Ordinary Kriging Research Articles

Accurate prediction of spatial distribution of soil heavy metal in complex mining terrain using an improved machine learning method.

Accurate prediction of spatial distribution of soil heavy metal in complex mining terrain using an improved machine learning method.

Spatial analysis and geostatistical characterization of nitrate pollution in Mahdia's shallow aquifers.

Industrial wastewater discharge, intensive livestock farming, and irrigation practices may be the primary groundwater pollution sources of nitrate in arid countries like Tunisia. Currently, this contamination is a real concern for the sustainable use of groundwater. In this regard, the primary difficulty facing Tunisia's water resource management is assessing the quality of the groundwater. To determine the spatial distribution of nitrate pollution in the groundwater system of the Mahdia region in Tunisia, 35 groundwater samples collected from shallow groundwater wells across the years 2006 and 2016 and their nitrate concentrations were determined by the Regional Commissariat for Agricultural Development of Mahdia (CRDA). The spatial distribution behavior of nitrate concentrations in groundwater was assessed by using GIS and ordinary kriging techniques. It was determined that exponential semi-variogram models were the most suitable to delineate the spatial dependence structure of nitrate concentration over the study area. Nitrate contour maps showed the high concentration of nitrates and the expansion of polluted groundwater areas in the last decade, especially in zones close to irrigated perimeters, textile industries, livestock farming, olive mills, and rural agglomerations without sanitation networks. The findings emphasize the critical necessity for formulating targeted groundwater management frameworks that prioritize the implementation of comprehensive monitoring and regulatory measures in high-risk zones. It is essential to deploy advanced nitrate removal technologies within industrial wastewater treatment systems and to expand rural sanitation infrastructure to mitigate further nitrate contamination from domestic sources. These results furnish essential data that can guide policymakers in devising precise, evidence-based interventions aimed at ensuring the long-term sustainability of groundwater resources.

Spatial epidemiology of lumpy skin disease: unraveling patterns in dairy farm clusters with short interfarm proximity

Lumpy skin disease (LSD) has caused economic losses in cattle, and Thailand experienced a nationwide outbreak in 2021. Spatial epidemiology plays a crucial role in identifying transmission patterns and high-risk areas for targeted disease control. This study examines the spatial epidemiology of LSD by analyzing clustering patterns, disease hotspots, and the directional spread of outbreaks in dairy farm networks with short interfarm proximities. LSD outbreak data from a large dairy farming area in northern Thailand were analyzed via multiple spatial analytical techniques. The standard deviation ellipse (SDE) approach, implemented with the Yuill and CrimeStat methods, was employed to determine the spatial-directional spread of outbreaks. Global and local Moran’s I statistics were used to assess spatial autocorrelation, whereas kernel density estimation (KDE) was used to identify the density areas of the LSD outbreaks. Ordinary kriging was applied to interpolate high-intensity surfaces. The results from the SDE indicate that the LSD outbreaks predominantly followed a northeast-to-southwest trend. Global Moran’s I revealed no statistical significance, whereas local Moran’s I indicated significant local spatial autocorrelation. KDE revealed a high density of outbreaks in the upper northern part of the farming region. Additionally, ordinary kriging was used to quantify the likelihood of outbreaks across different areas, highlighting potential high-intensity surfaces. These results enhance the understanding of LSD spatial epidemiology, providing valuable insights into disease dynamics and transmission. Additionally, these findings support policymakers in making informed decisions on targeted prevention, control strategies, and resource allocation at the local and regional levels.

Phosphorus Requirement Mapping for Bread Wheat at Wachale District, North Shewa Zone, Oromia

