Dear Editor and Referees,

We thank both referees for their insightful comments and constructive suggestions. They have contributed to significantly improving the manuscript and its discussion. In the following, we address their comments point-by-point. We use text in italics to repeat the referees’ comments, normal text for our response, and the marked-up text from the manuscript showing the changes applied.

RC1

RC1: The manuscript delves into the pivotal task of independently estimating and verifying regional and national fossil CO2 emissions, employing the Lund University Modular Inversion Algorithm (LUMIA) for assimilating in situ observations over Europe. The study's foundation lies in the assimilation of data from the Integrated Carbon Observation System (ICOS) network, a crucial aspect that warrants attention. However, the paper falls short in clearly articulating the novel contributions it brings to the existing body of inverse modeling studies. The reliance on the ICOS network is evident, but the connection to prior studies and the specific advancements provided by this investigation are not well-defined.

A critical aspect to address is the lack of clarity on why earlier studies might have failed or why they were conceptually, or due to data limitations, unable to address the question posed in the title. The reviewer suggests considering aspects such as bio and oceanic recycling of dispersed 14C, which could have been potential challenges or gaps in previous research. Providing insights into these aspects would enhance the reader's understanding of the study's significance in addressing potential limitations or gaps in existing literature.

We have extended the Introduction (L75-109 and 136-139 of the revised manuscript) and put the objective of our study in place with the existing body of inverse modeling studies focusing on the estimation of both fossil CO₂ emissions and terrestrial CO₂ fluxes using a multi-tracer (CO₂ and Δ¹⁴CO₂) approach. Clearly, the existing body of literature is very limited regarding such inversions on a continental scale, which is the specific objective of our study here. We do not claim that earlier studies have failed or could not address the problem of constraining fossil CO₂ emissions over Europe using Δ¹⁴CO₂ observations. In fact, to our knowledge, there is only one other existing study (Wang et al., 2018a) that addressed this specific problem. Our approach here is based on a very different modeling set-up than that of Wang et al. (2018) (e.g. transport model, resolution, Δ¹⁴CO₂ modeling approach), and hence contributes to the estimation of model uncertainty. Besides, this is the first time LUMIA is used in a multi-tracer approach and this manuscript serves as a model description reference paper for future studies on some of the important questions raised by the referee such as the impact of the terrestrial disequilibrium on the inferred fluxes.

Despite these concerns, the paper effectively demonstrates LUMIA's capabilities in well-sampled regions, showcasing its potential for accurate estimation of fossil CO2 budgets. The challenges faced in regions with low sampling coverage are acknowledged, shedding light on the limitations of the applied methodology in certain contexts.

Furthermore, the study underscores the importance of a reliable prior estimate of terrestrial isotopic disequilibrium, emphasizing the need to minimize uncertainties for robust posterior fossil CO2 flux estimates. This aspect adds valuable insights to the methodology used in estimating and verifying fossil CO2 emissions.

In summary, while the study contributes valuable information regarding fossil CO2 emissions, addressing the critique by explicitly stating the novel results in relation to prior studies, highlighting potential limitations, and discussing alternative explanations, particularly related to the bio and oceanic recycling of dispersed 14C, would significantly strengthen the paper.

As mentioned above we have extended the Introduction to put our study in context to previous studies and explained the novel aspect (LUMIA as a multi-tracer inversion system) of the manuscript. We also highlight the potential limitations and open questions of employing our system for estimating fossil CO₂ emissions. We believe this paper should serve as a model description reference and will address the open questions such as Δ¹⁴CO₂ sampling strategies, terrestrial disequilibrium, and Δ¹⁴CO₂ emissions from nuclear power plants in detail in a follow-up study. In addition to the adjustments made to the introduction in this regard, we updated the discussion to further comment on this regard.

Specific points:

Prior to publication, a number of points need to be addressed, both in terms of content and methodological description. In detail:

Help the reader to make the paper more self-consistent. References to previous LUMIA papers and others (e.g. Chatterjee and Michalak, 2013; Rayner et al., 2019; Scholze et al., 2017) should not be overused as a substitute for a more comprehensive description of the inversion model. See below for specific points.

RC1: L120: What is a shifted delta? This should be defined in a mathematically sound way.

