Linear Coregionalization Research Articles

Co-simulation of hydrofacies and piezometric data in the West Thessaly basin, Greece: A geostatistical application using the GeoSim R package

In the present study, we co-simulate hydrofacies and piezometric data in order to construct geostatistical realizations of underground geology in an area of the West Thessaly basin. This basin is of great importance in terms of sustainable water management and environmental perspective in Greece. Through Plurigaussian modeling, the hydrofacies are first transformed into Gaussian Random Fields. Then, a Linear Coregionalization Model is established to account for the dependencies between hydrofacies and the Normal scores of piezometric data. The effect of co-simulation shows an improvement of the facies transition probabilities in comparison with those of Plurigaussian simulation. For the purpose of this study, we use the GeoSim package in R developed by our team for the implementation of Plurigaussian simulation and co-simulation.

Multivariate Modeling for Spatio-Temporal Radon Flux Predictions.

Nowadays, various fields in environmental sciences require the availability of appropriate techniques to exploit the information given by multivariate spatial or spatio-temporal observations. In particular, radon flux data which are of high interest to monitor greenhouse gas emissions and to assess human exposure to indoor radon are determined by the deposit of uranium and radio (precursor elements). Furthermore, they are also affected by various atmospheric variables, such as humidity, temperature, precipitation and evapotranspiration. To this aim, a significant role can be recognized to the tools of multivariate geostatistics which supports the modeling and prediction of variables under study. In this paper, the spatio-temporal distribution of radon flux densities over the Veneto Region (Italy) and its estimation at unsampled points in space and time are discussed. In particular, the spatio-temporal linear coregionalization model is identified on the basis of the joint diagonalization of the empirical covariance matrices evaluated at different spatio-temporal lags and is used to produce predicted radon flux maps for different months. Probability maps, that the radon flux density in the upcoming months is greater than three historical statistics, are then built. This might be of interest especially in summer months when the risk of radon exhalation is higher. Moreover, a comparison with respect to alternative models in the univariate and multivariate context is provided.

GeoSim: An R-package for plurigaussian simulation and Co-simulation between categorical and continuous variables

Plurigaussian simulation is widely used to model geological facies in geosciences and is predominantly applied in mineral deposits and petroleum reservoirs exploration. GeoSim package builds geostatistical models of categorical regionalized variables via conditional or unconditional Plurigaussian simulation and co-simulation. Co-simulation between Gaussian Random Fields representing the geological facies and other numerical variables accounting for auxiliary hydrological or geophysical quantities, is also available in this package with the definition of a linear coregionalization model. The algorithm is not restricted by the number of simulated facies and the number of truncated Gaussians, while parts of the code requiring heavy computations are compiled in C++ taking benefits of the integration between R and C++. In this work, we introduce the GeoSim package and demonstrate its capabilities. We present a 3D application focused on a lignite mine in Greece, where we investigate the Plurigaussian simulation and co-simulation of five geological facies (categorical variables) and the lower calorific value (continuous variable). The findings of our study highlight the significant benefits of Plurigaussian and co-simulation to capture the geological spatial heterogeneity.

Silver origins and spatial distribution in soils of southwestern France: A first assessment with cokriging

Ag is a rare, non-essential metal with many applications in different forms, including nanoparticles (Ag-NPs). Its release into the soil environment can represent a risk, due to the toxicity of silver. This study is based on the analysis of samples from the two sampling campaigns of the Soil Quality Measurement Network, RMQS1 and RMQS2, to establish the spatial distribution and origin of Ag in the soils of southwestern France (area of 90,293 km2). Given the progress of the second campaign (completion in 2027), we chose to do a cokriging to estimate the spatial distribution, with Pb, Cd or Cu as potential secondary variables. The results of the analyzes of the 35 samples of the RMQS2 indicate a range of Ag concentration in the soils going from 0.015 to 0.222 mg kg−1 and a median of 0.081 mg kg−1. Correlations were found between the concentrations of Ag Cd, Cu and Pb, and also between Ag concentrations and distance to Pb mines and deposits (r = 0.62). This suggests a common origin for Ag and Pb linked to historical exploitations of argentiferous galena. No link could be demonstrated between the concentration of Ag and the high density of anti-hail ground generators in the region. The analysis of the (auto-)variograms showed that Pb is the most relevant variable compared to Cd and Cu to achieve a cokriging. Thus, based on Ag concentrations from RMQS2 and Pb concentrations from RMQS1 (356 samples), a linear co-regionalization model was established and the co-kriging of Ag concentrations in soils calculated. Compared to ordinary kriging, cokriging improved the quality of the interpolation (decrease in the root mean square error).

