the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Timescale dependence of airborne fraction and underlying climatecarbon cycle feedbacks for weak perturbations in CMIP5 models
Abstract. The response of the global climatecarbon cycle system to anthropogenic perturbations happens differently at different time scales. The unraveling of the memory structure underlying this timescale dependence is a major challenge in climate research. Recently the widely applied αβγ framework proposed by Friedlingstein et al. (2003) to quantify climatecarbon cycle feedbacks has been generalized to account also for such internal memory. By means of this generalized framework, we investigate the timescale dependence of the airborne fraction for a set of Earth System Models that participated in CMIP5 (Coupled Model Intercomparison Project Phase 5); the analysis is based on published simulation data from C^{4}MIPtype experiments with these models. Independently of the considered scenario, the proposed generalization describes at global scale the reaction of the climatecarbon system to sufficiently weak perturbations. One prediction from this theory is how the timescale resolved airborne fraction depends on the underlying feedbacks between climate and carbon cycle. These feedbacks are expressed as timescale resolved functions depending solely on analogues of the α, β, and γ sensitivities, introduced in the generalized framework as linear response functions. In this way a feedbackdependent quantity (airborne fraction) is predicted from feedbackindependent quantities (the sensitivities). This is the key relation underlying our study. As a preparatory step, we demonstrate the predictive power of the generalized framework exemplarily for simulations with the MPI Earth System Model. The whole approach turns out to be valid for perturbations up to about 100 ppm CO_{2} rise above preindustrial level; beyond this value the response gets nonlinear. By means of the generalized framework we then derive the timescale dependence of the airborne fraction from the underlying climatecarbon cycle feedbacks for an ensemble of CMIP5 models. Our analysis reveals that for all studied CMIP5 models (1) the total climatecarbon cycle feedback is negative at all investigated time scales; (2) the airborne fraction generally decreases for increasing time scales; and (3) the land biogeochemical feedback dominates the model spread in the airborne fraction at all these time scales. Qualitatively similar results were previously found by employing the original αβγ framework to particular perturbation scenarios, but our study demonstrates that, although obtained from particular scenario simulations, they are characteristics of the coupled climatecarbon cycle system as such, valid at all considered time scales. These more general conclusions are obtained by accounting for the internal memory of the system as encoded in the generalized sensitivities, which in contrast to the original α, β, and γ are scenarioindependent.
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RC1: 'Comment on bg2023101', Ian Enting, 07 Nov 2023
Review of TorresMendonca et al, bg2023101, by I. G. EntingThis paper (hereafter denoted TM23) is suitable for publication. The authors may wish to consider the following comments.
References cited below refer to those listed in the bibliography of TM23 or additional references listed below.Some of my points are, for convenience, illustrated by citing my own publications. This not in itself a suggestion that
these papers should be cited, nor that they are the most appropriate references.
OVERVIEWThis paper considers feedbacks in the carbon cycle, expressing the gain 1/(1f) from feedbacks in terms of the total feedback, f.
A generic result for linear feedbacks is that the combined loop feedback from multiple feedbacks is the sum of the feedbacks so
that linear feedback contributions can be partitioned or aggregated without restriction.The analysis is in terms of the alpha, beta, gamma description (Friedlingstein et al, 2003), where carbon cycle feedback, f, is partitioned
into a concentration feedback, beta, and a radiative feedback alpha*gamma. Each of these is often further partitioned into
Land and Ocean contributions (and may be further partitioned, cf Enting and Clisby 2019 appendix).The description by (Friedlingstein et al, 2003) is historydependent. A scenarioindependent form requires generalising each of alpha, beta, gamma to functions which can be combined by convolutions over time.
These combinations take a simpler form as Laplace transforms. (In this regard, as in the study by Enting and Clisby (2019), Laplace transform
expressions can be regarded as a compact expression for calculations that are actually performed in the time domain, analogous to the
way that vector expressions for electromagnetic fields define expressions that are ultimately calculated using specific components).
The paper uses results from the CMIP intercomparison to estimate the generalised alpha, beta, gamma (in each case
partitioned into Land and Ocean) and combines them into a generalised airborne fraction.COMMENTS
Line 71. This notes characterising climate change in terms of a single global temperature change. This is the basis for "pattern scaling" which is
often used in impact assessments (Mitchell 2003, which should be cited). However this is not what is done later (see comment on eqn 4,5).At some point between lines 155 and 175, it could be worth combining all the terms into one equation and write down the original
alpha, beta, gamma formalism for comparison Delta C [1+ beta +alpha * gamma ] = integrated emissions.
