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AbstractThe severity of multivariate extreme events is driven by the dependence between the largest marginal observations. The Hüsler–Reiss distribution is a versatile model for this extremal dependence, and it is usually parameterized by a variogram matrix. In order to represent conditional independence relations and obtain sparse parameterizations, we introduce the novel Hüsler–Reiss precision matrix. Similarly to the Gaussian case, this matrix appears naturally in density representations of the Hüsler–Reiss Pareto distribution and encodes the extremal graphical structure through its zero pattern. For a given, arbitrary graph we prove the existence and uniqueness of the completion of a partially specified Hüsler–Reiss variogram matrix so that its precision matrix has zeros on non-edges in the graph. Using suitable estimators for the parameters on the edges, our theory provides the first consistent estimator of graph structured Hüsler–Reiss distributions. If the graph is unknown, our method can be combined with recent structure learning algorithms to jointly infer the graph and the corresponding parameter matrix. Based on our methodology, we propose new tools for statistical inference of sparse Hüsler–Reiss models and illustrate them on large flight delay data in the United States, as well as Danube river flow data. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work. [ABSTRACT FROM AUTHOR]

Simulation of random fields is widely used in Earth sciences for modeling and uncertainty quantification. The spatial features of these fields may have a strong impact on the forecasts made using these fields. For instance, in flow and transport problems the connectivity of the permeability fields is a crucial aspect. Multi-Gaussian random fields are the most common tools to analyze and model continuous fields. Their spatial correlation structure is described by a covariance or variogram model. However, these types of spatial models are unable to represent highly or poorly connected structures even if a broad range of covariance models can be employed. With this type of model, the regions with values close to the mean are always well connected whereas the regions of low or high values are isolated. Substitution random functions (SRFs) belong to another broad class of random functions that are more flexible. SRFs are constructed by composing ( Z = Y ∘ T ) two stochastic processes: the directing function T (latent field) and the coding process Y (modifying the latent field in a stochastic manner). In this paper, we study the properties of SRFs obtained by combining stationary multi-Gaussian random fields for both T and Y with bounded variograms. The resulting SRFs Z are stationary, but as T has a finite variance Z is not ergodic for the mean and the covariance. This means that single realizations behave differently from each other. We propose a simple technique to control which values (low, intermediate, or high) are connected. It consists of adding a control point on the process Y to guide every single realization. The conditioning to local values is obtained using a Gibbs sampler. [ABSTRACT FROM AUTHOR]

The purpose of this work is to analyze and validate the gravimetric information in the study area in order to highlight the valid data for mapping the gravity anomaly from a grid generated by interpolation. In this context, we have performed a first step of data validation using the Kriging interpolation, this method was applied to a set of free-air anomalies reduced by the EGM08 geopotential model, from the test zone located in the north of Algeria, but the data were considered as values of an isotropic variable whose the variographic analysis depends only on the distances between points. In the second step the anisotropic character was taken into account, where the main tasks were to calculate the directional variograms for validating data and to perform interpolation by directional models. Finally, we have performed a computer application for interpolation and validation of gravimetric data by Kriging method, based on the algorithm for validation and grid generation of free-air anomalies, proposed in this article, where the results showed that the free-air anomaly is an isotropic variable which only depends on the distances between positions, and that the values resulting from interpolation are validated by the BGI grid, where the two maps generated one by interpolation and the other by the BGI grid are very similar to confirm the validity of the Kriging algorithm developed in this context. [ABSTRACT FROM AUTHOR]

Spatial neighborhood definitions are a consistent source of disagreement among geographic scholars. This research will focus on the implementation and evaluation of spatiotemporal variograms (STVs) as a source of spatial neighborhood definition. STVs show the similarity, measured by semivariance, of spatial events to each other when separated by time and space. Over both time and space, there should exist distances over which pairs of points become "independent" of one another. We seek seeks to answer two questions in relation to STVs and their use as neighborhood definitions: (1) What data and process adjustments are necessary to implement STVs to provide neighborhood search definitions in time and space, and (2) Given that there are many ways to define a neighborhood, STVs may provide a comprehensive method that uses the data themselves to inform the size and scope of neighborhoods, with the added advantage to evaluating both spatial and temporal axes at once. We demonstrate a well‐defined neighborhood that accounts for temporal variation as well as spatial and will be a needed addition as the tools incorporating simultaneous spatial and temporal neighborhoods are implemented. [ABSTRACT FROM AUTHOR]

During the construction of underground engineering, the prediction of groundwater distribution and rock body permeability is essential for evaluating the safety of the project and guiding subsequent design and construction. This article proposes an objective function that solves an underdetermined inverse analysis problem based on the least-squares theory and regularization method and uses geostatistics theory and the variogram function to describe the spatial characteristics of the actual engineering system. It also establishes an optimization model of the analysis stratum seepage field and puts forward the method of using on-site test observation data to solve the stratum penetration coefficient. Relying on the foundation pit project of the Lingshanwei Station of Qingdao Metro, the on-site pumping and packer permeability test was conducted for different strata venues in the foundation pit, and the on-site water-head observation value was obtained. Physical detection of the influence area of foundation pit excavation confirms the correctness of the model from the region and verifies the accuracy of the model on the value through the on-site pumping test. Results show that the accuracy of the use of this objective function to solve the underdetermined inverse problem is above 85%, which proves the effectiveness of the method. The stratigraphic geological information obtained by the inverse analysis model provides an important basis for engineering design and security construction. [ABSTRACT FROM AUTHOR]

In the Algerian-Tunisian border region lies the Tebessa plain, which belongs to the Saharan Atlas domain and is characterized by a semi-arid climate and high salinity, often rendering the groundwater unusable. The objective of this research is to highlight groundwater salinization using statistical and geostatistical methods to model this spatial heterogeneity. A total of 30 data points were used, and the statistical analysis revealed a strong association between chlorides, sulfates, electrical conductivity, and values significantly exceeding the standards. Surface and directional variograms for EC (electrical conductivity), SO4 (sulfates), and Cl (chlorides) exhibit anisotropy, with the major axis showing a hole effect and a periodicity of 5 km. Geostatistical mapping through ordinary kriging allowed us to identify areas of high salinity and deduce the underlying cause; which is the dissolution of gypsiferous formations. These results were confirmed by thermodynamics and Sr2+/Ca2+ ratio. [ABSTRACT FROM AUTHOR]

