At the ecosystem level, we define Net Ecosystem Exchange (NEE) as the simple difference between Gross Primary Production (GPP) and Total Ecosystem Respiration (TER). These terms are measured as the flux density µmolCO2m-2s-1. The sign of NEE indicates the direction of the flux. When NEE is negative, GPP exceeds TER and the flux is a local sink of CO2. If NEE is positive, TER exceeds GPP and the ecosystem is a local source. NEE represents the net carbon exchange of the vegetation and does not include disturbance fluxes, fires, or out-gassing from freshwater ecosystems.

The core ecosystem-level meteorological and NEE data was collected between 2001 and 2020 at 294 globally distributed eddy-covariance (EC) towers (see appendix for a full list of sites). These data were then processed using the ONEFLUX processing pipeline . Variables from the eddy-covariance sites are air temperature, vapor pressure deficit, incoming shortwave radiation, the computed potential shortwave incoming radiation, and the computed wind speed.

During training, at pixels co-located with eddy-covariance towers we use the surface reflectances (MCD43A4, ) and land surface temperature (LST) from the MOderate Resolution Imaging Spectroradiometer (MODIS) collection v006 (http://daac.ornl.gov/MODIS/, last access: 11 January 2024). The surface reflectances are used to compute two vegetation indices, the enhanced vegetation index (EVI) , and the spectral reflectance of vegetation in the near-infrared (NIRv) . A water index, the normalized difference water index (NDWI) with MODIS band 7 was also included. MODIS data has a spatial resolution of 0.05°, and a 1 d temporal resolution. The full set of variables is described in Table .

When producing our atmospheric constraint, and when estimating global NEE, the drivers for our bottom-up model are the same set of remotely-sensed and eddy-covariance quantities, extracted from remotely-sensed and meteorological reanalysis data. The RS data are derived from the 0.05° global MODIS product (MCD43C4, ), and the global meteorological variables are derived from the European Center for Medium-Range Weather Forecasting (ERA5) atmospheric reanalysis data (https://www.ecmwf.int/en/forecasts/dataset/ecmwf-reanalysis-v5, last access: 11 January 2024). ERA5 is provided at 0.25° spatial resolution, and hourly temporal resolution . Air temperature at 2 m height, incoming shortwave radiation, and vapor pressure deficit (computed from air temperature, relative humidity, and surface pressure) are used to create the global meteorological data.

All data are accessed through the FLUXCOM-X code base using the preprocessing, and gap-filling from and the procedures of FluxnetEO data version 2 , and the quality flagging described in . Please see for a full description of the FLUXCOM-X data and processing environment.

The model in this study uses observations of atmospheric CO2 mole fraction from tall-tower observatories measured at three sites: ATTO , Hegyhatsal and Zottino . These sites are chosen to include one tropical, one extra-tropical and one boreal domain. The mole fraction data used for the Amazon Tall Tower Observatory in Brazil (ATTO), collected at 79 m are the same used and fully described in . Mole fraction data for the Hegyhátsál tower in Hungary (HUN), collected at 115 m are provided by the Integrated Carbon Observation System (ICOS). The observations for the Zotino Tall Tower Observatory in Siberia (ZOTTO), collected at 301 m are provided by . The data for all three towers was provided hourly, for one calendar year (2019 for ZOTTO and Hegyhátsál, 2015 for ATTO). Only daytime observations from 13:00–17:00 (local time) were used. The selected times represent a daytime planetary boundary layer (PBL) with assumed well-mixed convective conditions .

For global and regional comparison we use the land flux corrected for fossil fuel and cement production from the ensemble of N=14 atmospheric inversions from GCB23 (10.18160/4M52-VCRU, , last access: 4 March 2024). The atmospheric inversions are provided as monthly data with a 1° spatial resolution. See for a full description of the ensemble members. These fluxes are adjusted for fire by removing the fire emissions from Global Fire Emissions Database, Version 4.1 (GFEDv4) (last access: 13 February 2024) .

For comparison with the current state-of-the-art bottom-up NEE model, we use the global NEE product from the latest version of FLUXCOM X-BASE (, last access: January 2025). The FLUXCOM X-BASE NEE model (hereafter X-BASE) is trained using the same data processing and data quality flagging, including gap-filling. X-BASE uses a different data-driven algorithm, XGBoost , instead of a neural network. Comparison with X-BASE allows us to understand the performance of our system relative to a highly optimized bottom-up model. So, despite differences in structure, we consider X-BASE to be the most appropriate bottom-up model for comparison.

