Search Documents

By P. J. Le Roux

"Mining companies are under constant pressure to reduce their cost structures to sustain profitability. In mines using an open stoping mining method, dilution often has a significant effect on profitability. At Target mine in South Africa, dilution in some open stopes was found to exceed 10%, falls of ground being a major contributor to the problem. This dilution can reduce the recovered grade from 5.5 to 4.5 g/t, resulting in a potential loss of about R21 million per month based on a gold price of R240 000 per kilogram. In addition to the cost of dilution, the cost of damage to, or loss of, trackless equipment as a direct result of falls of ground in open stopes is significant. Other associated costs include transport, hoisting, secondary blasting, milling, and plant treatment of waste material. Stable open stopes are essential in order to reduce falls of ground. Back-analyses of stope instability at Target mine have indicated that conventional rock mass failure criteria are unsuitable for stope design. An alternative strain-based criterion has been developed, and has proved to be very successful, allowing the stability of open stopes to be calculated reliably. Since its implementation in 2010, dilution and equipment damage have decreased markedly, creating value for the mine. IntroductionMines that make use of an open stoping mining method aim to extract only the ore, leaving the waste behind. Many orebodies have well-defined boundaries between the ore and the waste rock. In such orebodies, the introduction of waste into the ore due to overbreak dilutes the grade. In massive disseminated orebodies, dilution is less problematic. Overblasting, falls of ground, or other causes may result in dilution. A study undertaken in Canada (Pakalnis et al., 1995) found that approximately 51% of all underground metal mines utilized open stoping mining methods and, from surveys conducted at these mines, open stoping operations experienced dilutions of up to 20%, and occasionally more. Dilution of this magnitude has a significant economic impact on any mining venture. Research carried out in Australia by Capes (2009) came to the same conclusion."

Jan 1, 2017

1. Book review Technology of metal powders- recent developments, edited by L. H. Yaverbaum. New Jersey, Noyes Data Corporation, 1980. 400 pp. (Reviewer: B. K. Loveday) 2. New books The use of microprocessors, by M. Aumiaux. New York, Wiley, 1980. 176 pp. $25,50. Aluminium smelter technology, a pure and applied approach, by K. Grojtheim and B. J. Welch. Diisseldorf, Aluminium-Verlag GmbH (P.O. Box 1207, D-4000, Federal Republic of Germany). 146 pp. DM48. Coal-bridge to the future. New York, Harper & Row, 1980. $35. System modeling and response.. theoretical and experimental approaches, by O. Doebelin. New York, Wiley, 1980. 576 pp. $26. Environmental and climatic impact of coal utilization, edited by J. J. Singh and A. Deepak. New York, Academic Press, 1980. 646 pp. $39,50. Proceedings of the Third European Conference on Mixing, York, England, April 1979, edited by H. S. Stephens and C. A. Stapleton. BHRA Fluid Engineering (Cranfield, Bedford Mk43 OAJ, U.K.), 1980. 2 volumes. 500 pp. $50. Lignite technology, edited by P. Nowacki. New Jersey, Noyes Data Corporation, 1980. 228 pp. $42. The application of rock mechanics to cut-and jill mining. London, Institution of Mining and Metallurgy, 1980. Papers presented at the Fifth International Symposium on Jet Cutting Technology, Hanover, Federal Republic of Germany, June 1980, edited by H. S. Stephens and B. Jawis. BHRA Fluid Engineering (Cranfield, Bedford Mk 43 OAJ, U.K.), 1980. 438 pp. £38. World coal resources.. methods of assessment and results, by G. Fettweis. Amsterdam, Elsevier, 1979. 425 pp. $80,50. Geophysical and geochemical techniques for exploration of hydrocarbons and minerals, edited by M. Sittig. New Jersey, Noyes Data Corporation, 1980. 300 pp. $40. Financial Times mining international year book 1980. Longman Group Ltd (Periodicals and Directories Division, Longman House, Burnt Mill, Harlow, Essex CM20 2JE, England). £24 surface, £34 airmail. National Symposium on the Availability of Strategic Minerals, edited by M. J. Jones. London, Institution of Mining and Metallurgy, 1980. 109 pp. 3. NIM reports Report No. 1941 The recovery of chromite and platinum from the UG-2 Upper Chromitite Layer .of the Bushveld Complex at the Pandora Mine (first published 10th Mar., 1978). Report No. .2012 A mineralogical investigation of borehole cores from the Trompsburg Complex (first published 6th Apr., 1979). Report No. 2038 The leaching of sphalerite in acidic ferric sulphate media in the absence of elemental oxygen. Report No. 2052 An investigation of the impurities in native gold by neutron-activation analysis. Report No. 2056 The mineralogy and geochemistry of the copper, lead, and zinc sulphides of the Otavi Mountainland. Report No. 2058 A n elementary introduction to image analysis - a new field of interest at the National Institute for Metallurgy. Report No. 2059 Distributed computer control of industrial processes. Report No. 2060 The analysis, by atomic-absorption spectrophotometry, of activated charcoal.

