An efficient and accurate algorithm for generating spatially-correlated random fields (original) (raw)
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Generating spatially correlated fields with a genetic algorithm
Computers & Geosciences, 1998
AbstractÐGenerated realizations of random ®elds are used to quantify the natural variability of geological properties. When the realizations are used as inputs for simulations with a deterministic model, it may be desirable to minimize dierences between statistics of sequential realizations and make the statistics close to ones speci®ed at generating the realizations. We describe the use of a genetic algorithm (GA) for this purpose. In unconditioned simulations, statistics of the GA-generated realizations were signi®cantly closer to the input ones than those from sequential Gaussian simulations. Distributions of generated values at a particular node over sequential realizations were close to the normal distribution. The GA is computationally intensive and may not be suitable for ®ne grids. The sequential Gaussian algorithm conditioned with GA-generated values on a coarse grid can produce a set of realizations with similar statistics for the ®ne grids embedding the coarse one. #
Joint High-Order Simulation of Spatially Correlated Variables Using High-Order Spatial Statistics
Joint geostatistical simulation techniques are used to quantify uncertainty for spatially correlated attributes, including mineral deposits, petroleum reservoirs, hydrogeological horizons, environmental contaminants. Existing joint simulation methods consider only second-order spatial statistics and Gaussian processes. Motivated by the presence of relatively large datasets for multiple correlated variables that typically are available from mineral deposits and the effects of complex spatial con-nectivity between grades on the subsequent use of simulated realizations, this paper presents a new approach for the joint high-order simulation of spatially correlated random fields. First, a vector random function is orthogonalized with a new decorrelation algorithm into independent factors using the so-termed diagonal domination condition of high-order cumulants. Each of the factors is then simulated independently using a high-order univariate simulation method on the basis of high-order spatial cumulants and Legendre polynomials. Finally, attributes of interest are reconstructed through the back-transformation of the simulated factors. In contrast to state-of-the-art methods, the decorrelation step of the proposed approach not only considers the covariance matrix, but also high-order statistics to obtain independent non-Gaussian factors. The intricacies of the application of the proposed method are shown with a dataset from a multi-element iron ore deposit. The application shows the reproduction of high-order spatial statistics of available data by the jointly simulated attributes.
Conditional Latin Hypercube Simulation of (Log)Gaussian Random Fields
Mathematical geosciences, 2017
In earth and environmental sciences applications, uncertainty analysis regarding the outputs of models whose parameters are spatially varying (or spatially distributed) is often performed in a Monte Carlo framework. In this context, alternative realizations of the spatial distribution of model inputs, typically conditioned to reproduce attribute values at locations where measurements are obtained, are generated via geostatistical simulation using simple random (SR) sampling. The environmental model under consideration is then evaluated using each of these realizations as a plausible input, in order to construct a distribution of plausible model outputs for uncertainty analysis purposes. In hydrogeological investigations, for example, conditional simulations of saturated hydraulic conductivity are used as input to physically-based simulators of flow and transport to evaluate the associated uncertainty in the spatial distribution of solute concentration. Realistic uncertainty analysis via SR sampling, however, requires a large number of simulated attribute realizations for the model inputs in order to yield a representative distribution of model outputs; this often hinders the application of uncertainty analysis due to the computational expense B Stelios Liodakis
Spatial Random Fields (SRFs) are used to characterize the behavior of variables which are difficult to measure everywhere. In the case of hydraulic conductivity used in groundwater models, heterogeneity can be modeled through structural parameters of the random fieldsparameters that define the field's spatial distribution. Characterization of these structural parameters using a stochastic inverse method requires the evaluation of thousands of potential realizations of the SRF which is a time consuming and challenging taskreducing the adoption of stochastic models. To address this issue, this paper presents the integration of high throughput computing using HTCondor with the MAD# frameworkan uncertainty characterization tool that uses the Method of Anchored Distributions (MAD). MAD# is coupled with HTCondor allowing users to search for the optimal location of anchorsstatistical devices located in the fieldusing the parallelization capabilities of HTCondor. As expected, using this approach, the simulation processing time decreases as the number of instances (CPUs) used in the HTCondor Network increases. Also, weekend results show marked improvement over weekday results likely due to fewer interruptions and reassignment of processing tasks between nodes. This demonstration and implementation of an SRF characterization process in a parallel environment shows potential for broader use of the method in environmental modeling.