Equations 2 and 3.
These have several issues created by working with separate land and ocean
temperatures:
* it contradicts the initial statement about using a single global temperature
* the numbers are going to differ from other work that has a single
global alpha and absorbs the landocean differences into gamma
* working with a single alpha simplifies the inclusion of other radiative forcing (CH4, volcanoes etc) which would be needed for possible
future work. Even within the context of the present work, including these terms helps make the important point that the carbonclimate
coupling amplifies these forcings by the same factor that applies to amplification of the CO2 response determined by the beta term
(Gregory 2009, eqns 8a,b of Freidlingstein 2003).The results shown in panels c and f of figure 2, suggest that for most, but not all, models the results are consistent with the assumptions
of pattern scaling, with the land response about twice the ocean response on all timescales. It would be interesting to analyse this
in more detail than what can be gleaned from low resolution plots eg a plot of the ratios of the two responses as a function of timescale.Eqn 6. Introducing CO2 concentrations (rather than CO2 content as originally done by Friedlingstein et al 2003) (and thus introducing
the factor k) seems an unnecessary complication which hinders comparisons with other work. (Of course k is given by the mass
of the atmosphere scaled by ratio of molecular weights).Line 188 seems poor wording of the situation. You can work in the time domain (and in many cases do so), but you have to
incorporate history and not just single times (which is what Oeschger and Heimann pointed out in 1983).Circa line 295. At around this point it could be helpful to have one or two sentences summarising the key aspects
of the RFI technique (and maybe a longer summary in the relevant appendix).Eqn 15, (and associated definitions). What this means is that chi is the CO2 impulse response function that is widely used in the definition of
GWP and was the subject of an extensive intercomparisons by Joos et al (2013). This should be noted and Joos et al cited at this point.Eqn 16, goes back at least to Enting (1990), as a Laplace transform.
As a relation for growth at single timescale it is implicit in the results of Oeschger et al 1980.The overall response for CO2 is given by 1/p/(1+beta(p) +alpha(p) * gamma(p) ) Enting (2009) suggested that models could give similar fits to
20th century changes (p approx 0.02) whether or not the gamma feedback was included, simply by changing beta (specific examples were given).
While the CMIP data presented here do not give details of how the various models were calibrated, the results suggest that a similar intermodel
flexibility appears here.Panels d and e of figure 2 suggest that on timescales of about 50 years, the models with smaller positive beta have smaller negative gamma.
This interpretation is supported by figure 6a, where the difference between the true AF spread and that from eqn (22) (which assumes that
the spreads are independent), suggests that the spreads of beta and gamma are not independent.
* POTENTIAL FUTURE EXTENSIONS
An indication of the quality of the work is the extent that it suggests potential future studies. Some possibilities (realising that such work
may already be in progress) are:1: Calculate the airborne fraction, dC/dt/E, as a function of time for various cases of RCPSSP (Representative
Concentration Profiles  Shared Socioeconomic Pathways). Nonlinearity will limit the accuracy, but the least affected will
be the ones that are of most interest as likely to have the greatest change from nearconstant airborne fraction.
2: Using the formalism to analyse the CMIP historical runs.These two extensions would require extending eqn (4,5) to include nonCO2 radiative forcing.
3: A more speculative possibility is the extension to response functions describing radiocarbon (C14). As noted by Enting and Clisby (2021)
and Enting (2022), expressions for the responses to exponential forcing were described by Oeschger et al (1980).Oeschger et al also described corresponding responses for C14 perturbations and how these relate to total carbon (see also Enting 1990).
Since such responses to exponential forcing are the Laplace transforms of the impulse response
functions (eg see eqn 2.11 of Enting 2022), the Oeschger et al results suggest the possibility of defining generalised
sensitivities for C14. This would be of most interest in the analysis of historical trends.MINOR POINTS
Line 182: the PgC/ppm CO$_2$ should be in upright font, since these are not symbols representing mathematical variables.
