Quantifying and reducing uncertainties in estimated soil CO₂ fluxes with hierarchical data-model integration

Nonsteady state chambers are often employed to measure soil CO₂ fluxes. CO₂ concentrations (C) in the headspace are sampled at different times (t), and fluxes (f) are calculated from regressions of C versus t based on a limited number of observations. Variability in the data can lead to poor fits and unreliable f estimates; groups with too few observations or poor fits are often discarded, resulting in “missing” f values. We solve these problems by fitting linear (steady state) and nonlinear (nonsteady state, diffusion based) models of C versus t, within a hierarchical Bayesian framework. Data are from the Prairie Heating and CO₂ Enrichment study that manipulated atmospheric CO₂, temperature, soil moisture, and vegetation. CO₂ was collected from static chambers biweekly during five growing seasons, resulting in >12,000 samples and >3100 groups and associated fluxes. We compare f estimates based on nonhierarchical and hierarchical Bayesian (B versus HB) versions of the linear and diffusion-based (L versus D) models, resulting in four different models (BL, BD, HBL, and HBD). Three models fit the data exceptionally well (R² ≥ 0.98), but the BD model was inferior (R² = 0.87). The nonhierarchical models (BL and BD) produced highly uncertain f estimates (wide 95% credible intervals), whereas the hierarchical models (HBL and HBD) produced very precise estimates. Of the hierarchical versions, the linear model (HBL) underestimated f by ~33% relative to the nonsteady state model (HBD). The hierarchical models offer improvements upon traditional nonhierarchical approaches to estimating f, and we provide example code for the models.