Jan 1, 1980

By S. C. Maji, U. S. Chattopadyay, S. Chaudhuri, T. Gouri Charan

India is devoid of coking coal resources, which is a vital raw material for coke making. As a result it has to import about 80% coking coal for steel making. Indian National Steel Policy, 2017 has envisaged crude steel production around 300 million tonnes Mt) by the year 2030-31 and forecasts coking coal demand of 161 Mtpa (million tonnes per annum). One way of improving the indigenous resources is to study the performance of existing washeries and to improve the clean coal recovery by adopting suitable technology. A study was carried out, mainly to focus on the fine coal circuit of an existing coking coal washery, to generate the desired quality of clean coal. The operating coking coal washery being studied is presently feeding coal from nine different sources with varying feed ash contents. The coal fines varied in ash content ranging from 19.5% - 43.6%. For treating the coal fines, the washery has operated a spiral for the past few years. Performance studies were carried out, and it was found that the spiral is not efficient in recovering the cleans. Detailed investigations were carried out on the coal fines using flotation techniques, and it was found that to achieve a low ash clean coal, the froth flotation is a much better beneficiation option than a spiral, The studies showed that the recovery of clean coal through a spiral is 67.7% at ash content of 27.0%, while flotation tests showed a recovery of 68.1% at 16.5% ash content. Keywords: Coking coal washery, near gravity material, spiral washer, froth flotation

Jan 1, 2020

By K. J. Craig, M. A. Swart, Q. G. Reynolds, L. Hockaday

A solar reflector soiling study was carried out at a ferromanganese smelter in South Africa to assess the soiling rates at different locations around the plant. Several meteorological parameters were monitored to give insight into the conditions that lead to increased soiling. Mineralogical characterization of dust samples collected from the reflectors and the atmosphere revealed that only a certain size fraction is of importance with regard to soiling, and that the dust can be attributed to both raw materials and smelter products. Proximity to the dust source was the primary driver for increased soiling. The site that experienced the most soiling was very close to raw material heaps; this was deemed an outlier and was excluded from the summary statistics. The secondary driver for increased soiling was location relative to the smelter dust sources and the wind’s direction and speed. The reflector set at the best location experienced 13.1% less soiling than the set at the ‘worst’ (but still feasible) location, represented by an averaged mean daily reflectance loss of 0.0186. The study revealed that while there are periods of intense soiling at this particular site, proper planning of reflector location in relation to the smelter dust sources can significantly mitigate the soiling rate.

