Summary response to all three reviewers

We thank three referees for taking the time to comment on our manuscript, and we appreciate their willingness to review a second draft. 

We understand concerns that the findings here are not entirely new, and we regret that we did not make the intention and goals of the manuscript clear enough. Given the importance of understanding and quantifying time-scale-dependent noise in proxy records for their use in reconstructions and data assimilation—along with the insights gained in the last decade from other groups (e.g., proxy system frameworks, proxy modeling insights on time-scale dependent noise) and our own very recent work (empirical estimates of time-scale dependent noise)—we believe it is timely and useful to synthesize these results in a single manuscript and to provide a perspective on open questions and next steps.

Our aim is to present, side by side, the results we obtained for different proxies in more specialized manuscripts in a unified way and to offer a conceptualization of how non-climatic noise and error are integrated into diverse proxy types. We regret that we did not make this sufficiently clear and hope to better articulate our approach in a revised manuscript.

One of the reviewers’ concerns was that the results we present here may already be known from earlier literature. While we respectfully disagree with this assessment, as we will detail below, we recognize the importance of clearly articulating how our work builds upon these key studies, as well as where it diverges and, in some cases, presents contrasting findings. We will ensure that these distinctions are made more explicit in the revised manuscript.

We suggest expanding the text and results to review more methods of proxy noise estimation, their pros and cons, and what the current understanding is of colored noise across proxy types. This would involve a significant rewriting of the paper to incorporate additional background on a variety of approaches to noise estimation, including our direct empirical approach alongside proxy system modeling.

In a revision, we will also remove details regarding the specific method we rely on for noise estimation, or the data that was used (i.e. lines 90-100). We will instead simply cite the existing papers and move any necessary methods descriptions to the appendix. By doing this we aim to make clear that this is primarily a reproduction and extension of published results. We will need to maintain some data and methods descriptions in the appendix as (the result from the North Greenland Traverse, Fig 3 c,f is a new result),

 As one referee points out, colored noise models have appeared in various places in the literature of specific proxy communities, often assumed to be either white or red, but not estimated directly. As such, we feel strongly that the paleoclimate community would benefit from an overview on this topic drawing from multiple sources to provide a definitive reference for conceptualizing and modeling frequency-dependent noise in proxies including, but not limited to, those described here. We emphasize that the audience for this piece extends beyond those already familiar with noise modeling in frequency space, but a more general paleoclimate research community. In a revision, we will incorporate additional lines of evidence and a more expansive review of prior literature in the service of this goal. 

Specific comments to Reviewer #1 (Responses in bold) 

Summary: The manuscript discusses the spectral characteristics of non-climatic variability in climate proxies, so-called 'proxy noise'. The manuscript formalizes much of the empirical knowledge about the effect of noise on climate reconstructions based on tree-rings, corals and ice-cores.

Recommendation: The information presented in the manuscript is not completely new. It is scattered among the different proxy communities and usually impolitely assumed when reconstructing uncertainty bounds in climate reconstructions. Nevertheless, I found the manuscript interesting and worth publishing, as it presents more formal characterizations and definitions of noise and discusses the effects that, in general, proxy retrieval and post-processing methods have on the final proxy time series. I think this can be a nice contribution to the different proxy communities.

The manuscript does not touch upon one important source of 'noise', namely dating uncertainty, which can be substantial for some proxies and negligible for others (e.g. tree-rings). One appendix briefly indicates that it lies outside the scope of the study, but I think this should be mentioned in the main text and possibly also briefly discussed.

My recommendation is, therefore, that the manuscript needs a few minor revisions. The author may want to consider my suggestions below.

Authors: We thank the referee for taking the time to comment on our manuscript.

We understand the referee’s concern that the findings presented here are not entirely new. Our aim was to synthesize recent findings across proxy types and to offer an accessible conceptualization of different correlation structures of proxy noise. In a revised manuscript we would expand on how these empirical results contrast with the correlation structures assumed in existing proxy system modelling approaches. We also will integrate more discussion of how dating uncertainty affects the spectral power, and address a number of small technical comments to improve clarity.

1) Dating uncertainty is not considered in this manuscript, but many readers would precisely expect to read what the effects of this uncertainty could be. Could the authors include a brief discussion, perhaps without formal treatment, pointing to a follow-up study?

Drawing on evidence from the literature (e.g. Rhines & Huybers, 2011; Comboul et al. 2014; Münch & Laepple, 2018), we  propose to include a description of how dating uncertainty appears as frequency-dependent error in these proxy types. We plan to include this alongside measurement and observation errors in the introduction (~lines 30-40), and in the discussion along with our overview of how smoothing affects the noise/signal spectra. 

