Assessment of the Impact of Non-Additivity in the Estimation of Iron Ore Attributes - with narration
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For the AusIMM Iron Ore Conference, Perth, 2013.
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Assessment of the Impact of Non-Additivity in the Estimation of Iron Ore Attributes - with narration
- 1. Assessment of the Impact of Non- Additivity in the Estimation of Iron Ore Attributes Iron Ore 2013, 12th – 14th August 2013 Alastair Cornah
- 2. Outline
- 3. Planning, development and execution Value add Heterogeneous in situ chemical and physical attributes Homogeneous product chemical and physical attributes Product marketing Deposit level sequencing and blending Life of mine planning and sequencing Production planning and sequencing Mining and processing Stockpiling and blending Data collection, but incomplete knowledge Sparse drillholes Infill drilling Blast holes / grade control drilling Local block models Global models, local block models Grade control models Spatial estimation models
- 4. Mean in situ grades Mean product yields Mean product grades Mean in situ grades Mean product yields Mean product grades Destination 2 Destination 1 In situ grades Product yields Product grades variable Response Linear average of data understates responseResponse variable Linear average of data overstates response
- 5. Rock Response Properties Primary Rock Properties Energy or Process Primary – Response Framework, modified from Coward et al., 2009. In situ geochemical grades Bulk and particle density Mineral abundances Material type abundances Texture Primary Response Davis Tube concentrate recovery and grades Porosity Dry crush and screen / beneficiation product yields and grades Material handling, degradation metrics Sinter performance metrics (TI, RDI, RI)
- 6. Compare between: 1. The average of the ratios; and 2. The ratio of the averages
- 7. Additive variables are shown in blue, non additive variables are shown in red using the following notation:
- 8.
- 9. Direct (illegitimate) approach Samples or nodes Averaged samples or nodes Indirect (legitimate) approach
- 10. Global estimation approach % Fe units recovered to concentrate % Fe units lost to tails Direct (illegitimate non- additive procedure) 76.36 23.64 Indirect (legitimate additive procedure) 76.60 23.40
- 11. Direct (illegitimate non-additive procedure) Realisation 5 / 25
- 12. Direct (illegitimate non-additive procedure) Realisation 10 / 25
- 13. Direct (illegitimate non-additive procedure) Realisation 15 / 25
- 14. Direct (illegitimate non-additive procedure) Realisation 20 / 25
- 19. VARIABLE Mean (biased - unbiased) Realisation 1 -0.2 Realisation 2 -0.2 Realisation 3 -0.21 Realisation 4 -0.23 Realisation 5 -0.35 Realisation 6 -0.29 Realisation 7 -0.28 Realisation 8 -0.24 Realisation 9 -0.34 Realisation 10 -0.23 Realisation 11 -0.17 Realisation 12 -0.67 Realisation 13 -0.24 Realisation 14 -0.11 Realisation 15 -0.4 Realisation 16 -0.65 Realisation 17 -0.33 Realisation 18 -0.43 Realisation 19 -0.12 Realisation 20 -0.19 Realisation 21 -0.86 Realisation 22 -0.57 Realisation 23 -0.04 Realisation 24 -0.39 Realisation 25 -0.1 Residuals, % (biased – unbiased)
- 20. Conclusions and Recommendations Key References: Carrasco P., Chiles J.P. and Seguret S., 2008. Additivity, Metallurgical Recovery and Grade. In Geostats 2008: VIII International Geostatistics Congress (eds: J. Ortiz and X. Emery), Santiago, pp.465-476. Springer. Coward, S., Vann J., Dunham S., and Stewart M., 2009, The Primary-Response Framework for Geometallurgical Variables. In Proceedings Seventh International Mining Geology Conference, pp.109-113. The Australasian Institute of Mining and Metallurgy: Melbourne. Correspondence: ac@qgroup.net.au
Editor's Notes
- The paper firstly discuss the role of spatial models in the iron ore value chain. Many different types of variables are spatially estimated during iron ore project evaluation and production, generally under the assumption of additivity. For some of these attributes the additive assumption will be perfectly valid, for others potentially not. The primary-response framework provides some conceptual guidelines around modelling of variables and some iron ore variables are discussed within this context. Finally, an additivity testing mechanism is used to quantify the bias resulting from direct linear estimation of a non-additive attribute from a magnetite project.
