**Model description paper**
17 Apr 2018

**Model description paper** | 17 Apr 2018

# Assimilating solar-induced chlorophyll fluorescence into the terrestrial biosphere model BETHY-SCOPE v1.0: model description and information content

Alexander J. Norton Peter J. Rayner Ernest N. Koffi and Marko Scholze

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**Alexander J. Norton et al.**Alexander J. Norton Peter J. Rayner Ernest N. Koffi and Marko Scholze

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^{1}School of Earth Sciences, University of Melbourne, Melbourne, Australia^{2}European Commission Joint Research Centre, Ispra, Italy^{3}Department of Physical Geography and Ecosystem Science, Lund University, Lund, Sweden

^{1}School of Earth Sciences, University of Melbourne, Melbourne, Australia^{2}European Commission Joint Research Centre, Ispra, Italy^{3}Department of Physical Geography and Ecosystem Science, Lund University, Lund, Sweden

**Correspondence**: Alexander J. Norton (alexander.norton@climate-energy-college.org)

**Correspondence**: Alexander J. Norton (alexander.norton@climate-energy-college.org)

Received: 08 Feb 2017 – Discussion started: 21 Feb 2017 – Revised: 05 Jan 2018 – Accepted: 01 Mar 2018 – Published: 17 Apr 2018

The synthesis of model and observational information using data assimilation
can improve our understanding of the terrestrial carbon cycle, a key
component of the Earth's climate–carbon system. Here we provide a data
assimilation framework for combining observations of solar-induced
chlorophyll fluorescence (SIF) and a process-based model to improve estimates
of terrestrial carbon uptake or gross primary production (GPP). We then
quantify and assess the constraint SIF provides on the uncertainty in global
GPP through model process parameters in an error propagation study. By
incorporating 1 year of SIF observations from the GOSAT satellite, we find
that the parametric uncertainty in global annual GPP is reduced by 73 %
from ±19.0 to ±5.2 Pg C yr^{−1}. This improvement is
achieved through strong constraint of leaf growth processes and weak to
moderate constraint of physiological parameters. We also find that the
inclusion of uncertainty in shortwave down-radiation forcing has a net-zero
effect on uncertainty in GPP when incorporated into the SIF assimilation
framework. This study demonstrates the powerful capacity of SIF to reduce
uncertainties in process-based model estimates of GPP and the potential for
improving our predictive capability of this uncertain carbon flux.

The productivity of the terrestrial biosphere forms a key component of
Earth's climate–carbon system. Estimates show that the terrestrial biosphere
has removed about one quarter of all anthropogenic CO_{2} emissions, thus
preventing additional climate warming (Ciais et al., 2013). Much of the
interannual variability in atmospheric CO_{2} concentration is also
driven by terrestrial productivity. Despite this significance, an understanding
of the underlying mechanisms of terrestrial productivity is still lacking.
This results in large uncertainties in predictions of terrestrial
productivity and thus predictions of future atmospheric CO_{2} and
temperature (Friedlingstein et al., 2006).

A key challenge is disaggregating the observable net CO_{2} flux into
its component fluxes: gross primary production and ecosystem respiration.
Gross primary production (GPP) is the rate of CO_{2} uptake through
plant photosynthesis and the largest natural surface-to-atmosphere flux of
carbon on Earth (Ciais et al., 2013). Estimating spatiotemporal patterns of GPP
at the scales required for global change and modelling studies has proven
difficult. This is primarily for two reasons: the complexity of the
processes involved and the difficulty in observing those processes
(Baldocchi et al., 2016; Schimel et al., 2015). Remote-sensing observations of
solar-induced chlorophyll fluorescence (SIF) offer a novel constraint on GPP
and the potential to partly address these two issues (Schimel et al., 2015).

At the leaf scale chlorophyll fluorescence is emitted from photosystems I and
II during the light reactions of photosynthesis. These photosystems are
pigment–protein complexes that form the reaction centres for converting light
energy into chemical energy. It is in photosystem II (PSII) where
photochemistry, the process initiating photosynthetic electron transport and
leading to CO_{2} fixation, is initiated. The link between chlorophyll
fluorescence and photochemistry is confounded by a third key process,
however: heat dissipation, also termed non-photochemical quenching (NPQ). Both
photochemistry and NPQ are regulated processes, responding to changing
physiological and environmental conditions (Porcar-Castell et al., 2014).
Changes in the rates of photochemistry and NPQ, and electron sinks other than
CO_{2} fixation, lead to a non-trivial but direct link between
chlorophyll fluorescence and photosynthetic rate
(Flexas et al., 1999; Magney et al., 2017). Because chlorophyll fluorescence is tied in
with these physiological processes, it has become a highly useful indicator of
the physiological state of leaves (see reviews
by Baker, 2008; Porcar-Castell et al., 2014).

At the canopy scale and beyond, the link appears simpler, exhibiting ecosystem-dependent linear relationships (Guanter et al., 2013). The slope of this linear relationship can change as the light-use efficiency of either SIF or GPP changes, for example due to water stress (Daumard et al., 2010) or changing light conditions (Yang et al., 2015). SIF also seems to outperform traditional remote-sensing methods, such as the Normalized Difference Vegetation Index (NDVI) and the Enhanced Vegetation Index (EVI), which use reflectance to derive vegetation indices, in tracking changes in GPP at this scale (Yang et al., 2015; Walther et al., 2016). This is in part because the SIF emission originates exclusively from plants; thus, the retrieval is not contaminated by background materials like soil or snow. It is expected, however, that complicating factors such as the retrieval wavelength, temporal scaling, chlorophyll content, three-dimensional canopy structure, and stress will also play a role in the GPP–SIF link (Damm et al., 2015; Guanter et al., 2012; Rossini et al., 2015; Zhang et al., 2016). Using high-resolution spectrometers onboard satellites, global maps of SIF have been produced. A number of existing (GOME-2, GOSAT, OCO-2, TROPOMI, SCHIAMACHY) and planned (FLEX, GEOCARB) satellite missions are capable of measuring SIF. Utilizing these remotely sensed SIF observations directly to track changes in GPP has already proven useful even without the addition of ancillary data or model information (Lee et al., 2013; Parazoo et al., 2013; Walther et al., 2016; Yang et al., 2015).

Data assimilation enables the use of observations and model information
together to produce a best estimate of the state and function of the system.
In the case of mechanistic models this is done by constraining the simulated
processes and their parameters. Such an approach has been applied to
terrestrial biosphere models to optimize model parameters and constrain the
uncertainty in carbon flux estimates in a number of studies
(see Kaminski et al., 2013; Koffi et al., 2013; Macbean et al., 2016; Peylin et al., 2016). The Carbon
Cycle Data Assimilation System (CCDAS) is one such system, and it has incorporated
observations such as atmospheric CO_{2} concentration and/or the
fraction of absorbed photosynthetically active radiation (FAPAR),
demonstrating the benefit of combining model and observations in a
regularized approach (Rayner et al., 2005; Kaminski et al., 2012). The use of SIF
observations within a data assimilation framework may provide a highly
useful, complementary constraint on GPP. While one study by
Parazoo et al. (2014) utilized SIF in a data assimilation system to
redistribute multiple model estimates of GPP, no optimization of model
process parameters was performed. Koffi et al. (2015) incorporated a
mechanistic model for SIF into the CCDAS system and then conducted
sensitivity tests and compared model-simulated SIF and observed SIF from
GOSAT, demonstrating that the model is capable of incorporating the data. However, SIF
has not yet been used on a global scale in a data assimilation system to
optimize process parameters.

