Fuzzy Kriging Research Articles

Performance of Spatial Distribution Quality by Ordinary Fuzzy Kriging for Soil Properties under Uncertainty

Performance of Spatial Distribution Quality by Ordinary Fuzzy Kriging for Soil Properties under Uncertainty

Uncertainty Analysis of Thermodynamic Variables of Fluid Inclusions: A Deposit-Scale Spatial Exploratory Data Modeling through Fuzzy Kriging

Uncertainty Analysis of Thermodynamic Variables of Fluid Inclusions: A Deposit-Scale Spatial Exploratory Data Modeling through Fuzzy Kriging

Reservoir Geostatistical Estimates of Imprecise Information Using Fuzzy-Kriging Approach

SummaryThis paper presents a methodology for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. Given certain known well-log data in a geological region, the Kriging methodology is used to estimate or predict spatial phenomena at nonsampled locations from the estimated random function. The approach assumes that the data are accurate and precise, and the random function is generated from a thorough descriptive analysis of the known data set. Regarding the assumptions considered in classic Kriging, it is realistic to assume that spatial data contain a certain amount of imprecision, mostly because of measurement errors, and information is lacking to properly assess a unique random-function model. A methodology is presented for the geostatistical estimation of reservoir properties to handle uncertainties in observation and modeling. A combination of regular, or classic, Kriging and the fuzzy-logic method is proposed. As such, imprecise input data and variogram parameters are modeled on the basis of fuzzy-logic theory, while the predictions and variances are computed from Kriging analysis characterized by membership functions. Last, an optimization method is included to solve the constrained fuzzy-nonlinear-equation system. The proposed methodology was implemented, and a user-friendly integrated tool was developed, which enables the user to create a grid structure on the basis of the input data, conduct statistical analysis, and run fuzzy Kriging for various problems. We used the tool to run a test case using the SPE 10 (SPE Comparative Solution Project, Model-II 2000) porosity data. With the fuzzy-Kriging methodology, two maps are generated with upper-bound values and lower-bound values. Compared with true data, the upper-bound map trends to include higher values better, while the lower-bound map trends to include lower-value parts better. In addition, a case study has been conducted using measured core-permeability data in a heterogeneous reservoir to demonstrate the viability of the technology.

Fuzzy tonnage-average grade model based on extension principle

Due to the nature of the geological and mining activities, different input parameters in the grade estimation and mineral resource evaluation are always tainted with uncertainties. It is possible to investigate the uncertainties related to the measurements and parameters of the variogram model using the fuzzy kriging method instead of the kriging method. The fuzzy kriging theory has already been the subject of relatively various research studies but the main weak point in such studies is that the results of the fuzzy estimations are not used in decision-making and planning. A very common, but key, tool of decision-making for mining engineers is the tonnage-average grade models. Under conditions where measurements or/and variogram model parameters are tainted with uncertainties, the tonnage-average grade model will be uncertain as well. Therefore, it is necessary to use the fuzzy tonnage-grade model instead of the crisp ones, and the next analysis steps and decision-makings are done accordingly. In this paper, the computational principles of the fuzzy tonnage-average grade curve and a case study regarding its usage are presented.

Improvement of water table interpolation and groundwater storage volume using fuzzy computations.

The water table is an important piece of data for hydrogeological studies, particularly as input data to groundwater simulation models. Since the accuracy of groundwater simulation models significantly depends on input data, this study highlights the application of fuzzy kriging to improve the accuracy of water table interpolation. The results of the fuzzy kriging approach are compared with common methods in water table interpolation like ordinary kriging, inverse distance weighting (IDW), and Thiessen polygon methods to justify the suitability of the fuzzy kriging. The Gilan and Zanjan plains, located in the northwest of Iran, are used as case study areas. The Gilan Plain is characterized by a dense and regular piezometric network and gentle hydraulic gradient. The longitudinal plain of Zanjan has a sparse and irregular piezometric network and steep hydraulic gradient. Since these plains have different piezometric network configurations, the sensitivity of the interpolation methods to the monitoring point configuration is analyzed. The cross-validation method is employed to validate the accuracy of interpolation methods in water table interpolation. In control points, the average of root-mean-square errors associated with groundwater water table values estimated using fuzzy kriging, ordinary kriging, IDW, and Thiessen polygon methods are obtained to be respectively 1.36, 1.93, 3.49, and 9.10 in the Gilan Plain and 13.60, 22.86, 32.30, and 59.81 in the Zanjan Plain. The results indicate that the fuzzy kriging technique has greater precision in comparison with other methods, especially under the conditions of the sparse piezometric network and steep hydraulic gradient. The results also demonstrate that the used methods generally have higher accuracy in the Gilan Plain with a regular piezometric network than in the Zanjan Plain. Furthermore, Thiessen polygon, IDW, and ordinary kriging methods overestimated water table in comparison with the fuzzy kriging method in our cases. This overestimation may cause large error values in subsequent calculations such as water budget and aquifer storage which play a major role in the appropriate management of water resources.