For regional analysis we use the set of 11 large regions from the TransCom 3 intercomparison project (TransCom) project . See Fig.  for the coverage and names of the 11 land regions used in this study. For analysis of model function by climate zone, we use the Köppen-Geiger (KG) classification system using maps from , for the years 1991–2020. The Köppen-Geiger classification groups the land-surface into regions by their annual climate (Fig. ).

The ecosystem-level model takes as input observations of meteorological drivers (from eddy-covariance or reanalysis data) and remotely-sensed drivers and predicts NEE in µmolCO2m-2s-1 (Fig. , red lines, orange lines) at an hourly tempo. As a machine-learning system, the ecosystem-level model can be described as a feed-forward neural network trained using standard backpropagation techniques . The network is a set of fully-connected layers which consist of nodes or “neurons”. The neurons are exposed to the output of all neurons in the previous layer. Between each layer of neurons, the output is passed through a non-linear activation function. Additionally, the output of the activation function is passed through a normalization function, which scales the output to a mean of 0 and a standard deviation of 1. Our network is a set of three fully-connected layers with the Rectified Linear Unit (ReLU) activation function and batch normalization. Each layer in a neural network is a complex, non-linear embedding of the previous inputs into a new latent space. This series of embedding functions is the learning space of the network. These latent spaces allow the model to learn complex non-linear relationships between driver variables and NEE. Because these relationships are discovered in these abstract non-geographic spaces, they can represent complex space-for-time and time-for-space substitutions. We refer to the process of learning these non-linear, non-geographic mappings between the input distribution and NEE as learning in “environmental response” space.

The computational link from this ecosystem-level model to the atmospheric constraint (Fig. , purple lines) is built around the STILT atmospheric transport model . STILT is a Lagrangian particle dispersion model (LPDM) which uses meteorological forcing data to compute the transport of sources and sinks to the time and location of an observation. The STILT footprint represents this atmospheric transport as a set of sensitivities of the final observation to fluxes at different times and locations. This footprint can be multiplied with a set of local estimates of NEE, which transports them and converts them into a simulated mole fraction. EC-STILT uses a set of pre-computed STILT footprints at the three tall tower locations, for midday observations, for one year (Fig. ). For each tower, the STILT model is run for each selected observation over the year, producing footprints covering the tower domain hourly for the 10 d prior to the observation time. STILT domains (Fig. ) are selected to capture the relevant near field over the time of analysis based on regional meteorological conditions .

For the atmospheric constraint, we must isolate the biogenic contribution (ΔPPMNEE) of an hourly observation of CO2 (PPMObs). Conceptually, the biogenic contribution is the residual of the observation and the non-biogenic terms: the lateral boundary condition (LBC), non-biogenic fluxes (NBF) and Ocean contributions. This is presented in Eq. (), with spatially explicit terms on the right. 1PPMObs-PPMLBC-ΔPPMOcean=ΔPPMNEE+ΔPPMNBF The NBF contribution (ΔPPMNBF) consists of fossil fuel fluxes and fire emissions. For fossil fuel emissions, we use fossil fuel/energy signal across sectors, using GridFED (version 2022.2, last access: 15 February 2024) , provided at a 0.1° resolution. Fire emissions are provided by the Global Fire Emissions Database, Version 4.1 (GFEDv4) (last access: 13 February 2024) provided at a 0.25°. For the GFEDv4 data, temporal profiles were used to add monthly, daily, and diurnal cycles to the fire signal. The GridFED data are interpolated to the 0.25° spatial resolution of the analysis. We do not include lateral transport fluxes in the current analysis. Crop and wood harvest are important for regional and long term accounting , but are not critical for the instantaneous carbon budget that is represented in Eq. (). The potential impact of riverine transport are discussed below in Sect. .

