Cokriging Model Research Articles

Geostatistical Approach for Groundwater Quality Evaluation in Zarin Abad Plain, Iran

Background and Purpose: This study was undertaken, first, to investigate the hydrogeological setting of the study area and geophysical data, second to examine the general nature of the groundwater quality. In this regard, ordinary Kriging, Co-Kriging, and Inverse Weighted Distance (IWD) strategies were applied to develop spatial variability maps, and study the fluctuations in groundwater quality parameters in Zarin Abad plain, Zanjan Province, Iran in 2017-2018.
 Materials and methods: To inquire the groundwater quality parameters, samples were provided from 61 shallow and deeply drilled observed wells in Zarin Abad Goltapeh plain. The studies were carried out by using geostatistical methods to find out the most applicable method, which can be used to develop spatial variability maps in order to study the changes in groundwater quality parameters (Na+, K+, Ca2+, Mg2+, SO42-, HCO3-, Cl- and EC). The local geophysical, geological, and hydrogeological surveys were precisely accomplished to specify the architecture of various subsurface geological horizons. In addition, a geophysical investigation with a Schlumberger configuration was performed in the study region for the purpose of field data generation.
 Results: Based on key results, the values of electrical conductivity (EC) were recorded within the range of 480 and 6580 μS/cm. The order of major cations and anions were Na+>Ca2+>Mg2+ and SO42->Cl->HCO3-, respectively. It is worthwhile mentioning that groundwater salinity was found to be dependent upon factors, such as water long residence time and minerals dissolution.
 Conclusion: To assess the spatial distribution in groundwater parameters, the variable mode was used. The results obtained from Kriging, Co-Kriging, and IDW methods were then evaluated by the error indices of RMSE and MAE. Co-Kriging Model was the most optimal approach in studying the spatial variation of groundwater quality parameters.

Cokriging based statistical approximation model for forecasting ionospheric VTEC during high solar activity and storm days

The ionospheric storm is a space climate based event that takes place as a function of time, season and location. In spite of that, it is not yet possible to state how the ionosphere will react to a given space climate event since the measurements of the location, time of observation, amount of disturbances, and type of storm vary from event to event. Nevertheless, statistical analysis of observations in terms of Total Electron Content (TEC) enables one to calculate and forecast a most likely upcoming disturbed TEC, based on predictions of geomagnetic activity. The forecast of disturbed TEC is important and it helps users to know how much it affects the satellite communication systems. In this paper, GPS Vertical TEC (VTEC) data during ionospheric storm has been analyzed and storm-influenced features in the VTEC have been studied. Input parameters such as, location (Latitude & Longitude), Time, F10.7, Ap and Dst indices are collected from NASA website and its corresponding TEC values are used for construction of Cokriging (COK) based statistical approximation model. Prediction of VTEC is performed during storm and high solar activity days and for this purpose previous one-week data of different regions are used. The VTEC prediction is performed using COK model during high solar activity (2015) for IISC station. From the prediction, it is found that, our proposed COK model gives good prediction with RMSE of 6.5325 TECU, Correlation Coefficient as 0.9643 and Mean accuracy of 5.1752 TECU. Also VTEC prediction is performed over three different latitude regions during storm days (2003) and from the predicted results it is noted that, in high latitude region RMSE is 6.1848 TECU, correlation coefficient is 0.3958 and mean accuracy is 4.8105 TECU, in mid latitude RMSE is 12.7957 TECU, correlation coefficient is 0.9249 and mean accuracy is 9.2674 TECU and in low latitude RMSE is 11.5348 TECU, correlation coefficient is 0.9299 and mean accuracy is 9.2116 TECU. The results show that COK model provides good forecasting at low and mid latitude regions. Finally, prediction results of COK model is compared with the Empirical Orthogonal Function (EOF) and IRI-Plas-2017 models over Sutherland (−32.21°N, 20.81°E) station in South Africa during storm conditions. It is found that the average RMSE of EOF model is 5.91 TECU, average RMSE of the IRI-Plas-2017 model is 8.2079 TECU and average RMSE of COK model is 5.5971 TECU. Hence the prediction results show that COK model performs 5.6% better than the EOF model and 46.64% better than the IRI-Plas-2017 model. These results of the proposed COK model is encouraging and useful for indicating storm and high solar activity.

