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1.
A number of criteria based on kriging variance calculations may be used for infill sampling design in geologic site characterization. Searching for the best new sample locations from a set of candidate locations can result in excessive computation time if these criteria and the naive rekriging are used. The relative updated kriging estimate and variance for universal kriging estimation are demonstrated as a simple kriging estimate and variance, respectively. The updated kriging variance is demonstrated as the multiplication of two kriging variances. Using these updated kriging variance equations can increase the computational speed for selecting the best new sample locations. The application results for oil rock thickness in an oilfield indicate that minimizing the average relative updated kriging variance is a useful alternative to the other criteria based on kriging variance in optimal infill sampling design for geologic site characterization.  相似文献   

2.
An Alternative Measure of the Reliability of Ordinary Kriging Estimates   总被引：4，自引：0，他引：4  
This paper presents an interpolation variance as an alternative to the measure of the reliability of ordinary kriging estimates. Contrary to the traditional kriging variance, the interpolation variance is data-values dependent, variogram dependent, and a measure of local accuracy. Natural phenomena are not homogeneous; therefore, local variability as expressed through data values must be recognized for a correct assessment of uncertainty. The interpolation variance is simply the weighted average of the squared differences between data values and the retained estimate. Ordinary kriging or simple kriging variances are the expected values of interpolation variances; therefore, these traditional homoscedastic estimation variances cannot properly measure local data dispersion. More precisely, the interpolation variance is an estimate of the local conditional variance, when the ordinary kriging weights are interpreted as conditional probabilities associated to the n neighboring data. This interpretation is valid if, and only if, all ordinary kriging weights are positive or constrained to be such. Extensive tests illustrate that the interpolation variance is a useful alternative to the traditional kriging variance.  相似文献   

3.
Ordinary kriging and non-linear geostatistical estimators are now well accepted methods in mining grade control and mine reserve estimation. In kriging, the search volume or ‘kriging neighbourhood’ is defined by the user. The definition of the search space can have a significant impact on the outcome of the kriging estimate. In particular, too restrictive neighbourhood, can result in serious conditional bias. Kriging is commonly described as a ‘minimum variance estimator’ but this is only true when the neighbourhood is properly selected. Arbitrary decisions about search space are highly risky. The criteria to consider when evaluating a particular kriging neighbourhood are the slope of the regression of the ‘true’ and ‘estimated’ block grades, the number of kriging negative weights and the kriging variance. Search radius is one of the most important parameters of search volume which often is determined on the basis of influence of the variogram. In this paper the above-mentioned parameters are used to determine optimal search radius.  相似文献   

4.
A key problem in the application of kriging is the definition of a local neighborhood in which to search for the most relevant data. A usual practice consists in selecting data close to the location targeted for prediction and, at the same time, distributed as uniformly as possible around this location, in order to discard data conveying redundant information. This approach may however not be optimal, insofar as it does not account for the data spatial correlation. To improve the kriging neighborhood definition, we first examine the effect of including one or more data and present equations in order to quickly update the kriging weights and kriging variances. These equations are then applied to design a stepwise selection algorithm that progressively incorporates the most relevant data, i.e., the data that make the kriging variance decrease more. The proposed algorithm is illustrated on a soil contamination dataset.  相似文献   

5.
In this study, machine learning methods such as neural networks, random forests, and Gaussian processes are applied to the estimation of copper grade in a mineral deposit. The performance of these methods is compared to geostatistical techniques, such as ordinary kriging and indicator kriging. To ensure that these comparisons are realistic and relevant, the predictive accuracy is estimated on test instances located in drill holes that are different from the training data. The results of an extensive empirical study in the Sarcheshmeh porphyry copper deposit in Southeastern Iran illustrate that specially designed Gaussian processes with a symmetric standardization of the spatial location inputs and an anisotropic kernel yield the most accurate predictions. Furthermore, significant improvements are obtained when, besides location, information on the rock type is included in the set of predictor variables. This observation highlights the importance of carrying out detailed studies of the geological composition of the deposit to obtain more accurate ore grade predictions.  相似文献   

