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Geophysical Research Letters
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Supporting Information for
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An analytical model for relating global terrestrial carbon assimilation with climate and
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surface conditions using a rate-limitation framework
Yuting Yang1,*, Randall J. Donohue1,2, Tim R. McVicar1,2 and Michael L. Roderick2,3
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CSIRO Land and Water, Canberra, Australia
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Australian Research Council Centre of Excellence for Climate System Science, Sydney,
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Australia
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University, Canberra, Australia
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(Surnames are underlined)
Research School of Biology and Research School of Earth Sciences, Australian National
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Email: Yuting.yang@csiro.au;
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Tel:+61 (2) 62183448
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Address: CSIRO Land and Water, Black Mountain, Canberra, ACT 2602, Australia
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Manuscript submitted to
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Geophysical Research Letters
Contents of this file
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Figures S1 to S12
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Tables S1
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Additional Supporting Information
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Supplementary Figure fs01 Relationship between mean annual incident solar radiation and
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mean annual air temperature at the (a) catchment (32 global large catchments) and (b) grid-
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cell (0.5o spatial resolution) scales over the period of 1982-2010.
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Supplementary Figure fs02 Relationship between mean annual GPP_MTE and mean annual
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P at the grid-cell scale (0.5o spatial resolution) over the period of 1982-2010.
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Supplementary Figure fs03 Comparison between (a) observed mean annual GPP and GPP
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estimated from the linear function (solid blue lines in Figure 3a) at the flux-site scale, and
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between (b) mean annual GPP_MTE and GPP estimated from the linear function (solid blue
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lines in Figure 3b) at the grid-cell scale. The red dash lines are 1:1 lines.
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Supplementary Figure fs04 Relationship between A/Am(I) and Am(P)/Am(I) within the rate-
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limitation framework using GPP estimates from 13 ecosystem models. (a) Multi-model
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ensemble mean, (b) Biome-BGC, (c) CLASS, (d) CLM, (e) CLMVIC, (f) DLEM, (g) GTEC,
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(h) ISAM, (i) ORCHIDEE, (J) LPJ, (k) SiBCASA, (l) TRIPLEX, (m) VEGAS and (m)
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VISIT. The red ellipses highlight situations where the GPP estimation from some individual
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models do not align with the rate-limitation framework.
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Supplementary Figure fs05 Calibration of the model with solely fPAR at the flux-site level. (a)
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Relationship between inverted α and fPAR and (b) comparison of GPP estimated by Eq. (6) in
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combination with α estimated from the best-fit non-linear relationship between α and fPAR
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shown in (a) (i.e., 𝛼 = 7.49𝑓PAR
− 6.47𝑓PAR + 2.02) against site-level observations. The red
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solid line in (a) indicate the best-fit function based on the least-square criteria, and the red
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dashed line in (b) is 1:1 line.
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Supplementary Figure fs06 Calibration of the model with solely fPAR at the grid-cell scale. (a)
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Relationship between inverted α and fPAR and (b) comparison of GPP estimated by Eq. (6) in
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combination with α estimated from the best-fit non-linear relationship between α and fPAR
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0.5
shown in (a) (i.e., 𝛼 = 0.98𝑓PAR
) against GPP_MTE. The red solid line in (a) indicate the
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best-fit function based on the least-square criteria, and the red dashed line in (b) is 1:1 line.
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Supplementary Figure fs07 Comparison of estimated GPP and GPP_MTE at the grid-cell
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scale. In part (a) α is estimated from the model calibrated at the flux-site scale (Eq. 8) and in
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part (b) α is estimated from the model calibrated at the grid-cell scale (Eq. 9). Please note that
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in (a) we only applied the model for grid-cells with fPAR larger than 0.3, as we do not have
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low fPAR flux-sites (i.e., fPAR < 0.3) for calibrating the model (see Supplementary Table ts01).
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In both parts the dashed red line is the 1:1 line.
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Supplementary Figure fs08 Relationship between fPAR/I and P/I at the grid-cell scale across
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the globe.
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Supplementary Figure fs09 Location of the 32 catchments used in the supplementary
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material.
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Supplementary Figure fs10 Relationship between A/Am(I) and Am(P)/Am(I) within the rate-
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limitation framework at the catchment (regression, y = 0.4295 x, R2 =0.97, blue line). The
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dashed lines show the analytical solution for different parameter values.
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Supplementary Figure fs11 Leave-one-out cross validation of estimated α from Eq. (7)
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against inverted α at the catchment scale. The red dashed line is 1:1 line.
