Traditional Geostatistics Research Articles

Transitive kriging for modeling tailings deposits: A case study in southwest Finland

Global mining generates a large amount of mine tailings, which can produce negative effects on the environment. To counteract this, government guidelines and scientific interest have emerged to reuse tailings deposits in innovative ways, converting them from an environmental liability to an economic asset. Thus, the characterization of a tailings deposit is of great importance to analyze the content of critical raw materials, as well as for a possible revaluation of the deposit, facilitating the mining process to be carried out in a more sustainable way. To characterize their chemical composition, information from drilling campaigns can be used. However, the evaluation of resources in tailings deposits has several complexities such as the poor grade spatial continuity and its narrow geometry, reasons for which traditional geostatistics is not adapted to model this type of deposit. In contrast, transitive geostatistics can be an opportunity to tackle this taking advantage of the fully delimited domain in a tailings dam and a different structural analysis. By achieving a better characterization of tailings composition, it is possible to make better decisions for their reprocessing, thus supporting cleaner mining production.This study aims to provide a framework for the geostatistical modeling and prediction of remaining metal resources of the Haveri tailings in southwestern Finland. The modeled variables are the gold, cobalt, copper and iron grades, with cobalt being of special interest as a critical material. The grades have been measured at 165 drill holes, totaling 1201 samples composited at 1 meter.An exploratory data analysis is performed first to clean the database and to identify the statistical and spatial distributions of the data. Then a structural analysis is applied to model the grade spatial continuity. Leave-one-out cross-validation is subsequently used to validate the fitted model and to quantify the prediction errors. Finally, 3D block models of the gold, cobalt, copper and iron grade are constructed with ordinary kriging and transitive kriging and are compared.Cross-validation shows that both kriging methods yield a good precision of the predictions and perform equally well for copper and iron, but transitive kriging significantly outperforms ordinary kriging for gold and cobalt and also provides less smoothed block models than ordinary kriging. Transitive kriging thus appears as an effective alternative for assessing resources in tailings and other narrow deposits. Recommendations on the sampling design to optimize the coverage of the target area and to ease the covariogram inference are also given.

An Improved Method of Reservoir Facies Modeling Based on Generative Adversarial Networks

As the reservoir and its attribute distribution are obviously controlled by sedimentary facies, the facies modeling is one of the important bases for delineating the area of high-quality reservoir and characterizing the attribute parameter distribution. There are a large number of continental sedimentary reservoirs with strong heterogeneity in China, the geometry and distribution of various sedimentary microfacies are relatively complex. The traditional geostatistics methods which have shortage in characterization of the complex and non-stationary geological patterns, have limitation in facies modeling of continental sedimentary reservoirs. The generative adversarial network (GANs) is a recent state-of-the-art deep learning method, which has capabilities of pattern learning and generation, and is widely used in the domain of image generation. Because of the similarity in content and structure between facies models and specific images (such as fluvial facies and the images of modern rivers), and the various images generated by GANs are often more complex than reservoir facies models, GANs has potential to be used in reservoir facies modeling. Therefore, this paper proposes a reservoir facies modeling method based on GANs: (1) for unconditional modeling, select training images (TIs) based on priori geological knowledge, and use GANs to learn priori geological patterns in TIs, then generate the reservoir facies model by GANs; (2) for conditional modeling, a training method of “unconditional-conditional simulation cooperation” (UCSC) is used to realize the constraint of hard data while learning the priori geological patterns. Testing the method using both synthetic data and actual data from oil field, the results meet perfectly the priori geological patterns and honor the well point hard data, and show that this method can overcome the limitation that traditional geostatistics are difficult to deal with the complex non-stationary patterns and improve the conditional constraint effect of GANs based methods. Given its good performance in facies modeling, the method has a good prospect in practical application.

