Multiple Indicator Kriging Research Articles

Copula-based multiple indicator kriging for non-Gaussian random fields

In spatial statistics, the kriging predictor is the best linear predictor at unsampled locations, but not the optimal predictor for non-Gaussian processes. In this paper, we introduce a copula-based multiple indicator kriging model for the analysis of non-Gaussian spatial data by thresholding the spatial observations at a given set of quantile values. The proposed copula model allows for flexible marginal distributions while modeling the spatial dependence via copulas. We show that the covariances required by kriging have a direct link to the chosen copula function. We then develop a semiparametric estimation procedure. The proposed method provides the entire predictive distribution function at a new location, and thus allows for both point and interval predictions. The proposed method demonstrates better predictive performance than the commonly used variogram approach and Gaussian kriging in the simulation studies. We illustrate our methods on precipitation data in Spain during November 2019, and heavy metal dataset in topsoil along the river Meuse, and obtain probability exceedance maps.

One Step at a Time: The Origins of Sequential Simulation and Beyond

In the mid-1980s, still in his young 40s, André Journel was already recognized as one of the giants of geostatistics. Many of the contributions from his new research program at Stanford University had centered around the indicator methods that he developed: indicator kriging and multiple indicator kriging. But when his second crop of graduate students arrived at Stanford, indicator methods still lacked an approach to conditional simulation that was not tainted by what André called the ‘Gaussian disease’; early indicator simulations went through the tortuous path of converting all indicators to Gaussian variables, running a turning bands simulation, and truncating the resulting multi-Gaussian realizations. When he conceived of sequential indicator simulation (SIS), even André likely did not recognize the generality of an approach to simulation that tackled the simulation task one step at a time. The early enthusiasm for SIS was its ability, in its multiple-indicator form, to cure the Gaussian disease and to build realizations in which spatial continuity did not deteriorate in the extreme values. Much of Stanford’s work in the 1980s focused on petroleum geostatistics, where extreme values (the high-permeability fracture zones and the low-permeability shale barriers) have much stronger anisotropy, and much longer ranges of correlation in the maximum continuity direction, than mid-range values. With multi-Gaussian simulations necessarily imparting weaker continuity to the extremes, SIS was an important breakthrough. The generality of the sequential approach was soon recognized, first through its analogy with multi-variate unconditional simulation achieved using the lower triangular matrix of an LU decomposition of the covariance matrix as the multiplier of random normal deviates. Modifying LU simulation so that it became conditional gave rise to sequential Gaussian simulation (SGS), an algorithm that shared much in common with SIS. With nagging implementation details like the sequential path and the search neighborhood being common to both methods, improvements in either SIS or SGS often became improvements to the other. Almost half of the contributors to this Special Issue became students of André in the classes of 1984–1988, and several are the pioneers of SIS and SGS. Others who studied later with André explored and developed the first multipoint statistics simulation procedures, which are based on the same concept that underlies sequential simulation. Among his many significant intellectual accomplishments, one of the cornerstones of André Journel’s legacy was sequential simulation, built one step at a time.