Similarly, throughout, the labels A, O, L should be in upright font.In the figure captions, it may be helpful to the reader if the symbol (and maybe number of defining equation)
was included after the text description.Note that actual (as opposed to CMIP model) human CO2 emissions include cement production and not just fuel use and also land use
change and forestry.ADDITIONAL REFERENCES
\bibitem[Enting(1990)]{enting90}
\newblock Ambiguities in the calibration of carbon cycle models.
\newblock \emph{Inverse Problems}, 6:\penalty0 L39L46, 1990.\bibitem[Enting((on line, 2009))]{enting09}
\newblock Seeking carbonconsistency in the climatesciencetopolicy
interface.
\newblock \emph{Biogeochemistry}, pages DOI 10.1007/s1053300993517, (on
line, 2009).\bibitem[Mitchell(2003)]{mitchell03}
\newblock Pattern scaling: An examination of the accuracy of the technique for
describing future climates.
\newblock \emph{Climatic Change}, 60\penalty0 (3):\penalty0 217242, 2003.
\newblock ISSN 01650009.
\newblock \doi{10.1023/A:1026035305597}.\bibitem[Oeschger et~al.(1980)Oeschger, Siegenthaler, and Heimann]{oeschger80}
\newblock The carbon cycle and its perturbations by man.
\newblock In W.~Bach, J.~Pankrath, and J.~Williams, editors, \emph{Interactions
of Energy and Climate}, pages 107127. Reidel, Dordrecht, 1980.Citation: https://doi.org/10.5194/bg2023101RC1  AC1: 'Reply on RC1', Guilherme Torres Mendonça, 22 Nov 2023

RC2: 'Comment on bg2023101', Vivek Arora, 03 Jan 2024
Authors present a new framework for representing the carbon feedbacks in the climate system that takes into account the history/memory of the system by using a convolution function based on Volterra series. This is indeed a new development that is welcome. The paper is written extremely well and should be published. I only have minor comments to improve the readability/clarity of the paper. I note the background of the first author in math. This may not be the case for a lot of carbon cycle folks, including myself. Hence a lot of math related questions in the following minor comments and my request to simplify/clarify things for a more general audience.
I also apologize for taking such a long time to review. This is a long paper. Unfortunately, I still haven’t made my way through the entire appendix, but I don’t want to hold this process on any longer.
Minor comments Recall that the carbon feedbacks framework can use results from any two of the three runs (COU, RAD, and BGC). Please note this in your manuscript and clarify that this manuscript uses the RAD and BGC runs.
 Lines 28 and 29. Please changes “reaction” to “response”.
 Lines 680. This sentence is too long. Please also reword “the negative biogeochemical feedback is in terms of radiative forcing more than four times stronger than the positive radiative feedback” to make it more clear.
 Equation (1) – I think E(s)ds should be changed to E(t)dt for easy interpretation.
 Line 255. Why is there is square in CAF(t)^2?
 Line 267. I am not a math expert but I didn’t follow what the plus(+) sign in “lim(t>0+)” means.
 Line 304 needs rewording – “Accordingly, when studying in the next section also these other CMIP5 models, we …”
 Line 319. Does “definition (12)” actually mean “equation (12)”?
 Why does the Laplace transform of equation (12) yields a p in the denominator in equation (13), and the Laplace transform of equation (15) doesn’t (in equation 16).
 Lines 379381 are somewhat difficult to follow. Can you simply say a delta CO2 of how many ppm is considered a linear regime?
 The results in Figure 1 correspond to which scenario?
 Equation (17) has a lot of meaning. It implies that airborne fraction in the frequency domain is not a function of emissions but rather a function of the feedbacks. Is this correct interpretation? If yes, please bring out this message more clearly.
 Equation (18) has a lot going on. In particular, what does the term between the last two “=” means physically?
 I felt, Section 3.3 needs more description of the experiments/simulations.
 Line 405. “In addition, because the two curves were obtained from very different simulations …”. Sorry, what simulations are being referred to here.
 In the context of the airborne fraction, A, is it correct interpretation that A(t) depends on the emissions scenario while A(p) does not. If yes, again this is profound and should be brought out more clearly.
 Please make it clear that you have assumed T*=0, i.e. the temperature change in the BGC simulation is ignored.
 Lines 511512. “In contrast, for all models the predicted beta(O)(t) is for times larger than 15 years systematically too high, and …”. This sentence is unclear.
 Note that typically we want the perturbation to be larger to enhance the response. In the usual carbon feedbacks analysis the feedback metrics are highly variable when c’ and especially T’ are small. Only when c’ and T’ have increased sufficiently then the feedback metrics settle down.
