The framework we use in this paper is called ICLASS-can, which is an extension of the ICLASS framework extensively described in. The ICLASS framework can be used to study the exchange of gases, moisture, heat, and momentum between the land surface and the lower atmosphere. The general aim of the framework is to allow the assimilation of various streams of observations (fluxes, mole fractions at multiple heights, etc.) to estimate model parameters, thereby obtaining a model that is consistent with a diverse set of observations . In an optimisation, a cost function is minimised. This cost function usually contains two parts. One part contains the difference between model output and observations, the other (optional) part contains the difference between the parameter values and the prior estimates of these parameters more details in Sect. 3.1 of. In an optimisation, the framework aims to obtain values of the parameters to optimise (the state) that minimise the cost function. This is done starting from an initial guess of parameter values, after which the state is improved iteratively, thereby calculating the cost function and its gradient. For calculating the gradient of the cost function, an analytical gradient is available using the model adjoint. ICLASS is a variational Inverse modelling framework for the (slightly adapted) Chemistry Land-surface Atmosphere Soil Slab model CLASS,. We have added a relatively simple canopy model, including its adjoint, to the ICLASS framework (SiLCan, see Sect. ). This enables the simulation of trace gases, moisture and energy exchange, and atmospheric conditions within forest canopies in more detail. The resulting coupled forward model consists of an atmospheric mixed layer model, coupled to a new canopy model, which is in turn coupled with a simple soil model. For temperature and moisture, the soil model distinguishes between upper soil and deeper soil. It has a dynamic upper soil temperature and moisture content, used for calculating soil respiration. Upper soil temperature and moisture are simulated based on a force–restore model and references therein. For COS, the soil is treated in more detail, using a multi-layer soil diffusion-reaction model for COS . The atmospheric mixed layer model assumes that turbulence is vigorous enough to result in a well-mixed layer see also Sect. 2 of. Therefore, there is just one value for scalars such as mixed-layer potential temperature and CO2 mole fraction. Because of this assumption, we do not model night-time conditions in this study. The mixed-layer height is dynamic. There is exchange taking place between the mixed layer and both the underlying canopy and the free troposphere above. Above the mixed layer, a discontinuity occurs in the scalar quantities, representing an infinitely thin inversion layer. Above the inversion, the scalars are normally assumed to follow a linear profile with height in the free troposphere. The information on the free troposphere is used for calculating the exchange between the mixed layer and the free troposphere. Above-canopy downward shortwave radiation is calculated as a function of time, location, and cloud cover. In our configuration, above-canopy surface layer scalars are calculated employing Monin–Obukhov similarity theory . The coupled model has no horizontal dimension, but a constant mixed-layer advection can be prescribed. The mixed-layer model provides the best results on days with prototypical mixed layer behaviour, i.e. days on which advection is either absent or uniform in time and space, deep convection and precipitation are absent, and sufficient incoming shortwave radiation heats the surface . We provide a brief description of the new canopy model in the next section, and a more detailed description can be found in the Supplement.

SiLCan stands for Simplified Layered Canopy. The model simulates three different gases in the canopy, namely CO2, COS, and H2O. The canopy is layered, and the user defines the number of layers. Temperature and wind speed is also calculated in all layers. Exchange between the layers (calculated for heat and gases) is parameterised in a simple way, with eddy-diffusivity exchange coefficients. For photosynthesis, the A–gs approach at leaf scale is followed, thereby explicitly simulating leaf scale CO2 fluxes, separately for sunlit and shaded leaves. The leaf CO2 uptake is calculated from the difference between the CO2 concentration outside the leaf boundary layer and the concentration inside the leaf, divided by the total uptake resistance. Although A–gs is not the most widely used photosynthesis model , it is still a well-established model, e.g. it has been used in land surface models of the European Centre for Medium-Range Weather Forecasts and in the Earth system model operated by the Centre National de Recherches Météorologiques CNRM-ESM1,. recently made a comparison between the photosynthetic responses to light and CO2 predicted by the leaf photosynthesis models of and . The latter model is used in the A–gs approach. They found both models calculate near-similar responses of photosynthesis to changes in intercellular CO2. They also found near-similar responses to light (at different fixed levels of intercellular CO2). To illustrate the behaviour of our A–gs model, response curves are shown in Fig. .

