Kriging Variance Research Articles

Drill Hole Spacing Analysis for Evaluation of Quartz Sand Resources as Residual Material from On-Land Alluvial Tin Processing based on Global Estimation Variance and Kriging Variance, with Case Study in the Bangka and Belitung Islands, Indonesia

On-land alluvial tin mining activities produce residual materials from mineral processing or mineral washing; the major one of these is quartz sand, which can be utilized in the manufacturing industry. As time goes by, the reserves of alluvial tin are decreasing, which is in line with the increasing residual material from abundant mineral processing. Mining procedures are carried out following the Indonesian government regulations, covering mining as well as reprocessing. This motivated us to evaluate the potential of quartz sand at two mining sites in Bangka and Belitung Islands. This study evaluated alluvial tin processing residual quartz sand through drill hole spacing analysis (DHSA) by comparing two geostatistics parameters, namely global estimation variance (GEV) and kriging variance (KV). Drill hole samples were taken with varying spacing, ranging from 50 to 200 m, after which geostatistical analysis was carried out. With several simulations, the incorporation of GEV and KV was able to produce the optimal drill hole spacing with measured resource categories in the range of 40 to 55 m, indicated resources in the range of 55 to 85 m, and inferred resources >85 m. Accordingly, the total estimated quartz sand resources of both sites were obtained.

Optimising the placement of additional drill holes to enhanced mineral resource classification: a case study on a porphyry copper deposit

ABSTRACT In this study, we compared outcomes of optimising the placement of five additional drill holes using three geostatistical cost functions (AKV, WAKV, and CV) and the Particle Swarm Optimisation algorithm (PSO). WAKV identified locations with higher average copper grades compared to AKV. Conversely, CV suggested sites with high kriging variance and copper grade variation. Initial holes, alongside those determined by each cost function, were used to classify mineral resources. Findings underscored the effectiveness of optimising drill hole placement based on cost functions in reducing uncertainty and improving mineral resource classification.

Spatial-temporal Trends Mapping and Geostatistical Modelling of Groundwater Level Depth Over Northern Parts of Indo-Gangetic Basin, India

Haryana-Punjab is a vast part of the Indo-Gangetic basin (India), recognised globally as a major hotspot of groundwater abstraction and agricultural economic reliance. A lack of information is present in this region regarding spatiotemporal changes in groundwater levels. Geostatistical anisotropic processes are reliable, especially when the monitored regions are extensive. The modelling study involves a twofold objective. First, it estimates and evaluates anisotropic spatial variations in groundwater level depth (GWD, Surface to water level) using geostatistics for all four seasons of Indian cropping patterns—pre-monsoon, monsoon, post-monsoon (rabi) and post-monsoon (Kharif), and parameters of point kriging cross-validation (PKCV) are optimum, acceptable, and support the unbiasedness hypothesis of kriging. Based on the PKCV, five essential parameters were computed to accept the anisotropy semi-variogram fitted model. These are (1) kriging mean error (KME), ideally close to zero so that there are no over or underestimates, (2) goodness of fit (R2), (3) the ratio of estimated variance (EV) to kriging variance (KV) lies between 0.95-1.05, (4) good eye visualisation fit and (5) significant t-test on the correlation coefficient. The second objective is spatiotemporal modelling, a pixel-based Mann-Kendall trend testing (at 95% Confidence Interval) on kriged raster surfaces of GWD (For all four seasons) at 1km grid resolution. The trend, significant or not, is determined by the Mann-Kendall test, while Sen's slope estimator determines the slope magnitude of the trend. Results revealed that the study area's east-central to the central-northern region comes under a high depletion zone for groundwater levels. GWD significantly increased by 120cm/year-80cm/year in this region, and the mean kriged GWD for the entire study area increased by 30cm/year-29cm/year over 25 years. Seasonal climatological mean maps of kriged surfaces of GWD and mean rainfall surfaces in two different time phases for all four seasons have been evaluated. It is observed from these maps that wherever the rainfall is increasing, GWD is decreasing. Results clarified that anisotropic semi-variogram modelling with kriging, pixel-based trend analysis, and regression studies is a valuable tool for identifying the critical region of groundwater levels, and the central region of the study area falls under the severe depletion condition of the groundwater level.