2) 'whereas the removal of variance constitutes error, but not ‘noise’ per se'

I struggled to understand this sentence. I could get the meaning in the end, but it needs a clearer phrasing.

We will change this sentence. 

3) ' represented with a positive slope value (β>0)'

I think it is clear here that the authors are referring to the function of power density as a function of frequency spectral_power(f). However, some readers may also think in terms of period instead of frequency. The function spectral_density(T) as a function of period is not just a variable substitution since there is an additional factor:

P(f)df = P'(T)dT

Since dT and df are not linearly related, there is an additional multiplicative factor that depends on T as well. Perhaps the authors may want to include a warning for those readers. This would also include Figure 2 and its caption.

The reviewer is right that many readers may think in terms of period rather than frequency. The two conventions are equivalent and indeed, if one would like to calculate the variance, or another metric involving integration over a frequency/period band, then indeed the substitution f=1/T implies df=-dT/T2 and leads to P(f)df=-P(T)/T2. However, the definition of β is not affected since it is defined as the negative of the slope w.r.t. to log of frequency such that P(f)~f-β and as the positive of the slope w.r.t to the log of the period such that P(T)~Tβ since f-β=(1/T)-β=Tβ, leaving power spectral density and the slope values calculated from individual spectra unchanged.

In the revised manuscript, we will better introduce the equivalency between frequency and period, and further revise and harmonize the caption to figure 2.

4) 'Alternatively, estimation of noise spectra can be done with relying solely on proxies by'

by relying

We will correct this error.

5) 'on ice sheets as wind redistributes snow causing blue noise in noise in annual layer thickness records from ice-cores #

blue noise in noise sounds harder to understand than it should

We will remove the repeated word.

References

Comboul, M., Emile-Geay, J., Evans, M. N., Mirnateghi, N., Cobb, K. M., and Thompson, D. M.: A probabilistic model of chronological errors in layer-counted climate proxies: applications to annually banded coral archives, Clim. Past, 10, 825–841, https://doi.org/10.5194/cp-10-825-2014, 2014.

Münch, T. and Laepple, T.: What climate signal is contained in decadal- to centennial-scale isotope variations from Antarctic ice cores?, Climate of the Past, 14, 2053–2070, https://doi.org/10.5194/cp-14-2053-2018, 2018.

Rhines, A. and Huybers, P.: Estimation of spectral power laws in time uncertain series of data with application to the Greenland Ice Sheet Project 2 δ18O record, Journal of Geophysical Research: Atmospheres, 116, https://doi.org/10.1029/2010JD014764, 2011.

Response to Anonymous Referee #2 (Author responses in bold)

Review of the manuscript "The Colors of Proxy Noise"  by Mara McPartland and colleagues submitted for publication in Climate of the Past

General:

The authors present set out to conceptualize the colors of proxy noise for different proxy archives. For this they use results that are already published or soon will be, in conjunction with already published data sets. They conclude that their models can be used by the community to account for the range of specific biological and physical processes influencing the proxy system.

In general the manuscript is prepared in a very superficial and simplistic style. The abstract does not present any specific hypotheses or questions that are addressed in the body of the manuscript., i.e. why their study is important and which are the specific new results found ?

Authors: We thank referee #2 for taking the time to review our work, and appreciate their willingness to read a new draft. 

 

We understand the referee’s concerns, and regret that we did not make the intention and goals of the manuscript clear enough. Given the importance of understanding and quantifying time-scale-dependent noise in proxy records for their use in reconstructions and data assimilation—along with the insights gained in the last decade from other groups (e.g., proxy system frameworks, proxy modeling insights on time-scale dependent noise) and our own very recent work (empirical estimates of time-scale dependent noise)—we believe it is timely and useful to synthesize these results in a manuscript and to provide a perspective on open questions and next steps. This includes to present, side by side, the results we obtained for different proxies in more specialized manuscripts in a unified way and to offer a conceptualization of how non-climatic noise and error are integrated into diverse proxy types. We hope to better articulate our approach in a revised manuscript.

In particular, the power-law like noise we identify in both coral and dendro temperature reconstructions is a recent development and is fundamentally different to the noise structure in proxy system models often applied to coral and tree-ring proxies. We also think it is useful to note the similarity in the noise structure of these two bioaccumulative proxies and contrast this with the very different noise spectrum from ice cores, and draw on the literature on both types of proxies as supporting evidence alongside our result. 

While the framework of proxy forward modelling is well-established, the method used in the three main studies we synthesise is not yet widely used, nor is power-law like noise typically suggested by existing proxy forward modelling implementation or frameworks.