Multi-fidelity uncertainty propagation using polynomial chaos and Gaussian process modeling

The polynomial chaos (PC) method has been widely studied and applied for uncertainty propagation (UP) due to its high efficiency and mathematical rigor. However, the straightforward application of PC on the computationally expensive and highly complicated model for UP might be too costly. Therefore, a multi-fidelity PC approach using the Gaussian process modeling theory is developed in this work, by which the classic multi-level co-kriging multi-fidelity modeling framework in the deterministic domain is extended to the stochastic one. Meanwhile, taking advantage of the Gaussian process modeling theory, the strategies for response models with hierarchical and non-hierarchical fidelity are both addressed within the proposed multi-fidelity PC approach. The effectiveness and relative merit of the proposed method are demonstrated by comparative studies on several numerical examples for UP. It is noticed that the proposed approach can significantly improve the accuracy and robustness of UP compared to the commonly used addition correction-based multi-fidelity PC method; compared to co-kriging, the accuracy and robustness are generally also improved, especially for problems with unsymmetric distributed random input and large variation. An engineering robust aerodynamic optimization problem further verifies the applicability of the proposed multi-fidelity PC method.

A Co-kriging Multi-fidelity Surrogate Model Assisted Robust Optimization Approach

A Co-kriging Multi-fidelity Surrogate Model Assisted Robust Optimization Approach

A radial basis function-based multi-fidelity surrogate model: exploring correlation between high-fidelity and low-fidelity models

In computational simulation, a high-fidelity (HF) model is generally more accurate than a low-fidelity (LF) model, while the latter is generally more computationally efficient than the former. To take advantages of both HF and LF models, a multi-fidelity surrogate model based on radial basis function (MFS-RBF) is developed in this paper by combining HF and LF models. To determine the scaling factor between HF and LF models, a correlation matrix is augmented by further integrating LF responses. The scaling factor and relevant basis function weights are then calculated by employing corresponding HF responses. MFS-RBF is compared with Co-Kriging model, multi-fidelity surrogate based on linear regression (LR-MFS) model, CoRBF model, and three single-fidelity surrogates. The impact of key factors, such as the cost ratio of LF to HF models and different combinations of HF and LF samples, is also investigated. The results show that (i) MFS-RBF presents a better accuracy and robustness than the three benchmark MFS models and single-fidelity surrogates in about 90% cases of this paper; (ii) MFS-RBF is less sensitive to the correlation between HF and LF models than the three MFS models; (iii) by fixing the total computational cost, the cost ratio of LF to HF models is suggested to be less than 0.2, and 10–80% of the total cost should be used for LF samples; (iv) the MFS-RBF model is able to save an average of 50 to 70% computational cost if HF and LF models are highly correlated.

Multi-Level and High-Resolution Fracture Modeling, Simulation and History Matching in Field-scale Reservoir Study

Carbonate formations are usually highly fractured. The fractures exist at all length scales ranging from microscopic fissures to kilometer-size structures. Consequently, it is vitally important to rigorously characterize the multi-scale fracture system and to accurately model the fluid flow in such highly fractured porous media. In this paper, we will present an integrated workflow for characterization of natural fracture system and accurate numerical modeling of fluid flow in fractured rocks.The fracture characterization usually involves the integration of multiple data sources including core analysis, image logs and seismic interpretations etc., at different scales. The fracture mappings are thus required to perform at different length scales and resolutions. In this study, we present a systematic approach that characterizes the actual fracture distribution with seismic and log interpretations. Firstly, the small-scale fractures are interpreted using image logs and seismic anisotropy analysis with a co-Kriging and stochastic modeling approach. The small-scale fractures are initially constructed in an explicit fashion, and are then incorporated in the matrix medium using a flow-based upscaling procedure. Secondly, the large-scale fractures are mapped out from post-stack seismic data with ant-tracking and maximum curvature analysis. The reservoir is then gridded into unstructured tetrahedral or prism meshes to fully resolve these large fractures. This discrete fracture model (DFM) is employed in a reservoir simulation study based on such grid system.We applied the procedure for simulation study and history match on a portion of a carbonate gas reservoir in Longwangmiao formation in Southwest China. Currently, there are 7 horizontal and 2 vertical wells in operation. Due to the connection of some large-scale fractures to edge water, early water breakthrough and drastic water invasion were observed. We partitioned the reservoir block into 0.6 million cells, in which 153 large-scale and thousands of small-scale fractures are fully resolved. The simulation study was performed and automated history match was successfully achieved using our extended Ensemble Karman Filter algorithm based on a parameterized description of the fractures that are explicitly represented in DFM. This proposed overall procedure is able to efficiently capture such extreme flow dynamics and achieve significantly better history match results.