We are not certain what exactly the referee means by ‘shifted delta’ because we do not use this terminology in the manuscript. We explained the meaning of capital delta (Δ) in lines 110 to 113 of the original manuscript. Nevertheless, we have rephrased the explanation of Δ¹⁴CO₂ and clarified the delta notation in L152-153 of the revised manuscript.

RC1: There is further confusion in the transition from the rather traditional formulation of the cost function (6) (line 172) to the use of a space-time covariance matrix B (see also sloppy use of simple or bolded notation of B). Some specific details:

We updated Equation 6 and the subsequent use of the matrix terms B, H, and R to straight bolded notation, consistent with other studies. The use of a space-time covariance matrix is completely standard in the research field, and our paper does not depart from the norm in that respect (see e.g. Broquet et al., (2011), Monteil & Scholze, (2021), Munassar et al., (2023)).

RC1: L192-195: Confusing description of eqn (7): Adding a matrix TTXTH to a vector Fc0? Please rephrase and give more explanatory details. As it stands, this is of little use. Define T and X more precisely!

There is indeed a small error:  with shape (, ) is a matrix and not a vector. We corrected this in L236 of the revised manuscript. The remainder of the text is correct.

RC1: L202: Why spatio-temporal for the diagonal matrix elements? This is just autocorrelation. Spatial correlations between two different locations are off-diagonal elements. Is this a block diagonal construction?

The sentence refers to the spatiotemporal pattern of the uncertainties (i.e. the spatiotemporal distribution of the variances), not of their correlations. The text seems clear enough to us (and to the second referee), so we chose not to modify it.

RC1: L203: Explain the temporal formulation together with the related segmentation of the vectors x / F (see also eq 6) for the cause of confusion, where x is used in a traditional space phase vector.

We believe that what the referee asks is exactly what is provided in L211-216 of the original manuscript. Lines 202-210 describe in a generic way the process used to achieve a target's overall uncertainty, independently of what they are based upon.

RC1: L207: Is the scaling controlled by some a posteriori technique? For systematic approaches see Talagrand, O. 1997 Assimilation of observations; an introduction. J. Meteorol. Soc. Japan, 75; Desroziers et al, 2005, QJRMS, Diagnosis of observation, background and analysis-error statistics in observation space.

No, the scaling is set (prescribed) by us. The target uncertainties are given in Table 2 (as indicated in line 223 of the original manuscript).

RC1: L214: This optimisation time step of one week should be explained in more detail above, along with the optimisation concept. Again, the implications for the construction of the state vector x are not sufficiently addressed to be understood.

The optimization time step (now replaced in the text by ‘optimization interval’ to not be confused with the model time step) refers to the temporal resolution of the flux adjustments: the state vector contains weekly offsets to the prior emissions, whereas the prior emissions are hourly: the emissions within an optimization interval will be adjusted by the same value. This is already described in Section 2.3.1 (L185-191 of the original manuscript). There is no implication on the construction of the state vector (rather, the opposite: the way the state vector is constructed determines how the B matrix should be constructed).

We would like to point out that the overall approach is very standard in the field, similar to what has been used in other studies, both with the same inversion system (e.g. Monteil et al. (2020), Monteil & Scholze (2021), Munassar et al. (2023)) and with other inversion systems (e.g. (Basu et al., 2016) (TM5), Broquet et al. (2011, 2013) (PYVAR-CHIMERE)), therefore, it does not seem appropriate to re-describe this in details. We describe what makes the specificity of our implementation of that approach, but we think that we do not need to explain the fundamentals of inversions here and refer to other studies for a more fundamental understanding.

RC1: Figure 3: Caption and subscript of the left panel show 1-hour integration backward in time. Is there evidence of hypersonic winds?

The caption indicating a 1-hour sampling integration for CO₂ does not imply the presence of hypersonic winds. Instead, the 1-hour integration time refers to the period over which atmospheric data are collected and synthesized to represent the CO₂ sample at a specific moment. The plume or back trajectory displayed in the maps by the black to orange colors represents the sensitivity of the sampled atmospheric CO₂ to the surface fluxes over the 14 days before the sampling time (or the starting sampling time in the case of Δ¹⁴CO₂). This methodology is standard in atmospheric inversions for capturing the influence of regional fluxes on a sampling site and does not suggest unusual wind speeds.

We updated Figure 3 and removed the text “Integration: 1h” from the left panel to not generate further confusion. We added the text “Continuous samples” to the left panel and “Integrated samples” to the right panel, and updated the caption.

