the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 09 Sep 2024
Impulse response functions as a framework for quantifying ocean-based carbon dioxide removal
Abstract. Limiting global warming to 2 °C by the end of the century requires dramatically reducing CO2 emissions, and also implementing carbon dioxide removal (CDR) technologies. A promising avenue is marine CDR through ocean alkalinity enhancement (OAE). However, quantifying carbon removal achieved by OAE deployments is challenging because it requires determining air-to-sea CO2 transfer over large spatiotemporal scales–and there is the possibility that ocean circulation will remove alkalinity from the surface ocean before complete equilibration. This challenge makes it difficult to establish robust accounting frameworks suitable for an effective carbon market. Here, we propose using impulse response functions (IRFs) to address such challenges. We perform model simulations of a short-duration alkalinity release (the “impulse”), compute the resultant air-sea CO2 flux as a function of time, and generate a characteristic carbon uptake curve for the given location (the IRF). Applying the IRF method requires a linear and time-invariant system. We attempt to meet these conditions by using small alkalinity forcing values and creating an IRF ensemble accounting for seasonal variability. The IRF ensemble is then used to predict carbon uptake for an arbitrary-duration alkalinity release at the same location. We test whether the IRF approach provides a reasonable approximation by performing OAE simulations in a global ocean model at locations that span a variety of dynamical and biogeochemical regimes. We find that the IRF prediction can typically reconstruct the carbon uptake in continuous-release simulations within several percent error. Our simulations elucidate the influences of oceanic variability and deployment duration on carbon uptake efficiency. We discuss the strengths and possible shortcomings of the IRF approach as a basis for quantification and uncertainty assessment of OAE, facilitating its potential for adoption as a component of the carbon removal market’s standard approach to Monitoring, Reporting, and Verification (MRV).
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CC1: 'Comment on egusphere-2024-2697', Benoit Pasquier, 16 Sep 2024
I just stumbled on this article and noticed a few issues with the maths that I thought I should point out to the authors in case it is useful:
- Eq. (1) misdefines the Dirac delta function δ, of which the value at 0 is not 1. The Dirac δ does not have a finite value at 0 and is not technically a function. The Dirac δ is instead a "generalised" function, or more specifically a "distribution". It cannot be defined the way Eq. (1) is written. While the first line of Eq. (1) is correct (the Dirac δ is 0 everywhere except at 0), the second line is incorrect and should be replaced with its integral over the entire real line being equal to 1: ∫ δ(t) dt = 1.
- Eq. (2) uses the discrete summation symbol ∑, which seems to suggest that t is an integer, which does not seem correct to me. Why not use the integral notation, x(t) = ∫ δ(t') x(t') dt', instead?
- Eq. (3) could work without the intermediate equality (the one with the summation symbol ∑)
- Eq, (3) and throughout the paper, parentheses must be placed around the functions being convoluted, as in (x ∗ h)(t) instead of x(t) ∗ h(t), the latter being easily confused for simple multiplication otherwise.
This looks like a timely and worthy article otherwise!
Benoît PasquierCitation: https://doi.org/10.5194/egusphere-2024-2697-CC1 -
AC1: 'Reply on CC1', Elizabeth Yankovsky, 16 Sep 2024
Hello Benoît, thank you very much for your comments. We will make these corrections to the manuscript.
Citation: https://doi.org/10.5194/egusphere-2024-2697-AC1 -
CC2: 'Reply on AC1', Benoit Pasquier, 18 Sep 2024
No worries!
Apologies, there is a typo in my comment for Eq. (2): it should be x(t) = ∫ δ(t - t') x(t') dt' instead of x(t) = ∫ δ(t') x(t') dt'.
Good luck with the review!
Citation: https://doi.org/10.5194/egusphere-2024-2697-CC2
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CC2: 'Reply on AC1', Benoit Pasquier, 18 Sep 2024
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AC1: 'Reply on CC1', Elizabeth Yankovsky, 16 Sep 2024
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RC1: 'Comment on egusphere-2024-2697', Anonymous Referee #1, 02 Oct 2024
Review for Yankovsky et al.
The authors have explored nuances of a possible path forward that the community has been collectively been considering for model-based MRV for marine CDR: impulse response functions. IRFs can be used to estimate efficiency losses from incomplete air-sea gas exchange following an intervention. The authors present some mathematics and schematics explaining how this might be able to be done. They do this using a model that captures some seasonal and interannual variability in ways that alternative approaches to deriving IRFs (e.g., from transport matrices) cannot.
