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1.
It is common in geostatistics to use the variogram to describe the spatial dependence structure and to use kriging as the spatial prediction methodology. Both methods are sensitive to outlying observations and are strongly influenced by the marginal distribution of the underlying random field. Hence, they lead to unreliable results when applied to extreme value or multimodal data. As an alternative to traditional spatial modeling and interpolation we consider the use of copula functions. This paper extends existing copula-based geostatistical models. We show how location dependent covariates e.g. a spatial trend can be accounted for in spatial copula models. Furthermore, we introduce geostatistical copula-based models that are able to deal with random fields having discrete marginal distributions. We propose three different copula-based spatial interpolation methods. By exploiting the relationship between bivariate copulas and indicator covariances, we present indicator kriging and disjunctive kriging. As a second method we present simple kriging of the rank-transformed data. The third method is a plug-in prediction and generalizes the frequently applied trans-Gaussian kriging. Finally, we report on the results obtained for the so-called Helicopter data set which contains extreme radioactivity measurements.  相似文献   

2.
This paper presents new ideas on sampling design and minimax prediction in a geostatistical model setting. Both presented methodologies are based on regression design ideas. For this reason the appendix of this paper gives an introduction to optimum Bayesian experimental design theory for linear regression models with uncorrelated errors. The presented methodologies and algorithms are then applied to the spatial setting of correlated random fields. To be specific, in Sect. 1 we will approximate an isotropic random field by means of a regression model with a large number of regression functions with random amplitudes, similarly to Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997). These authors make use of the Karhunen Loeve approximation of the isotropic random field. We use the so-called polar spectral approximation instead; i.e. we approximate the isotropic random field by means of a regression model with sine-cosine-Bessel surface harmonics with random amplitudes and then, in accordance with Fedorov and Flanagan (J Combat Inf Syst Sci: 23, 1997), apply standard Bayesian experimental design algorithms to the resulting Bayesian regression model. Section 2 deals with minimax prediction when the covariance function is known to vary in some set of a priori plausible covariance functions. Using a minimax theorem due to Sion (Pac J Math 8:171–176, 1958) we are able to formulate the minimax problem as being equivalent to an optimum experimental design problem, too. This makes the whole experimental design apparatus available for finding minimax kriging predictors. Furthermore some hints are given, how the approach to spatial sampling design with one a priori fixed covariance function may be extended by means of minimax kriging to a whole set of a priori plausible covariance functions such that the resulting designs are robust. The theoretical developments are illustrated with two examples taken from radiological monitoring and soil science.  相似文献   

3.
Moving window kriging with geographically weighted variograms   总被引：2，自引：2，他引：0  
This study adds to our ability to predict the unknown by empirically assessing the performance of a novel geostatistical-nonparametric hybrid technique to provide accurate predictions of the value of an attribute together with locally-relevant measures of prediction confidence, at point locations for a single realisation spatial process. The nonstationary variogram technique employed generalises a moving window kriging (MWK) model where classic variogram (CV) estimators are replaced with information-rich, geographically weighted variogram (GWV) estimators. The GWVs are constructed using kernel smoothing. The resultant and novel MWK–GWV model is compared with a standard MWK model (MWK–CV), a standard nonlinear model (Box–Cox kriging, BCK) and a standard linear model (simple kriging, SK), using four example datasets. Exploratory local analyses suggest that each dataset may benefit from a MWK application. This expectation was broadly confirmed once the models were applied. Model performance results indicate much promise in the MWK–GWV model. Situations where a MWK model is preferred to a BCK model and where a MWK–GWV model is preferred to a MWK–CV model are discussed with respect to model performance, parameterisation and complexity; and with respect to sample scale, information and heterogeneity.  相似文献   