In this paper, we assess the ability of satellite SIF observations to constrain the parametric uncertainty in simulated GPP in a terrestrial biosphere model within a data assimilation system. This is termed an error propagation study and is similar in concept to an observing system simulation experiment or quantitative network design study (Hungershoefer et al., 2010; Kaminski et al., 2010; Koffi et al., 2013). Parameters and simulated GPP are therefore optimized only for their uncertainty and not for their absolute quantities. Considering that SIF is a novel observational constraint, this is an important first step toward a full assimilation of the data as it allows us to test whether an assimilation of SIF data will be beneficial for reducing uncertainty in GPP. This is performed by estimating the constraint that SIF provides on the uncertainty in model parameters and the parametric uncertainty in model-simulated GPP.

Under the linear Gaussian assumption, the uncertainty in a target quantity (here, GPP) following the assimilation of the measured data (here, SIF) is conditional only on the prior uncertainty, the uncertainty in the measured data and the sensitivity of simulated quantities (SIF and GPP) to changes in the parameters (Tarantola, 2005). Given that we apply this assumption to estimate posterior uncertainties, this linear problem can be performed independently of the optimization of the parameter values. The model used for determining the sensitivity of simulated observations to changes in the parameters is run at a relatively low spatial resolution which provides high computational efficiency. We note that subsequent work to assimilate the data should be performed at a higher spatial resolution in order to better represent the heterogeneity of the land surface.

We formulate this error propagation study in two stages: (i) optimization of parameter uncertainties and (ii) projection of the parameter uncertainties onto uncertainty in diagnostic GPP. Here, we outline the model used to simulate the observation (SIF) and the target quantity (GPP). We also outline the model parameter set describing these processes, the uncertainty in the observations and model forcing, and the general experimental set-up.

## 2.1 Model description

In order to incorporate an observation into a data assimilation system, we require a model or “observation operator” that can simulate SIF, ideally providing a process-based relationship between SIF and GPP. There are a few ways one might formulate the observation operator. Evidence shows a strong linear relationship between SIF and GPP at large spatial scales and relatively long temporal scales (Frankenberg et al., 2011b; Guanter et al., 2012), suggesting relatively simple scaling between GPP and SIF. However, it is known that the link is more complex than this, and it is expected to differ at finer spatial and temporal scales due to, for example, land surface heterogeneity or the time of day of the measurements. To ensure the model has these capabilities, we have opted for a process-based observation operator.

In this section we describe the newly developed terrestrial biosphere model
for simulating and assimilating SIF. The model is an integration of the
existing models BETHY (Biosphere Energy Transfer HYdrology)
(Rayner et al., 2005; Knorr et al., 2010) and SCOPE (Soil Canopy Observation,
Photosynthesis and Energy fluxes) (van der Tol et al., 2009) and builds upon the
developments of Koffi et al. (2015). The coupling of BETHY and SCOPE enables
spatially explicit, plant-type-dependent, global simulations of GPP and SIF.
This model may be run on a computationally efficient, low spatial-resolution
grid of 7.5^{∘} × 10^{∘} or a high spatial-resolution
grid of 2^{∘} × 2^{∘}.

BETHY is a process-based terrestrial biosphere model at the core of the Carbon Cycle Data Assimilation System (CCDAS) (Rayner et al., 2005; Scholze et al., 2007). Full model description details can be found elsewhere (e.g. Rayner et al., 2005; Scholze et al., 2007; Knorr et al., 2010). Briefly, BETHY simulates carbon assimilation and plant and soil respiration within a full energy and water balance. The version used here also incorporates a leaf area dynamics module for prognostic leaf area index (LAI) as described in Knorr et al. (2010). This module includes parameters for leaf development, phenology, and senescence processes (hereby collectively termed leaf growth) to determine LAI in a scheme that incorporates temperature, water, and light limitations on growth and is capable of representing the major global phenology types (Knorr et al., 2010). This scheme also enables the representation of subgrid variability in leaf growth, representing the likely variability in growth triggers across a grid cell and the necessary mathematical form for differentiability between process parameters and state variables. The full BETHY model consists of four key modules: (i) energy and water balance; (ii) photosynthesis; (iii) leaf growth; and (iv) carbon balance. It represents variability in physiology and leaf growth of plant classes by 13 plant functional types (PFTs) (see Table 1) originally based on classifications by Wilson and Henderson-Sellers (1985). Each model grid cell may consist of up to three PFTs as defined by their grid cell fractional coverage.

SCOPE is a vertical (1-D) integrated radiative transfer and energy balance model with modules for photosynthesis and chlorophyll fluorescence (van der Tol et al., 2009). At present it is the only process-based model capable of simulating canopy-scale chlorophyll fluorescence. SCOPE incorporates the current understanding of chlorophyll fluorescence processes including canopy radiative transfer, reabsorption of fluorescence within the canopy, and the non-linear relationship between chlorophyll fluorescence quantum yield and other quenching processes (van der Tol et al., 2009, 2014). Leaf level chlorophyll fluorescence is coupled to the commonly used Farquhar and Collatz models for C3 and C4 photosynthesis, respectively (van der Tol et al., 2009). A current limitation of SCOPE is that there is no link between leaf level biochemistry and soil moisture. This is partly compensated for by changes in LAI due to soil moisture as simulated by BETHY.

The canopy radiative transfer and photosynthesis schemes of BETHY have been
replaced by the corresponding schemes in SCOPE, including the components
required for the calculation of chlorophyll fluorescence at leaf and canopy
scales. The spatial resolution, vegetation (PFT) characteristics, leaf
growth, and carbon balance are handled by BETHY. SCOPE therefore takes in
climate forcing (meteorological and radiation data) and LAI from BETHY and
returns GPP. BETHY calculates the canopy water balance, leaf growth, and net
carbon fluxes, which will prove useful in future when assimilating other data
streams (e.g. atmospheric CO_{2} concentration). Importantly, SCOPE
provides a process-based link between SIF and GPP allowing the transfer of
information from observations of SIF to simulated GPP. Subsequently,
information from SIF may also be transferred to carbon fluxes resulting from
GPP such as net ecosystem productivity.

## 2.2 Model process parameters

In this error propagation study, information from the SIF observations is
used to constrain the uncertainty in the model process parameters. Parameters
can either be global or differentiated by PFT. Global parameters apply to
plants or soils everywhere, while PFT-dependent parameters enable
differentiation between physiological and leaf growth traits. Some key
parameters for this study such as the maximum carboxylation capacity
(*V*_{cmax}) and chlorophyll *a*/*b* content (*C*_{ab}) are considered
PFT-dependent. From an ecophysiological perspective, there are other
parameters specific to SCOPE that may be considered PFT-dependent such as the
vegetation height and leaf angle distribution parameters. However, we have
assumed them to be global to simplify the problem. GPP is relatively
insensitive to these parameters, so this is not expected to impact the GPP
uncertainty reduction results. Despite this, in a full assimilation with the
SIF data, it may be necessary to make these PFT-dependent to improve the
model-observed fit.

We expose 53 parameters from BETHY-SCOPE to the error propagation system (see Table A1). Each parameter is represented by a probability density function (PDF) which is assumed to be Gaussian. The mean and standard deviation for the prior parameters are shown in Table A1. The choice of the prior mean and uncertainty for parameters follows those used in previous studies (Kaminski et al., 2012; Knorr et al., 2010; Koffi et al., 2015). For new parameters that are not well characterized (e.g. SCOPE parameters), we assign relatively large prior uncertainties and mean values in line with the default SCOPE parameters and with Koffi et al. (2015). The choice of the prior may be considered important here, considering that we are using a linear approximation of the model around the prior and that the model is known to be non-linear. Therefore, sensitivities can differ depending upon the choice of the prior parameter values (Koffi et al., 2015).

