Why RBF?

01 · The problemA section through the deposit

You drilled eight holes along a section line and assayed the core. Eight numbers. But the orebody is continuous — grade exists at every point between your holes, and you paid for none of it.

Every resource model, every pit design, every "where do we drill next" starts with the same question: what happens between the data?

Distance along the section on the horizontal axis, gold grade (g/t) on the vertical. These eight samples are the only truth we get.

02 · False startsThe obvious ideas fail

Connect the dots? Nature doesn't have corners — a kink at every drillhole is geological nonsense, and the slope changes are artefacts of where you happened to drill.

Nearest neighbour? Now grade jumps off cliffs halfway between holes. Mining engineers call these polygons; nobody believes the edges.

One big polynomial? Smooth, passes through every sample — and beyond the data it invents negative grade, plunging to −37 g/t at the edge of the section. This is Runge's phenomenon, and it gets worse with more data.

Leapfrog's constraints are brutal: honour every assay exactly, stay smooth, behave sensibly away from data — and do it in 3D with millions of irregularly-spaced samples.

03 · The ideaA bump per sample

Start over with one modest claim: a sample tells you the most about grade right where it is, and less and less as you move away.

Draw that claim: a smooth bump centred on the hole — its zone of influence. This shape is the radial basis function φ. Radial, because it only cares about distance; the same in every direction.

This is the exact spheroidal function the explorer uses: near its full value up close, ~96% faded at the base range, effectively gone beyond it.

04 · The chorusA weighted sum of bumps

Give every sample its own bump, and give each bump its own volume knob — a weight w_i. Some weights turn out negative: those samples push the curve down. Then just add them up.

That sum is the interpolant — one smooth curve, threading every sample, inventing no drama between them.

Watch each term light up: one bump, one weight, one sample. The curve is nothing more than their sum (plus a constant the model relaxes to far from data).

05 · The machineWhere the weights come from

The weights aren't guessed — they're forced. At each hole, the bumps that reach it must add up to the assay measured there. Watch the amber line visit hole after hole: the little ladder beside it stacks each bump's contribution at that hole, and the stack has to land exactly on the sample.

One demand per hole gives eight equations for eight unknown weights — a linear system A·w = v. Row i is hole i's demand. Cell (i, j) is just φ at the distance between holes i and j: brightest on the diagonal (a hole is at distance zero from itself), fading with separation, zero beyond the base range. Solve the system and the weights are whatever makes every ladder land.

These are the real numbers for this section — brighter cells are bigger. The extra row and column carry the constant c. Leapfrog's FastRBF is, at heart, a very fast way to solve exactly this with millions of rows.

06 · The dialsNugget and range

Real assays carry noise — duplicate samples from the same interval disagree. Adding a nugget relaxes the "exactly through every sample" demand: the curve is allowed to miss by a little, and smooths out.

The base range sets how far each bump reaches. Short range: influence dies fast and the curve collapses to the background between holes — bullseyes. Long range: broad, confident trends.

These are the exact dials you're about to turn in the explorer. Everything you'll see there is this page's mathematics, in 3D.

07 · To spaceThe same sum, in 3D

Nothing about the sum cared that x was a single number. Measure distance in three dimensions and each bump becomes a sphere of influence — or an ellipsoid, once you let direction matter. The weighted sum becomes a surface.

That's the tool you're about to use: the same bumps, the same system A·w = v, the same nugget and range — over a drilled-out gold bench.