4.
 This paper deals with the problem of spatial data mapping. A new method based on wavelet interpolation and geostatistical prediction (kriging) is proposed. The method – wavelet analysis residual kriging (WARK) – is developed in order to assess the problems rising for highly variable data in presence of spatial trends. In these cases stationary prediction models have very limited application. Wavelet analysis is used to model large-scale structures and kriging of the remaining residuals focuses on small-scale peculiarities. WARK is able to model spatial pattern which features multiscale structure. In the present work WARK is applied to the rainfall data and the results of validation are compared with the ones obtained from neural network residual kriging (NNRK). NNRK is also a residual-based method, which uses artificial neural network to model large-scale non-linear trends. The comparison of the results demonstrates the high quality performance of WARK in predicting hot spots, reproducing global statistical characteristics of the distribution and spatial correlation structure.  相似文献   

5.
Abstract

The present research study investigates the application of nonlinear normalizing data transformations in conjunction with ordinary kriging (OK) for the accurate prediction of groundwater level spatial variability in a sparsely-gauged basin. We investigate three established normalizing methods, Gaussian anamorphosis, trans-Gaussian kriging and the Box-Cox method to improve the estimation accuracy. The first two are applied for the first time to groundwater level data. All three methods improve the mean absolute prediction error compared to the application of OK to the non-transformed data. In addition, a modified Box-Cox transformation is proposed and applied to normalize the hydraulic heads. The modified Box-Cox transformation in conjunction with OK is found to be the optimal spatial model based on leave-one-out cross-validation. The recently established Spartan semivariogram family provides the optimal model fit to the transformed data. Finally, we present maps of the groundwater level and the kriging variance based on the optimal spatial model.

Editor D. Koutsoyiannis; Associate editor A. Montanari

Citation Varouchakis, E.A., Hristopoulos, D.T., and Karatzas, G.P., 2012. Improving kriging of groundwater level data using nonlinear normalizing transformations—a field application. Hydrological Sciences Journal, 57 (7), 1404–1419.  相似文献   

6.
In many studies, the distribution of soil attributes depends on both spatial location and environmental factors, and prediction and process identification are performed using existing methods such as kriging. However, it is often too restrictive to model soil attributes as dependent on a known, parametric function of environmental factors, which kriging typically assumes. This paper investigates a semiparametric approach for identifying and modeling the nonlinear relationships of spatially dependent soil constituent levels with environmental variables and obtaining point and interval predictions over a spatial region. Frequentist and Bayesian versions of the proposed method are applied to measured soil nitrogen levels throughout Florida, USA and are compared to competing models, including frequentist and Bayesian kriging, based an array of point and interval measures of out-of-sample forecast quality. The semiparametric models outperformed competing models in all cases. Bayesian semiparametric models yielded the best predictive results and provided empirical coverage probability nearly equal to nominal.  相似文献   

7.
Obtaining new and flexible classes of nonseparable spatio-temporal covariances and variograms has resulted a key point of research in the last years. The goal of this paper is to introduce and develop new spatio-temporal covariance models taking into account the problem of spatial anisotropy. Recent literature has focused on the problem of full symmetry and the problem of anisotropy has been overcome. Here we propose a generalization of Gneiting’s (J Am Stat Assoc 97:590–600, 2002a) approach and obtain new classes of stationary nonseparable spatio-temporal covariance functions which are spatially anisotropic. The resulting structures are proved to have certain interesting mathematical properties, together with a considerable applicability.Work partially funded by grant MTM2004-06231 from the Spanish Ministry of Science and Education.  相似文献   

8.
Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.  相似文献   

9.
Simulation of multigaussian stochastic fields can be made after a Karhunen-Loéve expansion of a given covariance function. This method is also called simulation by Empirical Orthogonal Functions. The simulations are made by drawing stochastic coefficients from a random generator. These numbers are multiplied with eigenfunctions and eigenvalues derived from the predefined covariance model. The number of eigenfunctions necessary to reproduce the stochastic process within a predefined variance error, turns out to be a cardinal question. Some ordinary analytical covariance functions are used to evaluate how quickly the series of eigenfunctions can be truncated. This analysis demonstrates extremely quick convergence to 99.5% of total variance for the 2nd order exponential (‘gaussian’) covariance function, while the opposite is true for the 1st order exponential covariance function. Due to these convergence characteristics, the Karhunen-Loéve method is most suitable for simulating smooth fields with ‘gaussian’ shaped covariance functions. Practical applications of Karhunen-Loéve simulations can be improved by spatial interpolation of the eigenfunctions. In this paper, we suggest interpolation by kriging and limits for reproduction of the predefined covariance functions are evaluated.  相似文献   