Use of Geostatistics for Multi-Scale Spatial Modeling of Xylella fastidiosa subsp. pauca (Xfp) Infection with Unmanned Aerial Vehicle Image

In recent years, the use of Unmanned Aerial Vehicles (UAVs) has been spreading widely, as in plant pest control. The collection of huge amounts of spatial data raises various issues including that of scale. Data from UAVs generally explore multiple scales, so the problem arises in determining which one(s) may be relevant for a given application. The objective of this work was to investigate the potential of UAV images in the fight against the Xylella pest for olive trees. The data were a multiband UAV image collected on one date in an olive grove affected by Xylella. A multivariate geostatistics approach was applied, consisting firstly of estimating the linear coregionalization model to detect the scales from the data; and secondly, of using multiple factor kriging to extract the sets of scale-dependent regionalized factors. One factor was retained for each of the two selected scales. The short-range factor could be used in controlling the bacterium infection while the longer-range factor could be used in partitioning the field into three management zones. The work has shown the UAV data potential in Xylella control, but many problems still need to be solved for the automatic detection of infected plants in the early stages.

Response envelopes for linear coregionalization models

Dimension reduction provides a useful tool for statistical data analysis with high-dimensional data. In this paper, we develop a parsimonious multivariate spatial regression model with a non-separable covariance function. The efficacy of this new solution is illustrated through simulation studies and a real data analysis. We show that for cases where the marginal spatial correlations are different from each other, the proposed non-separable model provides better estimation and inference than the related separable model, and provides tighter inference than a non-separable spatial model without dimension reduction when there is immaterial variation in the data.

Computational advances for spatio-temporal multivariate environmental models

In multivariate Geostatistics, the linear coregionalization model (LCM) has been widely used over the last decades, in order to describe the spatial dependence which characterizes two or more variables of interest. However, in spatio-temporal multiple modeling, the identification of the main elements of a space–time linear coregionalization model (ST-LCM), as well as of the latent structures underlying the analyzed phenomenon, represents a tough task. In this paper, some computational advances which support the selection of an ST-LCM are described, gathering all the necessary steps which allow the analyst to easily and properly detect the basic space–time components for the phenomenon under study. The implemented algorithm is applied on space–time air quality data measured in Scotland in 2017.

Spatio-temporal modeling of an environmental trivariate vector combining air and soil measurements from Ireland

In environmental sciences, it is very common to observe spatio-temporal multiple data concerning several correlated variables which are measured in time over a monitored spatial domain. In multivariate Geostatistics, the evaluation of their behavior is often based on the knowledge of the spatio-temporal multivariate covariance structure. Since this last is often unknown it has to be estimated and modeled. In this paper, a spatio-temporal multivariate analysis of three relevant environmental indicators, which include 10-centimeter soil temperature, minimum and maximum air temperature, is proposed. This study is of particular interest for its reflection in ecology and the lack of information due to the presence of monitoring networks for soil and air variables characterized by different levels of spatial and temporal detail. A space–time linear coregionalization model (ST-LCM) with suitable models for the latent components of the variables under study is selected by using a simple procedure.

Choosing suitable linear coregionalization models for spatio-temporal data

In multivariate spatio-temporal Geostatistics, direct and cross-correlations among the variables of interest are measured by the matrix-valued covariance function. In this paper, a new and complete procedure useful for selecting an appropriate spatio-temporal linear coregionalization model (ST-LCM) with suitable models for the basic components is proposed. Thus, after detecting the spatio-temporal correlation of the latent components, through simultaneous diagonalization of the sample covariance matrices, some essential characteristics of each component are tested so that an aware choice for basic covariance models can be made. In the literature, some statistical tests to assess separability and symmetry of the covariance matrix, as well as the adequacy of the LCM were proposed; however in this paper further aspects on the basic components are investigated. All steps of the proposed procedure for analyzing and modeling the components of an ST-LCM are discussed in a case study where a very large spatio-temporal data set, concerning two environmental variables, is considered.

