Parameters of a process-based forest growth simulator are difficult or
impossible to obtain from field observations. Reliable estimates can be
obtained using calibration against observations of output and state
variables. In this study, we present a Bayesian framework to calibrate the
widely used process-based simulator Biome-BGC against estimates of gross
primary production (GPP) data. We used GPP partitioned from flux tower
measurements of a net ecosystem exchange over a 55-year-old Douglas fir stand
as an example. The uncertainties of both the Biome-BGC parameters and the
simulated GPP values were estimated. The calibrated parameters leaf and fine
root turnover (LFRT), ratio of fine root carbon to leaf carbon (FRC : LC),
ratio of carbon to nitrogen in leaf (C : N

Forest ecosystems play an important role in the global carbon cycle by
controlling the atmospheric CO

Simulating a PBS requires input parameters that distinguish different
vegetation types by their physiological and morphological characteristics.
Implementation of a PBS for specific sites is complicated by the large number
of parameters for plants, the soil and the atmosphere. Field measurements of
PBS parameters are difficult or impossible to obtain, leading to incomplete
knowledge of site-specific parameters for the occurring species. In practice,
practitioners often rely on the literature for values of the PBS parameters

A systematic adjustment of PBS parameters is required within the margins of
the uncertainty so that the simulated outputs (e.g. GPP) satisfy pre-agreed
criteria. This adjustment of simulator parameters is called calibration.
Calibration is often performed to obtain single optimized values of the
parameters without the quantification of uncertainty in the parameters and
the simulated outputs. Quantification of uncertainty is important for both
scientific and practical purposes

A Bayesian framework provides a coherent method for calibrating a PBS

In this study, a widely used simulator, Biome-BGC

Biome-BGC simulates GPP at a daily time step and it updates its memory between
days

The objective of this study was to quantify the uncertainty in Biome-BGC input parameters and simulated GPP by integrating flux tower GPP into Biome-BGC in a Bayesian framework. We obtained the posterior Biome-BGC parameters (a) by calibrating the Biome-BGC to the data of entire study period (growing season) and (b) by calibrating the Biome-BGC to 1 month of GPP data and repeating the calibration for all months in the growing season. The main novelty of this paper is the presentation of a Bayesian framework for Biome-BGC parameter estimation. The simulator itself is left unaltered. Additionally, investigation of temporal variation in Biome-BGC input parameters would also reinforce the reconsideration of the assumption of constant parameters of other process-based simulators for photosynthesis.

Calibration of Biome-BGC was performed at the Speulderbos Forest site, which
is located at 52

The 35 ecophysiological parameters needed to run Biome-BGC for Douglas
fir (evergreen needleleaf species). Mean values/distributions were taken from

Biome-BGC simulates biogeochemical processes including carbon, water, and
nitrogen fluxes within the vegetation, litter, and soil compartment of
terrestrial ecosystem at daily time steps

Biome-BGC requires site characteristics, daily meteorological data, and
ecophysiological parameters as inputs. The site characteristics include soil
texture (percentage of sand, silt, and clay), elevation, latitude, shortwave
albedo, wet and dry atmospheric deposition of nitrogen, symbiotic and
asymbiotic fixation of nitrogen, and the effective soil rooting depth. We
took the site characteristics data at Speulderbos from

In this study, initial states of water, carbon, and nitrogen variables of the
Biome-BGC were prescribed with very low value (

We used observed data of NEE to predict GPP at Speulderbos for the growing
season (April to October) of 2009. To predict GPP, half-hourly GPP values
were separated from flux tower measurements of half-hourly net ecosystem
exchange at Speulderbos site using the non-rectangular hyperbola (NRH) model

Bayesian calibration begins with Bayes' rule

The likelihood function is determined by the probability distribution of the
residuals

The posterior pdf in Eq. (

We adopted the DREAM algorithm proposed by

For each chain

A simulator is run at the starting points and the likelihood

The choice of likelihood and prior pdf of

For

A candidate point

and

where

The simulator is run at the candidate point

The Metropolis ratio is given as

The candidate point

If the candidate point is accepted:

All

The choice of

The computational load of Bayesian calibration of a simulator can be reduced
by excluding those input parameters that have negligible influence on the
simulated output

Recall from Sect.

