The application of the jackknife in geostatistical resource estimation (in conjunction with kriging) is shown to yield two significant contributions. The first one is a robust new estimator, called jackknife kriging, which retains ordinary kriging's simplicity and global unbiasedness while at the same time reduces its local bias and oversmoothing tendency. The second contribution is the ability, through the jackknife standard deviation, to set a confidence limit for a reserve estimate of a general shape. Jackknifing the ordinary kriging estimate maximizes sample utilization, as well as information of sample spatial correlation. The jackknife kriging estimator handles the high grade smearing problem typical in ordinary kriging by assigning more weight to the closest sample(s). The result is a reduction in the local bias without sacrificing global unbiasedness. When data distribution is skewed, log transformation of the data prior to jackknifing is shown to improve the estimate by making the data behave better under jackknifing. The technique of block kriging short-cut, combined with jackknifing, are shown as an easy-to-use solution to the problem of grade estimation of a general three-dimensional digitized shape and the uncertainty associated with the estimate. The results are a single jackknife kriging estimate for the shape and its corresponding jackknife variance. This approach solves the problem of combining independent block estimation variances, and provides a simple way to set confidence levels for global estimates. Unlike the ordinary kriging variance, which is a measure of data configuration and is independent of data values, the jackknife kriging variance reflects the variability of the values being inferred, both on an individual block level and on the global level. Case studies involving two exhaustive (symmetric and highly skewed) data sets indicates the superiority of the jackknife kriging estimator over the original (ordinary kriging) estimator. Some instability of the log-transformed jackknife estimate is noted in the highly skewed situation, where the data do not generally behave well under standard jackknifing. A promising solution for future investigations seems to lie in the use of weighted jackknife formulation, which should better handle a wider spectrum of data distribution.

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Three forms of multivariate analysis, one very classical and the other two relatively new and little-known, are showcased and enhanced: the first is the Singular Value Decomposition (SVD), which is at the heart of many statistical, and now geostatistical, techniques; the second is the method of Variogram Analysis, which is one way of investigating spatial correlation in one or several variables; and the third is the process of interpolation known as cokriging, a method for optimizing the estimation of multivariate data based on the information provided through variogram analysis. The SVD is described in detail, and it is shown that the SVD can be generalized from its familiar matrix (two-dimensional) case to three, and possibly n, dimensions. This generalization we call the "Tensor SVD" (or TSVD), and we demonstrate useful applications in the field of geostatistics (and indicate ways in which it will be useful in other areas). Applications of the SVD to the tools of geostatistics are described: in particular, applications dependent on the TSVD, including variogram modelling in coregionalization. Variogram analysis in general is explored, and we propose broader use of an old tool (which we call the "corhogram ", based on the variogram) which proves useful in helping one choose variables for multivariate interpolation. The reasoning behind kriging and cokriging is discussed, and a better algorithm for solving the cokriging equations is developed, which results in simultaneous kriging estimates for comparison with those obtained from cokriging. Links from kriging systems to kernel systems are made; discovering kerneIs equivalent to kriging systems will be useful in the case where data are plentiful. Finally, some results of the application of geostatistical techniques to a data set concerning nitrate pollution in the West Salt River Valley of Arizona are described.

