Measurements are points; phenomena are continuous. Spatial interpolation bridges the gap, estimating values everywhere from samples somewhere — rainfall between gauges, groundwater levels between wells, pollution between monitors. Methods range from simple inverse distance weighting to geostatistical kriging, and the honest differences lie less in smoothness of output than in assumptions made and uncertainty acknowledged.
The deterministic family
Inverse Distance Weighting (IDW) averages nearby samples, weighted by inverse distance — intuitive, fast, assumption-light, but blind to directional structure and prone to bull’s-eyes around samples. Splines fit smooth flexible surfaces, well-suited to gently varying fields; natural neighbour interpolation offers robust, artefact-resistant behaviour within the data hull.
All deterministic methods share a silence: they provide estimates without error measures, leaving users to guess where the surface is trustworthy.
Kriging: interpolation with statistics
Kriging first models the data’s spatial structure through the semivariogram — how similarity decays with distance and direction — then computes best linear unbiased estimates honouring that structure. Its gift is the companion surface of standard errors: a map of confidence, showing exactly where sampling is too thin.
Variants (ordinary, universal, co-kriging with covariates, regression kriging) extend the idea; the price is statistical care — stationarity assumptions, variogram fitting and validation are not optional decorations.
Validation and craft
Cross-validation is the standard honesty test: withhold samples, predict them, compare. Report the errors alongside the map. Sample design dominates method choice — no algorithm rescues clustered, biased or sparse sampling — and covariates (elevation for temperature, land use for pollution) often improve estimates more than method sophistication does.
Finally, interpolate only what is continuous: population counts and categorical classes need different tools entirely.