Spatiotemporal forecast with local temporal drift applied to weather patterns in Patagonia

Geostatistics was developed to generate maps or 3D models interpolating observed values in space. The so-called spatiotemporal geostatistic applies the same principles to estimate observed values that have both spatial and temporal distribution. Moreover, time series analysis can decompose and extrapolate its main trends and seasonality, preparing data for geostatistical assumptions. Using this principle, this study aims to decompose the time series of a spatiotemporal dataset as external drifts and estimate its residuals by spatiotemporal kriging. Since each observation point is a time series, it is possible to decompose its trend and seasonality locally and map its parameters, preferable, by traditional geostatistics. Aftermath, it is possible to extrapolate the trend and seasonality at each pixel. This procedure can achieve great long-term forecasting maps even in regions with poor sampling due to its time series analysis. As well as, the geostatistics guarantee that the spatio-temporal correlation is maintained. This method is especially good for prediction in regions that the time series pattern depends on its location, which is a common problem in large areas and the problem is worsened in poorly sampled regions. This study presents a 10 years map forecast (2008–2017) comparison by spatiotemporal geostatistics, the first with original data, with ARIMA Models Panels, then with global decomposition, finally, with the local decomposition approach. The target variable is temperature captured by the 18 active weather stations in Patagonia between 1973 and 2007. To validate the results, they are compared to Land Surface Temperature (LST), which is an image product MOD11C3 derived from the MODIS sensor onboard on Terra/Aqua satellites. The proposed method can make long-term forecasts with low error, low smoothing effect and similar spatiotemporal statistics (mean and variance) of the stations and the LST product. Finally, its results are comparable with the ARIMA Models Panels with the advantage that it can generate maps with spatiotemporal correlation and better than the often-used methods (st-kriging and global decomposition) to forecast large areas maps.

A spatial model for a stream networks of Citarik River with the environmental variables: potential of hydrogen (PH) and temperature

Application of existing geostatistical theory of stream networks provides a number of interesting and challenging problems. Most of statistical tools in the traditional geostatistics have been based on a Euclidean distance such as autocovariance functions, but for stream data is not permissible since it deals with a stream distance. To overcome this autocovariance developed a model based on the distance the flow with using convolution kernel approach (moving average construction). Spatial model for a stream networks is widely used to monitor environmental on a river networks. In a case study of a river in province of West Java, the objective of this paper is to analyze a capability of a predictive on two environmental variables, potential of hydrogen (PH) and temperature using ordinary kriging. Several the empirical results show: (1) The best fit of autocovariance functions for temperature and potential hydrogen (ph) of Citarik River is linear which also yields the smallest root mean squared prediction error (RMSPE), (2) the spatial correlation values between the locations on upstream and on downstream of Citarik river exhibit decreasingly

Process-Driven Direction-Dependent Asymmetry: Identification and Quantification of Directional Dependence in Spatial Fields

Natural processes generate spatial fields which reflect their specific properties. In this paper the effect of the direction of processes on the resulting spatial fields is investigated. This is done by extending the concept of reversibility used for time series to space. A novel copula based measure of asymmetry is defined which is an indicator of directional dependence. Contrary to traditional geostatistics where all points separated by a vector are considered irrespective of its sign, in this study the direction of the vector is also taken into consideration, leading to differences in the dependence corresponding to the vector \({\mathbf {h}}\) and \({-}{\mathbf {h}}\). The concept of directional dependence and the corresponding measure of asymmetry are defined using spatial copulas, and are thus independent of the scale of measurement. The result is a bivariate directional third-order moment based measure which can identify the direction in which the processes generating the spatial field acted. A statistical test to find the statistical significance of the asymmetry indicating directional dependence is presented. The methodology is tested on a number of synthetic and observed cases. Precipitation and groundwater quality parameters obtained using numerical models are first investigated. Regular dense grids obtained by numerical simulations show good correspondence between properties of the modeled processes and the new measure introduced. Measured variables observed on sparse irregular networks show similar behavior to the theoretical examples. Mean flow directions in groundwater and advection directions of precipitation fields can be detected from single snapshots. As a further example, dominant wind directions in the Sahara are found by investigating the digital terrain model.

Characterization of Aquifer Multiscale Properties by Generating Random Fractal Field with Truncated Power Variogram Model Using Karhunen–Loève Expansion

The traditional geostatistics to describe the spatial variation of hydrogeological properties is based on the assumption of stationarity or statistical homogeneity. However, growing evidences show and it has been widely recognized that the spatial distribution of many hydrogeological properties can be characterized as random fractals with multiscale feature, and spatial variation can be described by power variogram model. It is difficult to generate a multiscale random fractal field by directly using nonstationary power variogram model due to the lack of explicit covariance function. Here we adopt the stationary truncated power variogram model to avoid this difficulty and generate the multiscale random fractal field using Karhunen–Loève (KL) expansion. The results show that either the unconditional or conditional (on measurements) multiscale random fractal field can be generated by using truncated power variogram model and KL expansion when the upper limit of the integral scale is sufficiently large, and the main structure of the spatial variation can be described by using only the first few dominant KL expansion terms associated with large eigenvalues. The latter provides a foundation to perform dimensionality reduction and saves computational effort when analyzing the stochastic flow and transport problems.