10.
This study is an extension of the stochastic analysis of transient two-phase flow in randomly heterogeneous porous media (Chen et al. in Water Resour Res 42:W03425, 2006), by incorporating direct measurements of the random soil properties. The log-transformed intrinsic permeability, soil pore size distribution parameter, and van Genuchten fitting parameter are treated as stochastic variables that are normally distributed with a separable exponential covariance model. These three random variables conditioned on given measurements are decomposed via Karhunen–Loève decomposition. Combined with the conditional eigenvalues and eigenfunctions of random variables, we conduct a series of numerical simulations using stochastic transient water–oil flow model (Chen et al. in Water Resour Res 42:W03425, 2006) based on the KLME approach to investigate how the number and location of measurement points, different random soil properties, as well as the correlation length of the random soil properties, affect the stochastic behavior of water and oil flow in heterogeneous porous media.  相似文献   

11.
In this paper, a certain bivariate exponential distribution is used for the spatial prediction. The unobserved random variable is predicted by the projection onto the space of all linear combinations of the powers, up to degree m, of the observed random variables plus the constant 1. We obtain a solution by assuming that all the bivariate distributions follow Gumbel’s type III or logistic form of bivariate exponential. The method is implemented on two data sets and the results are presented. The predictions are compared with the original values through Mean Structural Similarity (MSSIM) index of Wang et al. (IEEE Trans Image Process 13(4):600–612, 2004). Using the MSSIM index the proposed method is also compared with Ordinary Kriging and with Simple Kriging after normal score transform.  相似文献   

12.
The spatial distribution of residual light non-aqueous phase liquid (LNAPL) is an important factor in reactive solute transport modeling studies. There is great uncertainty associated with both the areal limits of LNAPL source zones and smaller scale variability within the areal limits. A statistical approach is proposed to construct a probabilistic model for the spatial distribution of residual NAPL and it is applied to a site characterized by ultra-violet-induced-cone-penetration testing (CPT–UVIF). The uncertainty in areal limits is explicitly addressed by a novel distance function (DF) approach. In modeling the small-scale variability within the areal limits, the CPT–UVIF data are used as primary source of information, while soil texture and distance to water table are treated as secondary data. Two widely used geostatistical techniques are applied for the data integration, namely sequential indicator simulation with locally varying means (SIS–LVM) and Bayesian updating (BU). A close match between the calibrated uncertainty band (UB) and the target probabilities shows the performance of the proposed DF technique in characterization of uncertainty in the areal limits. A cross-validation study also shows that the integration of the secondary data sources substantially improves the prediction of contaminated and uncontaminated locations and that the SIS–LVM algorithm gives a more accurate prediction of residual NAPL contamination. The proposed DF approach is useful in modeling the areal limits of the non-stationary continuous or categorical random variables, and in providing a prior probability map for source zone sizes to be used in Monte Carlo simulations of contaminant transport or Monte Carlo type inverse modeling studies.  相似文献   

13.
A regression model is used to study spatiotemporal distributions of solute content ion concentration data (calcium, chloride and nitrate), which provide important water contamination indicators. The model consists of three random and one deterministic components. The random space/time component is assumed to be homogeneous/stationary and to have a separable covariance. The purely spatial and the purely temporal random components are assumed to have homogenous and stationary increments, respectively. The deterministic component represents the space/time mean function. Inferences of the random components involve maximum likelihood and semi-parametric methods under some restrictions on the data configuration. Computational advantages and modelling limitations of the assumptions underlying the regression model are discussed. The regression model leads to simplifications in the space/time kriging and cokriging systems used to obtain space/time estimates at unobservable locations/instants. The application of the regression model in the study of the solute content ions was done at a global scale that covers the entire region of interest. The variability analysis focuses on the calculation of the spatial direct and cross-variograms and the evaluation of correlations between the three solute content ions. The space/time kriging system is developed in terms of the space direct and cross-variograms, and allows the separate estimation of the regression model components. Maps of these components are then obtained for each one of the three ions. Using the estimates of the purely spatial component, spatial dependencies between the ions are studied. Physical causes and consequences of the space/time variability are discussed, and comparisons are made with previous analyses of the solute content dataset.  相似文献   