A fully non-stationary linear coregionalization model for multivariate random fields

This paper introduces an extension of the traditional stationary linear coregionalization model to handle the lack of stationarity. Under the proposed model, coregionalization matrices are spatially dependent, and basic univariate spatial dependence structures are non-stationary. A parameter estimation procedure of the proposed non-stationary linear coregionalization model is developed under the local stationarity framework. The proposed estimation procedure is based on the method of moments and involves a matrix-valued local stationary variogram kernel estimator, a weighted local least squares method in combination with a kernel smoothing technique. Local parameter estimates are knitted together for prediction and simulation purposes. The proposed non-stationary multivariate spatial modeling approach is illustrated using two real bivariate data examples. Prediction performance comparison is carried out with the classical stationary multivariate spatial modeling approach. According to several criteria, the prediction performance of the proposed non-stationary multivariate spatial modeling approach appears to be significantly better.

Generalized linear latent models for multivariate longitudinal measurements mixed with hidden Markov models

This article presents a generalized linear latent variable model for analyzing multivariate longitudinal data within the hidden Markov model framework. The relationships among multiple items are captured by several common latent factors. The linear coregionalization method is adopted to model the temporal processes of latent variables. The merit of this modeling strategy lies in the fact that the processes among latent variables are nonseparate and codependent from each other. To account for possible heterogeneity and interrelationship among the longitudinal data, a hidden Markov model is introduced to model the transition probabilities across different latent states over time. The Monte Carlo expectation conditional maximization (MCECM) algorithm is developed to estimate unknown parameters in the proposed model. The Wald- and score-type statistics are proposed to test the related dependence of processes. A simulation study is conducted to investigate the performance of the proposed methodology. An example from a longitudinal study of cocaine use is taken to illustrate the proposed methodology.

Characterizing Geothermal Surface Manifestation Based on Multivariate Geostatistics of Ground Measurements Data

Mt. Wayang Windu is one of geothermal field located in West Java, Indonesia. The characterization of steam spots at surface manifestation zones based on the soil physical measurements of the area is presented in this study. The multivariate geostatistical methods incorporating the soil physical parameter data were used to characterize the zonation of geothermal surface manifestations. The purpose of this study is to evaluate the performance of spatial estimation method of multivariate geostatistics using Ordinary Cokriging (COK) to characterize the physical properties of geothermal surface manifestations at Mt. Wayang Windu. The COK method was selected because this method is favorable when the secondary variables has more number than the primary variables. There are four soil physical parameters used as the basis of COK method, i.e. Electrical Conductivity, Susceptibility, pH, and Temperature. The parameters were measured directly at and around geothermal surface manifestations including hot springs, fumaroles, and craters. Each location of surface manifestations was measured about 30 points with 30 x 30 m grids. The measurement results were analyzed by descriptive statistics to identify at the nature of data. The correlation among variables was analyzed using linear regression. When the correlation coefficient among variables is higher, the estimation results is expected to have better Linear Coregionalization Model (LCM). LCM was used to analyze the spatial correlation of each variable based on their variogram and cross-variogram model. In oder to evaluate the performance of multivariate geostatistical using COK method, a Root Mean Square Error (RMSE) was performed. Estimation result using COK method is well applicable for characterizing the surface physics parameters of radar images data.

Using informative priors in facies inversion: The case of C-ISR method

Inverse problems involving the characterization of hydraulic properties of groundwater flow systems by conditioning on observations of the state variables are mathematically ill-posed because they have multiple solutions and are sensitive to small changes in the data. In the framework of McMC methods for nonlinear optimization and under an iterative spatial resampling transition kernel, we present an algorithm for narrowing the prior and thus producing improved proposal realizations. To achieve this goal, we cosimulate the facies distribution conditionally to facies observations and normal scores transformed hydrologic response measurements, assuming a linear coregionalization model. The approach works by creating an importance sampling effect that steers the process to selected areas of the prior. The effectiveness of our approach is demonstrated by an example application on a synthetic underdetermined inverse problem in aquifer characterization.

