Many of the key processes represented in global terrestrial carbon models
remain largely unconstrained. For instance, plant allocation patterns and
residence times of carbon pools are poorly known globally, except perhaps at
a few intensively studied sites. As a consequence of data scarcity, carbon
models tend to be underdetermined, and so can produce similar net fluxes with
very different parameters and internal dynamics. To address these problems,
we propose a series of ecological and dynamic constraints (EDCs) on model
parameters and initial conditions, as a means to constrain ecosystem variable
inter-dependencies in the absence of local data. The EDCs consist of a range
of conditions on (a) carbon pool turnover and allocation ratios, (b) steady-state proximity, and (c) growth and decay of model carbon pools. We use
a simple ecosystem carbon model in a model–data fusion framework to determine
the added value of these constraints in a data-poor context. Based only on
leaf area index (LAI) time series and soil carbon data, we estimate net
ecosystem exchange (NEE) for (a) 40 synthetic experiments and (b) three
AmeriFlux tower sites. For the synthetic experiments, we show that EDCs lead
to an overall 34 % relative error reduction in model parameters, and
a 65 % reduction in the 3 yr NEE 90 % confidence range. In
the application at AmeriFlux sites all NEE estimates were made independently
of NEE measurements. Compared to these observations, EDCs resulted in
a 69–93 % reduction in 3 yr cumulative NEE median biases
(

Terrestrial ecosystem carbon exchange is a fundamental part of the global
carbon cycle link to biosphere processes. Atmospheric

In recent years a growing volume of data from flux towers, satellites and
plant trait databases has been used to constrain some of the key components
of the terrestrial carbon cycle

Although a range of process-based models have been used to represent the
dynamics of the terrestrial carbon cycle and land–atmosphere

Previous MDF studies have invariably relied on net ecosystem exchange (NEE)
measurements (real and synthetic), along with other site-level observations

NEE, the difference between photosynthesis and ecosystem respiration, is
a function of the dynamics of all carbon pools over a range of timescales. In
the absence of NEE observations, model NEE estimates depend on a knowledge of
carbon pool sizes and model parameter values. In reality, carbon pools and
model parameters (especially those related to plant allocation fractions and
pool turnover rates) are poorly constrained, and therefore NEE estimates are
subject to a comparably large uncertainty. Nonetheless, fundamental knowledge
on ecosystem behaviour can potentially be used to overcome the lack of
location-specific data or parameter values.
For example, while parameters
related to phenology,

In this paper we propose a series of ecological and dynamic constraints (EDCs) on model parameters: these include turnover and allocation parameter inter-relations, carbon pool dynamics and steady-state proximity conditions (Sect. 2). We quantify the added value of imposing EDCs in synthetic and real-data MDF contexts using a simple ecosystem carbon model, by measuring bias and confidence interval reductions of carbon cycle analyses relative to independent data (Sect. 3). Finally, we discuss the prospects and limitations of our approach, as well as the implications of a wider EDC implementation in terrestrial carbon cycle MDF methods (Sect. 4).

DALEC2 model parameters, descriptions, and minimum–maximum parameter values: the corresponding DALEC2 equations are fully described in Appendix A.

Here we present a series of EDCs for
a daily box budget terrestrial

DALEC has been extensively used in MDF frameworks

In previous work, DALEC MDF approaches

Here we propose a sequence of ecological and dynamic constraints (EDCs) on
DALEC2 parameters and pool dynamics. For any given DALEC2 parameter vector

We impose the following constraints on the relative sizes of turnover rates:

Strong correlations are expected between foliar and fine root carbon pools

While we expect pools to potentially grow through time, we assume no recent
disturbance and therefore limit the relative growth rate of pools. We
constrain pool growth as follows:

While carbon pools are expected to grow and contract through time, in the
absence of major and recent disturbance events carbon pool trajectories are
expected to exhibit gradual changes on inter-annual timescales

DALEC2 pool trajectories are rejected if the half-life of carbon pool changes is less than 3 years, i.e.

Exponential decay test (EDC 8) performed on nine example normalized
C

For ecosystems with no recent disturbance events, we propose that each pool
is within an order of magnitude of its steady-state attractor. We use mean
gross primary production (

The 12 presented EDCs are what we believe to be the most ecologically suitable constraints on DALEC2 parameters and state variables, and are based on broader ecological knowledge of carbon dynamics. We discuss the advantages and the limitations of the proposed EDCs in Sect. 4 of this paper.

