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Applying Geostatistics to Geometallurgy (6 replies)
We use conventional geostats approaches (kriging, inverse distance etc). Where appropriate stochastic methods are applied. The latter is useful in ascertaining an error of estimation.
We try to leverage correlations to our more dense geological information (incl. assay). Other advancements include 'other' physical characterisation of core to improve both domaining and the correlations.
A major advancement would be the propagation of error through the entire process, and the assessment of the error based on a mine production plan. The level of error over the production period of interest is critical to assess the value of the estimation techniques.
A major advancement would be reconciliation of forecast against plan with the intent to advance forensic geometallurgy. That is, combining source and destination information with accounts/costs and physical response as measured in the process. This would enable one to refine the mine plan with variable COGs to ensure your desired optimum is reached.
Additivity is poorly understood in the met discipline. I would advocate suitable use of transforms to modify conventional indices into their additive forms. I have to date, found only a few indices or rates that cannot readily be transformed into an additive variable. A major struggle for me has been chemical interaction of two pulps. NB: Any well thought out index will allow you to approximate a blend of two substances.
I am interested in including normative modal mineralogy (specifically host rock, and not just valuable mineral) in the process. I have not seen this evaluated across a relatively complex ore body, but I am eager to try this. Not quite an index, but I think it will assist in better modelling of met response. Also, I am interested how the geostats of single element population of a block model would compare to normative modal population of a block model. Does anyone have data/experience to share?
Not a subject matter expert.
2-6 weeks depending on complexity.
IDS is not a geostatistical method. Kriging can also give you a parameter (or two) on estimation error. The stochastic methods are used to add granularity to the estimation process (unlike kriging and IDS like interpolation techniques - which have a potential to smooth too much if not well constrained). More discussions you may find in the "mining geostatistics" group.
I'd add - investigate the relationship of particular geometallurgical properties with different types of mineralisations and rocks - see if that is "spatial". If that is true -then it is possible to establish the spatial model of the variable more easily - for use in estimation process.
Good point on IDS (not a subject matter expert)... I do think the kriging variance is interesting to use but I have been told (again not an SME) that it is not a good measure of the extent of the error. Agreed on the latter points. Q: what are your thoughts on non-additive properties and their estimation?
Like statistics can not be algebra or arithmetic or any other branch of mathematics: statistics is statistics - Geoststistics is already defined: It is based on the theory of "regionalized variable". I encourage reading the first few works by and about Matheron (father of geostatistics). There are principles and assumptions, we are not in a position to define it - it is there for more than 40 years.
As we know geometallurgy is in practice for many years: so it is new name to an old science. Example: modeling of recovery factors in Ni-laterite, bauxite deposits is a old practice. However Geometallurgy is recently being defined better - especially in spatial modeling: that's is very recent. That's is why this discussion is very relevant. Geosatistics is new to this area.
Geostatistics is applied in Geometallurgy the same way it is applied in geology, geochemistry, mining. In all these areas one thing is common: theory of regionalized variable. If the variable consider is not proven to be a regionalized variable then geostatistical modeling techniques should be avoided.
Two interesting articles / presentations:
History of geostatistics - Stanford School of Earth Sciences
http://ca.wiley.com/WileyCDA/WileyTitle/productCd-111866275X.html
http://cg.ensmp.fr/bibliotheque/public/MATHERON_Ouvrage_00167.pdf
You are looking to explore how mineral association patterns (texture) can be used to enhance geological or geometallurgical models. I have recently seen some work where textural information was used to modify (and improve?) gold grade estimates. If this is along the lines of your inquiry I will try to track down more details.
On the point James made regarding reconciliation, I would be interested to know where the gaps are that prevent the loop from being closed (might be the topic for another discussion lest this one become diverted).
Matheron has defined what geostatistics is; unfortunately I have seen some papers and books about the "application of statistical tools in the geosciences" named as geostatistics. Coming back to the main question, I did not understand which parameter of texture exactly needed; Degree of freedom, the shape or granular size? What is the aim of this work simulation of texture or estimation of a parameter?
The issue has come up on the issue of definition of Geostatistics. I would suggest there are three candidates:
- branch of Statistics focusing on Geology
- branch of statistics focusing on spatial or spatiotemporal datasets.
- The application of probabilistic methods to regionalized variables
But for comparative purposes I have also included a fourth.
- offers a way of describing the spatial continuity of natural phenomena and provides adaptations of classical regressiontechniques to take advantage of this continuity.” (Isaaks andSrivastava, 1989)
When was the word Geostatistics first used? I don’t know, but a very simple search indicates that the word Geostatsitics was used prior to Matheron’s definition. For example:
- It is suggested that def. 1 closely corresponds to the informal definition of geostatistics used before Matheron.
- Spatial interpolation was initially developed in the 1600s (numerous authors).
- Spatial interpolation for resource estimation was used and well established before both Matheron and Krige.
Take for example comments in the following link as well. This author regards 1. as the original definition, and presumes the definition of ‘Geo’ changed from geology to spatial as more diverse applications become evident.
Yet Geostatistical applications remain largely associated with geology rather than spatial data. This is clearly a result of the fact that Geostatistical courses are mainly designed for Geological application.
