Calibrated spatial uncertainty for Earth observation foundation models via Matérn-motivated latent stochastic regularization

Daniel Patrick Johnson 

Earth observation foundation models produce dense spatial embeddings that
support transfer learning across sensors and regions, yet these representations
carry no explicit spatial statistical model for covariance, smoothness,
or uncertainty. Existing deep learning uncertainty methods produce per-pixel
variance estimates that ignore spatial dependence, and their calibration
is rarely assessed beyond in-distribution data. This study introduces
a Matérn-motivated latent stochastic regularization layer for frozen foundation
model embeddings. The layer evolves embeddings through a learned Itô
stochastic di?erential equation whose linear-case Fokker?Planck stationary
distribution recovers the Matérn precision structure of the classical Whittle
SPDE; in the implemented nonlinear parameterization, this correspondence
serves as a design principle rather than as an explicit spatial di?erential operator.
The layer adds fewer than 800,000 parameters with no foundation
model retraining. Applied to OlmoEarth embeddings over a 10km × 10km
training patch in Indianapolis, USA, the layer achieves 90% prediction interval
coverage within 0.8 percentage points of nominal for both NDVI and
land surface temperature (LST). Under four-fold within-site spatial block
holdout, coverage degrades by 3.9 percentage points for NDVI and 7.7 for
LST, compared with 6.1?12.8 for matched-capacity baselines. Under deployment
to the full Marion County study area, the layer retains PICP@90 =
82.7% for LST while matched MLP and CNN baselines drop to 73.2?73.3%
despite comparable point predictions and similar mean predicted variances.
A tract-level analysis at the 38 ◦C summer LST threshold shows this calibration
advantage ?ags approximately 33,000 additional residents as warranting
heat-exposure investigation under a precautionary decision rule.