The lateral boundary condition of the region (PPMLBC) is precomputed using The Jena CarboScope (s04ocv4.3) , following . Optimized results from the CarbonTracker Data Assimilation Shell (CTDAS) (CTE2020) are used for the ocean flux and its conversion to mole fractions (ΔPPMOcean) with STILT. The LBC values are calculated for each tall tower for each observation time. We used the ensemble of STILT trajectories released at each data point to obtain the mean ending position (lat, lon) as well as a mean ending height above the ground. Using this information we sample global 3D fields, in this case the optimized CO2 mole fractions from CarboScope, and obtain a mole fraction associated with a measurement at the tower. We acknowledge that this method is sensible to biases in the global 3D fields, but for example at ATTO a bias-corrected version of the LBC yielded very similar results to the CarboScope LBC, please see for the full discussion.

During training, EC-STILT generates an objective function with two terms, or losses: the ecosystem loss term LEC, and the atmospheric loss term LATM. To generate the ecosystem loss term LEC from eddy-covariance, the EC-model F() is run using a batch of data randomly selected by time and site, xbatch collected at eddy-covariance tower sites (Eq. , along with remotely sensed (RS) data at pixels that are co-located with the eddy-covariance tower, producing an estimate of NEE at the ecosystem level (Eq. ). The ecosystem loss term is computed as the mean squared error (MSE) between this estimate and the observed NEE from the eddy-covariance observations (Eq. ). 2NEE^=F(xbatch)3LEC=MSE(NEEobs,NEEtower^)

To generate the atmospheric loss term LATM for each training step, an hourly daytime (13:00 to 17:00 local time) observation of the mole fraction of CO2 for each tower is selected. For each observation, a pre-computed STILT footprint, along with associated NBF, LBC and ocean data are retrieved. The observed mole fraction, LBC and ocean contributions are used to create the left-hand side of Eq. (). The ocean term represents the flux from any pixels under the footprint, transported using the footprint into concentration enhancements at the tower. To create the right-hand side of Eq. (), which is the change in CO2 mole fractions attributable to fluxes within the footprint (ΔPPMfoot). The ecosystem-level model is then run for each non-zero location in the STILT footprint, producing an estimate of local NEE (Fig. , NEEland). The NEE inferences and NBF values are transported with the footprint into concentration enhancements at the tower. This produces the two terms ΔPPMNEE, and ΔPPMNBF which are then added element-wise. These sums represent the total simulated change in PPM at the tower. The loss term Ltower,ATM is the squared difference between the simulated change in concentration ΔPPM^foot, and the observed change in concentration ΔPPMfoot. For a single training step, a loss is calculated for one observation for every tower. These three tower losses are averaged, producing the final atmospheric loss term (Eq. ). 4Ltower,ATM=(ΔPPMfoot-ΔPPM^foot)25LATM=∑tNum.TowersLt,ATMNum.Towers The two terms of the objective function, LEC and LATM are combined using a learned weighting scheme . This allows the model to learn the appropriate relative weight to assign each loss term during training, based on their internally estimated uncertainty. To achieve this, the model has a set of additional parameters which are used to learn an estimate of the homoscedastic uncertainty for each term of the objective function. This uncertainty is dependent on the inherent noise in the training data, rather than the quality or quantity of training data. For EC-STILT these mixing parameters, σEC2 and σATM2, are added to the processing chain of the model after its initialization. During training, using the normal backpropagation process that uses the chain rule to attribute and update the free parameters of the neural network according to their contribution to the loss value, the model also updates these mixing parameters. For the individual tasks L∈[EC,ATM], the σL2 parameter is used to create two terms; wL (Eq. ), and sL (Eq. ). These are then used to calculate the effective loss, balanced by the learned uncertainty of the terms (Eq. ). The evolution of these terms during training is provided in Fig. . After this final weighting calculation, the loss is combined in small batches to stabilize model training during backpropagation. 6wEC=12σEC2,wATM=12σATM27sEC=log⁡σEC2,sATM=log⁡σATM28Ltotal=(wEC×LEC)+(wATM×LATM)+sEC+sATM

EC-STILT is trained using 10-fold cross-validation. The full set of eddy-covariance observations are split by tower location into 10 equal subsets. Ten ensemble members are created, each trained on 8 of the subsets, with one held out for validation and one held out for testing. The ensemble is used to define the flux uncertainty as given in the Results section, as a 1σ of the member spread. The available footprints for each tower were randomized and 80 % are used for training, 20 % for validation on each member. The cross-validation results for the eddy-covariance level results below are the ensemble member results against the 10 % of sites held out for testing.