A new reliability method combining adaptive Kriging and active variance reduction using multiple importance sampling

A new reliability method combining adaptive Kriging and active variance reduction using multiple importance sampling

Complexity reduction of ordinary kriging algorithm for 3D REM design

This work has investigated the feasibility of using cluster ordinary kriging (OK) interpolation to speed up indoor three-dimensional (3D) radio environment map (REM) generation time than the conventional OK-based REM generation time in two-dimensional (2D) and 3D ways mainly. The REM has been generated on the real-time received signal strength (RSS) dataset of an ultra-high frequency active television (TV) channel to explore TV grey space (TVGS) volume for cognitive access inside a six-storied office building. It can support future home networks for high-speed data-based applications. The optimal number of clusters has been found at K=4 on both 3D and 2D ways through distortion % deviation analysis. The maximum distortion % deviation = 88.74%, has been found at K=4. The exponential semivariogram model has been selected as the best-fit model after a thorough cross-validated comparative analysis of the clustered and non-clustered OK algorithms separately on residual squared error (RSE), root mean square error (RMSE) and the ratio of expected variance (EV) and kriging variance (KV) results. The RSE and RMSE are minimum, and the EV: KV ratio is in between 0.95 to 1.05 for the exponential model. Significant observation of the analysis of the results shows that cluster OK algorithms outperform conventional OK algorithms of corresponding domains on time complexity. A comparative study on RMSE, correlation coefficient (CC), and relative recovery error (RRE) show that cluster OK improves the accuracy. Cluster 2D and 3D OK-based detected TVGS volume is 13450 m3, 5.634% of the whole building volume.

Effects of Confidentiality‐Preserving Geo‐Masking on the Estimation of Semivariogram and of the Kriging Variance

Geostatistical methods, such as semivariograms and kriging are well‐known spatial tools commonly employed in many disciplines such as health, mining, forestry, meteorology to name only few. They are based essentially on point‐referenced data on a continuous space and on the calculation of distances between them. In many practical instances, however, the exact point location, even if exactly known, is geo‐masked to preserve confidentiality. This typically happens when dealing with confidential data related to individuals‐health and their biometric parameters. In these situations, the estimation of the semivariogram and, hence, the spatial prediction can become biased and highly inefficient. This paper examines the extent of the bias in the particular case when the geo‐masking mechanism is known (called “intentional locational error”) and lays the ground to a full understanding of the phenomenon in more general cases. We also examine how the geo‐masking affects the estimation of the kriging variance thus reducing the efficiency of spatial prediction. We pursue our aims by developing some theoretical results and by making use of simulated and real data analysis.

Optimization of Monitoring Network to the Rainfall Distribution by Using Stochastic Search Algorithms: Lesson from Pakistan

Agricultural production is greatly influenced by environmental parameters such as temperature, rainfall, humidity, and wind speed. The accurate information about environmental parameters plays a vital and useful role when making policies for the agriculture sector as well as for other sectors. Pakistan meteorological department observed these environmental parameters at more than 90 stations. The allocation of these monitoring stations is not made systematically correct. This leads to inaccurate predictions for unobserved locations. The study aims to propose a monitoring network by which these prediction errors of the environmental parameters can be minimized. The well-known prediction techniques named, model-based ordinary kriging and model-based universal kriging (UK) with the known Matheron variogram model are used for prediction purposes. We investigate the monitoring network of Pakistan for rainfall and focus on both the optimal deletion/addition of monitoring stations from/to this network. The two stochastic search algorithms, spatial simulated annealing, and genetic algorithm are used for optimization purposes. Furthermore, the minimization of the Average Kriging Variance (AKV) is taken as the interpolation accuracy measure. The spatial simulated annealing exhibits a lower AKV as compared to the Genetic algorithm when adding/removing the optimal/redundant locations from the monitoring network.

