illite–smectite (I–S) samples (dioctahedral 2:1 phyllosilicates) using FT–IR and ²⁷Al MAS NMR spectroscopies. FT–IR allows determination of the nature and proportion of different cation pairs bound to OH groups measuring the intensities of OH-bending bands. ²⁷Al MAS NMR data provide information about cation configuration because ²⁷Al MAS NMR intensity depends on Fe distribution. MC calculations based on FT–IR data alone show Fe segregation by short-range ordering (Fe clusters within 9 to 15?Å from a given Fe atom). Fe segregation increases with illite proportion. MC calculations based on IR and ²⁷Al NMR simultaneously yield similar configurations in which Fe clusters are smaller. The latter calculations fail to build appropriate cation distributions for those samples with higher number of illite layers and significant Fe content, which is indicative of long-range Fe ordering that cannot be detected by FT–IR and ²⁷Al MAS NMR. The proportion of Mg–Mg pairs is negligible in all samples, and calculations, in which the number of Mg atoms, as second neighbours, is minimised, create appropriate configurations.