The STILT footprints were run for a 0.25° grid. All MODIS driver variables and NBF data were translated to this grid. MODIS variables were averaged. Where necessary, NBF fluxes were interpolated using the nearest neighbor method which reduced the overestimation of emissions in pixels surrounding urban hotspots. When run globally, EC-STILT is run at full MODIS 0.05° resolution.

We perform several analyses by KG geophysical region to understand how model performance is modified by the atmospheric information available during training. To discover the regions which contribute to the global IAV in EC-STILT, we use the covariance method described in . This method uses the row sum of the covariance matrix, scaled by the sum of the full covariance matrix to estimate the per-pixel contribution to the IAV, which is in turn additive. We then sum by KG region to determine the relative contribution.

To test our hypothesis that adding an atmospheric constraint improves the representation of the land surface, we estimate the probability distribution functions (PDF) of the model’s training data (such as temperature and VPD) by KG region. We also estimate the PDF of the full land surface, as represented by the full dataset used to create a global multi-year estimate of NEE. For the training data we estimate two different PDFs: one PDF of the data provided from the eddy-covariance towers, which is the training set for X-BASE, and one PDF representing additionally the areas of the land-surface which are under the STILT footprints, which is the training set of our model. The footprint data is weighted by the number of times that time and location is used during training. Global and regional PDFs are generated from a random subsample of the full dataset. The subsamples are the per-year time series for ten percent of randomly selected spatial locations. To quantify the relationship between the training PDFs and the full PDF, we use two metrics that describe the distance between two distributions; Kullback-Leibler and Jennsson-Shannon.

To understand the impact of the STILT data on our results, we try to isolate and test specific potential confounders by training nearly identical models with specific modifications. These models share the same architecture and random state. To test the role of model architecture on model performance, we train an EC-only version of our model which lacks the STILT operator, and so has no additional atmospheric information during training. To test for potential influences from the atmospheric inversion data which we use to calculate the LBC timeseries for each tall tower, we train a model which replaces the hourly LBC with the annual mean LBC for each time step. In this way this mean-LBC model has no time-varying information on the background state of the atmosphere.

EC-STILT produces an estimate of annual global NEE of -3.90±3.63PgCyr-1, compared to the ensemble mean of atmospheric inversions in GCB23, adjusted for fire, -4.29±0.65PgCyr-1 (Fig. B). The RMSE of the EC-STILT annual mean with the inversion ensemble annual mean is 0.69 PgCyr-1, compared with 1.81 PgCyr-1 for X-BASE. The uncertainty of the global annual flux is very high (Fig. A), being driven by uncertainties in tropical NEE (Fig. C).

EC-STILT has learned a relationship between the input variables and NEE, which captures a large part of the natural interannual variability of global annual NEE. When producing an estimate of global NEE, the model takes the driver variables across the full land surface as inputs, but does not access any STILT footprint data, LBC or NBF data, or mole-fraction data, which are only used in constraint during training. When the model is run globally for years 2001–2021, the standard deviation of annual NEE (IAV) of the EC-STILT member mean, 0.69 Pg C yr⁻¹ (1σ), is substantially closer to the GCB23 inversion ensemble IAV of 0.65 PgCyr-1 than that of X-BASE (Fig. C). The R2 of EC-STILT annual mean NEE with the GCB23 inversion ensemble is 0.42 (N=21 years), compared with an R2 of 0.02 for X-BASE. EC-STILT IAV is consistent across the model members despite the large uncertainty in the magnitude of the flux. The mean R2 between the 10 members and GCB23 inversion ensemble is 0.34 (1σ 0.08), and the mean IAV is 0.71 (1σ 0.14) Pg C yr⁻¹.

Spatially, the distribution of the mean annual NEE is largely consistent with X-BASE (Fig. B) with a spatial correlation of 0.71 for global mean NEE. EC-STILT has a stronger Amazonian sink and weaker boreal sink (Fig. C). The boreal reduction is a reduction in the length and intensity of the growing season. The EC-STILT results have removed several hotspots of source, in the Sahel and in the Indian subcontinent. These potentially unrealistic hotspots in X-BASE can be attributed to an incorrect learned relationship for crop-cover PFTs in certain dry conditions . The lack of specific Plant Functional Type (PFT) information in the EC-STILT training data may be responsible for the removal of these strong sources. The large model spread in the tropics (Fig. D) across the Amazon Basin, the Congo Basin, and Oceania is the major source of global model spread in Fig. A.