There are 12 SCOPE parameters exposed, two of which are PFT-dependent
(*C*_{ab} and *V*_{cmax}). These parameters were chosen due to their
importance in simulating SIF or GPP and due to sensitivity tests such as those
performed by Verrelst et al. (2015). They include *C*_{ab}, leaf dry matter
content (*C*_{dm}), leaf senescent material fraction (*C*_{s}),
two leaf distribution function parameters (LIDF_{a}, LIDF_{b}), vegetation
height (hc), and leaf width. Leaf physiological parameters include
*V*_{cmax}, Michaelis–Menten kinetic coefficients for CO_{2}
(*K*_{C}) and O_{2} (*K*_{O}), the ratio of the Rubisco
oxygenation rate to *V*_{cmax} (${\mathit{\alpha}}_{{V}_{\mathrm{o}},{V}_{\mathrm{c}}}$),
and the ratio of day respiration to *V*_{cmax}
(${\mathit{\alpha}}_{{R}_{\mathrm{d}},{V}_{\mathrm{c}}}$).

## 2.3 Uncertainty calculations

To calculate the uncertainty in parameter values following the constraint
provided by the observational information of SIF (i.e. the posterior
uncertainty), we propagate uncertainty from the observations onto the
parameters. In order to perform this, we utilize a probabilistic framework
where the state of information on parameters and observations is expressed by
their corresponding PDFs (see Tarantola, 2005). The probability
density of the errors in these quantities is assumed to be Gaussian; thus, they are describable by their mean and uncertainty. The prior information on
parameters is quantified by a PDF in parameter space and the observational
information by a PDF in observational space. The mean values for the
parameters and observations are denoted by *x* and *d*, respectively. The
uncertainty covariance matrices in parameter space and observational space
are denoted by **C**_{x} and **C**_{d}, respectively.

For linear and weakly non-linear problems we can assume that Gaussian
probability densities propagate forward through to Gaussian distributed
simulated quantities (Tarantola, 2005). This permits linear error
propagation from the input parameters to the model outputs. As mentioned
earlier, estimating posterior uncertainties of the parameters for these types
of problems can therefore be performed independently of the parameter
estimation, in other words without the need to constrain the mean values of
the parameters (Kaminski et al., 2010, 2012). This requires a matrix of
partial derivatives of a target quantity with respect to its variables, also
called a Jacobian matrix (**H**). This matrix represents the sensitivity of a
simulated quantity (e.g. SIF, GPP) to the parameters. With the linear
approximation, **H** is calculated around the prior parameter values (*x*_{0}).
This simplification of the model sensitivity brings limitations to the
accuracy of the method. However, with the aggregation of subgrid variability
across a model grid cell, sudden shifts in model sensitivity (e.g. step
functions) are less likely or realistic; the present model accounts for these
effects (Knorr et al., 2010). Additionally, because the parameter space can be
very large, the use of prior knowledge on *x*_{0} helps to limit the effect of
this problem as **H** at *x*_{0} likely provides a decent approximation of the
true **H** that would occur at the global optimum (Tarantola, 2005). The
simplification is also useful considering the high computational cost of
calculating **H**.

To calculate the posterior parameter covariance matrix
(${\mathbf{C}}_{{x}_{\mathrm{post}}}$) following constraint by observational
information, **C**_{d}, we use Eq. (1)
(Tarantola, 2005).

where **H** expresses the Jacobian for SIF and **H**^{T}
the Jacobian transposed. Comparing parameter uncertainties in the prior
(${\mathbf{C}}_{{x}_{\mathrm{0}}}$) and the posterior (${\mathbf{C}}_{{x}_{\mathrm{post}}}$)
allows us to quantify the improvement in parameter precision following the
observational constraint. The parameter uncertainties in ${\mathbf{C}}_{{x}_{\mathrm{0}}}$
and ${\mathbf{C}}_{{x}_{\mathrm{post}}}$ may be expressed as standard deviations
(*σ*) by calculating the square root of their diagonal elements. We
therefore assess the relative uncertainty reduction in parameters following
SIF constraint, or “effective constraint”, with $\mathrm{1}-({\mathit{\sigma}}_{\text{posterior}}/{\mathit{\sigma}}_{\text{prior}})$. This quantifies the effective
constraint of the prior uncertainty and may be represented as a percentage
decrease in *σ* uncertainty.

Formally, **C**_{d} represents the errors in the measurements
and in the model-simulated counterpart (i.e. model error)
(Scholze et al., 2016). As described further below in Sect. 2.4, we only
consider the contribution of measurement errors to **C**_{d} in
calculating posterior probabilities. However, to see if the assumptions that
we have made about uncertainties are consistent with the model–data mismatch,
we assess the reduced *χ*^{2} statistic (${\mathit{\chi}}_{r}^{\mathrm{2}}$) similar to a method
employed by Kuppel et al. (2013). While more formal approaches to optimally
estimate covariance parameters exist (Michalak et al., 2005), this metric can
highlight whether we are neglecting a significant source of error in
**C**_{d}, for example, model structural error. It also
provides an indication of whether the model is capable of reproducing the
measurements, given the assumed uncertainties. This is calculated by

where *N* is the number of degrees of freedom (equal to the number of
observations in this case), *M*(*x*_{0}) is the forward model-simulated SIF for
the prior case, and *d* is the SIF observations. A ${\mathit{\chi}}_{r}^{\mathrm{2}}$ greater than
1 would indicate that our assumptions around uncertainties may not be valid
given the model–data mismatch and that the model cannot simulate the
measurements (Michalak et al., 2005). Conversely, a ${\mathit{\chi}}_{r}^{\mathrm{2}}$ of less than 1
indicates overconfidence in the assumed uncertainties. A value of
approximately 1 is most desirable as it would indicate that our overall
assumptions of uncertainties are valid. At the low resolution applied in the
information content analysis, representation errors will be relatively large
and may dominate other sources of error in Eq. (2) and therefore mask
the actual ability of the model to simulate the measurements. For an
assimilation of the data, the model would not be used at such a low
resolution given the heterogeneity of the land surface and would instead be
run at a higher spatial resolution. To help reduce the representation error, we utilize unpublished work that compares the forward model at a higher
resolution (2^{∘} × 2^{∘}) and with SIF observations
from the OCO-2 (Orbiting Carbon Observatory-2) satellite for 2015. While this uses a slightly different
parameterization, it is more credible and helps minimize the effects of
representation error in determining whether the model can simulate the
measurements. The error propagation analysis, however, benefits from using
the low resolution as it greatly improves computational efficiency
considering the computational demand of the model simulations and subsequent
calculations.

The observational constraint introduces correlations into the posterior parameter distributions; thus, posterior parameter uncertainties are not wholly independent. Strong correlations in ${\mathbf{C}}_{{x}_{\mathrm{post}}}$ indicate parameters that cannot be resolved independently in an assimilation; however, their linear combinations can be. We calculate correlations in parameters by expressing the covariances as correlations as in Eq. (3) (see Tarantola, 2005, p. 71) by

where diagonal elements have a correlation equal to 1, while off-diagonals elements can range between −1 and 1. If large enough, these correlations can contribute significantly to the overall constraint of the target quantity (Bodman, 2013).

Using the parameter covariance matrix we can assess how parameter
uncertainties propagate forward through the model onto uncertainty in GPP
using the Jacobian rule of probabilities, the same method outlined in
Rayner et al. (2005). This is the second stage of our error propagation study.
Using ${\mathbf{C}}_{{x}_{\mathrm{0}}}$ we estimate the prior uncertainty in a vector of
simulated target quantities (i.e. GPP). Similarly, using
${\mathbf{C}}_{{x}_{\mathrm{post}}}$ we estimate the posterior uncertainty in a
vector of simulated target quantities. We calculate the uncertainty
covariance of GPP (*C*_{GPP}) using Eq. (4).

where **H**_{GPP} is the Jacobian matrix of GPP with respect to
the parameters. With this we can quantify the improvement in precision of
simulated GPP by using either ${\mathbf{C}}_{{x}_{\mathrm{0}}}$ or
${\mathbf{C}}_{{x}_{\mathrm{post}}}$ in Eq. (4). Therefore, using the
forward model, a statistical estimation scheme and a set of observational
uncertainties, we can assess the information content of the SIF observations
in the context of the model, its parameter set, and simulated GPP taking
explicit consideration of uncertainties.