14.
Data collected along transects are becoming more common in environmental studies as indirect measurement devices, such as geophysical sensors, that can be attached to mobile platforms become more prevalent. Because exhaustive sampling is not always possible under constraints of time and costs, geostatistical interpolation techniques are used to estimate unknown values at unsampled locations from transect data. It is known that outlying observations can receive significantly greater ordinary kriging weights than centrally located observations when the data are contiguously aligned along a transect within a finite search window. Deutsch (1994) proposed a kriging algorithm, finite domain kriging, that uses a redundancy measure in place of the covariance function in the data-to-data kriging matrix to address the problem of overweighting the outlying observations. This paper compares the performances of two kriging techniques, ordinary kriging (OK) and finite domain kriging (FDK), on examining unexploded ordnance (UXO) densities by comparing prediction errors at unsampled locations. The impact of sampling design on object count prediction is also investigated using data collected from transects and at random locations. The Poisson process is used to model the spatial distribution of UXO for three 5000 × 5000 m fields; one of which does not have any ordnance target (homogeneous field), while the other two sites have an ordnance target in the center of the site (isotropic and anisotropic fields). In general, for a given sampling transects width, the differences between OK and FDK in terms of the mean error and the mean square error are not significant regardless of the sampled area and the choice of the field. When 20% or more of the site is sampled, the estimation of object counts is unbiased on average for all three fields regardless of the choice of the transect width and the choice of the kriging algorithm. However, for non-homogeneous fields (isotropic and anisotropic fields), the mean error fluctuates considerably when a small number of transects are sampled. The difference between the transect sampling and the random sampling in terms of prediction errors becomes almost negligible if more than 20% of the site is sampled. Overall, FDK is no better than OK in terms of the prediction performances when the transect sampling procedure is used.  相似文献   

15.
A methodological approach for modelling the occurrence patterns of species for the purpose of fisheries management is proposed here. The presence/absence of the species is modelled with a hierarchical Bayesian spatial model using the geographical and environmental characteristics of each fishing location. Maps of predicted probabilities of presence are generated using Bayesian kriging. Bayesian inference on the parameters and prediction of presence/absence in new locations (Bayesian kriging) are made by considering the model as a latent Gaussian model, which allows the use of the integrated nested Laplace approximation ( INLA ) software (which has been seen to be quite a bit faster than the well-known MCMC methods). In particular, the spatial effect has been implemented with the stochastic partial differential equation (SPDE) approach. The methodology is evaluated on Mediterranean horse mackerel (Trachurus mediterraneus) in the Western Mediterranean. The analysis shows that environmental and geographical factors can play an important role in directing local distribution and variability in the occurrence of species. Although this approach is used to recognize the habitat of mackerel, it could also be for other different species and life stages in order to improve knowledge of fish populations and communities.  相似文献   

16.
Variation in disease risk underlying observed disease counts is increasingly a focus for Bayesian spatial modelling, including applications in spatial data mining. Bayesian analysis of spatial data, whether for disease or other types of event, often employs a conditionally autoregressive prior, which can express spatial dependence commonly present in underlying risks or rates. Such conditionally autoregressive priors typically assume a normal density and uniform local smoothing for underlying risks. However, normality assumptions may be affected or distorted by heteroscedasticity or spatial outliers. It is also desirable that spatial disease models represent variation that is not attributable to spatial dependence. A spatial prior representing spatial heteroscedasticity within a model accommodating both spatial and non-spatial variation is therefore proposed. Illustrative applications are to human TB incidence. A simulation example is based on mainland US states, while a real data application considers TB incidence in 326 English local authorities.  相似文献   