An active learning Kriging-based method combining the weight information entropy function and the adaptive candidate sample pool

The reliability analysis methods based on the Kriging model have been explored to update the design of experiments (DoE), but the number of calls to the performance function is high. In this paper, a new active learning method combining the weight information entropy function (WH) and the adaptive candidate sample pool is proposed to improve the reliability analysis efficiency. Based on the information entropy function (H), the learning function WH is constructed to consider Kriging variance and the joint probability density function (PDF). In this proposed method, the sample points can be assigned different weight by the learning function WH according to the important degrees of sample points. The point which not only nears the limit state function (LSF), but also has a high probability density function value and large Kriging variance is assigned more weight than others. To select the sample points with lower confidence level, the adaptive candidate sample pool is generated by Markov Chain Monte Carlo (MCMC) simulation. The Kriging model can be updated efficiently by the proposed method. Four numerical examples and an engineering example with implicit performance function are used to verify the efficiency and accuracy of the proposed method. The results show that the proposed method can significantly improve the computational efficiency of the reliability analysis without losing accuracy.

Application of a semivariogram based on a deep neural network to Ordinary Kriging interpolation of elevation data.

The Ordinary Kriging method is a common spatial interpolation algorithm in geostatistics. Because the semivariogram required for kriging interpolation greatly influences this process, optimal fitting of the semivariogram is of major significance for improving the theoretical accuracy of spatial interpolation. A deep neural network is a machine learning algorithm that can, in principle, be applied to any function, including a semivariogram. Accordingly, a novel spatial interpolation method based on a deep neural network and Ordinary Kriging was proposed in this research, and elevation data were used as a case study. Compared with the semivariogram fitted by the traditional exponential model, spherical model, and Gaussian model, the kriging variance in the proposed method is smaller, which means that the interpolation results are closer to the theoretical results of Ordinary Kriging interpolation. At the same time, this research can simplify processes for a variety of semivariogram analyses.

Nonmyopic Informative Path Planning Based on Global Kriging Variance Minimization

Remote, hazardous, and extreme exploration missions require robots to be equipped with on-board sensors for rich and heterogeneous information during deployment. In such tasks, path planning can directly affect the quality and quantity of the observations obtained under temporal and energetic constraints. While most informative path planners can only plan for a short horizon ahead of time (referred as to myopic planners), we propose a novel planner that is capable of planning global paths in which the nonmyopic exploration efficiency is optimized. To achieve this, a novel sampling algorithm named MPE is proposed to adaptively sample landmarks that are associated with high information capacity, in order to minimize the global Kriging variance. The traverse path for the landmarks is then obtained by the IPP-MPE algorithm for minimizing the overall traveling cost. The algorithm is flexible enough to be applied to various information acquisition tasks. The tractable computational cost allows the horizon to be long enough for scene coverage. The algorithm was deployed on a real robot for accomplishing tactile based object searching tasks, which shows superior efficiency compared to the myopic planner baseline. Last, the complexity and other theoretical analysis of the algorithm is provided.

Facilitating mapping and understanding of within‐vineyard variation in fruit composition using data pooled from multiple vineyards

Background and Aims A large number of fruit samples is required for adequate variogram estimation, making the development of prescriptive maps for vineyard management cost prohibitive for most growers. The project assessed the efficacy of aggregating samples from multiple vineyards, over multiple years, to estimate a ‘common’ variogram that could be generated and applied more efficiently. Methods and Results Fifteen hundred berry samples were collected over 3 years (2017–2019) in four vineyards in California for analysis of fruit composition and spatial variability. Maps were produced for anthocyanins, malic acid and β-damascenone in each vineyard using four separate aggregations of samples and showed only subtle changes in patterns of spatial variability in any of the three analytes assessed. A common variogram generated without points from the vineyard to be mapped indicated lower kriging variances over 100 simulations and was able to correctly classify up to 70% of sample values. Conclusions The use of a common variogram in describing spatial variability in vineyards adds important statistical support to the generation of robust maps that could be used for targeted vineyard management. Grower collaboration across multiple regional vineyards could therefore improve mapping support for all involved. Though high-density sampling may still be required in some cases, once stable zones of fruit quality have been characterised, the sample size could potentially be reduced in subsequent years. Significance of the Study Maps produced from combined datasets collected from multiple vineyards and years could provide growers and wineries more confidence in zonal management by showing the temporal stability of the spatial variability of several aspects of fruit quality.