Spatial variation of soil respiration in a coastal protection forest in southeastern China

Forest soil respiration is an important component of global carbon budgets, but its spatial variation is inadequately understood. This research aimed to measure soil respiration (R s), soil water content (M s-5), soil temperature (T), and carbon dioxide (M co2) in a coastal protection forest (CPF), which is one kind of man-made forests designed for coastal protection primarily along the coast in China, to determine the relationships among them, and to analyze their spatial distributions in a small scale. We measured R s, M s-5, T, and M co2 of 100 plots in an approximately flat grid (totally 4 hm2) by LI-8100A in a Casuarina equisetifolia L. forest on a state-owned forest farm of 326 hm2 in SE China. Traditional statistics and geo-statistics including semivariance, Moran’s I index, and fractal dimension were used to analyze data. Key findings were that (1) the spatial mean of R s, M s-5, T, and M co2 were 1.194 μmol m−2 s−1, 11.387 mmol mol−1, 14.153 °C, and 407.716 ppm, respectively, in the forest; (2) the relationship between soil respiration and the other three factors was weak, while M s-5, T, and M co2 have strong relationships with each other; and (3) the four factors, especially soil respiration, had strong autocorrelation within given limits and showed great heterogeneity with 95 % confidence intervals around the means in the study area, all of which can provide important value for the study of carbon cycling and for the sustainable management of coastal protection forests. According to geo-statistical analysis and field investigations, soil respiration in the coastal forest is less than in some broad-leaf forests but higher than in some conifers. Strong heterogeneity and autocorrelation are clear; however, its relation with other three factors is weak. CPF is a considerable potential forest for carbon conservation if it is well managed.

On the prediction of extreme flood quantiles at ungauged locations with spatial copula

The present study investigates the use of the spatial copula approach for predicting flood quantiles at ungauged basins. Spatial copulas are the formalization of traditional geostatistics by copulas. In regional flood frequency analysis (RFFA), the regression of flood quantiles is often carried out at the logarithmic scale. Consequently, traditional interpolation methods introduce a bias and provide suboptimal predictions. In this study, the copula framework is examined for offering proper corrections in this framework. Moreover, copula techniques separate the regional distribution of flood quantiles from spatial dependence. This provides a full probabilistic model that represents a more flexible framework where proper combinations of regional distribution and dependence can be adapted to various situations that are encountered in RFFA. The adequacy of the investigated methodology is evaluated on a real world case study involving hydrometric stations from southern Quebec, Canada. Results show that the spatial copula framework is able to deal with the problem of bias, is robust to the presence of problematic stations and may improve the quality of quantile predictions while reducing the level of complexity of the models used in RFFA.

Spatial-temporal character of exchangeable magnesium and its influencing factors at tobacco-growing soil region in Shaoyang

In order to understand the distribution of exchangeable magnesium content for tobacco-growing soils in Shaoyang, exchangeable magnesium content and its influencing factors such as pH, organic matter were analyzed using traditional statistics and geostatistics based on 1 790 observed soil samples. The results showed: ①The exchangeable magnesium content in the tobacco-growing soil was in lower level,with a mean of 0.78 cmol/kg and variation coefficient of 51.39%, and about 16.98% of exchangeable magnesium content in the observed soil samples was suitable for tobacco cultivation. ②The exchangeable magnesium content in tobacco-growing soils in Longhui and Shaoyang counties were significantly higher than that of in Xinning county; From the map interpolated by Kriging, the tendency of spatial variation of exchangeable magnesium decreased from north to south; Compared to the results of soil survey in 2002, the exchangeable magnesium in tobacco soils showed a decreased tendency. ③A quadratic curve relationship was found between exchangeable magnesium and soil pH value. If the pH value was lower than 5.0, the exchangeable magnesium was in the state of insufficiency or latent insufficiency; There was no apparent relationship between exchangeable magnesium and organic matter for tobacco-growing soils.

