Abstract: Despite the advances in computer capacity, the enormous computational cost of complex engineering simulations makes it impractical to rely exclusively on simulation for the purpose of design optimization. To cut down the cost, surrogate models, also known as metamodels, are constructed from and then used in lieu of the actual simulation models. In the paper, we systematically compare four popular metamodeling techniques —Polynomial Regression, Multivariate Adaptive Regression Splines, Radial Basis Functions, and Kriging —based on multiple performance criteria using fourteen test problems representing different classes of problems. Our objective in th is study is to investigate the advantages and disadvantages these four metamodeling techniques using multiple modeling criteria and multiple test problems rather than a single measure of merit and a single test problem.

Abstract: Summary The standard estimator of the variogram is sensitive to outlying data, a few of which can cause overestimation of the variogram. This will result in incorrect variances when estimating the value of a soil property by kriging or when designing a sampling grid to map the property to a required precision. Several robust estimators of the variogram, based on location and scale estimation, have been proposed as improvements. They seem to be suitable for analysis of soil data in circumstances where the standard estimator is likely to be affected by outliers. Robust estimators are based on assumptions about the distribution of the data which will not always hold and which need not be made in kriging or in estimating the variogram by the standard estimator. The estimators are reviewed. Simulation studies show that the robust estimators vary in their susceptibility to moderate skew in the underlying distribution, but that the effects of outliers are generally greater. The estimators are applied to some soil data, and the resulting variograms used for ordinary kriging at sites in a separate validation data set. In most cases the variograms derived from the standard estimator gave kriging variances which appeared to overestimate the mean squared error of prediction (MSEP). Kriging with variograms based on robust estimators sometimes gave kriging variances which underestimated the MSEP or did not differ significantly from it. Estimates of kriging variance and the MSEP derived from the validation data were generally close to estimates from cross-validation on the prediction set used to derive the variograms. This indicates that variogram models derived from different estimators could be compared by cross-validation.

Abstract: A new methodology for sampling plan design has been developed to reduce the costs associated with long-term monitoring of sites with groundwater contamination. The method combines a fate-and-transport model, plume interpolation, and a genetic algorithm to identify cost-effective sampling plans that accurately quantify the total mass of dissolved contaminant. The plume interpolation methods considered were inverse-distance weighting, ordinary kriging, and a hybrid method that combines the two approaches. Application of the methodology to Hill Air Force Base indicated that sampling costs could be reduced by as much as 60% without significant loss in accuracy of the global mass estimates. Inverse-distance weighting was shown to be most effective as a screening tool for evaluating whether more comprehensive geostatistical modeling is warranted. The hybrid method was effective for implementing such a tiered approach, reducing computational time by more than 60% relative to kriging alone.

Abstract: This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance

Abstract: Summary Variograms of soil properties are usually obtained by estimating the variogram for distinct lag classes by the method-of-moments and fitting an appropriate model to the estimates. An alternative is to fit a model by maximum likelihood to data on the assumption that they are a realization of a multivariate Gaussian process. This paper compares the two using both simulation and real data. The method-of-moments and maximum likelihood were used to estimate the variograms of data simulated from stationary Gaussian processes. In one example, where the simulated field was sampled at different intensities, maximum likelihood estimation was consistently more efficient than the method-of-moments, but this result was not general and the relative performance of the methods depends on the form of the variogram. Where the nugget variance was relatively small and the correlation range of the data was large the method-of-moments was at an advantage and likewise in the presence of data from a contaminating distribution. When fields were simulated with positive skew this affected the results of both the method-of-moments and maximum likelihood. The two methods were used to estimate variograms from actual metal concentrations in topsoil in the Swiss Jura, and the variograms were used for kriging. Both estimators were susceptible to sampling problems which resulted in over- or underestimation of the variance of three of the metals by kriging. For four other metals the results for kriging using the variogram obtained by maximum likelihood were consistently closer to the theoretical expectation than the results for kriging with the variogram obtained by the method-of-moments, although the differences between the results using the two approaches were not significantly different from each other or from expectation. Soil scientists should use both procedures in their analysis and compare the results.