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Supplementary Figure fs12 Comparison of mean annual GPP_MTE against mean annual
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modelled GPP at the catchment scale (using α model calibrated at the catchment scale, i.e.,
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𝛼cat = 1.587𝑓PAR + 0.006𝛥𝑇 − 0.084𝑧 + 0.094). The red dash line is 1:1 lines.
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Supplementary Table ts01 Summary of the flux site name (Site ID), mean annual GPP (g C
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m-2 yr-1), fPAR, precipitation (P in mm yr-1), photosynthetic active radiation (I in MJ m-2 yr-1)
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and references.
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Introduction
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Firstly, relationships between mean annual incident shortwave radiation and mean annual air
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temperature at grid-cell scale is shown in Supplementary Figure fs01, to support our
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statement that radiation and temperature is highly correlated at long-time scales in Section 2.1.
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Secondly, to determine the maximum rain use efficiency (θmax), we plot the relationship
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between mean annual GPP_MTE and mean annual P at the grid-cell scale (i.e., 0.5 degree
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spatial resolution) over the years 1982 to 2010 (Supplementary Figure fs02). We found that at
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low P, there is a distinct upper edge that represents the maximum GPP attainable for a given
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P, and the slope of this upper edge is taken to be the maximum rain use efficiency.
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Thirdly, we examined the performance of using the linear function in Figure 3(b) and (c) to
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estimate GPP (Supplementary Figure fs03), and we found that it results in reasonable GPP
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estimates.
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Fourthly, we show the relationship between A/Am(I) and Am(P)/Am(I) within the rate-
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limitation framework using GPP estimates from 13 ecosystem models during the Multi-scale
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Synthesis and Terrestrial Model Intercomparison Project (MsTMIP) [Huntzinger et al. 2013]
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(Supplementary Figure fs05). A similar result with Fig. 3b in the main text is found when
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using the multi-model Ensemble mean (Supplementary Figure fs07 (a)), further reinforcing
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the physical basis of the rate-limitation framework of the GPP model proposed herein.
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However, GPP estimates from 5 of the 13 ecosystem models do not always satisfy the first-
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order requirement (i.e., bounds between two GPP limits) based on visual examination, and
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these models are Biome-BGC (b), CLASS (c), CLM (d), DLEM (f) and GTEC (g).
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Fifthly, as fPAR plays a dominant role in determining α, we examined the relationship between
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fPAR and α and used the best-fit non-linear function to estimate α based solely on fPAR. The
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estimated α was then applied in Eq. (6) for estimating GPP at both the flux-site and grid-cell
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scales (Supplementary Figures fs05 and fs06).
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Sixthly, we also examined the difference and transferability of the two α models (i.e., Eq. (8)
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calibrated at the flux-site scale and Eq. (9) calibrated at the grid-cell scale). Our results show
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similar performances of Eq. (6) when applied with either Eq. (8) or Eq. (9) for predicting α at
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the flux-site (Figures 5a and 5b in main text) and grid-cell scales (Supplementary Figure
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fs07), suggesting that Eq. (8) and Eq. (9) are relatively conservative and transferable.
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Seventhly, we tested the developed framework with fPAR, as there is a close relationship
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between GPP and fPAR, and fPAR can be directly observed from satellite remote sensing. The
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relationship between fPAR/I and P/I is shown in Supplementary Figure fs08, which shows that
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fPAR also follows generally the Budyko-like framework. However, although fPAR is closely
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related with GPP, the relationship is non-linear (e.g., fPAR tends to saturated in tropical
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rainforest while GPP still shows large spatial variation due to variation in incoming solar
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radiation). A more fundamental difference is that GPP is a flux variable that is suitable to be
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analyzed with a rate-limitation framework, whereas fPAR is a state variable.
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Finally, our results in the main text suggest that the model tends to perform better as the
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spatial scale increases (i.e., better at the grid-cell scale than at the flux-site scale). This is not
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surprising, because the spatial variability of local effects can be largely reduced at coarser
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spatial scales. There have been similar findings regarding the Budyko hydroclimatological
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model. To further demonstrate this point, we test our model at the catchment scale – a spatial
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scale much larger than a 0.5o grid-cell. Another reason of testing the model at the catchment
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scale is that catchments are independent hydrological units, which ensure the precipitated
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water entirely consumed within the analysis unit. We chose here 32 global large catchments
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(supplementary Figure fs09), and did the same analyses based on GPP_MTE as what we have
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done for flux-sites and grid-cells. We found that the catchment-scale data also aligns with the
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proposed Budyko-like GPP framework (supplementary Figure fs10). We also found that
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calibration of the α model at the catchment scale results in the same three variables (i.e., fPAR,
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ΔT and z, 𝛼cat = 1.587𝑓PAR + 0.006𝛥𝑇 − 0.084𝑧 + 0.094) as for flux-sites and grid-cells,
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and fPAR at the catchment scale play an even more important role in determining α (i.e., using
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fPAR alone could explain 70% of the variance in αcat and adding ΔT explains a further 14%,
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with z explaining a further 3%) (see Supplementary Figure fs11 for the validation of the α
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model at the catchment scale using a leave-one-out cross validation approach). In addition,
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combining the α model at the catchment scale, the proposed theoretical GPP model
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performed well, and better than at the flux-site and grid-cell scales, in estimating GPP for the
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catchments (i.e., R2=0.97, RMSE=110 g C m-2 yr-1) (Supplementary Figure fs12).
