Kriging in a global neighborhood

The kriging estimator is usually computed in a moving neighborhood; only the data near the point to be estimated are used. This moving neighborhood approach creates discontinuities in mapping applications. An alternative approach is presented here, whereby all points are estimated using all the available data. To solve the resulting large linear system the kriging estimator is expressed in terms of the inverse of the covariance matrix. The covariance matrix has the advantage of being positive definite and the size of system which can be solved without encountering numerical instability is substantially increased. Because the kriging matrix does not change, the estimator can be written in terms of scalar products, thus avoiding the more time-consuming matrix multiplications of the standard approach. In the particular case of a covariance which is zero for distances greater than a fixed value (the range), the resulting banded structure of the covariance matrix is shown to lead to substantial computational savings in both run time and storage space. In this case the calculation time for the kriging variance is also substantially reduced. The present method is extended to the nonstationary case.     

Continuity for Kriging with Moving Neighborhood

By definition, kriging with a moving neighborhood consists in kriging each target point from a subset of data that varies with the target. When the target moves, data that were within the neighborhood are suddenly removed from the neighborhood. There is generally no screen effect, and the weight of such data goes suddenly from a non-zero value to a value of zero. This results in a discontinuity of the kriging map. Here a method to avoid such a discontinuity is proposed. It is based on the penalization of the outermost data points of the neighborhood, and amounts to considering that these points values are spoiled with a random error having a variance that increases infinitely when they are about to leave the neighborhood. Additional details are given regarding how the method is to be carried out, and properties are described. The method is illustrated by simple examples. While it appears to be similar to continuous kriging with a smoothing kernel, it is in fact based on a much simpler formalism.     

On the influence of kriging parameters on the cartographic output—A study in mapping subglacial topography

Geostatistics provides a suite of methods, summarized as kriging, to analyze a finite data set to describe a continuous property of the Earth. Kriging methods consist of moving window optimum estimation techniques, which are based on a least-squares principle and use a spatial structure function, usually the variogram. Applications of kriging techniques have become increasingly wide-spread, with ordinary kriging and universal kriging being the most popular ones. The dependence of the final map or model on the input, however, is not generally understood. Herein we demonstrate how changes in the kriging parameters and the neighborhood search affect the cartographic result. Principles are illustrated through a glaciological study. The objective is to map ice thickness and subglacial topography of Storglaciären, Kebnekaise Massif, northern Sweden, from several sets of radio-echo soundings and hot water drillings. New maps are presented.     

Dual Kriging with Local Neighborhoods: Application to the Representation of Surfaces

Ordinary kriging, in its common formulation, is a discrete estimator in that it requires the solution of a kriging system for each point in space in which an estimate is sought. The dual formulation of ordinary kriging provides a continuous estimator since, for a given set of data, only a kriging system has to be estimated and the resulting estimate is a function continuously defined in space. The main problem with dual kriging up to now has been that its benefits can only be capitalized if a global neighborhood is used. A formulation is proposed to solve the problem of patching together dual kriging estimates obtained with data from different neighborhoods by means of a blending belt around each neighborhood. This formulation ensures continuity of the variable and, if needed, of its first derivative along neighbor borders. The final result is an analytical formulation of the interpolating surface that can be used to compute gradients, cross-sections, or volumes; or for the quick evaluation of the interpolating surface in numerous locations.     

Kriging in the Presence of Locally Varying Anisotropy Using Non-Euclidean Distances

A stationary specification of anisotropy does not always capture the complexities of a geologic site. In this situation, the anisotropy can be varied locally. Directions of continuity and the range of the variogram can change depending on location within the domain being modeled. Kriging equations have been developed to use a local anisotropy specification within kriging neighborhoods; however, this approach does not account for variation in anisotropy within the kriging neighborhood. This paper presents an algorithm to determine the optimum path between points that results in the highest covariance in the presence of locally varying anisotropy. Using optimum paths increases covariance, results in lower estimation variance and leads to results that reflect important curvilinear structures. Although CPU intensive, the complex curvilinear structures of the kriged maps are important for process evaluation. Examples highlight the ability of this methodology to reproduce complex features that could not be generated with traditional kriging.     