The high-precision magnetic anomaly reference map is a prerequisite for magnetic navigation and magnetic target detection. However, it is difficult to reflect the detailed characteristics of magnetic anomaly changes by using conventional data interpolation and reconstruction in the areas where magnetic anomaly gradients vary drastically and the distribution of magnetic survey lines is sparse. To solve the problem, an improved variogram of the Kriging interpolation method is proposed to improve the spatial resolution of magnetic anomaly feature. This method selects the spherical variogram model and uses the third power of the lag distance to fit the trend of magnetic anomalies. Meanwhile, the second power of the lag distance is introduced to solve the problem of under-fitting between the lag distance and the value of the variation function near the origin of the sparse variogram graph of measured data. Hyperparameter λ is introduced to compensate for the unbalance caused by the introduction of quadratic lag in the spherical variogram model. The results of several sets of simulated and measured data show that the interpolation accuracy of the proposed method is improved by 30–50% compared with the traditional Gaussian, spherical, and exponential models in the region where the magnetic anomaly gradient changes drastically, and the proposed model provides an effective way to build a high-precision magnetic anomaly reference map of the complex magnetic background under the condition of sparse survey lines. [ABSTRACT FROM AUTHOR]

The geostatistical characterization of the subsurface is confronted with the double challenge of large uncertainties and high exploration costs. Making use of all available data sources is consequently very important. Bayesian inference is able to mitigate uncertainties in such a data-scarce context by drawing on available background information in the form of a prior distribution. To make such a prior distribution transparent and objective, it should be calibrated against a data set containing estimates of the target variable from available sites. In this study, we provide a collection of covariance and/or variogram functions of the subsurface hydraulic parameters from a large number of sites. We analyze this data set by fitting a number of widely used variogram model functions and show how they can be used to derive prior distributions of the parameters of said functions. In addition, we discuss a number of conclusions that can be drawn for our analysis and possible uses for the data set. [ABSTRACT FROM AUTHOR]

For complex and multi-layered clastic oil reservoir formations, modeling lithofacies and petrophysical parameters is essential for reservoir characterization, history matching, and uncertainty quantification. This study introduces a real oilfield case study that conducted high-resolution geostatistical modeling of 3D lithofacies and petrophysical properties for rapid and reliable history matching of the Luhais oil reservoir in southern Iraq. For capturing the reservoir's tidal depositional setting using data collected from 47 wells, the lithofacies distribution (sand, shaly sand, and shale) of a 3D geomodel was constructed using sequential indicator simulation (SISIM). Based on the lithofacies modeling results, 50 sets of porosity and permeability distributions were generated using sequential Gaussian simulation (SGSIM) to provide insight into the spatial geological uncertainty and stochastic history matching. For each rock type, distinct variograms were created in the 0° azimuth direction, representing the shoreface line. The standard deviation between every pair of spatial realizations justified the number of variograms employed. An upscaled version of the geomodel, incorporating the lithofacies, permeability, and porosity, was used to construct a reservoir-flow model capable of providing rapid, accurate, and reliable production history matching, including well and field production rates. [ABSTRACT FROM AUTHOR]

The aim of this work is to gain a better understanding of the diversity of groundwater resource quality in the Auvergne-Rhône-Alpes region (France) using the national Sise-Eaux database. Three matrices were extracted, which included a hollow matrix (approximately 120,000 observations and 21 variables) and two complete matrices (8078 observations with 13 variables each and 150 observations with 20 variables each, respectively). The mapping of these parameters, the chemical profiles of the water, and the characteristics of the variograms make it possible to estimate the importance of the temporal variance compared with the spatial variance. This distinction led to a typology separating 4 groups of chemical parameters and 2 groups of bacteriological parameters, highlighting the information redundancies linking several parameters. A PCA was used to considerably reduce the size of the hyperspace of the data. The study of the factorial axes combined with their distribution over the study area made it possible to discriminate and identify certain mechanisms for acquiring the physico-chemical and bacteriological characteristics of groundwater, the importance of lithology, the components of faecal contamination, and the role of environmental conditions. A typology of the parameters by hierarchical clustering on the major part of the information makes it possible to reduce the information to that carried by a few representative parameters. This work is a new step in understanding the diversity of groundwater resources in general, with a view to more targeted monitoring based on this diversity. [ABSTRACT FROM AUTHOR]

Pseudo cross-variograms appear naturally in the context of multivariate Brown–Resnick processes, and are a useful tool for analysis and prediction of multivariate random fields. We give a necessary and sufficient criterion for a matrix-valued function to be a pseudo cross-variogram, and further provide a Schoenberg-type result connecting pseudo cross-variograms and multivariate correlation functions. By means of these characterizations, we provide extensions of the popular univariate space–time covariance model of Gneiting to the multivariate case. [ABSTRACT FROM AUTHOR]