In a revised version of this manuscript we would make the synthesis format of this manuscript more clear from the start while also better integrating this new approach into the existing literature on proxy system modeling, and taking the time to distinguish among different types of noise estimation methods and what the current state of knowledge is with regards to noise structure among proxy types. We will expand the section ‘The colors of proxy noise’ by further justifying claims about the difference between bioaccumulative and depositional proxies with additional exposition and references. 

The main part also does not include any proper scientific setup with a clear description of methods and a concise description of results, justifying the added value of the manuscript. Therefore the content related to new results is in my opinion not enough to be published in a manuscript in CP, when large parts of the conceptually ideas are already published by other studies in the recent decade (c.f. Smerdon, 2012, Evans et al, 2013, Dee et al., 2017). What is also unclear to me is why the appendix is 30% of the overall length of the (very short) manuscript and does not form a regular method section??

We respectfully disagree with the reviewer’s assertion that a large part of the conceptual ideas presented in our work have been previously published. The seminal paper by Evans et al. (2013) introduced and formalized proxy system models, emphasizing the need for uncertainty analyses and the use of proxy models to quantify uncertainties. The important work of Dee et al. demonstrates how numerical proxy system models can improve model-proxy comparisons in the spectral domain. However, their conclusions are based solely on the processes implemented within their proxy system model. As we show in our studies by using empirical data, the dominant processes driving proxy variability were not included in the PRYSM model framework of Dee et al. Specifically, corals are strongly influenced by autocorrelated noise with a power-law structure, in contrast to the model-based findings of Dee et al., which suggest that the coral signal is ‘whiter’ than the climate signal. Ice cores, on the other hand, are significantly affected by ‘white’ noise, primarily driven by stratigraphic noise — a process not included in PRYSM. Similarly, tree rings, like corals, exhibit strong autocorrelated noise, contradicting the findings of Dee et al. More recent work by Zhu et al. (2023) features a red noise model for modeling the tree-ring records found in the PAGES database, reflecting a growing consensus within dendrochronology that tree rings contain autocorrelated, rather than white noise. In revisions, we will better describe the value of the mechanistic PSM approach in these existing works while making our point that empirical estimation of noise structure is an important complementary step. We will discuss the similarities and differences among these published findings.

With regards to the appendix being long and not forming a regular methods section, this was also intentional as these methods have mostly been published elsewhere, with the exception of the North Greenland traverse result. Because this is a new analysis, we need to maintain the description of this ice core data and analysis in the appendix. 

I suggest to reject the manuscript in its present form and the authors should completely re-think their setup and present substantially new results in a revised version.

Specific:

Abstract/Title:

The Title is very unspecific. Authors should be more specific. Actually it is more promised than the article really holds. The abstract also does not contain substantially new results compared to studies cited mentioned or used in the manuscript.

The title expresses the aim of the paper to review the structure or ‘color’ of noise in proxy-records. We would prefer to leave the title as-is.

Introduction:

The introduction is a summary of results achieved so far in the context of proxy forward and proxy system modelling. No clear question is formulated. As much as i can understand the study just contains a summary of results presented already elsewhere (c.f. „We show results from three studies that have applied this approach to ice core (Münch & Laepple 2018), tree ring (McPartland et al., 2024), and coral data (Dolman et al., in prep).“ on page 3.

In a revision, we will be more explicit regarding our intervention. For example, we suggest adding a paragraph discussing how noise models have previously been used in reconstructions and proxy system models, and showing that white noise or AR1 autoregressive red noise models are almost always assumed rather than tested. From there, we will articulate our perspective that empirical estimation is preferable where possible, acknowledging constraints arising from data availability. In this we will also discuss the implications of assuming that the signal-to-noise ratio increases with timescale, and what this does to low-frequency variability climate variations where the SNR decreases, rather than increases, with timescale. 

In addition, also the information contained in Text boxes and Fig 1 and 2 is published elsewhere and does not present any new result.

The schematic diagram (Fig 1) is modeled off of a similar one from Evans et al (2013), which presents a simple and straightforward overview of paleoclimate proxy formation. A modified version of the same diagram appears in Dee et al. (2015 and 2018). In contrast to these diagrams that show the parts of a proxy system model  we show the processes that affect the noise that are the topic of our manuscript. Fig 2 is certainly not ‘new’ but is a visualization of colored noise spectra - that in our view is useful for the readers not familiar with thinking in the time and frequency domain. The text box provides an overview of power-law scaling, which as this is intended for a general audience, is intended to provide a straightforward definition for a non-specialist. 