Application and comparison of the cokriging and the fractal model for identifying geochemical anomalies in Janja area, SE Iran

In this study, the cokriging model was used for determining the geochemical concentrations in Janja, SE Iran. Cokriging is a multivariate prediction method in spatial statistics which considers the spatial dependence as well as predicting the dependence between variables. In addition, fractal concentration-area (C-A) model was used for determining the geochemical anomalies in Janja region. Eight elements (Au, Cu, Mn, Zn, Fe, As, Mo and Pb) from 300 stream sediment samples were used to identify geochemical anomalies. The geochemical maps obtained from cokriging and fractal modelling showed that geochemical anomalies for copper and gold are located in the north, NW, and centre of the study area. Comparing the anomalies maps of the elements showed a correlation between elements of gold, copper, molybdenum, and iron in the region. Also, a negative correlation was determined between arsenic with gold and copper elements and there was not any correlation between the gold and copper elements of the region with manganese, lead, and zinc. For examining the results of the studies more closely, some of the data were analysed and evaluated using concentration-area (C-A) fractal model.

A multi-fidelity modelling approach to the statistical inference of the equivalent initial flaw size distribution for multiple-site damage

A new methodology for the statistical inference of the Equivalent Initial Flaw Size Distribution (EIFSD) using the Dual Boundary Element Method (DBEM) is proposed. As part of the inference, Bayesian updating is used to calibrate the EIFS based on data obtained from simulated routine inspections of a structural component from a fleet of aircraft. An incremental crack growth procedure making use of the DBEM is employed for the modelling of the simultaneous growth of cracks in the structure due to fatigue. Multi-fidelity modelling, in the form of Co-Kriging, is used to create surrogate models that act in place of the DBEM model for the expensive Monte Carlo sampling procedure required for the statistical inference of the EIFSD. The proposed methodology is applied to a numerical example featuring a long fuselage lap joint splice in the presence of Multiple Site Damage (MSD). Results show that the EIFSD can be accurately estimated within 10% error with data from just 50 inspections. The employed Co-Kriging models proved to be effective substitutes for the DBEM model, providing significant reductions in the computational cost associated with the implementation of the proposed statistical inference methodology.

A Novel Combination Co-Kriging Model Based on Gaussian Random Process

Co-Kriging (CK) modeling provides an efficient way to predict responses of complicated engineering problems based on a set of sample data obtained by methods with varying degree of accuracy and computation cost. In this work, the Gaussian random process (GRP) is introduced to construct a novel combination CK model (CK-GRP) to improve the prediction accuracy of the conventional CK model, in which all the sample information provided by different correlation models is well utilized. The features of the new model are demonstrated and evaluated for a numerical case and an engineering application. It is shown that the CK-GRP model proposed in this work is effective and can be used to improve the prediction accuracy and robustness of the CK model.

Enhancing the Spatial Variability of Soil Salinity Indicators by Remote Sensing Indices and Geo-Statistical Approach

Soil salinization is considered limiting factor for crop production and land management for dry land in Sudan, its spatial variation is affected by different factors of soil properties, vegetation and environment hence its interaction formulate the planning for successful sustainable agriculture in salt affected soils. This study aims to evolve the spatial prediction of soil salinity indicators by integrated remote sensing indices and geo-statistical cokriging model. Soil samples were collected from 476 square kilometer area in salt affected area, the samples were analyzed following standard procedures for electrical conductivity, sodium adsorption ratio, hydrogen ions and saturation percentage. Information of vegetation status identified by Normalized Difference Vegetation Index (NDVI) and soil salinization by Salinity index and brightness index were used and utilized for prediction of the soil parameters variability by cokriging model. It was found that the method was resulted in high accuracy based on RMSE and enhances the soil spatial variability assessment and provides significant interaction of different variables and indices in the landscape.

The rapid development of bespoke small unmanned aircraft

ABSTRACTThe ability to quickly fabricate small unmanned aircraft system (sUAS) through Additive Manufacturing (AM) methods opens a range of new possibilities for the design and optimisation of these vehicles. In this paper, we propose a design loop that makes use of surrogate modelling and AM to reduce the design and optimisation time of scientific sUAS. AM reduces the time and effort required to fabricate a complete aircraft, allowing for rapid design iterations and flight testing. Co-Kriging surrogate models allow data collected from test flights to correct Kriging models trained with numerically simulated data. The resulting model provides physically accurate and computationally cheap aircraft performance predictions. A global optimiser is used to search this model to find an optimal design for a bespoke aircraft. This paper presents the design loop and a case study which demonstrates its application.