Hence the term ‘Spatial statistics’ overlaps with Geostatistics but is much clearer in definition. As I previously said, as an Oceanographer, Kriging was commonly used. It wasn’t called geostatistics. It was just called ‘contouring’.
There are other authors who regard the formal definition of geostatistics so inflexible, they need to define different phrases (for example Def. 4).
I have already argued that Matheron’s definition is not the first use of the word; yet it is certainly fair to call Matheron the Founder of Geostatistics; and hence his definition demands some recognition.
But who was Matheron? And what exactly did he achieve?
Matheron was first and foremost a mathematician. Some sources refer to him as a Geologist. I can find no evidence that he ever studies Geology at Graduate level. Matheron was not just any mathematician; his specialty was probability. Yet he developed numerous methods in diverse fields: Probability, Geostatistics, Image analysis, Morphology, Sample Theory. And he wasn’t just a theorist. He was highly application-oriented.
Matheron was influenced by Krige’s work.
Krige’s work is described in Wikepedia as:
Kriging is a group of geostatistical techniques to interpolate the value of a random field (e.g., the elevation, z, of the landscape as a function of the geographic location) at an unobserved location from observations of its value at nearby locations.
The authors of this article do not consider Krige’s work to be mathematical original. i.e. in the statistical community the same technique is also known as Gaussian process regression, Kolmogorov Wiener prediction, or best linear unbiased prediction. Matheron saw the potential of Krige’s work and coined the phrase Kriging, but Matheron’s work in Geostatistics was also not considered mathematical original. One can argue the point on this, but it is indisputable that there were a number of other authors at the time using variograms for spatial analysis. Whether other mathematicians saw the originality or not, the point remained that the work had practical value, and despite being initially limited to the French-speaking community Matheron’s work started to gain popularity among the general mining industry.
In 1968, the Paris School of Mines created the Centre de Morphologie Mathématique, located in Fontainebleau, France, and named Matheron its first director.
Matheron presented his Stationary Random Function at the first colloquium on geostatistics in the USA (1970). He called on Brownian motion to conjecture the continuity of his Riemann integral but did not explain what Brownian motion and ore deposits have in common.
There is no evidence that Matheron instigated or founded this colloquium. Only in 1979 was Centre de Géostatistique et de Morphologie Mathématique created and thus Geostatistics was now formally recognised as a Centre.
So when did Matheron define Geostatistics? In time, Geostatistics became associated with Matheron. I see no reason why the two are synonymous.
Matheron (1963) actually defined Geostatistics as:
- Geostatistics is the application of the formalism of random functions to the reconnaissance and estimation of natural phenomena
- Definition 3 above is a simplification of this rather verbose definition. The two definitions have only limited agreement.
- But even if we read information from the Paris School of Mines we read:
Theoretically speaking, first and until roughly 1985, geostatistics mainly focused on Gaussian random functions (or linear methods, which take place in a much broader frame, but we know they have good properties in the Gaussian case). Works were then oriented towards non-Gaussian random functions, in particular for the conditional simulation of random sets, for example to represent lithologic facies.
In other words, the focus of geostatistics changes. But in summary I cannot see any reason that Definition 3 is deemed ‘correct’. I do not see how this definition in anyway ‘respects’ Matheron. Even had Matheron defined it this way there is no justification that it needs be treated as a Trademark rather than a ‘working definition’. I do not think Geostatistcs is distinctly separate to spatial statistics; other than its clear focus on geology.
- I do not think a limited definition of Geostatistics serves any useful purpose.
- So what then is the Matheron legacy?
- Matheron was a diverse mathematician who saw the implication of work being conducted by a practitioner (Krige).
Krige was able to see how conventional approaches could be improved using sophisticated mathematical methods. Matheron was able to develop long-term relationships with the industry; by adjusting near-purist mathematics into something practical.
Yet it was the mining industry itself that in this case was also responsible for the success of Geostatistics, through their appreciation and support of Matheron's work..
So how do we truly show respect for Matheron's work? By developing, adapting and utilising, mathematical approaches of value to the mining industry. This then is the far-reaching implication of Matheron's definition of geostatistics; and its relevance to geometallurgy.
My main area of interest is the development of a texture-based model for geostatistics. This is a radical alternative to the current variable by variable approaches. It is something I am aiming to pursue with one of my clients; but definitely not moving forward as fast as I would like.
A few areas:
1. What Geostatistical approaches do you use?
2. Where do you see major advancements?
3. Is the variable set for Geostatistics appropriate.
4. What variables should be included but aren't?
5. Is Geostatistics the same as spatial interpolation?
6. How long does it take to apply a Geostatistical model?
Am starting the discussion with the term 'Geostatistics' I was speaking generally. The view that some forms of analysis are 'not Geostatistics' does not eliminate them from the discussion. Yet this raises the subissue of whether Geostatistics has now become so narrow in scope that it encompasses only some forms of analysis; and autonomously excludes others. I like this issue to the one of 'Gys Sampling Theory' discussed in the forum 'Sampling Theory'. As Gy is synonymous with Sampling Theory; so too is Krig synonymous with Geostatistics.
Geometallurgy is a new area and therefore old methods have to be reviewed. This is a prime role of geometallurgy. Old methods may not need replacement. But on the other hand more modern methods (particularly advances in spatial analysis) should not be inadvertently be discarded.