Determination of soil phosphorus level is an important factor affecting crop production and P-use efficiency in the study area. Fertilizer requirement mapping is the way of determining fertilizer demanded by specific crop production type on the basis of soil sample testing results. In order to make efficient fertilizer application for wheat at Wachale district, the study was conducted to map phosphorus requirement for wheat crop. Starting from soil sample collection, laboratory analysis and interpretation and mapping standard methods and tools were used. The total soil samples collected for the wheat potential kebeles were 167 and phosphorus requirement mapping was done for analyzed soil samples. Based on the laboratory result, phosphorus requirement was calculated by subtracting initial soil phosphorus from phosphorus critical for wheat at the study area. Considering the minimum and maximum level of phosphorus requirement, the output of the result ranges from 18.04 to 45.98ppm for the areas under consideration with an average of 31.43ppm. Large areas (84393.29ha) of the district require 20 to 35ppm and the smallest area (348.64ha) require 15 to 20ppm of phosphorus fertilizer for wheat crop production. Consequently, phosphorus fertilizer application rate for wheat crop might be very efficient by using the map, otherwise using the mean rate (31.43ppm) appropriate than blanket application rate. The map of phosphorus requirement in ppm was predicted for unknown locations by using ordinary Kriging interpolation of ArcGIS10.3 software.

Spatial Distribution and Management of Trace Elements in Arid Agricultural Systems: A Geostatistical Assessment of the Jordan Valley

Sustainable land management in arid regions such as the Jordan Valley (JV) is essential as climate pressures and water shortages intensify. The extended use of treated wastewater (TWW) for irrigation, while necessary, brings potential risks related to the accumulation of trace elements and their impact on soil health and food safety. This study examined the spatial distribution, variability, and potential sources of five trace elements (Co, Hg, Mo, Mn, and Ni) in agricultural soils across a 305 km2 area. A total of 127 surface soil samples were collected from fields irrigated with either TWW or freshwater (FW). Trace element concentrations were consistently higher in TWW-irrigated soils, although all values remained below WHO/FAO recommended thresholds for agricultural use. Spatial modeling was conducted using both ordinary kriging (OK) and empirical Bayesian kriging (EBK), with EBK showing greater prediction accuracy based on cross-validation statistics. To explore potential sources, semivariogram modeling, principal component analysis (PCA), and hierarchical clustering were employed. PCA, spatial distribution patterns, correlation analysis, and comparisons between TWW and FW sources suggest that Co, Mn, Mo, and Ni are primarily influenced by anthropogenic inputs, including TWW irrigation, chemical fertilizers, and organic amendments. Co exhibited a stronger association with TWW, whereas Mn, Mo, and Ni were more closely linked to fertilizer application. In contrast, Hg appears to originate predominantly from geogenic sources. These findings provide a foundation for improved irrigation management and fertilizer application strategies, contributing to long-term soil sustainability in water-limited environments like the JV.

Comparison of Ordinary Kriging and Cokriging for Spatial Estimation Based on Simulated Data

This study compares the performance of Ordinary Kriging (OK) and Cokriging (CK) methods in spatial estimation based on simulated data. Twelve scenarios are arranged based on a combination of sample size (50, 250, 500) and correlation levels between variables (ρ=0.1, 0.6, 0.9), with each scenario repeated 30 times. Spatial data are generated randomly within the geographical boundaries of Indonesia, variables are generated based on spherical variograms with nugget or sill or dan range or ,, and model evaluation is carried out using Leave-One-Out Cross Validation (LOOCV) with RMSE and metrics. The results show that Cokriging consistently produces more accurate estimates than Ordinary Kriging in all scenarios. In the best configuration (CK, n=500), RMSE = 1.04 and = 0.945 were obtained, while the best performance of OK only reached RMSE = 1.06 and = 0.873. All levels of correlation in Cokriging showed good performance, especially when the amount of data is sufficient. Therefore, Cokriging is recommended as a superior spatial interpolation method in the context of multivariate and spatial data, especially when relevant secondary information is available.

Canada’s Fourth Generation of Homogenized Surface Air Temperature and its Trends for 1948–2023