Abstract: A GIS-based method for constructing high-resolution (in space) maps of mean seasonal temperature and precipitation is developed for the Mediterranean Basin. Terrain variables and geo- graphical location are used as predictors of the climate variables at all points on a grid with a 1 km res- olution, using a regression-based approach. Variables used for model development include: longitude, latitude, elevation, distance from the nearest coast, direction to the nearest coast, slope, aspect, and the ratio of land to sea within given radii. Seasonal mean temperature and precipitation data, for the obser- vation period 1952 to 1989, were assembled from 248 temperature sites and 285 precipitation sites in order to initialise the regression model. Temperature data from 36 stations and precipitation data from 35 stations were retained for model validation. Climate surfaces were constructed using the regression equations, and refined by kriging the residuals from the regression model and subtracting the result from the predicted 'observation' surface. Latitude, elevation and distance from the sea are found to be the most effective predictors of local seasonal climate. Validation determined that regression plus kriging predicts mean seasonal temperatures with a coefficient of determination (R 2 ), between the expected and observed values, of 0.87 (summer) and 0.97 (winter), and mean seasonal precipitation with an R 2 of 0.46 (autumn) and 0.94 (summer). A simple regression model without kriging yields less

Abstract: A full-Bayesian approach to the estimation of transmissivity from hydraulic head and transmissivity measurements is developed for two-dimensional steady state groundwater flow. The approach combines both Bayesian and maximum entropy viewpoints of probability. In the first phase, log transmissivity measurements are incorporated into Bayes' theorem, and the prior probability density function is updated, yielding posterior estimates of the mean value of the log transmissivity field and covariance. The two central moments are generated assuming that the prior mean, variance, and integral scales are “hyperparameters”; that is, they are treated as random variables in themselves which is contrary to classical statistical approaches. The probability density functions (pdfs) of these hyperparameters are, in turn, determined from maximum entropy considerations. In other words, pdfs are chosen for each of the hyperparameters that are maximally uncommitted with respect to unknown information. This methodology is quite general and provides an alternative to kriging for spatial interpolation. The final step consists of updating the conditioned natural logarithm transmissivity (ln(T)) field with hydraulic head measurements, utilizing a linearized aquifer equation. It is assumed that the statistical properties of the noise in the hydraulic head measurements are also uncertain. At each step, uncertainties in all pertinent hyperparameters are removed by marginalization. Finally, what is produced is a ln(T) field conditioned on measurements of both hydraulic heads and log transmissivity and covariances of the ln(T) field. In addition, we can also produce resolution matrices, confidence (credibility) limits, and the like for the ln(T) field. It is shown that the application of the methodology yields good estimates of transmissivities, even when hydraulic head measurements are noisy and little or no information is specified on mean values of ln(T), variance of ln(T), and integral scales.

Abstract: Methane emissions from a Swedish landfill were measured with a static chamber technique on three occasions during 1997. Methane flux rates ranged from −15.2 × 10-3 to 40 g of CH4 m-2 h-1, and the spatial variability was high (CV = 343−386%). The spatial distribution of the emissions was estimated with the help of ordinary kriging, which is a spatial interpolation method. Three different approaches to estimate the total amounts were used: kriging on logarithm-transformed data, kriging with extremes excluded, and linear interpolation of measurements. These were compared between themselves and with the flux rates measured with a tracer gas technique. While the latter gave an estimate of 41 kg of CH4 h-1 from the landfill (with small variations), the highest possible estimate obtained with static chambers and geostatistical methods was 9.7 kg of CH4 h-1. The conclusion is that static chambers can hardly be trusted for making more than small-scale estimates of landfill gas emissions.

Abstract: Monitoring the Comprehensive Nuclear-Test-Ban Treaty will require improved seismic location capability for small-magnitude events. The International Monitoring System (IMS) is well suited to locate events that are large enough to be recorded at teleseismic distances. However, small events are likely to be recorded on a sparse subset of IMS stations at regional- to upper-mantle distances (less than 30°), and sparse-network locations can be strongly effected by travel-time errors that result from path-specific velocity model inaccuracies. In an effort to improve sparse network location capability, we outline a procedure that applies empirical corrections to travel times determined with an appropriate velocity model. More specifically, Bayesian kriging and calibration events (constrained with a global network) are used to estimate epicenter-specific travel-time corrections. For a test (sparse) network of stations, we calculate travel-time residuals for the calibration events relative to the ak135 velocity model. Travel-time residuals are assigned to the respective calibration epicenter, forming a set of spatially varying travel-time correction points. The spatial set of correction points is declustered to reduce the dimension of the observations with minimal reduction in accuracy of the travel-time corrections. We then use the declustered set of calibration points and Bayesian kriging to form continuous travel-time correction surfaces for each station of the test network. The effectiveness of travel-time correction surfaces is evaluated by locating, with and without corrections, a subset of the 1991 Racha earthquake sequence (Caucasus Mountains), for which we have accurate locations that were independently determined with a dense local network. When no travel-time correction is applied, the mean horizontal distance between the local and test network locations is 42 km, and there is a distinct bias in sparse-network locations toward the north-northwest. The mean difference between local and sparse network locations is cut to 13 km when corrections are applied, and the bias in location is significantly reduced. When calibration events in the Racha vicinity are not used to make the correction surfaces, there is still a significant improvement in location, with mean mislocations of 15 km. When corrections are not applied, only one of the locally determined locations lies within the associated 90% coverage ellipse determined with the test (sparse) network. However, by using travel-time corrections and estimates of model uncertainty determined using kriging, representative error ellipses are obtained. This study demonstrates that kriging correction surfaces based on global-network-constrained calibration events can improve the ability to accurately locate lower magnitude events while providing representative coverage ellipses.

