It may seem like a mundane question, but do we report tonnage, grade and metal correctly and consistently in our public disclosures of mineral resources and reserves? I’ll venture that our public disclosures of mineral resources and reserves don’t always measure up. This post is about uncertainty and significant figures and how we largely misuse or ignore them. Let’s have a quick look at this, shall we?
The papers and regulatory guidance that I have found so far that discuss measurement and reporting of ore tonnages, grade, and metal give short shrift to significant figures and the rationale behind reporting guidance, if any.
Significant Figures Rules
To find the rules for significant figures I had to go to the online sources since I threw my poorly-bound algebra textbook away a couple of moves ago. LibreTexts, Chapter 1.7, provides a condensed summary of the principles and application of significant figures in calculations which is further abstracted here:
- Zeros between non-zero numbers are significant
- Leading zeros are not significant for fractional numbers
- Trailing zeros are not significant if used only as placeholders
Further, numbers which exceed the precision of the measuring device are not significant. Just because a calculator or spreadsheet will show nine decimal places, that doesn’t mean all the numbers are significant. With respect to calculations involving multiplication, division, addition and subtraction, the result should have the same number of significant figures as the quantity entering into the calculation that has the least number of significant figures.
Measurements
We often talk about estimation error, but in mining we never have a true value to measure against so I will mostly use ‘uncertainty’ in this discussion. Let’s look at significant figures separately for each of tonnes, grade and metal. For mineral resource estimations, tonnes are not measured directly; tonnage is calculated from volume and bulk density. Volume itself is calculated from X, Y, Z dimensions which in many cases equate to length x width x height. If the ratio Surface Area/Volume is relatively high, the case for a large, bulk tonnage deposit, then there isn’t much error in volume as a percentage of the deposit and volume can easily be reported with >3 significant figures. Let’s contrast that with a narrow vein or seam model that has a relatively high Surface Area/Volume and some uncertainty about local volume. In this case the accuracy of the volume calculation relative to the seam model might be a percent or more in error.
What about the bulk density measurement? Balances used in a core shack or commercial analytical laboratory setting are of two broad types:
- Beam balances with accuracy between 0.1 g and 0.001 g.
- Digital balances with accuracy up to 0.001 g
Let’s say we are measuring specimens for bulk density on a mechanical balance which likely has a stated accuracy of no better than 0.1 g. If we weigh a 10 cm piece of HQ core the mass is ~ 800 g. We can round our measured weights to the nearest gram and still realize only ~0.1% error by doing so. That is pretty good and we should be able to report bulk density to three or four significant figures, to be confirmed by replication in duplicate measurements. Thus, in general, it appears that we can justify 3-4 significant figures for volume and bulk density, therefore also tonnage, if considering only our measurement accuracy and precision in isolation.
Measurement certainty is even better with grade, where our laboratory generally provides us with established detection and accuracy limits. For base, precious and noble metals, most results at near-economic levels are reported on certificates with three or more signifcant figures.
Estimation
However, the reader will likely agree that there is more uncertainty besides measurement accuracy and precision that needs to be considered. The uncertainty involved in extrapolation of grade by estimation is harder, if not impossible, to objectively measure, but is always of significant magnitude compared to measurements at the sample level. It is dependent on many factors such as variogram quality, consistency of sample support, estimation method, etc. Maybe the most robust measurement of estimation uncertainty is the conditional simulation approach, but this can only be effectively undertaken for certain deposits. Dr. Harry Parker’s suggestion that Measured/Proven estimates should have uncertainties in the range of ± 10% and Indicated/Probable within ±20% has gained a lot of traction. What’s 10% of a billion-ton reserve? It’s 100M tons. Seems like any big copper deposit resource/reserve tonnage should report as 1.5 billion tons, not 1.51 billion tons. If the estimated tons are 5.42 M tons and the grade is 0.924% Cu, and the Measured resource has only a 90% certainty, doesn’t that suggest that the tons reported should be 5 M tons and the grade 0.9% Cu? The argument for rounding is even stronger for Indicated and Inferred resources.
Now, let’s look at our bulk density estimates, the multiplier of volume to report estimated tons, in the same way we look at estimated tons and grade. Our actual industry practice for measurement and extrapolation of bulk density is grossly substandard, most would agree. The issue is mostly given lip service with nothing done about it. We collect too few bulk density measurements, the measurement protocols are inadequate and/or inconsistent, and they can never fully represent the rock mass unless there are no fractures or crumbly zones, thus, they tend to have selection bias.
