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1.
Hristopulos Dionissios T. Pavlides Andrew Agou Vasiliki D. Gkafa Panagiota 《Mathematical Geosciences》2021,53(8):1907-1949
Mathematical Geosciences - Classical geostatistical methods face serious computational challenges if they are confronted with large spatial datasets. The stochastic local interaction (SLI) approach... 相似文献
2.
Dionissios T. Hristopulos 《Stochastic Environmental Research and Risk Assessment (SERRA)》2015,29(3):739-754
Random fields based on energy functionals with local interactions possess flexible covariance functions, lead to computationally efficient algorithms for spatial data processing, and have important applications in Bayesian field theory. In this paper we address the calculation of covariance functions for a family of isotropic local-interaction random fields in two dimensions. We derive explicit expressions for non-differentiable Spartan covariance functions in \({\mathbb{R}}^2\) that are based on the modified Bessel function of the second kind. We also derive a family of infinitely differentiable, Bessel-Lommel covariance functions that exhibit a hole effect and are valid in \({\mathbb{R}}^{d}\), where d > 2. Finally, we define a generalized spectrum of correlation scales that can be applied to both differentiable and non-differentiable random fields in contrast with the smoothness microscale. 相似文献
3.
G. Christakos D. T. Hristopulos 《Stochastic Environmental Research and Risk Assessment (SERRA)》1994,8(2):117-138
In earlier publications, certain applications of space transformation operators in subsurface hydrology were considered. These operators reduce the original multi-dimensional problem to the one-dimensional space, and can be used to study stochastic partial differential equations governing groundwater flow and solute transport processes. In the present work we discuss developments in the theoretical formulation of flow models with space-dependent coefficients in terms of space transformations. The formulation is based on stochastic Radon operator representations of generalized functions. A generalized spectral decomposition of the flow parameters is introduced, which leads to analytically tractable expressions of the space transformed flow equation. A Plancherel representation of the space transformation product of the head potential and the log-conductivity is also obtained. A test problem is first considered in detail and the solutions obtained by means of the proposed approach are compared with the exact solutions obtained by standard partial differential equation methods. Then, solutions of three-dimensional groundwater flow are derived starting from solutions of a one-dimensional model along various directions in space. A step-by-step numerical formulation of the approach to the flow problem is also discussed, which is useful for practical applications. Finally, the space transformation solutions are compared with local solutions obtained by means of series expansions of the log-conductivity gradient. 相似文献
4.
This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous
media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic
representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media.
A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic
terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the
selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression
is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective
hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained.
The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous
limit. 相似文献
5.
Dionissios T. Hristopulos George Christakos 《Stochastic Environmental Research and Risk Assessment (SERRA)》1997,11(5):369-395
This work presents a stochastic diagrammatic theory for the calculation of the effective hydraulic conductivity of heterogeneous
media. The theory is based on the mean-flux series expansion of a log-normal hydraulic conductivity medium in terms of diagrammatic
representations and leads to certain general results for the effective hydraulic conductivity of three-dimensional media.
A selective summation technique is used to improve low-order perturbation analysis by evaluating an infinite set of diagrammatic
terms with a specific topological structure that dominates the perturbation series. For stochastically isotropic media the
selective summation yeilds the anticipated exponential expression for the effective hydraulic conductivity. This expression
is extended to stochastically anisotropic media. It is also shown that in the case of non homogeneous media the uniform effective
hydraulic conductivity is replaced by a non-local tensor kernel, for which general diagrammatic expressions are obtained.
The non-local kernel leads to the standard exponential behavior for the effective hydraulic conductivity at the homogeneous
limit. 相似文献
6.
New anisotropic covariance models and estimation of anisotropic parameters based on the covariance tensor identity 总被引：2，自引：0，他引：2
Many heterogeneous media and environmental processes are statistically anisotropic. In this paper we focus on range anisotropy,
that is, stochastic processes with variograms that have direction dependent correlation lengths and direction 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 can be separated into a product of one-dimensional functions oriented
along the principal axes. We propose a new Class (A) model with multiscale properties that has applications in subsurface
hydrology. We also present a family of Class (B) models based on non-Euclidean distance metrics that are generated by superellipsoidal
functions. Next, we propose a new method for determining the orientation of the principal axes and the degree of anisotropy,
i.e., the ratio(s) 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 covariance tensor. The procedure is conceptually simple and numerically efficient.
It is more accurate for regular (on-grid) data distributions, but it can also be used for sparse (off-grid) spatial distributions.
In the case of non-differentiable random fields the method can be extended using generalized derivatives. We illustrate its
implementation with numerical simulations. 相似文献
7.