Abstract: Existing studies on spatial interpolation tend to overplay statistical perspective, paying little attention the locality and the visual performance of generated surface models. In an attempt to bridge these gaps in literatures, the authors compared the performance of five surface modelling methods, using a set of integrative criteria including absolute and relative statistical accuracy, visual pleasantness and faithfulness of generated surface models, sensitivity to changing sample size and search conditions, and computational intensity. The modeling methods used were: inverse distance, kriging, linear triangulation, minimum curvature, and radial basis functions. Because terrain relief is one of few environmental attributes whose continuous surfaces can be directly observed through appropriate procedures, we used as input data two sets of elevation points sampled irregularly from a USGS 1:24,000 topographical map covering a hilly area. We found that surface modeling methods, even if statistically accurate...

Abstract: The number of radiation data collection stations is limited due to economic reasons Hence, there is a need for the spatially continuous mapping of solar radiation by estimation This paper utilizes a geostatistical technique for the estimation of solar radiation in Saudi Arabia This technique includes five steps: (i) data collection, (ii) univariate analysis, (iii) experimental variogram calculations and model fitting, (iv) estimation using kriging, and (v) plotting contour maps Variogram models are fitted to measured variograms for each month of the year Estimates were obtained at 1500 grid points (30 50) between a longitude of 36588E and 50008E, and latitude of 17178N and 31338N for a grid resolution of 55 33 km These values were used to plot the contour maps of solar radiation for each month of the year To test the performance of the technique, estimates were obtained at the 41 known locations by systematically excluding one of these points from the known data The error analysis showed a maximum mean deviation between measured and estimated values of 00037 (January) and a minimum of 00013 (March and October) The mean percent errors were found to vary between a minimum of 05% and a maximum of 17% This technique may be expanded for the spatial estimation of solar radiation on regional and continental scales 7 2000 Published by Elsevier Science Ltd

Abstract: A method for obtaining, by means of an inversion process, an optimum model of a physical characteristic in a heterogeneous medium (the impedance of an underground zone in relation to waves transmitted in the ground for example), by taking as the starting point an a priori model of the physical characterized that is optimized by minimizing a cost function dependent on differences between the optimized model which is sought and the known data, considering the a priori model. Construction of the a priori model comprises correlation by kriging between values of the physical quantity known at different points of the medium along discontinuities (strata directions). Uncertainties about the values of the physical quantity in the a priori model in relation to the corresponding values in the medium follow a covariance model that controls the inversion parameters more quantitatively. The characteristics of the covariance model are defined in connection with the structure of the data observed or measured in the medium. An application of the optimum model is location of hydrocarbon reservoirs.

Abstract: The applicability and usefulness of Geostatistics (kriging) as a tool for optimum selection of sites for monitoring groundwater levels has been demonstrated through a case study. The criterion used is the estimation of error variance. Groundwater level data (pre-monsoon 1994) obtained from 32 observation wells of Upper Kongal basin, Nalgonda District, A.P. (India) has been stochastically analyzed. The spatial distribution of water levels and its associated error variance is computed and the locations having maximum error variance are selected as additional sites for augmenting the existing observational well network.

Abstract: Geostatistical simulations are globally accurate in the sense that they reproduce global statistics such as variograms and histograms. Kriging is locally accurate in the minimum local error variance sense. Building on the concept of direct sequential simulation, we propose a fast simulation method that can share these opposing objectives. It is shown that the multiple-point entropy of the resulting simulation is related to the univariate entropy of the local conditional distributions used to draw simulated values. Adding local accuracy to conditional simulations does not detract much from variogram reproduction and can be used to increase multiple-point entropy. The methods developed are illustrated using a case study.