Remote-sensing images of high spatial resolution (HSR) are valuable sources of fine-grained spatial information for various applications, such as urban surveys and governance. There is continuing research on positional errors in remote-sensing images and their impacts in geoprocessing and applications. This paper explores the combined use of multi-point geostatistics (MPS), machine learning—in particular, generalized additive modeling (GAM)—and computer-image correlation for characterizing positional errors in images—in particular, HSR images. These methods are employed because of the merits of MPS in being flexible for non-parametric and joint simulation of positional errors in X and Y coordinates, the merits of GAM in being capable of handling non-stationarity in-positional errors through error de-trending, and the merits of computer-image correlation in being cost-effective in furnishing the training data (TD) required in MPS. Procedurally, image correlation is applied to identify homologous image points in reference-test image pairs to extract image displacements automatically in constructing TD. To cope with the complexity of urban scenes and the unavailability of truly orthorectified images, visual screening is performed to clean the raw displacement data to create quality-enhanced TD, while manual digitization is used to obtain reference sample data, including conditioning data (CD), for MPS and test data for performance evaluation. GAM is used to decompose CD and TD into trends and residuals. With CD and TD both de-trended, the direct sampling (DS) algorithm for MPS is applied to simulate residuals over a simulation grid (SG) at 80 m spatial resolution. With the realizations of residuals and, hence, positional errors generated in this way, the means, standard deviation, and cross correlation in bivariate positional errors at SG nodes are computed. The simulated error fields are also used to generate equal-probable realizations of vertices that define some road centerlines (RCLs), selected for this research through interpolation over the aforementioned simulated error fields, leading to error metrics for the RCLs and for the lengths of some RCL segments. The enhanced georectification of the RCLs is facilitated through error correction. A case study based in Shanghai municipality, China, was carried out, using HSR images as part of generalized point clouds that were developed. The experiment results confirmed that by using the proposed methods, spatially explicit positional-error metrics, including means, standard deviation, and cross correlation, can be quantified flexibly, with those in the selected RCLs and the lengths of some RCL segments derived easily through error propagation. The reference positions of these RCLs were obtained through error correction. The positional accuracy gains achieved by the proposed methods were found to be comparable with those achieved by conventional image georectification, in which the CD were used as image-georectification control data. The proposed methods are valuable not only for uncertainty-informed image geolocation and analysis, but also for integrated geoinformation processing. [ABSTRACT FROM AUTHOR]

Riverbed roughness is an important aspect of river dynamics research, and its quantitative measurement has always been challenging. In this experiment, regardless of the macroscopic undulation of the riverbed, we laid the initial bed surface in a straight channel with non-uniform natural sand and thickened it by gradually increasing the flow rate of water. Laser scanning technology was used to obtain high-resolution bed elevation data after each bed roughening. Statistical theory and variogram function were used to investigate the surface roughness characteristics of riverbeds with varying degrees of coarseness, and to deepen the understanding of their statistical characteristics. The study's main finding is that artificially laid non-uniform sand presents an elevation frequency distribution that approximates a normal distribution. On the contrary, the frequency distribution of the stable rough bed surface formed by the continuous cumulative erosion of different flow intensities shows a slight positive deviation trend. The study found evidence that the increase of flow intensity leads strictly to an associated elevation standard deviation increase. This finding indicates that elevation standard deviation can be used as a reliable indicator to measure the characteristics of riverbed roughness in future studies. The study revealed that the kurtosis of the bed elevation data precisely de creased monotonically with the flow intensity, and that skewness and kurtosis showed a monotonic trend, indicating that they can be used to measure bed roughness. Additionally, the study found that the flow forming the bed surface is anisotropic, and as flow intensity increases, the rough layer breaks and becomes rougher again. This leads to an increase in the base value of the two-dimensional bed elevation variogram. Therefore, the study suggests that the base value of the two-dimensional variogram could serve as another reliable index for evaluating bed surface roughness. In addition, due to the influence of water flow, although the change trend of the characteristic parameters of the average variogram along the flow direction, vertical flow direction and radial profile elevation is consistent, the average variogram base value and golden point value of the transverse section are slightly larger than those of the transverse and longitudinal sections. Conversely, in the radial section, the abutment value and nugget value are considerably smaller than the values in transverse and longitudinal sections. In conclusion, this study enhances our understanding of the characteristics of bed roughness and provides crucial insights into statistical expressions for quantifying it. [ABSTRACT FROM AUTHOR]

Purpose. This paper aims to assess the distribution of uranium and thorium ore grade distribution to produce a uranium potential map and estimate of the uranium ore reserves in the Gattar-V area, Eastern Desert, Egypt. Methods. Multidisciplinary approach is applied to determine the equivalent uranium mineralization in the Gattar-V area. It includes geological (petrographical, mineralogical and geochemical) and geostatistical (kriging analysis and variogram models) methods. Findings. Geological studies show that the U-mineralization located along or near the contact between younger granites and Hammamat sediments exhibits episyenitization and bleaching alteration, respectively. Geochemical studies indicate a strong relationship between U-mineralization and Chemical Alteration Index (CIA), alteration features, and associated hydrothermal solution mineralization. The geostatistical method is used to study the behavior and distribution of U and Th in both younger granites and Hammamat sediments. The Total Gamma, eU, and eTh values are used in kriging analysis and variogram models to determine their spatial dependence and perform a spatial interpolation of sparse measurements and deposition level map. Originality. The use of a multidisciplinary method combining petrographical, mineralogical and geochemical investigations with geostatistical analysis allowed for a quantitative evaluation of the spatial location of geological objects such as uranium mineralization in the area. Practical implications. Variogram models and kriging analysis can also be used to assess the lithological composition of rocks and mineralogical phases, and they also provide a clear vision of the elements distributed in the ore, which is very useful during the planning and production stages. [ABSTRACT FROM AUTHOR]

Recent theories of cortical organisation suggest features of function emerge from the spatial arrangement of brain regions. For example, association cortex is located furthest from systems involved in action and perception. Association cortex is also 'interdigitated' with adjacent regions having different patterns of functional connectivity. It is assumed that topographic properties, such as distance between regions, constrains their functions, however, we lack a formal description of how this occurs. Here we use variograms, a quantification of spatial autocorrelation, to profile how function changes with the distance between cortical regions. We find function changes with distance more gradually within sensory-motor cortex than association cortex. Importantly, systems within the same type of cortex (e.g., fronto-parietal and default mode networks) have similar profiles. Primary and association cortex, therefore, are differentiated by how function changes over space, emphasising the value of topographical features of a region when estimating its contribution to cognition and behaviour. How spatial organisation in the brain affects function is not well understood. Here, the authors show that function changes gradually across sensory cortex, more rapidly across association cortex, and that these changes are related to variation in intracortical myelination. [ABSTRACT FROM AUTHOR]