Comparison of artificial neural networks, geographically weighted regression and Cokriging methods for predicting the spatial distribution of soil macronutrients (N, P, and K)

Soil macronutrients (i.e. nitrogen (N), phosphorus (P), and potassium (K)) are important soils components and knowing the spatial distribution of these parameters are necessary at precision agriculture. The purpose of this study was to evaluate the feasibility of different methods such as artificial neural networks (ANN) and two geostatistical methods (geographically weighted regression (GWR) and cokriging (CK)) to estimate N, P and K contents. For this purpose, soil samples were taken from topsoil (0–30 cm) at 106 points and analyzed for their chemical and physical parameters. These data were divided into calibration (n = 84) and validation (n = 22). Chemical and physical variables including clay, pH and organic carbon (OC) were used as auxiliary soil variables to estimate the N, P and K contents. Results showed that the ANN model (with coefficient of determination R 2 = 0.922 and root mean square error RMSE = 0.0079%) was more accurate compared to the CK model (with R 2 = 0.612 and RMSE = 0.0094%), and the GWR model (with R 2 = 0.872 and RMSE = 0.0089%) to estimate the N variable. The ANN model estimated the P with the RMSE of 3.630 ppm, which was respectively 28.93% and 20.00% less than the RMSE of 4.680 ppm and 4.357 ppm from the CK and GWR models. The estimated K by CK, GWR and ANN models have the RMSE of 76.794 ppm, 75.790 ppm and 52.484 ppm. Results indicated that the performance of the CK model for estimation of macro nutrients (N, P and K) was slightly lower than the GWR model. Also, the accuracy of the ANN model was higher than CK and GWR models, which proved to be more effective and reliable methods for estimating macro nutrients.

Gradient-Enhanced Hierarchical Kriging Model for Aerodynamic Design Optimization

AbstractA cokriging model incorporating gradient information and the function value of sample points can reduce the computational cost with a given level of accuracy. In this paper, the hierarchica...

Optimal Design of Winding Transposition of Power Transformer Using Adaptive Co-Kriging Surrogate Model

An adaptive co-Kriging surrogate model, which is numerically more efficient and accurate than a conventional co-Kriging model, is developed, and incorporated into a heuristic optimization algorithm to be applied to the optimal design of transposition of power transformer windings. The sampling data of the proposed adaptive co-Kriging consist of a few expensive and many cheap data to save the computational efforts while increasing modeling accuracy. A criterion on the minimum number of expensive sampling data is investigated to achieve a desired fitting accuracy.

Multi-objective Geometry Optimization of a Gas Cyclone Using Triple-Fidelity Co-Kriging Surrogate Models

Cyclone separators are widely used in a variety of industrial applications. A low-mass loading gas cyclone is characterized by two performance parameters, namely the Euler and Stokes numbers. These parameters are highly sensitive to the geometrical design parameters defining the cyclone. Optimizing the cyclone geometry therefore is a complex problem. Testing a large number of cyclone geometries is impractical due to time constraints. Experimental data and even computational fluid dynamics simulations are time-consuming to perform, with a single simulation or experiment taking several weeks. Simpler analytical models are therefore often used to expedite the design process. However, this comes at the cost of model accuracy. Existing techniques used for cyclone shape optimization in literature do not take multiple fidelities into account. This work combines cheap-to-evaluate well-known mathematical models of cyclones, available data from computational fluid dynamics simulations and experimental data to build a triple-fidelity recursive co-Kriging model. This model can be used as a surrogate with a multi-objective optimization algorithm to identify a Pareto set of a finite number of solutions. The proposed scheme is applied to optimize the cyclone geometry, parametrized by seven design variables.

A Critical Assessment of Kriging Model Variants for High-Fidelity Uncertainty Quantification in Dynamics of composite Shells

This paper presents a critical comparative assessment of Kriging model variants for surrogate based uncertainty propagation considering stochastic natural frequencies of composite doubly curved shells. The five Kriging model variants studied here are: Ordinary Kriging, Universal Kriging based on pseudo-likelihood estimator, Blind Kriging, Co-Kriging and Universal Kriging based on marginal likelihood estimator. First three stochastic natural frequencies of the composite shell are analysed by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The comparative assessment is carried out to address the accuracy and computational efficiency of five Kriging model variants. Comparative performance of different covariance functions is also studied. Subsequently the effect of noise in uncertainty propagation is addressed by using the Stochastic Kriging. Representative results are presented for both individual and combined stochasticity in layer-wise input parameters to address performance of various Kriging variants for low dimensional and relatively higher dimensional input parameter spaces. The error estimation and convergence studies are conducted with respect to original Monte Carlo Simulation to justify merit of the present investigation. The study reveals that Universal Kriging coupled with marginal likelihood estimate yields the most accurate results, followed by Co-Kriging and Blind Kriging. As far as computational efficiency of the Kriging models is concerned, it is observed that for high-dimensional problems, CPU time required for building the Co-Kriging model is significantly less as compared to other Kriging variants.