abstract This study presents a newly developed homogenized temperature dataset aimed at enhancing the reliability of the observed temperature records for computing long-term trends. The dataset includes two main improvements over its previous versions: an adjustment applied to correct a cold bias affecting daily minimum temperatures recorded after 1 July 1961, at all stations with daily and hourly observations, and a missing values and gaps filling procedure based on ANUSPLIN surface temperature estimates. The procedures can be summarized as follows. Daily minimum temperatures were first adjusted for the cold bias arising from changes in the timing of the observation window. Data from closely located stations were joined into single records to ensure observations for long periods. Comprehensive data quality assurance procedures were then applied to make sure that the data was of good quality. Daily and monthly ANUSPLIN surface estimates were used to fill missing values and gaps in both daily and monthly series. Finally, changepoints due to non-climatic changes were identified in monthly time series using a semi-automatic data homogenization procedure, and a quantile-matching procedure was applied to adjust the daily and monthly data for the changepoints identified by the procedure. The dataset contains homogenized daily maximum and minimum temperatures, as well as their monthly means, for 651 sites across the country, with all sites active and updated to 2023. Gap fillings start from 1948 for ensuring complete temporal coverage at each site and consistency when evaluating trends at regional and national levels. An ordinary kriging method was used to interpolate the daily and monthly temperature anomalies onto 10 km grids. This new homogenized temperature dataset indicates that warming continues across Canada, with an increase of 2.08°C in the annual mean of the daily maximum temperature and of 2.46 °C in the annual mean of the daily minimum temperature over the past 76 years (1948–2023); the findings also show that the warming continues to be more pronounced during the wintertime.

KMeans-Riemannian model for classification mineral resources in a copper deposit in Peru

ABSTRACT This study applies the KMeans clustering model with Riemannian geometric distance to classify mineral resources in a copper deposit in Peru. Covariance matrices of Ordinary Kriging estimates, kriging variance, and average sample distances are used to represent multivariate spatial structures for classification based on intrinsic geometry. The new automated method obtains similar results to the Qualified Person (QP), offering a reproducible and consistent framework aligned with geological variability and expert interpretation. The Riemannian approach improves spatial coherence and segmentation, making it suitable for deposits with complex geometries. This methodology supports objective, automated resource classification while preserving geological integrity.

Evaluating the Uncertainty and Predictive Performance of Probabilistic Models Devised for Grade Estimation in a Porphyry Copper Deposit

Probabilistic models are used to describe random processes and quantify prediction uncertainties in a principled way. Examples include geotechnical and geological investigations that seek to model subsurface hydrostratigraphic properties or mineral deposits. In mining geology, model validation efforts have generally lagged behind the development and deployment of computational models. One problem is the lack of industry guidelines for evaluating the uncertainty and predictive performance of probabilistic ore grade models. This paper aims to bridge this gap by developing a holistic approach that is autonomous, scalable and transferable across domains. The proposed model assessment targets three objectives. First, we aim to ensure that the predictions are reasonably calibrated with probabilities. Second, statistics are viewed as images to help facilitate large-scale simultaneous comparisons for multiple models across space and time, spanning multiple regions and inference periods. Third, variogram ratios are used to objectively measure the spatial fidelity of models. In this study, we examine models created by ordinary kriging and the Gaussian process in conjunction with sequential or random field simulations. The assessments are underpinned by statistics that evaluate the model’s predictive distributions relative to the ground truth. These statistics are standardised, interpretable and amenable to significance testing. The proposed methods are demonstrated using extensive data from a real copper mine in a grade estimation task and are accompanied by an open-source implementation. The experiments are designed to emphasise data diversity and convey insights, such as the increased difficulty of future-bench prediction (extrapolation) relative to in situ regression (interpolation). This work enables competing models to be evaluated consistently and the robustness and validity of probabilistic predictions to be tested, and it makes cross-study comparison possible irrespective of site conditions.

Land Price Distribution Prediction in Jakarta Using Support Vector Machine with Feature Expansion and Kriging Interpolation

Fluctuations in land prices over time are very significant, especially in big cities, one of which is Jakarta. The increase in land prices is influenced by high demand, location-related needs, ease of access to various public facilities and crowds. Uncontrolled prices and lack of information about the distribution of land prices cause buyers to get land that is not in accordance with their needs. This study develops a land price distribution prediction system for Jakarta for 2025-2026 using Support Vector Machine (SVM) with time-based feature expansion and spatial interpolation. The SVM model with RBF kernel demonstrated superior performance, achieving 93.14% accuracy for 2025 predictions using the t-1 model. For 2026 predictions, the t-2 model achieved 83.33% accuracy. This approach involves utilizing one to two years of historical data and systematically selected features, ensuring more accurate and relevant predictions. Ordinary kriging interpolation visualizations revealed a significant shift in land price distribution patterns, indicating a decline in affordable land availability and an increase in high-value properties across Jakarta. The integration of SVM and kriging interpolation, coupled with comprehensive evaluation metrics, provides a robust methodological framework for predicting urban land price distributions. This system offers practical implications for informed decision-making in Jakarta's dynamic land market, enabling stakeholders to make efficient, budget-based property decisions. The research contributes significantly to urban planning by providing a comprehensive tool for understanding and predicting land price trends, which can assist various stakeholders in making informed property investment decisions.