Leaf or plant area densities are used to scale up fluxes from the leaf scale to the canopy layer scale, depending on the considered flux. The stomatal conductances for CO2 are calculated by A–gs, and are linearly related to those for COS and H2O (different diffusivities). Stomatal conductances thus link the leaf fluxes of CO2, COS and H2O. Leaf boundary layer conductances are calculated for the three before-mentioned gases, taking differences in diffusivity into account. For COS, an internal resistance is calculated following. Incoming shortwave radiation at the top of the canopy is calculated by CLASS, and used to calculate (absorbed) PAR per leaf area in each layer, separately for sunlit and shaded leaves. Outgoing longwave radiation from a leaf surface is calculated based on the Stefan–Boltzmann law. In our configuration, absorbed incoming longwave radiation is calculated as the incoming longwave radiation at the top of the canopy (calculated by CLASS), multiplied with a constant leaf emissivity and a factor sLWin (constant in space and time) that we optimise. The energy balance is calculated at leaf level, leading to a leaf (skin) temperature which is used in the calculation of the sensible heat fluxes and H2O fluxes of leaves (sunlit and shaded separately). In our configuration we set heat storage in the leaves to zero, the energy balance is calculated using only modelled radiation and sensible and latent heat flux terms. A sketch of the canopy model is shown in Fig. , and elaborate details can be found in the Supplement.

Hyytiälä (Fluxnet ID FI-Hyy) is a forest location in Finland, at 61.85° N, 24.28° E, and 181 m above sea level. The forest is a Scots pine stand (Pinus sylvestris) sown in 1962 , with some other tree species present as well , supplementary material of. A measurement tower is present at the location. More details on the location can be found in . In all simulations, we use a time step of 60 s, and divide the canopy in 17 layers. The vertical canopy structure is represented by layers with a depth of ≈ 1 m except for the top and bottom canopy layers that have a depth of ≈ 1.5 m to properly resolve the exchange with the soil and atmospheric mixed layer (avoiding potential numerical problems with large fluxes in small layers). The soil COS model was run with a smaller time step of 10 s, to prevent numerical instability. Advection of all scalars was set to zero in all simulations.

We aimed to minimise the mismatch between model output and observations, albeit with the constraint of taking prior information about the parameters into account (minimising a cost function, containing an observation and prior information part). In a first optimisation, we used observations from July 2015, in total 26 different observation streams (CO2 mole fractions at different heights, above-canopy sensible and latent heat flux, …; full list in Table ). We selected the daytime window within 04:00 and 16:00 UTC (07:00 and 19:00 LT). Part of the observation streams we used are shown in Fig. . The observations represent averages over an hour, and we further averaged these observations over multiple days, to obtain a representative average and reduce the influence of processes such as time-varying advection. For this averaging, we selected the 8 d that have the highest mean PAR between 04:00 and 16:00 UTC. Measurement errors were estimated as the standard deviation of the observations over the 8 d we average (e.g. the measurement error of the CO2 mole fraction at 125 m at 10:30 UTC is the standard deviation of the 8 CO2 125 m mole fraction values at 10:30 UTC). For some measurement errors we used less than 8 data points, as there is some missing or insufficient quality data. Observational errors are constructed from these measurement errors and specified model errors .

The 25 parameters we optimised (the state) are listed in Table , together with prior and posterior values and the prior standard deviations. These parameters include e.g. free-tropospheric lapse rates of potential temperature and humidity, the initial soil water content of the top soil layer, and a constant that is important for soil respiration. We also included some parameters of the A–gs photosynthesis model and a parameter scaling the internal conductance of COS (αgiCOS). Thus, the state has an effect on the leaf relative uptake of COS and CO2. To limit the complexity of the optimisation problem, we did not optimise all parameters of the A–gs model. The prior parameter values were chosen as reasonable guesses. We included both parts of the cost function (Sect. ), i.e. also the deviations of the parameters from prior values are taken into account. The prior variances for the leaf exchange parameters were chosen large (Table ), allowing sufficient freedom for the optimisation algorithm, and limiting the influence of the subjective prior guesses.

In subsequent optimisations we used data from August and September 2015, averaged in a similar way as the July 2015 data. For those optimisations we used the posterior parameters from the July optimisation as prior parameter guesses. We optimised the same variables as for July, apart from Am,max,ref,toc, ad, Kb, gm,ref, f0, ε0, αgiCOS, αwind scale and VSU,max (description in Table ). For these variables we stick to the optimised values from July, as we aim to find values for these parameters that are transferable to other months and locations. R10 (soil respiration at 10 °C without water stress) was kept in the state, as the observations suggest a stronger respiration flux in (the averaged periods in) August and September compared to July.

