Geostatistical data

Spatially continuous phenomenon measured at discrete sites
- We often believe the underlying thing we care about varies smoothly over space
- But we can only measure it at a limited number of locations
Examples
- Soil mercury level taken at a set of soil probes
- Relative humidity measured at a set of observatories
- Traffic measured at a set of traffic lights
$Z=Z(s_i),...,Z(s_n)$ at locations $s_1,...s_n$
A partial realisation of random process $Z(s):s\in D\subset \mathbb{R}^2$
Common goals are to
- Infer the characteristics of the spatial process
  - How quickly does similarity decay with distance?
- Predict the values at unsampled locations
  - Estimate $Z(s_x)$at a location $s_x$ where there is no measurement
- Construct a spatially continuous surface of the variable

Spatial interpolation

Taking known values at sample locations and using them to estimate unknown values at other locations
Guided by Tobler’s First Law

Values of the closes sampled locations
- For a prediction site $Z(s_x)$, find a sampled location $s_i$ that is closest to $s_x$, let $Z(s_x)=Z(s_i)$
Voronoi diagram (aka Dirichlet or Thiessen diagram)
- A region with $n$ points is partitioned into convex polygons
- Each polygon contains exactly one generating point
- Every point in a given polygon is closer to its generating point than any other

Example
- Imagine dividing a city so that each house belongs to the nearest fire station
- Each station’s service area is its Voronoi polygon
- Any house in that area will call that station first
Pros
- Very simple and fast
- Easy to understand
Cons
- Produces sudden jumps at polygon boundaries (no smooth transitions)
- Doesn’t use information from more than one neighbour