We ran separate CO2 simulations for the two cities, each nested offline within a larger central European domain. The central European domain was simulated at 6.5 km resolution using grid R3B8 (see Fig. ), while the inner domains were run at a resolution of 0.5 km for Zurich (R19B9 grid) and 1 km for Paris (R5B10 grid). Zurich required a higher resolution due to its complex topography and smaller size. The European simulation provided the initial and boundary conditions for the two nested domains and was itself nested into the Copernicus Atmospheric Monitoring Service (CAMS) global inversion-optimized CO2 simulation v24r3, which is based on assimilation of satellite observations . Meteorological initial and boundary conditions for the European simulation were taken from the ERA5 global reanalysis of the European Centre for Medium Range Weather Forecasts (ECWMF) . The intermediate European simulation was necessary because of the significant spatial resolution difference between the CAMS and ERA5 global products and the high-resolution simulations for the two cities. To keep the simulated meteorology close to the analyzed meteorology, the ICON-ART meteorological fields were weakly nudged towards the ERA5 data as described in .

The same parameters were also used in the study of , which compared four different biospheric flux models applied over the city of Zurich with each other and against observations of respiration fluxes, leaf area density and sap flow in urban parks. This limited and partially indirect evaluation (e.g. sap flow used as a proxy of GPP) showed VPRM to perform equally well as other, more complex biospheric models. A more extensive evaluation of different variants of VPRM including urban VPRM against measurements in Zurich and Munich is currently ongoing. It is clear that further developments are needed to improve biospheric flux models for urban areas, since they have mostly been developed for, and tuned to, natural environments.

All simulations were run with 60 vertical layers. The time step was 50 s in the European domain, 10 s for Paris, and 5 s for Zurich. Model output was saved hourly. The Tiedtke–Bechtold convection scheme (inwp_convection = 1) was used throughout, but in the city domains, only shallow convection was enabled (shallowconv_only = .TRUE.), while in the European domain both shallow and deep convection were active. The configuration for the city domains closely followed the setup of the Swiss weather service for its operational forecasts at 1 km resolution.

Assimilation of CO2 observations was only performed in the two nested domains, not in the European domain. Any potential CO2 mole fraction biases in the European run, which served as boundary conditions for the regional domains, were corrected for by the inversion as described later. To estimate the magnitude of these biases, the simulated mole fractions were compared against measurements from the European Integrated Carbon Observation System (ICOS) . The results will be presented in Sect. .

Different gridded emission inventories were used depending on the simulation domain. For the European domain, the TNO-GHGco inventory for the year 2021 at about 5 km resolution was used. This inventory relied on the 2023 official reporting of 2021 emissions from the AVENGERS project, except for shipping emissions, which were sourced from the 2021 TNO-GHGco inventory used in the CoCO2 project, reflecting an earlier reporting version for that sector. For the Zurich domain, three inventories were combined: the TNO-GHGco inventory was used for regions outside Switzerland; within Switzerland, the 2020 Swiss national inventory at 100 m resolution was applied; and for the city of Zurich specifically, this was replaced by a more detailed 2020 inventory as described in . The city inventory, originally provided as point, line, and area sources, was rasterized onto the ICON model grid before merging with the other inventories (see Fig. a). The three inventories were mapped to the ICON grid and merged into a single dataset using the Python package emiproc . For the Paris domain, the TNO-GHGco inventory was merged with a 500 m resolution inventory for the Île-de-France (see Fig. b) provided by the regional air quality agency AIRPARIF for 2022.

The temporal emission profiles were not always available, necessitating the use of generic time profiles in some cases. To account for these limitations and avoid introducing biases into the optimization, relatively large uncertainties were prescribed for all anthropogenic and biospheric sources, allowing the system to adjust the fluxes based on observations. At the same time, spatial correlations were introduced in the prior error covariance matrices to limit the degrees of freedom and prevent overfitting.