A new methodology to assess the maximum irrigation rates at catchment scale using geostatistics and GIS

Soil hydraulic parameters are important for irrigation scheduling. In the domain of “precision irrigation”, knowledge of the spatial distribution of these parameters is useful in determining the maximum irrigation rate for each field in a catchment. This study focuses on the development of a new methodology to assess the spatial distribution of the maximum irrigation rate depending on the available soil water holding capacity (ASWHC). This methodology combines geostatistical techniques with geographical information system (GIS) tools. A pilot zone of 12 400 ha in a Spanish Mediterranean area was selected to develop this methodology. The linear coregionalization model (LMCR), considering the percentage of sand, carbonates, and ASWHC at others soil depths as covariates, was the best option to model the ASWHC. Other required soil parameters were also spatially modeled. The percent of coarse fragments was modeled by regression kriging considering the soil map as an auxiliary variable. The bulk density was spatially modeled by LMCR, and extended to the rooting depth by linear regression. The spatial distributions modeled were implemented in a GIS with other spatial information layers of irrigation management parameters, such as the maximum allowable depletion of soil water content, the percent of wetted soil and the irrigation depth. The combination of these layers in the GIS was used to estimate the maximum irrigation rates for each field. A propagation error analysis was performed to know the uncertainties in the maximum irrigation rate estimation. Based on this information, the irrigation managers could optimize the irrigation rates for each field.

Geostatistical approach for identifying scale-specific correlations between soil thickness and topographic attributes

This paper investigates how the spatial correlations between topographic attributes and a soil thickness can be improved by focusing on the relationships between them at specific spatial scales. In addition, this paper examines the effects of the topographic attribute data sources that are used as explanatory variables for modeling the response variable, and considers the possibility of model extrapolation for mapping beyond the area where the model was established. Here, factorial kriging analysis (FKA) and partial least square regression (PLSR) analysis are used to separate nuggets and small- and large-scale structures in data including four topographic attributes and soil thickness (ST). These analyses were conducted at different scales to analyze the relationships between ST and the selected topographic attributes in the southwest region of the Parisian Basin. The structural correlation coefficients from the FKA show strong correlations between the variables. These correlations, which change as a function of spatial scale, are not revealed by the linear correlation coefficients. The Eigen vectors from the principal component analysis that was performed on the small-scale and large-scale structures of the linear co-regionalization model are used to obtain ST and the topographic attributes at both spatial scales over the study area. The ST models are built as a function of topographic attributes using PLSR. Results have shown that the models built using variables that were assessed at a specific scale are better at predicting the target variable than models that were built using raw data. Regarding the models that were built using raw data, the structural correlations that occur at different spatial scales are merged together and the variance–covariance matrix of the nugget that represents data noise is not filtered out. Measures of model performance that are based on a validation data set have shown that the model based on small-scale structure (Model-S) is better for predicting soil thickness than the model based on large-scale structure (Model-L). The effects of topographic attribute data sources as explanatory variables for modeling ST are less significant than the effects of the two models for mapping. Moreover, extrapolation of the model-S beyond the area where it was generated is appropriate. The decomposition process is associated with a modeling approach, such as the PLSR, which accounts for the collinearity between predictor variables and leads to an efficient prediction model. These results are important for modeling soil properties based on topographic attributes and for spatially generalizing models that have been established over small to large areas. Thus, in the presence of nested variogram models, the correlations between variables of interest and auxiliary information should be improved by filtering out some of the spatial structures by factorial kriging. The information filtered is associated with an appropriate approach for modeling when collinearity occurs between the predictor variables and provides a suitable model for predicting and spatially generalizing a locally established model.

Prediction of particle pollution through spatio-temporal multivariate geostatistical analysis: spatial special issue

Vehicular traffic, industrial activity and street dust are important sources of atmospheric particles, which cause pollution and serious health problems, including respiratory illness. Hence, techniques for analyzing and modeling the spatio-temporal behavior of particulate matter (PM), in the recent statistical literature, represent an essential support for environmental and human health protection. In this paper, air pollution from particles with diameters smaller than 10 \({\rm \mu}\)m and related meteorological variables, such as temperature and wind speed, measured during November 2009 in the south of Apulian region (Lecce, Brindisi, and Taranto districts) are studied. A thorough multivariate geostatistical analysis is proposed, where different tools for testing the symmetry assumption of the spatio-temporal linear coregionalization model (ST-LCM) are considered, as well as a recent fitting procedure of the ST-LCM, based on the simultaneous diagonalization of symmetric real-valued matrix variograms, is adopted and two non-separable classes of variogram models, the product–sum and Gneiting classes, are fitted to the basic components. The most significant aspects of this study are (a) the quantitative assessment of the assumption of symmetry of the ST-LCM, (b) the use of different non-separable spatio-temporal models for fitting the basic components of a ST-LCM and, more importantly, (c) the application of the spatio-temporal multivariate geostatistical analysis to predict particle pollution in one of the most polluted geographical area. Prediction maps for particle pollution levels with the corresponding validation results are given.

