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Environmental time series interpolation based on spartan random processes
(2008) D.T. Hristopulos; M. Zukovic
In many environmental applications, time series are either incomplete or irregularly spaced. We investigate the application of the Spartan random process to missing data prediction. We employ a novel modified method of moments (MMoM) for parameter inference. The CPU time of MMoM is shown to be much faster than that of maximum likelihood estimation and almost independent of the data size. We formulate an explicit Spartan interpolator for estimating missing data. The model validation is performed on both synthetic data and real time series of atmospheric aerosol concentrations. The prediction performance is shown to be comparable with that attained by the best linear unbiased (Kolmogorov-Wiener) predictor at reduced computational cost.
A model of machine-direction tension variations in paper webs with runnability applications
(2002) T. Uesaka; D.T. Hristopulos
We developed a model of web dynamics to investigate tension (strain) variations, which play a major role in runnability and paper breaks.We focus on a simplified version of the model, which is valid for low-frequency tension variations (<100 Hz). The model predicts abrupt changes in the web speed and tension at contact points where external forces are applied, high strain rates at nip contacts, and tension surges during start-up, as observed in many pressrooms. We prove that the mechanical draw is an accurate estimate of the strain increment for time-independent, but not fluctuating, web speeds. We also investigate the impact of out-of-round rolls on runnability by coupling the web dynamics model to a weak-link model of web failure. We find that, for moderate roll deformations (<5 mm), the out-of-roundness contributes statistically to the increase in break frequency. The risk of web breaks is significant for extreme deformations (>6 mm), especially when a new roll begins unwinding.
New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity
(Springer-Verlag, 2002) D.T. Hristopulos
Many heterogeneous media and environmental processes are statistically anisotropic, that is, their moments have directional dependence. The term range anisotropy denotes processes that have variograms characterized by direction-dependent correlation lengths and directionally independent sill. We distinguish between two classes of anisotropic covariance models: Class (A) models are reducible to isotropic after rotation and rescaling operations. Class (B) models are separable and reduce to a product of one- dimensional functions along the principal axes. We present a Class (A) model for multiscale processes and suggest applications in subsurface hydrology. This model is based on a truncated power law with short and long-range cutoffs. We also present a family of Class (B) models generated by superellipsoidal functions that are based on non- Euclidean distance metrics. We propose a new method for determining the orientation of the principal axes and the degree of anisotropy (i.e., the ratios of the correlation lengths). This information reduces the degrees of freedom of anisotropic variograms and thus simplifies the estimation procedure. In particular, Class (A) models are reduced to isotropic, and Class (B) models to one-dimensional functions. Our method is based on an explicit relation between the second-rank slope tensor (SRST), which can be estimated from the data, and the second-rank covariance tensor. The method is conceptually simple and numerically efficient. It is more accurate for regular (on-grid) data distributions, but it can also be used for irregular (off-grid) spatial distributions. We illustrate its implementation with numerical simulations.
Permissibility of fractal exponents and models of band-limited two-point functions for fGn and fBm random fields
(Springer-Verlag, 2003) D.T. Hristopulos
The fractional Gaussian noise (fGn) and fractional Brownian motion (fBm) random field models have many applications in the environmental sciences. An issue of practical interest is the permissible range and the relations between different fractal exponents used to characterize these processes. Here we derive the bounds of the covariance exponent for fGn and the Hurst exponent for fBm based on the permissibility theorem by Bochner. We exploit the theoretical constraints on the spectral density to construct explicit two-point (covariance and structure) functions that are band-limited fractals with smooth cutoffs. Such functions are useful for modeling a gradual cutoff of power-law correlations. We also point out certain peculiarities of the relations between fractal exponents imposed by the mathematical bounds. Reliable estimation of the correlation and Hurst exponents typically requires measurements over a large range of scales (more than 3 orders of magnitude). For isotropic fractals and partially isotropic self-affine processes the dimensionality curse is partially lifted by estimating the exponent from measurements along fixed directions. We derive relations between the fractal exponents and the one-dimensional spectral density exponents, and we illustrate the relations using measurements of paper roughness.