The colors of proxy noise

The (supposedly) main chapter also contains a loose collection of (qualitative) information that is presented without any context. No methods section is presented, nor a concrete formulation of an hypothesis that should be addressed in the study. Again, the concept authors motivate is already published in detail in former studies.

‘The colors of proxy noise’ section includes Fig. 3, which provides a clear quantitative illustration of the ideas discussed in the section. We support our result with evidence from the literature for why each type of proxy would present with certain types of noise. For example, much has been published in the dendro literature about medium and low-frequency biases in tree-ring records arising during the age-growth detrending process (e.g. Melvin & Briffa 2008). We use this background to discuss where red noise likely originates. We propose to expand this section and provide additional exposition and background on each proxy, making a more extensive use of the literature to support our claims. 

Implications/Conclusions

The conclusions contain a vague summary without any concrete answer to a previously formulated question, without any context to studies and literature published so far in the field of research.

In a revised version we will make it clearer that we are synthesising the results from previous studies. We know of no study that clearly compares empirical estimates of noise correlation structure among different proxy types. We will more clearly highlight the difference between these empirically obtained noise estimates and the noise structure assumed by current proxy system models, and discuss applications within paleoclimatology.

Referenced literature

Dee, S., Emile-Geay, J., Evans, M. N., Allam, A., Steig, E. J., and Thompson, D. M.: PRYSM: An open-source framework for PRoxY System Modeling, with applications to oxygen-isotope systems, J. Adv. Model. Earth Syst., 7, 1220–1247, https://doi.org/10.1002/2015MS000447, 2015.

Dee, S. G., Russell, J. M., Morrill, C., Chen, Z., and Neary, A.: PRYSM v2.0: A Proxy System Model for Lacustrine Archives, Paleoceanography and Paleoclimatology, 33, 1250–1269, https://doi.org/10.1029/2018PA003413, 2018.

Evans, M. N., Tolwinski-Ward, S. E., Thompson, D. M., and Anchukaitis, K. J.: Applications of proxy system modeling in high resolution paleoclimatology, Quaternary Science Reviews, 76, 16–28, https://doi.org/10.1016/j.quascirev.2013.05.024, 2013.

Melvin, T. M. and Briffa, K. R.: A “signal-free” approach to dendroclimatic standardisation, Dendrochronologia, 26, 71–86, https://doi.org/10.1016/j.dendro.2007.12.001, 2008.

Zhu, F., Emile-Geay, J., McKay, N. P., Stevenson, S., and Meng, Z.: A pseudoproxy emulation of the PAGES 2k database using a hierarchy of proxy system models, Sci Data, 10, 624, https://doi.org/10.1038/s41597-023-02489-1, 2023.

Summary response to all three reviewers

We thank three referees for taking the time to comment on our manuscript, and we appreciate their willingness to review a second draft. 

We understand concerns that the findings here are not entirely new, and we regret that we did not make the intention and goals of the manuscript clear enough. Given the importance of understanding and quantifying time-scale-dependent noise in proxy records for their use in reconstructions and data assimilation—along with the insights gained in the last decade from other groups (e.g., proxy system frameworks, proxy modeling insights on time-scale dependent noise) and our own very recent work (empirical estimates of time-scale dependent noise)—we believe it is timely and useful to synthesize these results in a single manuscript and to provide a perspective on open questions and next steps.

Our aim is to present, side by side, the results we obtained for different proxies in more specialized manuscripts in a unified way and to offer a conceptualization of how non-climatic noise and error are integrated into diverse proxy types. We regret that we did not make this sufficiently clear and hope to better articulate our approach in a revised manuscript.

One of the reviewers’ concerns was that the results we present here may already be known from earlier literature. While we respectfully disagree with this assessment, as we will detail below, we recognize the importance of clearly articulating how our work builds upon these key studies, as well as where it diverges and, in some cases, presents contrasting findings. We will ensure that these distinctions are made more explicit in the revised manuscript.

We suggest expanding the text and results to review more methods of proxy noise estimation, their pros and cons, and what the current understanding is of colored noise across proxy types. This would involve a significant rewriting of the paper to incorporate additional background on a variety of approaches to noise estimation, including our direct empirical approach alongside proxy system modeling.

In a revision, we will also remove details regarding the specific method we rely on for noise estimation, or the data that was used (i.e. lines 90-100). We will instead simply cite the existing papers and move any necessary methods descriptions to the appendix. By doing this we aim to make clear that this is primarily a reproduction and extension of published results. We will need to maintain some data and methods descriptions in the appendix as (the result from the North Greenland Traverse, Fig 3 c,f is a new result),

 As one referee points out, colored noise models have appeared in various places in the literature of specific proxy communities, often assumed to be either white or red, but not estimated directly. As such, we feel strongly that the paleoclimate community would benefit from an overview on this topic drawing from multiple sources to provide a definitive reference for conceptualizing and modeling frequency-dependent noise in proxies including, but not limited to, those described here. We emphasize that the audience for this piece extends beyond those already familiar with noise modeling in frequency space, but a more general paleoclimate research community. In a revision, we will incorporate additional lines of evidence and a more expansive review of prior literature in the service of this goal. 