The shape of the leaf area density profile for all simulations was first estimated based on Fig. 1 of . As all-sided leaf area index (LAI) in Hyytiälä is roughly between 6.5 and 7 in the period July–September 2015 based on Fig. 1 of , the leaf area densities were then scaled by a factor, identical for all layers, to make total LAI (summed over the model layers) equal to 6.5. Using observations of PAR at 0.6 m height above ground (at a few locations) and observed above-canopy PAR, we calculated relative PAR extinction at 0.6 m height, and compared it with the modelled extinction in the relevant canopy layer. The PAR observations indicate that at some locations, a substantial amount of PAR remains available at 0.6 m above the forest floor (due to openings in the canopy or sections with low plant area density). The comparison indicated that overall the relative extinction of PAR is fitted relatively well, although extinction is sometimes somewhat overestimated.

For analysing posterior correlations of the parameters we optimised, we performed an ensemble of parameter optimisations, consisting of 127 members. For each member, the prior parameters and the model-observation differences are perturbed. The prior information part of the cost function was disabled in the ensemble. Details of the correlation analysis procedure can be found in .

Mieming is a forest location in Austria (Fluxnet ID AT-Mmg, 47°18.9938^′ N, 10°58.2053^′ E) at 960 ma.g.l. A 30 m measurement mast is present at the location. Close to the site a mountain range is present, the station is located on a gently sloped plateau . This potentially complicates the thermodynamics and flow. Scots pine is the dominant tree species, with substantial Juniperus (Juniperus communis) in the understorey. More details on the site can be found in .

The leaf area density profile was estimated from a lidar scan of the site, and scaled such that the all-sided leaf area index becomes 4 (measurements indicate 3.4 ± 0.6). We use the same time resolution and a similar vertical resolution as for Hyytiälä, now dividing the (somewhat lower) canopy into 12 layers. We use averaged data for August 2023 in a first optimisation, and for July 2023 in a second optimisation. Data were again averaged over days with high mean PAR (the 8 d in the month with highest mean PAR between 07:00 and 19:00 LT, local time), such that we get hourly observations for one daytime period, from 07:30 to 18:30 LT. Measurement errors are estimated by the same approach as for Hyytiälä. As Scots pine dominates both Hyytiälä and Mieming, we use the A–gs-model parameters (and αgiCOS) obtained from the optimisation in Hyytiälä for simulating Mieming. Besides these parameters, the set of parameters we optimise (Table ) is to a large extent similar to those for Hyytiälä (Table ). The posterior parameters from the Hyytiälä optimisation for July 2015 serve as prior parameter estimates for the Mieming optimisations.

On some of the days with high PAR that were selected for averaging observations, we also have branch bag measurements, containing leaf COS and CO2 fluxes and mole fractions, for leaves in the upper canopy. These measurements are not assimilated in the optimisation, but we use these for comparing modelled leaf-scale relative uptake of COS and CO2 (Sect. ) with observations.

The leaf relative uptake at canopy scale (LRUcan), as introduced in Eq. (), cannot directly be derived from eddy covariance flux observations, as COS can also be taken up by the soil, the measured CO2 flux includes respiration and storage fluxes can be present. The ecosystem relative uptake, defined below, can however easily be derived from eddy covariance flux observations and observed concentrations (or mole fractions): 2ERU=FCOS,ecoFCO2,eco[CO2][COS] wherein FCOS,eco is the flux above the canopy, measured by eddy covariance. We use here eddy covariance COS and CO2 fluxes that are not storage corrected, given that the modelled COS flux to which we compare the FCOS,eco observations, is the instantaneous flux between the top canopy layer and the mixed layer. To calculate ERU in Hyytiälä, we use mole fractions at 125 m height, both for the observations and the model. This height is chosen since we have observations of both CO2 and COS mole fractions at this height. The mole fractions at the levels of the leaves might however be different from those at 125 m height. For calculating ERU in Mieming, we use mole fractions (measured and modelled) at 20 m height.

For Hyytiälä, we can derive LRUcan from the observations. To this end, we subtract the measured soil COS flux from the eddy covariance COS flux and the soil respiration flux from the eddy covariance CO2 flux. We assume respiration and COS emission from above-ground sources to be small, and neglect it in the calculation. Next to that, we correct the eddy covariance observations for storage below the sensor (in contrast to the calculation for ERU), as we are interested in plant fluxes. Note that for well-mixed daytime conditions the storage fluxes are thought to have generally a small influence . The formula for LRUcan is given in Eq. (), after rearranging. The LRUcan error bars are estimated from the spread in LRUcan values over the days we use data from. As an example, the error bar for LRUcan at 10:30 UTC indicates the standard deviation of the LRUcan values at 10:30 UTC for the different days we average the observations over (8 d, or less in case of missing or bad quality data). For Mieming, we do not have all required measurements to derive LRUcan. For calculating modelled LRUcan in Mieming we use modelled COS and CO2 mole fractions at 20 m height.

