Variography: Post II - The Nugget Effect

Welcome to 2025! Hopefully the new year for you all is starting out in the right direction. I know some years start and I sit patiently and quietly waiting for something crazy to happen, but as long as I'm not watching the news it seems to be going smoothly LOL!

Let's continue on with talking about Variography. If you need the first Variography post, go here: https://www.dhirubhai.net/pulse/variography-post-i-celeste-wilson-ce7xc/?trackingId=vZlUFS0UByseKYhlL3lsWQ%3D%3D

Variography is the process or practice of analyzing and constructing variograms. It's broader than a variogram as it includes the prep, computation, analysis and modeling steps. It can also be thought of as the process of creating the "tool" just like in cartography which involves making maps.

A variogram is the end product, just like the map for cartography, or specific tool. It is the result or output that is obtained from using variography. It's a mathematical function, graphical representation that quantifies the spatial correlation of the dataset. It shows (usually) how data similarity decreases as the separation distance (h) between sample points increases. It is, generally speaking, a graph of the semi-variance(γ(h)) versus that distance (h).

Now why did they choose gamma over some other Greek letter. Well I don't know this for certain, so hopefully someone can respond but what I think is that gamma has been used in fields like physics, statistics and engineering to represent energy or variance related quantities such as gamma radiation or gamma function. It's also been used as a function or parameter describing change or variability. It also probably avoids confusion with other well known statistic symbols such as:

σ2: (sigma) Variance

ρ(h): (rho) Spatial correlation coefficient

μ: (mu) Mean

You'll use it enough if you are building block models regularly that you can memorize what greek letter is used but if you are starting out, I always like to try to have a visual for memorization. The upward slope in some fonts for lower case gamma looks similar to the semi-variance rising on a variogram. The curved upper part of lower cased gamma looks like a plateau or "sill" of the variogram. This is very similar to trying to learn PEMDAS (does anyone remember what that means? Or what was used to remember it? Comment below).

What are the parts of a variogram?

There's actually many parts to a variogram: nugget, sill, range, partial sill, lag distance (h), semi-variance (γ(h)), experimental variogram and well the whole point you are doing this for...the model.

Let's talk about the Nugget for today.

The first time I heard the word nugget, I thought did a geologist name this part of the variogram because gold nuggets are random high-grade ore and are difficult to predict. The nugget effect is very similar in that it represents the variability that is: (1) random, (2) occurring at very small scales and (3) not captured by sampling resolution or caused by measurement error. So, small-scale, random variability that isn't captured by spatial patterns.

How can you determine what the nugget is. Now I'm going to share with you the ways that I have calculated the nugget or have chosen the nugget. I can tell you that at times this is very subjective and there are ramifications if a nugget is too high or too low that you should definitely test for yourself, but here are the ways I was shown and taught how to come up with the nugget:

(1) Local custom/common knowledge: It was always this way, in this domain, in this variable, at this grade and you can see if you trace back the variogram model to the y-axis that it about lines up or you can see that if you run a Downhole Variogram that it lines up.

(2) Downhole Variogram: Using the scale (composite to the spacing that is meaningful for your study whether it be the nominal sample interval downhole or if you are looking at a different composite size) you are interested in for your analysis, perform a downhole variogram (most software nowadays has this option) at that scale. You are mostly interested in the short range of this and looking only for the nugget that is calculated from this. I don't use semi-variogram nor variogram, I've always been taught to use the correlogram for this and for normal modeling practices. We can get into more of these later but, you will first plot the experimental downhole variogram and then fit a variogram model to it (or in my case a correlogram model) and the nugget is calculated as the y-intercept of the variogram so focus on the first few lags (as the separation distance h --> 0). I've used Sage2001, Supervisor, gslib and rmsp to do this but there are many many software that can do this including but not limited to Isatis, Maptek's Vulcan, etc. (Does Leapfrog do this now?)

(3) Core duplicate calculating: Just as it states, you take duplicates in core and then you calculate the nugget by hand. We had a handy little Excel spreadsheet that you could paste the data into and it would do it for you -- you would just have to ensure that the calculations covered all the rows of data you gave it and voila, it was calculated. You also have to remember to ensure that the nugget you calculate is applied/related to the domain you pulled the data from and that you only apply it to that data/domain. I would also check it with a downhole variogram. But this is super easy, can easily make a python script instead of using Excel nowadays.

(4) Estimating the nugget based upon similar geological domains. In a pinch especially when you don't have enough data and you are doing inventory "guesstimates" this helps when you assume that what you are estimating is very similar to something you already "know".

(5) Straight from a non-downhole experimental variogram. Look at the short distances and see where the line visually goes to with the model you have chosen. Where does it cross the y-axis.

Other approaches which I have not yet tried are:

(6) You could do a kriging/cross-validation at short scales using kriging errors and prediction variance.

(7) Simulation. Seems like a lot of work to get a nugget but let me tell you why the nugget is so important in a minute.

(8) Empirical Methods: using heuristic or known empirical formulas to estimate nugget - for example lets say you know that for a typical type of mineralization at a certain spacing that the nugget is roughly 15% of the total variance --- you can then calculate the Nugget = 0.15* Total Variance. The bad part here is its not data driven and its all assumptions...you know what they say about assuming? Makes an ...

So why go through all this work to get a nugget?

What does a nugget do in a kriged estimate?

If you choose a high nugget --> it will result in a smoother estimate because although it seems counterintuitive since high nugget means high variability at a short scale. This results in less spatial correlation between nearby locations. The kriging interpolator then relies on global trends and will use fewer local details for the predictions. This results in estimates that are more biased towards the global mean. There's just not much spatial structure to account for.

If you choose a low nugget --> more spatial structure is captured and variable, more influenced by local pattern. Data points are more correlated over distance. The kriging model will capture finer-scale spatial patterns in the data, leading to estimates that reflect the local spatial variability more accurately.

So I would spend some time running some sensitivities on the nugget, seeing what it does with your datasets and variables. Nowadays workflows and kriging are pretty fast so should be able to run many many different scenarios at the same time, to prove it to yourself. Also, I would look at different spacings of data against nuggets. I have seen where if you drill too much it sort of just washes out and forces the kriged estimate and the model doesn't matter much since you overloaded it with data so where you have lots of data, the estimate looks great but then outside of here... what do you think is going on?

Also, I should make note, the nugget is usually denoted by C0 in the models whether you choose spherical, exponential, gaussian, etc. We will get into this at a later date although I believe I already stated I've used spherical a lot but there is a story that I want to share about it.

I'll try to add a visual of a high nugget vs. a low nugget when I get a chance to use my fake dataset :) at a later date. Sorry ran out of time this time.

Until next time!

Please share your thoughts, methods and stories about nuggets.

#Nugget #Geostatistics #Mining #ResourceModeling #WisdomWednesdayWithCW