## 2.4 Uncertainty in observations and model forcing variables

The uncertainty in the measured data (hereafter, data) is a critical
component in assessing the potential impact of an observing system on the
estimation of carbon fluxes. Data uncertainties in SIF used here are
calculated from the GOSAT satellite observations for 2010. These data are
obtained from the ACOS (Atmospheric CO_{2} Observations from Space)
project at a grid resolution of 3^{∘} × 3^{∘}
(Frankenberg, 2018). As the model simulations are performed on a
low-resolution grid (7.5^{∘} × 10^{∘}), we aggregate these
uncertainties to this resolution using Eq. (5) as described below in
a way that conserves the information content from the original
3^{∘} × 3^{∘} observations.

We assume that the observations are independent and have uncorrelated errors, that is, they are distributed randomly. Assuming uncorrelated errors is, however, likely to overestimate the information content, particularly if using the standard error as the uncertainty. Although it has been used in recent studies with satellite SIF (e.g. Parazoo et al., 2014), the standard error under an assumption of uncorrelated errors is likely to be an overly optimistic approximation of the information content. For this study, we take a slightly conservative approach, scaling the calculated standard error by the square root of 2 as shown in Eq. (5). This effectively doubles the variance in an independent dimension and reduces the information content to compensate for the assumption of uncorrelated errors.

Through the aggregation of GOSAT grid cells to the model grid resolution, the
number of independent measurements is reduced. To account for this and
preserve the information content of the original GOSAT observations, the
uncertainty in a given model grid cell is, approximately, divided by the
square root of the number of GOSAT grid cells with SIF data that fall within
that model grid cell (*N*). More precisely, we apply an area-weighting term
in the equation (see Eq. A1 in the Appendix). This has the effect of
scaling the uncertainty by the $\mathrm{1}/\sqrt{N}$ law but takes into account the
fact that SIF is in physical units per unit area (i.e.
W m^{−2} µm^{−1} sr^{−1}) and that grid cells have
different areas over different latitudes. A full description of this
calculation and a detailed example is shown in the Appendix.

Therefore, the calculation of the SIF data uncertainties used here is
approximated by Eq. (5) (for further details see
Sect. A2 in the Appendix). For a given model grid
cell, the variance (*σ*^{2}) is approximately equal to the sum of the
standard error of each individual GOSAT grid cell (*σ*_{i})
squared and then scaled by the number of individual GOSAT grid cells with data
and the square root of 2.

The resulting annual observational uncertainties, shown in Fig. 3, appear to be much smaller than the uncertainties in individual GOSAT grid cells. In part this is due to the aggregation of multiple independent observations. Regions with more soundings across the year (e.g. the tropics) will also have smaller annual uncertainties.

Uncertainty in SIF observations may also have a systematic component. A
known, potential systematic error in SIF stems from the zero-level offset
calculated during the retrieval. Any error in the calculated zero-level
offset will add to the measurement error. This radiometric correction is done
to prevent biases in the SIF retrieval (Frankenberg et al., 2011a; Guanter et al., 2012),
and this is performed monthly in the present GOSAT retrieval of SIF. In this
case, it is systematic in the sense that it applies to multiple measurements.
This type of error is distinguished from a bias, which is a systematic error
with a precisely known magnitude and sign that should be corrected for. A
bias cannot be incorporated into the present error
propagation framework,
whereas an error in the zero-level offset can be, provided it is Gaussian. To
clarify, a retrieved measurement (*d*) of a quantity (e.g. SIF) at index
point *i* can be given by

where ${d}_{i}^{\mathrm{t}}$ is the true value at index point *i*, *ε*_{i}
is a random variable with a variance of *σ*^{2} at index point *i*, and
*ε*_{z} is a random variable that has some variance and is constant
for a subset of the measurements (e.g. across a particular region or time).
Based on previous analyses of the instruments, the error in zero-level offset
in the SIF retrieval may be considered small
(Frankenberg et al., 2011a, 2014). Here, we provide a more detailed
assessment and characterization of the in-orbit systematic error. This is
performed by assessing zero-level offset-corrected GOSAT SIF soundings over
the non-fluorescent regions of Antarctica and central Greenland during
January and July, respectively (see Fig. A2), in
order to sample the error distribution of *ε*_{z}. These systematic
errors appear quite small
(±0.06 W m^{−2} µm^{−1} sr^{−1}) and may vary
seasonally due to factors such as atmospheric conditions or
instrument-related causes (Guanter et al., 2012). We therefore assess the effect
of a conservative systematic random error of size
±0.1 W m^{−2} µm^{−1} sr^{−1} in the zero-level offset
seasonally. Practically, this means adding four (one for each season) extra
uncertainty terms to **C**_{x}, corresponding to the estimated error, and
adding four extra terms in **H**, which are scaling terms (equal to
1) applied to the corresponding season. Including these terms provides a
sensitivity test to indicate how an error in the zero-level offset propagates
through to uncertainty in GPP.

An additional source of uncertainty in model estimates of GPP is climate
forcing. As mentioned by Koffi et al. (2015), while uncertainty in forcing such
as incoming radiation is not considered in the current CCDAS set-up, it is
considered to be an important variable in driving SIF (Verrelst et al., 2015)
and GPP (Farquhar et al., 1980). Without a consideration of uncertainties in
forcing variables, the uncertainty in GPP may be underestimated. Studies that
use process-based models or empirically derived relationships do not
explicitly consider such uncertainties (e.g. Beer et al., 2010). One such
forcing variable is downward shortwave radiation (SWRad). Monthly means of
SWRad are suggested to have a random error of 12 W m^{−2} (6 % of the
mean) due mostly to uncertainty in clouds and aerosols (Kato et al., 2012). We
therefore investigate how this random error in SWRad may be considered in GPP
estimates. Furthermore, as SIF responds strongly to SWRad, there is the
potential to utilize SIF observations as a constraint on the uncertainty in
the forcing. We therefore conduct an additional experiment that incorporates
the uncertainty in SWRad in the error propagation system. For this experiment
an additional parameter representing SWRad is added to the inversion, which
acts as a scaling factor for SWRad globally. We investigate the level of
constraint SIF provides on this scaling factor and the subsequent effects of
incorporating uncertainty in SWRad on uncertainty in GPP.

## 2.5 Model and data set-up

In this study BETHY-SCOPE is run for the year 2010 on the computationally
efficient, low-resolution spatial grid (7.5^{∘} × 10^{∘}).
As the dynamical equations are the same for either low-resolution or
high-resolution scales, the use of the low-resolution set-up is appropriate for an
error propagation study as long as careful consideration is taken with
observational uncertainties. Climate forcing in the form of daily
meteorological input fields for running the model (precipitation, minimum and
maximum temperatures, and incoming solar radiation) were obtained from the
WATCH/ERA Interim data set (WFDEI; Weedon et al., 2014). Photosynthesis and
fluorescence are simulated at an hourly time step but forced by the
respective monthly mean diurnal cycle. Leaf growth and hydrology are
simulated daily.

SIF is simulated at 755 nm, the wavelength corresponding to the GOSAT retrieval frequency and near the OCO-2 retrieval frequency (757 nm). We focus upon the constraint by SIF measurements at 13:00 local time as it closely corresponds to the local overpass time of the SIF-observing satellites GOSAT and OCO-2. However, we also investigate the effect of using alternative SIF-observing times (e.g. the GOME-2 satellite overpass time) and multiple observing times simultaneously on the constraint of GPP.

## 3.1 Prior mismatch

First, we present the results from Eq. (2) that determines whether the assumed uncertainties allow for coverage of observed SIF. As described in the methods, in this case we use a model forward run using the high-resolution version of the model and compare this with SIF observations from the OCO-2 satellite. We find that ${\mathit{\chi}}_{r}^{\mathrm{2}}=\mathrm{0.97}$ in this high-resolution case, close to the optimal value of 1.