We define the leaf-scale relative uptake as: 3LRUleaf=FCOS,leafFCO2,leaf[CO2][COS] wherein FCOS,leaf [molm-2s-1] is the leaf-scale flux of COS and [COS] is the concentration of COS [molm-3] (or mole fraction) outside the leaf boundary layer. Similarly, FCO2,leaf is the leaf-scale flux of CO2 and [CO2] is the concentration of CO2. All four components are within the same canopy model layer (or for the same branch in case of observations). The calculation is performed in all model layers, and separately for sunlit and shaded leaves. For a theoretical analysis of the leaf-scale relative uptake, see . For our analysis of in-canopy LRUleaf differences in Sect. , we further expand the equation above. Using the model equations (for our configuration) given in Sect. S1.8, Eq. () can be written as: 4LRUleaf=rtot,CO2rtot,COS[CO2][CO2]-[CO2]int=rb,CO2+11rs,CO2+1rcut,CO2andCOS1.561.37rb,CO2+111.21rs,CO2+rint,COS+1rcut,CO2andCOS×11-[CO2]int[CO2]≈rb,CO2+rs,CO21.561.37rb,CO2+1.21rs,CO2+rint,COS×11-[CO2]int[CO2] wherein rcut is the cuticular resistance, rb is the leaf boundary layer resistance, rint,COS is the internal resistance for COS, rs is the stomatal resistance, rtot is the total resistance and [CO2]int is the internal CO2 concentration. The approximation in the last part of the equation concerns the assumption that the cuticular pathways for CO2 and COS uptake are negligible e.g.. When additionally neglecting the leaf boundary layer resistance, the equation above becomes equal to Eq. (8) of .

What is clear from the equation above is that any environmental variable that influences stomatal resistances, the COS internal resistance, the boundary layer resistance, or the internal CO2 concentration can influence LRUleaf (e.g. PAR, vapour pressure deficit, wind speed, …). We simulate the canopy profiles of environmental variables. We thereby take effects of canopy structure (leaf and plant area density distribution) on environmental variables into account.

The model variables that vary between sunlit and shaded leaves are the leaf temperature and the amount of absorbed PAR. To find out the relative importance of both factors in determining LRU differences between sunlit and shaded leaves, we performed the following model experiment: we took a sunlit leaf in the top layer, and prescribe for this leaf the leaf temperature of a shaded leaf in the same layer, without changing the absorbed PAR of the leaf. We subsequently investigated how the modelled LRU of the leaf changed. Next, we took again a sunlit leaf in the top model layer, and prescribed it the absorbed PAR of a shaded leaf, without changing the leaf temperature. Thus, we performed a univariate sensitivity analysis.

The elementary model variables that vary between leaves in the top and bottom layer, and have an influence on LRU, are the CO2 mole fraction, vapour pressure, air temperature, leaf temperature, amount of absorbed PAR, Am,max,ref (a leaf photosynthesis parameter, see Table ) and wind speed. Note that COS mole fraction is not included, as it cancels out in our LRU equation, given that in the model COS uptake at the leaf scale is a linear function of the COS mole fraction. We again performed a univariate sensitivity analysis, now replacing one-by-one the values of the 7 relevant variables from a shaded top-layer leaf in the model with those of a shaded leaf in the bottom layer.

We first take a look at the observations and the performance of the prior (i.e. before optimisation) and posterior (i.e. after optimisation) models. We than briefly analyse posterior state changes in the optimisation. To gain some insight in the relevant in-canopy physics, we analyse vertical canopy profiles of the posterior model simulation, before moving on to the relative uptake of COS and CO2.