The differential impact of physical disorder and collective efficacy: a geographically weighted regression on violent crime

While much of the research on neighborhood crime considers the neighborhood as a whole, this study utilizes spatial analysis techniques to examine how the presence of disorder and collective efficacy create unique pockets of opportunities for criminal behavior within neighborhoods. This spatial perspective reveals how the effect of disorder and efficacy upon crime patterns itself varies across a neighborhood. Physical disorder is measured through systematic social observations and the level of collective efficacy is evaluated through survey responses of neighborhood residents. The indices of disorder and efficacy are compared to crime data from police call logs using geographically weighted regression. Our findings demonstrate a complex spatial relationship between disorder and efficacy. The effects of disorder and efficacy are not consistent across an entire neighborhood, but rather display local variations in small geographical areas within neighborhoods, including some pockets of the neighborhood where the relationships between disorder, efficacy, and crime were contrary to the expected relationships. Based upon these findings, we conclude opportunity is central to understanding crime, and emphasize the role of informal social control in neighborhoods.     

Two key parameters when choosing the kriging neighborhood

In the stationary case, two parameters are especially interesting when choosing the kriging neighborhood: weight of the mean, which shows how kriging depends on the neighborhood, and slope of the regression, which indicates if the neighborhood is large enough.     

The Bi-Gaussian approach: A simple method for recovery estimation

A new and simple method is proposed to obtain estimates of recovery functions: the Bi-Gaussian approach. Existing methods estimate recovery functions with conditional distributions where the conditioning set is all the data available. Here instead the simple kriging estimate of the Gaussian transform is proposed to be used. Results in the point recovery case are identical to the multi-Gaussian approach of Verly (1983, 1984), whereas in the non-point-support situation, an approximation is derived which saves computer time as compared to employing the strict multi-Gaussian hypothesis. Two examples compare favorably with the well-established disjunctive kriging method (discrete Gaussian model).     

Cross validation of kriging in a unique neighborhood

Cross validation is an appropriate tool for testing interpolation methods: it consists of leaving out one data point at a time, and determining how well this point can be estimated from the other data. Cross validation is often used for testing “moving neighborhood” kriging models; in this case, each unknown value is predicted from a small number of surrounding data. In “unique neighborhood” kriging algorithms, each estimation uses all the available data; as a result, cross validation would spend much computer time. For instance, with ndata points it would cost at least the resolution of nsystems of n × nlinear equations (each with a different matrix).Here, we present a much faster method for cross validation in a unique neighborhood. Instead of solving nsystems n × n,it only requires the inversion of one n × nmatrix. We also generalized this method to leaving out several points instead of one.     

The kriging update equations and their application to the selection of neighboring data

A key problem in the application of kriging is the definition of a local neighborhood in which to search for the most relevant data. A usual practice consists in selecting data close to the location targeted for prediction and, at the same time, distributed as uniformly as possible around this location, in order to discard data conveying redundant information. This approach may however not be optimal, insofar as it does not account for the data spatial correlation. To improve the kriging neighborhood definition, we first examine the effect of including one or more data and present equations in order to quickly update the kriging weights and kriging variances. These equations are then applied to design a stepwise selection algorithm that progressively incorporates the most relevant data, i.e., the data that make the kriging variance decrease more. The proposed algorithm is illustrated on a soil contamination dataset.     

Application of universal kriging for estimation of earthquake ground motion: Statistical significance of results

Universal kriging is compared with ordinary kriging for estimation of earthquake ground motion. Ordinary kriging is based on a stationary random function model; universal kriging is based on a nonstationary random function model representing first-order drift. Accuracy of universal kriging is compared with that for ordinary kriging; cross-validation is used as the basis for comparison. Hypothesis testing on these results shows that accuracy obtained using universal kriging is not significantly different from accuracy obtained using ordinary kriging. Tests based on normal distribution assumptions are applied to errors measured in the cross-validation procedure;t andF tests reveal no evidence to suggest universal and ordinary kriging are different for estimation of earthquake ground motion. Nonparametric hypothesis tests applied to these errors and jackknife statistics yield the same conclusion: universal and ordinary kriging are not significantly different for this application as determined by a cross-validation procedure. These results are based on application to four independent data sets (four different seismic events).     

