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.

Table 1PFTs defined in BETHY and their abbreviations.

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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 CO₂ 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 CO₂ (K_C) and O₂ (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 (x₀). 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 x₀ helps to limit the effect of this problem as H at x₀ 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 χ² 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(x₀) 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 CO₂ 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 (σ²) 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 σ² 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.

Figure 1Effective constraint of BETHY-SCOPE model process parameters from SIF observations. Only the parameter numbers are given; for the corresponding descriptions, see Table A1.

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Figure 2Correlation coefficients (r value) between parameters in the posterior parameter covariance matrix (${\mathbf{C}}_{x_{\mathrm{post}}}$). This shows the magnitude and sign of correlations in posterior parameter uncertainties following constraint with SIF data. The axis labels show the parameter symbol and number as defined in Table A1. Only parameters with an absolute correlation coefficient > 0.25 with one or more other parameters are shown. Values above and below the diagonal are identical; therefore, those above are coloured grey.

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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 CO₂ (K_C) and O₂ (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).

Figure 3Annual observational uncertainty in SIF interpolated from GOSAT observations for 2010.

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Figure 4Prior parametric uncertainty in annual GPP.

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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.

Figure 5Posterior parametric uncertainty in annual GPP.

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Figure 6Relative uncertainty reduction (i.e. effective constraint) of parametric uncertainty in annual GPP from prior to posterior.

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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 CO₂ (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.

Figure 7Contribution of parameter classes to parametric uncertainty in monthly GPP for three regions (see Table A1 for details on these parameter classes). For each month, the bar on the left is the prior and the bar on the right is the posterior. Uncertainties are represented as variances; thus, the units are in Pg C yr⁻¹ squared, and, for clarity, the y axes are on a quadratic-transformed scale.

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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.

Figure 8Effective constraint on global annual GPP for different observing times and the two diurnal cycle configurations. Values at the top of the bars correspond to the posterior uncertainty (σ) in global annual GPP.

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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.

Table 2Parametric uncertainty and effective constraint of global annual GPP for each of the SWDown experiments. Prior and posterior values shown are the 1 standard deviation (σ) uncertainty in global annual GPP.

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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.