Correcting errors from spatial upscaling of nonlinear greenhouse gas flux models

Ecological models are used to quantify processes over large regions. When the model is nonlinear and input variables are spatially averaged, the regional mean will be in error.

A formula for estimating the upscaling error can be derived from Taylor expansion of the model (Bresler and Dagan 1988). We test this for simple models under three different input distributions (Gaussian, exponential, lognormal). In several cases the formula is exact, in others it provides a reasonable approximation.
We then study models for emissions of methane, ammonia, and nitrous oxide across the UK. We scale from 1 × 1 km to 32 × 32 km. The UK-average upscaling errors are −12%, −48% and −3%, well estimated using the formula.
The formula is a useful tool for modellers desiring to correct upscaling error for their application. Calculation of second-order partial derivatives of model output is required, for which we provide R-code.