GEOSTATISTICAL APPROACH FOR THE ESTIMATION OF SHEAR-HOSTED GOLD DEPOSIT: A CASE STUDY OF THE OBUASI GOLD DEPOSIT, GHANA

Underground mining at Obuasi in Ghana has been in operation since 1947. This paper uses geostatistical methods to evaluate gold ore blocks to ensure reliable grades for mining large tonnage and low-grade resources. Historically, the principal ores were low tonnage, high grade and relatively homogeneous quartz stockwork with simple geometry and average bulk grades in the range of 20-30 g/t that were evaluated using conventional polygonal methods and mined by semi-mechanized means. Currently, the ore is a shear-hosted mixed quartz vein and disseminated sulphide type deposit of low grade that is mined using highly mechanized means. The need therefore arises for a re-assessment of the estimation procedures to ensure prolonged and more profitable mining. Both diamond drill (DD) core and stope/cross-cut channel samples were taken from Block 1 at the mine for analyses and re-assessment. A wireframe model was used to constrain the three dimensional (3D) block model of the deposit. Ordinary kriging (OK) and multiple indicator kriging (MIK) geostatistical methods were used to estimate gold grades. Grade distribution is positively skewed with high spatial variability and extreme values while background values are established as <0.6 g/t. The Spatial variability is characterized by fitting models on experimental variograms. The MIK approach mitigates the effects of outliers and establish grades that are consistently lower than the OK and the weighted average method that are widely used at the mine. The MIK method, a non-linear, non-parametric method of local grade estimation are applicable to the deposit architecture. Profoundly, the MIK method is a more reliable approach considering the fact that the MCF based on the estimates at the mine are high despite operational deficiencies on the mine. The results from this study demonstrates usefulness of geostatistics to determine the architecture of Au mineralization at the deposit scale.

Geochemical distribution mapping by combining number-size multifractal model and multiple indicator kriging

The Mirgeh-Naghshineh region located at the northwest of Iran has a high potential for Au mineralization due to its adjacency to shear zone as a plausible environment responsible for ore-bearing occurrences. In this study, geochemical samples were collected over promising regions of the study area to delineate favorable zones for exploratory drillings. A number-size (NS) fractal approach was taken into account to separate anomalous regions from regional and background ones, and the prospect zone was divided into some indicators based on the extracted thresholds. Such results were incorporated in the multiple indicator kriging (MIK) approach to augment the interpolation of Au distribution map. It was observed that the highest favorable zones cover about 1.6% of the area, while the drilled holes prove such Au mineralization. The applicability of the proposed method was also compared with the conventional interpolation techniques, such as Inverse Distance Weighting (IDW), Ordinary Kriging (OK), and Median Indicator Kriging (Median-IK). The Mean Error (ME) and Root Mean Square Error (RMSE) values for the proposed method were smaller than those of other methods. Moreover, cross validation indicated the superiority of the proposed method.

Comparative Study on Linear and Non-Linear Geostatistical Methods: A Case Study on Kalsaka Hill Gold Deposit, Burkina Faso

Selecting an appropriate method to evaluate an ore deposit is imperative in resource estimation since it becomes the basis for reliable planning and development of a mine. Even though linear geostatistical methods such as Ordinary Kriging (OK) give reasonable estimates, there may be instances where recoverable resource estimates are difficult to obtain, particularly when the deposit is characterised by a positively skewed grade distribution with some outliers. Multiple Indicator Kriging (MIK), which is a non-linear estimation technique, is not based on any assumption about the distribution underlying the data and offers realistic solutions to problems associated with skewness and outliers. This paper looks at the use of MIK technique as an alternative method of recoverable resource estimation to OK by comparing the resource estimates obtained from MIK and OK models. OK model showed more smoothing effect on its estimates than the MIK model as evidenced in swath plots. Underestimated grades and tonnages were observed when OK was used to estimate a gold deposit at Kalsaka, according to the grade and tonnage reconciliation. MIK model yields estimates which are higher and closer to the actual than the OK model estimates. Keywords : Gold, Multiple Indicator Kriging, Ordinary Kriging, Outlier, Variography

Investigation of linear and non-linear estimation methods in highly-skewed gold distribution

The purpose of this work is to compare the linear and non-linear kriging methods in the mineral resource estimation of the Qolqoleh gold deposit in Saqqez, NW Iran. Considering the fact that the gold distribution is positively skewed and has a significant difference with a normal curve, a geostatistical estimation is complicated in these cases. Linear kriging, as a resource estimation method, can be problematic and gives an unrealistic gold grade. In order to check and correct the errors in the linear methods, the non-linear kriging method has been deployed. One of the applicant's non-linear estimation methods is Indicator Kriging (IK). The IK method converts grade values into binary units of 0 and 1 using multiple thresholds that can be selected by the number-size (N-S) fractal model. The N-S model identifies important and critical thresholds based on the grade distribution. In IK, the Multiple Indicator Kriging (Multiple IK) and Median Indicator Kriging (Median IK) methods could be involved due to the number of indicator thresholds. IK is not sensitive to high values. Here, we make a comparison between Median IK and Multiple IK as well as those with ordinary kriging (OK), which is a linear kriging method. Overall, we conclude that all of these methods are suitable for resource estimation among these methods, although the IK method is better for estimation in different categories of gold grades.