My main criticism for this manuscript is that it addresses a “cat” in the room rather than the “elephants.” Specifically, the questions that are addressed by the model are:
- Can an IRF be effectively discretized using 4 distributed seasonally-specific IRFs? (This question is adequately addressed.)
- To what degree does the IRF break down due to interannual variability as the length of time between its definition and use grows? (This question is not well addressed or posed, but some of the results speak to the issue.)
The authors do not address the main problems with IRFs, which swirl around the question of whether the model is adequately representing the true Earth system (are the resolution; parameterizations of biological, atmospheric, and terrestrial feedbacks; parameterizations for mixing, parameterizations of gas exchange; initialization; and forcing adequate to resolve the signal of interest?). Normally that would be okay, and a nice paper can be written that does a targeted analysis of a limited question, except that the authors present their analysis as an assessment of the viability of IRFs for MRV generally. This makes the central argument of the paper feel a bit like a “strawman” argument. When a subset of these other issues are raised, they are mostly dismissed using the logic that (Paraphrased to make a point... this is not a quote) “We don’t have to worry about some challenges to the IRF framework because they only become relevant if we do mCDR in a way that might affect the Earth system.” If this is a fair argument, then the paper is making itself irrelevant by arguing that these approaches to mCDR are not viable at a meaningful scale. It would be better if these issues were more quickly brought up and listed as issues that are not addressed at present rather than presented as issues that can be dismissed.
Another limitation of the paper is that it seems to rely on access to IRFs that are specific to a both a location and a time of release. Several recent studies have blanketed the global surface ocean with IRF estimates and the great Zhou et al. study referenced indeed provides seasonally varying global IRFs. However, it seems unlikely that most people using IRFs will have estimates that are specific to the same year as the release, as is assumed in this study. It would therefore be helpful if the authors could use their analysis to propose a more quantitative approach for assessing uncertainty in IRFs when they are used in different years from when they are determined (or better yet, from another year in another model entirely). The author’s results speak to interannual variability, but the presentation feels anecdotal and doesn’t provide actionable recommendations for quantifying this uncertainty at a general location.
I also would criticize the presentation on two accounts: the complexity and the organization.
Regarding complexity: The main ideas of the paper are reasonably simple, but they are presented in unnecessarily complicated ways. Admittedly, for a formal MRV approach it is important to explicitly present every step in a calculation and drill down into the details to make certain that the calculations are being done in ways that are both practical and defensible. This paper therefore has appropriate ambitions despite the simplicity of the underlying calculations provided the authors can make the underlying math both exactingly correct and highly accessible. I worry that at present the manuscript seems to do neither, and manages to make the simple math behind the proposed idea appear complicated. The presented math also seems to have errors, as noted by an earlier public comment. A solution to both would be to keep the summation notation rather than switching to an integral notation and just accepting that any OAE intervention can be approximated as a sequence of discrete releases rather than an infinite number of infinitesimal releases. This seems likely to be how the IRFs will be implemented in practice in any event, and it seems strange to worry about discretizing the release much finer than the 4x/year IRF functions that will need to be interpolated. Indeed, it is not clear that further discretization of the releases beyond 4x 3-month-long releases could even potentially result in any disagreement whatsoever from the instance with 4x 3-month-long releases because the authors have only modeled releases with no temporal variability between the beginning and end of the release (except in one schematic, which implies a great deal of short-timescale variability). It is possible that linear interpolation of response functions results in a non-linear effective response function, but that math wasn't fully explored.
Regarding organization: The treatment of the various issues that are not addressed (aside from dismissal as mentioned above) and the introduction of the natural “thermostat” hypothesis are mentioned at an unhappy medium level of detail: just enough to convince the readers that these topics will be addressed by the manuscript, but not enough to address them. This makes the presentation somewhat confusing and longer than it needs to be. Similarly, there is some repetition of ideas (e.g., in captions and text) and scattering of methods text throughout the document that makes the paper longer than it needs to be.
In summary, this paper has a worthy if modest aim and good "ingredients" (that is, analysis and simulations). However, it is sufficiently miss-marketed and occasionally overstated that I believe it needs to be rewritten to be shorter, more focused, simpler, and more straightforward in its aims.
Line by line comments are transcribed as going through the paper for the first time... some questions raised in these comments are answered later in the manuscript, but are kept in these comments because they were questions or objections that should likely have been dealt with before that point in the paper.
1-5: why is the abstract focused on OAE when the title is not and the issues raised are not specific to OAE vs. e.g., DOC?
