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## Spatial Interpolation (7 replies)

I know that in different companies such Anglo American (Copper), BHP Billiton (Base metals), Codelco, etc., is usual the aplying of geostatistical tools for the interpolation of geometallurgical parameters. Even I know some experiences using conditional simulation for some orebodies. Obviously it is necesary to take in acount non additive effects, but that is usual when is using linear interpolators for non additive parameteres

Widely popular and free sources of geostatistical softwares:

But industry standard and the best geostatistical software package:

Thanks for this. Now am wondering what do you mean at your last question about differences between estimation and simulation techniques. Personally I think both of them provide the same (complete) information about the characteristics of the uncertain spatial variable as both of them are based on the same data set. Now, we know that regardless of the technique to be used (e.g., parametric such as kriging and SGS or non-parametric techniques) there is always a trade off between accuracy in mean or variability, so, estimation provides information about the mean or expectation while simulation provides information about the variability. For this reason I certainly believe that both techniques should be used when assessing and evaluating natural resource projects and in general any project related to spatial distribution.

Agree in that spatial analysis should be taught in an easy fashion at universities so not to scare students about learning the topic, as it was my case when being at uni.

I was hoping others would provide comment on this. Have to emphasise I am not a Geostatistician!

Obviously I have a strong maths background, and many of the mathematical approaches used in Geostatistics I am familiar with. I understand that the end-objective of 'simulation' and 'estimation' are the same; and I could have chosen my words better. But to be frank I am quite worried that I might inadvertently using a wrong 'word' and cause major offense; so perhaps wasn't direct enough.

I am aware there is a deep issue regarding Kriging vs. simulation (conditional simulation); and I was rather hoping someone else would offer an explanation. I am actually trying to open up the discussion in such a manner many of the concepts can be explained in simple language on the basis many of the members of this group are not Geophysical (or spatial analysis) experts.

SGS is sequential Gaussian. I know this is obvious to you; but given I once worked for SGS I thought it important to differentiate the difference.

My personal end objective is the include of texture models in spatial or Geostatistical models; I don't see that I will be able to do this other than independently. I don't want to work independently.

So I am hoping to find out what is accepted practice; what are the current 'boundaries' and whether these boundaries are technical or cultural. For example, there might be true technical advances that can be made, but there might be perceived lack of benefit.

I have, in the last couple of weeks, had to deal with metallurgical data spatially interpolated on an inapproriate basis. The science of geostatistics as used by geologists has, I am sure, been through many development pathways, quite a few of which were dead ends. Over the years/decades this has led to a set of accepted outcomes that we now accept as valid geostatistical methods.

My problem is that the application in metallurgy is in a relative state of infancy and there are practicioners out there that use geostatistics as a camoflage of legitimacy. I am sure that geostatisticians would be horrified at the abuse and misuse of geostatistics that is going on in some circles.

The problem I recently dealt with is gold project where about 30 comminution measurements were Kriged across about 100k mine blocks. The resultant outcome was persented as a geostatistically valid orebody picture that was then going to be the basis for plant design and project NPV (value) prediction. I smelled a rat when the comminution energy was showing an erratic shape by year, including a major fall in the final year when hard ore should be dominant and the energy requirement should have been going up (and mill throughput going down).

The problem was that the metallurgical data had not been looked at from a pattern point of view before being blindly used for geostatistical estimation. In geology this is called preparing a geological model of the orebody and this model constrains any subsequent geostatistics. Typically this means that ore type (usually lithological) boundaries are recognised within the geostatistics package and calculations do not progress in a blind manner across these boundaries. I am not sure that even the lithological model was used as a constraint in my recent metallurgical example.

As metallurgists attempting geometallurgy we need to be responsible for seting up a metallurgical model (usually substantially different to the geological model) before attempting any sort of data distribution through an ore zone. We also need to be conscious of the limited amount of data at our disposal and not attempt to make too much out of nothing.

My re-examination of the small metallurgical data set was done in a spreadsheet with reference to geologically defined lithology, spatial location and other factors such as alteration and oxidation as reported in the geologists drill database. It rapidly emerged that the major determinant of metallurgical properties across the orebody was simply the depth of the sample below surface. This is reasonably logical for this particular orebody as it is mainly in the oxidized zone. I then produced an equation to estimate each of the metallurgical properties by depth and applied this to the available block model information. The resulting outcome was a much more sensible looking comminutuion energy progression through the life of the orebody as defined by the ore delivery schedule. It was also a much more appropriate use of a limited data set than the metallurgically blind attempt at geostatistical distribution.

My overriding message is that as metallurgists it is our responsibility that a metallurgical model of the property in question be developed before any consideration is given to distributing it across an orebody. It is not good enough to simply distribute the property unbounded or to assume that the geological model (usually lithological) has any metallurgical relevance. I have even see the lithological approach abused in literature where broadly spread properties were found across two lithologies so these were simply combined into one lithology (~80% of the orebody), rather than searching for an alternative control mechanism for the property. Perhaps in 10 or 20 years time such practices will have been refined by bitter experiences and through real world feedback as the folly of blind metallurgical geostatistics is proven.

I thought is best to clarify above with technical examples. To a large extent, Matheron's work focused on utilisation of the variogram. (How the variance of measured values changes with distance). Yet this raises an immediate obvious extension. What about the covariogram (or correlogram)?