Abstract: Digital images are rich in data, but in many instances they are so complex as to require spatial filtering to distinguish the structures in them and facilitate interpretion. The filtering can be done geostatistically by kriging analysis. It proceeds in two stages. The first involves modelling the correlation structure in the imagery by decomposing the variogram into independent spatial components. The second takes each component in turn and kriges it, thereby filtering it from the others. The paper describes the theory and illustrates it with an example of an analysis of a SPOT image in a forested landscape of the south-eastern United States. Variograms of the three wavebands, originally recorded as digital numbers and for the red and infrared transformed to the logarithms, revealed spatial variation on two distinct scales with effective ranges of 300m and 3km. These variograms and that of the Normalized Difference Vegetation Index (NDVI) were fitted by nested (double) exponential models. The two spatial ...

Abstract: Spatial prediction and uncertainty assessment of ecological modeling and simulation systems are a difficult task because of system complexities that include multi components, their interaction and variability over space and time. Developing a general methodology and framework of uncertainty assessment for the systems9 users has become very important. As the first part of a large study addressing these issues, the focus of this paper is on spatial prediction and uncertainty assessment of topographic factors involved in the Revised Universal Soil Loss Equation (RUSLE). The spatial variability of these topographic factors including slope steepness factor S, slope length factor L, and their combined LS factor were modeled with semivariogram models. Three geostatistical methods, including ordinary kriging, indicator kriging, and sequential indicator simulation, were applied and compared. The predicted value maps of these factors, their error variance or conditional variance maps, and probability maps for the predicted values larger than a given threshold value were derived. The comparison of the geostatistical methods suggests that sequential indicator simulation better than ordinary and indicator kriging.

Abstract: Editor's Note: The Geologic Column, which appears monthly in TLE , is (1) produced cooperatively by the SEG Interpretation Committee and the AAPG Geophysical Integration Committee and (2) coordinated by M. Ray Thomasson and Lee Lawyer. This is the first of two articles intended to describe petroleum geostatistics for the nongeostatistician. There are many misconceptions about geostatistics, what it is, and what it can or can't do for the petroleum industry. The first article defines geostatistics, examines its origins, and reviews the spatial model and the kriging interpolation algorithm. The second article describes geostatistical conditional simulation and its use for uncertainty (risk) analysis. Earth science data exhibit spatial correlation to greater or lesser degrees. As the distance between two data points increases, the similarity between the two measurements decreases. Geostatistics is a rapidly evolving branch of applied statistics and mathematics that offers a collection of tools which quantify and model spatial variability. Spatial variability includes scales of variability (heterogeneity) and directionality within data sets. The origins of geostatistics are found exclusively in the mining industry. D. G. Krige, a South African mining engineer, and H. S. Sichel, a statistician, developed a new estimation method in the early 1950s when “classical” statistics was found unsuitable for estimating disseminated ore reserves. Georges Matheron, a French engineer, developed Krige's innovative concepts and formalized them within a single framework with his Theory of Regionalized Variables . Matheron, at the Centre de Geostatistique, pioneered the use of mining geostatistics in the early 1960s. The word kriging was coined in recognition of D. G. Krige. It is interesting that geostatistics was not originally developed to solve interpolation problems (kriging) but to address what is called the support effect. In ore mining, this refers to the difference between the variance of average values measured from large samples …

Abstract: The probability of pollutant concentrations greater than a cutoff value is useful for delineating hazardous areas in contaminated soils. It is essential for risk assessment and reclamation. In this study, three nonparametric kriging methods [indicator kriging, probability kriging, and kriging with the cumulative distribution function (CDF) of order statistics (CDF kriging)] were used to estimate the probability of heavy-metal concentrations lower than a cutoff value. In terms of methodology, the probability kriging estimator and CDF kriging estimator take into account the information of the order relation, which is not considered in indicator kriging. Since probability kriging has been shown to be better than indicator kriging for delineating contaminated soils, the performance of CDF kriging, which we propose, was compared with that of probability kriging in this study. A data set of soil Cd and Pb concentrations obtained from a 10-ha heavy-metal contaminated site in Taoyuan, Taiwan, was used. The results demonstrated that the probability kriging and CDF kriging estimations were more accurate than the indicator kriging estimation. On the other hand, because the probability kriging was based on the cokriging estimator, some unreliable estimates occurred in the probability kriging estimation. This indicated that probability kriging was not as robust as CDF kriging. Therefore, CDF kriging is more suitable than probability kriging for estimating the probability of heavy-metal concentrations lower than a cutoff value.

Abstract: This paper presents a simple least squares procedure for estimating the spatial covariance and compares it with the numerically more difficult restricted maximum likelihood procedure. Thereafter, it compares design-based and kriging techniques for the estimation of spatial averages in the context of double sampling, as used in forest inventory, where terrestrial sample plots are combined with auxiliary information based on aerial photographs. A case study illustrates the theory.