Simulating Large Gaussian Random Vectors Subject to Inequality Constraints by Gibbs Sampling

The Gibbs sampler is an iterative algorithm used to simulate Gaussian random vectors subject to inequality constraints. This algorithm relies on the fact that the distribution of a vector component conditioned by the other components is Gaussian, the mean and variance of which are obtained by solving a kriging system. If the number of components is large, kriging is usually applied with a moving search neighborhood, but this practice can make the simulated vector not reproduce the target correlation matrix. To avoid these problems, variations of the Gibbs sampler are presented. The conditioning to inequality constraints on the vector components can be achieved by simulated annealing or by restricting the transition matrix of the iterative algorithm. Numerical experiments indicate that both approaches provide realizations that reproduce the correlation matrix of the Gaussian random vector, but some conditioning constraints may not be satisfied when using simulated annealing. On the contrary, the restriction of the transition matrix manages to satisfy all the constraints, although at the cost of a large number of iterations.     

Geostatistical Mapping with Continuous Moving Neighborhood

An issue that often arises in such GIS applications as digital elevation modeling (DEM) is how to create a continuous surface using a limited number of point observations. In hydrological applications, such as estimating drainage areas, direction of water flow is easier to detect from a smooth DEM than from a grid created using standard interpolation programs. Another reason for continuous mapping is esthetic; like a picture, a map should be visually appealing, and for some GIS users this is more important than map accuracy. There are many methods for local smoothing. Spline algorithms are usually used to create a continuous map, because they minimize curvature of the surface. Geostatistical models are commonly used approaches to spatial prediction and mapping in many scientific disciplines, but classical kriging models produce noncontinuous surfaces when local neighborhood is used. This motivated us to develop a continuous version of kriging. We propose a modification of kriging that produces continuous prediction and prediction standard error surfaces. The idea is to modify kriging systems so that data outside a specified distance from the prediction location have zero weights. We discuss simple kriging and conditional geostatistical simulation, models that essentially use information about mean value or trend surface. We also discuss how to modify ordinary and universal kriging models to produce continuous predictions, and limitations using the proposed models.     

Comparison of surrogate models with different methods in groundwater remediation process

Surrogate modelling is an effective tool for reducing computational burden of simulation optimization. In this article, polynomial regression (PR), radial basis function artificial neural network (RBFANN), and kriging methods were compared for building surrogate models of a multiphase flow simulation model in a simplified nitrobenzene contaminated aquifer remediation problem. In the model accuracy analysis process, a 10-fold cross validation method was adopted to evaluate the approximation accuracy of the three surrogate models. The results demonstrated that: RBFANN surrogate model and kriging surrogate model had acceptable approximation accuracy, and further that kriging model’s approximation accuracy was slightly higher than RBFANN model. However, the PR model demonstrated unacceptably poor approximation accuracy. Therefore, the RBFANN and kriging surrogates were selected and used in the optimization process to identify the most cost-effective remediation strategy at a nitrobenzene-contaminated site. The optimal remediation costs obtained with the two surrogate-based optimization models were similar, and had similar computational burden. These two surrogate-based optimization models are efficient tools for optimal groundwater remediation strategy identification.     

Addressing issues of spatial spillover effects and non-stationarity in analysis of residential burglary crime

The aim of this paper is to address two critical but largely neglected issues in the spatial analysis of urban crime which are spatial spillover effects of crime penetrating neighborhood boundaries and non-stationarity regarding the relationships between contextual factors and neighborhood crime. We use a GIS-based spatial approach to normalize the estimate of burglary crime at block group level and use the geographically weighted regression (GWR) to investigate the correlates of neighborhood crime. Results suggest that the use of normalized measure of neighborhood crime helps better reveal the spatial patterns of burglary crime and the use of GWR accounts for the spatial variations of relationships between contextual factors and crime. In particular, the normalized measure of crime has implications for improving the measurement accuracy of the risk of crime across urban neighborhoods and can be applied to the spatial analysis of other socioeconomic issues such as housing foreclosures and environmental hazards which are also plagued by the spatial spillover issue when geographically contiguous data are analyzed.     