- The iron ore value chain fundamentally involves the extraction of a mineral deposit, or a portfolio of complimentary deposits whose in situ chemical and physical attributes are heterogeneous, as can be seen in this drill core. The deposit or deposits are mined, processed and blended into relatively homogeneous products. Essentially product homogeneity is engineered from in situ geological heterogeneity within a number of business and system constraints. The value maximising sequencing of production, processing and blending that is required to achieve this is generally determined through a multi-scaled hierarchy of planning which may range from global sequencing of deposits over the 50 year plus timescale, through to daily tactical planning of production, stockpiling and blending.Importantly, at all scales within that hierarchy of planning, knowledge of the mineral deposit’s in situ attributes and the expected outcomes of mining and processing it are based upon non-exhaustive data which ranges from sparse drillholes to grade control information. Global models, local block models, and grade control models are therefore required to inform the entire production planning process.The hierarchy of planning processes can only truly maximise value if the models upon which they are based represent unbiased estimates.
- Spatial estimates are typically built either using classical geostatistical techniques, or through some other form of linear averaging. Herein lies an implicit assumption that the attributes of interest behave additively; that is that the arithmetic mean can legitimately be calculated by a linear combination of the available data, without creating bias.In iron ore spatial estimation of geometallurgical attributes in addition to grade attributes is well established. It is fundamental to the challenge of engineering product homogeneity from in situ geological heterogeneity. For example product yields and grades are sometimes predicted in grade control. Some form of averaging is used to determine mean values for the attributes over blockouts, allowing them to be assigned to processing or stockpiling destinations. However, in the broader geometallurgy field, non-additivity of these types of variables is seen as a key challenge. Geomet variables generally result from non-linear process and often behave non-additively. If the attributes behave non-additively, linear averaging will result in some degree of positive bias, or negative bias, depending upon the nature of the non-linearity. This may potentially result in misclassifications and sub-optimal decision making, depending upon the materiality of the bias.The pertinent question therefore is which iron ore attributes might behave non-additively and what is the materiality of impact resulting from treating them as additive?
- An excellent way to address these questions is to consider the iron attributes within the context of the Primary-Response framework which was proposed by my colleague Steven Coward and his co-authors in 2009. Spatial attributes are classified based upon the degree to which they represent either an intrinsic property of the rock mass (‘primary variables’) or a response to an energy or a process applied to the rock mass (‘response variables’).In situ grade attributes represent a concentration per unit mass of the in situ rock and are thus clearly primary attributes. Density, porosity, texture, and mineral or material type abundances per unit of volume can be considered as primary attributes, even though they may be measured indirectly. Primary variables are inherent properties of the rock mass and are independent of mining methods, plant operating parameters and material handling; they generally represent additive quantities which can be directly estimated.Even in the simplest dry crush and screen processing cases for hematite projects, the yields and grades reporting to products are not an inherent property of the rockmass, but rather a response of that rockmass to the various energies that are applied to it during blasting, mining, and processing. Product yields and grades including those derived through Davis Tube Analysis in magnetites are response variables. Other examples include material handling and degradation metrics, as well as sinter performance metrics. See the full paper for further comments on each.Caution is suggested in directly estimating response variables into spatial models; they typically represent responses to non-linear processes and may behave non-additively. Best practice is to estimate the primary variables that underlie the responses and develop transfer functions between the two.But supposing we must directly estimate a response variable; how can the impact of potential non-additivity be quantified?
- One approach is to use a mechanism which was proposed by Pedro Carrasco and co-authors in 2008 to test the impact of treating the recovery of copper metal from a sulphide ore as an additive variable.This is effectively the ratio of the metal within the concentrate to the in situ metal and is probably non-additive, but it can be expressed as the ratio of two additive variables.The test amounts to comparing between the average of this ratioagainst the ratio of the averageof the two additive variables. If the metal recovery is an additive quantity the two calculations will yield exactly the same results.The same mechanism can be applied to test the impact of directly estimating the recovery of in situ iron units into a lumps or fines product, or as in the following case study, into a magnetite concentrate.
- The case study is based upon a Davis Tube Analysis dataset from an early stage magnetite project which comprises 172 determinations for sample head grade, concentrate mass recovery, and recovered concentrate grade (only considering iron in this case).Using the established notation, with additive variables coloured blue and non-additive variables red; we have firstly the additive variables:in situ Fe grade ZH, which represents a typical negatively skewed distributionin situ sample quantity THand from these can be calculated in situ iron units QH.The non-additive variables include product grade ZPProduct quantity TPAnd from these we can calculate product iron units QP.
