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  • Articles  (6)
  • random fields  (6)
  • 2010-2014
  • 1990-1994  (5)
  • 1985-1989  (1)
  • 1950-1954
  • Architecture, Civil Engineering, Surveying  (6)
  • 1
    Electronic Resource
    Electronic Resource
    Solution of stochastic groundwater flow by infinite series, and convergence of the one-dimensional expansion (1994)
    Springer
    Stochastic environmental research and risk assessment 8 (1994), S. 139-155 
    Details
    ISSN: 1436-3259
    Keywords: Porous media ; random media ; random fields ; groundwater flow ; stochastic hydrology ; stochastic partial differential equations ; perturbation methods ; Taylor expansions ; hierarchical systems ; Green's functions ; effective conductivity ; homogenization
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract This paper investigates analytical solutions of stochastic Darcy flow in randomly heterogeneous porous media. We focus on infinite series solutions of the steady-state equations in the case of continuous porous media whose saturated log-conductivity (lnK) is a gaussian random field. The standard deviation of lnK is denoted ‘σ’. The solution method is based on a Taylor series expansion in terms of parameter σ, around the value σ=0, of the hydraulic head (H) and gradient (J). The head solution H is expressed, for any spatial dimension, as an infinite hierarchy of Green's function integrals, and the hydraulic gradient J is given by a linear first-order recursion involving a stochastic integral operator. The convergence of the ‘σ-expansion’ solution is not guaranteed a priori. In one dimension, however, we prove convergence by solving explicitly the hierarchical sequence of equations to all orders. An ‘infinite-order stochastic solution is obtained in the form of a σ-power series that converges for any finite value of σ. It is pointed out that other expansion methods based on K rather than lnK yield divergent series. The infinite-order solution depends on the integration method and the boundary conditions imposed on individual order equations. The most flexible and general method is that based on Laplacian Green's functions and boundary integrals. Imposing zero head conditions for all orders greater than one yields meaningful far-field gradient conditions. The whole approach can serve as a basis for treatment of higher-dimensional problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589894
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  • 2
    Electronic Resource
    Electronic Resource
    Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations (1994)
    Springer
    Stochastic environmental research and risk assessment 8 (1994), S. 117-138 
    Details
    ISSN: 1436-3259
    Keywords: Stochastic hydrology ; random fields ; space transformations ; generalized functions
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract 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.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01589893
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  • 3
    Electronic Resource
    Electronic Resource
    Stochastic perturbation analysis of groundwater flow. Spatially variable soils, semi-infinite domains and large fluctuations (1993)
    Springer
    Stochastic environmental research and risk assessment 7 (1993), S. 213-239 
    Details
    ISSN: 1436-3259
    Keywords: Stochastic hydrology ; perturbation ; random fields ; graph theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract As is well known, a complete stochastic solution of the stochastic differential equation governing saturated groundwater flow leads to an infinite hierarchy of equations in terms of higher-order moments. Perturbation techniques are commonly used to close this hierarchy, using power-series expansions. These methods are applied by truncating the series after a finite number of terms, and products of random gradients of conductivity and head potential are neglected. Uncertainty regarding the number or terms required to yield a sufficiently accurate result is a significant drawback with the application of power series-based perturbation methods for such problems. Low-order series truncation may be incapable of representing fundamental characteristics of flow and can lead to physically unreasonable and inaccurate solutions of the stochastic flow equation. To support this argument, one-dimensional, steady-state, saturated groundwater flow is examined, for the case of a spatially distributed hydraulic conductivity field. An ordinary power-series perturbation method is used to approximate the mean head, using second-order statistics to characterize the conductivity field. Then an interactive perturbation approach is introduced, which yields improved results compared to low-order, power-series perturbation methods for situations where strong interactions exist between terms in such approximations. The interactive perturbation concept is further developed using Feynman-type diagrams and graph theory, which reduce the original stochastic flow problem to a closed set of equations for the mean and the covariance functions. Both theoretical and practical advantages of diagrammatic solutions are discussed; these include the study of bounded domains and large fluctuations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01585600
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  • 4
    Electronic Resource
    Electronic Resource
    The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology (1993)