Indicator-Based Geostatistical Models For Mapping Fish Survey Data

Marine research survey data on fish stocks often show a small proportion of very high-density values, as for many environmental data. This makes the estimation of second-order statistics, such as the variance and the variogram, non-robust. The high fish density values are generated by fish aggregative behaviour, which may vary greatly at small scale in time and space. The high values are thus imprecisely known, both in their spatial occurrence and order of magnitude. To map such data, three indicator-based geostatistical methods were considered, the top-cut model, min–max autocorrelation factors (MAF) of indicators, and multiple indicator kriging. In the top-cut and MAF approaches, the variable is decomposed into components and the most continuous ones (those corresponding to the low and medium values) are used to guide the mapping. The methods are proposed as alternatives to ordinary kriging when the variogram is difficult to estimate. The methods are detailed and applied on a spatial data set of anchovy densities derived from a typical fish stock acoustic survey performed in the Bay of Biscay, which show a few high-density values distributed in small spatial patches and also as solitary events. The model performances are analyzed by cross-validating the data and comparing the kriged maps. Results are compared to ordinary kriging as a base case. The top-cut model had the best cross-validation performance. The indicator-based models allowed mapping high-value areas with small spatial extent, in contrast to ordinary kriging. Practical guidelines for implementing the indicator-based methods are provided.

Risk reduction in Sechahun iron ore deposit by geological boundary modification using multiple indicator Kriging

Uncertainty on the geological contacts and the block volumes of the models along boundaries is often a major part of the global uncertainty of reserve estimation. This work introduces a geostatistical technique that has been developed and tested in an iron ore deposit at Bafq mining district, in central Iran, and that, based on a probability criterion, helps to objectively model the geometry of this iron ore deposit. The main problem in reserve estimation of this ore body is its geometrical modeling and uncertainty in geological boundaries. This work deals with the geostatistical method of multiple indicator kriging, which is used to determine the real boundaries of ore body in different categories. This approach has potential to improve project performance and decrease operational risk. For this purpose, the ore body is separated into two categories including rich iron zone (w(Fe)>45%) and poor iron zone (20%<w(Fe)<45%). It significantly benefits to decrease the risk of reserve evaluation in the deposit. This case study also highlights the value of multiple indicator kriging as a tool for estimates the position of grade boundaries within the deposit. Comparison of the resultant probability maps with the real ore/waste contacts on the extracted levels shows that the first indicator model could separate the whole ore body (poor plus rich) from the waste zone by probability of more than 0.35, which concludes the total reserve of 53 million tons. The second indicator model applied to separate the rich and poor domains and the results show that the blocks with the estimated probability of equal to or more than 0.4 lay within the rich ore zone consisting of 15.8 million tons reserve.

Application of multiple indicator Kriging for RMR value estimation in areas of new drift excavation during mine site redevelopment

Rock mass rating (RMR) values along additional drifts that are to be constructed during mine site redevelopment (Gagok mine, Korea) were estimated using multiple criteria indicator Kriging with borehole RMR and electrical resistivity tomography data. This study outlines two potential correction methods that supplement indicator Kriging and that can reduce error, allowing more accurate estimation of RMR values. The correction methods used a filtering of indirect data affected by drifts (FIED) method to reduce uncertainties and errors within electrical resistivity tomography data caused by changes in setup and the location of existing drifts. In addition, a correction indicating accurate representation of indirect data (CARI) method was used to resolve distortions in drilling log results. This meant that 73 % of the estimated RMR values were assigned to the correct RMR class, with only 1 % of these values being overestimated. The sensitivity of the FIED method was 4 %, with the linear CARI method having a sensitivity of 56–60 % and the logarithmic CARI method having a sensitivity of 61–65 %.

