Aggregating Covariates in Spatial Downscaling
## The problem
Downscaling is hard because going from coarse to fine is underdetermined. But aggregating a covariate from fine to coarse is well-posed — it’s just a weighted average. This asymmetry is worth understanding.
How covariate aggregation works
[The key idea: a high-resolution covariate like landcover can be averaged down to the observation footprint straightforwardly, using weighted averages over the footprint geometry]
The A matrix connection
[The elegant result: the same change-of-support operator A from the residual discretization provides a consistency-preserving approximation to covariate aggregation. So you don’t need a separate aggregation scheme — the geometry is already encoded in A]
\[X^{(o)} \approx A X^{(\ell)}\]
## Why this matters practically
[You can exploit high-resolution covariates without paying the computational cost of a fine latent grid. The two discretizations — fine for covariates, coarser for the residual — can be handled separately and cleanly]
In SpatialBasis
[How to pass covariates, what resolution they should be at]
Example
[Code example with a real covariate]