RC1: Lines 266-269: The statement is difficult to reconcile with the displayed extension in the graphics. Please clarify the meaning of 'm(?) 3'.

As explained before, the purpose of the statement and the figure is to show and describe the simulated sensitivity of the two types of samples used in the study for a specific sampling time: continuous for CO₂ and integrated for Δ¹⁴CO₂. ‘m(?) 3’ is a typo and was removed from the text (see L313 of the revised manuscript).

RC1: Lines 282-283: Do you conduct experiments with identical twins instead of OSSEs?

In our study, we performed OSSEs. To clarify this, we modified the text in L330-337 of the revised manuscript. OSSEs are the reference term in the field of study for this kind of sensitivity as shown in other studies such as Basu et al. (2016, 2020), Philip et al. (2019), Wang et al. (2018).

RC1: Please declare sources for the values estimated along external information in L300-301.

The text was modified in L357-365 of the revised manuscript to give more clarity. The same methodology was implemented by Basu et al. (2016), in which they defined the uncertainty for the fossil fuel fluxes as the spread among a series of emission inventories and products in the U.S. (CarbonTracker, VULCAN, and ODIAC). However, we need to point out that, while this is a reasonable value, we intend to demonstrate that our system works and can perform multi-tracer inversions using observations of CO₂ and Δ¹⁴CO₂.

RC1: In L305-309, are there errors of representativity?

In our inversion system, the observation error represents both the instrumental error and the model representation error. In the case of the CO₂ observations, the observation error is mainly composed of the error of representativity, since the instrumental error is very small (in the order of 0.1 ppm), and by calculating the moving standard deviation in a weekly window we calculate an error proportional to how rapidly CO₂ varies (for background sites it will be small while for polluted sites it will be larger). For Δ¹⁴CO₂, the opposite is true. The instrumental error is larger than the error of representativity, therefore, we pick a value of 0.8 ppm in CΔ¹⁴C units (1.91±0.05‰ Δ¹⁴CO₂).

Note: There was confusion generated by the value used as the observation error of Δ¹⁴CO₂ and the unit conversion from ppm to ‰ that was pointed out by RC2 and was included in the manuscript.

We modify the text in L367-375 of the revised manuscript to clarify this.

RC1: The caption in Figure 4 is of little value, please add a significantly extended description to the 8 panels.

We extended the description and updated the figure to enhance its value.

RC1: In Section 4.2.1: The subsection could benefit from additional analyses on the air mass pathways for other stations, following the scheme provided in Fig. 3 for a single station. The following footprints aim to explain the reasons for the varying performances across the individual regional domains that were analyzed.

Figure 3 is a “snapshot” of the sensitivity for one specific hourly CO₂ and integrated Δ¹⁴CO₂ sample at one sampling station. Figure 2 is a more statistical representation of the overall sensitivity in the study. In regions where there is a high sensitivity (as shown in Figure 2 for Western/Central Europe where most yellow clusters are located), there is a better constraint of the emissions as shown in Figure 6. An expanded analysis, as suggested, would require evaluating individual timesteps for each station to construct a detailed understanding of the air mass pathways. This would involve a substantial increase in computational resources and time, as each station's footprints need to be analyzed for each timestep to ascertain the variances in air mass influences. Moreover, the spatial and temporal resolution of the data, along with the model's inherent limitations in resolving complex atmospheric processes, may introduce additional uncertainties. It is not possible, and to our understanding also not needed, to provide the footprint for every observation and every site.

RC1: In line 382, please define the term 'pixel-level'.

We replaced the term “pixel-level” with “grid cell level” (L441) which is the correct term.

RC1: Additionally, in line 386, please specify which samples were used to obtain the statistics. It would be clearer to state 'average values of...' and 'standard deviation' with respect to the underlying samples.

We updated Equation 9 to make it consistent with the metrics used in the previous subsection (4.2.1 Retrieval of the monthly and regional time series). We additionally updated Figure 7 to show the individual maps of the posterior RMSE for both experiments and both categories and updated the text to include the changes in the equation and the figure.

RC1: Lastly, please revise line 400. In the northern part of

The sentence was changed to “[…] the northern part of Western/Central Europe, Denmark, and southern Sweden, as well as some areas in Eastern Europe.” (see L456 of the revised manuscript).

