Geostatistical Conditional Simulation Research Articles

Approaches to reporting grade uncertainty in high nugget gold veins

High nugget effect gold veins are generally considered to be one of the most challenging of deposit types to evaluate and exploit. To potential investors and mining companies, they are viewed as high risk because of the associated uncertainties in the grade estimate and general paucity of reserves at production start-up. Despite the well-known risks, these deposits are often high grade and have a grade upside which makes them potentially attractive. The reporting of resource grade is of importance and perhaps even more so is the reporting of the associated uncertainty attached to the grade estimate. The quoting of the estimated grade within a grade range is recommended to achieve more complete and useful disclosure. The definition of the grade range can be somewhat subjective, ranging from opinion-based through to data-based estimates and geostatistical conditional simulation. Whichever approach is used, the over-riding issue must be the provision of enough data of appropriate quality and full disclosure to the public of relevant uncertainties.

Toward a reliable decomposition of predictive uncertainty in hydrological modeling: Characterizing rainfall errors using conditional simulation

This study explores the decomposition of predictive uncertainty in hydrological modeling into its contributing sources. This is pursued by developing data‐based probability models describing uncertainties in rainfall and runoff data and incorporating them into the Bayesian total error analysis methodology (BATEA). A case study based on the Yzeron catchment (France) and the conceptual rainfall‐runoff model GR4J is presented. It exploits a calibration period where dense rain gauge data are available to characterize the uncertainty in the catchment average rainfall using geostatistical conditional simulation. The inclusion of information about rainfall and runoff data uncertainties overcomes ill‐posedness problems and enables simultaneous estimation of forcing and structural errors as part of the Bayesian inference. This yields more reliable predictions than approaches that ignore or lump different sources of uncertainty in a simplistic way (e.g., standard least squares). It is shown that independently derived data quality estimates are needed to decompose the total uncertainty in the runoff predictions into the individual contributions of rainfall, runoff, and structural errors. In this case study, the total predictive uncertainty appears dominated by structural errors. Although further research is needed to interpret and verify this decomposition, it can provide strategic guidance for investments in environmental data collection and/or modeling improvement. More generally, this study demonstrates the power of the Bayesian paradigm to improve the reliability of environmental modeling using independent estimates of sampling and instrumental data uncertainties.

Bayesian inverse problem and optimization with iterative spatial resampling

Measurements are often unable to uniquely characterize the subsurface at a desired modeling resolution. In particular, inverse problems involving the characterization of hydraulic properties are typically ill‐posed since they generally present more unknowns than data. In a Bayesian context, solutions to such problems consist of a posterior ensemble of models that fit the data (up to a certain precision specified by a likelihood function) and that are a subset of a prior distribution. Two possible approaches for this problem are Markov chain Monte Carlo (McMC) techniques and optimization (calibration) methods. Both frameworks rely on a perturbation mechanism to steer the search for solutions. When the model parameters are spatially dependent variable fields obtained using geostatistical realizations, such as hydraulic conductivity or porosity, it is not trivial to incur perturbations that respect the prior spatial model. To overcome this problem, we propose a general transition kernel (iterative spatial resampling, ISR) that preserves any spatial model produced by conditional simulation. We also present a stochastic stopping criterion for the optimizations inspired from importance sampling. In the studied cases, this yields posterior distributions reasonably close to the ones obtained by a rejection sampler, but with a greatly reduced number of forward model runs. The technique is general in the sense that it can be used with any conditional geostatistical simulation method, whether it generates continuous or discrete variables. Therefore it allows sampling of different priors and conditioning to a variety of data types. Several examples are provided based on either multi‐Gaussian or multiple‐point statistics.

Geostatistical conditional simulation for the assessment of contaminated land by abandoned heavy metal mining

Abandoned mine workings can undoubtedly cause varying degrees of contamination of soil with heavy metals such as lead and zinc has occurred on a global scale. Exposure to these elements may cause to harm human health and environment. In the study, a total of 269 soil samples were collected at 1, 5, and 10 m regular grid intervals of 100 x 100 m area of Carsington Pasture in the UK. Cell declustering technique was applied to the data set due to no statistical representativity. Directional experimental semivariograms of the elements for the transformed data showed that both geometric and zonal anisotropy exists in the data. The most evident spatial dependence structure of the continuity for the directional experimental semivariogram, characterized by spherical and exponential models of Pb and Zn were obtained. This study reports the spatial distribution and uncertainty of Pb and Zn concentrations in soil at the study site using a probabilistic approach. The approach was based on geostatistical sequential Gaussian simulation (SGS), which is used to yield a series of conditional images characterized by equally probable spatial distributions of the heavy elements concentrations across the area. Postprocessing of many simulations allowed the mapping of contaminated and uncontaminated areas, and provided a model for the uncertainty in the spatial distribution of element concentrations. Maps of the simulated Pb and Zn concentrations revealed the extent and severity of contamination. SGS was validated by statistics, histogram, variogram reproduction, and simulation errors. The maps of the elements might be used in the remediation studies, help decision-makers and others involved in the abandoned heavy metal mining site in the world.