Biome-BGC simulates the time series of GPP at daily time steps. We relaxed
the assumption that the temporal profile of simulated GPP perfectly follows
the flux tower GPP and modelled the temporal correlation in the residuals. We
adopted a likelihood that assumes the residuals follow an autoregressive
process of an order of 1

Equation (

If the error residuals are assumed to be uncorrelated, Eq. (

We also checked the changes in the results using the likelihood function not
accounting for correlation in the residuals (Eq.

We implemented the DREAM algorithm in MatLab version R2015b. The DREAM
toolbox was provided by its developer, Jasper A. Vrugt, from the University of
California, Davis, USA. Technical details of the DREAM toolbox are provided
by

We used

Gelman–Rubin potential scale reduction factor (PSRF) of each
Biome-BGC parameter selected for calibration and

The post-burn-in samples created 50 000 vectors of

We conducted two experiments to obtain the posterior samples of

Experiment 1: We used daily mean of flux tower GPP for 5 months in the growing season
(April to August 2009) to calibrate Biome-BGC for the growing season. For
the calculation of the likelihood using Eq. (

Experiment 2: We used daily mean of flux tower GPP for 1 month only,
e.g. April, in the growing season to calibrate Biome-BGC. For the
calculation of likelihood using Eq. (

For both experiments, we followed the same procedure explained in the second and third paragraphs of this section.

We determined the performance of the calibration using two criteria that
evaluate efficiency with which the calibrated Biome-BGC reproduces the flux
tower GPP. Both criteria provide a single measure of Biome-BGC efficiency in
simulating daily GPP over the selected period. The first criterion was the
root mean square error (RMSE) between the simulated and flux tower GPP:

Trace plot of each calibrated Biome-BGC parameter and

We evaluated the performance of Biome-BGC for the following cases:

For Experiment 1, we obtained RMSE and NSE for the two periods:
the calibration period of 5 months (April to August) and the validation period
of 2 months (September and October). For each period, the calculations were
made for 2.5 percentiles, 97.5 percentiles, and medians. Note that the RMSE
and NSE are typically evaluated at the median of the posterior predictive
distribution; however, this does not evaluate the posterior uncertainty

For Experiment 2, we obtained RMSE and NSE for the same two periods and percentiles as stated in point 1 (above), to make a direct comparison with the results of Experiment 1.

To show the performance of uncalibrated Biome-BGC, we obtained the daily
simulated GPP with 95 % credible intervals at the prior distributions of
six selected parameters (Table

The value of the Gelman–Rubin PSRF was close to 1 for each

Figure

Median (solid lines) and 95 % credible intervals (dashed lines)
of the posterior distributions of each calibrated Biome-BGC parameter
obtained from Experiment 2 for each month during the growing season of 2009.
The grey shade and dotted–dashed line represent median and 95 % credible
intervals obtained for Experiment 1. The range of the

Figure

A Bayesian calibration also allowed us to obtain correlation between the
calibrated parameters. Figure

For Experiment 2, the uncertainties in LFRT, FRC : LC,

Correlation coefficient and scatterplot between the posterior
distributions of each pair of calibrated Biome-BGC parameters obtained from
Experiment 1. Information about the Biome-BGC parameters is given in Table

We evaluated the performance of calibrated Biome-BGC by comparing the daily
posterior GPP and the daily flux tower GPP for the calibration period of
April to August (Fig.

Overall, daily posterior GPP was close to flux tower GPP during the
calibration period (Fig.

The posterior GPP was improved compared with the prior GPP, as indicated by
the drop of RMSE for the median as well as the 2.5 and 97.5 percentiles for
both calibration and validation periods (Table

We also evaluated the performance of calibrated Biome-BGC using the
likelihood function without the temporal correlation in the residuals
(Eq.

Temporal profile of daily posterior GPP, obtained from Experiment 1,
and daily flux tower GPP for the calibration period of 5 months (April to
August, Julian days 91 to 243). Daily medians and 95 % credible intervals
of posterior GPP, obtained using likelihood function of Eq. (

Temporal profile of daily posterior GPP, obtained from Experiment 1,
and daily flux tower GPP for the validation period of 2 months (September
and October, Julian days 244 to 304). Other details are as for
Fig.

Root mean square error (RMSE) and Nash–Sutcliffe efficiency (NSE)
between the prior (before calibration)/posterior GPP and flux tower GPP for
different experiments (see Sect.

Temporal profile of daily posterior GPP, obtained from Experiment 2,
and daily flux tower GPP for 5 months (April to August, Julian days 91 to
243). Other details are as for Fig.