Robust and efficient uncertainty quantification for extreme events that deviate significantly from the training dataset using polynomial chaos-kriging

This study presents the strengths of polynomial chaos-kriging (PCK), a new surrogate model that merges polynomial chaos extension (PCE) and Gaussian process with kriging variance. This combination enabled streamflow prediction for extreme events that deviated significantly from the trained data space, and allowed for quantifying predictive uncertainty robustly and efficiently. The uncertainty quantification results to eight testing flood events through a modeling framework that applies generalized likelihood uncertainty estimation (GLUE) to surrogate models are as follows. (1) PCK outperformed PCE and ordinary kriging (OK) in mimicking predictive and sensitive behaviors of the original model with a smaller-sized training dataset. (2) Three surrogate models trained on the identical dataset exhibited equivalent predictability with the original model for six smaller events similar to their training data space. However, for two extreme events, which differed significantly from the training set, only PCK was found to accurately predict the hydrograph and flood peaks. (3) Since two types of acceptance thresholds, defined here as “accuracy-aimed” or “efficiency-aimed” threshold, have their own pros and cons, the type and size of the threshold should be determined depending on the availability of computational resources and the degree of accuracy needed. (4) A new “performance score” is proposed here to assess the overall performance of the surrogate models. This compensates for situations in which the performance of a surrogate model can be misjudged through individual indices of efficiency or accuracy in the process of uncertainty quantification.

Fair train-test split in machine learning: Mitigating spatial autocorrelation for improved prediction accuracy

Machine learning supports prediction and inference in multivariate and complex datasets where observations are spatially related to one another. Frequently, these datasets depict spatial autocorrelation that violates the assumption of identically and independently distributed data. Overlooking this correlation result in over-optimistic models that fail to account for the geographical configuration of data. Furthermore, although different data split methods account for spatial autocorrelation, these methods are inflexible, and the parameter training and hyperparameter tuning of the machine learning model is set with a different prediction difficulty than the planned real-world use of the model. In other words, it is an unfair training-testing process. We present a novel method that considers spatial autocorrelation and planned real-world use of the spatial prediction model to design a fair train-test split. Demonstrations include two examples of the planned real-world use of the model using a realistic multivariate synthetic dataset and the analysis of 148 wells from an undisclosed Equinor play. First, the workflow applies the semivariogram model of the target to compute the simple kriging variance as a proxy of spatial estimation difficulty based on the spatial data configuration. Second, the workflow employs a modified rejection sampling to generate a test set with similar prediction difficulty as the planned real-world use of the model. Third, we compare 100 test sets' realizations to the model's planned real-world use, using probability distributions and two divergence metrics: the Jensen-Shannon distance and the mean squared error. The analysis ranks the spatial fair train-test split method as the only one to replicate the difficulty (i.e., kriging variance) compared to the validation set approach and spatial cross-validation. Moreover, the proposed method outperforms the validation set approach, yielding a minor mean percentage error when predicting a target feature in an undisclosed Equinor play using a random forest model. The resulting outputs are training and test sets ready for model fit and assessment with any machine learning algorithm. Thus, the proposed workflow offers spatial aware sets ready for predictive machine learning problems with similar estimation difficulty as the planned real-world use of the model and compatible with any spatial data analysis task.