Coalseam Thickness Analysis with Geostatistics

When we treat coalseam which is thicker and complicated, we can use kriging modeling coalseam and interlayer, then we minus prediction surface; by comparing different effect between traditional method and geostatistics to reserves calculation and coalseam thickness prediction, the paper shows advantage of using of geostatistics in analysis of coalseam thickness and bring a new method of treating coalseam containing many dirt bands.

Comparison of maps of spatial variability of soil resistance to penetration constructed with and without covariables using a spatial linear model

A study about the spatial variability of data of soil resistance to penetration (RSP) was conducted at layers 0.0-0.1 m, 0.1-0.2 m and 0.2-0.3 m depth, using the statistical methods in univariate forms, i.e., using traditional geostatistics, forming thematic maps by ordinary kriging for each layer of the study. It was analyzed the RSP in layer 0.2-0.3 m depth through a spatial linear model (SLM), which considered the layers 0.0-0.1 m and 0.1-0.2 m in depth as covariable, obtaining an estimation model and a thematic map by universal kriging. The thematic maps of the RSP at layer 0.2-0.3 m depth, constructed by both methods, were compared using measures of accuracy obtained from the construction of the matrix of errors and confusion matrix. There are similarities between the thematic maps. All maps showed that the RSP is higher in the north region.

Combining Geostatistics with Moran’s I Analysis for Mapping Soil Heavy Metals in Beijing, China

Production of high quality interpolation maps of heavy metals is important for risk assessment of environmental pollution. In this paper, the spatial correlation characteristics information obtained from Moran’s I analysis was used to supplement the traditional geostatistics. According to Moran’s I analysis, four characteristics distances were obtained and used as the active lag distance to calculate the semivariance. Validation of the optimality of semivariance demonstrated that using the two distances where the Moran’s I and the standardized Moran’s I, Z(I) reached a maximum as the active lag distance can improve the fitting accuracy of semivariance. Then, spatial interpolation was produced based on the two distances and their nested model. The comparative analysis of estimation accuracy and the measured and predicted pollution status showed that the method combining geostatistics with Moran’s I analysis was better than traditional geostatistics. Thus, Moran’s I analysis is a useful complement for geostatistics to improve the spatial interpolation accuracy of heavy metals.

Multiple-point Simulation and Its Application and Development in RS Image Classification

Multiple-point simulation(MPS) is one of the applications of multiple-point statistics in geology.It can simulate the complex geometry-structure by getting the conditional probability from training image through search-tree while the traditional geostatistics has little ability to capture the complex geometry-structure.When MPS is applied into remote sensing classification,it can help us to improve the traditional-pixel-by-pixel-classification accuracy by combining the structural and spectral information.This article firstly introduces the development of new classification method which is called CCSSM,and then discusses the revision of this methods.Furthermore,it extends CCSSM's application to the classification of the multiple-class from in the perspects of a theoretical and feasible way.

Multiple Point Metrics to Assess Categorical Variable Models

Geostatistical models should be checked to ensure consistency with conditioning data and statistical inputs. These are minimum acceptance criteria. Often the first and second-order statistics such as the histogram and variogram of simulated geological realizations are compared to the input parameters to check the reasonableness of the simulation implementation. Assessing the reproduction of statistics beyond second-order is often not considered because the “correct” higher order statistics are rarely known. With multiple point simulation (MPS) geostatistical methods, practitioners are now explicitly modeling higher-order statistics taken from a training image (TI). This article explores methods for extending minimum acceptance criteria to multiple point statistical comparisons between geostatistical realizations made with MPS algorithms and the associated TI. The intent is to assess how well the geostatistical models have reproduced the input statistics of the TI; akin to assessing the histogram and variogram reproduction in traditional semivariogram-based geostatistics. A number of metrics are presented to compare the input multiple point statistics of the TI with the statistics of the geostatistical realizations. These metrics are (1) first and second-order statistics, (2) trends, (3) the multiscale histogram, (4) the multiple point density function, and (5) the missing bins in the multiple point density function. A case study using MPS realizations is presented to demonstrate the proposed metrics; however, the metrics are not limited to specific MPS realizations. Comparisons could be made between any reference numerical analogue model and any simulated categorical variable model.