Specific comments to Reviewer #1 (Responses in bold) 

Summary: The manuscript discusses the spectral characteristics of non-climatic variability in climate proxies, so-called 'proxy noise'. The manuscript formalizes much of the empirical knowledge about the effect of noise on climate reconstructions based on tree-rings, corals and ice-cores.

Recommendation: The information presented in the manuscript is not completely new. It is scattered among the different proxy communities and usually impolitely assumed when reconstructing uncertainty bounds in climate reconstructions. Nevertheless, I found the manuscript interesting and worth publishing, as it presents more formal characterizations and definitions of noise and discusses the effects that, in general, proxy retrieval and post-processing methods have on the final proxy time series. I think this can be a nice contribution to the different proxy communities.

The manuscript does not touch upon one important source of 'noise', namely dating uncertainty, which can be substantial for some proxies and negligible for others (e.g. tree-rings). One appendix briefly indicates that it lies outside the scope of the study, but I think this should be mentioned in the main text and possibly also briefly discussed.

My recommendation is, therefore, that the manuscript needs a few minor revisions. The author may want to consider my suggestions below.

Authors: We thank the referee for taking the time to comment on our manuscript.

We understand the referee’s concern that the findings presented here are not entirely new. Our aim was to synthesize recent findings across proxy types and to offer an accessible conceptualization of different correlation structures of proxy noise. In a revised manuscript we would expand on how these empirical results contrast with the correlation structures assumed in existing proxy system modelling approaches. We also will integrate more discussion of how dating uncertainty affects the spectral power, and address a number of small technical comments to improve clarity.

1) Dating uncertainty is not considered in this manuscript, but many readers would precisely expect to read what the effects of this uncertainty could be. Could the authors include a brief discussion, perhaps without formal treatment, pointing to a follow-up study?

Drawing on evidence from the literature (e.g. Rhines & Huybers, 2011; Comboul et al. 2014; Münch & Laepple, 2018), we  propose to include a description of how dating uncertainty appears as frequency-dependent error in these proxy types. We plan to include this alongside measurement and observation errors in the introduction (~lines 30-40), and in the discussion along with our overview of how smoothing affects the noise/signal spectra. 

2) 'whereas the removal of variance constitutes error, but not ‘noise’ per se'

I struggled to understand this sentence. I could get the meaning in the end, but it needs a clearer phrasing.

We will change this sentence. 

3) ' represented with a positive slope value (β>0)'

I think it is clear here that the authors are referring to the function of power density as a function of frequency spectral_power(f). However, some readers may also think in terms of period instead of frequency. The function spectral_density(T) as a function of period is not just a variable substitution since there is an additional factor:

P(f)df = P'(T)dT

Since dT and df are not linearly related, there is an additional multiplicative factor that depends on T as well. Perhaps the authors may want to include a warning for those readers. This would also include Figure 2 and its caption.

The reviewer is right that many readers may think in terms of period rather than frequency. The two conventions are equivalent and indeed, if one would like to calculate the variance, or another metric involving integration over a frequency/period band, then indeed the substitution f=1/T implies df=-dT/T2 and leads to P(f)df=-P(T)/T2. However, the definition of β is not affected since it is defined as the negative of the slope w.r.t. to log of frequency such that P(f)~f-β and as the positive of the slope w.r.t to the log of the period such that P(T)~Tβ since f-β=(1/T)-β=Tβ, leaving power spectral density and the slope values calculated from individual spectra unchanged.

In the revised manuscript, we will better introduce the equivalency between frequency and period, and further revise and harmonize the caption to figure 2.

4) 'Alternatively, estimation of noise spectra can be done with relying solely on proxies by'

by relying

We will correct this error.

5) 'on ice sheets as wind redistributes snow causing blue noise in noise in annual layer thickness records from ice-cores #

blue noise in noise sounds harder to understand than it should

We will remove the repeated word.

References

Comboul, M., Emile-Geay, J., Evans, M. N., Mirnateghi, N., Cobb, K. M., and Thompson, D. M.: A probabilistic model of chronological errors in layer-counted climate proxies: applications to annually banded coral archives, Clim. Past, 10, 825–841, https://doi.org/10.5194/cp-10-825-2014, 2014.

