Dynamic stochastic estimation of physical variables

A fundamental problem facing the physical sciences today is analysis of natural variations and mapping of spatiotemporal processes. Detailed maps describing the space/time distribution of groundwater contaminants, atmospheric pollutant deposition processes, rainfall intensity variables, external intermittency functions, etc. are tools whose importance in practical applications cannot be overestimated. Such maps are valuable inputs for numerous applications including, for example, solute transport, storm modeling, turbulent-nonturbulent flow characterization, weather prediction, and human exposure to hazardous substances. The approach considered here uses the spatiotemporal random field theory to study natural space/time variations and derive dynamic stochastic estimates of physical variables. The random field model is constructed in a space/time continuum that explicitly involves both spatial and temporal aspects and provides a rigorous representation of spatiotemporal variabilities and uncertainties. This has considerable advantages as regards analytical investigations of natural processes. The model is used to study natural space/time variations of springwater calcium ion data from the Dyle River catchment area, Belgium. This dataset is characterized by a spatially nonhomogeneous and temporally nonstationary variability that is quantified by random field parameters, such as orders of space/time continuity and random field increments. A rich class of covariance models is determined from the properties of the random field increments. The analysis leads to maps of continuity orders and covariances reflecting space/time calcium ion correlations and trends. Calcium ion estimates and the associated statistical errors are calculated at unmeasured locations/instants over the Dyle region using a space/time kriging algorithm. In practice, the interpretation of the results of the dynamic stochastic analysis should take into consideration the scale effects.     

On-line estimation of nonlinear physical systems

Recursive algorithms for estimating states of nonlinear physical systems are presented. Orthogonality properties are rediscovered and the associated polynomials are used to linearize state and observation models of the underlying random processes. This requires some key hypotheses regarding the structure of these processes, which may then take account of a wide range of applications. The latter include streamflow forecasting, flood estimation, environmental protection, earthquake engineering, and mine planning. The proposed estimation algorithm may be compared favorably to Taylor series-type filters, nonlinear filters which approximate the probability density by Edgeworth or Gram-Charlier series, as well as to conventional statistical linearization-type estimators. Moreover, the method has several advantages over nonrecursive estimators like disjunctive kriging. To link theory with practice, some numerical results for a simulated system are presented, in which responses from the proposed and extended Kalman algorithms are compared.     

Stochastic perturbation analysis of groundwater flow. Spatially variable soils,semi-infinite domains and large fluctuations

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.     

A BME solution of the inverse problem for saturated groundwater flow

In most real-world hydrogeologic situations, natural heterogeneity and measurement errors introduce major sources of uncertainty in the solution of the inverse problem. The Bayesian Maximum Entropy (BME) method of modern geostatistics offers an efficient solution to the inverse problem by first assimilating various physical knowledge bases (hydrologic laws, water table elevation data, uncertain hydraulic resistivity measurements, etc.) and then producing robust estimates of the subsurface variables across space. We present specific methods for implementing the BME conceptual framework to solve an inverse problem involving Darcys law for subsurface flow. We illustrate one of these methods in the case of a synthetic one-dimensional case study concerned with the estimation of hydraulic resistivity conditioned on soft data and hydraulic head measurements. The BME framework processes the physical knowledge contained in Darcys law and generates accurate estimates of hydraulic resistivity across space. The optimal distribution of hard and soft data needed to minimize the associated estimation error at a specified sampling cost is determined. This work was supported by grants from the National Institute of Environmental Health Sciences (Grant no. 5 P42 ES05948 and P30ES10126), the National Aeronautics and Space Administration (Grant no. 60-00RFQ041), the Army Research Office (Grant no. DAAG55-98-1-0289), and the National Science Foundation under Agreement No. DMS-0112069.     

Spatial statistics of clustered data

Modern spatial statistics techniques are widely used to make predictions for natural processes that are continuously distributed over some convex domain. Implementation of these techniques often relies on the adequate estimation of certain spatial correlation functions such as the covariance and the variogram from the data sets available. This work studies the practical estimation of such spatial correlation functions in the case of clustered data. The coefficient of variation of the dimensionless spatial density of the point pattern of sample locations is suggested as a useful metric for degree of clusteredness of the clustered data set. We show that the common variogram estimator becomes increasingly unreliable with increasing coefficient of variation of the dimensionless spatial density of the point pattern of sample locations. Moreover, we present a modified form of the variogram estimator that incorporates declustering weights, and propose a scheme for estimating the declustering weights based on zones of proximity. Finally, insight is gained in terms of a numerical application of the common and modified methods on piezometric head data collected over an irregular network.Acknowledgments. This work has been supported by grants from the National Institute of Environmental Health Sciences (P42 ES05948-02), the Army Research Office (DAAG55-98-1-0289), and the National Aeronautics and Space Administration (60-00RFQ041). Some of the calculations conducted in support of this work were done on the SGI Origin 2400 at the North Carolina Supercomputing Center, RTP, NC.     

Improved space–time sea surface salinity mapping in Western Pacific ocean using contingogram modeling

Stochastic Environmental Research and Risk Assessment - In the context of statistical correlation theory and geostatistics, the covariance function has been widely used to characterize the...     

The development of stochastic space transformation and diagrammatic perturbation techniques in subsurface hydrology

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.     

Stochastic space transforms in subsurface hydrology — Part 2: Generalized spectral decompositions and plancherel representations

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.     

Assimilation of fuzzy data by the BME method

Modern spatiotemporal geostatistics provides a powerful framework for generation of predictive maps over a spatiotemporal domain by accounting for general knowledge to define a space of plausible events and then restricting this space of plausible events to be consistent with available site-specific knowledge. The Bayesian maximum entropy (BME) method is one of the most widely used modern geostatistics methods. BME results from assigning probabilities of plausible events based on general knowledge through information maximization and then applying operational Bayesian conditionalization that can explicitly assimilate stochastic representations of various uncertain (soft) data bases. The paper demonstrates that fuzzy data sets can be indirectly assimilated by BME through a two-step process: (a) reinterpretation of the fuzzy data as probabilistic through a generalized defuzzification procedure, and (b) efficient assimilation of the probabilistic results of generalized defuzzification by the BME method. A numerical demonstration involves site-specific probabilistic results obtained from the generalized defuzzification of a simulated fuzzy data set and general knowledge that includes the spatial mean trend and correlation structure models. The parameters of these models can be inferred from the hard data equivalent values of the probabilistic results. Accordingly, details of inference based on probabilistic soft data are also considered.     

Analysis and Estimation of Natural Processes with Nonhomogeneous Spatial Variation Using Secondary Information

Natural processes encountered in mining, hydrogeologic, environmental, etc. applications usually are poorly known because of scarcity of data over the area of interest. Therefore, stochastic estimation techniques are the tool of choice for a careful accounting of the heterogeneity and uncertainty involved. Within such a framework, a better utilization of all available data concerning the process of interest and all other natural processes related to it, is of primary importance. Because many natural processes show complicated spatial trends, the hypothesis of spatial homogeneity cannot be invoked always, and the more general theory of intrinsic spatial random fields should be employed. Efficient use of secondary information in terms of the intrinsic model requires that suitable permissibility criteria for the generalized covariances and cross-covariances are satisfied. A set of permissibility criteria are presented for the situation of two intrinsic random fields. These criteria are more general and comprehensive than the ones currently available in the geostatistical literature. A constrained least-square technique is implemented for the inference of the generalized covariance and cross-covariance parameters, and a synthetic example is used to illustrate the methodology. The numerical results show that the use of secondary information can lead to significant reductions in the estimation errors.