Temporal-spatial variability and fractal characteristics of soil nitrogen and phosphorus in Xinji District, Hebei Province, China

Traditional statistics, geostatistics, fractal dimensions, and geographic information systems (GIS) were employed to study the temporal-spatial variability of soil total nitrogen (TN) and total phosphorus (TP) levels in Xinji District, Hebei Province area of the North China Plain from 1980 to 2007. The results indicate that nutrient levels follow normal or lognormal distributions. The TN content was 0.59 ± 0.155 g kg( -1) in 2007, an increase of 0.44 g kg( -1) compared with that of 1980. In 2007, the TP content was 1.21 ± 0.227 g kg( -1), an increase of 0.01 g kg( -1) from 1980. The geostatistical analysis showed that the distribution of these soil nutrients in the study area exhibits a trend and anisotropy. The range and [C (0)/(C (0) +C)] of TN and TP in 1980 were all less than in 2007. The ordinary kriging interpolation method was used to analyze the nutrient contents differences between 1980 and 2007. The results indicate that soil TN levels have increased over the 27-year period, and the area where the TN level had increased by at least 0.4 g kg( -1) was about 61.7% of the district. The area where the TP content increased covered about 58.4% of the district. The variance analysis indicated that land-use type had a clear influence on the distribution and change in TN and TP content. Using the 3-D box-counting dimension method combined with GIS, the fractal dimension of soil nutrient spatial distribution over the two periods showed that in 27 years, the fractal dimension of TN increased from 1.95 to 2.02, and the fractal dimension of TP increased from 1.89 to 2.01, indicating that the complexity of the spatial distribution of all nutrient contents had increased. This study can provide a basis for accurate fertilizing and to enhance the conversion of soil characteristics under different spatial scales.

Using landscape characteristics to define an adjusted distance metric for improving kriging interpolations

Interpolation of point measurements using geostatistical techniques such as kriging can be used to estimate values at non-sampled locations in space. Traditional geostatistics are based on the spatial autocorrelation concept that nearby things are more related than distant things. In this study, additional information was used to modify the traditional Euclidean concept of distance into an adjusted distance metric that incorporates similarity in terms of quantifiable landscape characteristics such as topography or land use. This new approach was tested by interpolating soil moisture content, pH and carbon-to-nitrogen (C:N) ratio measured in both the mineral and the organic soil layers at a field site in central Sweden. Semivariograms were created using both the traditional distance metrics and the proposed adjusted distance metrics to carry out ordinary kriging (OK) interpolations between sampling points. In addition, kriging with external drift (KED) was used to interpolate soil properties to evaluate the ability of the adjusted distance metric to incorporate secondary data into interpolations. The new adjusted distance metric typically lowered the nugget associated with the semivariogram, thereby better representing small-scale variability in the measured data compared to semivariograms based on the traditional distance metric. The pattern of the resulting kriging interpolations using KED and OK based on the adjusted distance metric were similar because they represented secondary data and, thus, enhanced small-scale variability compared to traditional distance OK. This created interpolations that agreed better with what is expected for the real-world spatial variation of the measured properties. Based on cross-validation error, OK interpolations using the adjusted distance metric better fit observed data than either OK interpolations using traditional distance or KED.

Multiple‐point geostatistics for modeling subsurface heterogeneity: A comprehensive review

For more than half a century, geostatistics has been developed and has increasingly been used for modeling subsurface heterogeneity. Traditionally, geostatistical simulations are based on a random function model defined according to the specificities of the geological formation under investigation. Unlike traditional geostatistics, multiple‐point (MP) geostatistics avoids the explicit definition of a random function but directly infers the necessary multivariate distributions from training images. This confers on MP geostatistics a potential applicability to any geological environment, provided that there is a training image representative of the geological heterogeneity and that the essential features of this training image can be characterized by statistics defined on a limited point configuration. This paper presents a comprehensive review of MP geostatistics. If the principle of MP geostatistics is straightforward and attractive, its industrial applicability largely depends on the implementation methods. The use of a search tree for storing MP statistics is a great step that made MP geostatistics actually applicable in industry. In the meantime, other methods such as multiple grids and image postprocessing are introduced and allow enhancing the reproduction of patterns observed in training images. Because of the sequential procedure, MP simulations can easily honor local hard data. There are methods available for constraining MP simulations to target global statistics (facies proportions) and also for integrating spatial auxiliary constraints that can tremendously improve the spatial features of MP simulations. Furthermore, both the gradual deformation method and the probability perturbation method are compatible with MP geostatistics and allow integrating hydrodynamic data into MP simulations. More recent developments of MP geostatistics include sequentially simulating patterns instead of points and using different geological scenarios (training images) for dynamic data inversion.