10-15: this has a stronger statement than warranted, as the approach only tests the fidelity of an IRF for a continuous release within the same model, and the real world should be considered a different “model” entirely. The statement is mostly okay, but should be qualified and moderated.
30: the sentence beginning with “A thermostat…” is possibly missing multiple words or is just incorrect.
39: Jumping around timescales here is problematic, as the premature subduction of TA is irrelevant on the timescales that are relevant for the Earth system feedbacks that were the focus of the beginning of the paragraph. Consider dropping the discussion of the natural thermostat to save length and to focus on how mCDR methods tend to create pCO2 deficits in the surface ocean relative to unmodified conditions, how mCDR doesn’t happen until the deficit is eliminated by air-sea exchange, and how subduction slows this equilibration (to timescales that are too slow for climate mitigation strategies).
53: a strategy can consider multiple scales.
57: This implies that computation is our main limitation for modeling. I would argue it is process parameterization and understanding.
58: finish the point by explaining why we don’t do counterfactual experiments
65: There is a disconnect in the community currently with some researchers deliberately avoiding the use of the word “efficiency” to refer to nu with others continuing to use the term to refer to nu. The argument against this term is that there is not a 1:1 equivalence between DIC and TA so the DeltaDIC excess relative to DeltaTA does not fit within the “wasted work” paradigm typically reserved for the term (in)efficiency. Several recent publications have instead taken to using efficiency to refer to the fraction of the expected DIC increase from thermodynamic equilibria that has been achieved, e.g., https://iopscience.iop.org/article/10.1088/1748-9326/ad7477/meta, https://essopenarchive.org/doi/full/10.22541/essoar.170957083.34212619, https://agupubs.onlinelibrary.wiley.com/doi/full/10.1029/2022EF002816
It would be helpful if the authors would adopt this practice or remark on why they do not.
The first schematic left me confused. Figure 1C, why is the Alk input varying over time if the alk pulse was instantaneous at t=1 in the subplot in A (the answer is implied in the main text, but opaque from the figure and its caption)? What is the color representing in A? Why does t sometimes have a prime?
79-88: Why are these conditions important? I can guess, but it should be stated.
106: it is not clear how this is addressed in Figure 2.
125-130: don’t IRFs typically include some degree of subduction and re-emergence?
Figure 3: how different would this figure and figure 4 be if the atmospheric pCO2 were not fixed?
175: if we don’t expect an impact on pCO2atm then why are we doing mCDR? (partially addressed a few lines later, but this should be addressed immediately)
215: this argument cuts both ways. Having a small impact means that a small carbon cycle change from a small and local perturbation to the ecosystem function could result in a significant fractional loss in the expected impact.
256: there are a lot of methods mixed in these results with a fair bit of repetition. It would be best to bring them together and remove them from the results.
293: that appears to be a 50% increase in nu if I am reading it correctly?
309: On the contrary, to my eye, there appears to be more seasonal variability in 4 realizations than in 14 interannual ensemble members. Please explore this point quantitatively rather than visually. Visually, the point might be more easily seen without the 3 sigma envelope. The two sources of variability appear quantitatively dissimilar at this point in the manuscript... they only appear similar in the context of the later figures that the reader has not yet encountered. It might be better to do a comparison across regions initially.
310: presumably, all variants converge over infinite time, though it is interesting that the seasonal variations seem to converge more slowly
313: this has started to address one of many concerns for the IRF method. This claim is too strong.
324: This seems problematic unless the authors feel that in practice it is likely that IRFs will be computed from the same year as the release. If such simulations are available, then why bother with IRFs at all? Wouldn’t it be better to define an ensemble of IRFs for these locations and then test them against an ensemble of releases in various years? The strength of IRFs is that they can be "precomputed" and used later.
Figure 12... nice figure!
345: most of this belongs (and is repeated in) in the caption
346: This seems to significantly undercut the utility of these results.
360: generally this section is well written, but the phrasing of this initial statement is too strong
375: or coastal processes, which may be significant for the many proposed coastal mCDR approaches.
376: lead to or prevent
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC1 -
RC2: 'Comment on egusphere-2024-2697', Anonymous Referee #2, 22 Oct 2024
This manuscript explores the use of impulse response functions (IRFs) to quantify carbon dioxide removal (CDR) through ocean alkalinity enhancement (OAE), a promising marine CDR method. The authors test the IRF approach across various oceanic conditions, finding that it can reliably approximate carbon uptake with minimal error.