Measures of spatial association are important to reveal the spatial structures and patterns in geographical phenomena. They have utility for spatial interpolation, stochastic simulation, and causal inference, among others. Such measures are abundantly available for continuous spatial variables, whereas for categorical spatial variables they are less well developed. In this research, we developed a measure of spatial association for categorical spatial variables coined the entropogram, quantifying its spatial association using mutual information. Mutual information concerns information shared by pairs of random variables at different locations as revealed by their observed joint frequency distribution and marginal frequency distributions. The developed new measure is modeled as a function of lag in analogy to the variogram. Whereas existing measures focus mainly on interstate relationships, the entropogram models the spatial correlation in categorical spatial variables holistically. In this way, the entropogram imparts multiple advantages, for example, simplifying the representation of spatial structure for categorical variables and facilitating communication. The entropogram also reflects variation in the spatial correlation between different states. We first explored the properties of the entropogram in a simulation study. Then, we applied the entropogram to analyze the spatial association of land cover types in Qinxian, Shanxi, China. We conclude that the entropogram provides a suitable addition to existing measures of spatial association for applications in a wide range of disciplines where the categorical spatial variable is of interest. [ABSTRACT FROM AUTHOR]

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Geographic registration (geo-registration) is a crucial foundation for augmented reality (AR) map applications. However, existing methods encounter difficulties in aligning spatial data with the ground surface in complex outdoor scenarios. These challenges make it difficult to accurately estimate the geographic north orientation. Consequently, the accuracy and robustness of these methods are limited. To overcome these challenges, this paper proposes a rotation-invariant estimation method for high-precision geo-registration in AR maps. The method introduces several innovations. Firstly, it improves the accuracy of generating heading data from low-cost hardware by utilizing Real-Time Kinematic GPS and visual-inertial fusion. This improvement contributes to the increased stability and precise alignment of virtual objects in complex environments. Secondly, a fusion method combines the true-north direction vector and the gravity vector to eliminate alignment errors between geospatial data and the ground surface. Lastly, the proposed method dynamically combines the initial attitude relative to the geographic north direction with the motion-estimated attitude using visual-inertial fusion. This approach significantly reduces the requirements on sensor hardware quality and calibration accuracy, making it applicable to various AR precision systems such as smartphones and augmented reality glasses. The experimental results show that this method achieves AR geo-registration accuracy at the 0.1-degree level, which is about twice as high as traditional AR geo-registration methods. Additionally, it exhibits better robustness for AR applications in complex scenarios. [ABSTRACT FROM AUTHOR]

The suitability of several low-labor geostatistical procedures in the interpolation of highly positively skewed seismic data distributions was tested in the Baltic Basin. These procedures were acombination of various estimators of the model of spatial variation (theoretical variogram) and kriging techniques, together with the initial data transformation to normal distribution or lack thereof. This transformation consisted of logarithmization or normalization using the anamorphosis technique. Two variations of the theoretical variogram estimator were used: the commonly used classical Matheron estimator and the inverse covariance estimator (InvCov), which is robust with regard to non-ergodic data. It was expected that the latter would also be resistant to strongly skewed data distributions. The kriging techniques used included the commonly used ordinary kriging, simple kriging useful for standardized data and the non-linear median indicator kriging technique. It was confirmed that normalization (anamorphosis) is the most useful and less laborious geostatistical procedure of those suitable for such data, which results in a standardized normal distribution. The second, not obvious statement for highly skewed data distributions suggests that the non-ergodic inverted covariance (InvCov) estimator of variogram has an advantage over the Matheron’s estimator. It gives a better assessment of the C0 (nugget effect) and C (sill) parameters of the spatial variability model. Such a conclusion can be drawn from the fact that the higher the estimation of the relative nugget effect L = C0 /(C0 + C) using the InvCov estimator, the weaker the correlation between the kriging estimates and the observed values. The values of the coefficient L estimates obtained by using the Matheron’s estimator do not meet this expectation. [ABSTRACT FROM AUTHOR]

Current literature dealing with the theory of sampling process flows is limited in scope, a situation addressed by papers presented at conferences by the author. This paper provides a summary of methods to quantify process sampling variance in a sound statistical manner. Two methods of calculation of the process sampling variance are presented here with models for robust estimation of the process covariance/variogram. The strategies for process sampling to acquire the data for process characterisation are described. The method for process characterisation can be based on special sampling campaigns, on the analysis of on-line measurement data, or on the analysis of historical data, which may be based on composite samples, such as shift samples. These scenarios are all dealt with as well as methods based on analysis of process dynamics. The latter analysis demonstrates that use of covariance functions drawn from geostatistical practice can be inappropriate for process analysis. The work is supported with plant data examples and quantitative analyses of process conditions. [ABSTRACT FROM AUTHOR]

The regularisation on a segment of a stationary or intrinsic variogram model is usually expressed as an integral. By introducing an auxiliary function, it is shown that this integral is a linear combination of elementary terms. This paper shows how these terms are arranged and provides a simple and safe procedure to calculate them. An important application of this procedure is the deregularisation problem, that is, the conversion of a regularised variogram to a point variogram. This ill-posed problem is tackled using an iterative optimisation algorithm designed to produce a family of point variograms that best fit the initial regularised variogram. Subsequently, each of these point variograms can be regularised on any segment, which results in a fully coherent family of simple and cross regularised variograms. [ABSTRACT FROM AUTHOR]

This paper presents a case study in the application of geostatistics in the analysis and adaptation of representative values of the geotechnical parameters at coal deposits. In order to select the most reliable geotechnical values for the design process of open cast pit excavations of large heights, different pools of laboratory data sets have been processed. The following critical geotechnical parameters were investigated: volume weight, angle of internal friction, and moisture content. By applying the correct geostatistical tools, and in comparison with practical experience and removal of obvious outliers from the datasets, for one of the deposit are prepared the so-called "geotechnical domains" are prepared, shown in the form of spatial distribution maps of the analysed parameters. Advantages of the applied geostatistical approach are seen in enabling higher confidence when performing slope stability analysis, selection of the most optimal geometry of the mines, lesser overburden quantities, protection of the environment, production cost cutting, and time saving. [ABSTRACT FROM AUTHOR]