Münch, T. and Laepple, T.: What climate signal is contained in decadal- to centennial-scale isotope variations from Antarctic ice cores?, Climate of the Past, 14, 2053–2070, https://doi.org/10.5194/cp-14-2053-2018, 2018.

Rhines, A. and Huybers, P.: Estimation of spectral power laws in time uncertain series of data with application to the Greenland Ice Sheet Project 2 δ18O record, Journal of Geophysical Research: Atmospheres, 116, https://doi.org/10.1029/2010JD014764, 2011.

Response to Anonymous Referee #3

The authors discuss the characteristics of the noise spectrum in climate proxies—tree rings, corals, and ice cores—and argue that the spectral characteristics of noise differ depending on whether the originating materials are formed through biological growth or deposition. The topic itself is important for paleoclimate research, but the characteristics and limitations of these proxies have long been discussed already. As the paper lacks both a quantitative evaluation of the uncertainty of the proxies across the frequency range and an in-depth assessment of the cause of the noise characteristics, I feel that the addition of new insights is limited. I therefore suggest a major revision. The following are the detailed comments.

We thank Referee #3 for taking the time to review our work. 

While there is a large body of work discussing the characteristics and limitations of these proxies, much of this work builds mechanistic proxy system models from first principles. While this is a very valuable approach, which we have ourselves used, the empirical results we synthesise here indicate that the true noise spectrum for corals, dendro proxies and ice cores differs substantially from the noise generated by these PSMs.

Our intention was to synthesize recent results, to emphasize the importance of checking whether the noise assumed by mechanistic PSMs matches empirical estimates, and to highlight one way this can be achieved.

In the revised manuscript, we will clarify that we are synthesising existing results across proxy types, and that the details of the results for each proxy type are found in existing publications.

1. About data

It is important to indicate the details of the data used in this study, including the lengths and periods, to assess the validity of the methods and the limitations. Perhaps the authors could plot the time series used in this study in appendix?

We will expand this to include the time periods covered, but also make clearer what is new analysis and what is reproduced for comparison (for example the coral study does not show SNR, so this is calculated here for comparison with the tree and ice based proxies). Since there are very many time series involved in this project (450 tree ring series, 54 coral series), it would be very hard to read any individual series. We propose to add instead some density plots to the appendix showing the time periods covered by each time series. 

For the comparison of the behavior of proxies, it is quite important to have as wide range of spectrum as possible (it is particularly important to assess the reliability of tree rings for a frequency range from centennial to one thousand years). While the ice-core spectrum is provided for a few years to 500 years, the range is quite limited for tree rings and corals. Particularly, there should be tree-ring records available to discuss P > 100 years.

There is a tradeoff in our analysis between how many sites are included in the analysis and how long the final spectrum can be since the analysis is limited by the shortest record shared by all sites within a cluster. Here we opted to include all sites and limit the timespan to improve replication, but we can change the analysis to include fewer, longer clusters. McPartland et al. (2024) contains a figure with fewer replicates but a longer timespan which we can also reproduce here, although the lower replication creates more uncertainty in the final estimate. 

The authors mention on Line 234 (Appendix B) that they have 186 clusters of data. I assume that the analyses were conducted independently, as the authors also mention that they analyzed “co-located records” on Line 91. Further details are needed on how Figure 3 (a) and (d) were derived from the 186 clusters.

We will add in these details from the referenced published study. 

1. Discussion

Sections 2 and 3 contain only a small portion of the authors' own results, while a significant amount is dedicated to reviewing previous work, and it is not clear what new insights this study provides. The text may be reorganized to emphasize the author’s findings.

Line 170 “if the power of the noise rises more steeply than the signal”: For the comprehensive understanding of the behavior of the noise and signals, it is desired to show signal PSD as done in Muench and Laepple (2018), and even the original PSD of the data, in addition to the noise PSD and SNR, either in the main text or in the appendix.

We will add the original and corrected (i.e. signal) PSD in the Appendix, as our focus is on the noise component rather than the climate signal. 

Line 174 “The color of the noise thus determines at which timescales a robust climate signal can be reconstructed”: Shouldn’t the amplitude of the noise be much more critical than its color?

Agreed, ultimately the amplitude of the noise relative to the amplitude of the signal is what is important and the color introduces a nontrivial timescale-dependence to their relative amplitude, thereby determining at which timescales a proxy is more or less useful.

