Why RBF?

01 · The problemA section along the alignment

You drilled eight boreholes along an alignment that crosses the site's buried channel and logged where each one hit rock. Eight numbers. But the ground is continuous — bedrock exists at every chainage between your holes, and you paid for none of it.

Every excavation depth, every pile founding level, every "do we need another hole" starts with the same question: what happens between the boreholes?

Chainage along the alignment on the horizontal axis, depth to bedrock (m) increasing downward — drawn the way you'd draw the section. For teaching clarity these picks are deterministic samples of the hidden synthetic ground; the explorer adds logging noise when you resample.

02 · False startsThe obvious ideas fail

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

Nearest neighbour? Now bedrock jumps off cliffs halfway between holes; nobody would sign off on those edges.

One big polynomial? Smooth, passes through every pick — and between the data it invents bedrock about 8 m above the ground surface. This is Runge's phenomenon, and it gets worse with more data.

Leapfrog Works' constraints are brutal: honour every logged pick exactly, stay smooth, behave sensibly away from data — and do it in 3D with thousands of irregularly-spaced observations.

03 · The ideaA bump per borehole

Start over with one modest claim: a borehole tells you the most about bedrock 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, with about 4% influence left at the base range before its tail continues to decay. On our section it hangs downward — this hole pulling the bedrock line toward its measured depth.

04 · The chorusA weighted sum of bumps

Give every borehole its own bump, and give each bump its own volume knob — a weight w_i. Some weights turn out negative: those holes pull the line back toward the surface. Then just add them up.

That sum is the interpolant — one smooth bedrock line, threading every pick, inventing no drama between them.

Watch each term light up: one bump, one weight, one borehole. The line 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 depth logged 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 pick.

Eight interpolation constraints, plus one constant-drift side constraint, solve eight RBF weights and the constant c in a 9×9 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, small at long range (values below the display threshold are shown as zero). 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 logs carry uncertainty — through a weathered transition, two loggers can pick bedrock a metre apart in the same core. A nugget represents that short-scale, variance-like uncertainty. As it grows relative to the sill, the line relaxes away from the exact picks and smooths out.

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

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 · Range sensitivityWhat the assumptions change

"How confident can I be between the boreholes?" Here are three bedrock lines. Same eight holes, same method — every one honours every pick exactly. They differ only in the range, and between the holes they disagree by metres.

That spread is a range sensitivity view: between holes the model is an assumption, not a measurement. It is not a statistical confidence interval. What it does make explicit is the range assumption. A short range says "this ground varies quickly"; a long range says "this stratum is continuous"; a nugget says "my picks carry uncertainty".

That's what makes the interpolant a good way to model between boreholes: the assumptions sit in named dials you can see, defend, and tune to match the geology — not in a draftsman's wrist.

A wide band marks ground worth reviewing or investigating; it does not automatically prescribe the next borehole. Understanding the dials matters more than admiring the surface.

08 · 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, and the same nugget and range dials — over a drilled-out site with the same buried channel this alignment crossed in section.