Major Concerns:
1. Simplified Assumptions in IRFs
The authors used IRFs to predict OAE-CDR efficiency from a chemical perspective, relying on assumptions of linearity and time invariance. These IRF predictions were then compared to the outcomes of a coarse Earth System Model version 2 (ESMV2) simulation. However, this comparison is effectively circular (at least to me): the IRFs do not account for water mass exchanges or mesoscale processes, and the coarse ESM also omits these physical dynamics. As a result, the match between the two datasets is unsurprising, as both primarily reflect chemical hydrodynamics without fully considering the physical processes. It's important to emphasize that without capturing fine-scale physical variability—such as water currents, downwelling, and other dynamic features—the comparison may be less robust than it appears.
2. Figure 12c and the Overlap with ηmax
The close match in Figure 12c is also expected. After five years of TA release, most of the ocean surface has reached or close to equilibrium (based on Figures 9–11) through sea-air exchange, either quickly or slowly. Essentially, this comparison in Figure 12 is like comparing ηmax after considering seasonal and interannual dynamics, which could be determined by simpler methods using datasets like GLODAPv2 (as shown in Figure 3a). Thus, I question the added value of using IRFs to predict CDR efficiency over longer timescales (years) and across much larger spatial coverage (~100 km), when not accounting the physical processes that prevent reaching ηmax.
3. Limited Application for Mesoscale Processes
Given the limitations of IRFs, I also question the claim that IRFs can guide future regional modeling efforts aimed at resolving mesoscale turbulence or submesoscale dynamics (as suggested in lines 400 of the text). Since the IRFs were not designed to capture these more complex physical processes, their applicability to regional models aiming to resolve such variability seems quite constrained.Additionally, Equation (5) appears incorrect. As currently written, η(t) would never approach ηmax but reach to 0 as time increases , which is problematic for the model.
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC2 -
RC3: 'Comment on egusphere-2024-2697', Anonymous Referee #3, 01 Nov 2024
Review of Yankovsky et al.
General comments
The authors developed the use of impulse response functions (IRFs) for predicting the carbon uptake from ocean alkalinity enhancement (OAE) interventions and tested the accuracy of their approach against simulations of continuous-release OAE scenarios using a Community Earth System Model. They found that IRFs sufficiently meet the requirements of linearity and time invariance that they can predict the carbon uptake from continuous alkalinity release scenarios within several percent error and suggest that IRFs may be a viable approach for Monitoring, Reporting and Verification (MRV) of OAE interventions. A key advantage of the IRF approach is that the pre-computed functions can greatly simplify the estimation of carbon uptake from OAE without the need for full biogeochemical simulations. Although the manuscript is generally worthy and well-written, there are some major issues that the authors need to address before publication.
Major issues:
Lack of independent model validation:
As noted by another reviewer, the IRFs were developed and tested using the same ESM and thus have many of the same biases in their representation of physical and biogeochemical processes. The resulting agreement between the IRFs and model simulations should therefore be interpreted cautiously. The authors should address this issue and explicitly discuss the physical/biogeochemical processes and feedbacks represented and not represented in the models and potential sources of biases in the IRFs.
Period of validity of IRFs:
Can the authors speak on the performance of IRFs older than 5 years and provide recommendations on their reliable application? When should IRFs be recomputed?
Confusing mathematical notation:
As noted by other reviews and a public comment, the mathematical notation is confusing and there are some errors. The switch between δ and h(t) is confusing, and it is not clear what h(t) is referring to. The manuscript refers to both h(t) and y(t) = η(t) as “impulse response functions.”
Equation 5 should be η(t) = ηmax – exp(-t/τ) as in Zhou et al. (2024).
What does the prime notation in t’ refer to?
Minor comments:
Calculation of ηmax (Lines 148-151): Please provide a complete description of how ηmax is calculated (i.e., that ΔDIC is the difference between the initial DIC and the final DIC after the alkalinity perturbation and complete equilibration with atmospheric p(CO2)). What are the assumptions made about atmospheric p(CO2) in the calculations and the values used?
Fig. 4: Is this computed at a constant temperature and salinity or with the in situ values from the GLODAP dataset?
Fig. 6 and Lines 272-274: What is the reason for the 5 year curve in the lower right panel for η(t) being different from the others, and why aren’t there visible kinks on the other curves at the 1 year and 1 month mark?
Line 136: It may be helpful to the reader to include the range of equilibration timescales in the ocean.
Fig. 7: The color scheme makes it difficult to distinguish the curves for different months.
Citation: https://doi.org/10.5194/egusphere-2024-2697-RC3
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