10.3 Focal and Zonal Statistics

Key takeaways

• Focal statistics compute per-cell values from a moving-window neighbourhood.
• Zonal statistics summarise raster values per polygon (or categorical raster zone).
• Both are core building blocks of terrain analysis, landscape ecology, and environmental modelling.

Introduction

If local map algebra is arithmetic on rasters, focal and zonal operations are where geography asserts itself. Focal looks at a cell's neighbours; zonal summarises cells within a region. This lesson covers both in practical terms.

Focal statistics

A focal operation applies a function over a neighbourhood (window / kernel) centred on each cell, producing an output cell value. Typical functions:

• Mean — smoothing / low-pass filtering.
• Standard deviation — roughness.
• Maximum / minimum — local extremes.
• Sum — density (population per window).
• Majority — per-cell most common class (denoising classified rasters).

Window shapes

• Square (3×3, 5×5, 11×11) — most common.
• Circle — rotationally symmetric, better for natural phenomena.
• Custom kernel — weighted by distance or direction.

Implementation

from scipy.ndimage import generic_filter

Or with a convolution kernel:

from scipy.ndimage import convolve

Specific focal operations

• Slope — first derivative of elevation (Module 11.2).
• Aspect — direction of steepest descent.
• Curvature — second derivative.
• Hillshade — shaded relief from slope + aspect (Module 11.3).
• Terrain Ruggedness Index (TRI) — mean absolute difference from neighbours.

Edge effects

At the raster edge, the neighbourhood extends off the array. Strategies:

• Pad with nodata — edge cells inherit nodata.
• Pad with edge value — reflect or repeat.
• Shrink output — drop edge cells.

Document the choice; it matters for mosaicking.

Zonal statistics

A zonal operation summarises a raster's cells within each zone, where zones are defined by another layer:

• A polygon feature class (each polygon is a zone).
• A categorical raster (each class code is a zone).

Common statistics: mean, sum, count, min, max, median, standard deviation, majority.

With rasterstats

1from rasterstats import zonal_stats
2[object Object]
3[object Object]
4

Join the results back to the polygon layer:

import geopandas as gpd

In QGIS

Processing → Zonal Statistics or the Zonal Statistics Plugin.

In PostGIS raster

1SELECT d.id, d.name, (ST_SummaryStats(ST_Clip(r.rast, d.geom))).*
2FROM districts d
3JOIN rasters r ON ST_Intersects(r.rast, d.geom);

In Google Earth Engine

1var stats = image.reduceRegions({
2  collection: districts,
3  reducer: ee.Reducer.mean(),
4  scale: 30
5});

GEE scales to continents trivially.

Coverage (fractional zonal stats)

Strict "a cell is fully inside a zone" ignores partial overlaps. For small zones and coarse rasters, this matters. Fractional / area-weighted zonal statistics weight each cell by the fraction inside the zone:

1stats = zonal_stats('zones.gpkg', 'population.tif',
2                    stats='mean',
3                    all_touched=True)

Or use exactextract for exact fraction-weighted stats.

Majority / diversity

For categorical rasters (land cover):

• Majority — most common class per zone.
• Diversity — count of unique classes.
• Shannon / Simpson indices — diversity metrics from ecology.

1stats = zonal_stats('zones.gpkg', 'landcover.tif',
2                    categorical=True, stats=['count'])

Performance

Zonal stats on millions of zones and continent-scale rasters is slow naively. Tips:

• Reproject zones to raster CRS (not vice versa — reprojecting polygons is much cheaper).
• Pre-filter zones by bounding box intersection.
• Tile the raster and process in parallel.
• Earth Engine / Planetary Computer for large jobs.

Self-check exercises

Summary

• Focal ops compute per-cell statistics from a neighbourhood window.
• Zonal ops summarise raster values per polygon or categorical zone.
• Both scale from hectares to continents with the right tools.
• Edge effects, alignment, and fractional cells are the real-world wrinkles.

Further reading

• Tomlin, C. D. — Geographic Information Systems and Cartographic Modeling.
• rasterstats and exactextract Python packages.
• Earth Engine reduceRegions documentation.
• Geocomputation with R — raster analysis chapter.