## 3.2 Parameter uncertainties

As described in the “Methods” section, a key metric for assessing the relative uncertainty reduction, or effective constraint, is defined as $\mathrm{1}-({\mathit{\sigma}}_{\mathrm{posterior}}/{\mathit{\sigma}}_{\mathrm{prior}})$. The effective constraint for all 53 parameters following constraint by SIF is shown in Fig. 1 and in Table A1. We define weak, moderate, and strong effective constraint as the relative uncertainty reduction from 1 to 10, 10 to 50, and > 50 %, respectively.

Parameters describing leaf composition (*C*_{ab}, *C*_{dm},
*C*_{sm}) generally achieve strong effective constraint from SIF. For
11 of the 13 *C*_{ab} parameters the uncertainty is strongly
constrained, between 50 and 84 %. SIF is highly sensitive to *C*_{ab}, and
we assign a relatively large prior uncertainty to these parameters, so
considerable constraint is expected. For the tropical broadleaved evergreen
tree PFT, however, the effective constraint on *C*_{ab} is much lower at
7 %. For other leaf composition parameters *C*_{dm} and
*C*_{sm}, SIF effectively constrains the uncertainty by 1 and
< 1 %, respectively.

Varied effective constraint is seen for the leaf growth parameters
(parameters 18–34 in Table A1) that control phenology and
leaf area. Four out of the seventeen leaf growth parameters exhibit strong
uncertainty reductions. These parameters describe a variety of processes
including the temperature at leaf onset, day length at leaf shedding, leaf
longevity, and the expected length of dry spell before leaf shedding
(*τ*_{W}) (see Table A1). The parameter *τ*_{W} is
important in controlling leaf area, and it sees strong effective constraint
from SIF, from 38 to 65 % depending upon which class of PFT it pertains
to. For the parameters that are PFT-specific, there is generally a larger
constraint seen when they relate to the C3Gr, C4Gr, and crops. For example,
uncertainty in *τ*_{W} for grasses and crops
$\left({\mathit{\tau}}_{W}^{\mathrm{Gr}}\right)$ is effectively constrained by 65 %.

Leaf physiological parameters (parameters 1–17 in
Table A1) see a weak to moderate level of effective
constraint. Of particular importance for simulating GPP is the PFT-specific
parameter *V*_{cmax}. Effective constraint on *V*_{cmax} varies
from < 1 up to 31 % depending upon the PFT of interest. Five PFTs
that, combined, represent about 65 % of the land surface have their
*V*_{cmax} parameters constrained by > 10 %. The global
physiological parameters include the ratio of the maximum rate of oxygenation
(*V*_{omax}) to *V*_{cmax} (${a}_{{V}_{\mathrm{o}},{V}_{\mathrm{c}}}$),
the ratio of dark respiration (*R*_{d}) to *V*_{cmax}
(${a}_{{R}_{\mathrm{d}},{V}_{\mathrm{c}}}$), and the Michaelis–Menten enzyme kinetic
constants of Rubisco for CO_{2} (*K*_{C}) and O_{2}
(*K*_{O}). These all see very weak effective constraint from SIF
(< 1 %). Across all PFT-specific parameters, those that pertain to
more dominant PFTs in terms of land surface coverage (e.g. C3 grass) tend to
see stronger uncertainty reductions. This is largely due to them being
exposed to more SIF observations.

Global canopy structure parameters (parameters 50–53 in
Table A1) also see a weak to moderate constraint from SIF.
In particular, the structural parameters LIDF_{a} and LIDF_{b} see their
uncertainty reduced by 22 and 9 %, respectively. The parameters for
vegetation height and leaf width, which are used to calculate the
fluorescence “hot-spot” variable (see van der Tol et al., 2009), are
effectively constrained by 7 and < 1 %, respectively.

With the observational constraint, correlations are introduced into the
posterior parameter distributions. We assess these correlations using
Eq. (3), shown in Fig. 2. We find
strong (*R*≥0.5) positive correlations between nine of the PFT-specific
*C*_{ab} parameters. These are also negatively correlated with the leaf angle distribution
parameter LIDF_{a}. Thus, during a full assimilation with SIF data, only the
sum of *C*_{ab} and LIDF_{a} can be resolved, not their individual values.
Two leaf growth parameters are also strongly correlated: *T*_{ϕ} with
*T*_{r}. Smaller correlations are also present between the subset of
parameters shown in Fig. 2.

To assess the effect of incorporating a systematic error from the
observations into this analysis, we apply a seasonal *σ* error of
0.1 W m^{−2} µm^{−1} sr^{−1} (equivalent to
*ε*_{z} in Eq. 6). This is incorporated as four additional
parameters, one for each season, that scale the SIF signal across the globe.
We find that the inclusion of this systematic error has a negligible effect
on posterior uncertainties of the parameters. The difference in effective
constraint between this sensitivity test case and the standard case above is
< 1 % for any given parameter.

## 3.3 Uncertainty in GPP

To assess the constraint imposed by SIF on simulated GPP, we compare the prior
and posterior uncertainty in GPP as calculated using Eq. (4).
Similar to the assessment of parameter uncertainty reductions, to assess the
effective constraint of SIF on GPP, we use a metric that measures the relative
uncertainty reduction in *σ* from the prior to the posterior.

Global GPP from the prior model is approximately 164 Pg C yr^{−1}
with a prior uncertainty (*σ*) of 19.0 Pg C yr^{−1}. Utilizing
SIF observations at 13:00 results in a 73 % reduction in the prior
uncertainty, giving a posterior of 5.2 Pg C yr^{−1}. Spatially, the
prior uncertainty in GPP varies across the globe, with particularly large
uncertainties in regions with high productivity
(Fig. 4). This is to be expected, considering GPP
uncertainty will typically correlate with GPP. In the posterior, it is clear
that uncertainty in GPP is strongly reduced across the globe
(Fig. 5). The relative uncertainty reduction
(Fig. 6) appears to show smaller constraint of
uncertainty in the boreal regions, likely due to the relatively large SIF
uncertainty (Fig. 3) and low prior uncertainty in GPP
(Fig. 4).

To assess which parameters contribute to the uncertainty in GPP for the prior
and posterior, we can conduct a linear analysis of the uncertainty
contributions. Typically this technique can only be used for the prior as the
correlations in posterior parameter uncertainties, excluded from the linear
analysis, also contribute toward the overall constraint. However, we can
assess the contribution of these correlations to the constraint of GPP by
setting the off-diagonal elements in ${\mathbf{C}}_{{x}_{\mathrm{post}}}$ to zero
and using it in Eq. (4); the difference between this and the
standard case that uses the full ${\mathbf{C}}_{{x}_{\mathrm{post}}}$ equates to
the contribution of correlations. We find that the contribution of these
correlations to the constraint of GPP is small (0.16 Pg C yr^{−1}
or < 1 %); thus, we can assume that the linear analysis technique holds for
the posterior as well. This finding is supported by the correlation analysis
in posterior parameter uncertainties which showed few significant
correlations in parameters relevant for GPP. This result is encouraging as it
indicates that the parameters in a SIF assimilation system contributing most
to the constraint of GPP are capable of being resolved independently.

Using a linear analysis of the uncertainty, we find that uncertainty in global
annual GPP in the prior and posterior stems from different processes. For the
prior we find that the uncertainty in GPP is dominated, at 89 %, by
parameters describing leaf growth processes. Of these, a single parameter, *τ*_{W}, for C3 grass, C4 grass, and crops $\left({\mathit{\tau}}_{W}^{\mathrm{Gr}}\right)$
makes up 74 % of the uncertainty in global annual GPP. Parameters
representing physiological processes account for about 9 % of prior
uncertainty, most of which stems from the *V*_{cmax} parameters.
Parameters for *C*_{ab} only account for 2.5 % of the prior uncertainty.