Now the Covariance measures association of variables. It may have limited value to geostatstics, but is totally applicable to Geometallurgy. i.e. the use of 'proxies' is a natural extension of multivariate analysis. Now once I went to a course on Geostatistics and the lecturer explained variograms.

I asked the question (which I regard as obvious) as to whether the covariogram is also used. At this the lecturer indicated that it was necessary to go this level of 'depth'. I simply could not (and still do not) understand this.

If we look at the 'Mathematical Morphology' (also founded by Matheron, and also Serra) utilisation of the covariogram is integral. In fact Serra went deeper and suggested the use of the two dimensional covariogram, where we do not confine the covariance to a distance parameter, but to both distance and angle; hence to avoid the assumption of isotropy.

When I was at JKMRC I worked with George Leigh, who developed a texture model based on wavelets. Yet not only could wavelets be used for texture modelling, it also had application to two other important areas:

- Mineral identification (ie. from image data such as Qemscan, MLA or even hyperspectral data).
- Geostatistics (spatial analysis).

Now because of various imposed 'resistances' , this work really didn't advance as far as it could have. But it doesn't mean I have lost interest. The texture model I developed used a cooccurence matrix only (this is related to 'grain size' and mineral 'composition' but is a more mathematically defined approach) It is different to George's wavelet approach.

Using the cooccurence matrix the 'similarity' between textures can be measured. Using the 'similarity' measure and a spatial model, one can mode the 'texture' thoughout the orebody; hence extract mineralogical information. I don't think the spatial model I am envisaging should be independent of other variables; hence I just want to know current approaches. As said, if this work does process with a client, it is obviously useful if I know the status quo.

This does indeed discuss the advantages of covariance, and their modelling approach is consistent with what I considered to be a reasonable method. That is, convert muivariate information to principal components, and then apply geostatistics (Kriging) on each factor.

There is an interesting parallel between geostatistics and 'mixture models'. There remains technical issues that are still worth pursuing, but nice to know my question re: multivariate analysis was perfectly valid. I would very much appreciate your opinion as to why you consider Geovariances to be the industry standard and best?

Also, is this opinion in the context of Geostatistics (resource estimation), or also in the context of Geometallurgy?

I made comment last year regarding A*b in a Geostats model. But the main issue that the parameters were used in a model without reference to their dimensionality or additivity.

Here I simply stated that a simple transformation of the variables 1/(A*b) would convert the data to an additive form. I only read Coward's paper only briefly, and he raised the issue of additivity, but I didn't see any comment re: simple transforms. I am starting to think I might be a lone voice on this issue. With a very simple understanding of what additivity means (and generally means some measure per unit tonne) many of the additive problems become non-existent. Hence 'recovery' is not additive. But recovery*grade is additive (recoverable mineral per tonne).

But of course recovery may be correlated with grade, and therefore one should use a covariogram rather than a variogram.

It was the discussion about recovery by Mohammed that was the main motivation for why I started this discussion. Which again brings me back to one of the original questions: what metallurgical parameters are used in a Geomet. model and how are the parameters subject to spatial or geostatistical analysis?

But I think you are saying that all too often not enough care is taken is applying a Geostat. model.

This then raises issues of code of practice. Are there any? Should geometallurgy establish a code of practice? And I don't mean code of practice for resource evaluation (which clearly exist).

Please visit the website of Geovariance- particularly Isatis software and their publications. If you have a good general statistics package (like JMP) and use Isatis - you can do almost any type of resource modeling. It is quite relevant to geometallurgy. This group comes from the school where Matheron started the geostatistics research. They are active researchers with huge industry support. Simply searching for Isatis(r) capability - you may find a lot of works being done these days using Isatis.

Discussion on Geostatistics wasn't going to work as a number consider Geostatics synonymous with the work of Krig. Hence I have changed the topic to spatial interpolation (which is now inclusive of Geostatistics).

What we can see from history is that spatial modelling can be sourced to the 1600s. But more modern spatial interpolation originated in about 3 separate fields. Various authors are commonly referred to, and the area of mining this mainly includes Krig and Matheron.

The work of Matheron in particular lead to its acceptance in mining.

One thing which is weird was that I learnt 'Kriging' via application of Golden Software in the 80s. But the context was Oceanography. Certainly there was no mention of this being Geostatistics. Yet when I went to mineral processing, the realm of Geostatistics was near-taboo because it was regarded as a highly specialised field. So in the area of Oceanography it was user-friendly but in Mineral Processing it was mystified. Can anyone explain why?

I still do not understand why Geostatistical courses don't start by providing a simple example using a user-friendly software software package such as Golden Software?

Yet another thing I find interesting is that now there is an abundance of algorithms available using the R language. R is a public-domain mathematical language similar to Matlab. Has anyone in this forum used this? James said 6 weeks to run a Geostatistical model? What are the accuracy/cost/benefit tradeoffs?

A form of spatial simulation was developed by Kolmogorov in the 40s. Kolmogorov was a mathematician, famous for his work in probability theory, but he also developed spatial methods in areas as diverse Oceanography.

I notice a group called 'Mathematical Geology'. Has anyone studied this? Is this different to Geostatistics (I notice it included Geostatistics)?

In simple language what is the difference between conditional simulation and Kriging? and what level of a depth should a Geometallurgist be required to understand the subject area?