A comparison of multiple indicator kriging and area-to-point Poisson kriging for mapping patterns of herbivore species abundance in Kruger National Park, South Africa

Kruger National Park (KNP), South Africa, provides protected habitats for the unique animals of the African savannah. For the past 40 years, annual aerial surveys of herbivores have been conducted to aid management decisions based on (1) the spatial distribution of species throughout the park and (2) total species populations in a year. The surveys are extremely time consuming and costly. For many years, the whole park was surveyed, but in 1998 a transect survey approach was adopted. This is cheaper and less time consuming but leaves gaps in the data spatially. Also the distance method currently employed by the park only gives estimates of total species populations but not their spatial distribution. We compare the ability of multiple indicator kriging and area-to-point Poisson kriging to accurately map species distribution in the park. A leave-one-out cross-validation approach indicates that multiple indicator kriging makes poor estimates of the number of animals, particularly the few large counts, as the indicator variograms for such high thresholds are pure nugget. Poisson kriging was applied to the prediction of two types of abundance data: spatial density and proportion of a given species. Both Poisson approaches had standardized mean absolute errors (St. MAEs) of animal counts at least an order of magnitude lower than multiple indicator kriging. The spatial density, Poisson approach (1), gave the lowest St. MAEs for the most abundant species and the proportion, Poisson approach (2), did for the least abundant species. Incorporating environmental data into Poisson approach (2) further reduced St. MAEs.

Constrained multiple indicator kriging using sequential quadratic programming

Multiple indicator kriging (MIK) is a nonparametric method used to estimate conditional cumulative distribution functions (CCDF). Indicator estimates produced by MIK may not satisfy the order relations of a valid CCDF which is ordered and bounded between 0 and 1. In this paper a new method has been presented that guarantees the order relations of the cumulative distribution functions estimated by multiple indicator kriging. The method is based on minimizing the sum of kriging variances for each cutoff under unbiasedness and order relations constraints and solving constrained indicator kriging system by sequential quadratic programming. A computer code is written in the Matlab environment to implement the developed algorithm and the method is applied to the thickness data.

Joint inversion of steady-state hydrologic and self-potential data for 3D hydraulic conductivity distribution at the Boise Hydrogeophysical Research Site

► We combine sedimentological and hydro-geophysical information to characterize a real aquifer. ► We characterize the distribution of materials at the site in the spirit of multiple- (overlapping) continua concepts. ► A Multiple Indicator Kriging is adopted to estimate the probability of occurrence of each geo-material. ► Self potential and hydraulic head are embedded jointly within a three-dimensional inverse groundwater model. ► We obtained a very detailed reconstruction of the heterogeneity of the site better than previous studies. We combine sedimentological, hydraulic and geophysical information to characterize the 3D distribution of transport properties of an heterogeneous aquifer. We focus on the joint inversion of hydraulic head and self-potential measurements collected during an extensive experimental campaign performed at the Boise Hydrogeophysical Research Site (BHRS), Boise, Idaho, and involving a series of dipole tests. While hydraulic head data obtained from piezometric readings in open wells represent a depth-averaged value, self-potential signals provide an estimate of the water table location. The aquifer is conceptualized as a multiple-continuum, where the volumetric fraction of a geo-material within a cell of the numerical flow model is calculated by Multiple Indicator Kriging. The latter is implemented on the basis of available sedimentological information. The functional format of the indicator variograms and associated parameters are estimated on the basis of formal model identification criteria. Self-potential and hydraulic head data have been embedded jointly within a three-dimensional inverse model of groundwater flow at the site. Each identified geo-material (category) is assumed to be characterized by a constant hydraulic conductivity. The latter constitute the set of model parameters. The hydraulic conductivity associated with a numerical block is then calculated as a weighted average of the conductivities of the geo-materials which are collocated in the block by means of Multiple Indicator Kriging. Model parameters are estimated by a Maximum Likelihood fit between measured and modeled state variables, resulting in a spatially heterogeneous distribution of hydraulic conductivity. The latter is effectively constrained on the sedimentological data and conditioned on both self-potential and borehole hydraulic head readings. Minimization of the Maximum Likelihood objective function allows estimating the relative weight of measurement errors associated with self-potential and borehole-based head data. The procedure adopted allowed a reconstruction of the heterogeneity of the site with a level of details, which was not obtained in previous studies and with relatively modest computational efforts. Further validation against dipole tests which were not used in the inversion procedure supports the robustness of the results.

