The Problem of Space

Your supervisor pins a printed map to the wall: fifteen coloured dots marking Bluewater Basin's groundwater monitoring wells, each with a number, the nitrate concentration measured last month. The dots range from 2.8 mg/L near the forest reserve to 13.2 mg/L at well GW-14, near the agricultural zone.

A policymaker across the table points to a gap between wells GW-14 and GW-06, roughly 8 kilometres of unmonitored aquifer. 'What's the nitrate level there?' she asks. Your regression model can't answer this: it predicts a number from other variables, not from spatial position. A different kind of model is needed, one that understands that nearby things tend to be more similar than distant things.

Spatial analysis exploits a powerful and empirically well-supported idea called Tobler's First Law of Geography: 'everything is related to everything else, but near things are more related than distant things.' In practice, this means a well 200 metres from GW-14 is far more likely to have similar nitrate levels than a well 8 kilometres away.

The field that formalises this intuition is geostatistics (the study of how measurements vary and relate to each other across physical space). Rather than ignoring spatial position, geostatistics treats location as a fundamental variable and models not just values but the spatial structure of their variation, how similarity decays with distance.

Think about how loud music sounds as you walk away from a speaker at a party. Right next to it, it is deafening. A few steps away, it is a bit quieter. Across the yard, you can barely hear it. Nitrate levels near a well behave the same way, values close to a known measurement tend to resemble it, and that resemblance fades the farther away you go.

Tobler's First Law is one of the most cited concepts in all of geography, and it underlies everything from weather forecasting (nearby weather stations are correlated) to remote sensing (adjacent pixels share information) to environmental modelling (contamination spreads in spatially structured patterns). Recognising when your data has spatial structure, and accounting for it, is one of the most frequently missed steps in environmental data analysis.

Explain Tobler's First Law of Geography using temperature as your example instead of nitrate.