multi-Gaussian Kriging Research Articles

Geostatistical mapping of precipitation: implications for rain gauge network design

This study examined four univariate kriging techniques; simple kriging (SK), ordinary kriging (OK), multi-Gaussian kriging (MGC), and log-normal kriging (LNK); and two multivariate kriging algorithms; kriging with external drift (KED) using elevation and slope in two different models for the estimation of daily rainfall in a 250 m x 250 m grid over a 750 km2 area in the Canadian Boreal forest. Multivariate kriging did not enhance daily rainfall predictions. SK, OK, and LNK produced statistically comparative results with OK being slightly better. MGC was the worst univariate estimator, mainly due to the high percentage of data spikes. Sequential Gaussian simulation (SGS) was then implemented to produce 100 equiprobable maps of rainfall. A multi-objective approach; that is based on overlaying the map of the kriging variance, the DEM, and land use/land cover maps in a GIS framework to identify the areas of commonly favourable features; was proposed to identify potential future sampling locations.

A disjunctive kriging program for assessing point-support conditional distributions

In geostatistical applications, the local distributions of the values of a regionalized attribute at unsampled locations can be assessed by nonlinear methods such as indicator or multigaussian kriging. Disjunctive kriging can also be applied in the framework of bivariate isofactorial models, for which there exists a complete family of functions (factors) with no spatial cross-correlations. This work focuses on the point-support models with polynomial factors and gives practical tips for the modeling of the univariate and bivariate distributions and for the implementation of disjunctive kriging, mainly in what refers to the convergence of the expansions into factors, the post-processing of the estimated statistics and the use of ordinary kriging. The tools and concepts are complemented by a set of computer programs and applied to two case studies. The first one consists of topsoil samples measuring the lead concentration at a smelter site in Dallas, Texas. A gamma isofactorial model is fitted to these data and disjunctive kriging is used to map the local probabilities that the actual concentrations exceed a toxic threshold and to divide the smelter site into a safe and a polluted area. The second case study concerns the infestation of field crops by a caterpillar. A negative binomial model is used to characterize the number of bored stalk internodes and to assess the risk that this number exceeds given values.

Multigaussian kriging for point-support estimation: incorporating constraints on the sum of the kriging weights

In the geostatistical analysis of regionalized data, the practitioner may not be interested in mapping the unsampled values of the variable that has been monitored, but in assessing the risk that these values exceed or fall short of a regulatory threshold. This kind of concern is part of the more general problem of estimating a transfer function of the variable under study. In this paper, we focus on the multigaussian model, for which the regionalized variable can be represented (up to a nonlinear transformation) by a Gaussian random field. Two cases are analyzed, depending on whether the mean of this Gaussian field is considered known or not, which lead to the simple and ordinary multigaussian kriging estimators respectively. Although both of these estimators are theoretically unbiased, the latter may be preferred to the former for practical applications since it is robust to a misspecification of the mean value over the domain of interest and also to local fluctuations around this mean value. An advantage of multigaussian kriging over other nonlinear geostatistical methods such as indicator and disjunctive kriging is that it makes use of the multivariate distribution of the available data and does not produce order relation violations. The use of expansions into Hermite polynomials provides three additional results: first, an expression of the multigaussian kriging estimators in terms of series that can be calculated without numerical integration; second, an expression of the associated estimation variances; third, the derivation of a disjunctive-type estimator that minimizes the variance of the error when the mean is unknown.

Ordinary multigaussian kriging for mapping conditional probabilities of soil properties

This paper addresses the problem of assessing the risk of deficiency or excess of a soil property at unsampled locations, and more generally of estimating a function of such a property given the information monitored at sampled sites. It focuses on a particular model that has been widely used in geostatistical applications: the multigaussian model, for which the available data can be transformed into a set of Gaussian values compatible with a multivariate Gaussian distribution. First, the conditional expectation estimator is reviewed and its main properties and limitations are pointed out; in particular, it relies on the mean value of the normal scores data since it uses a simple kriging of these data. Then we propose a generalization of this estimator, called “ordinary multigaussian kriging” and based on ordinary kriging instead of simple kriging. Such estimator is unbiased and robust to local variations of the mean value of the Gaussian field over the domain of interest. Unlike indicator and disjunctive kriging, it does not suffer from order-relation deviations and provides consistent estimations. An application to soil data is presented, which consists of pH measurements on a set of 165 soil samples. First, a test is proposed to check the suitability of the multigaussian distribution to the available data, accounting for the fact that the mean value is considered unknown. Then four geostatistical methods (conditional expectation, ordinary multigaussian, disjunctive and indicator kriging) are used to estimate the risk that the pH at unsampled locations is less than a critical threshold and to delineate areas where liming is needed. The case study shows that ordinary multigaussian kriging is close to the ideal conditional expectation estimator when the neighboring information is abundant, and departs from it only in under-sampled areas. In contrast, even within the sampled area, disjunctive and indicator kriging substantially differ from the conditional expectation; the discrepancy is greater in the case of indicator kriging and can be explained by the loss of information due to the binary coding of the pH data.