Spatial evaluation of the risk of groundwater quality degradation. A comparison between disjunctive kriging and geostatistical simulation

In some previous papers a probabilistic methodology was introduced to estimate a spatial index of risk of groundwater quality degradation, defined as the conditional probability of exceeding assigned thresholds of concentration of a generic chemical sampled in the studied water system. A crucial stage of this methodology was the use of geostatistical techniques to provide an estimation of the above-mentioned probability in a number of selected points by crossing spatial and temporal information. In this work, spatial risk values were obtained using alternatively stochastic conditional simulation and disjunctive kriging. A comparison between the resulting two sets of spatial risks, based on global and local statistical tests, showed that they do not come from the same statistical population and, consequently, they cannot be viewed as equivalent in a statistical sense. At a first glance, geostatistical conditional simulation may appear to represent the spatial variability of the phenomenon more effectively, as the latter tends to be smoothed by DK. However, a close examination of real case study results suggests that disjunctive kriging is more effective than simulation in estimating the spatial risk of groundwater quality degradation. In the study case, the potentially 'harmful event' considered, threatening a natural 'vulnerable groundwater system,' is fertilizer and manure application.

Application of kriging and simulated annealing for spatial variability modelling of a coal seam

Spatial variability models for characterisation of coal seams may be achieved by use of geostatistical techniques, namely kriging and conditional simulation. While kriging provides only one realisation of a true phenomenon, conditional simulation techniques generate a family of equally probable images, of which the ones closest to the reality are chosen by conditioning the simulated values to be equal to known values at sampling locations. The techniques are considered to be more appropriate and result oriented for spatial variability modelling as compared with kriging that produces smoothed estimates while minimising the estimation variance. Among various simulation techniques, simulated annealing with its powerful algorithm is a more well known technique for geostatistical imaging on a node-by-node basis. In the present study, geostatistical imaging of a coal seam in an Indian coalfield has been carried out for spatial variability modelling employing kriging as well as simulated annealing techniques. It is observed that the simulation technique provides a better framework to geostatistical imaging for spatial variability modelling of the coal seam and the results of simulation are more realistic as compared with kriging.

Real-Time Reservoir Model Updating Using Ensemble Kalman Filter With Confirming Option

Summary The ensemble Kalman Filter technique (EnKF) has been reported to be very efficient for real-time updating of reservoir models to match the most current production data. Using EnKF, an ensemble of reservoir models assimilating the most current observations of production data is always available. Thus, the estimations of reservoir model parameters, and their associated uncertainty, as well as the forecasts are always up-to-date. In this paper, we apply the EnKF for continuously updating an ensemble of permeability models to match real-time multiphase production data. We improve the previous EnKF by adding a confirming option (i.e., the flow equations are re-solved from the previous assimilating step to the current step using the updated current permeability models). By doing so, we ensure that the updated static and dynamic parameters are always consistent with the flow equations at the current step. However, it also creates some inconsistency between the static and dynamic parameters at the previous step where the confirming starts. Nevertheless, we show that, with the confirming approach, the filter shows better performance for the particular example investigated. We also investigate the sensitivity of using a different number of realizations in the EnKF. Our results show that a relatively large number of realizations are needed to obtain stable results, particularly for the reliable assessment of uncertainty. The sensitivity of using different covariance functions is also investigated. The efficiency and robustness of the EnKF is demonstrated using an example. By assimilating more production data, new features of heterogeneity in the reservoir model can be revealed with reduced uncertainty, resulting in more accurate predictions of reservoir production. Introduction The reliability of reservoir models could increase as more data are included in their construction. Traditionally, static (hard and soft) data, such as geological, geophysical, and well log/core data are incorporated into reservoir geological models through conditional geostatistical simulation (Deutsch and Journel 1998). Dynamic production data, such as historical measurements of reservoir production, account for the majority of reservoir data collected during the production phase. These data are directly related to the recovery process and to the response variables that form the basis for reservoir management decisions. Incorporation of dynamic data is typically done through a history-matching process. Traditionally, history matching adjusts model variables (such as permeability, porosity, and transmissibility) so that the flow simulation results using the adjusted parameters match the observations. It usually requires repeated flow simulations. Both manual and (semi-) automatic history-matching processes are available in the industry (Chen et al. 1974; He et al. 1996; Landa and Horne 1997; Milliken and Emanuel 1998; Vasco et al. 1998; Wen et al. 1998a, 1998b; Roggero and Hu 1998; Agarwal and Blunt 2003; Caers 2003; Cheng et al. 2004). Automatic history matching is usually formulated in the form of a minimization problem in which the mismatch between measurements and computed values is minimized (Tarantola 1987; Sun 1994). Gradient-based methods are widely employed for such minimization problems, which require the computation of sensitivity coefficients (Li et al. 2003; Wen et al. 2003; Gao and Reynolds 2006). In the recent decade, automatic history matching has been a very active research area with significant progress reported (Cheng et al. 2004; Gao and Reynolds 2006; Wen et al. 1997). However, most approaches are either limited to small and simple reservoir models or are computationally too intensive for practical applications. Under the framework of traditional history matching, the assessment of uncertainty is usually through a repeated history-matching process with different initial models, which makes the process even more CPU-demanding. In addition, the traditional history-matching methods are not designed in such a fashion that allows for continuous model updating. When new production data are available and are required to be incorporated, the history-matching process has to be repeated using all measured data. These limit the efficiency and applicability of the traditional automatic history-matching techniques.