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A Research Report submitted in partial fulfilment of the requirements for the degree of Master of Science in Engineering (Mining) to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, Johannesburg, South Africa, July 2017 / The project investigated the application of geostatistical techniques in evaluating a mechanically deposited platinum tailings resource. The project was undertaken on one of the Anglo American Platinum tailings dams, the identity of which cannot be revealed, due to the agreement in place or permission given. Remnant unrecovered minerals of economic potential still exist in tailings dams. These unrecovered minerals have influenced several mining companies to turn their attention to the economic potential that still exists in tailings, making them a key strategic component of their resources and reserves. Geostatistics has been developed and thoroughly tested or improved to address challenges experienced in estimating in situ geological ore bodies. The main aim of this Research Project is to test whether these fundamental principles and theories of geostatistics are relevant and appropriate in evaluating man-made ore bodies, such as a Platinum tailings dam, without any significant changes needed on the underlying principles or estimation algorithms. The findings on the Case Study tailings resource can be applied in the evaluation of other tailings dams, as well as any other man-made structures such as low grade rock dumps, muck piles, with related characteristics. A standard approach (methodology) was followed to evaluate the Case Study tailings resource. Drilling and sampling was conducted through sonic drilling. It is a dry drilling technique that is suitable for sampling unconsolidated particles such as tailings. Thereafter, 2 samples were sent to the laboratory to establish grade (concentration) of Platinum Group Metals (Platinum, Palladium and Rhodium), Gold and Base Metals (Copper and Nickel). Density was also measured, and comprehensively analysed as part of variables of interest in this research. Statistical analyses were performed on all variables of interest contained in the dam: which are Platinum (Pt), Palladium (Pd), Gold (Au), 3E (two PGMs plus Gold), Copper (Cu), Nickel (Ni) and Density. The underlying statistical distributions of all metals and density were found to be non-symmetrical and slightly positive skewed. The skewness of the distributions was established to be marginal. Differences between raw data (untransformed) averages and the log-normal estimates were analysed and found to be insignificant. As such Ordinary Kriging of untransformed data was concluded to be the appropriate geostatistical technique for Case Study tailings resource. Analysis of mineralisation continuity (variography), a pre-requisite for geostatistical techniques such as Ordinary Kriging applied on the case study tailings resource, was also performed. Reasonable and sufficient mineralisation continuity was established to exist in the Case Study tailings resource. Although characterised by high nugget effect, these spatial correlations were established to be continuous with ranges of influence well beyond 450 m in all variables. Anisotropic variograms were modelled for all variables and are comprised of nested structures with two to three spherical models. Resource estimation was conducted through Ordinary Kriging in Datamine. All the seven variables were successfully interpolated into each cell of the 5m x 5m x 5m block model. Rigorous validation of the resource model was performed to establish the quality and reliability of the estimation carried out. Estimated resource model was analysed against the original borehole data, through comparison of grade profiles, statistical analysis, QQ Plots and histograms. The grade profile was recognised to be similar between boreholes (5 m composites) and the adjacent cells that have been estimated. Furthermore, statistical analyses revealed minimal differences between means of the estimated model and the original borehole data: the highest difference being 1.7% realised on 3E, followed by 1.1% on Density and Gold (Au). The rest of the variables (Pt, Pd, Cu, and Ni) have differences that are below 1%. 3 QQ plots and histogram were plotted from resource model with 5m x 5m x 5m cells and 5 m composited boreholes. Although these data sets are of different (slightly incompatible) supports, the intended purpose of comparing distributions was achieved. QQ plots and histograms revealed approximately identical shaped distributions of the two data sets, with some minor deviations noticeable in graphs of only two variables (Au and Density) that are underlain by two populations. The validation process carried out gave a compelling assurance on the quality and reliability of the resource model produced. The Case Study tailings resource therefore is successfully estimated by Ordinary Kriging. The results achieved on the Case Study tailings dam has successfully proved that geostatistical principles and theories can confidently be applied, in their current form or understanding, to any man-made tailings resource / CK2018

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Thesis (MTech (Cartography))--Cape Peninsula University of Technology, 2016. / This case study investigates the change in the geographical boundaries by creating a Spatio-temporal mapping of Ciskei (one of the so-called Bantustans or Homelands) during the period of Apartheid. It examines the reasons for its establishment, and what impact the apartheid land legislation had on the geographical boundaries of Ciskei. GIS technology was used in this study to create time series animation and Static map to display the spatial change of the Ciskei boundaries. This investigation was split into quantitative and qualitative assessments. The aim of the quantitative assessments was to determine the amount of the spatial change of the Ciskei geographic boundary. The qualitative methods was used to investigate the map viewer’s understanding of the amount of the information in the static and animated maps. The results of qualitative assessments showed that static and animated maps have their respective advantages in the visualization of the map viewer. The importance of this research is to take advantage of time series mapping techniques to study the homeland areas in South Africa and see all the changes that have occurred as a result of a period of apartheid legislation. For this research, the following data were gathered: Attribute and metadata was the legislation and laws related to the land and the geographic data was the historical maps and coordinate data.