Comparative soil CO2 flux measurements and geostatistical estimation methods on Masaya volcano, Nicaragua

We present a comparative study of soil CO2 flux (\(F_{{\rm CO}_2 }\)) measured by five groups (Groups 1–5) at the IAVCEI-CCVG Eighth Workshop on Volcanic Gases on Masaya volcano, Nicaragua. Groups 1–5 measured \(F_{{\rm CO}_2 }\) using the accumulation chamber method at 5-m spacing within a 900 m2 grid during a morning (AM) period. These measurements were repeated by Groups 1–3 during an afternoon (PM) period. Measured \(F_{{\rm CO}_2 }\) ranged from 218 to 14,719 g m−2 day−1. The variability of the five measurements made at each grid point ranged from ±5 to 167%. However, the arithmetic means of fluxes measured over the entire grid and associated total CO2 emission rate estimates varied between groups by only ±22%. All three groups that made PM measurements reported an 8–19% increase in total emissions over the AM results. Based on a comparison of measurements made during AM and PM times, we argue that this change is due in large part to natural temporal variability of gas flow, rather than to measurement error. In order to estimate the mean and associated CO2 emission rate of one data set and to map the spatial \(F_{{\rm CO}_2 }\) distribution, we compared six geostatistical methods: arithmetic and minimum variance unbiased estimator means of uninterpolated data, and arithmetic means of data interpolated by the multiquadric radial basis function, ordinary kriging, multi-Gaussian kriging, and sequential Gaussian simulation methods. While the total CO2 emission rates estimated using the different techniques only varied by ±4.4%, the \(F_{{\rm CO}_2 }\) maps showed important differences. We suggest that the sequential Gaussian simulation method yields the most realistic representation of the spatial distribution of \(F_{{\rm CO}_2 }\), but a variety of geostatistical methods are appropriate to estimate the total CO2 emission rate from a study area, which is a primary goal in volcano monitoring research.

Simple and Ordinary Multigaussian Kriging for Estimating Recoverable Reserves

Multigaussian kriging is used in geostatistical applications to assess the recoverable reserves in ore deposits, or the probability for a contaminant to exceed a critical threshold. However, in general, the estimates have to be calculated by a numerical integration (Monte Carlo approach). In this paper, we propose analytical expressions to compute the multigaussian kriging estimator and its estimation variance, thanks to polynomial expansions. Three extensions are then considered, which are essential for mining and environmental applications: accounting for an unknown and locally varying mean (local stationarity), accounting for a block-support correction, and estimating spatial averages. All these extensions can be combined; they generalize several known techniques like ordinary lognormal kriging and uniform conditioning by a Gaussian value. An application of the concepts to a porphyry copper deposit shows that the proposed “ordinary multigaussian kriging” approach leads to more realistic estimates of the recoverable reserves than the conventional methods (disjunctive and simple multigaussian krigings), in particular in the nonmineralized undersampled areas.

Geostatistical modelling of uncertainty in soil science

This paper addresses the issue of modelling the uncertainty about the value of continuous soil attributes, at any particular unsampled location (local uncertainty) as well as jointly over several locations (multiple-point or spatial uncertainty). Two approaches are presented: kriging-based and simulation-based techniques that can be implemented within a parametric (e.g. multi-Gaussian) or non-parametric (indicator) frameworks. As expected in theory and illustrated by case studies, the two approaches yield similar models of local uncertainty, yet the simulation-based approach has several advantages over kriging: (1) it provides a model of spatial uncertainty, e.g. the probability that a given threshold is exceeded jointly at several locations can be readily computed, (2) conditional cumulative distribution function (ccdf) for supports larger than the measurement support (e.g. remediation units or flow simulator cells) can be numerically approximated by the cumulative distribution of block simulated values that are obtained by averaging values simulated within the block, and (3) the set of realizations allows one to study the propagation of uncertainty through global GIS operations or complex transfer functions, such as flow simulators that consider many locations simultaneously rather than one at a time. The other issue is the evaluation of the quality or “goodness” of uncertainty models. Two new criteria (exceedence probability plot and narrowness of probability intervals that include the true values) are presented to assess the accuracy and precision of local uncertainty models using cross-validation. According to the second criterion, multi-Gaussian kriging performs better than indicator kriging for the hydraulic conductivity (HC) data set. However, looking at the distribution of flow simulator responses, sequential indicator simulation (sis) yields better results than sequential Gaussian simulation (sGs) that does not allow for significant correlation of extreme values (destructuration effect).

Geostatistical Interpolation of Positively Skewed and Censored Data in a Dioxin-Contaminated Site

A correct delineation of hazardous areas in a contaminated site relies first on accurate predictions of pollutant concentrations, a task usually complicated by the presence of censored data (observations below the detection limit) and highly positively skewed histograms. This paper compares the prediction performances of four geostatistical algorithms (ordinary kriging, log-normal kriging, multi-Gaussian kriging, and indicator kriging) through the cross validation of a set of 600 dioxin concentrations. Despite its theoretical limitations, log-normal kriging consistently yields the best results (smallest prediction errors, least false positives, and lowest total costs). The cross validation has been repeated 100 times for a series of sampling intensities, which reduces the risk that these results simply reflect sampling fluctuations. Indicator kriging (IK), in the simplified implementation of median IK, produces good predictions except for a moderate bias caused by the underestimation of high dioxin concentrations. Ordinary kriging is the most affected by data sparsity, leading to a large proportion of remediation units wrongly declared contaminated when less than 100 observations were used. Last, decisions based on multi-Gaussian kriging estimates are the most costly and create a large proportion of false positives that cannot be reduced by the collection of additional samples.

The multigaussian approach and its applications to the estimation of local reserves

The performance of the geostatistical techniques of disjunctive kriging and multigaussian kriging for the estimation of local recovered reserves are compared using a simulated deposit.