    Springer
    Stochastic environmental research and risk assessment 7 (1993), S. 14-32 
    Details
    ISSN: 1436-3259
    Keywords: Stochastic hydrology ; random fields ; space transformation ; perturbation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract This paper develops concepts and methods to study stochastic hydrologic models. Problems regarding the application of the existing stochastic approaches in the study of groundwater flow are acknowledged, and an attempt is made to develop efficient means for their solution. These problems include: the spatial multi-dimensionality of the differential equation models governing transport-type phenomena; physically unrealistic assumptions and approximations and the inadequacy of the ordinary perturbation techniques. Multi-dimensionality creates serious mathematical and technical difficulties in the stochastic analysis of groundwater flow, due to the need for large mesh sizes and the poorly conditioned matrices arising from numerical approximations. An alternative to the purely computational approach is to simplify the complex partial differential equations analytically. This can be achieved efficiently by means of a space transformation approach, which transforms the original multi-dimensional problem to a much simpler unidimensional space. The space transformation method is applied to stochastic partial differential equations whose coefficients are random functions of space and/or time. Such equations constitute an integral part of groundwater flow and solute transport. Ordinary perturbation methods for studying stochastic flow equations are in many cases physically inadequate and may lead to questionable approximations of the actual flow. To address these problems, a perturbation analysis based on Feynman-diagram expansions is proposed in this paper. This approach incorporates important information on spatial variability and fulfills essential physical requirements, both important advantages over ordinary hydrologic perturbation techniques. Moreover, the diagram-expansion approach reduces the original stochastic flow problem to a closed set of equations for the mean and the covariance function.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01581564
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  • 5
    Electronic Resource
    Electronic Resource
    On the use of Empirical Orthogonal Function (EOF) analysis in the simulation of random fields (1991)
    Springer
    Stochastic environmental research and risk assessment 5 (1991), S. 125-134 
    Details
    ISSN: 1436-3259
    Keywords: Empirical Orthogonal Function analysis ; random fields ; simulation ; non-homogeneous fields
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract In several fields of Geophysics, such as Hydrology, Meteorology or Oceanography, it is often useful to generate random fields, displaying the same variabilitity as the observed variables. Usually, these synthetic data are used as forcing fields into numerical models, to test the sensitivity of their outputs to the variability of the inputs. Examples can be found in subsurface or surface Hydrology and in Meteorology with General Circulation Models (GCM). Different techniques have already been proposed, often based on the spectral representation of the random process, with, usually, assumptions of stationarity. This paper suggests that Empirical Orthogonal Function (EOF) analysis, which leads to the decomposition of the covariance kernel on the set of its eigen-functions, is a possible answer to this problem. The convergence and accuracy of the method are shown to depend mainly on the number of EOFs retained in the expansion of the covariance kemel. This result is confirmed by a comparison with the turning band method and a matrix technique. Furthermore, a synthetic example of non-homogencous fields shows the interest of EOF analysis in the direct simulation of such fields.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01543054
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  • 6
    Electronic Resource
    Electronic Resource
    Fitting a stochastic partial differential equation to aquifer data (1989)
    Springer
    Stochastic environmental research and risk assessment 3 (1989), S. 85-96 
    Details
    ISSN: 1436-3259
    Keywords: Stochastic equations ; irregularly spaced observations ; prediction, interpolation ; random fields
    Source: Springer Online Journal Archives 1860-2000
    Topics: Architecture, Civil Engineering, Surveying , Energy, Environment Protection, Nuclear Power Engineering , Geography , Geosciences
    Notes: Abstract The steady state two dimensional groundwater flow equation with constant transmissivities was studied by Whittle in 1954 as a stochastic Laplace equation. He showed that the correlation function consisted of a modified Bessel function of the second kind, order 1, multiplied by its argument. This paper uses this pioneering work of Whittle to fit an aquifer head field to unequally spaced observations by maximum likelihood. Observational error is also included in the model. Both the isotropic and anisotropic cases are considered. The fitted field is then calculated on a two dimensional grid together with its standard deviation. The method is closely related to the use of two-dimensional splines for fitting surfaces to irregularly spaced observations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01544074
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