The atmospheric constraint has a strong impact on the EC-STILT estimate of regional fluxes. Similar to the global results, EC-STILT estimates regional magnitudes of NEE which are largely inconsistent (Fig. , Table ) when compared with the ensemble of atmospheric inversions used in corrected for fire using GFED4.1 fire emissions (Fig. , purple lines). The regional IAV is more in agreement with atmospheric inversions, producing similar or better R2 and magnitude when compared with X-BASE regional NEE (Fig. , inset text, Table ).

Regionally, there appears to be a relationship between the coverage of the eddy-covariance and atmospheric observational networks, and the impact of the atmospheric information (Table ). For example, in Europe (Fig. ), both the atmospheric and eddy-covariance networks are extensive. It appears that in limited cases, the atmospheric information provides some additional information on the magnitude. EC-STILT is closer to the inversion mean with an RMSE of 0.2 PgCyr-1 compared with 0.37 PgCyr-1 for X-BASE. However, the IAV is very strongly conditioned on the information in eddy-covariance data; the IAV for Europe from EC-STILT is still very similar to X-BASE with a R2 of 0.86 compared with X-BASE, and 0.31 compared with the inversion mean (Table ). The spatial pattern of summer NEE during the peak growing season (mean JJA) is very consistent between EC-STILT and X-BASE (Fig. ) with a spatial correlation of 0.86, demonstrating the strong influence eddy-covariance information retains when used in dual-constraint. The reduced sink in the region estimated by EC-STILT (Fig. ), appears in Fig.  to be largely an overall bias, rather than a learned difference in the spatial distribution of NEE.

In regions such as South American Tropical where there is limited eddy-covariance coverage to adequately constrain the model, additional atmospheric information does not improve the modeled NEE, and increases the model uncertainty. EC-STILT has moved away from the inversion mean, compared with X-BASE (RMSE of 3.73 PgCyr-1 compared with 2.05 PgCyr-1). The low R2 for EC-STILT compared with both the inversion mean and X-BASE annual time series (0.10 compared with the inversions, 0.13 compared with X-BASE) indicate that neither the eddy-covariance data nor the atmospheric signal provide sufficient constraint for EC-STILT to find similar IAV to the atmospheric inversion mean, which itself might be wrong.

The increase in the IAV for the South American Temperate region demonstrates the potential for complex effects from the atmospheric constraint. The region is only directly impacted by the ATTO tower observations in unusual meteorological conditions, which is evident in the mean ATTO footprint in Fig. . The eddy-covariance record for the region is sparse, but may have similarities to other, more densely sampled regions. We see a small move away from the inversion mean compared with FLUXCOM X-BASE in the magnitude of NEE (RMSE of 0.74 PgCyr-1 compared with 0.59 PgCyr-1 for X-BASE), but an increase in the magnitude of IAV compared with FLUXCOM X-BASE (0.14 PgCyr-1 compared with 0.07 PgCyr-1). The atmospheric constraint has added new information about this region despite very limited direct information available in training.

The atmospheric constraint also has strong effects in TransCom regions that do not directly contain or overlap the STILT regions used during training. This is because EC-STILT learns its land-surface response in environmental space of the features instead of in geographic space like an inversion. Across the globe, EC-STILT produces consistent changes over regions with similar latitude ranges (Fig. , right column) with a dominance of the global north in observational networks . The magnitude of the northern hemispheric land carbon sink is part of ongoing discrepancies in the Global Carbon Budget .

We find that EC-STILT attributes IAV to tropical drylands in regions Aw (tropical savanah) and Bsh (semi-arid) (Fig. ). This contrasts with the X-BASE NEE, which attributes IAV to regions Dfb (humid continental), and Dfc (sub-arctic). Both figures show the regional contribution of IAV, relative to the overall IAV of the dataset. This metric captures only the magnitude of the contribution, not the relative accuracy of the inferred IAV, which varies strongly by region (Fig. ). Recent studies also suggest that arid regions are the dominant source of the IAV in global NEE. The constraints that drive this result in our system must come from atmospheric CO2 data, and we will try to trace the source of this information in Sect. .

