We almost never measure the whole world.
We place weather stations at selected points, drill boreholes at selected sites, record pollution at selected sensors and sample soil at selected locations. Yet the map we want is continuous: rainfall across a region, elevation across a landscape, temperature across a city, contamination across a field.
Spatial interpolation is the family of methods used to estimate values at places we did not directly measure, using the spatial pattern of values we did measure.
Interpolation turns scattered observations into a surface—but the surface is an inference, not a photograph of reality.
Quick Read: The Interpolation Mechanism
MEASURED POINTS → ASSUMPTION ABOUT SPATIAL CONTINUITY → WEIGHTING / MODEL → PREDICTED VALUES BETWEEN POINTS → UNCERTAINTY
1. Interpolation Exists Because Sampling Is Sparse
A weather station tells us what happened at its location. A map of temperature across an entire island requires us to estimate what happened between stations. Spatial interpolation fills those gaps.
2. The Basic Geographic Assumption
Many interpolation methods rely on some form of spatial continuity: nearby locations are often more similar than distant ones. This connects interpolation to Spatial Autocorrelation.
3. Inverse Distance Weighting
Inverse distance weighting, or IDW, estimates a value using nearby measured points, giving closer points more influence than farther ones. The method is intuitive and useful when distance is a reasonable proxy for similarity.
4. IDW Has Limits
Because IDW is based on weighted averages, it cannot invent values beyond the observed range. If an unsampled ridge is actually hotter, colder, higher or lower than all measured points, IDW may miss the extreme completely.
5. Kriging Goes Further
Kriging uses a statistical model of spatial correlation to estimate values and prediction uncertainty. Instead of using distance alone, it examines how similarity changes with distance and sometimes direction.
6. The Variogram Matters
In geostatistics, a variogram describes how dissimilarity tends to increase with separation. Its shape helps determine how much weight nearby observations receive and over what distance spatial dependence remains meaningful.
7. Direction Can Matter
Wind, slope, geology and coastline can make spatial relationships anisotropic: similarity may extend farther in one direction than another. A method that assumes identical behaviour in every direction can smooth across important geographic structure.
8. Barriers Can Matter
Two points may be physically close but separated by a mountain ridge, river or coastline. Interpolation that relies only on straight-line distance may incorrectly blend values across a real barrier.
9. More Points Do Not Guarantee Better Predictions
Dense sampling helps, but only if the points capture the relevant variation. Fifty sensors clustered in one neighbourhood may tell us less about the whole city than twenty strategically distributed sensors.
10. Interpolation Is Not Extrapolation
Interpolation estimates between or within the spatial support of observations. Predicting far beyond the sampled area is extrapolation and usually much riskier because the model is operating where it has little evidence.
11. Cross-Validation Is Essential
One practical test is to hide known observations one at a time, predict them from the remaining points and compare prediction with reality. Large errors reveal where the method or assumptions are failing.
12. Primary Geography: Estimate Between Thermometers
Place a few temperature readings on a simple map and ask children what they expect between them. Then reveal an additional measurement. The lesson is not to guess perfectly, but to understand that unmeasured space requires reasoning.
13. Secondary Geography: Compare Two Surfaces
Students can generate or inspect two interpolated maps from the same observations using different methods. Where do they disagree? Those disagreement zones are where assumptions matter most.
14. Advanced Geography: Prediction and Uncertainty Belong Together
A beautiful surface can hide large uncertainty. Advanced geostatistics therefore asks for both predicted values and an estimate of prediction variance or error. A map without uncertainty can look more certain than the evidence deserves.
15. Singapore Example: Urban Heat
Temperature sensors can measure selected streets or sites, but planners often want a continuous heat surface. Interpolation can help reveal hotter and cooler zones, provided sensor placement captures differences in vegetation, building form, coast, traffic and ventilation.
16. Rainfall Example
Rainfall can vary sharply across short distances in convective storms. A sparse network may smooth local extremes. Interpolated rainfall maps therefore depend heavily on station density, storm scale and method.
17. Pollution Example
Air pollution follows roads, wind and urban form rather than spreading evenly. Interpolation that ignores those mechanisms may create plausible-looking but misleading gradients.
18. Soil and Geology Example
Soil properties can change with geology, slope and land use. Kriging and related methods are often useful precisely because nearby observations may show measurable spatial correlation.
19. Hostile Test: “The Smoothest Map Must Be the Best Map”
No. Smoothness can be an artefact of the method. A surface may look elegant while erasing real discontinuities, extremes or barriers. Predictive performance and mechanism matter more than visual beauty.
20. Where Interpolation Reasoning Breaks
- Pretty-surface fallacy: trusting visual smoothness as evidence of accuracy.
- Distance-only assumption: ignoring barriers and directional processes.
- Sparse-sample blindness: predicting confidently where observations are too thin.
- Extreme suppression: forgetting some methods cannot create values outside the sampled range.
- Extrapolation confusion: predicting far beyond the observed spatial support.
- Uncertainty deletion: publishing a prediction surface without showing where confidence weakens.
21. Ten Questions for Spatial Interpolation
- What was actually measured?
- Where are the sample points?
- How dense is the sampling?
- What spatial continuity is assumed?
- Does direction matter?
- Do barriers matter?
- Which interpolation method is used?
- How was it validated?
- Where is uncertainty highest?
- Are we interpolating or extrapolating?
22. Research Anchors
Esri describes interpolation as estimating values at unsampled locations from known surrounding values. Its documentation distinguishes deterministic approaches such as IDW from geostatistical approaches such as kriging, which explicitly model spatial autocorrelation and can characterise prediction uncertainty.
23. Where This Fits
Spatial Data owns how geographic observations are collected and represented broadly. Spatial Autocorrelation owns dependence among nearby observations. This article owns the inference from measured points to unmeasured space.
The Idea to Keep
Every interpolated map contains two layers: the world we measured and the world we inferred between the measurements.