Seismic stochastic inversion method has received much attention because of its considerable advantage of having higher vertical resolution than deterministic inversions. However, due to the lack of cross-well data, the inversion results typically exhibit poor lateral continuity. Furthermore, the inversion efficiency is low, and the inversion result is highly random. Therefore, this study proposes a geostatistical seismic inversion method constrained by a seismic waveform. The correlation coefficient of seismic data is used to measure the similarity of the seismic waveforms, replacing the traditional variogram for sequential Gaussian simulation. Under the Bayesian framework, the Monte Carlo-Markov Chain (MCMC) algorithm is combined with the constraints of seismic data to randomly perturb and optimize the simulation results for obtaining the optimized parameter inversion results. The model data tests show that the initial model based on seismic waveform constraints can accurately describe the spatial structure of the subsurface reservoir. In addition, perturbing and optimizing initial model can increase the convergence speed of the Markov chain and effectively improve the accuracy of the inversion results. In this paper, the proposed geostatistical inversion method is applied to the actual seismic data of an oil field. Under the constraints of the stochastic simulation process and objective function, the geological information contained in the seismic waveforms is fully mined, and a theoretical foundation is provided for realizing the multidata joint-constrained seismic inversion. [ABSTRACT FROM AUTHOR]

The estimation of rainfall depth–duration–frequency (DDF) curves is necessary for the design of several water systems and protection works. These curves are typically estimated from observed locations, but due to different sources of uncertainties, the risk may be underestimated. Therefore, it becomes crucial to quantify the uncertainty ranges of such curves. For this purpose, the propagation of different uncertainty sources in the regionalisation of the DDF curves for Germany is investigated. Annual extremes are extracted at each location for different durations (from 5 min up to 7 d), and local extreme value analysis is performed according to. Following this analysis, five parameters are obtained for each station, from which four are interpolated using external drift kriging, while one is kept constant over the whole region. Finally, quantiles are derived for each location, duration and given return period. Through a non-parametric bootstrap and geostatistical spatial simulations, the uncertainty is estimated in terms of precision (width of 95% confidence interval) and accuracy (expected error) for three different components of the regionalisation: (i) local estimation of parameters, (ii) variogram estimation and (iii) spatial estimation of parameters. First, two methods were tested for their suitability in generating multiple equiprobable spatial simulations: sequential Gaussian simulations (SGSs) and simulated annealing (SA) simulations. Between the two, SGS proved to be more accurate and was chosen for the uncertainty estimation from spatial simulations. Next, 100 realisations were run at each component of the regionalisation procedure to investigate their impact on the final regionalisation of parameters and DDF curves, and later combined simulations were performed to propagate the uncertainty from the main components to the final DDF curves. It was found that spatial estimation is the major uncertainty component in the chosen regionalisation procedure, followed by the local estimation of rainfall extremes. In particular, the variogram uncertainty had very little effect on the overall estimation of DDF curves. We conclude that the best way to estimate the total uncertainty consisted of a combination between local resampling and spatial simulations, which resulted in more precise estimation at long observation locations and a decline in precision at unobserved locations according to the distance and density of the observations in the vicinity. Through this combination, the total uncertainty was simulated by 10 000 runs in Germany, and it indicated that, depending on the location and duration level, tolerance ranges from ± 10 %–30 % for low-return periods (lower than 10 years) and from ± 15 %–60 % for high-return periods (higher than 10 years) should be expected, with the very short durations (5 min) being more uncertain than long durations. [ABSTRACT FROM AUTHOR]

Geostatistics has become a powerful method for investigating complex spatial variations of prehistoric settlements in floodplains and other geomorphological settings. A geoarchaeological drilling program that covers most of the Sha‐Ying River Basin provides a rare opportunity with unusually detailed environmental data to contest and develop the geostatistics method, which proves to be essential, in combination with archaeological data, to understand long‐term (9000–2500 B.P.) patterns of human inhabitation and adaption to volatile floodplain environments in eastern Central China. We analysed the variography and multivariate ordination of the borehole data and explored the complexities of landform evolution, with reference to sedimentation processes and soil development in the floodplain of the Sha‐Ying River. The recurrent impact of river floods on regional landforms is manifested by spatial‐autocorrelated properties over distances up to 10 km, sometimes with directional trends. We then developed a model of landform evolution through kriging and compared the model with detailed reconstruction of archaeological settlement patterns. Our results illustrate long‐term socio‐environmental dynamics by which human communities first populated and then adapted in diverse ways to the changing floodplain environments from the early to middle Holocene. This improved method will have far‐reaching implications for future studies on similar geomorphological settings across vast floodplains of Central China and other global regions. [ABSTRACT FROM AUTHOR]

The analysis of spatiotemporal changes of landscape patterns is of great significance for forest protection. However, the selection of landscape metrics is often subjective, and existing composite landscape metrics rarely consider the effects of spatial correlation. A more objective approach to formulating composite landscape metrics involves proper weighting that incorporates spatial structure information into integrating individual conventional metrics selected for building a composite metric. This paper proposes an integrated spatial landscape index (ISLI) based on variogram modeling and entropy weighting. It was tested through a case study, which sought to analyze spatiotemporal changes in the landscape pattern in the Changbai Mountains over 30 years based on six global land-cover products with a fine classification system at 30 m resolution (GLC_FCS30). The test results confirm: (1) spatial structure information is useful for weighting conventional landscape pattern metrics when constructing ISLI as validated by correlation analysis between the incorporated conventional metrics and their variogram ranges. In terms of the range parameters of different land cover types, broadleaf forest and needleleaf forest have much larger range values than those of other land cover types; (2) DIVISION and PLAND, two of the conventional landscape metrics considered for constructing ISLI, were assigned the greatest weights in computing ISLI for this study; and (3) ISLI values can be used to determine the dominant landscape types. For the study area, ISLI values of broadleaf forests remained the largest until 2020, indicating that forest landscape characteristics were the most prominent during that period. After 2020, the dominance of needleleaf forest gradually increased, with its ISLI value reaching a maximum of 0.91 in 2025. Therefore, the proposed ISLI not only functions as an extension and complement to conventional landscape metrics but also provides more comprehensive information concerning landscape pattern dynamics. [ABSTRACT FROM AUTHOR]

