Classical Geostatistical Methods Research Articles

Analyse géostatistique des niveaux piézométriques dans un secteur minier avec trois mines en activité

AbstractMining activities can significantly impact groundwater reservoirs in their vicinity. Different approaches have been employed, with varying success, to investigate the spatial variability of groundwater levels in mining areas. Typical problems include the small sample size, the non-Gaussian distribution of the data, and the clustering of sample locations near the mines. These conditions complicate the estimation of spatial dependence. Under sparse and irregular sampling conditions, stochastic methods, which can provide estimates of prediction uncertainty, are preferable to deterministic ones. This research focuses on the comparison of two stochastic methods, stochastic local interactions (SLI) and universal Kriging (UK), using water level data from 72 locations around three mines in Northern Greece. UK is a well-known, variogram-based geostatistical method, while SLI is a computationally efficient kernel-based method that can cope with large spatial datasets. The non-Gaussian distribution of the data is handled by means of a flexible, data-driven Gaussian anamorphosis method that uses kernel functions. The spatial prediction performance of both methods is assessed based on cross-validation. UK performs better than SLI, due to the fact that the former incorporates a linear trend function. On the other hand, a comparison of the two methods using data from a single mine that contains only 28 measurement locations shows that SLI performs slightly better than UK. The prediction uncertainties for both methods are also estimated and compared. The results suggest that SLI can provide better estimates than classical geostatistical methods for small sample sizes that do not allow reliable estimation of the variogram model.

Distribution Characteristics and Drivers of Soil Carbon and Nitrogen in the Drylands of Central Asia

Soil organic carbon (C) and soil total nitrogen (N) show different degrees of spatial variability at different scales. Both are important components of soil nutrients and essential elements for plant growth and development, and are closely related to biogeochemical cycles. However, there is limited information on the regional spatial validity of SOC and TN and the associated drivers at the scale of the Central Asian drylands. Therefore, this study uses the ISRIC-WISE (International Soil Reference and Information Centre-Word Inventory of Soil Property Estimates) database to conduct soil sampling at the raster level, combined with relevant climatic and environmental datasets, to investigate the spatial distribution characteristics and drivers of soil C and N in the drylands of Central Asia using classical geostatistical methods and structural equation modelling (SEM). The results of this study show that the distributions of soil C and N contents in the dry zone of Central Asia have greater similarity, with C content mainly concentrated in the ranges of 0–5.5 g/kg and 11.1–15.9 g/kg; soil N content mainly concentrated in the range of 0.4–1.1 g/kg, and the soil C:N ratio mainly concentrated in the range of 12.2–28.9. Structural equation modelling showed that the main driver of soil C change was Aridity (−0.51); the main driver of soil N change was Mean Annual Temperature (MAT) (−0.44); and soil C:N change was most influenced by Aboveground biomass (AGB) (−0.25). An analysis of the relative importance contribution showed that Aridity had the highest relative importance with regard to the change in C (32%); MAT had the highest relative importance with regard to the changes in N and C:N (29% and 40%, respectively). The above findings provide a reference for the use of soil resources in drylands and provide a scientific basis for regional differences in the response of arid ecosystems to climate change.

Stochastic Local Interaction Model: An Alternative to Kriging for Massive Datasets

Classical geostatistical methods face serious computational challenges if they are confronted with large sets of spatially distributed data. We present a simplified stochastic local interaction (SLI) model for computationally efficient spatial prediction that can handle large data. The SLI method constructs a spatial interaction matrix (precision matrix) that accounts for the data values, their locations, and the sampling density variations without user input. We show that this precision matrix is strictly positive definite. The SLI approach does not require matrix inversion for parameter estimation, spatial prediction, and uncertainty estimation, leading to computational procedures that are significantly less intensive computationally than kriging. The precision matrix involves compact kernel functions (spherical, quadratic, etc.) which enable the application of sparse matrix methods, thus improving computational time and memory requirements. We investigate the proposed SLI method with a data set that includes approximately 11500 drill-hole data of coal thickness from Campbell County (Wyoming, USA). We also compare SLI with ordinary kriging (OK) in terms of estimation performance, using cross validation analysis, and computational time. According to the validation measures used, SLI performs slightly better in estimating seam thickness than OK in terms of cross-validation measures. In terms of computation time, SLI prediction is 3 to 25 times (depending on the size of the kriging neighborhood) faster than OK for the same grid size.

