Gross primary production (GPP) can be separated from flux tower measurements
of net ecosystem exchange (NEE) of

Net ecosystem exchange (NEE) is a terrestrial component of the global carbon
cycle. It is the exchange of

Flux partitioning methods (FPM) are used to partition NEE into its component
flux (GPP and

A night-time-based FPM assumes that NEE is equal to

A quantification of uncertainty in partitioned GPP provides an associated
credible interval that can be used for proper implementation of calibration
and validation of a process-based simulator against partitioned
GPP

In this study, we adopted the NRH model to partition half-hourly GPP from NEE
data. In the past, numerical optimization has been used to estimate a single
optimized value of each model
parameters

The objective of this study was to implement a Bayesian approach for quantification of the uncertainty in half-hourly partitioned GPP using the NRH model given the availability of half-hourly NEE and other meteorological data. The time series of empirical distributions of half-hourly GPP values also allowed us to estimate the uncertainty in GPP at daily time steps. Data were available from a flux tower in the central Netherlands at the Speulderbos forest. This will provide relevant and important information for the validation of process-based simulators.

The Speulderbos forest is located at

The CSAT3, Campbell Sci, LI7500 LiCor Inc, and CR5000 instruments were
installed in June 2006 and have been maintained, and the data processed
(software AltEddy, Alterra) by C. van der Tol (University of Twente,
co-author) and A. Frumau (Energy Centre Netherlands). We examined half-hourly
NEE data (measured at the flux tower) for the growing season (April to
October) of 2009. The quality of the NEE data were assessed using the Foken
classification system, which provides a flag to each half-hourly NEE datum
from 1 through 9

NEE is given as

The light response curve is represented using the NRH
model

Both daytime and night-time half-hourly NEE, PPFD, and

List of symbols with unit.

Bayesian inference treats all parameters as random
variables

MCMC is a method for conducting inference on

We treated Eq. (

As is usual in regression modelling, we assumed normally distributed errors,
hence

In Bayesian analysis it is usual to refer to precision, which is the inverse
of the variance, hence

We made two choices for the prior distribution for each

We assumed a normal distribution for each

Below we justify choices for the informative prior
distributions on

The quantum yield,

A value of

We assumed a normal distribution of

The curvature parameter

The photosynthetic capacity at light saturation

The parameters for temperature sensitivity

The

where

We identified values of

Informative prior distribution of the NRH model parameters:

We used WinBUGS software version 1.4.3

We obtained the posterior distribution of each

We identified the appropriate length of the burn-in for both informative and
non-informative prior distributions. We calculated the Gelman–Rubin
potential scale reduction factor (PSRF) to evaluate the convergence of Markov
chains for each

To perform prediction for a given PPFD

Prediction was performed for each 10-day sample for

We examined the trace plots of the three Markov chains for each

Figure

Median (solid lines) and 95 % credible intervals (dashed lines)
of the posterior distribution of NEE together with
half-hourly NEE measurements (solid points) for a 10-day block (1 May to 10 May 2009, Julian days 121 to 130):

The 10-day block shown in Fig.

Figure

Histograms of half hourly GPP (Morning and afternoon) and daily sum
of GPP when using the following:

Median (solid line) and 95 % credible intervals (dashed lines) of daily GPP distributions during the growing season of 2009 (1 April to 31 October 2009, Julian days 91 to 304) for the choice of informative prior distributions.

We tested whether within the posterior half-hourly GPP distributions, the
non-rectangular hyperbolic relationship of GPP with PPFD had been preserved.
Figure

We concluded that the posterior predictions of half-hourly and daily GPP were
reliable. We used the posterior distribution of the NRH parameters to predict
half-hourly NEE and the 95 % credible intervals bracketed 94 % of the
available half-hourly NEE measurements (Sect.

Median (solid line) and 95 % credible intervals (dashed lines) of half-hourly gross primary production (GPP) with photosynthetic photon flux density (PPFD) for a 10-day block (1 May to 10 May 2009, Julian days 121 to 130) for the choice of informative prior distributions.

Figures

A clear seasonal pattern in the posterior distribution of

We calculated the sum of daily GPP for each of the above mentioned 10-day
blocks (91–100, 281–290, and 291–300) for both choices of prior (Fig. S5).
We found no significant difference in the range of GPP for each block. For
example, the range of daily-summed values for 10-day block 281–290 was
26–38

Median (solid lines) and 95 % credible intervals (dashed lines)
of the posterior distributions of the NRH parameters when using informative
prior distributions for each 10-day block during the growing season in 2009.
The

The plot of

The Bayesian approach applied to the NRH model is a solid method to quantify
the model parameters and their uncertainty. The 10-day block although suited
for the purpose of this study, is insufficient to incorporate the effects of
more rapid changes (day to day) in soil moisture and nutrient levels in the
NRH model. In principle, these rapid changes could be incorporated by daily
estimation of the NRH parameters

As Fig.

The residual term

Systematic errors also result in uncertainty in NEE
measurements

Variation of gross primary production (GPP) with the variation
of photosynthetic capacity (

The implementation of the NRH model assumed that PPFD and

We focused on the growing season in 2009. This short period was chosen to
illustrate the implementation of the Bayesian approach to quantify the
uncertainty in half-hourly partitioned GPP using the NRH model. The study
could be extended towards multiple years, allowing a multi-year comparison
although that was outside the scope of our methodological focus. Further,
different models have been investigated previously to partition
GPP

The study concluded that the choice of informative and non-informative prior
distributions of the NRH model parameters led to similar posterior
distributions for both GPP and NEE. Obtaining informative priors is time
consuming because the values of each parameter are not explicitly mentioned
in the literature. Informative priors also require the acquisition of
information on species or site-specific values of photosynthetic capacity at
light saturation (

The estimates of the NRH model parameters were obtained for 10-day blocks. The values of the posterior parameters and their variation over time could provide further understanding of how the forest responds to factors not included in the model, such as soil moisture, nutrition or tree age.

Quantifying uncertainty estimates as empirical distributions in half-hourly
gross primary production (GPP) was implemented in the Bayesian framework
using the non-rectangular hyperbola (NRH) model. These uncertainty estimates
were provided at daily time steps. The approach could be extended to include
the uncertainty in meteorological forcing, in particular photosynthetic
photon flux density and air temperature. The distributions in half-hourly GPP
can be further used to obtain distributions at any desired time steps, such
as 8-day and monthly. The uncertainty in GPP estimated in this study can be
used further to quantify the propagated uncertainty in the validation of
satellite GPP products such as MODIS 17 or process-based simulators such as
BIOME-BGC. Although we focussed on quantifying the uncertainty in GPP
partitioning, our approach could also be used to either estimate

The authors thankfully acknowledge the support of the Erasmus Mundus mobility grant and the University of Twente for funding this research. Edited by: A. Ito