By: Erik Ronald, PG
Mining Geology HQ
9 March 2016
A Mineral Resource or Reservoir model is a compilation of geologic data, interpretation, and numeric modeling. Data is collected, typically through drilling, at various spacings and used to infer characteristics across a large volume, zone, or deposit. The first key component to modelling, which I’ll call the Foundation, is the fundamental robustness of the underlying sample data including correct sampling methods, QA/QC, representation, spatial distribution, and appropriate sample mass. The second key is an accurate interpretation of the geological features, typically in 3-dimensions, that requires an understanding of geologic relationships, relative timing, structure, and alteration amongst other features. I like to think of this as the Framework. Lastly, numerical modelling, or Finishing, must be based on fundamental assumptions of the data and utilise geostatistics to interpolate the various attributes or variables of interest (i.e. Au, density, porosity…). These three components of a model allow an easy-to-understand analogy of a house with a solid foundation, sturdy frames and walls, and the finishing touches that makes it a home.
In the mining and oil & gas sectors, it’s typically the geologist’s role to create and maintain Resource/Reserve and Reservoir models with assistance from a variety of discipline specialists including geochemists, geophysicists, metallurgists, hydrologists, petrophysicists, and geostatisticians. This article will focus on the Finishing component including geostatistical contributions to a model, the understanding of stationarity, and a few important considerations with estimation.
To the average Geologist, Geostatisticians tend to communicate in unnecessarily complex pseudo-English interspersed with equations and a healthy sprinkling of jargon. It’s understandable why most in the natural resources sectors tend to see resource estimation as a specialist sub-field. I’m not suggesting Geologists aren’t guilty of their own complex language as well with statements like “penecontemporaneous intrusive events of leucotrondhjemite and anorthosite juxtaposed by sinistral displacement”. Or roughly translated to “those hard, white rocks next to each other are slightly different, but about the same age”. One can only assume geostatisticians communicate in the way they do in order to keep their numbers low in regard to demand (the author has calculated this to be between 8 and 12 globally) or merely to ensure they appear brilliant and insightful in front of management. I’m obviously poking a bit of fun but in all seriousness, quality estimation work can increase a model’s reliability and accuracy while dismissing geostatistics can drastically destroy the value of an ore body or reservoir. There are fundamental concepts that should be appreciated when estimating minerals, petroleum, or environmental contamination that can, if performed incorrectly, result in a fundamentally flawed and highly biased model.
The concept of Stationarity
Geologists involved with the modelling of geologic data must understand and appreciate the concept of “stationarity”. It’s a good place to start as it is highly influenced by the geologist’s interpretation and should be assessed collaboratively between the geologist and geostatistician. The term stationarity can be thought of simply as how well your data is pooled or grouped. This aspect of resource estimation is critical as it relies on both the geological understanding of how different units should be either grouped or split (Framework) and the geostatistical properties of those groups in order to understand the parameters that will be used in estimation (Finishing). In mining, the term “domain” is commonly used when describing a discrete unit or grouping of data so I’ll refer to domains going forward. In geological terms, a domain may be a lithologic unit, a grade shell, a stratigraphic formation, an individual coal seam, or a volume of payzone sandstone between impermeable fault zones just to name a few. In the house analogy, a domain may be thought of as a room in the house.
In order for estimation and simulation techniques to be effective, an assumption is made that the statistical and geological properties are the same throughout a particular domain. This is called the intrinsic hypothesis. Ordinary kriging, along with other estimation and simulation methods, work on the assumption that the intrinsic hypothesis is true. It assumes the expected value of the mean and the variogram are not changing between locations within the domain (termed a “robust domain”). As we only have a limited set of sample data for any ore body or reservoir, we are therefore required to infer the underlying characteristics of the dataset apply to the entire domain in which we want to interpolate values. This is where our old friend, the semivariogram (commonly referred to simply as a variogram), comes into the mix.
A variogram is calculated from the samples within a domain, then that domain’s parameters are interpreted from the variogram (or correlogram or other similar technique). This includes the spatial continuity, iso- or anisotropy, correlation range, amount of inherent geologic variability and sampling error (i.e. nugget), and the total variance or data spread (i.e. sill). Nearly all of these parameters are geologically related, so collaboration between the geologist and the geostatistician is critical to ensure the domain parameters make geologic sense. If they do not, chances are the domain is not robust resulting in a breakdown of the intrinsic hypothesis thus estimation methods are not reliable, leaving you with a biased and flawed model. No amount of “advanced” estimation techniques can make up for poor domaining of data. There’s no point is hanging pictures and moving in furniture to your house if your walls are crooked and incomplete!