Our aim is to obtain a parameterisation for LRUcan that is applicable to needleleaf forests in general, and is based on independent variables that are relatively easy to estimate. Using also the information from Sect. , we select the canopy-integrated amount of absorbed PAR (PARabs) and the vapour pressure deficit of sunlit top (upper canopy model layer) leaves (VPDsun,top) as independent variables for our regression model. These variables can be estimated or approximated based on remote sensing products e.g. or global re-analysis products e.g.. Note that in the within-canopy analysis above, leaf temperature was also identified as an important variable. However, leaf temperature is used in the calculation of VPD, and we can expect leaf temperature to correlate in time with absorbed PAR. For simplicity, we chose a linear regression model. As training data we use the model output from the optimised model for July 2015 in Hyytiälä. We obtain the following linear regression equation: 5LRUcan=2.52-1.07×10-3PARabs-0.399VPDsun,top wherein PARabs has the units Wmground-2 and VPDsun,top is in kPa. The equation above has an R2 value (coefficient of determination score) of 0.98 on the training data. The two mentioned variables thus capture most of the variability in LRUcan. Note that we use PARabs and VPDsun,top from the model output as input variables for our regression model.

To test to what extent the regression equation holds outside the training data set, we re-optimised our model using observations from August 2015. We re-optimised variables relating to the time-specific meteorological situation of the month, but did not re-optimise photosynthesis or leaf COS uptake parameters (we keep those parameters as the July-optimised values, see Sect. ).

In this new optimisation, we obtain again a satisfactory fit to many observation streams, with a cost function that reduces from about 565 to 125 (χ2=0.66). The performance of our regression model trained on the July data applied to the August data, is shown in Fig. b. Note that we use PARabs and VPDsun,top from the posterior model output as input variables for our regression model, as we do throughout this manuscript. Comparing the linear regression LRUcan model with the physical model, the R2 is 0.96 and the root mean squared error is 0.04. The regression model thus provides a very good approximation to the results of the physical model. Comparing with the (LRUcan derived from) observations, the R2 is 0.23 and the root mean squared error is 0.85. There is generally a positive bias, although limited when taking the spread in observations (error bars) into account. In Fig. , we also show the results of the leaf-scale-based parameterisation from , used in (referred to as Lai24). This parameterisation, to which we provide observed PAR at 18 m height as only input variable, clearly performs well also on the canopy scale for this data. Comparing with the observational LRUcan, the R2 is 0.48 and the root mean squared error is 0.70.

To test how well the regression equation performs in somewhat more different conditions, we performed a new optimisation for September 2015. The (averaged) data indicate a lower temperature and less incoming shortwave radiation compared to July. We optimised the same variables as for the August optimisation. In this optimisation, we obtain an acceptable fit to many observation streams, with a cost function that reduces from about 594 to 148 (χ2=0.78). The CO2 and COS fluxes are generally slightly underestimated by the model, although the fit is for most data points within the 1σ error bars.

The performance of the regression model for September is shown in Fig. c. Comparing the linear regression model with the physical model, the R2 is 0.76 and the root mean squared error is 0.15, indicating a good fit. Comparing with the observations, the R2 is -0.20 and the root mean squared error is 0.57. The regression model from performs very well in September as well. Comparing with the observational LRUcan, the R2 is -6.52 and the root mean squared error is 1.42 (strongly influenced by the first data point in the morning).

The regression model for LRUcan thus approximates the physical model well, also for months outside the training data. Comparing with (LRUcan derived from) observations, differences are larger, although still limited when taking the spread in observations into account. In the next section we will analyse the results of applying the framework and parameterisation to a different location.

We now apply our inverse modelling framework to a needleleaf forest at a more southerly location (Mieming, Austria) to check how universal the optimised photosynthesis parameters and LRUcan parameterisation are. We therefore use the leaf exchange parameters we obtained for July 2015 in Hyytiälä, and do not include these parameters in the state we optimise (Table ). We first discuss the optimisation that uses data from August 2023.