Temporal and spatial variability of soil organic matter in a county scale agricultural ecosystem

In agricultural ecosystems, soil organic matter (SOM) is a major determinant and indicator of soil fertility and quality, and is closely related to crop productivity. This paper examines the temporal and spatial variation of SOM in Pinggu district at a county scale using data for 1980 and 2005 and traditional statistics, geostatistics, and geographic information system (GIS) analyses. Pinggu district, Beijing, China is an area characterised by changing farming practices and land uses. A lognormal distribution feature was observed for both datasets, and the difference between them was significant (t‐test). The geostatistical characteristics were analysed using a GIS platform. The distributions of SOM content in 1980 and 2005 both exhibited a linear trend and anisotropy. Spherical semivariogram models and relative parameters were developed. The spatial distribution maps for 1980 and 2005 were generally similar, with the SOM content increasing gradually from south‐west to north‐east. The difference between the two maps was determined using map algebraic functions and was found to show considerable spatial change. Changes in farming practices and land use are thought to be the main reason for the temporal change. Therefore, a combination of geostatistics and GIS map algebra provides a useful tool for examining spatiotemporal changes in the environmental sciences and may detect features that are not discernible when only conventional statistics are used.

COMBINING GEOSTATISTICS AND MULTI‐ATTRIBUTE TRANSFORMS: A CHANNEL SAND CASE STUDY, BLACKFOOT OILFIELD (ALBERTA)

In this paper, we combine the methods of geostatistics and multi‐attribute reservoir parameter prediction (the multi‐attribute transform) for the integration of seismic and well log data, and illustrate this new procedure with a case study involving the prediction of porosity at the Blackfoot oilfield, central Alberta. The objectives of the survey were to delineate incised, valley‐fill sediments within the Early Cretaceous Glauconitic Formation at this field and to distinguish between sand‐fill and shale‐fill. The input consisted of twelve porosity logs together with a 3D seismic volume and the inversion of this volume. Although an excellent correlation was found between porosity and the initial inverted acoustic impedance volume, the combination of traditional geostatistics and the multi‐attribute transform produced an improved final result.Our approach uses well logs to “train” the multi‐attribute transform algorithm. We first extract average porosity values over the depth zone of interest, and compare these values to average seismic attributes over the same zone. Cross‐validation is used to show which attributes are significant. We then apply the results of the training and cross‐validation to data slices derived from both the seismic data cube and the inverted cube to produce an initial porosity map. Finally, we improve the fit between the well‐log values and the porosity map using cokriging.

Bayesian Maximum Entropy Analysis and Mapping: A Farewell to Kriging Estimators?

The Bayesian Maximum Entropy (BME) method of spatial analysis and mapping provides definite rules for incorporating prior information, hard and soft data into the mapping process. It has certain unique features that make it a loyal guardian of plausible reasoning under conditions of uncertainty. BME is a general approach that does not make any assumptions regarding the linearity of the estimator, the normality of the underlying probability laws, or the homogeneity of the spatial distribution. By capitalizing on various sources of information and data, BME introduces an epistemological framework that produces predictive maps that are more accurate and in many cases computationally more efficient than those derived by traditional techniques. In fact, kriging techniques can be derived as special cases of the BME approach, under restrictive assumptions regarding the prior information and the data available. BME is a more rigorous approach than indicator kriging for incorporating soft data. The BME formulation, in fact, applies in a spatial or a spatiotemporal domain and its extension to the case of block and vector random fields is straightforward. New theoretical results are presented and numerical examples are discussed, which use the BME approach to account for important sources of knowledge in a systematic manner. BME can be useful in practical situations in which prior information can be used to compensate for the limited amount of measurements available (e.g., preliminary or feasibility study levels) or soft data are available that can be combined with hard data to improve mapping significantly. BME may be then viewed as an effort towards the development of a more general framework of spatial/temporal analysis and mapping, which includes traditional geostatistics as its limiting case, and it also provides the means to derive novel results that could not be obtained by traditional geostatistics.