In contrast, your analysis requires c’ < 95 ppm to keep things in the linear regime.
Can these two statements be reconciled?  First sentence of section 4.2 – “Before in the next section finally the main question of this study on the role of feedbacks …” needs rewording.
 In Figure 5b what is the yaxis unit for “Feedback function”?
 Lines 594595, “These results are in particular at short time scales in contrast with previous estimates (Gregory et al., 2009; Arora et al., 2013) using Friedlingstein’s framework, which suggested that the biogeochemical feedback is about 4 times larger than the radiative feedback”.
Note that the 4 times number was in the context of C units (Pg C). Hence my question in bullet 21 (what are the yaxis units in Figure 5b).  Lines 676679. This last sentence of this paragraph is unclear. Please consider rewording.
 Line 813. I do not follow how X_beta_ln(O)(t) = X_beta(O)(t).
 What are the units of prediction error in Figure A1a on yaxis?
Citation: https://doi.org/10.5194/bg2023101RC2  AC2: 'Reply on RC2', Guilherme Torres Mendonça, 21 Jan 2024
Status: closed

RC1: 'Comment on bg2023101', Ian Enting, 07 Nov 2023
Review of TorresMendonca et al, bg2023101, by I. G. EntingThis paper (hereafter denoted TM23) is suitable for publication. The authors may wish to consider the following comments.
References cited below refer to those listed in the bibliography of TM23 or additional references listed below.Some of my points are, for convenience, illustrated by citing my own publications. This not in itself a suggestion that
these papers should be cited, nor that they are the most appropriate references.
OVERVIEWThis paper considers feedbacks in the carbon cycle, expressing the gain 1/(1f) from feedbacks in terms of the total feedback, f.
A generic result for linear feedbacks is that the combined loop feedback from multiple feedbacks is the sum of the feedbacks so
that linear feedback contributions can be partitioned or aggregated without restriction.The analysis is in terms of the alpha, beta, gamma description (Friedlingstein et al, 2003), where carbon cycle feedback, f, is partitioned
into a concentration feedback, beta, and a radiative feedback alpha*gamma. Each of these is often further partitioned into
Land and Ocean contributions (and may be further partitioned, cf Enting and Clisby 2019 appendix).The description by (Friedlingstein et al, 2003) is historydependent. A scenarioindependent form requires generalising each of alpha, beta, gamma to functions which can be combined by convolutions over time.
These combinations take a simpler form as Laplace transforms. (In this regard, as in the study by Enting and Clisby (2019), Laplace transform
expressions can be regarded as a compact expression for calculations that are actually performed in the time domain, analogous to the
way that vector expressions for electromagnetic fields define expressions that are ultimately calculated using specific components).
The paper uses results from the CMIP intercomparison to estimate the generalised alpha, beta, gamma (in each case
partitioned into Land and Ocean) and combines them into a generalised airborne fraction.COMMENTS
Line 71. This notes characterising climate change in terms of a single global temperature change. This is the basis for "pattern scaling" which is
often used in impact assessments (Mitchell 2003, which should be cited). However this is not what is done later (see comment on eqn 4,5).At some point between lines 155 and 175, it could be worth combining all the terms into one equation and write down the original
alpha, beta, gamma formalism for comparison Delta C [1+ beta +alpha * gamma ] = integrated emissions.
Equations 2 and 3.
These have several issues created by working with separate land and ocean
temperatures:
* it contradicts the initial statement about using a single global temperature
* the numbers are going to differ from other work that has a single
global alpha and absorbs the landocean differences into gamma
* working with a single alpha simplifies the inclusion of other radiative forcing (CH4, volcanoes etc) which would be needed for possible
future work. Even within the context of the present work, including these terms helps make the important point that the carbonclimate
coupling amplifies these forcings by the same factor that applies to amplification of the CO2 response determined by the beta term
(Gregory 2009, eqns 8a,b of Freidlingstein 2003).The results shown in panels c and f of figure 2, suggest that for most, but not all, models the results are consistent with the assumptions
of pattern scaling, with the land response about twice the ocean response on all timescales. It would be interesting to analyse this
in more detail than what can be gleaned from low resolution plots eg a plot of the ratios of the two responses as a function of timescale.Eqn 6. Introducing CO2 concentrations (rather than CO2 content as originally done by Friedlingstein et al 2003) (and thus introducing
the factor k) seems an unnecessary complication which hinders comparisons with other work. (Of course k is given by the mass
of the atmosphere scaled by ratio of molecular weights).Line 188 seems poor wording of the situation. You can work in the time domain (and in many cases do so), but you have to
incorporate history and not just single times (which is what Oeschger and Heimann pointed out in 1983).Circa line 295. At around this point it could be helpful to have one or two sentences summarising the key aspects
of the RFI technique (and maybe a longer summary in the relevant appendix).Eqn 15, (and associated definitions). What this means is that chi is the CO2 impulse response function that is widely used in the definition of
GWP and was the subject of an extensive intercomparisons by Joos et al (2013). This should be noted and Joos et al cited at this point.Eqn 16, goes back at least to Enting (1990), as a Laplace transform.