Jun 5, 2023

By B. P. Watson, A. Carpede, W. O. Kekana, N. Fernandes, W. Theron, M. P. Mahlangu, G. Betz

The research described in this paper was done to confirm the Upper Group 2 (UG2) PlatMine peak pillar strength formula (Watson et al., 2007), which was determined from a back-analysis of failed and unfailed pillars. Underground measurements were made on a stable pillar that was loaded by firstly reducing it’s length and then by mining the surrounding pillars until pillar failure took place. The pillar was instrumented with suitably positioned strain cells and closure meters, which allowed both the average pillar stress and strain to be determined. The paper describes the methodology applied to identify a suitable position for the instrumentation, as well as the results. A stress/strain curve is presented for a UG2 pillar with a w/h ratio of 2.0, at Booysendal Platinum Mine. The measured pillar strength was similar to the predicted strength using the PlatMine pillar strength formula for UG2 pillars. The PlatMine formula has been successfully implemented on Booysendal Platinum Mine, and about 3 670 pillars have been cut without a single failure. An additional revenue of US$1.3 billion was calculated for the 25-year life of the mine as a direct result of the improved pillar design, given the January 2020 platinum group metals basket price. An extended life of mine and better mining efficiencies will also be realized.

Aug 1, 2021

By A. Yañez, J. Yianatos, P. Vallejos, R. Grau

This paper presents an evaluation of the effect that the mineral feed characteristics have on the metallurgical performance in an industrial flotation circuit. Variables such as mineralogy, particle size, solid content and mineral grade were evaluated using a flotation simulator, which was previously developed and validated using industrial data from different concentrators. The metallurgical performance was evaluated not only from the point of view of recovery and concentrate grade, but also by analysing less common variables such as froth carrying rate, froth and collection recovery, water recovery and others. Then, new data from the industrial flotation circuit was used to complement this analysis and evaluate the simulator predictability by comparing simulated and measured data. The results allowed for the quantification of the significant effect that mineralogy and particle size have on recovery and/or concentrate grade. Therefore, the grinding stage and flotation cells design become important factors to avoid a significant loss in metallurgical performance when feed mineral characteristics change in an industrial operation. In this context, an evaluation of cells design and arrangement should be performed to compensate the loss in recovery and concentrate grade in order to find an optimal operating condition. Keywords: Flotation circuits, mineral feed, modelling, particle size, liberation, mineralogy

Jan 1, 2020

By Q. Liu, S. Liu, Y. Cui

To reduce the frequency of mining incidents and improve safety and production efficiency, it is important to develop technology for the autonomous positioning and navigation of mining equipment, without the assistance of satellite navigation signals. In this study we review the current technological environment for positioning and navigation underground and present several integrated navigation strategies. Existing underground positioning and navigation technology is described, and the performances of different methods based on this technology analysed and compared. The combination types and positioning effects of underground positioning and navigation strategies are discussed. Recent research directions for filtering methods and positioning estimation algorithms for underground positioning and navigation are reviewed. Finally, we summarize the future direction of underground positioning and navigation technology and related algorithms.

Jun 1, 2021

By L. C. Woollacott, A. K. Tripathy, J. H. Potgieter

This paper presents the second of a two part study on density stratification in a laboratory batch jig. The first study, reported elsewhere, investigated the effect of operating conditions on the stratification profile achieved at equilibrium. The study reported in this paper investigated their effect on the kinetics of density stratification under 18 different sets of operating conditions - five for binary systems with different particle densities; and 13 for a ternary system with different configurations of the jig cycle. Although the study was academic in nature (it involved simple systems of 8 mm glass beads) it has a number of practical implications. It found that the most influential operating variables affecting stratification kinetics are the amplitude of water displacement, the pulsion hold time, and, to a lesser extent, the pulsion time. It also found that reasonable rates of stratification can be expected in systems with density differences as low as 237 kg/m3 but below this the kinetics deteriorate significantly. The two studies together suggest that, within a broad range of operating conditions, the equilibrium stratification profile is independent of control variables and it is the control of stratification kinetics that has the primary influence on the quality of density separations in a jig. Keywords: Jigging kinetics, stratification, equilibrium stratification, batch jig