Spatial Interpolation in Applied Insect Ecology: A Review, Including Guidelines and Datasets for Practical Use

ABSTRACTSpatial interpolation represents a fundamental approach in applied insect ecology, offering insight into species distributions and supporting biodiversity analysis, pest management and disease vector mapping. Insects—including important pollinators—face escalating threats due to habitat loss, climate change and anthropogenic pressures. As data‐driven decisions become more critical in addressing these ecological challenges, spatial interpolation techniques such as kriging and regression‐based models have become essential for estimating insect abundance in unsampled areas. This paper offers an in‐depth review of both geostatistical and non‐geostatistical methods employed in insect ecology, including ordinary kriging, universal kriging and machine learning‐based methods such as random forests and maximum entropy. We present a structured overview of their applications in pest management, disease vector mapping and biodiversity monitoring, and we provide practical guidelines for selecting appropriate spatial interpolation methods. In addition, we present several datasets that can support case studies in spatial modelling for insect ecology. Our findings underscore the advantages of integrating geostatistical approaches with environmental variables to enhance the accuracy of species distribution models. This review serves as a resource for entomologists and researchers seeking to advance ecological monitoring and management through spatial interpolation techniques.

Spatiotemporal Analysis of Available Freshwater Resources in Watersheds Across Northern New Jersey

Groundwater is a critical freshwater resource, yet its quality is increasingly threatened by anthropogenic activities, particularly in urbanized regions. This study employs geospatial analysis to evaluate the spatiotemporal variability of groundwater quality across 11 Watershed Management Areas (WMAs) in northern New Jersey, from 1999 to 2016. Using specific conductance (SC) as a proxy for salinity, we applied Ordinary Kriging interpolation to estimate SC values in unmonitored locations, leveraging data from 295 shallow wells within the New Jersey Ambient Groundwater Quality Monitoring Network. The results reveal significant spatial heterogeneity in groundwater quality, strongly associated with land use and road density. The Northeast water region, characterized by high urbanization and extensive road networks, exhibited the poorest water quality, with salinity levels exceeding the 750 μS/cm threshold for freshwater in WMAs such as Lower Passaic (WMA-4) and Hackensack (WMA-5). In contrast, the Northwest region, dominated by agricultural and undeveloped land, maintained better water quality. Temporal analysis showed a worrying decline in freshwater coverage, from 80% in 1999–2004 to 74% in 2014–2016, with deicing salts and aging sewer infrastructure identified as major contamination sources. The study highlights the efficacy of Kriging and GIS tools in mapping groundwater quality trends and highlights the urgent need for targeted water management strategies in vulnerable regions. These findings provide policymakers and stakeholders with actionable insights to mitigate groundwater degradation and ensure long-term freshwater sustainability in northern New Jersey.

Comparison of interpolation methods in the production of rainfall-induced soil erosion maps in the urban area of Torres Novas, Portugal

Abstract. The evaluation of rainfall-induced soil erosion risk is fundamental for territorial planning and takes into account parameters such as rainfall erosivity, soil erodibility and the topographic factor. The Triangular Irregular Network (TIN) is the most frequently used interpolator in the production of digital elevation models (DEM) but is considered unsuitable by several authors for the calculation of soil erosion. Therefore, the DEM created for the city of Torres Novas, Portugal, using interpolation methods such as Inverse Distance Weighting, Ordinary Kriging, and Empirical Bayesian Kriging (EBK) were evaluated to determine which one was the most accurate. The best interpolator was EBK, from which a rainfall-induced soil erosion map was created. A map was also produced from TIN and both were compared with historical cartography. The EBK method was found to be the most effective interpolator for rainfall-induced soil erosion as well. Therefore, the authors recommend its use in future studies in the municipality of Torres Novas.

Urbanization, socioeconomic status, and exposure to PM2.5, associated with township-based cerebrovascular disease (CBD) mortality.