As a relation for growth at single timescale it is implicit in the results of Oeschger et al 1980.The overall response for CO2 is given by 1/p/(1+beta(p) +alpha(p) * gamma(p) ) Enting (2009) suggested that models could give similar fits to
20th century changes (p approx 0.02) whether or not the gamma feedback was included, simply by changing beta (specific examples were given).
While the CMIP data presented here do not give details of how the various models were calibrated, the results suggest that a similar intermodel
flexibility appears here.Panels d and e of figure 2 suggest that on timescales of about 50 years, the models with smaller positive beta have smaller negative gamma.
This interpretation is supported by figure 6a, where the difference between the true AF spread and that from eqn (22) (which assumes that
the spreads are independent), suggests that the spreads of beta and gamma are not independent.
* POTENTIAL FUTURE EXTENSIONS
An indication of the quality of the work is the extent that it suggests potential future studies. Some possibilities (realising that such work
may already be in progress) are:1: Calculate the airborne fraction, dC/dt/E, as a function of time for various cases of RCPSSP (Representative
Concentration Profiles  Shared Socioeconomic Pathways). Nonlinearity will limit the accuracy, but the least affected will
be the ones that are of most interest as likely to have the greatest change from nearconstant airborne fraction.
2: Using the formalism to analyse the CMIP historical runs.These two extensions would require extending eqn (4,5) to include nonCO2 radiative forcing.
3: A more speculative possibility is the extension to response functions describing radiocarbon (C14). As noted by Enting and Clisby (2021)
and Enting (2022), expressions for the responses to exponential forcing were described by Oeschger et al (1980).Oeschger et al also described corresponding responses for C14 perturbations and how these relate to total carbon (see also Enting 1990).
Since such responses to exponential forcing are the Laplace transforms of the impulse response
functions (eg see eqn 2.11 of Enting 2022), the Oeschger et al results suggest the possibility of defining generalised
sensitivities for C14. This would be of most interest in the analysis of historical trends.MINOR POINTS
Line 182: the PgC/ppm CO$_2$ should be in upright font, since these are not symbols representing mathematical variables.
Similarly, throughout, the labels A, O, L should be in upright font.In the figure captions, it may be helpful to the reader if the symbol (and maybe number of defining equation)
was included after the text description.Note that actual (as opposed to CMIP model) human CO2 emissions include cement production and not just fuel use and also land use
change and forestry.ADDITIONAL REFERENCES
\bibitem[Enting(1990)]{enting90}
\newblock Ambiguities in the calibration of carbon cycle models.
\newblock \emph{Inverse Problems}, 6:\penalty0 L39L46, 1990.\bibitem[Enting((on line, 2009))]{enting09}
\newblock Seeking carbonconsistency in the climatesciencetopolicy
interface.
\newblock \emph{Biogeochemistry}, pages DOI 10.1007/s1053300993517, (on
line, 2009).\bibitem[Mitchell(2003)]{mitchell03}
\newblock Pattern scaling: An examination of the accuracy of the technique for
describing future climates.
\newblock \emph{Climatic Change}, 60\penalty0 (3):\penalty0 217242, 2003.
\newblock ISSN 01650009.
\newblock \doi{10.1023/A:1026035305597}.\bibitem[Oeschger et~al.(1980)Oeschger, Siegenthaler, and Heimann]{oeschger80}
\newblock The carbon cycle and its perturbations by man.