Geostatistical Conditional Simulation for the Assessment of the Quality Characteristics of Cayırhan Lignite Deposits

Traditional estimation techniques based on block models with interpolation such as inverse distance and kriging methods do not take into account the uncertainty associated with the estimates and variability of a deposit. These methods are also inadequate for short range mine planning. However, conditional simulation models (Sequential Gaussian Simulation) reproduce the actual variability (histogram) and spatial continuity (variogram) of the attributes of interest. Sequential Gaussian Simulation (SGS) method was used to address the problem of measuring the uncertainty associated with an estimate for Çayırhan coal deposits. The coal seam is split by a 0.5–1 m thick tuffaceous siltstone and claystone into the upper (Tv), and lower (Tb) seams, which contain essentially different mineral matter and quality parameters. Directional and omni-directional experimental semivariograms of the variables for the transformed data showed that neither geometric nor zonal anisotropy exist in the data. The most evident spatial dependence structure of the continuity for omnidirectional experimental variogram, characterised by spherical and experimental models of the quality variables were obtained. The case study addresses the spatial distribution and uncertainty of quality variables at the study area using a probabilistic approach. This approach was based on SGS used to yield a series of conditionally images characterised by equally probable spatial distributions of the quality variables across the area. The advantages of SGS were illustrated through the case study. The simulated models can be incorporated into mine planning and scheduling.

English

The estimation of actual or potential acid rock drainage (ARD) at mine sites is usually accomplished by sampling specific parameters that allow detection and prediction of the potential for ARD. The use of block models to estimate and describe the spatial extent of relevant variables is becoming more common, although quantification of the uncertainty associated with the problem is generally not available, yet it can be critical in an ARD characterization study. Uncertainties in sampling and analytical processes, in the characterization of the volumes and areas affected or potentially affected by ARD, in the interpolation of sampled values, and in the characterization of physical processes that allow prediction of fate and transport, are always present. It is unrealistic to pretend that the estimation process is error-free, and thus it follows that it is important to provide adequate models of uncertainty, in addition to reasonable estimates of ARD potential. The model of uncertainty can then be used to develop technical risk assessments, including false positives or negatives of certain variables exceeding (or not) certain thresholds. This paper outlines a stochastic method based on geostatistical conditional simulations that allows assessment and modeling of uncertainty in spatial modeling. This assessment is then translated into risk levels, allowing for a decision-making process that is based on levels of uncertainty. The concept of Loss Functions is illustrated with an example drawn from a porphyry Cu-Mo deposit in South America.