6.
Most significant iron ore deposits in Iran are located in Central Iran Zone. These deposits belong to the Bafq mining district. The Bafq mining district is located in the Early Cambrian Kashmar-Kerman volcanic arc of Central Iran. Linear estimation of regionalized variables (for example by inverse distance weighting or ordinary Kriging) results in relatively high estimation variances, i.e. the estimates have very low precision. Assessment of project economics (or other critical decision making) based on linear estimation is therefore risky. Non-linear estimation methods like disjunctive kriging perform better and the lower estimation variance allows less risky economic decision-making. Another advantage of disjunctive kriging is that it allows estimation of functions of the primary variable, which here is the grade (Fe %) of the ore. In particular it permits estimation of indicator functions defined using thresholds on the primary variable. This paper is devoted to application of disjunctive kriging method in Choghart North Anomaly iron ore deposit in Central Iran, Yazd province, Iran. In this study, the Fe concentration of Choghart North Anomaly iron ore deposit was modelled and estimated. The exploration data consists of borehole samples measuring the Fe concentration. A Gaussian isofactorial model is fitted to these data and disjunctive kriging was used to estimate the regionalized variable (Fe %) at unsampled locations and to assess the probabilities that the actual concentrations exceed a threshold value at a given location. Consequently a three dimensional model of probability of exceeding a threshold value and the estimated value are provided by disjunctive kriging to divide the ore into an economic and uneconomic part on the basis of estimation of indicator functions using thresholds grades defined on point support. The tools and concepts are complemented by a set of computer programs that are applied to the case study. The study showed that disjunctive kriging can be applied successfully for modeling the grade of an ore deposit. Results showed that the correlation between the estimated value and real value at locations close to each other is 81.9%.  相似文献   

7.
《Applied Geochemistry》1999,14(1):133-145
Three univariate geostatistical methods of estimation are applied to a geochemical data set. The studied methods are: ordinary kriging (cross-validation), factorial kriging, and indicator kriging. These techniques use the probabilistic and spatial behaviour of geochemical variables, giving a tool for identifying potential anomalous areas to locate mineralization. Ordinary kriging is easy to apply and to interpret the results. It has the advantage of using the same experimental grid points for its estimates, and no additional grid points are needed. Factorial kriging decomposes the raw variable into as many components as there are identified structures in the variogram. This, however, is a complex method and its application is more difficult than that of ordinary or indicator kriging. The main advantages of indicator kriging are that data are used by their rank order, being more robust about outlier values, and that the presentation of results is simple. Nevertheless, indicator kriging is incapable of separating anomalous values and the high values from the background, which have a behaviour different to the anomaly. In this work, the results of the application of these 3 kriging methods to a set of mineral exploration data obtained from a geochemical survey carried out in NW Spain are presented. This area is characterised by the presence of Au mineral occurrences. The kriging methods were applied to As, considered as a pathfinder of Au in this area. Numerical treatment of Au is not applicable, because it presents most values equal to the detection limit, and a series of extreme values. The results of the application of ordinary kriging, factorial kriging and indicator kriging to As make possible the location of a series of rich values, sited along a N–S shear zone, considered a structure related to the presence of Au.  相似文献   

8.
A common issue in spatial interpolation is the combination of data measured over different spatial supports. For example, information available for mapping disease risk typically includes point data (e.g. patients’ and controls’ residence) and aggregated data (e.g. socio-demographic and economic attributes recorded at the census track level). Similarly, soil measurements at discrete locations in the field are often supplemented with choropleth maps (e.g. soil or geological maps) that model the spatial distribution of soil attributes as the juxtaposition of polygons (areas) with constant values. This paper presents a general formulation of kriging that allows the combination of both point and areal data through the use of area-to-area, area-to-point, and point-to-point covariances in the kriging system. The procedure is illustrated using two data sets: (1) geological map and heavy metal concentrations recorded in the topsoil of the Swiss Jura, and (2) incidence rates of late-stage breast cancer diagnosis per census tract and location of patient residences for three counties in Michigan. In the second case, the kriging system includes an error variance term derived according to the binomial distribution to account for varying degree of reliability of incidence rates depending on the total number of cases recorded in those tracts. Except under the binomial kriging framework, area-and-point (AAP) kriging ensures the coherence of the prediction so that the average of interpolated values within each mapping unit is equal to the original areal datum. The relationships between binomial kriging, Poisson kriging, and indicator kriging are discussed under different scenarios for the population size and spatial support. Sensitivity analysis demonstrates the smaller smoothing and greater prediction accuracy of the new procedure over ordinary and traditional residual kriging based on the assumption that the local mean is constant within each mapping unit.  相似文献   

9.
Grade estimation is very important in designing open pits. In the process of grade estimation, underestimation can result in loss of economic ore, whereas overestimation would unnecessarily increase stripping ratio. Normally, kriging method, which suffers from underestimation and/or overestimation due to smoothing effect, is used for grade estimation. To overcome drawbacks of the kriging method, more efficient techniques such as conditional simulation can be applied. In this paper, utilizing sequential Gaussian conditional simulation, grade models were constructed for Sungun copper deposit situated in the North West of Iran. According to the obtained results, it was observed that conditional simulation can effectively cope with the weakness of kriging method. Also, it was observed that as compared to the kriging method, grade distribution, resulted from the conditional simulation, is almost identical to that of the real exploration data. Accordingly, using conditional simulation, the amount of mineable ore was significantly increased, and also, average net present value as the mines’ most important economic indicator was improved by 40%.  相似文献   

