Implicit modelling explained: from drillhole logs to 3D domains

Step 1: turn categories into numbers

Drillhole logs record categories: lithology code A from 0 to 10m, code B from 10m to 15m, etc. A smooth function cannot be fitted to a category, so the first step is to convert the intervals into points with numeric values, one unit at a time.

Where the point lies	Indicator value	What it means
Inside the unit	Positive	The sample is logged as the unit you are modelling
On the boundary	0	The contact where a drillhole passes from the unit into something else
Outside the unit	Negative	The sample is logged as any other unit

The boundary points are the most informative: the contact is exactly where the model surface has to pass. Points inside and outside tell the function which side is which.

Step 2: add the distance to the nearest boundary

A plain indicator, +1 inside and −1 outside, throws information away. A sample just inside a contact and a sample deep in the unit would both be +1. GEOREKA therefore also estimates the distance from each point to the nearest boundary and uses that signed distance as the value: again with positive inside, negative outside and zero on the boundary.

An implicit function fitted to the distances returning the estimated distance to a boundary, positive on one side and negative on the other, is a signed distance function (SDF). So the numbers in Step 2 are samples of an SDF, and the model you fit next is an estimate of that SDF everywhere else. The signed distance functions guide explains why this makes implicit models flexible and fast.

Step 3: fit one implicit function through all the points

The function is fitted so that it passes through the values at the data points and varies smoothly between them. Radial basis functions (RBFs) are the usual choice, and the fitted function has the form:

f(x) = Σ aᵢ · φ(|x − xᵢ|)

Each data point contributes a bump φ centred on it, and the weights aᵢ come from solving a linear system so that f reproduces the data. The choice of φ, and its range, controls how smooth the surface is and how far each point’s influence reaches. See the covariance function guide for how that choice behaves.

Step 4: the boundary is where the function equals zero

Once fitted, the function can be evaluated at any location, on a grid or in a section. Because the boundary is defined by a threshold on a function and not by a mesh, you can change it by changing function values. Adjust the threshold to move the surface, or add points to correct it locally, without redrawing anything.

For how this applies to real drillhole data, with several units and hard or soft contacts, see domains from drillholes and soft and hard domain boundaries.

How it relates to Kriging of Grades

Fitting an RBF, ML regression and Kriging are close relatives. Kriging estimates a value at a location as a weighted sum of nearby data, with weights from a covariance model. Written in its dual form, Kriging is also a sum of one function per data point with weights from a linear system, which is the RBF form above. With a suitable covariance function the two give a similar estimate. The kriging, RBF and ML guide shows the mathematics.

	Implicit modelling of a boundary	Kriging a grade
What is estimated	Signed distance to a boundary	Grade or another numeric attribute
Input values	Indicator points and distances, built from categories	Assays and other measurements
Treatment of data	Usually honoured exactly, because contacts are logged positions	Often smoothed with a nugget effect, because assays are noisy
Result used as	The zero level is the domain boundary	Estimated grade in each block
Shared machinery	Covariance or kernel, range, anisotropy, linear system	The same

How it ties in with grade estimates

• Domains control where grade is estimated. The implicit model gives true 3D domain volumes, and grade is then estimated inside each one, so high grades are not smeared across a geological contact. Whether samples from the neighbouring domain are used depends on whether the boundary is hard or soft (see soft and hard boundaries).
• Grade shells use the same trick. Apply a cut-off to a numeric attribute: values above it become positive, values below it negative, and the zero level is the shell at that cut-off. Change the cut-off and the shell changes with it.
• Anisotropy applies to both. Direction of continuity that follows the structure improves the domain surface and the grade estimate alike (see local varying anisotropy).
• One engine. In GEOREKA the same interpolation methods are used for boundaries and for grades, so the domain model and the resource estimate come from one project.

What to check in your own model

• Contacts are logged and imported at the right depths, because boundary points drive the surface.
• Enough points on both sides of the unit, so the function knows which side is which.
• The range is large enough to bridge drillhole spacing but not so large that it connects separate bodies.
• The surfaces look geologically reasonable in sections, and match drillholes you left out of the fit.

Further reading

• Resource estimation explained (part 1)
• From kriging to machine learning (part 2)
• Signed distance functions
• Glossary of geological modelling terms