EC-STILT maintains good overall performance for NEE at the eddy-covariance site level, particularly at daily time scale. We use R2 and the Nash Sutcliffe model efficiency (NSE) to evaluate model performance with regard to the observed eddy-covariance observations, and for comparison with X-BASE. In the EC-STILT's 10-fold cross-validation for all sites (Table ), the model achieves a median NSE of 0.66, and median R2 of 0.77 for hourly fluxes (X-BASE: NSE 0.71, R2 0.71), and a median NSE of 0.76 and median R2 of 0.87 for diurnal mean NEE (X-BASE: NSE 0.81, R2 0.81), when comparing the observed and simulated time series. The data used for these performance metrics are the test data, held out from each cross-validation fold, and these tests are performed using the model member which did not see these data in training.

As with regional results, the site-level performance of EC-STILT appears to be strongly influenced by the level of local constraint from eddy-covariance. Figure  shows the hourly and daily averaged time series of NEE for three EC sites directly impacted by the atmospheric domains, one in boreal Russia (RU-Ha1), one in a German mixed forest (DE-Hai), and one Brazilian site in the Amazon basin (BR-Sa1). In the Russian site, EC-STILT (blue line) slightly improves the estimate of the mean daily cycle of the EC observations (red line) compared with X-BASE (RMSE of 0.54 compared with 0.79). When averaged to the daily scale, both EC-STILT and X-BASE produce a similar RMSE compared with the observations. EC-STILT appears to better represent the daily signal than X-BASE with a higher R2 (0.43 compared with 0.18). EC-STILT produces a range in NEE which is slightly larger than the EC record and much greater than FLUXCOM X-BASE (yellow line). Boreal regions are only moderately constrained in EC-STILT by eddy-covariance given the large difference in annual NEE between EC-STILT and X-BASE (Fig. ). This may allow for larger differences at the site level, as the boreal observations do not sufficiently weigh against the atmospheric information.

In the German site, DE-Hai, where the constraint from eddy-covariance is strongest, EC-STILT underestimates the hourly magnitude (Fig. , center left) with a mean RMSE of 0.92 for EC-STILT, 0.22 for X-BASE), but still largely captures the daily NEE, but with a larger bias than X-BASE (Fig. , center right) with an R2 of 0.61 and RMSE of 2.00 compared with an R2 of 0.83 and RMSE of 1.04 for X-BASE. In the northern extra-tropics, as seen above, the marginal impact of the atmospheric information is minimized, and EC-STILT performs most similarly with X-BASE. This mismatch in the site level results is consistent with the slightly weaker regional sink seen in the northern temperate regions.

In the Brazilian site, over the mean daily cycle EC-STILT strongly underestimates the range of NEE (RMSE of 2.52), but infers a daily value when averaged, that is comparable with X-BASE (R2 of 0.47 for EC-STILT and X-BASE). This may indicate that the eddy-covariance and atmospheric constraints are providing confounding information about the hourly flux, which is discussed below.

We analyse in Fig.  the coverage of the EC-only and EC-STILT training data in terms of climate and ecological space by approximating the PDF of the two joint distributions in feature-space (TA/VPD, and EVI/NDWI). We compare the relative difference in the Kullback-Leibler distance (KLD) and Jennsen-Shannon metric (JS) between the PDF of the full, or natural distribution of a random subset of all pixels, either globally or by KG class, and the two training sets (Fig. , Tables –). The two variable pairs were chosen to create easily interpretable visualizations. We use subsets of the full 10-dimensional distribution to save on computational costs.

At the global scale, the distribution of the EC training set is more concentrated in cooler, and moderately productive regions than the EC-STILT set (Fig. , top panel, compare the higher density in the warmer, more productive regions). In the second row, we can see specifically how the training data changes the representation of a region which is not directly observed. As with the global distributions, the EC-STILT distribution covers the warmer, more water-stressed regions (Fig. , middle panel TA/VPD) and in the wetter, more highly vegetated regions (Fig. , middle panel EVI/NDWI).

This distributional approach also explains why the model under-performs in regions that are directly constrained by the atmospheric towers. In the third row, the Köppen-Geiger class Cfb (Temperate Oceanic) which covers most of western Europe, we can see that in both variable sets, the EC-only distribution is closer to the natural distribution (Fig. , bottom row). Because of the environmental learning of EC-STILT, this means that the inclusion of atmospheric towers may reduce the model's skill in this region.