Biospheric fluxes were computed online using the Vegetation Photosynthesis and Respiration Model VPRM, integrated into ICON-ART. As input, it requires shortwave radiation and two-meter temperature, which were provided by ICON, as well as satellite measurements of two indices, the enhanced vegetation index (EVI) and the land surface water index (LSWI) obtained from the MODIS instrument . These indices describe the influence of phenology and water stress on photosynthetic uptake of CO2 by vegetation. VPRM independently simulates gross photosynthetic production (GPP) and ecosystem respiration (RE) for seven vegetation classes. The net ecosystem exchange (NEE) is then the difference RE–GPP. The vegetation class-specific parameters of VPRM were taken from Table F1 of , obtained through an optimization procedure that compared VPRM simulated fluxes with Eddy covariance flux tower observations in Europe.

Another important input for calculating biospheric fluxes is land cover, which provides information about the spatial distribution and type of vegetation in the model domain. We used the CORINE 2018 land cover dataset from the European Environment Agency's Copernicus Land Monitoring Service at 100 m resolution . However, CORINE fails to resolve small vegetation patches or individual trees in urban areas; thus, a high-resolution vegetation dataset was created for Zurich as described in and merged with CORINE data outside the city. Since no such tailored high-resolution product was developed for Paris, vegetation cover in Paris is likely underestimated. A 10 m resolution version of the land cover map for Zurich is presented in the Supplement.

Land cover in ICON-ART is represented using a tile approach, where multiple land cover types are included in a grid cell proportional to their fractional area. We used the default setting of three tiles, i.e., only the three dominating land cover types were represented. In metropolitan areas, these typically consist of one urban land cover (non-vegetated) and two of the seven vegetation classes considered by VPRM, often grassland and deciduous forest.

To estimate CO2 fluxes, we used the Carbon Tracker Data Assimilation Shell (CTDAS), which is a flexible framework that employs an Ensemble Square Root Filter (EnSRF) to optimize a state vector of flux scaling factors and other elements . CTDAS has been coupled with various Eulerian and Lagrangian atmospheric transport models, including ICON-ART . The ensemble approach requires the model to simulate a large ensemble of CO2 tracers, each representing a different perturbation of the state vector elements being optimized.

Our setup closely followed that in . In short, we jointly optimized scaling factors for fluxes and background CO2 mole fractions over individual 1 week assimilation windows. Each week was optimized twice: first with observations from the current week and a second time with observations from the following week. This assumes that CO2 emitted in the current week also influenced CO2 in the next week.

Estimating the fluxes for a full year requires 52 individual assimilation cycles, each cycle consisting of three week-long simulations (except for the first cycle, which simulates only two weeks). A schematic illustrating the workflow and information transfer between weeks and assimilation cycles was presented in . Figure  presents an alternative view illustrating the ensemble of CO2 mole fractions simulated by the system at an arbitrary observation location.

The first two weeks comprise a continuous simulation with a restart after the first week to set different flux and boundary condition scaling factors for the second week (see Fig. a). ICON-ART provides a restart capability, enabling simulations to be paused, checkpointed, and continued from the same meteorological and tracer fields. Simulated mole fractions are written out hourly and interpolated to station locations using inverse distance weighting, with vertical interpolation based on the two nearest levels and horizontal interpolation from the five closest ICON cells. Only daytime values (11:00–16:00 UTC) are then averaged and used for assimilation. Each week is optimized twice using observations from two weeks, except for the first week of the first cycle (see Fig. a) which is optimized only once using the observations from the first two weeks.

The second cycle starts with another simulation for the first week but now using the optimized scaling factors returned by CTDAS to propagate the optimized boundary conditions and fluxes forward in time to serve as optimized CO2 initial conditions for the next week. Only one single CO2 tracer is simulated in this case as seen in the left part of Fig. b. The second week of the second cycle is then restarted from these optimized initial conditions and a new ensemble is generated based on the once optimized scaling factors for week 2 from the previous cycle. Consequently, the spread among ensemble members in week 2 is smaller in the second cycle compared to the first (compare panels a and b). The third week restarts from the checkpoint at the end of the second week, generating a new ensemble with a larger spread. Only observations from the third week are assimilated in this cycle, ensuring no observations are assimilated twice. Finally, panel (c) illustrates the third cycle, where the first two weeks are now fully optimized. This process continues for all subsequent cycles, following the same procedure as outlined for the second cycle.

