Spatial Analysis & Geostatistics

Variography, kriging, spatial autocorrelation, hotspot analysis, and geospatial interpolation methods

Overview

Geostatistics provides a mathematical framework for analyzing spatially correlated data. Founded on the regionalized variable theory of Matheron (1963), geostatistical methods model spatial dependence through variograms and use kriging to produce optimal, unbiased predictions at unsampled locations with associated uncertainty estimates. In ocean science, geostatistics is applied to map seafloor properties, interpolate water quality parameters, estimate fish abundance from acoustic surveys, and analyze spatial patterns of environmental variables. Tobler's First Law — "everything is related to everything else, but near things are more related than distant things" — is the foundational principle.

Key Methods

Variogram Analysis

The (semi)variogram γ(h) measures spatial dissimilarity as a function of lag distance h. Key parameters: nugget (measurement error + micro-scale variation), sill (total variance), and range (distance beyond which correlation vanishes). Models: spherical, exponential, Gaussian, Matérn.

γ(h) = ½ E[(Z(x+h) − Z(x))²]
Spherical: γ(h) = C₀ + C₁[1.5(h/a) − 0.5(h/a)³]

Kriging

Ordinary kriging is the Best Linear Unbiased Predictor (BLUP). It estimates Z at unsampled locations as a weighted sum of observed values, with weights determined by the variogram model. The kriging variance provides prediction uncertainty. Variants: simple, universal, indicator, co-kriging.

Ẑ(x₀) = Σ λᵢ Z(xᵢ) , Σ λᵢ = 1
Minimize: Var[Ẑ(x₀) − Z(x₀)]

Spatial Autocorrelation

Moran's I and Geary's C quantify global spatial autocorrelation. Local indicators (LISA, Getis-Ord Gi*) identify hotspots and coldspots. Positive autocorrelation (clustered) is common in environmental data.

IDW & Nearest Neighbor

Inverse Distance Weighting (IDW) is a deterministic interpolation that weights observations by inverse distance raised to a power. Simpler than kriging but does not provide uncertainty estimates or account for spatial structure.

Indicator Kriging & Probability Maps

Indicator kriging estimates the probability that a variable exceeds a threshold (e.g., P(DO < 2 mg/L)). This produces risk maps essential for environmental management and spatial decision-making.

Software & Tools

R: gstat, sp, geoR; Python: scikit-gstat, PyKrige, GSTools; GIS: ArcGIS Geostatistical Analyst, QGIS. GSLIB remains the classic reference implementation.

Interactive Visualizations

Experimental Variogram with Fitted Model

Kriging Interpolation — Simulated Spatial Field

Spatial Autocorrelation (Moran's I Scatterplot)

Key References

  1. Cressie, N.A.C. (1993). Statistics for Spatial Data. Wiley.
  2. Goovaerts, P. (1997). Geostatistics for Natural Resources Evaluation. Oxford University Press.
  3. Webster, R. & Oliver, M.A. (2007). Geostatistics for Environmental Scientists. 2nd ed. Wiley.
  4. Petitgas, P. (1993). Geostatistics for fish stock assessments: a review and an acoustic application. ICES J. Marine Science, 50, 285–298.
  5. Anselin, L. (1995). Local indicators of spatial association — LISA. Geographical Analysis, 27(2), 93–115.