Estimation of rock mass classes using the 3-dimensional multiple indicator kriging technique

For optimum tunnel designs, a considerable amount of ground investigations should be thoroughly performed. In general, geophysical investigations are rather preferred over drilling investigations because of their relative cheap cost. Also, there are many cases in which drilling investigations are basically impossible because of civil appeals or mountainous topology etc. Under the circumstances, in this study, 3 dimensional multiple indicator kriging (3D-MI kriging), which can incorporate both geophysical exploration data and drilling data off a tunnel center line, is proposed. To this end, two dimensional multiple indicator kriging, which is a geostatistical technique, is extended for 3 dimensions. Also, the proposed 3D-MI kriging was applied to determine the rock mass classes in terms of the RMR system for the design of a road tunnel before and during the construction. During the construction, the tunnel face was observed at a 3 to 3.5 m interval over a certain length of the tunnel. RMR index values were estimated based upon the face mapping observations and they were compared with the estimated RMR using 3D-MI kriging. (A) This paper was presented at Safety in the underground space - Proceedings of the ITA-AITES 2006 World Tunnel Congress and the 32nd ITA General Assembly, Seoul, Korea, 22-27 April 2006. For the covering abstract see ITRD E129148. Reprinted with permission from Elsevier.

Mapping of salinity risk in the lower Namoi valley using non-linear kriging methods

Dryland salinity is on the increase in the upper catchments of central and northern New South Wales, Australia. Consequently, salts may be exported downstream, which could adversely affect irrigated-cotton farming systems. To assess the potential threat of salinity downstream, a simple salt-balance model is used to simulate the potential impact of salinisation due to the farming systems, by using progressively saline water (i.e. EC w 0.4, 1.4, 4.0 and 9.0 dS/m). We studied the problem in the lower Namoi valley of northern New South Wales and compared the various non-linear (indicator, multiple indicator and disjunctive) kriging methods to determine an optimal method for assessing the risk. Our results indicate that potential salinisation due to application of the water currently used for irrigation is minimal and may not pose any problems to sustainability of irrigated agriculture. The same results were obtained by simulation based on irrigation using slightly saline water (EC w 1.4 dS/m). However, simulations based on irrigation using saline water (i.e. EC w of 4 and 9.0 dS/m) shows potential salinisation, which will require management inputs for sustainable cropping systems, especially legumes and wheat, which are used extensively in rotation with cotton. In terms of producing the fewest misclassifications disjunctive kriging was found to be optimal compared to multiple indicator kriging and indicator kriging. Using non-linear geostatistical methods to spatially distribute the results of a simple salt-balance model it is possible to identify the locations where the risks of salinisation are greatest in the lower Namoi valley from the application of saline water.