One design choice for the inversion system involved transferring information between different assimilation cycles. One option would be to use the optimized scaling factors from the previous week as prior values for the next week. Alternatively, one could revert to the original prior values, all set to 1. In the first case, the scaling factors optimized for the previous week are assumed to be a good first guess for the present week. In the second case, the weekly scaling factors are assumed to be independent of each other such that each week should restart from the original prior values (λ=1). We followed the same approach as in , which blends these two extremes: We computed the new priors as a weighted mean with a weight of 1/3 for the original prior and 2/3 for the posterior from the previous cycle. This approach propagates information from the previous cycle while ensuring that, in the absence of new observations, the fluxes gradually relax back to the prior within a few weeks.

Another aspect that could be transferred between optimization cycles is the state vector uncertainty or ensemble spread. However, we decided to keep the uncertainty for a given cycle unaffected by the optimized uncertainties of previous cycles. Transferring the optimized uncertainties to the next week tends to excessively reduce the ensemble spread. Keeping the value unchanged allows the system to remain stable and adapt quickly to sudden changes in flux intensity or background mole fraction errors.

To estimate anthropogenic and biospheric CO2 fluxes separately, the state vector included separate scaling factors for anthropogenic fluxes, gross photosynthetic uptake (GPP), and total ecosystem respiration (RE). Each factor applies to the fluxes within a group of four neighboring grid cells. This grouping reduces the spatial resolution of the inversion but was necessary to save computation time and memory. In addition, eight scaling factors were included to adjust the background CO2 mole fractions in eight inflow regions following the method outlined in . Since an assimilation cycle includes two time windows, each scaling factor is required twice: once for each of the two weeks. For Zurich, the state vector size s is 2×(2926×3+8)=17572. For Paris, which has a larger domain with 7680 regions, the size was 2×(7680×3+8)=46126 elements.

The state vector elements are scaling factors x=(λ1,λ2,…,λs), so the initial prior state vector xb simply consisted of ones, xb=(1,1,…,1). As mentioned before, from week 3 onwards (see Fig. ), the prior values were defined by a weighted mean between the initial prior and the posterior λi values from the previous week.

Treating GPP and RE as separate components in the state vector is a new approach to urban inverse modeling. Separating these two components is generally difficult, and it is particularly challenging when only daytime observations are assimilated. Nevertheless, the study by suggested that optimizing scaling factors for GPP and RE separately performs better than optimizing scaling factors for NEE only even when daytime observations are used. Their conclusion was based on evaluating six different regional CO2 flux inversion approaches in synthetic model experiments. Similar conclusions were drawn also by . By estimating GPP and RE separately, we provide the inversion more flexibility to adjust the biogenic fluxes. Optimizing only NEE would have required applying the same scaling factors to GPP and RE, which would preserve any prior errors in the relative magnitudes of the two components. We will present separate results for GPP and RE, but we will also discuss the degree to which they can be separated by analyzing their correlations in the posterior covariances.

In order to optimize fluxes of CO2, the following Bayesian cost function J of observation-model residuals and differences between prior and posterior fluxes is minimized : J(x)=12(x-xb)TP-1(x-xb)1+12(Hx-yo)TR-1(Hx-yo), where yo denotes the observed mole fractions, H the observation operator (implemented via the ICON-ART model sampled at station locations), x the state vector (flux and background scaling factors), xb the “background” (or prior) state vector, Hx the simulated mole fraction, P the prior error covariance matrix, and R the observation error covariance matrix. The optimal posterior solution is 2xa=xb+PHTHPHT+R-1yo-Hxb.

CTDAS implements an Ensemble Square Root Filter EnSRF,, or an Ensemble Square Root Smoother if more than one window per cycle is used, to solve this problem. A detailed description of the theory and implementation of EnSRFs for inverse modeling is provided by .

