the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Celestial Mechanics and Estimating the Termination of the Holocene
Abstract. This paper addresses several issues concerning Milankovitch Theory and its relationship to paleoclimate data over the last 800,000 years. A model is presented that deconvolutes the precession index (precession modulated by the eccentricity) and the obliquity contributions to the percentage change between successive meandailyinsolation minima and maxima. The sum of these contributions is in close agreement with the corresponding benchmark calculation of J. Laskar et al. The model predictions indicate that the precession index contribution dominates such insolation changes, and its timedependent behavior correlates with the occurrence of interglacial and glacial periods and temperature trends during these periods. Best fit curves to the separate contributions appear as quasiperiodic waves that correlate with interglacial initiations and terminations through their constructive and destructive interference. However, a comparison of model predictions with the EPICA Dome C (EDC) data indicates delayed inceptions for Marine Isotope Stages 18d and 13c, which have also been noted by Parrenin et al. through a comparison of LR04 benthic δ^{18}O and EDC ice core datasets. Finally, the model enables the classification of interglacial periods into two distinct types that approximately account for their durations. This classification also enables a lowresolution estimation of the Holocene termination based solely on celestial mechanical forcing.
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RC1: 'Comment on egusphere2022569', Anonymous Referee #1, 09 Aug 2022
Review of "Celestial Mechanics and Estimating the Termination of the Holocene"
# Summary
The article claims (in the abstract) to address several issues regarding the Milankovitch theory and "its relationship to palaeoclimate data over the last 800,000 years". The author observes that a substantial number of papers have connected astronomical parameters with "palaeoclimate data" but still fail to completely account for the interglacial and glacial durations, and the "timing" of prominent excursion in palaeoclimate data. He also remarks that some form of "pattern recognition" has enabled "quantitative estimates of the Holocene termination" but implies that this pattern recognition has not been successfully been applied to other terminations.
The originality of the approach presented here is to examine the percentage change between successive meandailyinsolation maxima and minima at 65N. This metric is, in principle defined only at successive insolation maxima, but it is being interpolated and then partitioned into its precession and obliquity components, which themselves are characterised by cycle durations (e.g. his Figures 4, 5, 6).
Figure 12 suggests a match between the precession component of the variations in the metric, and the duration of glacialinterglacial cycles. The match is admittedly intriguing (if not perfect) but the article does not provide a physical interpretation. A second result (Figures 13 and 14) suggests a (less clear) match between the obliquity component of the metric and the amplitude of interglacial maxima. Timings match at the cost of a 10,000 year temporal shift, which the author attributes to timing inaccuracies.
The author then comments on the timing of the next glacial inception.
# Comments## Position with respect to the state of the art
Quite a number of phenomenological models connecting astronomical forcing have been reasonably convincing at predicting the timing of glacialinterglacial transitions (many of these models take the form a dynamical system) but it is fair that the duration of interglacials, and the interglacial/glacial transitions may have received a bit less attention than the timing of terminations. The argument about "pattern recognition" brought about by the author is not obvious to me because, precisely, such rules are derived by examining all interglacials, see, e.g.,
Tzedakis P. C., J. E. T. Channell, D. A. Hodell, H. F. Kleiven and L. C. Skinner (2012), Determining the natural length of the current~interglacial, Nature Geoscience, (5) 138141 doi:10.1038/ngeo1358 and Tzedakis P. C., E. W. Wolff, L. C. Skinner, V. Brovkin, D. A. Hodell, J. F. McManus and D. Raynaud (2012), Can we predict the duration of an interglacial?, Climate of the Past, (8) 1473–1485 doi:10.5194/cp814732012
## No physical modelling
It should be made clear that the current contribution focuses entirely on the one hand, on insolation at 65N and astronomical components, and, on the other hand, the EPICA Dome C Deuterium curve. There are no explicitly physical assumption linking both. When the author uses the word 'model', it is to name the "deconvolution model" that decomposes the metric into its obliquity and precession components. One decomposition is based on arithmetic developments (eqs 1 to 10) and compared with a numerical deconvolution.
I could not read a justification for using 'percent change between two successive maxima' as a reference metric. There is no argument about its physical relevance, and there is no empirical evidence (here or in the literature) that using this metric provides superior results to other, more standard, metrics. For example, the obliquity imprint on the EDC curve would be more straightforwardly explained as an influence of annual mean insolation (controlled by obliquity) on Southern Ocean seasurface temperatures.