10.
This paper presents a modified ordinary kriging technique referred to as the “Area Influence Kriging” (AIK). The method is a simple and practical tool to use for more accurate prediction of global recoverable ore resources in any type of deposit. AIK performs well even in deposits with skewed grade distributions when the ordinary kriging (OK) results are unreasonably smooth. It is robust and globally unbiased like OK. The AIK method is not intended to replace OK, which is a better estimator of the average grade of the blocks. Rather it aims to complement OK with its excellent performance in predicting recoverable resources that have been the major pitfalls of OK in many resource estimation cases. The paper details the methodology of AIK with a couple of examples. It also reports the results from its application to a gold deposit.  相似文献   

11.
Conditioning realizations of stationary Gaussian random fields to a set of data is traditionally based on simple kriging. In practice, this approach may be demanding as it does not account for the uncertainty in the spatial average of the random field. In this paper, an alternative model is presented, in which the Gaussian field is decomposed into a random mean, constant over space but variable over the realizations, and an independent residual. It is shown that, when the prior variance of the random mean is infinitely large (reflecting prior ignorance on the actual spatial average), the realizations of the Gaussian random field are made conditional by substituting ordinary kriging for simple kriging. The proposed approach can be extended to models with random drifts that are polynomials in the spatial coordinates, by using universal or intrinsic kriging for conditioning the realizations, and also to multivariate situations by using cokriging instead of kriging.  相似文献   

12.
This paper presents a modified ordinary kriging technique referred to as the Area Influence Kriging (AIK). The method is a simple and practical tool to use for more accurate prediction of global recoverable ore resources in any type of deposit. AIK performs well even in deposits with skewed grade distributions when the ordinary kriging (OK) results are unreasonably smooth. It is robust and globally unbiased like OK. The AIK method is not intended to replace OK, which is a better estimator of the average grade of the blocks. Rather it aims to complement OK with its excellent performance in predicting recoverable resources that have been the major pitfalls of OK in many resource estimation cases. The paper details the methodology of AIK with a couple of examples. It also reports the results from its application to a gold deposit.  相似文献   

13.
This paper presents the results of disjunctive kriging applied to a supergene iron ore deposit of Bailadila Range of India. Disjunctive kriging is applied firstly to compare estimates of the blocks by ordinary kriging and secondly to estimate benchwise local recoverable reserves of the orebody. Good agreement exists between block estimates by ordinary kriging and disjunctive kriging except for peripheral blocks with less borehole information. Estimation of benchwise reserves shows that the behavior of the distribution of grades is different in various benches. The study shows that disjunctive kriging can be applied successfully for estimation of local recoverable reserves in the case of a good grade hematite iron ore deposit.  相似文献   

14.
Restricted kriging for mixture of grade models   总被引：2，自引：0，他引：2  
A modified type of kriging, referred to as restricted kriging (RK), is proposed in this study. The method incorporates constraints on different grade classes to restrict the influence of the samples having different likelihoods in estimation. RK is motivated by the estimation of mineral reserves when grades have highly skewed distributions. Ordinary kriging tends to produce an overly smoothed interpolated surface by underestimating high grades and overestimating low grades. The fact that ordinary kriging gives a uniform prior treatment to all samples independent of their values is a major factor associated with this smoothing effect. The new approach differentiates each grade portion by preselected cutoffs. RK is developed for a single cutoff and then extended into a general form for any finite number of cutoffs. Restricted cokriging (RCK) is also formulated to simultaneously estimate a set of random functions with restriction conditions. Methods are suggested for determination of the probabilities of occurrence of different grade portions. Finally, the new approach is demonstrated on a case study of an epithermal gold deposit.  相似文献   

15.
In geostatistics, factorial kriging is often proposed to filter noise. This filter is built from a linear model which is ideally suited to a Gaussian signal with additive independent noise. Robustness of the performance of factorial kriging is evaluated in less congenial situations. Three different types of noise are considered all perturbing a lognormally distributed signal. The first noise model is independent of the signal. The second noise model is heteroscedastic; its variance depends on the signal, yet noise and signal are uncorrelated. The third noise model is both heteroscedastic and linearly correlated with the signal. In ideal conditions, exhaustive sampling and additive independent noise, factorial kriging succeeds to reproduce the spatial patterns of high signal values. This score remains good in presence of heteroscedastic noise variance but falls quickly in presence of noise-to-signal correlation as soon as the sample becomes sparser.  相似文献   