17.
There is a great demand for statistical modeling of phenomena that evolve in both space and time, and thus, there is a growing literature on correlation function models for spatio-temporal processes. In particular, various properties of these correlation functions have been studied only for the merely spatial or temporal case, fact that constitutes a strong motivation for our work. The goal of this paper is to inspect some properties, obtained with respect to partial differentiation and integration, of stationary spatio-temporal correlation functions for which anisotropy is obtained through isotropy between components as in Fernández-Casal et al. (Stat Comput 13(2):127–136, 2003). We show that through partial differentiation and integration it is possible to obtain permissible spatio-temporal correlation functions in the space–time domain. Other new results regard specific classes of space–time correlations introduced in recent literature. A curious result arises by differentiating scale mixtures of Euclid’s hat. Work partially funded by grant MTM2004-06231 from the Spanish Ministery of Science and Education.  相似文献   

18.
In this study, the KLME approach, a moment-equation approach based on the Karhunen–Loeve decomposition developed by Zhang and Lu (Comput Phys 194(2):773–794, 2004), is applied to unconfined flow with multiple random inputs. The log-transformed hydraulic conductivity F, the recharge R, the Dirichlet boundary condition H, and the Neumann boundary condition Q are assumed to be Gaussian random fields with known means and covariance functions. The F, R, H and Q are first decomposed into finite series in terms of Gaussian standard random variables by the Karhunen–Loeve expansion. The hydraulic head h is then represented by a perturbation expansion, and each term in the perturbation expansion is written as the products of unknown coefficients and Gaussian standard random variables obtained from the Karhunen–Loeve expansions. A series of deterministic partial differential equations are derived from the stochastic partial differential equations. The resulting equations for uncorrelated and perfectly correlated cases are developed. The equations can be solved sequentially from low to high order by the finite element method. We examine the accuracy of the KLME approach for the groundwater flow subject to uncorrelated or perfectly correlated random inputs and study the capability of the KLME method for predicting the head variance in the presence of various spatially variable parameters. It is shown that the proposed numerical model gives accurate results at a much smaller computational cost than the Monte Carlo simulation.  相似文献   

19.
Compositional Bayesian indicator estimation   总被引：1，自引：1，他引：0  
Indicator kriging is widely used for mapping spatial binary variables and for estimating the global and local spatial distributions of variables in geosciences. For continuous random variables, indicator kriging gives an estimate of the cumulative distribution function, for a given threshold, which is then the estimate of a probability. Like any other kriging procedure, indicator kriging provides an estimation variance that, although not often used in applications, should be taken into account as it assesses the uncertainty of the estimate. An alternative approach to indicator estimation is proposed in this paper. In this alternative approach the complete probability density function of the indicator estimate is evaluated. The procedure is described in a Bayesian framework, using a multivariate Gaussian likelihood and an a priori distribution which are both combined according to Bayes theorem in order to obtain a posterior distribution for the indicator estimate. From this posterior distribution, point estimates, interval estimates and uncertainty measures can be obtained. Among the point estimates, the median of the posterior distribution is the maximum entropy estimate because there is a fifty-fifty chance of the unknown value of the estimate being larger or smaller than the median; that is, there is maximum uncertainty in the choice between two alternatives. Thus in some sense, the latter is an indicator estimator, alternative to the kriging estimator, that includes its own uncertainty. On the other hand, the mode of the posterior distribution estimator, assuming a uniform prior, is coincidental with the simple kriging estimator. Additionally, because the indicator estimate can be considered as a two-part composition which domain of definition is the simplex, the method is extended to compositional Bayesian indicator estimation. Bayesian indicator estimation and compositional Bayesian indicator estimation are illustrated with an environmental case study in which the probability of the content of a geochemical element in soil being over a particular threshold is of interest. The computer codes and its user guides are public domain and freely available.  相似文献   

20.
An approach to the simulation of spatial random fields is proposed. The target random field is specified by its covariance function which need not be homogeneous or Gaussian. The technique provided is based on an approximate Karhunen–Loève expansion of spatial random fields which can be readily realized. Such an approximate representation is obtained from a correction to the Rayleigh–Ritz method based on the dual Riesz basis theory. The resulting numerical projection procedure improves Rayleigh–Ritz algorithm in the approximation of second-order random fields. Simulations are developed to illustrate the convergence and accuracy of the method presented.
J. C. Ruiz-MolinaEmail:
  相似文献   

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