A wireless soil network can only sample a handful of points across a field. Ordinary kriging is the standard geostatistical method (Krige 1951, formalised by Matheron) for turning those sparse readings into a full continuous map, plus an honest estimate of how much to trust each pixel. Because the estimate lives on the field's own X-Z plane, this top-down map is the native representation — the 3D version's height and camera orbit are cosmetic, not part of the mechanism.
Spatial correlation between two points a distance h apart is modelled by a spherical semivariogram:
γ(h) = nugget + sill·[1.5(h/range) − 0.5(h/range)³] for h < range
γ(h) = nugget + sill for h ≥ range
For a target point p, the kriging weights w solve the linear system that guarantees the estimate is unbiased (Σw = 1) and minimises estimation variance, using a Lagrange multiplier μ:
Σⱼ wⱼ·γ(xᵢ,xⱼ) + μ = γ(xᵢ,p) for every sensor i
Σᵢ wᵢ = 1
ẑ(p) = Σᵢ wᵢ·zᵢ (kriged estimate)
σ²(p) = Σᵢ wᵢ·γ(xᵢ,p) + μ (estimation variance)
The (N+1)×(N+1) system matrix depends only on sensor positions and the variogram, so it is inverted once per layout; every grid cell then reuses that inverse to solve for its own right-hand side. The strip below the map plots the model curve γ(h) against the empirical variogram cloud — every sensor pair's actual squared half-difference — so you can see directly whether the chosen range/nugget fit the data or over/under-smooth it.
- Sensors deployed — more sensors shrink σ² everywhere and pull the kriged surface closer to ground truth.
- Variogram range — how far a reading's influence reaches; too short and gaps between sensors go untrusted (high σ²), too long and the map over-smooths real local variation.
- Nugget effect — measurement noise / sub-metre variability that no amount of kriging can resolve; raises the σ² floor everywhere.
- Redeploy / New field — re-randomises sensor placement or the underlying (normally hidden) soil-moisture pattern the network is trying to reconstruct. Dragging a sensor by hand does the same thing a real operator relocating a probe would: the whole map re-solves instantly.
This is exactly the math behind real precision-agriculture IoT deployments: an operator picks sensor count and spacing to keep σ² below a budget everywhere in the field, instead of guessing.