Effective probability distribution approximation for the reconstruction of missing data

Spatially distributed processes can be modeled as random fields. The complex spatial dependence is then incorporated in the joint probability density function. Knowledge of the joint probability density allows predicting missing data. While environmental data often exhibit significant deviations from Gaussian behavior (rainfall, wind speed, and earthquakes being characteristic examples), only a few non-Gaussian joint probability density functions admit explicit expressions. In addition, random field models are computationally costly for big datasets. We propose an “effective distribution” approach which is based on the product of univariate conditional probability density functions modified by local interactions. The effective densities involve local parameters that are estimated by means of kernel regression. The prediction of missing data is based on the median value from an ensemble of simulated states generated from the effective distribution model. The latter can capture non-Gaussian dependence and is applicable to large spatial datasets, since it does not require the storage and inversion of large covariance matrices. We compare the predictive performance of the effective distribution approach with classical geostatistical methods using Gaussian and non-Gaussian synthetic data. We also apply the effective distribution approach to the reconstruction of gaps in large raster data.

Space–time forecasting of groundwater level using a hybrid soft computing model

ABSTRACTForecasting of space–time groundwater level is important for sparsely monitored regions. Time series analysis using soft computing tools is powerful in temporal data analysis. Classical geostatistical methods provide the best estimates of spatial data. In the present work a hybrid framework for space–time groundwater level forecasting is proposed by combining a soft computing tool and a geostatistical model. Three time series forecasting models: artificial neural network, least square support vector machine and genetic programming (GP), are individually combined with the geostatistical ordinary kriging model. The experimental variogram thus obtained fits a linear combination of a nugget effect model and a power model. The efficacy of the space–time models was decided on both visual interpretation (spatial maps) and calculated error statistics. It was found that the GP–kriging space–time model gave the most satisfactory results in terms of average absolute relative error, root mean square error, normalized mean bias error and normalized root mean square error.

Improving the analysis of well-logs by wavelet cross-correlation

The concept of wavelet cross-correlation is used to provide a new approach to identify similar patterns in related data sets, which largely improves the confidence of the results. The method amounts to decompose the data sets in the wavelet space so that correlations between wavelet coefficients can be analyzed in every scale. Besides the identification of the scales in which two independent measures are correlated, the method makes it possible to find patches of data sets where correlations exist simultaneously in all scales. This allows to extend the information of a small number of spots to larger regions. Well-log data sets from two neighboring oil wells are used. We compare similar measures at different probe sites, and also measurements of different physical quantities taken on the same place. Although this is a typical scenario for the application of classical geostatistical methods, it is well known that such methods erase out local differences in favor of smoother variability. In contraposition, this wavelet cross-correlation takes advantage of the fluctuations to give information about the continuity of the geological structures in space. It works even better if no filtering procedure has been applied to the original raw data.

Towards soil geostatistics

In a brief survey of some issues in the application of geostatistics in soil science it is shown how the recasting of classical geostatistical methods in the linear mixed model (LMM) framework has allowed the more effective integration of soil knowledge (classifications, covariates) with statistical spatial prediction of soil properties. The LMM framework has also allowed the development of models in which the spatial covariance need not be assumed to be stationary. Such models are generally more plausible than stationary ones from a pedological perspective, and when applied to soil data they have been found to give prediction error variances that better describe the uncertainty of predictions at validation sites. Finally consideration is given to how scientific understanding of variable processes in the soil might be used to infer the likely statistical form of the observed soil variation.

An application of Spartan spatial random fields in environmental mapping: focus on automatic mapping capabilities

This paper investigates the potential of Spartan spatial random fields (SSRFs) in real-time mapping applications. The data set that we study focuses on the distribution of daily gamma dose rates over part of Germany. Our goal is to determine a Spartan spatial model from the data, and then use it to generate “predictive” maps of the radioactivity. In the SSRF framework, the spatial dependence is determined from sample functions that focus on short-range correlations. A recently formulated SSRF predictor is used to derive isolevel contour maps of the dose rates. The SSRF predictor is explicit. Moreover, the adjustments that it requires by the user are reduced compared to classical geostatistical methods. These features present clear advantages for an automatic mapping system. The performance of the SSRF predictor is evaluated by means of various cross-validation measures. The values of the performance measures are similar to those obtained by classical geostatistical methods. Application of the SSRF method to data that simulate a radioactivity release scenario is also discussed. Hot spots are detected and removed using a heuristic method. The extreme values that appear in the path of the simulated plume are not captured by the currently used Spartan spatial model. Modeling of the processes leading to extreme values can enhance the predictive capabilities of the spatial model, by incorporating physical information.

Canopy Influence on Rainfall Fields' Microscale Structure in Tropical Forests

Abstract This study of tropical rainfall fields was conducted in French Guyana in the moist tropical forest. Two experimental catchments were monitored. The area of these catchments is about 10 000 m2; one was deforested and is now a meadow, and the second is still in the state of primary forest. For both catchments, rainfall was measured by a dense network of rain gauges. Classical geostatistical methods and tools, especially variograms, were used to characterize the spatial structure of rainfall fields. Over the deforested catchment the spatial rainfall field is continuous. On the contrary, the rainfall field under the canopy of the forested catchment is unstructured. The vegetation there is completely responsible for the random rainfall distribution. However, when the canopy retention capacity is saturated, the spatial throughfall field can become continuous.