All sorts of shortcuts are standard practice to assign, extrapolate, or interpolate bulk density. For example, bulk density sample means within domains are used to apply a single value to the entire domain volume. Another practitioner will use the median value in some way. Others will use an adjusted central value after culling “outliers”. Still another will make block-by-block estimates using the point samples but with some very basic method because there are rarely enough good, consistently measured samples to do more. In the worst case, mostly seen in old operating mines, “plug” numbers are used, these calculated from “experience” or adjustments based on mine-mill production comparisons. The only good thing about these latter is that they generally are expressed with only one or two significant figures. Again, our actual practice of applying bulk density measurements to large block volumes suggests the application of very conservative significant figure assumptions in reporting.
One could argue for deriving the metal content significant figures from the significant figures of either of either reported tons or grade, whichever is less. The reported metal should be reported to one significant figure in this example: 5 M tons of 0.9% Cu for 50 K st Cu, or 100 M lbs Cu.
Classification
Resource confidence thresholds depend on a reliable classification scheme. The most common classification method that I observe in resource estimate studies is some variation on distance to the nearest composite applied as a multiple of the nominal, and imprecisely calculated drill hole spacing, an arbitrarily chosen selection in the first place. There are cases where drill hole spacing impact on estimation error has been tested with conditional simulation, or where classification thresholds have been evaluated with tools like the single-block kriger. But most resources documented by technical reports do not have this level of due diligence, and in any case, uncertainty is still a significant percentage around the reported totals. Thus, the uncertainty inherent in classification methods lends further support to limiting significant figures of grade estimates beyond measurement uncertainty considerations.
Regulatory and Other Guidance
Regulatory guidance from AUSIMM, SME, and CIM seems pretty casual and high-level. It doesn’t define the term “significant figure”. Published papers concerned with precision and accuracy of measurement, estimation, and reporting show a characteristic reluctance to advocate a prescriptive approach (e.g., Lipton and Horton, 2014). It’s left for you to decide based on this and that, so anything goes, right? The discussion in the AUSIMM guide includes detailed descriptions of bulk density methods most of which nobody in hard-rock mining actually uses, and they lack enough detail for practical implementation. A paper on the subject of bulk density measurements by a M.S. student, Crawford (2013), is more useful, but still, there isn’t a discussion of significant figures.
The JORC Code tells us that “rounding to the second significant figure should be sufficient” , or even to as little as one significant figure for Inferred Resources. It’s hard to argue with the JORC code, based on the preceding discussion.
The SME guidance is looser, allowing up to three significant figures, “Depending on the accuracy of the estimate…”. How the accuracy is determined and how it should be applied to either two or three figures is left to the Competent Persons. That leaves a barn door-size opening for qualitative judgement.
This is the general and non-prescriptive advice of CIM (2019):” Reporting of tonnage and grade figures should reflect the order of accuracy or precision of the estimate by rounding off to an appropriate number of significant figures.”
Ditto SEC’s CFR Title 17 blah blah, “The qualified person preparing the mineral resource estimates must round off, to appropriate significant figures chosen to reflect order of accuracy, any estimates of quantity and grade or quality.” That’s it. And what does “chosen to reflect order of accuracy” mean exactly. So Qualified Persons writing a NI 43-101 or SK-1300 report can decide what “appropriate” means. Or in Clinton-speak, “the appropriate meaning of “appropriate” is. At least the SK-1300 author could fall back on the SME guidance which is somewhat prescriptive (2 or 3).
Based on a cursory and somewhat random search of technical reports I’ve downloaded on my computer, two 2024 NI 43-101 Technical Reports list mineral resources reported to five significant figures in tonnage, three in grade, and four for metal content. For these, significant figures should be three for metal for consistency with the rules presented above.
Another recent report shows five significant figures for tons, two for grade and three for metal–should be two for metal. And one more with four significant figures for tons, two for grade and three for ounces–should be two for ounces.
I also find a few that followed significant figure rules, at least at the reporting level. It seems like reporting is not always wrong but is not very consistent. I plead guilty on this, by the way. All that work on the project, and the estimates, and that validation, and I have to round everything off? It doesn’t seem fair….
A few references below:
Abzolov, M.Z., 2013. Measuring and modelling of dry bulk rock density for mineral resource estimation. Applied Earth Science (Trans. Inst. Min. Metall. B) Vol 122 No. 1.
CIM, 2019. CIM Estimation of Mineral Resources & Mineral Reserves Best Practice Guidelines. CIM Mineral Resource & Mineral Reserve Committee. 75 p.
Crawford, K.M., 2013. Determination of bulk density of rock core using standard industry methods. MS Thesis, Michigan Technological University. 209 p.
Lipton, I.T. and Horton, J.A., 2014. Measurement of bulk density for resource estimation – methods, guidelines and quality control. Mineral Resource and Ore Reserve Estimation – The AusIMM Guide to Good Practice. 2nd edition, Monograph 30.
SME, 2014. The SME guide for reporting exploration results, mineral resources, and mineral reserves (The 2014 SME Guide). The Society for Mining, Metallurgy and Exploration, Inc. 65 p.