Jan 1, 2020

By U. Atici, R. Comakli

"Understanding rock properties is of vital importance in mining and civil engineering. Rock properties are generally determined through laboratory tests, which require a large number of uniformly shaped samples and technicians who use indirect methods to determine rock properties. The texture coefficient (TC), which represents various textural parameters including the size, shape, orientation, and relative proportions of grains and the matrix, is an efficient means for indirectly evaluating the mechanical behaviour of rocks. In this study, 12 different rocks (ten granites, one diorite, and one gabbro) were collected and their engineering properties. including specific gravity, porosity, P- and S-wave velocities, Schmidt rebound hardness, uniaxial compressive strength, and elasticity modulus, were investigated. Relationships between TC and these measured properties were assessed. Statistically significant (at the 95% confidence level) correlations were found between the TC and compressive strength, elasticity modulus, sound velocity (both compression and shear waves for dry and saturated samples), but no significant correlation was found with Schmidt rebound hardness. The results suggest that it is possible to use the relationships between TC and the mechanical properties of rocks in rock classification. IntroductionIn every aspect of rock engineering, knowing rock properties is essential for determining and predicting the behaviour of rocks and rock masses. Mechanical properties of rocks are important parameters in mining, geology, and engineering, and they are affected by textural characteristics such as grain size, grain shape, cement type, degree of cementation, mineral content, and the ratio of the grain area to the matrix area. The effects of textural properties on the physical and mechanical properties of rocks have been investigated for a long time, and previous studies revealed close relationships between single parameters related to rock texture and mechanical properties of rocks (Akesson et al. 2001; Gunsallus and Kulhawy 1984; Merriam, Rieke, and Kim, 1970: Onodera and Kumara 1980; Prikryl 2006; Ulusay, Tureli, and Bider, 1994; Tugrul and Zarif 1998). Rock properties are usually determined using mechanical tests, which require uniformly shaped and relatively large samples. However, it is not always possible to obtain such samples. Furthermore, rocks with similar textural and structural properties may exhibit different mechanical behaviour. Howarth and Rowlands (1987) developed a texture coefficient (TC) that expresses all the main textural parameters of a rock in a single dimensionless quantity based primarily on statistical assessment. The TC makes it possible to understand the effects of a rock’s textural properties on its mechanical properties using thin sections instead of large, uniform samples. Ersoy and Waller (1995) investigated the relationship between strength and TC for limestone, sandstone, and siltstone; they also examined the effect of rock properties and TC on drilling performance. Ozturk, Nasuf, and Bilgin (2004) explored the relationship between rock quality and TC by evaluating the uniaxial compressive strength (UCS) values of limestone, basaltic andesite, and tuff. Gupta and Sharma (2012) evaluated TCs as well as the petrophysical and mechanical properties of quartzite from the lower and higher Himalayas and found a strong relationship between TC and UCS. Similarly, Ozcelik, Bayram, and Yasitli (2013) investigated the relationship between TC and various parameters in marble by applying the statistical analyses developed by Ozturk, Nasuf, and Kahraman (2014)."

Jan 1, 2019

By K. Karaman

This study investigated the effect of surface roughness on Schmidt rebound hardness (RL). Four different test surfaces of rock samples were studied: natural, ground, cut surfaces, and core samples. There was significant variability of standard deviation based on the RL on the natural surface, which indicated high roughness of the rock surface, whereas surface polishing caused a significant decrease in standard deviation. ISRM and ASTM methods were compared to estimate unconfined compressive strength (UCS) for different testing surfaces. RL obtained from the cut surface was found to be more reliable than those obtained from other testing surfaces for the prediction of UCS; however, regression and ANOVA analyses revealed that the ISRM method gave a more accurate UCS estimation of rocks with highly rough surfaces. It was also shown that RL values obtained from a cut surface were significantly higher than those obtained from core samples. Therefore, a comparison between RL values obtained from core samples and cut surfaces was made based on previous studies. This study statistically showed that estimated UCS values are not statistically significant if Schmidt rebound tests are not performed on similar surfaces. In addition, the mineral grain sizes of the studied rocks, different testing surfaces compared with those in literature, and standard deviation from RL are evaluated and discussed. The Schmidt hammer technique is a rapid, inexpensive, and straightforward method for estimating UCS for preliminary assessment; however, roughness of the surface should be eliminated if variations are shown in the surface rebound hardness.