Another way to think of proper domaining is “always use like data to estimate like material”. Geologically you wouldn’t estimate a coal zone with data from a granite just like you wouldn’t estimate the average height of Norwegians by using data from southern Spain or include toddlers in the data. The robustness of the domain must always be tested with thorough exploratory data analysis (EDA) to ensure it is stationary while representing a single population as outliers can have a dramatically detrimental effect on a variogram. Further groupings or sub-domaining may be required in order to separate out the various populations, if possible. Geologically, think of coal seams with significant interbedded sand zones or portions of unmineralised waste within your ore body. There are a variety of techniques available to handle highly skewed data, outliers, and even multi-populations including masking and normal-score transforms to name a few. A combination of geologic sub-domaining and geostatistics can usually improve the estimate in these cases.
Sometimes geological characteristics of a domain result in the variogram exhibiting a trend. This effect will occur when your mean varies within the domain such as porosity with depth or increasing grade toward a mineralised centre. When a variable exhibits a trend, it is termed as non-stationary. A trend can be interpreted in the variogram at long separation intervals where the variogram never stabilizes at the total data variance or sill (see Fig. 1). To make matters more complicated, a variable may exhibit localised zones of stationarity (Fig. 1R up to ~ 15m range) but globally be a non-stationary variable or the trend may only be in one direction, such as vertical versus horizontal permeability. In each of these cases, the variogram computation requires understanding and addressing or removing the trend. If these trends are not addressed and the variable is of economic importance, the model may be fundamentally flawed. In cases like this, it is quite handy to have a geostatistician on speed-dial.
Fig. 1: (L) Experimental and modelled semivariogram displaying classic structure and stationarity and (R) experimental semivariogram displaying non-stationarity trend in data.
Soft or Hard Boundaries
In the simplified world of stationary domains, ideally you’ll want to be using sample data from the white rocks of Domain A to estimate Domain A in your model and sample data from the black rocks of Domain B to estimate Domain B in your model. This would be termed a “hard boundary”. In reality, it would certainly be nice to draw a line and state “everything on this side is white and everything on that side is black”. We know this is not always possible in geology as there’s bound to be a bit of grey in there somewhere. We can address that grey zone by using what is called a soft boundary. A soft boundary is essentially using the variogram from either domain and data from both the black and white rocks to inform the “grey” zone thus creating a gradational transition between Domain A and B (Fig. 2). Of course you can be smarter by limiting the data to a select distance from the boundary or some other discernable characteristic that makes geologic sense. This is common practice in ore bodies such as disseminated porphyry copper deposits where there’s a gradual increase in grades toward a central high-grade zone or in different depositional environments such as a submarine fan. Think of trying to paint or put up unique decorations in each room of your house; bedrooms are easy but what do you do about that “great” room which incorporates your living, dining, and kitchen areas?
Fig. 2 – Contact plots (L) showing “hard” boundary for silica and (R) soft boundary for phosphorus.
Typically, a model will include both hard and soft boundaries based on the geological understanding of the deposit evolution or paragenesis and is validated through EDA such as contact analysis (Fig. 3) showing non-stationary characteristics near contacts. Whether using hard or soft boundaries, the contacts can have significant impacts on the total Resources due to dilution, data spacing, and uncertainty of characteristics in these zones. Additionally, “edge-effects” when using inappropriate domains such as a high-grade ore shell may show artificially higher grades up to the boundary then a dramatic drop once the boundary is crossed biasing the total metal in your deposit above your cut-off grade. These cases of sharp boundaries are rarely true in reality but it doesn’t take a PhD to guess where the mining engineer will place the edge of your pit shell due to incorrect estimation in your model.
Fig. 3: An example of “edge effect” of grade distribution differences using boundary types (image from University of Alberta, Applied Geostatistics course notes: C. Deutsch 2007).
1) Get the basics right! The most important parts of your home are a solid foundation, strong walls & roof, and a logical layout of rooms. Ensure your underlying sample data has minimum sampling errors and is unbiased. The geologic interpretation must be sound and robust in regards to the data. If you’ve got those aspects right, you’re 80-90% there!
2) An understanding of geostatistics is important to assess the potential for conditional bias into your interpolation. Understand and interrogate the data by domain and variable to know which exhibit stationarity and which are non-stationary. Estimation techniques assume intrinsic stationarity, so before you can make any assumption on deposit uncertainty, make predictions (i.e. the mine plan) and confidently perform classification (reported Mineral Resources), this assumption needs to be checked. You cannot make business decisions from a model in which the geologist who created it does not understand how the numbers got into the boxes nor do you want your in-laws to be in charge of your interior decorating.
3) The lower the concentration of your economic variable (%, ppm, ppb), the greater the importance geostatistics can have on impacting the accuracy of your model and ensuring your mine plan is based on the best data available.
4) Validate your domaining in order to have confidence in your estimation.
5) Understand the areas of your ore body or reservoir model where you can confidently assess the uncertainty and areas where you cannot. Classify and communicate risk appropriately.
I hope you’ve enjoyed my take on the importance of stationarity in modeling and assessing Resources. I look forward to follow-up articles in the future providing more details on EDA, interpreting variograms, and any other topics you may be interested in related to Resource Geology. I welcome your comments and feedback. Cheers!