Previous research has shown an association between socioeconomic status (SES) and mortality, particularly in chronic diseases. However, limited studies simultaneously examined the relationship between urbanization, SES, exposure to PM2.5, and cerebrovascular disease (CBD) mortality at a township level from 2011 to 2020 in Taiwan. Township-level SES data (percentages of low-income and education with college and above) and seven levels of urbanization from 2011 to 2020 were obtained from data sources in Taiwan's central government. Age-standardized CBD mortality rates in 358 townships were calculated using the Geographic Information System (GIS) provided by the Research Center for the Humanities and Social Sciences (RCHSS) at Academia Sinica. Exposure to PM2.5 concentration was estimated using a combination of land-use regression and Ordinary Kriging to enhance the robustness of PM2.5 concentration estimates at the township level. Panel regression and structural equation modeling (SEM) was employed to analyze the association between urbanization, SES, exposure to PM2.5, and township-based CBD mortality rates. There are significant differences in SES variables and exposure to PM2.5 among townships with seven levels of urbanization (P < 0.001). Even after controlling for other covariates (SES and PM2.5 concentration) through multivariate analysis, the associations between CBD mortality rates and urbanization areas persisted. SEM analysis revealed a negative correlation between age-standardized CBD mortality rate and education levels (β = -0.22), but a positive correlation with the proportion of low-income individuals (β = 0.41). There was no significant association between exposure to PM2.5 and CBD mortality. The panel regression analysis revealed that socioeconomic variables had different effects on CBD mortality rates across the three models (pooled ordinary least squares, fixed-effects, and random-effects) in both urban and rural areas. Notably, the level of urbanization was observed to modify the relationship between socioeconomic variables and CBD mortality rates. Our findings suggest that township-based CBD mortality is significantly associated with SES variables and levels of urbanization, despite a reduction in CBD mortality from 2011 to 2020. Therefore, targeted intervention programs should be implemented to reduce CBD mortality in different levels of urbanization, particularly in remote townships. It is necessary to assess the disparities in socioeconomic status to achieve a fair allocation of resources at the township level.

Impacts of area closure on selected soil physicochemical properties at Meldam watershed, Sekota area, North Western Ethiopia

Area closure, by excluding human and animal interference, has proven to be an effective conservation strategy that facilitates natural regeneration, improves soil health, and enhances ecosystem resilience in degraded landscapes. This study evaluated how area closure influences key soil physicochemical properties critical for restoring degraded lands in the Meldam sub-watershed, Northwestern Ethiopia. Eighteen plots (9 from grazing land and 9 from area closures), each 20 × 20 m, were established across three slope positions with three replications along six transects. From each plot, one composite soil sample (for TN, avP, CEC, EC, exchangeable bases, pH, and SOM) and one undisturbed sample (for BD) were collected from a 0–20 cm depth. In total, 18 soil samples (9 from open grazing and 9 from area closures) were analyzed. Paired t-tests (at 1% significance) using SAS 9.3 assessed differences across land use and slope positions, while Pearson correlation evaluated relationships among soil properties. Ordinary kriging was used to map the spatial distribution of soil physical and chemical properties across different land uses. The results indicated that most soil properties improved following the conversion of open grazing land to area closure. Compared to grazing lands, bulk density and sand content decreased in area closures and lower slope positions, while clay content and soil moisture increased. Soil organic carbon, total nitrogen, available phosphorus, and cation exchange capacity rose by 73.15%, 34.78%, 64.13%, and 63.32%, respectively, in area closure and lower watershed areas. Base cations also showed similar improvements under closed land conditions. Soil fertility restoration was more pronounced on lower slope positions than upper ones. These findings highlight area closure as an effective approach for rehabilitating degraded soils, supporting its expansion within the study area and in similarly affected regions.