RC1: L475: the conclusion drawn that the inversion process has effectively adjusted the model outputs, bringing them closer to the true observations, is scientifically poor. It is recommended to provide more quantitative evaluation to support this claim.

The whole section was completely modified following this comment and comments from RC2. We included the analysis of a polluted station (Saclay), along with JFJ (background), and added Figures 13 and 14, and Table 5 showing the performance metrics.

RC1: In line 480, the text asks how the Chi2 validation is performed by designing B. It is unclear what is meant by 'designing B'. The text could be improved by providing more context and explanation for the question being asked.  

We do not understand the referee’s comment. The term ‘designing B’ does not appear in this section, nor the whole document.

RC1: The options for the validation method are Talagrand, O. 1997 Assimilation of observations and Desroziers et al, 2005, QJRMS, Diagnosis of observation, background and analysis-error statistics in observation space.

We find his suggestions out of the scope of this study.

RC1: “To improve numerical accuracy, it may be necessary to normalize the fluxes or implement regional scaling factors.” A sound approach involves preconditioning techniques from minimization methods. Please provide your comments on this suggestion.

The preconditioning technique was implemented since the initial development of LUMIA (see Monteil & Scholze (2021)). We realized that this was not mentioned anywhere in the text and we included it in section “2.3.1 Construction of the control vector (x)” (see L240-242 of the revised manuscript). This suggestion was completely removed from the discussion by the recommendation of the second referee.

RC2

The main question of the paper by Gomez-Ortiz et al. is already in the title: "Can ∆14CO2 observations assist atmospheric inversions in constraining Europe's fossil CO2 emission budget?" The authors employ the Lund University Modular Inversion Algorithm (LUMIA) to estimate fossil CO2 emissions and natural fluxes by simultaneously inverting in-situ observations of CO2 and ∆14CO2 across Europe. They evaluate the system's performance through a series of Observing System Simulation Experiments (OSSEs). Their main result is that in regions with dense observation networks, like Western/Central Europe, LUMIA, with the inclusion of ∆14CO2 observations, can reconstruct the emissions' time series and the fossil and biogenic CO2 fluxes. However, in regions with lower observation coverage, such as Southern Europe and the British Isles, estimating fossil CO2 emissions is less successful.

The paper discusses a current and significant topic, showcasing the LUMIA system's capability to utilize both CO2 and 14CO2 data simultaneously. However, the manuscript's discussion of the results is often too descriptive and fails to address the underlying processes. As a result, the study requires major revisions before it can be published.

Overarching comments:

There are certain parts in the manuscript where the language needs to be more accurate and specific. For instance, the national emission data that is reported to the UNFCCC cannot be compared with the emission inventories that are distributed spatially and temporally. However, both are referred to as "inventories" in the text. For experienced readers, the context might be clear, but for new readers, the language needs to be more precise and differentiated in many places.

We updated the manuscript to use the correct language.

The manuscript contains many estimates of posterior CO2 emissions at regional and national levels, but it fails to mention the uncertainties associated with these estimates. To address this issue, an ensemble approach can be used that takes into account different realizations of synthetic observations and various prior uncertainties.

We performed a Monte Carlo ensemble of 100 members to calculate the posterior uncertainties and updated Figures 8 and 10 and the corresponding results and discussion.

The OSSE assumes a constant nuclear 14C contamination. However, as the authors write in their summary, this does not reflect reality. An OSSE study on 14CO2 in Europe is predestined to analyse the influences of variable nuclear contamination. The authors should calculate an additional scenario for this purpose.

We mentioned this in the ‘Discussion’ and ‘Conclusions and future perspectives’ sections in the original manuscript. We feel that adding a scenario for this purpose is beyond the scope of this manuscript (which serves as a model description reference paper). A more detailed analysis of the influence of a range of influencing variables (including nuclear contamination) will be investigated in a follow-up study.

Below are general remarks on specific sections. Note that these comments do not cover all minor issues such as grammatical errors, missing words, or imprecise wording. After addressing the general remarks, a second review should take care of these smaller issues.

Section 1:

RC2: The literature cited lacks clarity and omits fundamental publications, particularly regarding the basic 14C cycle, or is citing them only indirectly.

The Introduction section was thoroughly updated to satisfy the concerns of both referees regarding existing literature in the field of using radiocarbon observations for estimating fossil CO₂ emissions (L75-109 and 136-139 of the revised manuscript).