Apr 14, 2024

By G. Akdogan, S. M. Bradshaw, M. I. Isaacs, P. Van Wyk

In the mineral processing industry, Ni-Cu-Fe-S converter matte is leached in order to remove base metals from a concentrate containing platinum group metals (PGMs). This paper investigates the atmospheric leaching process, in order to develop a better understanding of the leaching behaviour of the base metals; in particular Cu, Fe, Ni, and the PGMs in particular Ru and Ir with key process variables, acid concentration, copper concentration under oxidative leaching conditions. Higher initial acid concentration resulted in a higher copper, iron and nickel extractions, as well as higher leaching rates together with faster Cu, Ru, Ir precipitation occurring. On the other hand, a higher initial copper concentration resulted in more nickel but less iron being extracted. Keywords: PGM, converter matte, leaching, precipitation

Jan 1, 2020

By O. Koatlhai

Mine closures have traditionally prioritised environmental rehabilitation, often overlooking the deep socio-economic disruptions that follow, particularly in mining-dependent communities. This paper argues that mine closure should not mark the end of prosperity but rather the beginning of a sustainable transition. The coproduction model, which integrates community participation, governance, and economic planning, offers a transformative approach to closure planning. Drawing on case studies from Ghana, Zambia, and Indonesia, this research examines the successes and failures of past closures, highlighting the critical role of stakeholder engagement, skills development, and economic diversification. By incorporating emerging technologies such as AI-driven economic forecasting, blockchain for governance transparency, and decentralised finance (DeFi) models, mining companies can ensure a just and sustainable post-mining future. This study challenges outdated closure models and presents a forward-looking strategy that sees mine closure not as an endpoint, but as a catalyst for social resilience and economic transformation. It underscores the need for policy reforms, stronger governance, and investment in alternative industries to prevent mining towns from becoming economic wastelands. By shifting the paradigm from remediation to regeneration, this research calls for a new era of mine closure planning that puts people and prosperity at its core.

Apr 25, 2025

By K. B. Le Bron, T. van Aard

The empirical pillar strength formulae have been widely, and in most cases successfully, applied to design underground pillars for mines in the Bushveld Igneous Complex (BIC). This paper presents the case for combining the empirical method with numerical modelling when it comes to crush pillar stability analyses. The benefits include an understanding of potential failure mechanisms, which may then be accounted for in the designs. In addition, the impact of weak layers, such as shear zones, the impact of the dip of the orebody, differences in layouts, such as holing widths and the presence of advanced strike gulleys (ASGs) immediately adjacent to the crush pillars, and stress regime (k-ratio) may be studied using numerical modelling, as it is not accounted for in the empirical method.

Mar 5, 2025

Slope stability is a fundamental part of a successful mining operation. It impacts directly on the safety of personnel and the economics of the mine. The Jwaneng Mine in Botswana is planning a push back of 644m depth in the south east wall. Mining will commence in 2010 and be completed in 2024. This will put the mine in the deep pit category. The mine has invested in extensive geotechnical data gathering and design programs that started as far back as 2003 for the south east wall. The south east wail of Jwaneng Mine is characterized by foliation that dips adversely into the mining faces. The orienta1ion of the foliation is variable due to intense tectonic movements that have also caused the occurrence of faults that are sub-vertical. A considerable amount of information on the characteristics of structural patterns and the rock mass has been collected. This paper presents an overview of the feasibility level geotechnical design that is about to be concluded.

