Why exact fractional coverage

The single most important modelling choice in geohalo is how a cell on the polygon boundary is weighted. Three approaches are common, and they disagree most exactly where it matters: along the perimeter.

The same polygon over a 5×5 mesh, scored three ways. Centroid keeps a cell only if its centre is inside; all-touched keeps any grazed cell whole; exact fractional weights each cell by the true overlap area.

The three methods

CentroidAll-touchedExact fractional

A cell counts (weight 1) if and only if its centre falls inside the polygon.

\[ W_{ij} = \begin{cases} 1 & \text{centre}_j \in P_i \\ 0 & \text{otherwise} \end{cases} \]

Cheap, but lossy: it ignores how much of each boundary cell is actually covered, and any polygon smaller than a single cell that doesn't happen to contain a centre collapses to zero cells — no estimate at all.

A cell counts (weight 1) if it intersects the polygon at all.

\[ W_{ij} = \begin{cases} 1 & \text{cell}_j \cap P_i \neq \varnothing \\ 0 & \text{otherwise} \end{cases} \]

Also cheap, but bloated: a cell the polygon barely grazes is counted as if fully inside, inflating the footprint by roughly half a cell all the way around the perimeter.

A cell's weight is the fraction of its area the polygon covers.

\[ W_{ij} = \frac{\text{area}(\text{cell}_j \cap P_i)}{\text{area}(\text{cell}_j)} \in [0, 1] \]

This is what geohalo uses, via exactextract. A boundary cell that is 30 % inside contributes 0.30; a fully interior cell contributes 1.0. The weight is proportional to true overlap, so the estimate is unbiased.

The bias, quantified

Method	Bias	Cost
Centroid	High; polygons smaller than a cell collapse to 0	Cheap
All-touched	Overcounts by ~½-cell around the perimeter	Cheap
Exact fractional	Unbiased — weight is the true overlap fraction	One-time only

The errors from centroid and all-touched are systematic, not random: centroid consistently under-samples the boundary, all-touched consistently over-samples it. They do not average out across polygons, and they grow as cells get larger relative to the polygons — exactly the regime of coarse grids over small administrative areas.

Why the cost stops mattering

Exact coverage is more expensive to compute than a centroid test — but in geohalo it is computed once, baked into the stencil, and cached. Every grid afterwards pays the same fast matmul regardless of which method built the weights. You buy unbiasedness at precompute time and never pay for it again, which is why "cheap" no longer recommends the lossy options.

Sub-cell polygons

Exact coverage gives a sensible answer even when a polygon is smaller than one cell: it returns that cell's value (the area-weighted mean of the one cell it sits in). That is the best estimate under a "uniform within a cell" assumption — but geohalo can do better by refining the grid first with mean-preserving downscaling, which uses neighbouring cells to give sub-cell polygons a sharper, still-unbiased answer.