Further details are needed for the discussions regarding the cause of the noise spectrum characteristics. For example, the authors seem to suggest that large seasonal variability contributes to the white-noise characteristics of ice-core records; however, the noise PSD is relatively high across the whole range from 10 years to 100 years (Figure 3c), although the authors mention that they “fail in many regions to reconstruct interannual to decadal changes”. The frequency range should be specified when proposing the hypothesis for the cause of the noise.

The mechanism here is that precipitation intermittency and post-depositional redistribution break up the signal of the large seasonal cycle that would appear as a large spike in the spectrum at period = 1-year if the signal were recorded without disruption. Instead the spike is redistributed as white noise across lower frequencies. We will expand on this in a revised manuscript.

Lines 114-117: The relatively lower SNR for corals appears on the timescale of 10-30 years, and it is not a “slow” change. I feel that the proposed reasons suggested here are not for this short timescale (here I assume the lifespan of corals are much longer, as the lengths of the data are not indicated in Appendix B). The proposed causes of the noise should specifically correspond to the frequency range under discussion.

In the case of corals we are not thinking about ageing/ontogenic effects as in trees, rather shifts in biology in response to stress events. For example, shifts in the Sr/Ca - temperature relationship in response to thermal stress are well established and shown in the papers cited (D’Olivo & McCulloch 2017; D’Olivo et al., 2019). The timescale of significant thermal stress events is not well quantified, but regional scale bleaching events now occur with a return time of approximately 6 years (Hughes et al 2018). In a revision we will make this link clearer.]

References 

D’Olivo, J. P. and McCulloch, M. T.: Response of coral calcification and calcifying fluid composition to thermally induced bleaching stress, Sci Rep, 7, 2207, https://doi.org/10.1038/s41598-017-02306-x, 2017.

D’Olivo, J. P., Georgiou, L., Falter, J., DeCarlo, T. M., Irigoien, X., Voolstra, C. R., Roder, C., Trotter, J., and McCulloch, M. T.: Long-Term Impacts of the 1997–1998 Bleaching Event on the Growth and Resilience of Massive Porites Corals From the Central Red Sea, Geochemistry, Geophysics, Geosystems, 20, 2936–2954, https://doi.org/10.1029/2019GC008312, 2019.

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Summary response to all three reviewers

We thank three referees for taking the time to comment on our manuscript, and we appreciate their willingness to review a second draft. 

We understand concerns that the findings here are not entirely new, and we regret that we did not make the intention and goals of the manuscript clear enough. Given the importance of understanding and quantifying time-scale-dependent noise in proxy records for their use in reconstructions and data assimilation—along with the insights gained in the last decade from other groups (e.g., proxy system frameworks, proxy modeling insights on time-scale dependent noise) and our own very recent work (empirical estimates of time-scale dependent noise)—we believe it is timely and useful to synthesize these results in a single manuscript and to provide a perspective on open questions and next steps.

Our aim is to present, side by side, the results we obtained for different proxies in more specialized manuscripts in a unified way and to offer a conceptualization of how non-climatic noise and error are integrated into diverse proxy types. We regret that we did not make this sufficiently clear and hope to better articulate our approach in a revised manuscript.

One of the reviewers’ concerns was that the results we present here may already be known from earlier literature. While we respectfully disagree with this assessment, as we will detail below, we recognize the importance of clearly articulating how our work builds upon these key studies, as well as where it diverges and, in some cases, presents contrasting findings. We will ensure that these distinctions are made more explicit in the revised manuscript.

We suggest expanding the text and results to review more methods of proxy noise estimation, their pros and cons, and what the current understanding is of colored noise across proxy types. This would involve a significant rewriting of the paper to incorporate additional background on a variety of approaches to noise estimation, including our direct empirical approach alongside proxy system modeling.

In a revision, we will also remove details regarding the specific method we rely on for noise estimation, or the data that was used (i.e. lines 90-100). We will instead simply cite the existing papers and move any necessary methods descriptions to the appendix. By doing this we aim to make clear that this is primarily a reproduction and extension of published results. We will need to maintain some data and methods descriptions in the appendix as (the result from the North Greenland Traverse, Fig 3 c,f is a new result),

 As one referee points out, colored noise models have appeared in various places in the literature of specific proxy communities, often assumed to be either white or red, but not estimated directly. As such, we feel strongly that the paleoclimate community would benefit from an overview on this topic drawing from multiple sources to provide a definitive reference for conceptualizing and modeling frequency-dependent noise in proxies including, but not limited to, those described here. We emphasize that the audience for this piece extends beyond those already familiar with noise modeling in frequency space, but a more general paleoclimate research community. In a revision, we will incorporate additional lines of evidence and a more expansive review of prior literature in the service of this goal. 