Evaluating Spatio-Temporal Kriging with Machine Learning Considering the Sources of Spatio-Temporal Variation

Integrating spatio-temporal kriging with machine learning improves estimation accuracy by addressing complex spatial and temporal variations in spatio-temporal phenomena. The improvement can be attributed to the enhanced flexibility of machine learning in capturing non-linear global trends, which traditional methods struggle to model, while kriging remains effective in representing spatio-temporal interactions. However, differences in the estimated global trends and spatio-temporal interactions resulting from applying machine learning may influence the spatio-temporal variation patterns of the kriging results. Therefore, this study evaluates the effectiveness of machine learning in spatio-temporal kriging using NO2 concentrations in Seoul, focusing on its impact on overall accuracy and the contributions to global trends and spatio-temporal interactions. The results show that integrating machine learning enhances overall accuracy relative to ordinary spatio-temporal kriging. Global trend estimates differ by the models, with polynomial regression producing smoother patterns but larger errors, while random forest and boosting yield more abrupt patterns with smaller errors. These differences lead to smoother kriging outcomes in the polynomial model and more discrete patterns in the ensemble-based models. This study highlights the importance of considering both overall estimation accuracy and spatio-temporal patterns when selecting kriging methods.

Comparison of Neural Network, Ordinary Kriging, and Inverse Distance Weighting Algorithms for Seismic and Well-Derived Depth Data: A Case Study in the Bjelovar Subdepression, Croatia

In subsurface geological mapping, it is more than advisable to compare different solutions obtained with neural and other algorithms. Here, for such comparison, we used the previously published and well-prepared dataset of subsurface data collected from the Bjelovar Subdepression, a 2900 km2 large regional macrounit in the Croatian part of the Pannonian Basin System. Data on depth were obtained for the youngest (the shallowest) Lonja Formation (Pliocene, Quaternary) and mapped using neural network (NN), inverse distance weighting (IDW), and ordinary kriging (OK) algorithms. The obtained maps were compared based on square error (using k-fold cross-validation) and the visual interpretation of isopaches. Two other algorithms were also tested, namely, random forest (RF) and extreme gradient boosting (XGB) algorithms, but they were rejected as inappropriate for this purpose solely based on the visuals of the obtained maps, which did not follow any interpretable geological structures. The results showed that NN is a highly adjustable method for interpolation, with adjustment for numerous hyperparameters. IDW showed its strength as one of the classical interpolators, and its results are always located close to the top if several methods are compared. OK is the relative winner, showing the flexibility of variogram analysis regarding the number of data points and possible clustering. The presented variogram model, even with a relatively high sill and occasional nugget effect, can be well fitted into OK, giving better results than other methods when applied to the presented area and datasets. This was not surprising because kriging is a well-established method used exclusively for interpolation. In contrast, NN and machine learning algorithms are used in many fields, and these algorithms, particularly the fitting of hyperparameters in NN, simply cannot be the best solution for all.

Model description of combined numerical and stochastic groundwater flow in Bandung-Soreang Groundwater Basin, West Java, Indonesia.

Model description of combined numerical and stochastic groundwater flow in Bandung-Soreang Groundwater Basin, West Java, Indonesia.

Low Prediction Error Model-Free Predictive Control on PMSM Drives With Ordinary Kriging Time-Shift

Low Prediction Error Model-Free Predictive Control on PMSM Drives With Ordinary Kriging Time-Shift

Improving the estimation precision of the mapping of groundwater salinity by employing the Indicator Kriging Technique

In tropical and semiarid areas, saline sub-surface water irrigation presents difficulties, resulting in salinized soil and reduced agricultural yields. In the pre-monsoon and post-monsoon seasons of 2022, this study evaluated electrical conductivity (EC) in 30 wells in Yamunanagar and Ambala, Haryana. The results showed that the danger of sub-surface water salinity, which could cause soil salinization, ranged from moderate (Group C2) to high (Group C3). Additionally, the data showed typical fluctuations. In the absence of field data, the Ordinary Kriging Technique (OKT) and Indicator Kriging Technique (IKT) were used to estimate salinity using the semivariogram approach to groundwater salinity levels. OKT tends to underestimate high salt levels and overestimate low salinity. By using nonlinear and nonparametric techniques with electrical conductivity thresholds, the IKT improved the salinity estimates’ spatial distribution accuracy. IKT was useful for public health, water management, and agricultural planning since it offered a more accurate and probabilistic prediction of high salinity levels. The shortcomings of conventional kriging are addressed by this development in geostatistical modeling, which also aids in environmental management, especially in areas vulnerable to contamination and seawater intrusion.