In general, we can fit the Mieming observations well (Fig. , showing the 10 assimilated observation streams for the August optimisation). As for Hyytiälä, the posterior fit with observations is improved compared to the prior. The cost function reduces from a value of about 45 to about 25 (χ2=0.20). The relative reduction in cost function is a lot smaller as for the July 2015 Hyytiälä optimisation (prior 1317, posterior 77). The prior model already predicts the above-canopy CO2, COS and H2O fluxes relatively well (Fig. e, f, and j), suggesting reasonable transferability of the Hyytiälä leaf exchange parameters to Mieming. This optimisation is less constrained as those for Hyytiälä, given the lack of e.g. soil flux observations. The optimised parameters are shown in Table . Since we do not have soil fluxes, we also do not have observations of LRUcan, albeit we can derive the ERU from the observations (Fig. ). The fit between our modelled ERU (based on optimised parameters) and observed ERU is acceptable, given the huge spread in observations. We also applied the parameterisation for LRUcan, obtained from the July Hyytiälä optimisation, to Mieming. We find a satisfactory agreement between the linear regression model and the physical model (Fig. a). Except for the early morning, the difference between the physical model and our linear regression model is always smaller than 0.3. For the optimisation using data from July, this is also the case (Fig. b). For both months, the parameterisation from Lai24 (to which we now provide observed PAR at 30 m height as only input variable) underestimates the physical model LRUcan, and performs less well than our regression model in this respect. As for Hyytiälä, the difference between modelled LRUcan and the modelled LRUleaf of a sunlit top leaf is rather small (difference somewhat larger in the early morning). In Fig.  we also compare the modelled LRUleaf for a sunlit top leaf with observations of LRUleaf for (mostly) sunlit leaves in the middle to top crown layer of the canopy. For July, there is generally an underestimation by the model, although not as strong as the underestimation by the Lai24 parameterisation. For August, the fit is relatively good for 10:00 to 13:00 LT, in the afternoon the underestimation is relatively large. There was however only one day of measurements available for August, which might not be fully representative for the modelled period.

We performed an additional optimisation for Mieming to test the generality of the leaf exchange parameters obtained with Hyytiälä data. In this optimisation we included the set of leaf exchange parameters that was included in the July 2015 Hyytiälä optimisation (see Table ) in the state that we optimise. We find relatively strong adjustments in some of those parameters, the strongest relative changes occur for ε0 (maximum initial quantum use efficiency, +42 %) and ad (description in Table , +33 %). However, the optimisation results in a χ2 value that is only slightly lower than the one for the posterior model with Hyytiälä parameters (0.17 vs 0.20), described earlier. Re-optimising therefore seems to provide limited benefit for simulating the observations well. The prediction of LRUcan generally only slightly changes in this optimisation (Fig. ), with largest changes in the early morning.

For Hyytiälä, we have optimised a set of parameters making use of not less than 26 different observation streams. Because of this, we obtain model parameters that are consistent with a large number of observations, instead of over-focusing on one observation stream. This more holistic approach of modelling achieves a good fit with observations, despite the simplifications present in the model. Major simplifications are the exchange between canopy layers and the exchange between the canopy and the overlying mixed layer. The model uses exchange coefficients, thereby assuming that the local gradients drive the turbulent exchange. This neglects the influence of larger scale turbulent eddies see e.g.. Sweep and ejection events are important for canopy flow and exchange see e.g., e.g. CO2 and H2O can be ejected from the understory into the atmosphere above by ejection events . We tried to smooth out the influence of these processes (acting on short timescales), by using averaged observations.

The above-mentioned simplifications are a likely reason why the specific humidity was not fitted well (Table ) near the bottom of the canopy. The change in LRUleaf due to vertical within-canopy vapour pressure differences (Fig. ) might consequently be overestimated. However, the exact values of the changes in LRUleaf are not relevant for our analysis that served to better understand variability in LRU, and to identify variables relevant for inclusion in the LRUcan parameterisation.

We have tested the effect of our choice to set advection to zero. For Hyytiälä, we performed an additional optimisation in which we included advection of COS, CO2, H2O and heat in the state that we optimise. We found only a small change in the resulting cost function. Even without advection, the model already has quite some capabilities for fitting temperature observations etc., for example by adjusting the free tropospheric lapse rates. To disentangle the effects of advection and free-tropospheric entrainment, more specific observations such as vertical soundings might be useful, but this was not the focus of this study see also Sect. 9.6 of.

A close inspection of the A–gs model equations (Sect. S1.8.1) reveals that the internal CO2 concentration ([CO2]int) does not directly respond to photosynthesis and thus PAR. From a physiological point of view, [CO2]int is linked to PAR, as PAR supplies the energy and reduction equivalents for the carboxylation process, and thus influences the “photosynthetic demand” for CO2, which reflects in the internal CO2 concentration. Under sufficient amounts of radiation, this should be of limited importance and references therein. Under low radiation conditions, this might be more important, and thus it would be interesting for future studies to investigate how the modelled LRU would change when using a different photosynthesis model that represents the joint effects of diffusional supply of and photosynthetic demand for CO2 on [CO2]int.

The measurement tower in Mieming is located on a gently sloping plateau with mountains nearby, complicating the thermodynamics and flow situation. We left out the observations of the CO2 mole fraction at 2 m height, as they showed a very different time evolution compared to those at 20 m height. Such contrasting evolutions are very difficult to reproduce with our 1d model, when using realistic parameter values. Despite the simplification of the in-canopy physical processes in our model, the COS and CO2 mixing ratios at 20 m and above-canopy fluxes (important for LRUcan) are reproduced relatively well (Fig. ). We therefore believe that our relatively simple model adequately represents most key processes relevant for the combined daytime exchange of COS, CO2, and H2O between the canopy and the atmosphere.