A Dissertation submitted to the Faculty of Engineering and the Built Environment, University of the Witwatersrand, in fulfilment of the requirements for the Degree of Masters in Engineering by research only and to the Stochastic Mine Planning Laboratory, McGill University as part of the COSMO scholarship in Mine Planning under uncertainty Johannesburg, 2017 / This dissertation discusses the use of stochastic orebody modelling techniques for assessing geological uncertainty associated with gold mineralisation at Geita Gold Mine in Tanzania, and proposes a practical methodology that can be applied to similar studies. As part of the pre-feasibility stage studies for underground mining at Geita, stochastic simulations were required to assess the geological uncertainty associated with isolating (modelled) high grade lenses that occur within the known low grade mineralisation currently targeted for underground mining. Two different simulation techniques are applied in this research: Sequential Indicator Simulation to generate lithofacies realisations from which to assess ore category boundaries and shapes for use in quantifying volumetric uncertainty; and Direct Block Simulations to simulate gold grade realisations from which to assess grade uncertainty. This study identified potential upside and downside mine planning scenarios for volumes and total metal content from the ore category and grade simulations respectively. The findings of the results demonstrated that the high grade zones are much more broken up and discontinuous than the currently modelled high grade shape. The current business case uses a probabilistic high grade shape based on a single grade indicator and a probability choice of 50 percent as the threshold for high grade. The results of the study consider a simulation of possible outcomes based on the same threshold grade indicator and hence quantify the uncertainty or total geological risk. This geological risk may be introduced to mine designs, production schedules and NPV predictions The stochastic workflow developed can be applied to analogous deposit types to assess the risk related to geological uncertainty. The work includes a description of practical considerations to be accounted for when applying the techniques. / MT 2017

Orientador: Morel de Passos e Carvalho / Banca: Marcelo Andreotti / Banca: Cassiano Garcia Roque / Resumo: A cultura da cana-de-açúcar representa hoje grande fonte de divisas para o Brasil, seja pela produção de açúcar quanto pela produção de álcool etílico. A modelagem geoespacial permite a descrição quantitativa da variabilidade espacial dos atributos do solo e a estimativa não tendenciosa da variância mínima de valores desses atributos em locais não amostrados. Acessar essa variabilidade faz da geoestatística uma eficiente ferramenta de suporte na decisão do manejo do solo para incremento de produtividade das culturas. Este trabalho teve como objetivo analisar as correlações, lineares e espaciais, entre a produtividade da cana de açúcar e os atributos químicos do solo, determinando aqueles que melhor se relacionaram com o aumento da produtividade agrícola em questão. O trabalho foi realizado no município de Rubinéia (SP), cujo solo da área é um Argissolo Vermelho Eutrófico. O local das coletas de dados de fertilidade do solo e da produtividade da planta foram através da alocação de uma malha geoestatística com 121 pontos amostrais em duas profundidades (0-0,20 e 0,20-0,40m), numa área de 1,30 ha, com a distância entre os pontos de 10 x 13 metros. As análise de produtividade foram realizadas a campo por meio de contagem de número de plantas e pesagem de colmos em uma área de 9 m² por ponto amostral. A série de dados foi normal para todos os atributos tecnológicos da cana-de-açúcar, enquanto que, para os atributos do solo na profundidade de 0-0,20 m os atributos que apresentaram normalidade dos dados foram o pH, o potássio e a saturação por bases (V%), e na profundidade de 0,20-0,40 foram o pH, o teor de magnésio e a saturação por bases. A média da produtividade de colmos foi de 94,6 t ha-1 e a número de colmos com média de 11,48 colmo m2. Para os atributos do solo, os que apresentaram maiores... (Resumo completo, clicar acesso eletrônico abaixo) / Abstract: The culture of sugarcane today represents big source of hard currency for Brazil, either by production of sugar or by production of ethyl alcohol. The geospatial modeling allows the quantitative description of the spatial variability of soil attributes and unbiased estimate of the minimum variance values of these attributes in locations non-sampled. Access this variability makes geostatistical an efficient decision support tool in of the soil management to increase crop productivity. This work had as objective analyze the correlations, linear and space between the productivity of sugarcane (Saccharum spp.) and soil chemical attributes, determining those that best correlated with the increase in agricultural productivity at issue. The work was conducted in the municipality of Rubinéia (SP), the soil in the area is an Red Ultisol eutrophic. Installed a geostatistical grid for data collection of soil and plant, with 121 sampling points in two depths (0-0.20 and 0.20-0.40 m), an area of 1.30 ha, with distance between dots of 10 x 13 meters. The analysis was conducted field productivity means for counting number of plants and weighing of stems in an area of 9 m² per sample point, analysis of the technological plants were performed in the laboratory of power plant Vale do Parana S/A using the methodology of CONSECANA. The data series was normal for all the technological attributes of sugarcane, whereas for the soil attributes in the 0-0.20 m depth the attributes of the data were presented normality pH, potassium (K) and base saturation (V%), and in depth from 0.20-0.40 the pH, magnesium (Mg) and V%. The average production was found in 94.6 t ha-1 and plant population obtained an average 11.48 p. m-2. For the attributes of the soil, to the highest averages were potassium layers 0-0.20 and 0.20-0.40 (K1 and K2), Magnesium layers... (Complete abstract click electronic access below) / Mestre