Practical calculation of non-gaussian multivariate moments in spatiotemporal BME analysis
(Kluwer Academic Publishers-Plenum Publishers, 2001) George Christakos; D.T. Hristopulos
During the past decade, the Bayesian maximum entropy (BME) approach has been used with considerable success in a variety of geostatistical applications, including the spatiotemporal analysis and estimation of multivariate distributions. In this work, we investigate methods for calculating the space/time moments of such distributions that occur in BME mapping applications, and we propose general expressions for non-Gaussian model densities based on Gaussian averages. Two explicit approximations for the covariance are derived, one based on leading-order perturbation analysis and the other on the diagrammatic method. The leading-order estimator is accurate only for weakly non-Gaussian densities. The diagrammatic estimator includes higher-order terms and is accurate for larger non-Gaussian deviations. We also formulate general expressions for Monte Carlo moment calculations including precision estimates. A numerical algorithm based on importance sampling is developed, which is computationally efficient for multivariate probability densities with a large number of points in space/time. We also investigate the BME moment problem, which consists in determining the general knowledge-based BME density from experimental measurements. In the case of multivariate densities, this problem requires solving a system of nonlinear integral equations. We refomulate the system of equations as an optimization problem, which we then solve numerically for a symmetric univariate pdf. Finally, we discuss theoretical and numerical issues related to multivariate BME solutions.
Renormalization group methods in subsurface hydrology: Overview and applications in hydraulic conductivity upscaling
(2003) D.T. Hristopulos
The renormalization group (RG) approach is a powerful theoretical framework, more suitable for upscaling strong heterogeneity than low-order perturbation expansions. Applications of RG methods in subsurface hydrology include the calculation of (1) macroscopic transport parameters such as effective and equivalent hydraulic conductivity and dispersion coefficients, and (2) anomalous exponents characterizing the dispersion of contaminants due to long-range conductivity correlations or broad (heavy-tailed) distributions of the groundwater velocity. First, we review the main ideas of RG methods and their hydrological applications. Then, we focus on the hydraulic conductivity in saturated porous media with isotropic lognormal heterogeneity, and we present an RG calculation based on the replica method. The RG analysis gives rigorous support to the exponential conjecture for the effective hydraulic conductivity [38]. Using numerical simulations in two dimensions with a bimodal conductivity distribution, we demonstrate that the exponential expression is not suitable for all types of heterogeneity. We also introduce an RG coarse-grained conductivity and investigate its applications in estimating the conductivity of blocks or flow domains with finite size. Finally, we define the fractional effective dimension, and we show that it justifies fractal exponents in the range 1−2 d ≤α <1 (where d is the actual medium dimension) in the geostatistical power average.
Spartan gibbs random field models for geostatistical applications
(SIAM, 2003) D.T. Hristopulos
The inverse problem of determining the spatial dependence of random fields from an experimental sample is a central issue in Geostatistics. We propose a computationally efficient approach based on Spartan Gibbs random fields. Their probability density function is determined by a small set of parameters, which can be estimated by enforcing sample-based constraints on the stochastic moments. The computational complexity of calculating the constraints increases linearly with the sample size. We investigate a specific Gibbs probability density with spatial dependence derived from generalized gradient and Laplacian operators, and we derive permissibility conditions for the model parameters.
Stochastic diagrammatic analysis of groundwater flow in heterogeneous porous media
(1995) G. Christakos; D.T. Hristopulos; C.T. Miller
The diagrammatic approach is an alternative to standard analytical methods for solving stochastic differential equations governing groundwater flow with spatially variable hydraulic conductivity. This approach uses diagrams instead of abstract symbols to visualize complex multifold integrals that appear in the perturbative expansion of the stochastic flow solution and reduces the original flow problem to a closed set of equations for the mean and the covariance functions. Diagrammatic analysis provides an improved formulation of the flow problem over conventional first-order series approximations, which are based on assumptions such as constant mean hydraulic gradient, infinite flow domain, and neglect of cross correlation terms. This formulation includes simple schemes, like finite-order diagrammatic perturbations that account for mean gradient trends and boundary condition effects, as well as more advanced schemes, like diagrammatic porous media description operators which contain infinite-order correlations. In other words, diagrammatic analysis covers not only the cases where low-order diagrams lead to good approximations of flow, but also those situations where low-order perturbation is insufficient and a more sophisticated analysis is needed. Diagrams lead to a nonlocal equation for the mean hydraulic gradient in terms of which necessary conditions are formulated for the existence of an effective hydraulic conductivity. Three-dimensional flow in an isotropic bounded domain with Dirichlet boundary conditions is considered, and an integral equation for the mean hydraulic head is derived by means of diagrams. This formulation provides an explicit expression for the boundary effects within the three-dimensional flow domain. In addition to these theoretical results, the numerical performance of the diagrammatic approach is tested, and useful insight is obtained by means of one-dimensional flow examples where the exact stochastic solutions are available.

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