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).