Multivariate geostatistical modeling of seismic data: Case study of the Late Pleistocene paleodelta architecture (SW off-shore Hainan Island, south China sea)

This study investigates a Late Pleistocene paleodelta located on the northwestern shelf of South China Sea, in Beibu Gulf southwest offshore Hainan Island, and details data acquisition to generate reliable 3D geometric models. High-resolution seismic profiles and geostatistical interpolation tools, combined with assumptions about paleo-sea-level changes, allow examination of paleodelta geometrical features. This paper presents methods and procedures that generate morphological models of seismic reflector surfaces and sediment thicknesses. Reliable model-based maps require several trials to optimize various kriging parameters, in particular semivariogram interpretation. Two semivariogram models, the widely-used spherical and nested Gaussian-power methods, are described and applied. Uncertainty among directionally undersampled seismic data is here improved by reduced kriging variance and validation by “goodness-of-fit”. As the investigated paleodelta extends beyond the profile area, an innovative method for data extrapolation is applied based on paleogeographic and paleoeustatic data. Obtained results allow estimation of paleodelta volume and mass and riverine annual sediment discharge to reconstruct Gulf paleogeographic evolution and pathways of sediment distribution.

An efficient hybrid reliability analysis method based on active learning Kriging model and multimodal‐optimization‐based importance sampling

AbstractA novel method is proposed, which aims to solve rare‐event hybrid reliability problems with random and interval variables, where the performance function has various failure zones. It combines the active learning Kriging (ALK) model with importance sampling (IS) and evolutionary multimodal‐based multiobjective optimization (EMO‐MMO). The surrogate limit state surfaces (LSS) for the upper and lower failure probability bounds are respectively defined considering the Kriging variance. Failure candidate solutions located in different failure regions are generated by the EMO‐MMO method. Subsequently, all the most probable failure points (MPPs) are identified from those candidate solutions. The IS samples are simulated around the MPPs using the MPP‐based IS method. The IS samples located in unimportant regions are removed in order to improve the efficiency of approximating the surrogate LSSs. The optimal training points are selected from the truncated IS samples to update the Kriging model. After several training iterations, the surrogate LSSs are convergent. Ultimately, a reliable and unbiased estimation of the upper and lower failure probability bounds is provided. The performance of the ALK‐EMO‐IS‐HRA approach is verified through five application examples.

ESTIMATING THE UNCERTAINTY OF KRIGING ESTIMATES: A PRACTICAL REVIEW AND THE PROPOSAL OF TWO NOVEL-APPROACHES

This paper presents two novel kriging-based methods to measure the uncertainty associated with geostatistical estimates. Both approaches combine the Kriging Variance (KV), which is a good summary of the spatial configuration of the data, with a second component which locally measures the data dispersion. The first proposed method, referred to as the Combined Index (CI), merges KV with the local conditional distribution estimated by median Indicator Kriging. The other method is named Combined Variance II (CVII) and it is an adaptation of the originally proposed Combined Variance, where the equation used to merge KV and the Interpolation Variance was replaced. The two proposed methods are compared to other approaches available in the literature in a case study using the widely known Walker Lake dataset. The results indicate that both proposed methods outperformed the other tested solutions. The CVII index outperformed its original formulation in all the tested scenarios.

Optimal spatial sampling design for monitoring potentially toxic elements pollution on urban green space soil: A spatial simulated annealing and k-means integrated approach

Sampling design in soil science is critical because the lack of reliable methods and collecting samples requires tremendous work and resources. The aims were to obtain an optimal sampling design for assessing potentially toxic elements pollution using pilot Pb soil samples from the urban green space area of Shanghai, China. Two general steps have been used. The first step is to determine the optimum sample size against improving the prediction accuracy and monitoring costs using the spatial simulated annealing (SSA) algorithm. Secondly, we evaluated their likely placement of new extra sampling points by integrated SSA with k-means (SSA+ k-means) and expert-based (SSA+ expert-based) sampling methods. The improvement of sampling design by the integrated sampling approaches was evaluated using mean kriging variance (MKV), root mean square error (RMSE), and mean absolute percentage error (MAPE). The findings indicated that adding and placing 350 new monitoring points upon the existing sampling design by SSA increased the prediction accuracy by 64.35%. The MKV for the optimized SSA+ k-means sample was lower than by 4.12 mg/kg, 9.46 mg/kg compared with locations optimized by SSA and SSA+ expert-based method, respectively. Optimizing new sampling locations by SSA+ k-means sampling method was reduced MAPE by 9.26% and RMSE by 7.13 mg/kg compared to optimizing by SSA alone. However, there was no improvement in placing the new sampling points in SSA+ expert-based sampling method; instead, it increased the error by 8.11%. This paper shows integrating optimization approaches to evaluate the existing sampling design and optimize a new optimal sampling design.

