2.7 | Statistics
There was no replicate of the R and F treatments so it was impossible to perform analyses of variance to infer any fertilisation effect. However, we could not ignore the effect of the fertilisation on the F plot (Lim et al., 2015). We therefore presented the plot differences recognising that they may include a pre-existing plot effect as well as a fertiliser effect.
However, because 15 trees were sampled at each site for δ13Cp estimate, we did analyse a ‘plot effect’. We performed the same analyses of variance with WUEi which could be estimated for all of the 15 trees at each date. When necessary, δ13Cp and WUEi data were log-transformed to meet normality and homoscedasticity requirements. Temporal variations of δ13Cp and WUEi were analysed with a linear mixed model to take into account the repeated δ13Cp sampling within individual trees in 2012. ‘Sampling date’, ‘plot’ and ‘plot × sampling date’ were assigned as fixed factors whereas the ‘tree identity’ was considered as a random factor. Similarly, we determined the variance between the different annual sums of GPPiso/SF (according to the three gm assumptions) and with GPPPRELES: ‘plot’ and ‘method’ (three gmassumptions + PRELES) factors were tested on the mean value in 2012-2013. Daily GPP regressions were run with a first-order autoregressive structure, applying the corAR1correlation option. The analyses were performed with R nlme package (Pinheiro, Bates, DebRoy, & Sarkar, 2016). The anova function from ‘car’ library and multiple pairwise comparisons (library ‘lsmeans’ and ‘multcompView’) were performed.
Finally, we applied a Monte Carlo method to analyse the error propagation in our GPPiso/SF model. This approach was already used in a previous study estimating GPP over a few days (Hu et al., 2010). We randomly sampled from the uncertainty ranges of Δ, Ecd, and gm/\(g_{C\hat{\alpha}}\) to calculate GPPiso/SF in an iterative manner (1000 times). The seasonal pattern of Δ was modeled with the loess method (Cleveland, Grosse, & Shyu, 1992). The uncertainty of daily Δ was estimated based on the residual variance in the curve fitting. Uncertainty of Ecd (from Eqn 4 and 5) was calculated based on the original regression analysis of the transpiration model in Tor-Ngern et al. (2017). Uncertainty of gm/\(g_{C\hat{\alpha}}\) was estimated based on the field measurements in Stangl et al. (2019). Uncertainty of Γ* (from Eqn. 13) was estimated based on the mismatches in the original model fitting in Bernacchi, Singsaas, Pimentel, Portis Jr, & Long (2001). Errors in those inputs were assumed to follow normal distributions or truncated normal distributions (see Table S1). The 95% confidence intervals were calculated to illustrate the predictive uncertainty in our GPPiso/SF estimate (Figure S2). The Sobol indices (Saltelli et al., 2008) were also calculated to partition the variance into these uncertainty sources (Table S1). This method allows us to deal with the absence of replicate sites.
Using Bayesian calibration, we adjusted parameters of PRELES according to their ability to reproduce EC observations (Tian et al., 2020). The Bayesian framework treated all terms in the model calibrations and predictions as probability distributions (Clark, 2007; Dietze, 2017). The joint posterior distribution of parameters was obtained using Markov chain Monte Carlo sampling techniques (Hastings, 1970; Metropolis, Rosenbluth, Rosenbluth, Teller, & Teller, 1953). Meanwhile, the probability density distribution of measurement error was estimated. Based on the parametric uncertainty from the joint posterior distribution and the measurement uncertainty from the error distribution, we estimated the 95% confidence intervals of daily GPP predictions, which describes the ranges of eddy covariance observations that could possibly occur.
All analyses were conducted with R software, version 3.5.1 (R Core Team, 2016).