Using the optimised leaf exchange (A–gs model and αgiCOS) parameters we obtained for July, we fitted the observations for August and September in Hyytiälä, and the observations in Mieming, to a satisfactory level. This suggests a level of general applicability of these parameters.

As described in Sect. , we performed an additional optimisation for Mieming in which we included (as state parameters) the set of leaf exchange parameters that was included in the July 2015 Hyytiälä optimisation. We found strong changes in the parameters. However, given the correlations present between the A–gs-model parameters (derived from an ensemble of optimisations, Fig. , Sect. ), it is however difficult to precisely determine the values of these parameters, it is likely that parameter equifinality causes multiple sets of parameters to perform comparably. Re-optimising seems to provide limited benefit for simulating the observations well, as indicated by the small difference in χ2 between the optimisations with and without the leaf exchange parameters in the state. This supports the applicability of the A–gs-model parameters obtained for Hyytiälä to Mieming, and the applicability is likely to hold for more needleleaf forest locations.

The problem of parameter equifinality could be tackled by reducing the complexity of the A–gs model, thereby reducing the parameter count. This would ideally lead to a more uniquely defined A–gs parameter set that is still transferable to multiple locations. used an optimality model that requires fewer parameters than our A–gs model, which can be expected to lead to a more robust parameter set.

The linear regression model for LRUcan approximates the physical model very well, both for Hyytiälä and Mieming. The uncertainty in the assimilated observations, and in the physical model, also leads to uncertainty in the LRU values predicted by the physical model. The fit of the regression (and physical) model with Hyytiälä observations is acceptable, although there is a positive bias and the variation is somewhat underestimated. Note that we do not directly optimise LRUcan, but we assimilate the ecosystem CO2 and COS fluxes and molar fractions above canopy, which directly or indirectly link to LRUcan (Eq. ). Given the complexity of this variable (Eq. ), some extent of mismatch between model and observations seems logical. As is clear from the error bars (Fig. ), the differences in LRUcan between the 8 d over which we average are substantial, complicating the comparison with physical and regression models. Analysing the error bars of the 4 LRUcan-related components in Fig. c-f for Hyytiälä, makes clear that the COS flux causes an important part of the variation.

Our new regression model is based on canopy absorbed PAR and VPD of top sunlit leaves. A theoretical analysis by also indicated a dependence of LRU on PAR and humidity. The shapes of our Hyytiälä LRUleaf canopy profiles (Fig. ) correspond well to the shape of the LRU profiles for a hypothetical canopy in Fig. 8 of .

LRUcan is a canopy integrated quantity, and we have seen (Sect. ) that strong within-canopy variations occur. Theoretically, LRUcan should be larger than the LRUleaf of sunlit top leaves, as also shaded leaves, which have a higher LRUleaf (Fig. ), contribute to LRUcan. The difference between LRUcan and LRUleaf of sunlit top leaves should increase when the contribution of shaded leaves to the total COS and CO2 exchange of the canopy increases. However, as is clear from our model results in Figs.  and , the LRU of sunlit leaves is not very different from the total LRU of the canopy, especially when omitting the (early) morning and evening. This is related to the limited contribution of shaded leaves to the COS and CO2 exchange (Fig.  for Hyytiälä). The similarity between LRUcan and LRUleaf of sunlit top leaves is supporting the use of canopy COS fluxes to estimate canopy CO2 uptake, as it suggests that leaf-scale measurements of LRU on sunlit leaves could be used to estimate LRUcan. This also likely explains why the parameterisation used in , derived in , performs remarkably well for Hyytiälä, given that it was derived using (top of canopy, and thus (mostly) sunlit) leaf-scale observations from this site. For locations like Hyytiälä, leaf-scale measurements can be used for upscaling, provided these reflect what sunlit leaves are doing. It however remains to be seen whether this also works for tall canopies with large LAI (and consequently likely a larger shaded leaf area fraction), e.g. tropical rain forests.