Stochastic management modeling of a pump and treat system at the Rocky Mountain Arsenal near Denver, Colorado

Past activities at the Rocky Mountain Arsenal in Colorado, now the Rocky Mountain Arsenal Urban Wildlife Refuge, have resulted in groundwater contamination of the shallow alluvial aquifer. In response the US Army and Shell Chemical Company, the responsible parties, have implemented a series of pump and treat systems to intercept the contaminated groundwater. This paper discusses a stochastic groundwater model for the management of a pump and treat system located in the Offpost operable unit at the Rocky Mountain Arsenal. The local hydrogeology at the site is complex, which resulted in an intricate array of pumping wells, recharge wells and trenches. The goal of the modeling effort was to develop operational guidelines for the pump and treat system, which took into account uncertainties in hydraulic conductivity while minimizing the likelihood that contaminated groundwater would bypass the system. The modeling approach treated hydraulic conductivity as a stochastic variable and incorporated its uncertainty in system performance by using geostatistical conditional simulations. The Turning Bands and Kriging methods were used to generate a series of equally probable hydraulic conductivity distributions, which preserved the measured hydraulic conductivity values at sampled locations, as well as maintained the same statistics and spatial correlation structure of the known data. Generated hydraulic conductivity distributions were implemented as input to a finite element flow and transport model in order to simulate the performance, within a Monte Carlo framework of management scenarios, of the operation of the pump and treat system. Particle tracking simulations were utilized to determine the effectiveness of the system to intercept the plume under varying uncertainties in the hydraulic conductivity distribution in order to identify an optimal pumping and recharge strategy to prevent contaminated groundwater bypassing the system. The analyses were carried out statistically based upon the probability to capture the plume and to reduce the concentration levels to regulatory standards for a given operational time period. Results allowed improvement in the operation of the system and provided an excellent tool in the decision-making process for the management of contaminated groundwater.

Quantitative integration of hydrogeophysical data: Conditional geostatistical simulation for characterizing heterogeneous alluvial aquifers

High-resolution geophysical parameter information, as it can be provided, for example, by crosshole georadar and seismic tomography, has proven to provide useful spatial information to complement traditional hydrological methods such as core analyses, logging techniques, and tracer or pumping tests. Quantitative integration of these diverse database components is one of the major challenges in the field of high-resolution hydrogeophysics because of their different scales of measurement and the usually weak petrophysical relations among the measurements. In this study, we systematically explore the usefulness of a conditional stochastic simulation approach based on simulated annealing for this purpose. First, we generate a realistic model of an alluvial aquifer consisting of a 2D scale-invariant porosity field. On the basis of this model, we generate synthetic neutron porosity logs and crosshole georadar tomographic surveys. We then use the proposed geostatistical simulation approach to integrate this hydrogeophysical database. The effectiveness of this approach to characterize the detailed porosity distribution in heterogeneous alluvial aquifers is assessed by comparing the results for a variety of simulated porosity fields that differ fundamentally in terms of their conditioning information. Our results indicate this approach has the potential to allow for a realistic hydrogeophysical characterization in the submeter range of the porosity distribution in heterogeneous alluvial aquifers.

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.

Geostatistical Simulation of the Total Organic Carbon Values: An Example from Petroleum Source Rocks on the Coastal Area of Western Taurus Region, Turkey

Geostatistical conditional simulation is widely used and accepted as a method of generating stochastic models of hydrocarbon reservoirs. There are many applications described in the literature using one or more of the range of methods now available, and apart from the petroleum reservoirs, simulated annealing technique is applied to the total organic carbon values of the potential hydrocarbon source—rocks which has been penetrated by seven wells drilled on the western Taurus region. The main source rock intervals are mostly encountered in the autochthonous Beydaglari carbonates. After a description of the geology of the area a geostatistical study based on 440 composite samples is presented. The composite samples gave a positively skewed distribution with a mean of 0.27% total organic carbon values. Experimental variograms representing three main directions—strike plunge, cross dip, and downdip directions—were calculated a spherical type model was fitted to variograms. A model validation and a number of conditional simulation were applied to the data, respectively and the best simulation senario was chosen. Thus, a total of 1 440 000 grid nodes were conditonally simulated using the set of 440 data values. Simulation results indicate that the amount of organic matter that has generated hydrocarbons increases with increasing depth and significant source rocks rich in organic matter has concentrated on the coastal area of the western Taurus region.

Three-dimensional acoustic mapping and simulation of krill distribution in the Saguenay—St. Lawrence Marine Park whale feeding ground

Geostatistical conditional simulations are used to get a family of non-smoothed three-dimensional (3D) images of the St. Lawrence krill aggregation from echointegration data at 38 and 120 kHz on a systematic grid of transects. These maps respect the inherent patchiness of krill. Their ensemble gives an histogram of the krill density estimates at any point of the 3D grid, not only the mean density as in kriging. The spatially consistent simulations are conditional to match the observations and their histogram and variogram. The 3D problem is by-passed by transposing to a short series of 2D ones, which simplifies the modelling of the autocorrelation function and has the additional advantage of reducing by a factor of 5 the number of points to simulate. The spatially structured krill density profile is summarised by the first few factors of a principal component analysis, where the variables are the volume backscattering strength at 120 kHz for the integrated vertical bins along the profiles and the observations are the different profiles. The principal component scores are simulated over a 2D-grid and then used to reconstruct the full 3D-image. The method adequately reproduces the histogram, variogram and mean profile of krill density, and cross-validations replicate the observations. It is as precise as an alternative kriging approach to estimate the mean density, but has the additional advantages of requiring less computing time for our particular application. These conditional simulations enable estimating the probability density function of the krill density at any point, an essential information for predator-prey interactions and other ecosystem studies.