Fresh drinking water ensures healthy life while contaminated water causes waterborne diseases and other detrimental health impacts. This study evaluated the spatial distribution of fluoride and some other significant water quality parameters (WQPs). Hydrological spatial data of 127 water samples collected from district Multan were tested for 12 WQPs including fluoride which is among those chemicals which affect the health of the human beings and provides destructive as well as protecting impacts on the teeth. For the analysis, initially, descriptive statistical measures are computed along with normality tests. Due to multivariate nature of data, multivariate correlation analysis, cluster and factor analysis are considered to group the relatively aligned variables. Eight significant WQPs selected by cluster and factor analysis are further processed for the coregionalization where the distance based pairwise association was evaluated. Fluoride showed a relatively week distance based association with rest of the seven WQPs as compared to total dissolved solids (TDS). The parameters of the variogram model estimated by Restricted maximum likelihood, Weighted least square and Maximum likelihood estimation are compared on the basis of root mean square error and are considered in the final cokriging estimation. As suggested in literature, we considered the TDS as primary WQP while rest of the WQPs as secondary variables in the cokriging estimation. Finally, the prediction results are processed for the contour plots which showed the estimation of unmeasured location. Results showed that the area between longitude 71.2°-71.5° and latitude 29.5°-29.8° has relatively more contamination in drinking water of district Multan which is alarming and suggests that special care of fluoride levels in drinking water is required for health policies. [ABSTRACT FROM AUTHOR]

A significant contribution to the economic growth of a nation comes from the mineral industries. Therefore, the concentration of metallic or nonmetallic minerals in different regions of Earth's crust is important to determine. The present paper studies the grade and thickness estimation of iron and coal deposits, respectively, by applying two-dimensional cellular automata (CAs). Krigging is a popular method for the estimation of mineral resources. However, krigging results in complex mathematical calculations if the number of sample points increases. Here, each cell of the cellular automaton (CA) is represented as a block. Using CAs, the grade values and thickness are estimated in a simpler and faster way. Two-dimensional CAs are used in this paper where the local rule is the ordinary krigging estimator function using the spherical variogram model. The total weight of iron as well as coal is calculated using the CA-based technique. A comparative analysis between the estimated weight of minerals and the actual extracted mineral is also given. [ABSTRACT FROM AUTHOR]

Understanding the local dynamics of COVID-19 transmission calls for an approach that characterizes the incidence curve in a small geographical unit. Given that incidence curves exhibit considerable day-to-day variation, the fractal structure of the time series dynamics is investigated for the Flanders and Brussels Regions of Belgium. For each statistical sector, the smallest administrative geographical entity in Belgium, fractal dimensions of COVID-19 incidence rates, based on rolling time spans of 7, 14, and 21 days were estimated using four different estimators: box-count, Hall-Wood, variogram, and madogram. We found varying patterns of fractal dimensions across time and location. The fractal dimension is further summarized by its mean, variance, and autocorrelation over time. These summary statistics are then used to cluster regions with different incidence rate patterns using k-means clustering. Fractal dimension analysis of COVID-19 incidence thus offers important insight into the past, current, and arguably future evolution of an infectious disease outbreak. [ABSTRACT FROM AUTHOR]

spmodel is an R package used to fit, summarize, and predict for a variety spatial statistical models applied to point-referenced or areal (lattice) data. Parameters are estimated using various methods, including likelihood-based optimization and weighted least squares based on variograms. Additional modeling features include anisotropy, non-spatial random effects, partition factors, big data approaches, and more. Model-fit statistics are used to summarize, visualize, and compare models. Predictions at unobserved locations are readily obtainable. [ABSTRACT FROM AUTHOR]

A new approach is proposed for simultaneous diagonalisation of variance-covariance matrices to generate spatially orthogonal factors using based on simulated annealing method. Simultaneous diagonalisation was applied to three quality attributes of a coal seam. The proposed method was successful in diagonalisation due to creating factors with cross-variograms that lie in tighter intervals compared with minimum/maximum autocorrelation factors (MAF). Simulation through the proposed approach showed variable dependent results in terms of reproducing mean values, variances, correlation coefficients, and auto-variograms of variables. However, the new approach showed advantage over MAF in reproducing the cross-variograms of variables, especially in the downhole direction. [ABSTRACT FROM AUTHOR]

Tropospheric delay error must be reduced during interferometric synthetic aperture radar (InSAR) measurement. Depending on different geographical environments, an appropriate correction method should be selected to improve the accuracy of InSAR deformation monitoring. In this study, surface deformation monitoring was conducted in a high mountain gorge region in Yunnan Province, China, using Sentinel-1A images of ascending and descending tracks. The tropospheric delay in the InSAR interferogram was corrected using the Linear, Generic Atmospheric Correction Online Service for InSAR (GACOS) and ERA-5 meteorological reanalysis data (ERA5) methods. The correction effect was evaluated by combining phase standard deviation, semi-variance function, elevation correlation, and global navigation satellite system (GNSS) deformation monitoring results. The mean value of the phase standard deviation (Aver) of the linear correction interferogram and the threshold value (sill) of the semi-variogram were reduced by –20.98% and –41%, respectively, while the accuracy of the InSAR deformation points near the GNSS site was increased by 58%. The results showed that the three methods reduced the tropospheric delay error of InSAR deformation monitoring by different degrees in low-latitude mountains and valleys. Linear correction was the best at alleviating the tropospheric delay, followed by GACOS, while ERA5 had poor correction stability. [ABSTRACT FROM AUTHOR]

Convolutional neural networks (CNNs) are gaining tremendous attention in subsurface studies due to their ability to learn from spatial image data. However, most deep learning studies in spatial context do not consider the impact of geostatistical nonstationarity, which is commonly encountered within the subsurface phenomenon. We demonstrate the impact of geostatistical nonstationarity on CNN prediction performance. We propose a CNN model to predict the variogram range of sequential Gaussian simulation (SGS) realizations. Model performance is evaluated for stationarity and three common forms of geostatistical nonstationarity: (1) large relative variogram range-related nonstationarity, (2) additive trend and residual model-related nonstationarity, and (3) mixture population model-related nonstationarity. Our CNN model prediction accuracy decreases in the presence of large relative variogram range-related nonstationarity, for the additive trend and residual model-related nonstationarity, the relative prediction errors increase for high trend variance proportions with a decrease in variogram range; regarding the mixture population model-related nonstationarity, the predictions are closer to the smaller variogram range. Common forms of geostatistical nonstationarity may impact CNN predictions, as with geostatistical estimation methods, trend removal and workflows with stationary residuals are recommended. [ABSTRACT FROM AUTHOR]