16.
In the last few years, an increasing number of practical studies using so-called kriging estimation procedures have been published. Various terms, such as universal kriging, lognormal kriging, ordinary kriging, etc., are used to define different estimation procedures, leaving a certain confusion about what kriging really is. The object of this paper is to show what is the common backbone of all these estimation procedures, thus justifying the common name of kriging procedures. The word kriging (in French krigeage) is a concise and convenient term to designate the classical procedure of selecting, within agiven class of possible estimators, the estimator with a minimum estimation variance (i.e., the estimator which leads to a minimum variance of the resulting estimation error). This estimation variance can be seen as a squared distance between the unknown value and its estimator; the process of minimization of this distance can then be seen as the projection of the unknown value onto the space within which the search for an estimator is carried out.  相似文献   

17.
For any distribution of grades, a particular cutoff grade is shown here to exist at which the indicator covariance is proportional to the grade covariance to a very high degree of accuracy. The name “mononodal cutoff” is chosen to denote this grade. Its importance for robust grade variography in the presence of a large coefficient of variation—typical of precious metals—derives from the fact that the mononodal indicator variogram is then linearly related to the grade variogram yet is immune to outlier data and is found to be particularly robust under data information reduction. Thus, it is an excellent substitute to model in lieu of a difficult grade variogram. A theoretical expression for the indicator covariance is given as a double series of orthogonal polynomials that have the grade density function as weight function. Leading terms of this series suggest that indicator and grade covariances are first-order proportional, with cutoff grade dependence being carried by the proportionality factor. Kriging equations associated with this indicator covariance lead to cutoff-free kriging weights that are identical to grade kriging weights. This circumstance simplifies indicator kriging used to estimate local point-grade histograms, while at the same time obviating order relations problems.  相似文献   

18.
《Applied Geochemistry》2005,20(1):157-168
In monitoring a minor geochemical element in groundwater or soils, a background population of values below the instrumental detection limit is frequently present. When those values are found in the monitoring process, they are assigned to the detection limit which, in some cases, generates a probability mass in the probability density function of the variable at that value (the minimum value that can be detected). Such background values could distort both the estimation of the variable at nonsampled locations and the inference of the spatial structure of variability of the variable. Two important problems are the delineation of areas where the variable is above the detection limit and the estimation of the magnitude of the variables inside those areas. The importance of these issues in geochemical prospecting or in environmental sciences, in general related with contamination and environmental monitoring, is obvious. In this paper the authors describe the two-step procedure of indicator kriging and ordinary kriging and compare it with empirical maximum likelihood kriging. The first approach consists of using a binary indicator variable for estimating the probability of a location being above the detection limit, plus ordinary kriging conditional to the location being above the detection limit. An estimation variance, however, is not available for that estimator. Empirical maximum likelihood kriging, which was designed to deal with skew distributions, can also deal with an atom at the origin of the distribution. The method uses a Bayesian approach to kriging and gives intermittency in the form of a probability map, its estimates providing a realistic assessment of their estimation variance. The pros and cons of each method are discussed and illustrated using a large dataset of As concentration in groundwater. The results of the two methods are compared by cross-validation.  相似文献   

19.
 The applicability and usefulness of Geostatistics (kriging) as a tool for optimum selection of sites for monitoring groundwater levels has been demonstrated through a case study. The criterion used is the estimation of error variance. Groundwater level data (pre-monsoon 1994) obtained from 32 observation wells of Upper Kongal basin, Nalgonda District, A.P. (India) has been stochastically analyzed. The spatial distribution of water levels and its associated error variance is computed and the locations having maximum error variance are selected as additional sites for augmenting the existing observational well network. Received: 15 june 1998 · Accepted: 14 December 1998  相似文献   

20.
    
For any distribution of grades, a particular cutoff grade is shown here to exist at which the indicator covariance is proportional to the grade covariance to a very high degree of accuracy. The name mononodal cutoff is chosen to denote this grade. Its importance for robust grade variography in the presence of a large coefficient of variation—typical of precious metals—derives from the fact that the mononodal indicator variogram is then linearly related to the grade variogram yet is immune to outlier data and is found to be particularly robust under data information reduction. Thus, it is an excellent substitute to model in lieu of a difficult grade variogram. A theoretical expression for the indicator covariance is given as a double series of orthogonal polynomials that have the grade density function as weight function. Leading terms of this series suggest that indicator and grade covariances are first-order proportional, with cutoff grade dependence being carried by the proportionality factor. Kriging equations associated with this indicator covariance lead to cutoff-free kriging weights that are identical to grade kriging weights. This circumstance simplifies indicator kriging used to estimate local point-grade histograms, while at the same time obviating order relations problems.This paper is based in part on a PhD thesis submitted to the Department of Applied Earth Sciences, Stanford University, Stanford, California 94305, in 1984 (unpublished).  相似文献   

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