- Given these variables, we can calculate:Mass recovery (which is non-additive) from TP and TH.The recovery of iron units into product (noted as upper case R) is the ratio of concentrate iron units to the in situ units.Given these calculations, R can be expressed as the ratio of product grade ZP (which is non additive) to in situ grade ZH (which is additive) multiplied by mass recovery.The in situ grade of recovered product (ZR which is an additive variable) is the mass recovery multiplied by the product grade;Therefore the non-additive variable R can be expressed as the ratio of two additive variables ZR and ZH.The direct average of R can then be computed and compared against the ratio of averages of ZR and ZH.
- Schematically, the additive variable ZR is calculated from mass recovery and product grade. In the direct approach, R is calculated as the ratio of ZR to ZH and is then averaged (R is therefore the average of the ratios). In the indirect approach the additive variables ZR and ZH are averaged and R is subsequently calculated as the ratio of these averages. As previously discussed, if the recovery of iron units into product is an additive quantity, these two calculation approaches will yield the same result.
- As an initial test these calculations were carried out at a global level on the sample dataset.The table shows that the recovery of iron units into concentrate is non-additive since the two calculations yield slightly different results. The direct calculation results in an absolute underestimation of 0.24% compared to the legitimate calculation. In some cases this quantum of global difference may be considered material; but in this early stage project, it is certainly outweighed by other geological and metallurgical uncertainties.
- In order to test local impacts of directly averaging a non-additive variable, 25 Conditional Cosimulations of the dataset attributes were generated at 5m spaced nodes. Further details on this are contained in the full paper and the following paper comprehensively covers simulation of multiple correlated attributes.The 𝑍𝑅 variable was calculated at each node for each of the realisations; the plots show punctual realisation index 5 for Zr and Zh across a single mining bench. In the direct approach, R is calculated at each node, and is then averaged into blocks. This approximately mimics the averaging of grade control data into a blast blocks, or the scale up from sample support to block support that occurs in resource estimation.This is repeated for all of the other simulation indices, showing 10, 15, and 20.In order to test local impacts of directly averaging a non-additive variable, 25 Conditional Cosimulations of the dataset attributes were generated at 5m spaced nodes. Further details on this are contained in the full paper and the following paper comprehensively covers simulation of multiple correlated attributes.The 𝑍_𝑅 variable was calculated at each node for each of the realisations; the plots show punctual realisation index 5 for Zr and Zh across a single mining bench. In the direct approach, R is calculated at each node, and is then averaged into blocks. This approximately mimics the averaging of grade control data into a blast blocks, or the scale up from sample support to block support that occurs in resource estimation.This is repeated for all of the other simulation indices, showing 10, 15, and 20.
- In the alternative indirect approach, simulated Zr and Zh are averaged into blocks and these are used to calculate R in the blocks (block R is the ratio of the averages). This is also carried out for each of the simulation indices, showing here index 10, 15 and 20.
- The table shows the differences in the global means of the two approaches for the 25 realisations in absolute percentage terms. A minor negative global bias exists in the direct approach for each of the simulation indices, mirroring the global calculation on the sample values shown earlier.The plots show block by block biases for 4 of the realisations across the bench; reds show positives, blues negatives. Some of the local biases are significant and may result in localised mis-allocations and sub-optimal decision making. Here are the biases shown as histograms. Individual realisations do not tell us exactly where biases will occur; however, local probability distributions for bias can be built based upon the full suite of realisations. Some further analysis of the factors that may affect the local materiality of bias is contained in the full paper, but the summary of the case study is that directly estimating this response variable may result in some local surprises.
- In conclusion, large numbers of spatial attributes are measured during iron ore project evaluation and production, these should be considered within the context of the Primary – Response framework.Response variables often behave non-additively and their direct averages within resource estimation, mine planning, and grade control should be treated with some caution and avoided if possible. The case study demonstrated that direct estimation of a response can sometimes result in material biases, and so is risky. Those risks should at least be understood and better still quantified.Wherever possible the primary variables that underlie the response should be identified, measured and estimated instead, with transfer functions between the primary variables and responses being developed.The specific recommendation in the iron ore industry is to enrich traditional spatial models of chemical grades with spatial models of mineral abundances, material type abundances, and textures. These are primary attributes of the mineral deposit which can be estimated directly. Unlike response variables, they are independent of mining methods, plant configuration and material handling, but they underlie the key responses such as product yields, product grades, and ultimately the sinter performance of products.