This research is focused on the analysis of the sequence stratigraphic units of F3 Block, within a wave‐dominated delta of Plio–Pleistocene age. Three wells of F3 block and a 3D seismic data, are utilized in this research. The conventional techniques of 3D seismic interpretation were utilized to mark the 11 surfaces on the seismic section. Integration of seismic sequence stratigraphic interpretation, using well logs, and subsequent 3D geostatistical modeling, using seismic data, aided to evaluate the shallow hydrocarbon traps. The resulting models were obtained using System Tract and Facies models, which were generated by using sequential stimulation method and their variograms made by spherical method, moreover, these models are validated via histograms. The CDF curve generated from upscaling of well logs using geometric method, shows a good relation with less percentage of errors (1 to 2 for Facies and 3 to 4 for System Tract models) between upscaled and raw data that complements the resulted models. These approaches help us to delineate the best possible reservoir, lateral extent of system tracts (LST and/or HST) in the respective surface, and distribution of sand and shale in the delta. The clinoform break points alteration observed on seismic sections, also validates the sequence stratigraphic interpretation. The GR log‐based Facies model and sequence stratigraphy‐based System Tract model of SU‐04‐2 showed the reservoir characteristics, presence of sand bodies and majorly LST, respectively, mainly adjacent to the main fault of the studied area. Moreover, on the seismic section, SU‐04‐2 exhibits the presence of gas pockets at the same location that also complements the generated Facies and System Tract models. The generated models can be utilized for any similar kind of study and for the further research in the F3 block reservoir characterization. [ABSTRACT FROM AUTHOR]

Various disciplines, engineering, humanities, and other sciences require interpolating many parameters. Geostatistics, with its structural analysis step, is widely used for this purpose. Variography is the valuable step used to assess the correlation and dependence of the data. However, the wrong choice of the variogram model encounter all the predictive attended results. This article illustrates how the use of inappropriate variogram models can seriously conduct to a misleading of predicted results for such analysis. The influence of the selection of the semi-variogram model is highlighted and illustrated by thematic maps developed using three different models (Gaussian, spherical and exponential). To avoid such a drawback, a methodical approach to select the most suitable model, based on the calculation and analysis of the mean error (ME), the mean square error (MSE), the root of the square error mean (REQM), mean standard error (ESM) and root of mean standard error (REQSM), is proposed in the present research study. Such contribution could reduce the negative effects of the choice of variogram model on the interpolation operation using the kriging technique. [ABSTRACT FROM AUTHOR]

This Special Issue (SI), titled "Probability Theory and Stochastic Modeling with Applications", is concerned with the theory and applications of stochastic models. Footnotes 1 B Disclaimer/Publisher's Note: b The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). It consists of sixteen papers, and we would like to thank all the authors for their positive answers to our call for papers for the SI, as well as for their efforts to provide a high-quality contribution. Using fuzzy variables and robust optimization, the authors provide the electricity transaction policy under different expected costs so that the expectation of the risk-averse distribution system operator is fulfilled. [Extracted from the article]

To predict the lithofacies distributions for modelling hydrocarbon reservoir heterogeneities, geostatistical tools like the indicator variogram- based methods provide an efficient solution. However, for the sparse datasets, they potentially encompass drawbacks like the ambiguity of variograms and generating unrealistic patterns. Nevertheless, Transition probabilities (TPs)-based techniques using Markov chains (MCs) have a notable potential to overcome these problems. Hence, the facies distributions were predicted in an Iranian hydrocarbon reservoir using three methods including a novel algorithm proposed here (CGCMC). Using generalized coupled Markov Chain (GCMC), sequential indicator simulation (SIS), and a new combination of GCMC with the transition probability Markov Chain (TP/MC) framework, 50 realizations were generated through each method. Then, the models were evaluated regarding the reproduction of their: facies proportions, variograms/TPs and geological soundness. By ensemble probability distributions (EPDs) the spatial uncertainties were represented. Accordingly, CGMC produced more acceptable results quickly and more conveniently with a higher data integration capacity. [ABSTRACT FROM AUTHOR]

Sensitivity analysis provides information to understand response models and helps make better decisions in many fields, including the petroleum and mining industries. A new approximate method for sensitivity analysis through extracting covariogram ranges is proposed in this paper. In this method, the covariogram range will be treated the same as the variogram range, and each factor in the response model will be looked as a separate coordinate in a pseudo-spatial space. The result indicates that the covariogram ranges calculated in this space for each variable have a strong relationship with the main effect sensitivity index in variance-based method, that is, a longer range has less sensitivity. Thus, the sensitivity of each factor is accessible through covariogram range calculation, which is a unique approach to sensitivity analysis compared to existing approaches. This approach provides a new perspective of understanding global sensitivity. Furthermore, a modified covariogram fitting algorithm is put forward to calculate the ranges through optimization. This iterative algorithm minimizes the squared error between the modified experimental covariogram and Gaussian covariogram model, which is useful in a high dimension with sparse sampling. The simulated data are also used to validate the automatic covariogram fitting, and the optimized ranges are very close to the true ranges even with a small sample size. Finally, this method is successfully applied in the parameter sensitivity analysis of reservoir simulation, and limitations or considerations for the practical application have been discussed. [ABSTRACT FROM AUTHOR]

This study assesses the inherent variability in the geomaterial parameter by quantifying the parameter uncertainty and develops a site investigation plan with a low degree of uncertainty. A key research motivation was using sparse borehole data to predict a site geomaterial configuration in order to determine the design of a site investigation plan. This study develops a systematic methodology for carrying out a study of inherent variability in light of the limitations posed by borehole data. The data in this study was provided by the Iowa Department of Transportation which consisted of eight boreholes from which 92 associated SPT N-values was considered as the geomaterial parameter of interest. The systematic methodology then involved the following steps. A general linear model was employed to fit and compare various spatial covariance models with and without a nugget. These spatial covariance models were also evaluated with variograms. Predicted SPT N-values were generated using universal kriging. Simulations were performed conditionally and unconditionally to identify optimal site investigation plans. The results identified site investigation plans with reduced parameter uncertainty. The proposed approach can produce site investigation plans that target any or all geomaterial layers to reduce uncertainty with respect to any geomaterial parameter of interest. [ABSTRACT FROM AUTHOR]