Field-scale assessment of deep drainage risk

Improving irrigation efficiency is of primary importance in arid and semi-arid regions of the world as a consequence of increasing incidences of soil and water salinisation. In the cotton-growing regions of Australia salinisation is generally a result of inefficient irrigation practices, which lead to excessive deep drainage (DD). There is therefore the need to apply a relatively inexpensive approach to assessing where inefficiencies occur and make prediction of suitability of existing and new water storage sites. However, physical methods of measuring DD, such as flux meters and lysimeters, are time-consuming and site-specific. In this paper we apply a rapid method for determining the spatial distribution of soil in an irrigated cotton field in the lower Gwydir valley. First, ECa data (using EM38 and EM31) were used to determine a soil-sampling scheme for determining soil information such as clay content and exchangeable cations to a depth of 1.2 m. The soil data and water quality information were input into the SaLF (salt and leaching fraction) model to estimate DD rate (mm/year). In developing the relationship between ECa and estimated DD, three exponential models (two-, three- and four-parameter) were compared and evaluated using the Aikakie information criteria (AIC). The three-parameter exponential model was found to be best and was used for further analysis. Using the geostatistical approach of multiple indicator kriging (MIK), maps of conditional probability of DD exceeding a critical cut-off value (i.e. 50, 75, 100 mm) were produced for various rates of irrigation (I=300, 600, 1,200 and 1,500 mm/year). The areas of highest risk were consistent with where water-use efficiency was problematic and thus leading to the creation of perched water tables. The advantage of this approach is that it is quick and is applicable to situations where efficient use of water is required. The results can be used for irrigation planning, particularly in the location of large irrigation infrastructure such as water reservoirs.

Kriging method evaluation for assessing the spatial distribution of urban soil lead contamination.

Describing contaminant spatial distribution is an integral component of risk assessment. Application of geostatistical techniques for this purpose has been demonstrated previously. These techniques may provide both an estimate of the concentration at a given unsampled location, as well as the probability that the concentration at that location will exceed a critical threshold concentration. This research is a comparative study between multiple indicator kriging and kriging with the cumulative distribution function of order statistics, with both local and global variograms. The aim was to determine which of the four methods is best able to delineate between "contaminated" and "clean" soil. The four methods were validated with a subset of data values that were not used in the prediction. Method performance was assessed by calculating the root mean square error (RMSE), analysis of variance, the proportion of sites misclassified by each method as either "clean" when they were actually "contaminated" or vice versa, and the expected loss for each misclassification type. The data used for the comparison were 807 topsoil Pb concentrations from the inner-Sydney suburbs of Glebe and Camperdown, Australia. While there was very little difference between the four methods, multiple indicator kriging was found to produce the most accurate predictions for delineating "clean" from "contaminated" soil.

Development of Three‐Dimensional Hydrostratigraphical Architecture of the Unsaturated Zone Based on Soft and Hard Data

AbstractThree‐dimensional geological maps that include the hydrostratigraphical heterogeneity and depositional architecture of the natural formation are needed in hydrogeology to understand flow and transport. This paper presents a practical procedure for the generation of such maps. The method includes multiple indicator kriging, where the estimation of the structural architecture is conditioned on ground penetrating radar data. Physical core samples are used as tie points. This study utilizes data obtained from the unsaturated zone of a highly heterogeneous, glaciofluvial delta formation in Norway.

Comparative performance of indicator algorithms for modeling conditional probability distribution functions

This paper compares the performance of four algorithms (full indicator cokriging. adjacent cutoffs indicator cokriging, multiple indicator kriging, median indicator kriging) for modeling conditional cumulative distribution functions (ccdf).The latter three algorithms are approximations to the theoretically better full indicator cokriging in the sense that they disregard cross-covariances between some indicator variables or they consider that all covariances are proportional to the same function. Comparative performance is assessed using a reference soil data set that includes 2649 locations at which both topsoil copper and cobalt were measured. For all practical purposes, indicator cokriging does not perform better than the other simpler algorithms which involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is found to be higher for cokriging algorithms, especially when constraints on the kriging weights are applied.

Indicator principal component kriging

An alternative to multiple indicator kriging is proposed which approximates the full coindicator kriging system by kriging the principal components of the original indicator variables. This transformation is studied in detail for the biGaussian model. It is shown that the cross-correlations between principal components are either insignificant or exactly zero. This result allows derivation of the conditional cumulative density function (cdf) by kriging principal components and then applying a linear back transform. A performance comparison based on a real data set (Walker Lake) is presented which suggests that the proposed method achieves approximation of the conditional cdf equivalent to indicator cokriging but with substantially less variogram modeling effort and at smaller computational cost.