For the posterior, which has a lower overall uncertainty in GPP, uncertainty
is dominated by parameters representing physiological processes.
Physiological parameters account for 67 % of the uncertainty in posterior
annual GPP, with *V*_{cmax} parameters accounting for 32 % and the
Michaelis–Menten constant of Rubisco for CO_{2} (*K*_{C})
accounting for 30 %. The relative contribution by leaf growth parameters
is reduced to 33 %, and for ${\mathit{\tau}}_{W}^{\mathrm{Gr}}$, it is 15 %. For
*C*_{ab} the relative contribution is smaller than the prior at < 1 %.
This shift in which parameters contribute to the relative uncertainty in GPP
between the prior and the posterior demonstrates how effectively SIF
constrains leaf growth processes. Uncertainties in physiological parameters
are constrained less than the leaf growth parameters, which results in them
contributing more in relative terms to the posterior uncertainty in GPP.

Regionally, we split the land into three regions, the Boreal region (above
45^{∘} N), the Temperate North (30 to 45^{∘} N), and the Tropics
(30^{∘} S to 30^{∘} N). SIF constraint on annual GPP varies
substantially across different regions of the globe, with a relative
uncertainty reduction in of 48, 82, and 79 % for the Boreal, Temperate
North, and Tropics regions, respectively. In
Fig. 7 we show the contribution of parameter
classes (leaf physiology, leaf growth, leaf composition, and canopy structure;
see Table A1 for details) to the parametric uncertainty in
GPP across the year for each of these regions. From
Fig. 7 it can be seen that the Boreal and
Temperate North regions exhibit seasonal differences in GPP uncertainty,
mostly due to the seasonal cycle in GPP, and in the constraint SIF provides.
This is caused by seasonal dependencies in the sensitivity of SIF and GPP to
certain processes (e.g. leaf development versus leaf senescence) as well as
seasonal differences in the density of observations in these regions. There
are far fewer GOSAT satellite observations during Boreal autumn and winter; thus, there are fewer observations to constrain processes controlling GPP
during this time.

During the start of the growing season leaf physiology, in particular
photosynthetic rate constants (*V*_{cmax}), plays a larger role in GPP
uncertainty, whereas later in the growing season during the warmest months
leaf growth, via water limitation on leaf area
$\left({\mathit{\tau}}_{W}^{\mathrm{Gr}}\right)$ of grasses, plays a larger role. Therefore
in the Boreal region, where the strongest seasonality in constraint is seen,
from July through to January SIF constrains GPP by > 60 %. Uncertainty
in GPP during these months is dominated by the leaf growth parameters
${\mathit{\tau}}_{W}^{\mathrm{Gr}}$ and *k*_{L} along with *C*_{ab} (for EvCn), all of
which receive considerable constraint from SIF. From February to June
however, SIF constrains GPP by less than 50 %, as a large proportion of
the uncertainty arises from the less constrained *V*_{cmax}
parameters. Following SIF constraint, uncertainty in Boreal GPP stems mostly
from uncertainty in leaf physiology, particularly for the EvCn PFT. Similar
differences between seasonal constraint are seen for the Temperate North,
although with a smaller seasonal variation in SIF constraint that ranges
between 74 and 87 % across the year.

For the Tropics uncertainty reduction in GPP is about 80 % across the
year. Uncertainty in the prior is dominated by the leaf growth parameters and
in particular the *τ*_{W} parameters controlling water-limited leaf area.
SIF constraint is primarily propagated through the *τ*_{W} parameters to GPP
resulting in a well-constrained posterior with a *σ* uncertainty
of 1.6 Pg C yr^{−1} in the
annual GPP of the Tropics. Although moderate constraint is seen in the key
PFT-specific parameter *V*_{cmax} for the dominant tropical PFTs (see
Fig. 1), in the posterior these
parameters contribute to roughly 35 % of the uncertainty in annual GPP.

## 3.4 Diurnal SIF constraint

With this set-up it is possible to test how the SIF constraint on GPP might
change with alternative observational times. Considering this, we test how
the constraint on GPP changes when assimilating observations of SIF from
alternative times of the day, assuming the same number of observations and
the same observational uncertainty as used above. From this we see that
different observing times yield differences in the posterior uncertainty and
the effective constraint of GPP (see
Fig. 8). The constraint on global
annual GPP when using SIF-observing times between 09:00 and 15:00 is quite similar, with the posterior uncertainty in global annual GPP ranging
from 5.0 Pg C yr^{−1} (effective constraint of 74 %) to
6.0 Pg C yr^{−1} (effective constraint of 68 %). The most
significant constraint on GPP is obtained when using SIF observations at
between 11:00 and 13:00, nearest to the peak in the diurnal cycle of both GPP
and SIF.

We also test the effect of utilizing SIF measurements at multiple times of the day simultaneously. We select the times 08:00, 12 noon, and 16:00, replicating a theoretical geostationary satellite. For this experiment we first test the effect of increasing the number of observations by a factor of 3, assuming the same uncertainty for the three observation times. Second, we also increase the number of observations by a factor of 3, but scale the variance of these observations by one third. Using this second test we can assess whether differences in parameter sensitivities of SIF and GPP at the different times of the day add value in the overall constraint.

Using a diurnal cycle of observations results in a posterior uncertainty of
4.6 Pg C yr^{−1} or an effective constraint of 76 % as in
Fig. 8. This is an extra 2 %
constraint on the uncertainty in GPP compared with observations at 12:00 noon
alone. If we use a diurnal cycle of observations with scaled uncertainties,
we see a slightly reduced constraint on GPP where the posterior uncertainty
is 5.9 Pg C yr^{−1}, equivalent to an effective constraint of
69 % (Fig. 8).

## 3.5 Incorporating uncertainty in radiation

In order to assess the effects of incorporating uncertainty in SWRad, we
conduct three experiments. First is a control run, equivalent to using SIF at
13:00 as before. The second includes uncertainty in SWRad by adding it into
the posterior uncertainty calculation; this might be done normally when
accounting for uncertainty in forcing. The third experiment incorporates
uncertainty in SWRad into the
error propagation system with SIF, such that the uncertainty in SWRad may be constrained.
This third experiment effectively treats SWRad as a model parameter by adding
an extra row and column to **C**_{x}.

Including the uncertainty in SWRad in the calculation of posterior
uncertainty in GPP results in an additional 0.03 Pg C yr^{−1} to
the prior uncertainty in global annual GPP. This is a small effect relative
to the parametric uncertainties. Moreover, if we incorporate SWRad
uncertainty into the error propagation system, we see that this additional
uncertainty is mitigated by the SIF constraint. With SWRad uncertainty
included, the posterior uncertainty in GPP remains at
5.15 Pg C yr^{−1}, equivalent to the case without accounting for
uncertainty in SWRad, in both cases resulting in a relative reduction in the
GPP uncertainty by 72.9 %. This mitigation of the additional uncertainty
from SWRad is possible because both SIF and GPP are strongly sensitive to it; thus, any constraint on SWRad from SIF is also propagated through to GPP.

By assessing the prior and posterior uncertainty in SWRad in
${\mathbf{C}}_{{x}_{\mathrm{0}}}$ and ${\mathbf{C}}_{{x}_{\mathrm{post}}}$, respectively, we can
assess the effective constraint following the use of SIF in the error
propagation system. We find that SIF constrains the SWRad uncertainty by
about 29 %. This gain in information on SWRad naturally results in less
information being available for other parameters. The relative uncertainty
reduction for most parameters decreases by a few percent. For example most
*C*_{ab} parameters see a decrease in effective constraint of around 1 %
and *V*_{cmax} parameters up to 3 %. With GPP exhibiting low
sensitivity to *C*_{ab} parameters and strong sensitivity to SWRad, the
transfer of information from *C*_{ab} to SWRad results in an overall
mitigated effect of SWRad uncertainty on GPP.