The specific impact of individual towers, and their interactions in feature space, are inconsistent across regions, variable pairs, and metrics. In Tables  to , we expand the analysis by quantifying the effect of adding or removing individual atmospheric towers from training. The optimal distribution, shown in bold, characterized by the minimum distance between the two PDFs, is enriched by adding 1–3 towers in 49 of 52 regions across both distributions (Tables , ). However when the full distribution from all towers are considered (Table ), the improvement of the tower-enriched distributions is more modest, with only 9 of 26 regions (including global) having improved coverage by the towers in both variable pairs.

This study finds improvement in a number of important metrics in a bottom-up NEE model that is additionally constrained by three atmospheric tall-towers over one year. As noted above, neither the eddy-covariance record nor the atmospheric observational record have strongly representative global coverage, and additional atmospheric information over new geographic regions could be a benefit. Therefore an important future development of a dual-constrained model is the inclusion of additional tower and non-tower data streams: Additional tall towers could provide stronger constraint in well-measured regions. Non-tower data could be leveraged to provide information on regions that are poorly covered by the current observational network. The current model could run, as-is, against aircraft observations. With a modification of the STILT transport system in the objective function, the structure of this study would allow a bottom-up model to be constrained by satellite retrievals of total column CO2 (XCO2). XCO2 observations could be selected according to their potential to include biomes of specific interest in their upwind footprints. This would enrich our capacity to model NEE across un- and under-sampled biomes. This extension into XCO2 introduces several possible issues: to model the full column of CO2, multiple STILT footprints must be run for every observation at different elevations , and modeled fluxes transported multiple times. This may be a technical challenge to build effective training systems. The inclusion of the full column CO2 may dilute the available information on the biogenic flux in the observation, reducing the information gain to the bottom-up model.

The performance of EC-STILT in the tropics show the limitations of adding atmospheric information without adequate support in the eddy-covariance record. In the harmonized model described above, the ecosystem-level information appears to provide the “backbone” of the model's performance. Future modeling efforts will benefit from the ongoing efforts to improve and enlarge the corpus of available eddy-covariance sites and data, such as a potential extension of the FLUXNET 2015 dataset . As the eddy-covariance record better describes and constrains key biomes, additional atmospheric constraint may be able to provide improved information on the larger dynamics of regional and global NEE, while reducing the uncertainty.

An important aspect of transport modeling which is not addressed in the current study is the impact of model and meteorological uncertainty. In our analysis, we treat the output of the STILT model as definitive, relative to the meteorological fields in ECMWF short-term forecast. Computational schemes for addressing model uncertainty, multi-LPDM ensembling, or in-model ensembling for STILT by sampling within the stochastic particles could be attempted, without a major increase in computational and I/O workload. In this study, STILT runs are forced with ECMWF short-term forecast data , and the model is trained using ERA5 reanalysis data. Using identical meteorological data for STILT and the model drivers may also reduce uncertainty.

Conceptually, the current work accounts for two major terms in the non-biogenic flux budget; fires and fossil fuel emissions. In the Amazon and in boreal Russia, fire is the largest and the most relevant term, as the term accounts for both fire, and also for part of the instantaneous flux from land-cover/land-use change associated with biomass burning . In the European domain, fossil fuels are the dominant non-biogenic flux signal. As noted in the Sect. , NEE at the eddy-covariance tower and atmospheric CO2 are not directly comparable. The dual-constraint in this study is created using the residual of an observation and non-biogenic fluxes. Any biogenic or non-biogenic flux terms which are not included, or adequately represented temporally and spatially will bias the atmospheric target that the model is trying to match in training. An example of this would be the non-fire disturbance fluxes, or regrowth after disturbance. Another potentially important term is the instantaneous riverine flux of CO2, coming from lateral riverine transport. These terms could push the relative carbon balance towards CO2 release (disturbances) or towards CO2 assimilation (regrowth), and during retraining the model would attempt to match this new local target balance. As seen above, because of the distributional nature of the model's learning and inference, this new local balance could modify the global NEE response. This reduces the effectiveness of the technique. Any additional spatially explicit NBF data may therefore, improve this comparison, and increase the fidelity of the overall magnitude and may reduce the uncertainty in the global result.