Improved uncertainty estimates for eddy covariance-based carbon dioxide balances using deep ensembles for gap-filling

Eddy covariance (EC) measurements of carbon dioxide (CO${}_2$) fluxes are commonly used to determine CO${}_2$ balances of ecosystems. However, comparisons between experimental treatments, environmental controls or measurement sites are not meaningful without proper uncertainty estimates for the balances. We studied how random and systematic errors depend on the amount of missing data and whether the uncertainty estimates produced by popular gap-filling methods, including tree-based machine learning methods, neural networks and marginal distribution sampling (MDS), are in line with these errors. Using synthetic data created for European forest sites, we found that when the proportion of missing data increased from 30% to 90%, the random uncertainty related to gap-filling (2${\sigma }_{\text{rnd}}$, computed from observed model errors) increased from approximately 10 g C m⁻² y⁻¹ up to 25–75 g C m⁻² y⁻¹ depending on the site and gap-filling method. Ensembles of neural networks (deep ensembles) had smaller random errors than the standard EC gap-filling method MDS, and also produced improved uncertainty estimates for the CO${}_2$ balances. Long gaps of up to one month caused random uncertainty of mostly less than 50 g C m⁻² y⁻¹; however, a long gap during a dry and warm period that was inadequately represented in the measurements caused random uncertainty of up to 99 g C m⁻² y⁻¹. Deep ensembles produced well-calibrated uncertainty estimates also for the long gaps, except for the most difficult cases when long gaps occurred during periods of active change in the ecosystem. The uncertainty estimates produced by MDS for long gaps were clearly too small. Tree-based machine learning methods produced well-calibrated uncertainty estimates for short-term fluxes but not for balances and, unlike deep ensembles, did not extrapolate outside the training data.