The results presented show that with 1 year of satellite SIF data observed at
the GOSAT and OCO-2 satellite overpass time and SIF retrieval wavelength, we
can constrain a large portion of the BETHY-SCOPE parameter space and
ultimately yield a parametric uncertainty in global annual GPP of
±5.2 Pg C yr^{−1}. The parametric uncertainty in the prior is
approximately 12 % of the global annual GPP. Following the addition of
SIF information, this is reduced to about 3 % of global annual GPP. This
constitutes a reduction in parametric uncertainty of 73 % relative to the
prior. Although this data-driven constraint is model dependent, it is an
improvement on the often reported uncertainty of ±8 Pg C yr^{−1} from the
empirical-model-based upscaled product of Beer et al. (2010).

We note that this analysis is likely to underestimate the constraint that SIF
could provide on GPP as it is performed with uncertainties calculated from
the GOSAT SIF 3^{∘} × 3^{∘} spatial-resolution
observations. With the use of higher-resolution observations such as those
from OCO-2, the constraint will get stronger. Similarly, with a longer
time series of data, there will be stronger constraint. This occurs because
the number of independent observations increases while the number of
parameters remain constant.

This error propagation analysis does not assess how model SIF compares with observed SIF. However, our finding that the ${\mathit{\chi}}_{r}^{\mathrm{2}}$ is near the optimal value of 1 provides evidence that the range of possible model SIF realizations, given our assumptions of parameter and data uncertainties, can provide coverage of the observed SIF. While this is not evidence that each specific uncertainty (e.g. parameters, model, measurement) is optimal (Michalak et al., 2005), it does suggest that overall the assumptions are valid and that we are not overconfident in or underestimating covariances. We reiterate that the ${\mathit{\chi}}_{r}^{\mathrm{2}}$ test is performed using a higher spatial-resolution model and measured data because this is the resolution that would be applied in an assimilation of the data and this reduces the effects of representation errors. The error propagation analysis, however, benefits from using a low resolution as it greatly improves computational efficiency considering the simulations, and calculations are computationally demanding.

We also find that the effect of incorporating the error in the zero-level offset correction in the SIF observations is negligible on posterior parametric uncertainties. This may be negligible because, for a given season, this systematic uncertainty applies across all data points; thus, it scales all of the SIF values and therefore the sensitivities as well. In any case, the systematic error in the zero-level offset-corrected data assessed here (Fig. A2) appears small.

The constraint on global GPP is similar when assimilating SIF at any time between 09:00 and 15:00. Assimilating observations at the daily maximum of SIF and GPP provides the strongest constraint as both quantities exhibit the strongest parameter sensitivities at these times. Depending upon the state of the vegetation and the environmental stress conditions, maximum SIF and GPP may occur anywhere between mid-morning and early afternoon. Therefore, we expect that the effective use of different satellite-retrieved SIF observations for assimilation studies will depend not so much on their observing time but more on the spatiotemporal resolution, measurement precision, and subsequent uncertainty.

A confounding factor in this expectation is the uncertain role of physiological stress on the diurnal cycle of SIF and GPP and on modelling capabilities of these processes. Multiple studies have shown that various forms of environmental stress result in the downregulation of PSII and changes in the fluorescence yield, particularly evident across the diurnal cycle (Carter et al., 2004; Daumard et al., 2010; Flexas et al., 1999, 2000, 2002; Freedman et al., 2002). By incorporating SIF observations at multiple times of the day, we hypothesized that there could be improvements in the overall constraint on GPP as the SIF observations would capture the vegetation in different states of stress. We saw only minor improvements in the constraint and less constraint if we assumed no additional information in the observations (i.e. with scaled uncertainty). Thus, the difference in model parameter sensitivities of SIF and GPP at other times across the diurnal cycle was not sufficient to add value to the constraint. Additionally, the constraint is worse with these scaled observational uncertainties as we are effectively removing some useful observational information at midday, the time that provides the highest sensitivities, and getting extra observational information at the lower-sensitivity times of 08:00 and 16:00. This may be due to limitations of the model. Although BETHY-SCOPE simulates light-induced downregulation of PSII, there is no mechanism present to simulate other forms of stress that might be expected to emerge across the diurnal cycle. However, even with a perfect model, the spatial footprint and spatiotemporal averaging of satellite observations may smooth over stress signals. Considering these confounding factors, incorporating individual SIF soundings could help remedy this problem, and there is no technical reason other than the high computational requirements that would prevent a data assimilation system from doing so.

The constraint of SIF on GPP occurs via multiple processes including leaf growth, leaf composition, physiology, and canopy structure. For the prior, uncertainty in global GPP is dominated by leaf growth processes. There is a clear and direct link between leaf growth processes and GPP (Baldocchi, 2008) as the dynamics of leaf area influences canopy absorbed photosynthetically active radiation (APAR), which in turn strongly influences GPP. Leaf growth parameter uncertainties are relatively large in the prior, with coefficients of variation up to 50 %. It is perhaps no surprise then that these parameters project a large uncertainty onto GPP. Regardless, both GPP and SIF respond similarly to the leaf growth parameters, so information from observations of SIF can provide a direct constraint on GPP in this way. Many leaf growth parameters, particularly for grasses, crops, and deciduous trees and shrubs, receive a constraint of > 40 % from SIF; thus, the overall contribution of leaf growth parameters in the posterior is considerably reduced.

Of particular importance is the parameter describing water limitation on leaf
growth (*τ*_{W}), which accounts for about 80 % of the prior uncertainty
in global GPP. Model SIF and GPP are highly sensitive to this parameter;
hence, there are large values in **H** and **H**_{GPP}
pertaining to *τ*_{W}. This relates to the model formulation as many of the
leaf growth parameters determine phenological processes such as temperature
or light-dependent growth triggers (i.e. temporal evolution of leaf area),
while *τ*_{W} is the only process parameter controlling leaf area other than
intrinsic maximum LAI ($\stackrel{\mathrm{\u0303}}{\mathrm{\Lambda}}$) (Knorr et al., 2010). Additionally, as
we assume little prior knowledge for *τ*_{W} (i.e. it is highly uncertain)
it projects a relatively large uncertainty onto GPP.

At the global scale, *τ*_{W} for crops, C3 grasses, and C4 grasses
$\left({\mathit{\tau}}_{W}^{\mathrm{Gr}}\right)$ is particularly important. Combined, these
three PFTs cover about 47 % of the land surface and account for just over
50 % of global annual GPP in the present model set-up. Although this
contribution to global GPP may seem high, it is based on the prior estimate.
In a recent study by Scholze et al. (2016), where atmospheric CO_{2}
concentration and SMOS (Soil Moisture and Ocean Salinity) soil moisture were assimilated into BETHY, the posterior
value for ${\mathit{\tau}}_{W}^{\mathrm{Gr}}$ shifted approximately 3 standard deviations
away from the prior, the result of which would have been a large change in
the GPP of these PFTs. This exposes a limitation of the present study as we
can predict and quantify how SIF will constrain the uncertainty in process
parameters and GPP, but we cannot predict how their values will change.

The constraint SIF provides on leaf growth processes is also perhaps achievable from other remote-sensing products such as FAPAR (e.g. Kaminski et al., 2012). A direct comparative study would be required to assess the advantages and disadvantages of each observational constraint. Nevertheless, issues arise with these alternative observations when observing dense canopies (Yang et al., 2015) or vegetation with high photosynthetic rates such as crops (Guanter et al., 2014). Information on maximum potential LAI ($\stackrel{\mathrm{\u0303}}{\mathrm{\Lambda}}$) and parameters pertaining to understorey shrubs and grasses are therefore also limited (Knorr et al., 2010). A strong benefit of SIF is that it shows minimal saturation effects (e.g. Yang et al., 2015), especially beyond 700 nm, where most current satellite SIF measurements are made.