Computationally efficient approach for identification of fuzzy dynamic groundwater sampling network.

In the present work, uncertainty-based dynamic sampling frameworks were tested for a sampling horizon with multiple time steps. Future concentration values were assumed as fuzzy numbers. Multiple realization-based simulations were used for generation of fuzzy numbers. The first framework considers fuzzy variance reduction. The second framework considers mass estimation error reduction with maximization of spatial coverage of the dynamic sampling network. It is a multi-objective optimization problem with a large number of objectives. In Dhar and Patil (2012), uncertainty-based optimal sampling design model was suggested using Nondominated Sorting Genetic Algorithm-II (NSGA-II) as its optimization algorithm. However, NSGA-II becomes computationally expensive while handling more than three objectives. We extend the previously suggested algorithm for multi-objective sampling network design problems based on NSGA-III framework. Two design frameworks were proposed: one incorporating a simulation model and a fuzzy covariance for minimizing the total contaminant-concentration variance and the other incorporating a simulation model and a fuzzy kriging model in conjunction with an optimization model to minimize the fuzzy mass estimation error and spatial coverage of spatiotemporal sampling locations. NSGA-III was used for solving the sampling network design model. Performances of the proposed frameworks were evaluated for two hypothetical illustrative examples. The results indicate that the proposed design frameworks perform satisfactorily under uncertain system conditions.

FuzzyKrig: a comprehensive matlab toolbox for geostatistical estimation of imprecise information

Using kriging has been accepted today as the most common method of estimating spatial data in such different fields as the geosciences. To be able to apply kriging methods, it is necessary that the data and variogram model parameters be precise. To utilize the imprecise (fuzzy) data and parameters, use is made of fuzzy kriging methods. Although it has been 30 years since different fuzzy kriging algorithms were proposed, its use has not become as common as other kriging methods (ordinary, simple, log, universal, etc.); lack of a comprehensive software that can perform, based on different fuzzy kriging algorithms, the related calculations in a 3D space can be the main reason. This paper describes an open-source software toolbox (developed in Matlab) for running different algorithms proposed for fuzzy kriging. It also presents, besides a short presentation of the fuzzy kriging method and introduction of the functions provided by the FuzzyKrig toolbox, 3 cases of the software application under the conditions where: 1) data are hard and variogram model parameters are fuzzy, 2) data are fuzzy and variogram model parameters are hard, and 3) both data and variogram model parameters are fuzzy.

Considering the Epistemic Uncertainties of the Variogram Model in Locating Additional Exploratory Drillholes

To enhance the certainty of the grade block model, it is necessary to increase the number of exploratory drillholes and collect more data from the deposit. The inputs of the process of locating these additional drillholes include the variogram model parameters, locations of the samples taken from the initial drillholes, and the geological block model. The uncertainties of these inputs will lead to uncertainties in the optimal locations of additional drillholes. Meanwhile, the locations of the initial data are crisp, but the variogram model parameters and the geological model have uncertainties due to the limitation of the number of initial data. In this paper, effort has been made to consider the effects of variogram uncertainties on the optimal location of additional drillholes using the fuzzy kriging and solve the locating problem with the genetic algorithm (GA) optimization method.A bauxite deposit case study has shown the efficiency of the proposed model.

Method for the Evaluation of the Reserve of a Metallurgical Quartz Deposit by Advanced Kriging Techniques

There are several different methodologies for the evaluation of the reserve of a mineral deposit. Traditionally, classic methods of volume computation for mineral deposits have been used to evaluate the reserve according to its commercial quality. Nevertheless, in previous works it has been demonstrated the functionality of the advanced kriging techniques applied to granite batholiths. Comparing fuzzy kriging techniques with universal classical kriging, it was concluded that the fuzzy techniques had good results for planning the selective mining of the ornamental rock deposit. Since geostatistical methods had good results, these techniques are applied to a metallurgical quartz deposit in this work. The objective of the investigation is to develop the most adequate technique for the determination of the different material qualities in this type of deposits, where the chemical composition of the mineral (aluminium oxide and impurity contents) mainly determines the grade. The obtained results show that the compositional kriging technique is more adequate than fuzzy methods for this type of deposits, allowing a sufficiently coherent prediction of the distribution of grades in the deposit. Based on the results, a 3D geostatistical model composed of cubed cells was done as well. The model provides a visual tool, easy to put into practice for the exploitation management. Keeping all this in mind, the selective mining of the reserves would be possible, being that fact necessary to compete in the rising, competitive world market.