Jan 1, 2010

By X. Chen, G. Wang, D. Wan

Computational fluid dynamics have been applied for detailed modelling of particle-bubble collisions. The most accurate approach in the previous work is using the point particle method to treat particles and bubbles. Nevertheless, the sizes of particles and bubbles are generally bigger than the Kolmogorov scale in the high turbulence of the flotation environment, which makes the point particle method inappropriate. This paper describes particle-resolved direct numerical simulation on the collisions between particles (denoted as heavy particles) and bubbles (denoted as light particles) in a homogeneous isotropic turbulent flow field, and a comparison with the point particle method.. A largescale stochastic forcing scheme was implemented within the mesoscopic multiple-relaxation-time LBM approach to maintain turbulence intensity at targetted levels. Both kinematic and dynamic collision kernels were evaluated to study the correlation of total collision events and turbulent kinetic energy (TKE) dissipation rate. The results show that collisions between particles increase with the TKE dissipation rate as radial relative velocities between particles increase with it. In the particle-resolved simulations, the dynamic collision kernels were found to not match with the kinematic collision kernels as in the point particle method. Quantitatively, they were approximately 4 to 10 times smaller than the corresponding dynamic collision kernels. Therefore, it is not reasonable to estimate the kinematic collision kernel in particle-resolved simulations as its rationales are not valid. However, the collision behaviour can still be accounted for by means of the dynamic collision kernel since it is obtained from the direct statistical analysis of the collision events. Keywords: Lattice Boltzmann Method; particle-resolved simulation; multiphase flow; collisions

Jan 1, 2020

By A. Y. Senchenkoa, Y. V. Kulikov

The relationships revealed between ore hardness and specific energy consumed by crushing and grinding are described in the paper. A comparison of key technical and economical parameters of SAG and HPGR circuits is made, based on testing of several ore types for greenfield projects. The comparison of results make it possible to estimate efficiency of the technologies application, depending on ore hardness and taking into account potential risks for a circuit under design. Keywords: Comminution circuits, SAG, HPGR, test results

Jan 1, 2020

By S. O. Bada

Coal is a complex mixture of organic and mineral constituents and is the most abundant resource of fossil energy in the world. In recent years, significant research into dry coal beneficiation has gained much attention, primarily due to the need to improve grades and reduce the environmental contaminants in coal without the use of water, and to achieve this in as cost-effective manner as possible, relative to wet beneficiation processes. This paper seeks to review the application of various electrostatic separators with their process principles, to draw comparisons between different dry beneficiation techniques with specific emphasis on the tribo-electrostatic separation method, and finally to report the results of tribo electrostatic separation conducted on various South African coals. Previous research conducted on Indian, European and American coals has indicated that this technique is likely to lead to significant economic benefit through the reduction of ash content, NOx and more specifically Sox by separating out the liberated Fe-S-bearing minerals prior to utilization. The removal of the latter suite of minerals is also likely to significantly reduce or eliminate the emissions of associated trace elements, including mercury and arsenic. The research results reported in this paper indicate that the rotary tribo-electrostatic process has the potential for significant upgrading of high ash pulverized South African coal. The impact of various operational parameters was investigated and key factors established for the optimum recovery of low ash and low sulphur fine coal.