In the EnSRF approach, the matrix P of dimension [s×s] is approximated using an ensemble of N state vectors. Let X denote the ensemble of state vector anomalies of size [s×N], representing deviations from the ensemble mean (e.g., the prior has as mean simply 1). Then P is estimated as 3P=ZZT≈1N-1ZG︸XGTZT︸XT, where Z is the lower Cholesky decomposition, i.e., one possible definition of the matrix square root, and G is a Gaussian random matrix of reduced size [s×N] drawn from a standard normal distribution (and GGT/(N-1)≈I). This provides a low-rank approximation of the full covariance matrix, such that the problem becomes tractable for high-dimensional inverse problems, as we can now re-write Eq. () as 4xa≈xb+XYTN-1YYTN-1+R-1yo-Hxb, where Y=HX, denotes the ensemble of N modeled mixing ratio deviations from the ensemble mean sampled at the observation locations and times. All these mixing ratios were simulated as separate CO2 tracers within the same forward ICON-ART run. For our inversions, we chose N=186, following .

Uncertainties in the differences between modeled and observed mole fractions are described by the observation error covariance matrix R. It accounts for measurement, transport, and representation errors. Representation errors occur when a model with limited spatial resolution cannot resolve measurements at a single point. In most cases, this matrix is considered to be diagonal, implying statistical independence of model–observation differences. This assumption may be less valid for dense urban networks where stations lie within a few kilometers. While more complex formulations including off-diagonal terms have been proposed e.g.,, short sensitivity experiments with such configurations in our system showed only minor effects on flux estimates. Therefore, due to the high computational cost of full EnSRF inversions, we used a diagonal R matrix (see Eq. ) and assimilated observations serially.

To estimate the diagonal elements, we first calculated centered (i.e., bias-corrected) weekly average root mean square errors (RMSEs) between modeled and observed mole fractions at each station (Fig. ) and then applied a centered 5 week moving average. The RMSEs were generally larger in Zurich than in Paris because most of the stations were mid-cost sites measuring above rooftop with high sensitivity to surrounding emissions. The largest RMSEs occurred in the period from November to March and the lowest in late summer. Additionally, elevated RMSEs were observed in the first week of April 2023. In Paris, the highest RMSEs were found for the sites gns, cds and jus. This was likely because cds and jus were located within the city, while gns had a data gap during the late summer months, when model-observation differences are typically smaller.

The smoothed weekly uncertainties for each station served as the model-data mismatch (MDM) thus generating the R matrix as follows: 5R=MDM112⋯0⋮⋱⋮0⋯MDMnn2, where n is the number of observations assimilated in a given cycle.

The prior error covariance matrix P describes uncertainties related to prior flux and background scaling factors, as well as their correlations. These correlations effectively reduce the degrees of freedom, which is necessary to avoid overfitting. The observation networks often lack the density to independently constrain all state vector elements. The correlation length defines the spatial scale of independently resolvable structures and how uncertainties aggregate across the domain.

The structure of P was standardized for both inversions to a common block-diagonal format, 6P=fa⋅e-D⊘La000fb⋅e-D⊘Lb000Pbg, where e⋅ is implied to be taken element-wise, and we use definitions

fa=1, fb=0.5: the prior uncertainty for the anthropogenic and biospheric categories, respectively. Factors between 0 and 1 correspond to standard deviations of 0 % and 100 % per flux region,

For Paris, the spatial correlation between flux scaling factors was fixed to length scales of Lij,a=La=20 km and Lij,b=Lb=50 km for anthropogenic and biospheric fluxes, respectively. The greater length scale for biospheric fluxes arises from the assumption that vegetation responds to environmental drivers in a correlated manner across neighboring regions in the model domain.