Figure  shows that the Lai24 parameterisation has a stronger variation between midday and evening/morning, especially in September (compared to our regression model). The observed large LRU values in early morning/late evening are (probably at least to a large extent) caused by open stomata while PAR is low. This leads to the CO2 flux becoming close to zero (as the CO2 uptake is strongly light dependent, given that PAR supplies the energy and reduction equivalents for the carboxylation process), while COS uptake continues as a diffusion process as long as stomata are open (destruction by carbonic anhydrase maintains a leaf–air gradient). The parameterisation from Lai24 seems to better capture this effect. As mentioned in Sect. , in the A–gs photosynthesis model that is part of our modelling framework, the internal CO2 concentration does not directly respond to photosynthesis and thus PAR. This “weakness” of A–gs might be a reason why our model (and thus also our LRUcan parameterisation) does not capture the above-mentioned low-PAR effect well. However, one should realise that very low PAR conditions only occur shortly during our simulation period, and CO2 uptake is very low in the early morning/evening. Therefore, this can be expected to be of limited importance when estimating CO2 uptake using LRU. We also have to keep in mind that the Lai24 parameterisation was obtained by directly fitting LRU measurements of sunlit leaves in Hyytiälä. We derived our LRUcan parameterisation by fitting components of LRUcan (which has a non-linear dependence on some of the components). This provides a potential reason why the Lai24 parameterisation provides better results for the specific location the parameterisation was derived for (taking into account that the LRU of sunlit leaves is not very different from LRUcan, as discussed above).

However, for Mieming the Lai24 parameterisation clearly underestimates the leaf-scale observations, and our parameterisation outperforms Lai24 at this location, based on limited observations and model output. One could hypothesise that the difference in performance between our and the Lai24 parameterisation could be (partly) related to the inclusion of VPD in our parameterisation. In Mieming the VPD can be expected to be higher, due to climate differences, including more intense solar radiation. In our posterior July optimisations, the modelled VPD of sunlit top leaves is about 40 % higher in Mieming as in Hyytiälä at 14:00 LT. Compared to Hyytiälä, the modelled larger VPD in Mieming reduces LRUcan as predicted by our linear regression model by about 0.29. Consequently, the higher VPD in Mieming cannot be the explanation for the lower LRUcan for Mieming in Lai24 compared to our regression model, as the inclusion of VPD in our regression model reduces the difference between both parameterisations. Within the modelled period, the difference in magnitude of LRUcan between Lai24 and our physical model is mostly small in the early morning and evening, the difference is much larger later in the morning and during midday, when incoming PAR is higher. This suggests that the response to PAR in Lai24 is too strong for Mieming. It also has to be noted that for the July observations in Mieming, PAR reaches values above 2000 µmolm-2s-1 around noon. This is outside the range of values used in to derive the Lai24 parameterisation at the Hyytiälä site (see their Supplement Fig. S6).

applied the Lai24 parameterisation globally, obtaining a GPP of 157 (± 8.5) PgCyr-1. However, for July and August in Mieming, the (derived) leaf-scale LRUleaf observations are on average about 75 % higher than the Lai24 parameterisation results (Fig. ). Applying a 75 % higher LRUcan to COS fluxes (Eq. ) translates in a 75 % smaller canopy net photosynthesis flux (≈ GPP) around midday at this location. Given that our model results and theory, see indicate that LRUcan is still somewhat higher than the LRUleaf of sunlit top leaves (Figs.  and ), the underestimation of LRUcan by Lai24 will be even larger. We therefore conjecture that the global GPP estimate provided by is very uncertain, as similar biases could be present in other ecosystems as well. This can be especially the case when vegetation structure, vegetation properties and environmental conditions are very different compared to Hyytiälä.

Our developed parameterisation, although not performing equally well as Lai24 in Hyytiälä, shows better transferability to Mieming, based on our model results and limited observational data. For the same reasons as mentioned above, one should be careful in applying our parameterisation to other locations, especially to ecosystems other than needleleaf forests. Given that the Lai24 parameterisation seems to perform better at Hyytiälä, while our parameterisation seems to perform better at Mieming, it is likely that different parameterisations should be used for different locations to reach the best accuracy. Furthermore, complete transferability of LRUcan parameterisations between vastly different biomes might not be achievable, as the responses to PAR and VPD might differ, and (additional) vegetation-specific physiological/biogeochemical drivers of COS to CO2 relative uptake might exist. However, the transferability between the Hyytiälä and Mieming locations suggests that at least within similar biomes, reasonable results can be obtained with a single parameterisation. Our inverse modelling framework is well-suited to improve knowledge on the relative uptake of COS and CO2 for different ecosystems. However, extensive measurement data, including COS and CO2 fluxes, need to be available, and currently these datasets are sparse.