Building accurate radio environment maps from multi-fidelity spectrum sensing data

In cognitive wireless networks, active monitoring of the wireless environment is often performed through advanced spectrum sensing and network sniffing. This leads to a set of spatially distributed measurements which are collected from different sensing devices. Nowadays, several interpolation methods (e.g., Kriging) are available and can be used to combine these measurements into a single globally accurate radio environment map that covers a certain geographical area. However, the calibration of multi-fidelity measurements from heterogeneous sensing devices, and the integration into a map is a challenging problem. In this paper, the auto-regressive co-Kriging model is proposed as a novel solution. The algorithm is applied to model measurements which are collected in a heterogeneous wireless testbed environment, and the effectiveness of the new methodology is validated.

Integrated workflow approach to static modeling of Igloo R3 reservoir, onshore Niger Delta, Nigeria

We constructed a 3D static model of the R3 reservoir at the Igloo Field, Onshore Niger Delta, by integrating the 3D seismic volume, geophysical well logs, and core petrophysical data. In this model, we used a combined petrophysical-based reservoir zonation and geostatistical inversion of seismic attributes to reduce vertical upscaling problems and improve the estimation of reservoir properties between wells. The reservoir structural framework was interpreted to consist of three major synthetic faults; two of them formed northern and southern boundaries of the field, whereas the other one separated the field into two hydrocarbon compartments. These compartments were pillar gridded into 39,396 cells using a [Formula: see text] dimension over an area of [Formula: see text]. Analysis of the field petrophysical distribution showed an average of 21% porosity, 34% volume of shale, and 680-mD permeability. Eleven flow units delineated from a stratigraphic modified Lorenz plot were used to define the reservoir’s stratigraphic framework. The calibration of acoustic impedance using sonic- and density-log porosity showed a 0.88 correlation coefficient; this formed the basis for the geostatistic seismic inversion process. The acoustic impedance was transformed into reservoir parameters using a sequential Gaussian simulation algorithm with collocated cokriging and variogram models. Ten equiprobable acoustic impedance models were generated and further converted into porosity models by using their bivariate relationship. We modeled the permeability with a single transform of core porosity with a correlation coefficient of 0.86. We compared an alternative model of porosity without seismic as a secondary control, and the result showed differences in their spatial distributions, which was a major control to fluid flow. However, there were similarities in their probability distribution functions and cumulative distribution functions.

A Comparative Study on the Spatial Statistical Models for the Estimation of Population Distribution

This study aims to accurately estimate population distribution more specifically than administrative unites using a RK (Regression-Kriging) model. The RK model is the areal interpolation technique that involves linear regression and the Kriging model. In order to estimate a population’s distribution using a sample region, four different models were used, namely; a regression model, RK model, OK (Ordinary Kriging) model and CK (Co-Kriging) model. The results were then compared with each other. Evaluation of the accuracy and validity of evaluation analysis results were the basis RMSE (Root Mean Square Error), MAE (Mean Absolute Error), G statistic and correlation coefficient (ρ). In the sample regions, every statistic value of the RK model showed better results than other models. The results of this comparative study will be useful to estimate a population distribution of the metropolitan areas with high population density

Spatial distribution of Pb in urban soil from Port Pirie, South Australia

Abstract Exposure to Lead (Pb) from soil and dust in urban areas continues to threaten human health especially that of children, who are more likely to ingest a higher Pb fraction from contaminated soil. Geostatistical approaches were evaluated for the prediction of soil Pb concentrations. Ordinary kriging (OK) and co-kriging (CoK) were used to predict spatial distributions of both soil Pb concentrations and bioaccessibility. The relative accuracies of the developed predictive models were assessed using of coefficients of determination ( R 2 ) and the root mean square error (RMSE) based on the cross validation method. The accuracy of the final model was evaluated via comparison between predicted and traditional soil measurements. While OK models ( n = 73 ) were unreliable for predicting a Pb concentration surface, correlation between Pb and Zn was a useful mechanism for obtaining better predictions of soil Pb using CoK. The CoK model of log transformed Pb with Zn resulted in the best fitted model ( R 2 = 0.63 ). The percentage relative improvement (RI) for this model was 39% which suggested relatively reliable prediction accuracy. A probability kriging (PK) surface was used to describe the probability of bioaccessibility exceeding a threshold of 17% as an indicator for potential human risk. The areas with highest probability of exceeding the threshold were in agreement with previous risk area divisions related to blood Pb (BPb) levels for children under 5 years of age. The results of this research confirmed that geostatistical methods had the ability to rapidly estimate soil Pb distributions for environmental health risk assessment.