Section 2.1:

RC2: The regional model is presented, but not a word is said about the boundary conditions and how these were realised.

We added an explanation of the boundary condition calculation to Section 3.3 (L331-337 of the revised manuscript) and commented on the perfect transport and perfect boundary conditions in the discussion (L630-652 of the revised manuscript).

RC2: Eq. 3: What assumptions are made here concerning the d13C?

Since the nuclear emissions have a resolution of 1 year, we assumed δ¹³C as the global atmospheric average value reported by NOAA, as used in studies such as Basu et al. (2016). We included this explanation in the revised manuscript in L176-177.

Section 2.3.2:

RC2: Please begin this section by pointing out that this construction of the B matrix is about determining their spatiotemporal structure and that the absolute magnitude of the uncertainty is afterwards scaled with the reported uncertainties.

The text was modified as suggested in L244-245 of the revised manuscript.

Section 3:

RC2: In the introduction of the OSSE, you should point out that this OSSE uses the same transport model, and thus, a perfect realisation of the atmospheric transport and mixing processes is assumed. This ignores one of the largest sources of uncertainty existing for inversions.

We included this in the introduction of Section 3 as suggested by the referee (L331-337 of the revised manuscript), and commented on this in the discussion (L630-652 of the revised manuscript).

Section 3.3.

RC2: Where do the synthetic background concentrations for CO2 and Δ14CO2 come from? Was TM5 used for the background? I can't find anything about this in the manuscript.

As mentioned in a previous answer to the referee, we added the explanation on the background concentration calculation in Section 3.3.

L274: There are no gaps during sampling that can be attributed to the calibration.

We agree with the referee and removed this from the text (see L321-322 of the revised manuscript).

L285: CΔ14C -> Δ14C as you also show the 14C in Fig 4 and not the C14C. How large was the random perturbation which was added to the data?

We updated CΔ¹⁴C with Δ¹⁴CO₂ and added a sentence explaining the calculation of the random perturbation added to the synthetic observations (see L341-343 of the revised manuscript).

Section 3.3.1:

What is the motivation behind the definition of the prior uncertainties?

For Fbio and Fbiodis, this is not motivated in detail. What is the rationale for setting the bio error as 10% of negative Fbio flows? Likewise, for the 30% for the Fbiodis?

We updated the text in L357-365 of the revised manuscript to better motivate the choice of our prior uncertainties. We also modified the explanation of the F_bio uncertainty to be more accurate. As described in the manuscript, our aim with this study is to demonstrate the capabilities of the multi-tracer LUMIA system. Therefore, we focused on selecting uncertainty values that are reasonable and consistent with other studies.

The definition of the prior Fff uncertainty as the difference between EDGAR and ODIAC, where EDGAR is used as truth and ODIAC as prior, is clearer. However, it FORCES the inversion in the BASE scenario to fully utilise the prior uncertainty budget to arrive at the “truth”. In the 01Base sensitivity run, it is then even more "expensive" for the inversion algorithm to return to the truth. Some discussion on this would be welcome.

Following the referee’s suggestion, we commented on this in the discussion in L539-600 of the revised manuscript.

Uncertainties of observations:

Defining the uncertainty of the observations as the standard deviation of the observations over a 7-day window is incorrect. Over such a long period, the standard deviation of the concentration is dominated by the variable transport and mixing processes in the atmosphere. This has nothing to do with the uncertainty of the measurements. In the manuscript (p.14 l.307), this leads to uncertainties of the CO2 measurements varying from 1 to 215ppm! (See also Table A1.)The selected constant measurement uncertainty of 1.5‰, however, is too optimistic and should be replaced by 2‰. How does the (0.8 ppm ‰) error in Table 2 result?

With regard to the 14C measurement error, it would also be extremely interesting for an OSSE to illuminate the difference between 1.5 and 2 ‰ measurement accuracy.

Indeed, our aim with this definition of the CO₂ observation error is to represent the variability of the transport and mixing processes in the atmosphere, which can be larger in polluted sites in contrast with background sites. In our inversion system, the observation error represents both the instrumental error and the model representation error. In the case of the CO₂ observations, the observation error is mainly composed of the error of representativity, since the instrumental error is very small (in the order of 0.1 ppm), and by calculating the moving standard deviation in a weekly window we calculate an error proportional to how rapidly CO₂ varies.