Spatial Distribution and Pollution Risk Analysis of Electromagnetic Radiation in Changchun City

To evaluate the electromagnetic radiation pollution in Changchun City and determine the risk of pollution, the probability theory of extreme value distribution is used to build the computational model of electromagnetic pollution risk and discusses their property according to situation that the current urban growth of electromagnetic energy mainly comes from mobile base stations and radio frequency radiation sources. According to the integrated electric field intensity which monitored from Changchun City’s 442 monitoring station by electromagnetic environment monitoring station in Jilin Province, the risk of electromagnetic pollution in Changchun City probability is calculated, ordinary Kriging method is used to draw the map of Changchun City electromagnetic radiation field strength and calculates electromagnetic pollution risk probability. The result indicates that the electromagnetic environment is safe. The value of electric field strength in the office area is higher than other regions. The map drawing by fuzzy Kriging method is more accurately than the map drawing by ordinary Kriging method.

Neuro fuzzy and punctual kriging based filter for image restoration

In this paper, we present a hybrid, image restoration approach. The proposed approach combines the geostatistical interpolation of punctual kriging, artificial neural networks (ANNs), and fuzzy logic based approaches. Images degraded with Gaussian white noise are restored by first utilizing fuzzy logic for selecting pixels that needs kriging. Three fuzzy systems are employed. Both type-I and type-II fuzzy sets in addition with neuro fuzzy classifier (NFC) have been used for the detection of noisy pixels. To avoid edge pixels, a post processing technique is used to check the edge pixel connectivity up to lag 5. If the pixel under consideration is an edge pixel, it is excluded from the fuzzy map and thus not estimated. The concept of punctual kriging is then used to estimate the intensity of a noisy pixel. ANN is employed to minimize the cost function of the kriging based pixel intensity estimation procedure. ANN, in contrast to analytical methodologies, avoids both matrix inversion failure and negative weights problems. Image restoration performance based comparison has been made against adaptive Weiner filter and existing fuzzy kriging approaches. Experimental results using 450 images are used to validate the effectiveness of the proposed approach. Different image quality measures are used to compare the efficacy of the proposed NFC and fuzzy type-II approaches for detecting noisy pixels in conjunction with ANN and kriging based estimation.

Modeling uncertainties in sodium spatial dispersion using a computational intelligence-based kriging method

Classical and geostatistical methods have been used to create continuous surfaces from sampled data. A common geostatistical method is kriging, which provides an accurate estimation based on the existing spatial structure of sample points. However, kriging is sensitive to errors in the input data, the dispersion of the sample points, and the fit of the model to the variogram. The purpose of this research is to develop a new method to address the uncertainties resulting from the input data and choice of model in the kriging method. In our approach, the existing uncertainties in the input data are modeled by fuzzy computations, and the variogram variables are optimized by a genetic algorithm. To test this new hybrid method, sodium contamination values in the Zanjan aquifer were used. The results show a general improvement in accuracy compared with the ordinary kriging method. Consideration of all equations and values in fuzzy computations highlights the complexity of the computation. Herein, the integration problems experienced by other researchers when trying to use fuzzy kriging are resolved.

Evaluation of the reserve of a granite deposit by fuzzy kriging

Abstract This study describes a method for estimating the reserves in an ornamental granite deposit. The method is adapted to the existing optimised mining method and provides a more realistic estimate of commercial quality than conventional methods. Using a batholith geology-mining map, a description of fracturing in the mining area, and underground information provided by continuous-core boreholes, an estimate of reserves was made that featured the following innovations: 1) the use of fuzzy set theory to assign qualities to each block, and 2) the inclusion of fracture anisotropy at the deposit level in the fuzzy mathematical formulation. These innovations enabled quality distributions in the rock mass to be assessed more realistic, by reducing the estimation error caused by assigning a single quality to the primary block and by improving the estimate of the distribution of commercial qualities defined by fracturing. This method produced a satisfactory estimate of the commercial qualities of one of the world's most important granite deposits (located in northwest Spain). The results have been stored in a database–linked to a graphic representation of the blocks (similar to a geographical information system)–which, by providing automatic access to information in regard to the location of the best quality commercial blocks, ultimately optimises mining planning.

