A spectral approach to simulating intrinsic random fields with power and spline generalized covariances (original) (raw)

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.

Efficiency and accuracy in simulation of random fields

Probabilistic Engineering Mechanics, 1996

A direct method for the conditional simulation of a stationary, Gaussian scalar random field is compared with an alternative formulation which uses frequency domain probability density functions. In both cases the random field is described by given correlation or spectral density functions, and no restrictions are placed on these functions, except that they must be positive definite. Efficient implementation techniques are investigated for both general methods. The major computational effort in the most efficient implementations of both procedures is in the solution of linear algebraic equations in which the coefficients are spectral densities. The direct method is shown to be significantly more efficient than existing methods for applying the probability density function technique. However, a new implementation method for the latter technique is also presented, and it equals the efficiency of the direct method. Problems of numerical accuracy due to ill-conditioned matrices are shown not to be severe except when using an inherently problematic form for the spectral density. Numerical examples demonstrate that either method can simulate highly coherent time histories.

A Numerical Technique for Simulating Linear Operations on Random Fields

2007

A unifled technique for generating lin- ear operations on homogeneous/non-homogeneous, Gaussian/non-Gaussian random flelds deflned on any subset of the multidimensional Euclidean space is pro- vided. This is based on an approximate series repre- sentation valid for spatial random flelds with arbi- trary covariance function which can be readily real- ized. Furthermore, its applicability as a simulation tool is examined numerically by considering an ex- ample that illustrates its feasibility and accuracy.

Implementation of the three-dimensional turning bands random field generator

Water Resources Research, 1989

Numerical techniques ta generate replicates of spatially correlated random fields are often used ta synthe'size sets of highly variable physical quantities in stochastic models of naturally heterogeneous systems. Within the realm ofhydrologic research, for example, such tools are widely used to develop hypothetical raïnfalI distributions, hydraulic conductivity fields, fracture set properties, and other surface or subsurface flow parameters. The turning bands method is one s;uch algorithm which generates two-and three-dimensional fields by combining values found from a series of onedimensional simulations along lines radiating outward from a coordinate origin. Previous work with two-dimensional algorithms indicates that radial lines evenly spaced about the unit cirde lead ta enhanced convergence properties. The same can be said for the three-dimensional models, but it is more difficult ta choose an arbitrary number of evenly spaced lines about the unit sphere. The current inves-tigation shows that the use of larger numbers of randomly oriented lines (100) can enhance the performance of the three-dimensional algorithm. This improved performance is needed to effectively simulate problems characterized by full three dimensionality and/or anisotropy in either Monte Carlo or single-realization applications. Use of a large number of lines will also reduce the presence of a distortion effect manifested as linelike patterns in the field. Increased computational costs can be reduc'ed by employing a fast Fourier transform technique ta generate the line processes.

An enhanced hybrid method for the simulation of highly skewed non-Gaussian stochastic fields

Computer Methods in Applied Mechanics and Engineering, 2005

In this paper, an enhanced hybrid method (EHM) is presented for the simulation of homogeneous non-Gaussian stochastic fields with prescribed target marginal distribution and spectral density function. The presented methodology constitutes an efficient blending of the Deodatis-Micaletti method with a neural network based function approximation. Precisely, the function fitting ability of neural networks based on the resilient back-propagation (Rprop) learning algorithm is employed to approximate the unknown underlying Gaussian spectrum. The resulting algorithm can be successfully applied for simulating narrow-banded fields with very large skewness at a fraction of the computing time required by the existing methods. Its computational efficiency is demonstrated in three numerical examples involving fields that follow the beta and lognormal distributions.

Simulation of Homogeneous and Partially Isotropic Random Fields

Journal of Engineering Mechanics, 1999

A rigorous methodology for the simulation of homogeneous and partially isotropic multidimensional random fields is introduced. The property of partial isotropy of the random field is explicitly incorporated in the derivation of the algorithm. This consideration reduces significantly the computational effort associated with the generation of sample functions, as compared with the case when only the homogeneity in the field is taken into account. The approach is based on the spectral representation method, utilizes the fast Fourier transform, and generates simulations with random variability in both their amplitudes and phases, or in their phases only. Spatially variable seismic ground motions experiencing loss of coherence are generated as an example application of the developed approach.

Simulation of simply cross correlated random fields by series expansion methods

Structural Safety, 2008

A practical framework for generating cross correlated fields with a specified marginal distribution function, an autocorrelation function and cross correlation coefficients is presented in the paper. The approach relies on well known series expansion methods for simulation of a Gaussian random field. The proposed method requires all cross correlated fields over the domain to share an identical autocorrelation function and the cross correlation structure between each pair of simulated fields to be simply defined by a cross correlation coefficient. Such relations result in specific properties of eigenvectors of covariance matrices of discretized field over the domain. These properties are used to decompose the eigenproblem which must normally be solved in computing the series expansion into two smaller eigenproblems. Such a decomposition represents a significant reduction of computational effort. Non-Gaussian components of a multivariate random field are proposed to be simulated via memoryless transformation of underlying Gaussian random fields for which the Nataf model is employed to modify the correlation structure. In this method, the autocorrelation structure of each field is fulfilled exactly while the cross correlation is only approximated. The associated errors can be computed before performing simulations and it is shown that the errors happen especially in the cross correlation between distant points and that they are negligibly small in practical situations.