For Δ¹⁴CO₂, in which the instrumental error is larger than the error of representativity, we selected a fixed value of 0.8 ppm CΔ¹⁴C which translates to a mean value of 1.91±0.05‰ Δ¹⁴CO₂ using Equation 8 (closer to the 2‰ error suggested by the referee), and not 1.5‰ as previously stated in the original manuscript. Nevertheless, we run new inversions using an observation error of 0.9 ppm CΔ¹⁴C (2.15±0.05‰ Δ¹⁴CO₂) and 1.0 ppm CΔ¹⁴C (2.38±0.06‰ Δ¹⁴CO₂), the last one to show the impact of an approximately 0.5‰ in the observation error, as suggested by the referee. There are small differences in the results, but we consider them not to be significant enough and do not change the discussion and conclusions of the manuscript. See the figures in the attached document. We modify the text in L367-375 of the revised manuscript to clarify this.

Section 4.2.1

This section is long and difficult to read... much is obvious and should be tried to be presented in a shorter and more concise way. For example, the numbers are all given in Table 4 and, therefore, do not need to be included in the text.

We updated the whole subsection following the referee’s recommendation.

Section 4.2.2:

L380: I thought the errors in the Z simulations were larger than indicated in Table 2?

As we mentioned in the caption of Table 2 and at the beginning of Section 3.3.1, the same uncertainty and error correlation values were used across all the inversions (experiments) in the study.

To Fig 7:

RC2: From Eq. 9 I understand that it is a relative RMSE reduction. If this is true, then why are the units in Fig 7.d g/(m2day)?

We updated Equation 9 to make it consistent with the metrics used in the previous subsection (4.2.1 Retrieval of the monthly and regional time series) and the units shown in Figure 7.

RC2: The multi-pole structure in the biospheric RMSE reduction in Fig. 7d is striking. However, this is not discussed in the text. In Fig 7c one can see that the prior is spatially relatively smooth. However, since the results of ZBASE and ZCO2 are only mixed in the RMSE_reduction (7d), it is not possible to recognise whether this multipole structure arises from an inversion or from the interaction of the two. In any case, the multipole structure should be discussed more, also regarding a possible overfitting. The formation of the multipole structure cannot be due to the station distribution alone, as there are no stations in South-East Europe, and similar RSME_reductions are achieved in Central-West Europe.

We updated Figure 7 to show the individual maps of the posterior RMSE for both experiments and both categories and updated the text to comment on the multipole structure. In general terms, the regions where the biosphere fluxes are poorly constrained show higher RMSE values conforming dipoles in the map. We include a comment on this in Section 4.2.2 (see L453-454 of the revised manuscript) and the Discussion (L607-609).

RC2: The first sentence of the caption of Fig7D refers to 8 images... but only 4 are shown.

We updated the figure and the corresponding caption in the revised manuscript as shown in the previous answer.

RC2: L389: …show for each location…?

We removed the whole sentence since we found it rather confusing.

Section: 4.2.3

RC2: All data in Figure 8 are presented without any uncertainty information. A suitable measure for determining uncertainty should be considered and added here.

We performed a Monte Carlo simulation ensemble of 100 members to calculate the posterior uncertainty. We added this uncertainty to Figures 8 and 10. We also removed the bar “Ref. (ODIAC)” from Figure 8 to not cause any confusion, and updated the caption accordingly.

RC2: What is the reason for the significant underestimation of the total CO2 fluxes in the study domain in both the ZBASE and the ZCO2only simulation (Fig 8e)? Is this an indication that the uncertainties in the prior fluxes are too small?

The main reason is under-sampling in the whole study domain. As mentioned in the manuscript (and as can be seen in Figure 8) for Western/Central Europe and its countries (Germany, France, and Benelux), where there is a better network coverage (and therefore more samples) the inversion is able to estimate posterior values close to the true values, while in Southern Europe (and Spain) with a sparse sampling network this is not the case.

Section 4.3.1

RC2: L437: The summer deviation in Western/Central and Eastern Europe are nearly of the same magnitude. The analysis of the annual cycles should go into a little more depth. Is it due to an incorrect BIO prior? Or is an incorrect assumption on the 14C signature of the heterotrophic respiration? Is it due to the stronger dilution of the fossil emissions and, therefore, the smaller signal-to-noise ratio of the 14C measurements? The analysis needs to go into more depth here.