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Supplementary Figure fs01 Relationship between mean annual incident solar radiation and
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mean annual air temperature at the grid-cell (0.5o spatial resolution) scales over the period of
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1982-2010.
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Supplementary Figure fs02 Relationship between mean annual GPP_MTE and mean annual
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P at the grid-cell scale (0.5o spatial resolution) over the period of 1982-2010.
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Supplementary Figure fs03 Comparison between (a) observed mean annual GPP and GPP
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estimated from the linear function (solid blue lines in Figure 3a) at the flux-site scale, and
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between (b) mean annual GPP_MTE and GPP estimated from the linear function (solid blue
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lines in Figure 3b) at the grid-cell scale. The red dash lines are 1:1 lines.
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Supplementary Figure fs04 Relationship between A/Am(I) and Am(P)/Am(I) within the rate-
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limitation framework using GPP estimates from 13 ecosystem models. (a) Multi-model
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ensemble mean, (b) Biome-BGC, (c) CLASS, (d) CLM, (e) CLMVIC, (f) DLEM, (g) GTEC,
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(h) ISAM, (i) ORCHIDEE, (J) LPJ, (k) SiBCASA, (l) TRIPLEX, (m) VEGAS and (m)
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VISIT. The red ellipses highlight situations where the GPP estimation from some individual
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models do not align with the rate-limitation framework.
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Supplementary Figure fs05 Calibration of the model with solely fPAR at the flux-site level. (a)
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Relationship between inverted α and fPAR and (b) comparison of GPP estimated by Eq. (6) in
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combination with α estimated from the best-fit non-linear relationship between α and fPAR
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shown in (a) (i.e., 𝛼 = 7.49𝑓PAR
− 6.47𝑓PAR + 2.02) against site-level observations. The red
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solid line in (a) indicate the best-fit function based on the least-square criteria, and the red
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dashed line in (b) is 1:1 line.
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Supplementary Figure fs06 Calibration of the model with solely fPAR at the grid-cell scale. (a)
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Relationship between inverted α and fPAR and (b) comparison of GPP estimated by Eq. (6) in
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combination with α estimated from the best-fit non-linear relationship between α and fPAR
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0.5
shown in (a) (i.e., 𝛼 = 0.98𝑓PAR
) against GPP_MTE. The red solid line in (a) indicate the
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best-fit function based on the least-square criteria, and the red dashed line in (b) is 1:1 line.
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Supplementary Figure fs07 Comparison of estimated GPP and GPP_MTE at the grid-cell
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scale. In part (a) α is estimated from the model calibrated at the flux-site scale (Eq. 8) and in
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part (b) α is estimated from the model calibrated at the grid-cell scale (Eq. 9). Please note that
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in (a) we only applied the model for grid-cells with fPAR larger than 0.3, as we do not have
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low fPAR flux-sites (i.e., fPAR < 0.3) for calibrating the model (see Supplementary Table ts01).
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In both parts the dashed red line is the 1:1 line.
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Supplementary Figure fs08 Relationship between fPAR/I and P/I at the grid-cell scale across
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the globe.
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Supplementary Figure fs09 Location of the 32 catchments used in the supplementary
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material.
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Supplementary Figure fs10 Relationship between A/Am(I) and Am(P)/Am(I) within the rate-
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limitation framework at the catchment scale (regression, y = 0.4295 x, R2 =0.97, blue line).
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The dashed lines show the analytical solution for different parameter values.
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Supplementary Figure fs11 Leave-one-out cross validation of estimated α from Eq. (7)
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against inverted α at the catchment scale. The red dashed line is 1:1 line.
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Supplementary Figure fs12 Comparison of mean annual GPP_MTE against mean annual
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modelled GPP at the catchment scale (using α model calibrated at the catchment scale, i.e.,
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𝛼cat = 1.587𝑓PAR + 0.006𝛥𝑇 − 0.084𝑧 + 0.094). The red dash line is 1:1 lines.
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