In Zurich, spatially varying correlation lengths were considered. Shorter correlation lengths were applied within the city, as the dense observation network can resolve flux gradients. Longer correlation lengths were utilized outside the city. The spatial dependence of Lij was defined as a linear function of the average distance between flux regions i and j to the city center dc,ij: 7Lij,a=dc,ijdmaxLa+1-dc,ijdmaxLurban,a, (and analogously defined Lij,b) where La and Lurban,a are the correlation length limits between which we would like Lij,a to vary outside and inside the city, and dmax is the maximum distance between flux regions. Using the mean distance dc,ij was important to preserve the symmetry of the P matrix. As in Paris, we set La=20 km for anthropogenic and Lb=50 km for biospheric fluxes, while we used Lurban,a=5 km for anthropogenic and Lurban,b=8 km for biospheric fluxes.

The prior and posterior flux uncertainties presented in the Sect.  were computed from the full covariance matrices to account for spatial covariances, similar to the approach described in . For each weekly inversion cycle, we calculated: 8U=gTPg, where P is the prior or posterior error covariance matrix and g is a vector of 0 and 1 (or a binary mask) corresponding to the selected region of model grid (e.g., selected grid cells inside Zurich, Paris or Île-de-France region). For seasonal or annual means, weekly uncertainties were aggregated using standard error propagation.

The main purpose of the European simulation was to provide initial and boundary conditions for the two nested domains. Comparisons with measurements in Europe thus provide information on the magnitude and temporal dynamics of errors in these boundary conditions.

Overall, the mean daily afternoon CO2 mole fractions from the European simulation showed very good agreement with measurements from the ICOS network. The list of sites included in this comparison is provided in Fig. S2 in the Supplement. Figure  presents monthly mean mole fractions, model biases, RMSEs, and correlation coefficients averaged across all sites. The annual mean bias was about 1.24 ppm and the RMSE about 4.19 ppm.

The high monthly correlation coefficients (mean value of 0.73) suggest that the model captured most of the day-to-day variability in the observations. However, the error statistics show significant variations between months. For instance, biases in simulated CO2 mole fractions exhibit a clear seasonal pattern. The largest biases, reaching up to 3 ppm, occurred in August and September. This overestimation is likely related to the challenge in accurately modeling biospheric CO2 fluxes, which dominate during the warm season. Photosynthetic uptake and respiration processes introduce uncertainties in the prior fluxes that are difficult to constrain at the European scale.

RMSEs were quite stable over the year with a tendency of higher values during the cold season (October–March). This increase may be associated with uncertainties in residential heating, which is one of the main anthropogenic sources. Another important factor that may lead to larger errors during the cold season is stronger vertical stratification and shallower boundary layers.

The lowest correlation of 0.5 occurred in April 2023. This could be related to transition-season effects, where shifts in both biospheric activity (onset of vegetation activity) and heating demand (end of heating period) introduce additional uncertainty. All correlations were calculated as Pearson correlation coefficient r.

A similar comparison of the prior simulation against measurements was conducted for the two nested simulations using instead the measurement networks in Zurich and Paris (two stations of the Paris network are also ICOS stations). The results are presented in Fig.  (blue bars) in terms of statistics per station rather than per month. To be comparable with the results for the European domain, correlation coefficients were first averaged by month and then averaged over all months. Otherwise, high correlation coefficients would be obtained due to the strong seasonal cycle in CO2, which the model is usually able to capture well. RMSEs and biases of the prior simulation (blue bars) were generally much larger for stations in Zurich than in Paris (see also Table ). This is due to the different network types with mostly rooftop mid-cost sensors in Zurich as opposed to high-precision measurements on tall towers in Paris. The sites in Zurich are mostly located inside the city and at a much lower altitude above the surface than in Paris. As a result, they are more sensitive to emissions from the city and therefore also to errors in these emissions. For the two background sites, Beromünster and Lägern-Hochwacht in the Zurich domain, the model showed very similar performance as for the sites in the Paris domain.

The biases and RMSEs in the Paris domain were of similar magnitude to the values of the comparison with ICOS sites in the European simulation. The correlation coefficients were also comparable for both simulations, with Paris showing only a slight improvement in the priors, indicating that higher resolution did not substantially increase correlation in this case.