The prior terrestrial isotopic disequilibrium flux is on purpose incorrect with the aim of showing the impact that it can have on the estimation of fossil CO₂ emissions. We commented on this in the discussion in L611-618 of the revised manuscript.

RC2: L444: Make a reference to Fig. 10.

The reference was included in the text (see L504 of the revised manuscript).

RC2: L447: ”… with BASE0.3 having the highest recovery of 92%.”. This sentence is slightly misleading. Even in this scenario, the improvement in the DIFFERENCE of the Prior is only around 50%. The figure of 92% probably refers to the total emission, of which approx. 80% will certainly already be "recovered" by the prior.

We agree with the referee. In the text, we are referring to the recovery of the difference between prior and truth, while the percentage corresponds to the recovery of the total budget. We correct this in L505-507 of the revised manuscript.

Section 4.3.2

This section lacks the desired discussion of the effects. The summer overestimation of the Fbiodis flux is probably reflected in the summer maximum of the fossil flux. However, this is not discussed. This is also clearly shown in the BASEnoBD control experiment. The fact that Eastern Europe does not improve in this control experiment should not come as a surprise given the assumed station distribution.

At this point, the fundamental question arises as to how the third unknown, Fbiodis (in addition to FffCO2 and FbioCO2), can be robustly derived from the two observed variables CO2 and 14C?

These suggestions, together with the ones for section 4.3., were added to the discussion in L611-618 of the revised manuscript

Section 4.3.3

Why was JFJ selected as a representative station? JFJ certainly has by far the lowest ffCO2 contribution in Central/Western Europe. Also, the fact that the prior correlation for JFJ is only 0.61, whereas it is 0.92 for all stations, shows that JFJ is not a representative station.

We kept JFJ and added Saclay (SAC) to the analysis to show the parallel between a background and a polluted station. This section was rewritten completely and we added Figures 13 and 14 and Table 5 to complement the results analysis.

Fig 10e: The Synth Obs cannot be seen in the way they are plotted. Please change the line style to allow every piece of information to be seen.

We updated the figure to improve the visualization as shown in the answer above.

The random perturbation of the synth CO2 also appears to be too large in this plot, at least for measurement errors (see comments above). If the uncertainties of a real transport model error should also be represented by the large uncertainties, then a fundamental revision of section 3.3 on error determination is required.

As mentioned in a comment before, the observation error is the aggregate of the measurement or instrumental error and the error of representativity.

The conclusions drawn at the end of section 4.3.3 are fairly trivial. They merely show that the inversion optimisation algorithm works as it should, but this does not imply that the results are correct.

We acknowledged this comment when rewriting this section.

Unfortunately, the authors do not address the interesting mismatch of the prior 14C observations. Fig 12c shows a pronounced overestimation in 14C. The JFJ plot suggests that this overestimation occurs in summer/autumn and, therefore, likely originates from Fbiodis. All of this reflects to the previous results, but the discussion was not very in-depth. Fig. 12f, suggests that the Fbiodis fluxes are strongly overestimated. Here it would be interesting to know whether this is due to the randomly altered 14C signature of heterotrophic respiration or to the very different fluxes of VRPM and ORCHIDEE.

We added Figure 14 to show the influence of each category on the total prior Δ¹⁴CO₂ content. The main reason for this overestimation indeed originated from the prior F_biodis, followed by the prior F_nuc, which we found to have a large impact on stations surrounded by nuclear facilities such as SAC.

Section 5: Discussion

The discussion section is more like a summary and a comparison with previous literature. Unfortunately, there is no real in-depth discussion of the results.

We updated the Discussion section including the referee’s concern about the assumption of perfect transport, perfect boundary condition, prior and posterior uncertainties, spatial distribution, and the impact of the prior terrestrial isotopic disequilibrium product.

L508: Fig10 -> Fig9, Fig9 -> Fig 10 , Fig 11-> ?

We updated the references to the figures.

L520: How will such a scaling workaround solve the problem of small signals? This approach does not change the measurement uncertainty or the observational signal-to-noise ratio. The authors mention that this approach might be problematic due to noise. Thus, I recommend not making this suggestion at all.

We removed this suggestion from the manuscript and updated the text in L586-592 of the revised manuscript.

L524: The realistic… -> The more realistic…

We modified this in the text.

Appendix A:

Table A1: The numbers for the CO2 Obs Error are unrealistic (see above).

We commented on this in the answers above.