Jan 1, 2010

By S. Bradshaw, A. Mabentsela

"Physical and numerical modelling methods were followed to identify mechanism(s) for tundish slag entrainment in a bare tundish and one with a flow control device (FCD).The physical and numerical models made use of water and paraffin to model steel and slag respectively. Observations from the physical model showed that the steel-slag interface remains immobile in both cases. Entrained paraffin formed droplets approximately 1 mm in diameter. Results from both models (numerical and physical) showed that in both cases (bare and FCD case), areas of high entrained slag concentration exist near the inlet region. The entrained slag concentration decreases towards the tundish endwalls.Flow patterns and velocities tangential to the steel-slag interface from the numerical model showed that slag entrainment in both the bare tundish and tundish with a FCD possibly takes place via two mechanisms. First, the slag moves across the steel-slag interface via mass transfer; secondly small velocities tangential to the interface at depths greater than 10 mm below the interface carry the already ‘entrained’ slag into the bulk steel phase. These tangential flow patterns are dominant in the inlet region, hence the high concentration of entrained slag in this region. IntroductionThe tundish serves as the last metallurgical vessel through which steel passes before solidifying in the moulds. It is in the tundish that the last traces of non-metallic inclusions should be removed, otherwise the inclusions carry over to the moulds and cause defects in the steel product. In the tundish, the slag is less dense than the melt and thus resides on the top of the melt. The slag provides a sink into which the non-metallic inclusions float to and dissolve. The slag also protects the melt from air and heat loss. However, when slag is entrained in the melt as a result of increased turbulence or shearing at the steel-slag interface, slag can become a source of nonmetallic inclusions.Numerous studies on flow patterns in the tundish have been carried out since the introduction of the tundish. These studies were aimed at improving the flow characteristics in the tundish (Chattopadhyay, Isac, and Guthri, 2010; Cloete, 2014; Jha, Dash, and Kumar, 2001; Jha, Rao, and Dewan, 2008;"

Jan 1, 2017

By M. S. Shishvan, S. Klerkx, J. Sattarvand

The inaccurately drilled blastholes from the planned locations cause changes in the spatial distribution of explosive energy; some areas of the blast may receive a surplus of explosive energy, while others see a reduction. This results in less uniform fragmentation, reduces the efficiency of mining operations and increases costs. Additionally, it can lead to slope stability issues, flyrock, and uneven benches. To mitigate these problems, this study attempts to reduce differences in the explosive energy distribution (EED) through an optimization approach that adjusts the height of explosive columns in blastholes. Two optimization algorithms, the Genetic Algorithm (GA) and Tabu Search (TS), are tested. The latter is preferred for its local search neighbourhood strategy that gives superior computation times. Applications on different blast patterns result in improvements of 0.53–1.54%, or 2.14–3.94% when only the affected blocks are considered.

Sep 1, 2021

[Consider the situation where we wish to estimate the value of a block of ore W from the values of n samples wt (i = 1, 2.... n). The samples can be located anywhere inside or outside W. The blocks and the samples can have any size or shape (Fig. 8.1). We define ws as the set of all the samples wi: WS = Iw1; W2; W3.. . wn and we wish to estimate the block W using the samples ws. Let xi be the value of the ith sample and µw the unknown average value of the block. We can estimate µw from the average value of all the samples. Let µw be the estimator of µw. As the samples need not be of equal size, the sample average is: µw = wi xi/~ wi. (8.1) The mean squared error made when estimating µw by µw is known as the variance of estimation of W by ws. It is the error made when giving the value of the sample set ws to the block W and is written o2E (WS to W). By definition: o2E (ws to W) = E [(µw - µw)2]. (8.2) If n = 1, this error is known as the extension variance. Since the sum of the weights given to the sample values is equal to 1,0, the estimator µw is unbiased (the question of the absence of bias is studied in more detail in Chapter 9): E [( µw - µw)1 = 0. (8.3) The error (µw - µw) has zero mean and a variance o2E = E (W8 to W). If we assume that this error is normally distributed, we can define confidence limits for µw. Let tP be the value of the standardized normal variance t such that the probability that t is less than tP is p: (t,) = .\/27T f exp (- t2/2) dt = p (8.4) The probability that µw is larger than µw + t1-P oE is p. The probability that µw is smaller than µw- t1-P oE is p. The following approximation formulae are often used to calculate 2p central confidence limits for the block value: 68% central confidence limits (p = 16%): µw - o2E, and µw + o2E. 95 % central confidence limits (p = 2,5 %): µw - 2 oE, and µw + 2 oE. Tables of the function 0 can be found in any textbook on statistics (for example, Fraser, 1958, p. 388) (see Table 7.2). [ ] The last line of Table 2.2, corresponding to f = 00, also contains the value of tP for p = 80, 90, 95 and 97,5 %.]

Jan 1, 1978