On the synthesis of random field sampling from the spectrum: An application to the generation of hydrologic spatial processes

Abstract

A procedure for the synthesis of processes exhibiting temporal and spatial variability is presented. The method involves the addition of harmonics of random frequencies that are sampled from the spectral density function or the radial spectral density function. The process obtained is asymptotically Gaussian and ergodic. An estimate of the error in time or space averages due to the nonergodicity of the process as a function of the number of harmonics is also included. Spatial patterns of variation of geophysical phenomena canbe studied by means of stochastic processes representing these variations in a continuous way over the space considered or at discrete points on it. The stochastic process associated with the second of these representations is called a multivariate process by some authors, whereas others prefer the name multidimensional process. It represents a family of random vectors X(t) = {X•, (t), .--, Xn(t)} v depending on the parameter t, where each one of the components is the value assumed by the process at specified points on the area at time t. In contrast, the continuous representation of such processes requires the theory of random fields (some authors prefer the name multidimensional processes, but it is confusing) in which the stochastic process X(u, t) represents a random variable depending on the parameter u = {X•, ..-, X,•} r, representing the coordinates of a point in an n-dimensional space. The applications of' the theory of multivariate processes to the description of' geophysical phenomena have increased significantly over the last few years, whereas random fields have been almost completely ignored by the applied hydrostatistician. However, the study of random fields is justified, since the effect of coarse discrete representations of the phenomenon under study may not capture adequately the physical reality, whereas the use of a larger number of points, especially over large areas, involves operations with matrices who• size may render the problem impractical, particularly for precipitation due to the highly variable nature of the phenomenon.

Design methodology accounting for the effects of porous medium heterogeneity on hydraulic residence time and biodegradation in horizontal subsurface flow constructed wetlands

“…The hydraulic conductivity fields were generated using a tool based on Mejia's algorithm (Mejia and Rodriguez-Iturbe, 1974) developed by Frenzel (1995) and included in the PMWIN software (Processing Modflow for Windows) (Chiang, 2005). Twelve cases were studied having different standard deviations and correlation lengths.…”

An Approach to Statistical Spatial-Temporal Modeling of Meteorological Fields

“…0, > 0 is a scale parameter controlling the spatial range of correlation, while the smoothness parameter d2 > 0 directly controls the smoothness of the random field. The exponential class corresponds to the subclass with smoothness parameter B2 = 4 ; that is, The subclass defined by 02 = 1 was introduced by Whittle ( 1954) as a model for two-dimensional fields and it is commonly used in hydrology (Creutin and Obled 1982;Jones 1989;Mejia and Rodriguez-Iturbe 1974). As O2 + co, KO( x , y ) + exp ( -I x -y I / 0 : ) , often called the "Gaussian" covariance function.…”

A study of mean areal precipitation and spatial structure of rainfall distribution in the Tsen‐Wen river basin

“…Due to spatial patterns of topographic variation in the study area, spatial rainfall distributions are considered as random fields so that the distributions can be studied by means of stochastic approaches over the field (Mejia and Rodriguez-Iturbe, 1974; Bras and Rodriguez-Iturbe, 1985; Chua and Bras, 1980). The method for using stochastic approaches for random fields is by analyzing the spatial structure of the fields (de Marsily, 1986;Bakr, 1976;Gelhar, 1986; Kemblowski and Wen, 1993).…”