Given LAI observations, soil organic carbon estimates, prior parameter ranges
(Table 1) and EDCs (Sect. 2.2), our aim for each experiment is to estimate
the probability distribution of parameters

We employ an adaptive Metropolis Hastings Markov Chain Monte Carlo (MHMCMC)
approach to draw

To quantify our ability to estimate synthetic DALEC2 ecosystem states, we
perform the MDF approach over a 3-year period using LAI and SOM
observations created from a synthetic DALEC2 truth, based on known DALEC2 parameters.
Our choice of synthetic DALEC2 states represents globally spanning data sets of satellite LAI retrievals and
soil carbon map data.
Based on 40 DALEC2 parameter combinations, we create 40 synthetic data sets representing typical
temperate forest carbon dynamics, with 3 years of semi-continuous LAI
data and one simulated soil organic carbon estimate.
We use the 3-year
meteorology drivers (temperate climate) from the REFLEX synthetic experiments

We select 40 synthetic parameter combinations
by randomly sampling parameter vectors

We perform the MHMCMC and label the posterior parameter ensemble (

For the flux-tower experiments, we constrain DALEC2 parameters using (a)
MODIS derived Leaf Area Index (LAI), and (b) total soil carbon from the
harmonized world soil database

For each AmeriFlux site, we extract the corresponding MODIS LAI retrievals
from the MOD15A2 LAI 8-

For each site we extract total soil carbon density from the nearest
Harmonized World Soil Database 30 arc seconds resolution total soil carbon
content

To limit our study to the use of globally spanning data sets, we extract
DALEC2 drivers from 0.125

As done for the synthetic experiments, we perform the MHMCMC approach at each
site – with and without EDCs – and label the posterior parameter ensembles
(4 chains

To determine the sensitivity of our results to EDCs 1–12, we repeat MDF estimates of

We compare 3 yr integrated DALEC2 NEE estimates and AmeriFlux NEE measurements at all three sites (AmeriFlux NEE measurement temporal gaps have been consistently excluded from DALEC2 3 yr NEE estimates). We determine the DALEC2 3 yr NEE 50% confidence range (50% CR: 25th–75th percentile interval) reduction as follows:

Synthetic experiment parameter error reduction, and AmeriFlux experiment 3 yr NEE 50% CR and bias reduction for MDF estimates using individual EDCs, relative to the standard MDF estimates.

Aggregated parameter estimates

The inclusion of EDCs resulted in substantial error reductions in posterior
DALEC2 parameter and state variable estimates. We found an overall reduction
in the posterior MHMCMC EDC parameter vector errors

Three-year mean DALEC2 net ecosystem exchange (NEE) biases (relative to synthetic truth)
aggregated across 40 synthetic experiments at 0.5

v

We compared EDC total

We found that incorporating EDCs resulted in a reduced mode and 90 %
confidence range (90% CR: 95th–5th percentile interval) for 3-year NEE
biases (Fig.

The DALEC2(

DALEC2 daily NEE ensemble estimates at three AmeriFlux sites: Sylvania Wilderness (US-Syv, mixed
forest, top two rows), Howland Forest (US-Ho1, evergreen needleleaf, middle two rows), and Morgan Monroe
State Forest (US-MMS, deciduous broadleaf, bottom two rows). For each site the DALEC2(

Cumulative AmeriFlux NEE observations are compared against corresponding
DALEC2(

Three-year mean DALEC2 cumulative NEE (

With the use of a simple model and globally available data, i.e. leaf area dynamics and soil carbon observations, we have demonstrated that the EDC approach provides an improved ability to infer the magnitude of carbon fluxes, live carbon pools and model parameters, in comparison to a standard parameter optimization approach (STA).

For ecologically relevant synthetic truths, EDCs provide improved estimates
of the DALEC2 parameters and state variables. The EDC approach resulted in
(a) parameter estimation error reductions, (b) NEE bias and confidence range
reductions, and (c) improved estimates of the live biomass

By comparing DALEC2 analyses against independent AmeriFlux NEE measurements
over real ecosystems, we further validated the advantages of using EDCs. At
each AmeriFlux site, we found that EDCs led to an increased confidence and
a largely reduced NEE bias; our DALEC2 model analyses suggests that the use
of EDCs regionally and globally could significantly enhance our ability to
estimate ecosystem state variables in the absence of direct observational
constraints. In light of the large differences between Earth system models

Together, EDCs 1–12 lead to overall improvements in parameter estimates and AmeriFlux site NEE confidence range/bias (Table 2): however, with the exception of EDC 10, when EDCs were tested individually, they did not lead to comprehensive improvements. For example, EDC 8 alone (no rapid exponential pool decay) resulted in large AmeriFlux site NEE confidence range reductions, as well as improved synthetic parameter estimates; however, EDC 8 resulted in higher AmeriFlux site NEE biases. Conversely, EDC 9 (steady-state proximity of the soil carbon pool) resulted in the largest AmeriFlux site bias reductions, while NEE confidence was lower. EDC 5 (comparable fine root and foliar/labile allocation) led to the largest parameter improvements; however, the associated changes in AmeriFlux site NEE estimates were relatively small. Our findings demonstrate that robust improvements in carbon cycling parameter and state variable estimates only arise when EDCs are used collectively.