The Turning Bands Method for simulation of random fields using line generation by a spectral method
Water Resources Research, 1982
The turning bands method (TBM) for the simulation of multidimensional random fields is presented. These fields commonly occur in the Monte Carlo simulation of hydrologic processes, particularly groundwater flow and mass transport. The general TBM equations for two-and three-dimensional fields are derived with particular emphasis on the more complicated two-dimensional case. For stationary two-dimensional fields the unidimensional line process is generated by a simple spectral method, a technique which can be generally applied to any two-dimensional covariance function and which is easily extended to anisotropic and areal averaged processes. Theoretically and by example the TBM is shown to be ergodic even for a finite number of lines, and it is demonstrated that it rapidly converges to the true statistics of the field. Guidelines are presented for the selection of model parameters which will be helpful in the design of simulation experiments. The TBM is compared to other methods in terms of cost and accuracy, demonstrating that the TBM is as accurate as and much less expensive than multidimensional spectral techniques and more accurate than the most expensive approaches which use matrix inversion, such as the nearest neighbor approach. The unidimensional spectral technique presented here permits, for the first time, the inexpensive and accurate TBM simulation of any proper two-dimensional covariance function and should be of some help in the stochastic analysis of hydrologic processes.
Estimation of correlation structure for a homogeneous isotropic random field: A simulation study
Computers & Geosciences, 1988
ln this paper we investigate the effect of sampling density on the estimation of the covariance and semivariogram for homogeneous, isotropic, random fields. Two methods based on the least-squares principle, and a third method known as the Minimum Interpolation Error method are studied when the analytic form of the covariance or semivariogram model is known a priori. The analysis is accomplished through a single realization simulation experiment which is felt to represent the type of conditions usually encountered in real world environmental and geophysical field problems. The Turning Bands method is used to generate the field at randomly distributed sampling points in a fixed field for three types of correlation structure: exponential, Besscl, and Gaussian models. The performance of the three estimation methods is evaluated for varying sampling densities and correlation distances. The main results are: the least-squares methods work best for preserving the pattern of correlation in most situations examined: for a domain of fixed size. the ratio of the correlation distance to the length scale of the field is a measure of the "information" contained in the field, and when this ratio exceeded ~ 0.2 the statistics of the process became inaccurate. On the other hand, when this ratio is < 0.2 reasonable estimates for the mean and variance were determined even for small sampling densities (~ 25 50). The implications for practical problems are di,~ussed.
A general parallelization strategy for random path based geostatistical simulation methods
2010
The size of simulation grids used for numerical models has increased by many orders of magnitude in the past years, and this trend is likely to continue. Efficient pixel-based geostatistical simulation algorithms have been developed, but for very large grids and complex spatial models, the computational burden remains heavy. As cluster computers become widely available, using parallel strategies is a natural step for increasing the usable grid size and the complexity of the models. These strategies must profit from of the possibilities offered by machines with a large number of processors. On such machines, the bottleneck is often the communication time between processors. We present a strategy distributing grid nodes among all available processors while minimizing communication and latency times. It consists in centralizing the simulation on a master processor that calls other slave processors as if they were functions simulating one node every time. The key is to decouple the sending and the receiving operations to avoid synchronization. Centralization allows having a conflict management system ensuring that nodes being simulated simultaneously do not interfere in terms of neighborhood. The strategy is computationally efficient and is versatile enough to be applicable to all random path based simulation methods.
Exploring Monte Carlo Simulation Strategies for Geoscience Applications
Computer simulations are an increasingly important area of geoscience research and development. At the core of stochastic or Monte Carlo simulations are the random number sequences that are assumed to be distributed with specific characteristics. Computer-generated random numbers, uniformly distributed on (0, 1), can be very different depending on the selection of pseudo-random number (PRN) or chaotic random number (CRN) generators. In the evaluation of some definite integrals, the resulting error variances can even be of different orders of magnitude. Furthermore, practical techniques for variance reduction such as importance sampling and stratified sampling can be applied in most Monte Carlo simulations and significantly improve the results. A comparative analysis of these strategies has been carried out for computational applications in planar and spatial contexts. Based on these experiments, and on some practical examples of geodetic direct and inverse problems, conclusions and recommendations concerning their performance and general applicability are included.