Combining the daily simulations of each month provided the temporal profile
of the medians and 95 % credible intervals of the daily posterior GPP
over the growing season. Figure

The Biome-BGC internal routines that simulate GPP, controlled by the meteorological data and the six
calibrated parameters. Rectangular boxes represent the Biome-BGC routines and
the parallelograms represent the input and output of the routine. Information
about the Biome-BGC parameters is given in Table

To explain our results, we identified the processes within Biome-BGC that are
controlled by the six calibrated parameters and relate to the simulation of
GPP (Fig.

Biome-BGC simulates the daily development of plant carbon pools

The photosynthesis routine converts the conductance to water vapour to the
conductance for CO

We presented the link between six calibrated parameters and the Biome-BGC
internal routines so that we could explain our results considering the
development of the state variables, principally such as LAI and

Biome-BGC accounts for dynamics in carbon stocks in the vegetation by means of
allocation. Hence, it uses parameters that are constant for the year of
simulation. Consider Experiment 1. The memory of Biome-BGC is updated between
days (Sect.

Experiment 1 showed that Biome-BGC was able to reproduce closely the flux
tower GPP. Further, the Bayesian calibration allowed daily posterior GPP
simulation as well as quantification of the associated uncertainty
(Figs.

Consider Experiment 2. Note that Biome-BGC actually simulated daily posterior
GPP for a whole year with the posterior distributions of the parameters of
each month. We selected only the daily posterior GPP of that month to which
the posterior distributions belong and we discarded the other 11 months
of simulations. The temporal profile in Fig.

We observed an improvement (Fig.

The previous studies have also highlighted the improvement in the performance
of simulator BEPS (Boreal Ecosystem Productivity Simulator)

The major metrics of the carbon cycle include GPP, ecosystem respiration
(

We performed our calibration based on six parameters (LFRT, FRC : LC,
C : N

This study presented a Bayesian calibration framework for the simulator
Biome-BGC. We illustrated the framework at the Speulderbos Forest site in the
Netherlands. Use of the framework led to the following conclusions:

The Bayesian framework allowed quantification of uncertainty in both the estimated parameters and the posterior (predictive) GPP, through the posterior (predictive) distribution. The uncertainty is important in the sense that it helps to determine how much confidence can be placed in the results of forest carbon-related studies based on GPP. A calibration based on optimization of Biome-BGC parameters, as done in earlier studies, can not capture the associated uncertainty in the simulated GPP.

We modelled the temporal correlation in the residuals through the nuisance
parameter,

We used the calibration results to gain further insights into the functioning (dynamic processes) of Biome-BGC through analysis of the monthly variation in posterior parameter distributions. Our study revealed the model deficiency of Biome-BGC for using constant parameters to simulate seasonality of state variables and thus the seasonality in daily GPP. The seasonality was captured more precisely by using monthly variation in the Biome-BGC parameters. In future, such model deficiency should receive attention from the Biome-BGC modelling communities. Nevertheless, our findings also suggest that the other modelling communities that use the similar process-based simulators may also consider to improve such model deficiency.

We implemented our calibration using the DREAM algorithm. DREAM offers considerable computational advantages and flexibility as compared to other MCMC implementations. It shows promise for biogeochemical and other environmental simulation applications. Specifically, future research could calibrate more parameters.

We provide a MatLab script and input data as
supplementary material to support the implementation of Bayesian calibration
of the Biome-BGC simulator. The MatLab script uses the functionality of the
DREAM toolbox, which can be obtained, on request, from its developer, Jasper
A. Vrugt, University of California, Davis, USA (Vrugt, 2016). The source code
of the Biome-BGC simulator can be downloaded from

Additional information on code and data can be found in

The description of each file in the supplementary material is given below:

MatLab scripts:

Input data files used to run Biome-BGC in our experiment (for details, see the Biome-BGC user guide that comes with the source code of
Biome-BGC):

Input flux tower GPP (for calibration and comparison with posterior simulated
GPP):

The authors declare that they have no conflict of interest.

Biome-BGC version 4.2 was provided by Peter Thornton at the National Center for Atmospheric Research (NCAR) and by the Numerical Terradynamic Simulation Group (NTSG) at the University of Montana, USA. NCAR is sponsored by the National Science Foundation. The authors thankfully acknowledge the support of the Erasmus Mundus mobility grant and the University of Twente for funding this research. The authors acknowledge Jasper A. Vrugt, from the University of California, Davis, USA for providing the DREAM toolbox. Edited by: Philippe Peylin Reviewed by: Thomas Wutzler and one anonymous referee