Specific comments to Reviewer #1 (Responses in bold) 

Summary: The manuscript discusses the spectral characteristics of non-climatic variability in climate proxies, so-called 'proxy noise'. The manuscript formalizes much of the empirical knowledge about the effect of noise on climate reconstructions based on tree-rings, corals and ice-cores.

Recommendation: The information presented in the manuscript is not completely new. It is scattered among the different proxy communities and usually impolitely assumed when reconstructing uncertainty bounds in climate reconstructions. Nevertheless, I found the manuscript interesting and worth publishing, as it presents more formal characterizations and definitions of noise and discusses the effects that, in general, proxy retrieval and post-processing methods have on the final proxy time series. I think this can be a nice contribution to the different proxy communities.

The manuscript does not touch upon one important source of 'noise', namely dating uncertainty, which can be substantial for some proxies and negligible for others (e.g. tree-rings). One appendix briefly indicates that it lies outside the scope of the study, but I think this should be mentioned in the main text and possibly also briefly discussed.

My recommendation is, therefore, that the manuscript needs a few minor revisions. The author may want to consider my suggestions below.

Authors: We thank the referee for taking the time to comment on our manuscript.

We understand the referee’s concern that the findings presented here are not entirely new. Our aim was to synthesize recent findings across proxy types and to offer an accessible conceptualization of different correlation structures of proxy noise. In a revised manuscript we would expand on how these empirical results contrast with the correlation structures assumed in existing proxy system modelling approaches. We also will integrate more discussion of how dating uncertainty affects the spectral power, and address a number of small technical comments to improve clarity.

1) Dating uncertainty is not considered in this manuscript, but many readers would precisely expect to read what the effects of this uncertainty could be. Could the authors include a brief discussion, perhaps without formal treatment, pointing to a follow-up study?

Drawing on evidence from the literature (e.g. Rhines & Huybers, 2011; Comboul et al. 2014; Münch & Laepple, 2018), we  propose to include a description of how dating uncertainty appears as frequency-dependent error in these proxy types. We plan to include this alongside measurement and observation errors in the introduction (~lines 30-40), and in the discussion along with our overview of how smoothing affects the noise/signal spectra. 

2) 'whereas the removal of variance constitutes error, but not ‘noise’ per se'

I struggled to understand this sentence. I could get the meaning in the end, but it needs a clearer phrasing.

We will change this sentence. 

3) ' represented with a positive slope value (β>0)'

I think it is clear here that the authors are referring to the function of power density as a function of frequency spectral_power(f). However, some readers may also think in terms of period instead of frequency. The function spectral_density(T) as a function of period is not just a variable substitution since there is an additional factor:

P(f)df = P'(T)dT

Since dT and df are not linearly related, there is an additional multiplicative factor that depends on T as well. Perhaps the authors may want to include a warning for those readers. This would also include Figure 2 and its caption.

The reviewer is right that many readers may think in terms of period rather than frequency. The two conventions are equivalent and indeed, if one would like to calculate the variance, or another metric involving integration over a frequency/period band, then indeed the substitution f=1/T implies df=-dT/T2 and leads to P(f)df=-P(T)/T2. However, the definition of β is not affected since it is defined as the negative of the slope w.r.t. to log of frequency such that P(f)~f-β and as the positive of the slope w.r.t to the log of the period such that P(T)~Tβ since f-β=(1/T)-β=Tβ, leaving power spectral density and the slope values calculated from individual spectra unchanged.

In the revised manuscript, we will better introduce the equivalency between frequency and period, and further revise and harmonize the caption to figure 2.

4) 'Alternatively, estimation of noise spectra can be done with relying solely on proxies by'

by relying

We will correct this error.

5) 'on ice sheets as wind redistributes snow causing blue noise in noise in annual layer thickness records from ice-cores #

blue noise in noise sounds harder to understand than it should

We will remove the repeated word.

References

Comboul, M., Emile-Geay, J., Evans, M. N., Mirnateghi, N., Cobb, K. M., and Thompson, D. M.: A probabilistic model of chronological errors in layer-counted climate proxies: applications to annually banded coral archives, Clim. Past, 10, 825–841, https://doi.org/10.5194/cp-10-825-2014, 2014.

Münch, T. and Laepple, T.: What climate signal is contained in decadal- to centennial-scale isotope variations from Antarctic ice cores?, Climate of the Past, 14, 2053–2070, https://doi.org/10.5194/cp-14-2053-2018, 2018.

Rhines, A. and Huybers, P.: Estimation of spectral power laws in time uncertain series of data with application to the Greenland Ice Sheet Project 2 δ18O record, Journal of Geophysical Research: Atmospheres, 116, https://doi.org/10.1029/2010JD014764, 2011.