A hybrid image restoration approach: Using fuzzy punctual kriging and genetic programming

AbstractWe present an intelligent technique for image denoising problem of gray level images degraded with Gaussian white noise in spatial domain. The proposed technique consists of using fuzzy logic as a mapping function to decide whether a pixel needs to be krigged or not. Genetic programming is then used to evolve an optimal pixel intensity‐estimation function for restoring degraded images. The proposed system has shown considerable improvement when compared both qualitatively and quantitatively with the adaptive Wiener filter, methods based on fuzzy kriging, and a fuzzy‐based averaging technique. Experimental results conducted using an image database confirms that the proposed technique offers superior performance in terms of image quality measures. This also validates the use of hybrid techniques for image restoration. © 2007 Wiley Periodicals, Inc. Int J Imaging Syst Technol, 17, 224–231, 2007

IMAGE ENHANCEMENT BY KRIGING AND FUZZY SETS

A kriging method is presented as a spatial filter for smoothing gray-scale images degraded by Gaussian white noise. The concepts are based on the analysis of semivariances, the linear combination scheme of kriging, and fuzzy sets. Application of fuzzy sets allows a gradual transition between two boundaries of semivariance levels as a criterion for smoothing the pixel values. This fuzzy thresholding also allows some degree of flexibility to suit various desired results for particular problems. Experimental results obtained by the fuzzy kriging filter are smoother and still preserve edges compared with those by the adaptive Wiener filter.

Designing sampling grids from imprecise information on soil variability, an approach based on the fuzzy kriging variance

It is shown how, given a fuzzy formulation of the variogram of a soil property, a fuzzy set of grid spacings that will achieve a target kriging variance can be derived. This fuzzy set of grid spacings can then be defuzzified in a more or less conservative way to define a sampling scheme. The method is illustrated in three examples. The first uses a hypothetical fuzzy variogram. The second is a fuzzy variogram for topsoil organic carbon content in a field, based on the crisp variogram of a related ancillary variable and generalized information on topsoil variability at field scales. The third fuzzy variogram is for topsoil pH, assuming no local knowledge but representing in fuzzy form the range of variability in a collection of variograms for this property in the soil science literature.

Geostatistical regionalization of glacial aquitard thickness in northwestern Germany, based on fuzzy kriging

In many instances hydrogeological parameters obtained by conventional methods for selected localities within an aquifer or an aquitard are not sufficient for adequate regionalization at the scale of the entire layer. Here, we demonstrate an application of the fuzzy kriging method in regionalization of hydrogeological data, in which the set of conventional, crisp values is supplemented by imprecise information subjectively estimated by an expert. It is believed that such an approach eventually may reflect the real-world conditions more closely than a traditional crisp-value approach, because the former does not impose exactness artificially on phenomena which are diffuse by their nature. Spatial interpolation was done for the thickness of one of the major aquitards (till and glaciolacustrine clay) in northwestern Germany. The dataset consists of 329 crisp values from boreholes supplemented by 172 imprecise values defined as fuzzy numbers. It is demonstrated that the reliability of regionalization was higher, compared to regionalization performed with the crisp dataset only. Fuzzy kriging was performed with FUZZEKS (Fuzzy Evaluation and Kriging System) developed at the Ecosystem Research Center at the University of Kiel.

A review of recent developments in geostatistics

This paper reviews developments in geostatistics in the period 1987 to mid-1991. The developments which are regarded as significant by the author fall broadly under six headings: simulation, indicator kriging, interval estimation, applications to hydrocarbon reservoirs and hydrology, incorporation of prior information in spatial estimation, and fuzzy kriging. A summary of significant contributions under each of these headings is given together with an assessment of their importance and application.

Geostatistics utilizing imprecise (fuzzy) information

Fuzzy set theory, specifically fuzzy number arithmetic and the extension principle, is employed to utilize imprecise data in a geostatistical analysis. This formulation is an alternative of soft kriging. Imprecise data are defined as fuzzy numbers and lead to fuzzy experimental and fuzzy theoretical variograms. Then a fuzzy geostatistical operation termed fuzzy kriging type 3, is performed and the kriged estimates are represented as fuzzy numbers. The width of the fuzzy kriged estimate is interpreted as an uncertainty measure which depends on the data and characterizes local inhomogeniety. The estimation variance is also a fuzzy number. The illustrative example refers to permeability field predictions for a soil liner, a typical case where imprecise measurement data may be available and should be utilized.

Imprecise (fuzzy) information in geostatistics

A methodology based on fuzzy set theory for the utilization of imprecise data in geostatistics is presented. A common problem preventing a broader use of geostatistics has been the insufficient amount of accurate measurement data. In certain cases, additional but uncertain (soft) information is available and can be encoded as subjective probabilities, and then the soft kriging method can be applied (Journel, 1986). In other cases, a fuzzy encoding of soft information may be more realistic and simplify the numerical calculations. Imprecise (fuzzy) spatial information on the possible variogram is integrated into a single variogram which is used in a fuzzy kriging procedure. The overall uncertainty of prediction is represented by the estimation variance and the calculated membership function for each kriged point. The methodology is applied to the permeability prediction of a soil liner for hazardous waste containment. The available number of hard measurement data (20) was not enough for a classical geostatistical analysis. An additional 20 soft data made it possible to prepare kriged contour maps using the fuzzy geostatistical procedure.