The strong constraint SIF provides on leaf growth processes indicates that it is likely to provide improved monitoring of key phenological processes such as the timing of leaf onset, leaf senescence, and growing season length as also suggested by Joiner et al. (2014). This will be highly useful in interpreting results from a full assimilation with SIF as the posterior process parameter values can be compared with independent ecophysiological data, taking spatial scale issues into consideration.

Beyond observing LAI dynamics SIF can also provide critical insights into
physiological processes (e.g. Walther et al., 2016). We see here that SIF
provides a weak to moderate constraint on a range of physiological parameters,
including up to 30 % constraint on *V*_{cmax} parameters. The
limited constraint on these parameters results in the posterior being
dominated by uncertainty in the parameters representing physiological
processes. This is in line with Koffi et al. (2015), who found limited
sensitivity of simulated SIF to *V*_{cmax}. We note that under certain
conditions, where other key variables are well known, SIF can be used to
retrieve *V*_{cmax} (Zhang et al., 2014). The ability of SIF to provide information on
physiological processes at all will provide researchers with a powerful new
insight into the spatiotemporal patterns of GPP. As was shown by
Walther et al. (2016) and Yang et al. (2015), this is particularly important for
evergreen vegetation as changes in photosynthetic activity are not always
reflected by changes in traditional vegetation indices.

Chlorophyll content here constitutes a classic nuisance variable. A nuisance
variable is one that is not perfectly known and impacts the observations we
wish to use but not the target variable (Rayner et al., 2005). However,
exploiting the well-documented correlation between leaf nitrogen content,
*V*_{cmax}, and *C*_{ab} may help curtail this problem
(Evans, 1989; Kattge et al., 2009). Houborg et al. (2013) demonstrated that by
including a semi-mechanistic relationship between these variables in the
Community Land Model and using satellite-based estimates of chlorophyll to
derive *V*_{cmax}, there is significant improvement in predictions of
carbon fluxes over a field site. Implementing such a semi-mechanistic link in
a data assimilation system would enable the strong constraint that SIF
provides on *C*_{ab} to feed more directly into GPP. However, in this study
it is assumed *C*_{ab} and *V*_{cmax} can be resolved independently,
which may not be the case considering that ecophysiological studies have
shown that the two parameters are commonly correlated.

Almost all terrestrial carbon cycle models use down-welling radiation at the
Earth's surface as an input variable. Any uncertainty in this forcing will
translate into uncertainty in carbon fluxes including GPP, and few studies
consider such uncertainties. A known systematic error (i.e. bias) in forcing
variables (e.g. Boilley and Wald, 2015) cannot be considered in the present
error propagation system; however, in such a case a correction to the data
should be performed as it will bias carbon flux estimates. For random errors
that cannot be removed, however, they may be considered in the uncertainty in
carbon flux estimates using error propagation. At the global scale,
Kato et al. (2012) used a perturbation study, along with modelled irradiance and
remotely sensed measurements to compute a random error (*σ*) of
12 W m^{−2} for monthly gridded downward shortwave radiation over the
land. We considered this uncertainty by incorporating it into the error
propagation system with SIF. While including this forcing uncertainty in the
prior increases the prior uncertainty in GPP, incorporating the former into
the error propagation analysis with the SIF observations mitigates the
downstream effect on GPP. SIF can therefore provide useful information on the
SWRad forcing via a data assimilation system. The consideration of
uncertainties in forcing variables such as SWRad on terrestrial carbon fluxes
is important when estimating the uncertainty in GPP. However, the effect on
uncertainty in GPP may be strongly reduced by using SIF observations.

The results presented here demonstrate how SIF observations may be utilized
to optimize a process-based terrestrial biosphere model and constrain
uncertainty in simulated GPP. These results are, however, model dependent.
The assumption is that the model simulates the most important processes
driving SIF and GPP. Some key, remaining unknowns include how processes such
as environmental stress, three-dimensional canopy structure effects, or
nitrogen cycling may affect the SIF signal. As better understanding is
developed of the role that these processes play, modelling capabilities will
also be improved. Additionally, a different set of prior parameter values
will alter the results due to changes in **H**. The use of prior
knowledge, based on ecophysiological data and its probable range, is critical
to curtailing this problem. The choice of how to spatially differentiate the
parameters will also affect results (Ziehn et al., 2011). Selecting an optimal
parameter set that has the fewest degrees of freedom yet provides the best
fit to the observational data is outside the scope of this study, however.
The implementation of a parameter estimation scheme in a full data
assimilation system with SIF and other observational data will help address
these challenges. Earlier work by Koffi et al. (2015) demonstrated that the
model can simulate the patterns of observed satellite SIF quite well,
indicating that the model can incorporate the data. Further work will be
needed to assess how well the model can simulate patterns of SIF with an
optimized, realistic parameter set.

We assessed the ability of satellite SIF observations to constrain
uncertainty in model parameters and uncertainty in spatiotemporal patterns of
simulated GPP using a process-based terrestrial biosphere model. The results
show that there is a strong constraint of parametric uncertainties across a
wide range of processes including leaf growth dynamics and leaf physiology
when assimilating just 1 year of SIF observations. Combined, the SIF
constraint on parametric uncertainties propagates through to a strong
reduction in uncertainty in GPP. The prior uncertainty in global annual GPP
is reduced by 73 % from 19.0 to 5.2 Pg C yr^{−1}. Although
model dependent, this result demonstrates the potential of SIF observations
to improve our understanding of GPP. We also showed that a data assimilation
framework with error propagation such as this allows us to account for
uncertainty in model forcing such as SWRad. Surprisingly, by including it
into this framework with SIF observations, there is a net-zero effect on
uncertainty in GPP due to the sensitivity of both SIF and GPP to radiation.
This study is a crucial first step toward assimilating satellite SIF data to
estimate spatiotemporal patterns of GPP. With the addition of other
observational constraints such as atmospheric CO_{2} concentration or
soil moisture there is also the possibility of accurately disaggregating the
net carbon flux into its component fluxes – GPP and ecosystem respiration.
Indeed, with these additional, complementary observations of the terrestrial
biosphere further constraint could be gained as other regions of parameter
space can be resolved (Scholze et al., 2016).

The BETHY-SCOPE model code is available upon request from the authors. The GOSAT satellite SIF data used in this paper are from the ACOS project (version B3.5) (Frankenberg, 2018).

## A1 Model process parameters

## A2 GOSAT SIF uncertainty calculations

To obtain the variance of a target grid cell at the model grid resolution
(*y*_{lat}, *x*_{lon}), we first determine the area-weighted variance
of each GOSAT grid cell (*i*_{lat}, *j*_{lon}) within that target
grid cell. The area weighting per GOSAT grid cell
(${\widehat{\mathrm{Area}}}_{{i}_{\text{lat}},\phantom{\rule{0.125em}{0ex}}{j}_{\text{lon}}}$) is calculated as
the area divided by the total area of the target grid cell. This enables us
to account for different

grid cell sizes considering that SIF is in physical units per unit area. We then sum the area-weighted variances and scale this uncertainty by the square root of 2 (see Eq. 5). Scaling the uncertainty in this way effectively doubles the variance in an independent dimension.

## A3 Systematic error in GOSAT SIF observations

The authors declare that they have no conflict of interest.

Alexander Norton was partly supported by an Australian Postgraduate Award
provided by the Australian Government and a CSIRO OCE Scholarship. The
research was funded, in part, by the ARC Centre of Excellence for Climate
System Science (grant CE110001028).

Edited by:
Tomomichi Kato

Reviewed by: Sylvain Kuppel and one anonymous
referee

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_{2}plants absorb via photosynthesis and even more difficult to model this for the whole globe. Here, we present a framework to combine a new satellite measurement "solar-induced chlorophyll fluorescence" with a global photosynthesis model. We then quantify how this new measurement constrains model uncertainties and find highly effective constraint. These results pave a novel pathway for improving estimates and modelling abilities of photosynthesis globally.

_{2}plants absorb via photosynthesis and even more...