In this research, we made use of a technique known as principal component analysis (PCA). PCA is an approach to statistics that takes into account several variables that converts a fixed number of correlated variables into a fixed number of orthogonal, uncorrelated axes known as principal components by making use of orthogonal transformation. In other words, PCA transforms correlated variables into uncorrelated axes. The principal component analysis (PCA) method, to put it another way, transforms correlated variables into uncorrelated axes. We used (PCA) technique in order to bring the dimensionality of a data collection down to a more tolerable level that had a wide variety of interconnected variables while yet preserving as much of the natural variation that existed within the data set. Because of this, we were able to examine eleven different steel components. To do this Each independent variable is combined with others to generate a third independent variable. Known as principal components, which are not associated with one another (PC). The order of the principle components is chosen in such a manner that they maintain the vast majority of the variety that exists in each and every one of the several variables. This is accomplished by using a variogram. This is accomplished by reworking the individual variables into a brand new group of variables called principle components, which are not associated with one another (PC). Because this percentage reflects the principal aspect that is best among all 11 principal components, we are able to reach the conclusion that the five principal components that collectively account for approximately sixty-seven percent of the Total variance in all of the data are the best principal components. This allows us to come to the conclusion that the best principal components are the five principal components that collectively account for approximately sixty-seven percent of the variance in all of the data. [ABSTRACT FROM AUTHOR]

The representation of rainfall in space is important for hydrological modelling. Accurate estimation of rainfall is particularly challenging in mountainous regions where observations are often sparse relative to the spatial variability of rainfall. In these regions, orographic processes lead to complex patterns of rainfall enhancement and rain shadow depletion. This study tests Natural Neighbour Interpolation (NNI), ordinary kriging (OK) and ordinary cokriging (CK), to determine if CK improves rainfall interpolation during three extreme rainfall events. Three different elevation indices were considered as secondary variables for CK. Preliminary analysis using long‐term annual average rainfall totals, including additional high elevation rainfall observations, showed that CK with an effective elevation index (a directionally smoothed elevation, corrected for degree of 'orographic processing' and shifted to account for 'wind‐drift' of rainfall) as a secondary variable performed better than NNI and OK with an overall improvement of around 40%. Using rainfall totals for long‐term wind direction and wind speed rainfall classes, CK performance was variable but provided an improvement of approximately 15% for wind direction classes without an easterly wind component. For 15‐min timesteps during extreme rainfall events, there were comparatively small differences (cross‐validation using RMSE) between interpolation methods, partly attributed to having only relatively low elevation rainfall observations, providing weak constraint. Using cross‐validation and mean bias did, however, show an improvement for both high and low elevation observation classes. Importantly, cross‐variogram estimation provided differing cross‐validation results when estimated for different rainfall accumulation periods: 15‐min, hourly, daily and long‐term. Variograms and cross variograms estimated at a 15‐min timestep frequency were robust for many timesteps, but were difficult to fit automatically for others. Variograms estimated from longer periods were more reliably estimated, but tended to have lower variance and cross‐variance and longer correlation ranges producing a smoother interpolated rainfall field. Given the weak cross‐validation constraint, care must be taken in identifying the most appropriate method and variogram estimation period. [ABSTRACT FROM AUTHOR]

Extremal graphical models are sparse statistical models for multivariate extreme events. The underlying graph encodes conditional independencies and enables a visual interpretation of the complex extremal dependence structure. For the important case of tree models, we develop a data‐driven methodology for learning the graphical structure. We show that sample versions of the extremal correlation and a new summary statistic, which we call the extremal variogram, can be used as weights for a minimum spanning tree to consistently recover the true underlying tree. Remarkably, this implies that extremal tree models can be learned in a completely non‐parametric fashion by using simple summary statistics and without the need to assume discrete distributions, existence of densities or parametric models for bivariate distributions. [ABSTRACT FROM AUTHOR]

A hydro-geophysical investigation was carried out in the northeastern parts of Botswana, in a wellfield called Dukwi, with the aim of estimating the aquifer hydraulic parameters employing the geo-electrical method. A Schlumberger electrode configuration of 16 vertical electrical soundings (VES) was carried out in the Dukwi Wellfield. The data were recorded with a maximum AB/2 spacing of 500 m and processed using the InterpexRESIXIPtm software. In addition, physicochemical parameter measurements were carried out for 19 groundwater samples in the field. Aquifer hydraulic parameters, like hydraulic conductivity (K) and transmissivity (T), were computed from the pumping test data. An interpretation was made using statistics, including linear regression analysis. The major aquifer is the Mea Arkose Formation, and its estimated hydraulic conductivity and transmissivity ranges from 15.4 to 47.3 m/day and 539.9–2113.9 m2/day, respectively. The aquifer transverse resistance (Tr) and longitudinal conductance (S) of the rocks overlying the aquifer ranges from 945 to 2440 Ωm2 and 1.5 to 9.7 Ω-1, respectively. The R2 value for the geo-electrical and pumping test K and T values are 0.689 and 0.896, respectively. The estimated K has a positive and strong correlation (R2 = 0.946) with the resistivity. A strong positive correlation (R2 = 0.892) is obtained between the estimated T and Tr of the aquifer. On the basis of these correlations, equations that relate the resistivity of the aquifer to its hydraulic parameters were formulated. These equations enable the evaluation of aquifer hydraulic parameters where there is a lack of pumping test data. Furthermore, these equations can be used in other parts of the country and in regions elsewhere that have similar geological and hydrogeological setups. • Mea Arkose Formation is the major aquifer in the Dukwi wellfields. • K and T of the major aquifer ranges from 15.4 to 47.3 m/day and 539.9–2113.9 m2/day, respectively. • The aquifer Tr and S of the rock overlying the aquifer ranges from 945 to 2440 Ωm2 and 1.5 to 9.7Ω−1, respectively. • Geo-electrically estimated transmissivity, T e , of the aquifer has strong positive correlation with the Tr of the aquifer. • Estimated hydraulic conductivity, K e , and measured aquifer bulk resistivity, ρ b , have a positive and strong correlation. [ABSTRACT FROM AUTHOR]