Relationships between correlation lengths and integral scales for covariance models with more than two parameters 总被引：2，自引：2，他引：0
Dionissios T. Hristopulos Milan ?ukovi? 《Stochastic Environmental Research and Risk Assessment (SERRA)》2011,25(1):11-19
In geostatistical applications, the terms correlation length and range are often used interchangeably and refer to a characteristic
covariance length ξ that normalizes the lag distance in the variogram or the covariance model. We present equations that strictly
define the correlation length (r
c
) and integral range (ℓ
c
). We derive analytical expressions for r
c
and ℓ
c
of the Whittle–Matérn, fluctuation gradient curvature and rational quadratic covariances. For these covariances, we show
that the correlation length and integral range for a given model are not fully determined by ξ. We define non-trivial covariance
functions, and we formulate an ergodicity index based on ℓ
c
. We propose using the ergodicity index to compare coarse-grained measures corresponding to non-trivial covariance functions
with different parameters. Finally, we discuss potential applications of the proposed covariance models in stochastic subsurface
hydrology. 相似文献
8.
Milan Žukovič Dionissios T. Hristopulos 《Stochastic Environmental Research and Risk Assessment (SERRA)》2013,27(4):785-806
This paper addresses the issue of missing data reconstruction for partially sampled, two-dimensional, rectangular grid images of differentiable random fields. We introduce a stochastic gradient–curvature (GC) reconstruction method, which is based on the concept of a random field model defined by means of local interactions (constraints). The GC reconstruction method aims to match the gradient and curvature constraints for the entire grid with those of the sample using conditional Monte Carlo simulations that honor the sample values. The GC reconstruction method does not assume a parametric form for the underlying probability distribution of the data. It is also computationally efficient and requires minimal user input, properties that make it suitable for automated processing of large data sets (e.g. remotely sensed images). The GC reconstruction performance is compared with established classification and interpolation methods for both synthetic and real world data. The impact of various factors such as domain size, degree of thinning, discretization, initialization, correlation properties, and noise on GC reconstruction performance are investigated by means of simulated random field realizations. An assessment of GC reconstruction performance on real data is conducted by removing randomly selected and contiguous groups of points from satellite rainfall data and an image of the lunar surface. 相似文献
9.
An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities 总被引：4，自引：2，他引：2
Samuel N. Elogne Dionissios T. Hristopulos Emmanouil Varouchakis 《Stochastic Environmental Research and Risk Assessment (SERRA)》2008,22(5):633-646
This paper investigates the potential of Spartan spatial random fields (SSRFs) in real-time mapping applications. The data
set that we study focuses on the distribution of daily gamma dose rates over part of Germany. Our goal is to determine a Spartan
spatial model from the data, and then use it to generate “predictive” maps of the radioactivity. In the SSRF framework, the
spatial dependence is determined from sample functions that focus on short-range correlations. A recently formulated SSRF
predictor is used to derive isolevel contour maps of the dose rates. The SSRF predictor is explicit. Moreover, the adjustments that it requires by the user are reduced compared to classical geostatistical methods. These features
present clear advantages for an automatic mapping system. The performance of the SSRF predictor is evaluated by means of various
cross-validation measures. The values of the performance measures are similar to those obtained by classical geostatistical
methods. Application of the SSRF method to data that simulate a radioactivity release scenario is also discussed. Hot spots
are detected and removed using a heuristic method. The extreme values that appear in the path of the simulated plume are not
captured by the currently used Spartan spatial model. Modeling of the processes leading to extreme values can enhance the
predictive capabilities of the spatial model, by incorporating physical information. 相似文献
10.
Renormalization group analysis of permeability upscaling 总被引：1，自引：1，他引：0
D. T. Hristopulos G. Christakos 《Stochastic Environmental Research and Risk Assessment (SERRA)》1999,13(1-2):131-161
The heterogeneity of the subsurface permeability is considered as the most influential factor in determining groundwater flow and the transport of toxic contaminants. Numerical simulators cannot handle the large grids required to represent the small-scale variability of permeability, and thus explicit estimates of the large-scale behavior in terms of coarse-grained parameters are often required. Perturbation formulations of the effective permeability are based on simplifying assumptions that are valid only for certain probability distributions and weak heterogeneity. A generalized perturbation ansatz that involves higher orders has been proposed (Gelhar and Axness, 1983), but to our knowledge its validity has not been rigorously proved before in three dimensions. In this work we propose a general upscaling formulation valid for strong heterogeneity, general permeability distributions, and media with impermeable zones. We show that the effective permeability is determined by the self-energy series of the permeability fluctuations at zero frequency. Using the diagrammatic representation, we obtain a Dyson equation that involves only irreducible diagrams of the proper self-energy series. We develop a renormalization group (RG) analysis for isotropic lognormal media that proves the generalized perturbation ansatz to all orders. We show that the RG result accurately estimates laboratory permeability measurements in limestone (strong heterogeneity) and sandstone (weak heterogeneity). We also propose an explicit RG estimate for the preasymptotic effective permeability. We compare our results with an approach based on a leading order Green's function expansion (Paleologos et?al., 1996), which, however, requires intensive numerical computations. Finally, we investigate the relation between the RG expression and the algebraic means used in numerical upscaling. 相似文献