Parameterization of Pilot Point Methodology for Supplementing Sparse Transmissivity Data

Pilot point methodology (PPM) permits estimation of transmissivity at unsampled pilot points by solving the hydraulic head based inverse problem. Especially relevant to areas with sparse transmissivity data, the methodology supplements the limited field data. Presented herein is an approach for estimating parameters of PPM honoring the objectives of refinement of the transmissivity (T) interpolation and the model calibration. The parameters are the locations and number of pilot transmissivity points. The location parameter is estimated by defining a qualifying matrix Q comprising weighted sum of the hydraulic head-sensitivity and the kriging variance fields. Whereas the former component of Q promotes the model calibration, the latter one leads to improved T interpolation by locating pilot points in un-sampled tracts. Further, a three-stage methodology is proposed for an objective determination of the number of pilot points. It is based upon sequential upgradation of the Variogram as the pilot points are added to the data base, ensuring its convergence with the head-based optimal Variogram. The model has been illustrated by applying it to Satluj-Beas interbasin wherein the pumping test data is not only sparse, but also unevenly distributed.

Using Spatial Uncertainty to Dynamically Determine UAS Flight Paths

In the aftermath of radiation or chemical accidents, responders must rapidly map out regions of contamination as quickly and accurately as possible. One important and relevant statistical method for this kind of disaster response is spatial kriging, which makes predictions based on incomplete knowledge of spatially referenced observations. Given an Unmanned Aerial System (UAS) equipped with radiation sensors, we develop a spatial statistics-based approach to optimally map out a contamination field over a geographic region. In this article, we evaluate three approaches to UAS mapping: a Variance Driven Sampling (VDS) approach that minimizes kriging variance, a more computationally intensive Hybrid Entropy Search (HES), and a baseline Levy Flight search. Considering limited UAS range, we also implement a restricted version of these approaches that only considers nearby points. We find that HES is optimal for small numbers of sampled points with the restricted versions of HES and VDS becoming optimal for larger samples. Ultimately, the best method is dependent on the number of samples to be taken, with each method providing clear benefits over a random search in terms of both mean squared error and path length. We demonstrate the advantages of our methodology using actual radiation field test data from the Idaho National Lab.

Quantificando a incerteza estratigráfica na modelagem de águas subterrâneas para projeto de infraestrutura

A methodology to quantify the stratigraphic uncertainty of the aquifer bottom is used in assessing the impact of sheet piles on the water table in the vicinity of a motorway. The method includes uncertainties on model parameters and depth of sheet piles. While parameter uncertainty dominates when simulating the existing groundwater conditions, the stratigraphic uncertainty dominates when predicting the impacts of the motorway sheet piles. The uncertainty of groundwater head predictions is a key element in risk assessments for infrastructure projects, where groundwater affects construction costs. While geological structural uncertainty often is a dominating source of uncertainty in large-scale water-resource studies, stratigraphic uncertainty (defined as the uncertainty of the location of geological boundaries between units/facies) becomes more important for small-scale infrastructure projects. The methodologies used so far for handling stratigraphic uncertainty typically use a somewhat simplistic approach basing the uncertainties of the geological surfaces on the kriging variance. The proposed methodology has two novel elements. Firstly, the uncertainties in the geological interpretation of the borehole data are explicitly included. Secondly, conditional sequential Gaussian simulation (sGs) is used to quantify the uncertainties. The advantage of sGs over standard kriging-based approaches is that sGs allows descriptions of small-scale variations that are crucial in some contexts. The methodology was tested on a case site in Denmark where a new motorway has been constructed below the water table and with sheet-pile walls designed to penetrate the aquifer down to 1 m above the aquifer bottom.