Using Simultaneous Diagonalization to Identify a Space–Time Linear Coregionalization Model

Although there are multiple methods for modeling matrix covariance functions and matrix variograms in the geostatistical literature, the linear coregionalization model is still widely used. In particular it is easy to check to ensure whether the matrix covariance function is positive definite or that the matrix variogram is conditionally negative definite. One of the difficulties in using a linear coregionalization model is in determining the number of basic structures and the corresponding covariance functions or variograms. In this paper, a new procedure is given for identifying the basic structures of the space–time linear coregionalization model and modeling the matrix variogram. This procedure is based on the near simultaneous diagonalization of the sample matrix variograms computed for a set of spatiotemporal lags. A case study using a multivariate spatiotemporal data set provided by the Environmental Protection Agency of Lombardy, Italy, illustrates how nearly simultaneous diagonalization of the empirical matrix variograms simplifies modeling of the matrix variograms. The new methodology is compared with a previous one by analyzing various indices and statistics.

Reducing the number of orthogonal factors in linear coregionalization modeling

The simulation of vector random fields whose spatial correlation structure is represented by a linear coregionalization model can be performed by decomposing the vector components into spatially orthogonal factors and by simulating each factor separately. However, when the number of basic nested structures is large, so is the number of factors, making simulation computationally demanding.This paper proposes a methodology to construct linear coregionalization models with as many nested structures as desired, together with as few orthogonal factors as possible. The construction rests on the decomposition of the model coregionalization matrices into pairwise commuting matrices, followed by a factorization by principal component analysis. The proposed approach is illustrated through a case study in mineral resources evaluation and compared to the traditional fitting procedure, obtaining an equally good fit of the direct and cross variograms but with significantly less factors.

Towards an automatic procedure for modeling multivariate space–time data

In many environmental sciences, several correlated variables are observed at some locations of the domain of interest and over a certain period of time. In this context, appropriate modeling and prediction techniques for multivariate space–time data as well as interactive software packages are necessary. In this paper, a new automatic procedure for fitting the space–time linear coregionalization model (ST-LCM) using the product–sum variogram model is discussed. This procedure, based on the simultaneous diagonalization of the sample matrix variograms, allows the identification of the ST-LCM parameters in a very flexible way. The fitting process is analytically described by a main flow chart and all steps are specified by four subprocedures. An application of this procedure is illustrated through a case study concerning the daily concentrations of three air pollutants measured in an urban area. Then the fitted space–time coregionalization model is applied to predict the variable of interest using a recent GSLib routine, named “COK2ST.”

ANÁLISE ESPACIAL MULTIVARIADA APLICADA NA AVALIAÇÃO DA FERTILIDADE DO SOLO

A variabilidade espacial de atributos do solo pode sofrer influências de diversos fatores, relativos à formação do solo: (material de origem, topografia, vegetação, clima e tempo), práticas de manejo (tipo de preparo do solo, rotação de culturas e adubação) e erosão. Nesse sentido, são necessárias técnicas apropriadas para acessar informações relativas a quais variáveis exercem maior influência sobre determinados aspectos do solo, a fim de aperfeiçoar o manejo e as práticas de correção de acidez e adubação, em propriedades agrícolas. O objetivo deste trabalho foi demonstrar a aplicação de técnicas geoestatísticas multivariadas na investigação do comportamento de um conjunto de variáveis químicas do solo. O trabalho foi realizado em uma área experimental de 70 hectares, no município de em Araguari-MG, onde foram analisadas as variáveis Fósforo (P), Potássio (K) e Matéria Orgânica (MO), sob o escopo do modelo linear de corregionalização. Nesse método, são ajustados p(p+1) variogramas diretos e cruzados, para as variáveis, posteriormente decompostas em componentes principais. Os resultados demonstraram uma maior influência espacial da variável P, em micro e média escala, e uma influência dividida entre as varáveis P e K, em longa escala. Devido ao manejo a que vem sendo submetida, a área apresenta uma fraca influência da MO, no comportamento conjunto dessas três variáveis do solo, e uma maior dependência espacial ligada às variáveis P e K, durante as adubações para plantio de soja e milho. Os resultados indicam ainda um limite mínimo de 320,18 metros para criação de zonas de manejo desses elementos, em sistemas de adubação à taxa variável.