FuzzyKrig: a comprehensive matlab toolbox for geostatistical estimation of imprecise information

Using kriging has been accepted today as the most common method of estimating spatial data in such different fields as the geosciences. To be able to apply kriging methods, it is necessary that the data and variogram model parameters be precise. To utilize the imprecise (fuzzy) data and parameters, use is made of fuzzy kriging methods. Although it has been 30 years since different fuzzy kriging algorithms were proposed, its use has not become as common as other kriging methods (ordinary, simple, log, universal, etc.); lack of a comprehensive software that can perform, based on different fuzzy kriging algorithms, the related calculations in a 3D space can be the main reason. This paper describes an open-source software toolbox (developed in Matlab) for running different algorithms proposed for fuzzy kriging. It also presents, besides a short presentation of the fuzzy kriging method and introduction of the functions provided by the FuzzyKrig toolbox, 3 cases of the software application under the conditions where: 1) data are hard and variogram model parameters are fuzzy, 2) data are fuzzy and variogram model parameters are hard, and 3) both data and variogram model parameters are fuzzy.     

Quantifying and analyzing neighborhood configuration characteristics to cellular automata for land use simulation considering data source error

Cellular Automata (CA) simulation models have been increasingly used in land use studies. However, neighborhood configuration, an essential element of CA model, remarkably impacts the accuracy of simulated results. Moreover, errors from data source may propagate through the CA modeling process. The objective of this study is to analyze the effect of neighborhood configuration to CA model and further on to explore its capacity of resisting disturbance from data source error. With statistic-based CA model and several neighborhood configurations respectively, the land use changes of Wuhan, China were analyzed. It is demonstrated that there are significant differences on the simulated results produced by different neighborhoods. Besides, different neighborhoods respond differently to data source error. In light of these results, we find out that (1) neighborhood configurations with larger neighborhood size and planar neighborhood type, introduced in this paper, contribute to higher prediction accuracy; and (2) the neighborhood configurations above also have higher capacity of resisting disturbance from data source error and give rise to more stable simulated results. This study provides a comprehensive basis for scale selection of CA model with a meaningful consideration of data source error and thus will improve the research on land use change.     

Optimal Interpolation of Gravity Maps Using a Modified Neural Network

This paper proposes an interpolation method based on a modified Kohonen artificial neural network, and is used to interpolate marine gravity data on a regular grid. This method combines accuracy comparable to that of kriging with a much shorter computing time than kriging. It is particularly efficient when both the size of the grid and the quantity of available data are large. Under some hypotheses similar to those of kriging with a trend, the unbiasedness and optimality of the method can be demonstrated. Comparison with kriging with a trend using marine gravity data shows similar results. Although neural interpolation is slightly less efficient, it is more robust outside of the marine data area.     

A surrogate-based simulation–optimization approach application to parameters’ identification for the HydroGeoSphere model

A systematic approach is needed to use water more productively, because water shortages limit socio-economic development in many parts of the world. The aim of this paper is to establish a surrogate-based simulation–optimization approach to identify parameter values for a fully integrated surface water and groundwater flow coupling simulation. A surface water and groundwater flow coupling simulation model was implemented using HydroGeoSphere (HGS) model and the parameter sensitivities in the model were analyzed using local sensitivity analysis method. The parameters that exerted a large influence on the output results of the HGS model were then selected as stochastic variables, and the stochastic variable data sets were generated using the latin hypercube sampling (LHS) method, which, thereby, were used as inputs in HGS model to obtain the corresponding outputs. On the basis of input and output data sets, a kriging surrogate model of the HGS model was then established and verified, and parameter values of HGS model were identified using a surrogate-based simulation–optimization approach. The results of this study show that parameters that exert a large influence on the simulation output results include hydraulic conductivity, porosity, the van genuchten parameter (\(\alpha\)), and channel manning coefficient. The established kriging surrogate model is an ideal alternative to the HGS model for simulating and predicting, while optimal parameter values can be identified effectively and accurately using the established approach. The results of this research reveal that huge computational loads can be mitigated while using the kriging surrogate as an alternative for a simulation model in the solution process of optimization model.