The Incorporation of Model Uncertainty in Geostatistical Simulation

A growing area of application for geostatistical conditional simulation is as a tool for risk analysis in mineral resource and environmental projects. In these applications accurate field measurement of a variable at a specific location is difficult and measurement of variables at all locations is impossible. Conditional simulation provides a means of generating stochastic realizations of spatial (essentially geological and/or geotechnical) variables at unsampled locations thereby quantifying the uncertainty associated with limited sampling and providing stochastic models for 'downstream' applications such as risk assessment. However, because the number of experimental data in practical applications is limited, the estimated geostatistical parameters used in the simulation are themselves uncertain. The inference of these parameters by maximum likelihood provides a means of assessing this estimation uncertainty which, in turn, can be included in the conditional simulation procedure. A case study based on t...

Stochastic optimization for global minimization and geostatistical calibration

This study proposes a stochastic optimization technique that uses a gradient-based method as the primary optimization method, as well as a geostatistical conditional simulation to perturb and calibrate parameters at every local minimum. If the optimization process is trapped at a local minimum due to the limitations of the gradient-based method, it generates equi-probable parameter fields using a geostatistical conditional simulation. Among the generated fields, the optimization process selects one that enables the objective function to be reduced below the value of that at the local minimum, and then reactivates the gradient-based optimization. In generating equi-probable parameter fields, a constrained number of points (noted as releasing points) are randomly selected, and spatially correlated values are generated at the releasing points, conditioned to optimum parameters at the local minimum. By applying the stochastic optimization to synthetic permeability fields, it is observed that an inversed permeability field reproduces not only global distribution but also local spatial variability of the reference fields. In addition, the pressure distributions of the inversed and the reference field were much alike. To investigate dynamic properties of the inversed field and the reference field, streamline simulation was performed on both fields. Streamlines of the inversed field showed similar trajectories to those of the reference field, and time of flight (TOF) distribution of the inversed field was analogous to that of the reference field. The stochastic optimization technique proposed in this paper enables an inverse process to converge to a global minimum while preserving geostatistical properties such as mean, standard deviation, and variogram of an original field. Therefore, the stochastic optimization will be efficient in predicting future performance of a field from constrained number of permeability and pressure observation data.

Three‐dimensional interference test interpretation in a fractured aquifer using the Pilot Point Inverse Method

A geostatistical conditional simulation routine that employs both parametric and nonparametric geostatistics is coupled to the pilot point inverse method to simulate the spatial distribution of conductivities in a dolomitic aquifer which is fractured in some areas and unfractured in others. An inversion is then conducted to obtain the conductivities within the fractured and nonfractured parts of the aquifer to match interference data from a series of three‐dimensional pumping tests for an ensemble of 100 conditional simulations. The calibrated and predicted drawdowns match the observed drawdowns well. A comparison to the transmissivity interpreted from a single‐well pumping test compared very well to the geometric mean transmissivity calculated from the calibrated realizations in this study. The results from this study indicate that conditioning the conductivity fields to the geologic facies data, in this case fractured‐unfractured categorical data, as well as to ample transient hydraulic data can lead to robust groundwater flow models which can adequately predict the response to other hydraulic interference tests.

Petroleum geostatistics for nongeostaticians

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 …

Petroleum geostatistics for nongeostatisticians

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 …

Uncertainty in groundwater flow and transport modelling?a stochastic analysis of well-protection zones

The paper reviews the problems of uncertainty in the definition of aquifer properties and the scale-dependence of dispersion with specific regard to well-capture zones. Stochastic methods have been developed in response to the problems of characterizing aquifer properties for contaminant transport modelling, as they provide a formal framework for the treatment of uncertainty. Their use for well-capture zone analysis, however, has been neglected so far. In this paper a stochastic numerical modelling procedure is presented, based on conditional geostatistical simulation of fine-scale aquifer structure, which is shown to reproduce stochastic analytical properties. The methodological approach is applied to the analysis of uncertainty in well-protection zones, and implications of the results for groundwater management are discussed. Copyright © 2000 John Wiley & Sons, Ltd.