Here we developed a group of EDCs suitable to ecosystems with no recent major disturbance. However, we note that our EDCs can be adapted for a wider range of ecosystem dynamics. For example, recently disturbed ecosystems may be (a) rapidly recovering and (b) growing towards a steady state where carbon pools are greater than one order of magnitude from the initial carbon pools. Therefore a subset of our EDCs (EDCs 7–12) can be adapted to better represent ecological “common sense” in recovering ecosystems.

Ultimately, EDCs can be adapted to best represent ecological knowledge in
a variety of ecosystem carbon model MDF applications, where the ecosystem
observations are insufficient to constrain all model state variables (e.g.
Fox et al., 2009). For example, on regional and global spatial scales, there
is often no explicit knowledge on various model parameter values and their
associated uncertainty. In such cases, our EDC approach imposes
inter-parameter constraints while simultaneously allowing a global parameter
exploration across several orders of magnitude (see Table 1). Hence EDCs
allow us to incorporate ecologically consistent relationships between
parameters (i.e. allocation ratios, turnover ratios), without the need to
constrain otherwise unknown parameter and state variables. Moreover, as an
alternative to imposing plant-functional-type priors, which risk being
subjective and over-rigid, ecosystem trait inter-relationships derived from
plant trait data

In this study we limited our observational constraints to globally spanning
MODIS LAI retrievals and the HWSD soil map.
Given these two data sets, we have demonstrated that EDCs lead to improved model
parameter estimates and reduced NEE bias and confidence ranges. Nonetheless,
based on the posterior NEE probability density function,
we are unable to determine whether sites are net carbon sinks or sources on
annual timescales.
However, an increasing number of
continental and global scale biospheric data sets are becoming available:
these include a global canopy height map by

We have addressed the underdetermined nature of the carbon cycle problem by applying a group of widely applicable ecological and dynamic constraints (EDCs) on an ecosystem carbon model in a model–data fusion (MDF) framework. Particularly where extensive in situ measurements are not available, EDCs can be used to incorporate ecological knowledge, such as parameter inter-relationships and pool dynamics constraints, into ecosystem carbon model analyses. In a synthetic data experiment, we found improved estimates of DALEC2 model parameters, live carbon pools and net ecosystem exchange (NEE) when using EDCs in DALEC2 MDF analyses. By validating our DALEC2 MDF analyses against independent AmeriFlux NEE measurements, we found that EDCs led to a 69–93 % reduction in 3-year NEE biases. We incorporated 12 EDCs in DALEC2 analyses of temperate forest ecosystem carbon cycling: these EDCs can potentially be adapted for a range of models and biomes. Moreover, additional EDCs can be derived to incorporate parameter inter-relationships derived from regional or global plant trait data sets into ecosystem carbon model analyses. Here we have shown that EDCs can be used to constrain the poorly resolved components of the carbon cycle: we therefore advocate the use of EDCs in future MDF analyses of the terrestrial carbon cycle.

The full DALEC2 model dynamics can be expressed as six equations:

The model is initiated with six initial carbon pool values
(C

The

Schematic of the carbon fluxes in DALEC2. The green arrow indicates the gross primary production
(GPP). Red arrows represent respiration fluxes: autotrophic respiration (

For AmeriFlux DALEC2 analyses we used daily meteorological drivers for DALEC2
from 0.125

Exponentially decaying C

For the standard MDF parameter estimates the normalized parameter probability is

Based on the

Run DALEC2(

If

The MHMCMC parameter sampling approach is then repeated four times (four
chains): to determine whether all four chains have converged to the same
parameter distributions, we use the Gelman–Rubin convergence criterion

The 40 synthetic experiments were created by searching for parameter vectors

For a given vector

We simplistically simulate the 8-daily MODIS LAI data and soil carbon map
HWSD products from DALEC2(

For LAI synthetic observations, we only kept one in eight LAI values, and created correlated gaps in the remaining LAI data of random lengths until at least 50 % of the 8 daily data is removed. Overall, between 65 and 68 LAI observations are kept for each 3

This project was funded by the NERC National Centre for Earth Observation,
UK. This work has made use of the resources provided by the Edinburgh
Compute and Data Facility (ECDF,