\newblock In W.~Bach, J.~Pankrath, and J.~Williams, editors, \emph{Interactions
of Energy and Climate}, pages 107127. Reidel, Dordrecht, 1980.Citation: https://doi.org/10.5194/bg2023101RC1  AC1: 'Reply on RC1', Guilherme Torres Mendonça, 22 Nov 2023

RC2: 'Comment on bg2023101', Vivek Arora, 03 Jan 2024
Authors present a new framework for representing the carbon feedbacks in the climate system that takes into account the history/memory of the system by using a convolution function based on Volterra series. This is indeed a new development that is welcome. The paper is written extremely well and should be published. I only have minor comments to improve the readability/clarity of the paper. I note the background of the first author in math. This may not be the case for a lot of carbon cycle folks, including myself. Hence a lot of math related questions in the following minor comments and my request to simplify/clarify things for a more general audience.
I also apologize for taking such a long time to review. This is a long paper. Unfortunately, I still haven’t made my way through the entire appendix, but I don’t want to hold this process on any longer.
Minor comments Recall that the carbon feedbacks framework can use results from any two of the three runs (COU, RAD, and BGC). Please note this in your manuscript and clarify that this manuscript uses the RAD and BGC runs.
 Lines 28 and 29. Please changes “reaction” to “response”.
 Lines 680. This sentence is too long. Please also reword “the negative biogeochemical feedback is in terms of radiative forcing more than four times stronger than the positive radiative feedback” to make it more clear.
 Equation (1) – I think E(s)ds should be changed to E(t)dt for easy interpretation.
 Line 255. Why is there is square in CAF(t)^2?
 Line 267. I am not a math expert but I didn’t follow what the plus(+) sign in “lim(t>0+)” means.
 Line 304 needs rewording – “Accordingly, when studying in the next section also these other CMIP5 models, we …”
 Line 319. Does “definition (12)” actually mean “equation (12)”?
 Why does the Laplace transform of equation (12) yields a p in the denominator in equation (13), and the Laplace transform of equation (15) doesn’t (in equation 16).
 Lines 379381 are somewhat difficult to follow. Can you simply say a delta CO2 of how many ppm is considered a linear regime?
 The results in Figure 1 correspond to which scenario?
 Equation (17) has a lot of meaning. It implies that airborne fraction in the frequency domain is not a function of emissions but rather a function of the feedbacks. Is this correct interpretation? If yes, please bring out this message more clearly.
 Equation (18) has a lot going on. In particular, what does the term between the last two “=” means physically?
 I felt, Section 3.3 needs more description of the experiments/simulations.
 Line 405. “In addition, because the two curves were obtained from very different simulations …”. Sorry, what simulations are being referred to here.
 In the context of the airborne fraction, A, is it correct interpretation that A(t) depends on the emissions scenario while A(p) does not. If yes, again this is profound and should be brought out more clearly.
 Please make it clear that you have assumed T*=0, i.e. the temperature change in the BGC simulation is ignored.
 Lines 511512. “In contrast, for all models the predicted beta(O)(t) is for times larger than 15 years systematically too high, and …”. This sentence is unclear.
 Note that typically we want the perturbation to be larger to enhance the response. In the usual carbon feedbacks analysis the feedback metrics are highly variable when c’ and especially T’ are small. Only when c’ and T’ have increased sufficiently then the feedback metrics settle down.
In contrast, your analysis requires c’ < 95 ppm to keep things in the linear regime.
Can these two statements be reconciled?  First sentence of section 4.2 – “Before in the next section finally the main question of this study on the role of feedbacks …” needs rewording.
 In Figure 5b what is the yaxis unit for “Feedback function”?
 Lines 594595, “These results are in particular at short time scales in contrast with previous estimates (Gregory et al., 2009; Arora et al., 2013) using Friedlingstein’s framework, which suggested that the biogeochemical feedback is about 4 times larger than the radiative feedback”.
Note that the 4 times number was in the context of C units (Pg C). Hence my question in bullet 21 (what are the yaxis units in Figure 5b).  Lines 676679. This last sentence of this paragraph is unclear. Please consider rewording.
 Line 813. I do not follow how X_beta_ln(O)(t) = X_beta(O)(t).
 What are the units of prediction error in Figure A1a on yaxis?
Citation: https://doi.org/10.5194/bg2023101RC2  AC2: 'Reply on RC2', Guilherme Torres Mendonça, 21 Jan 2024
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