## Lack of method description
Numerical deconvolution and estimates of instantaneous amplitude and frequencies have become fairly standard in astronomical theory of palaeoclimates and cyclostratigraphy. The author does not give much detail about the numerical methods used here. He explains that he uses "Laskar's tool", which is a reference to the IMCCE website. As far as I could tell the website generates insolation for usersupplied latitudes and seasons but (again as far as I could tell) does not make demodulation.
## Questions about equations equation (1) featuring a product between obliquity and precession is not justified. Insolation anomalies would have come much more naturally as as sum of obliquity and precession components, not a product.
 I suspect indices a and p are swapped in equations (4) and (6)## Lose semantics
 the author says that between maxima the curve "should not be trusted for numerical precision", but given that the percent change in successive maxima is only defined at maximum points, the meaning to be given to "numerical precision" is not clear
 some sentences are clumsy. E.g. l. 136: "According to the Milankovitch hypothesis, their determination [I understand, of the behaviours of astronomical parameters] provides a consistent temporal calibration that should correlate insolation changes with features of palaeoclimate data. Things could have been stated more clearly. The discussion about "ameliorating timing differences" through "constructive and destructive interference" is somewhat obscure. l. 214: "While the time scale of the precession index contribution to insolation is affected by eccentricity, its shortterm halfcycle is primarily due to precession" remained impenetrable I am afraid. l. 547: "both MIS 18d and 13c represent deep ice cores". With some good will one can see what the author is referring to, but semantics are inaccurate.
 aperiodic and quasiperiodic seem to be used interchangeably, while they should not.
Conclusions about the "catastrophic consequences to future of civilization from another ice age" really need to be recast and put in context given that the catastrophe we are all facing is definitely not that of a coming ice age.Relevance of Appendix A is arguable and seem to be mostly standard material about insolation but I would leave it to the editor.
# Conclusion
A lot of work has been put in this article and I feel sorry to come with a recommendation that sounds negative. Given the countless possibilities given by playing with insolation curves and data, it seems to me that new metrics have to be introduced with care, and the possible link with the palaeoclimate proxy for climate, which they are supposed to explain, have to be physically justified. This is not the case here.
Citation: https://doi.org/10.5194/egusphere2022569RC1 
AC1: 'Reply on RC1', John Parmentola, 22 Aug 2022
Review of "Celestial Mechanics and Estimating the Termination of the Holocene"
# Summary
The article claims (in the abstract) to address several issues regarding the Milankovitch theory and "its relationship to palaeoclimate data over the last 800,000 years". The author observes that a substantial number of papers have connected astronomical parameters with "palaeoclimate data" but still fail to completely account for the interglacial and glacial durations, and the "timing" of prominent excursion in palaeoclimate data. He also remarks that some form of "pattern recognition" has enabled "quantitative estimates of the Holocene termination" but implies that this pattern recognition has not been successfully been applied to other terminations.
The originality of the approach presented here is to examine the percentage change between successive meandailyinsolation maxima and minima at 65N. This metric is, in principle defined only at successive insolation maxima, but it is being interpolated and then partitioned into its precession and obliquity components, which themselves are characterised by cycle durations (e.g. his Figures 4, 5, 6).
The statements in the above paragraph are inconsistent with the paper. The deconvolution (partitioning) is not between the precession and obliquity components. It's between the precession index (PI), sometimes referred to as the climate precession, and the obliquity. There is a significant difference. The PI is the precession modulated by the eccentricity, which is shown in the paper to dominate the insolation. There are many papers that conclude the obliquity dominates. There is important physics in the PI because it provides the characteristic timescale relevant to the behaviors observed in the paleoclimate data and makes the largest amplitude contribution to the insolation primarily due to changes in the eccentricity over a precession halfcycle, which is not constant as the paper demonstrates.
Also, Figures 4,5, and 6 represent the halfcycle behaviors of the three celestial parameters, the eccentricity, precession, and obliquity, and not their cyclical behaviors. The reason as stated in the paper for this is to make clear that these celestial parameters are quasiperiodic, and the quasiperiodicity is important for making a time series comparison between the deconvolution model predictions and the paleoclimate proxy data.
I believe a major point has been missed here. The insolation is inherently a wave phenomenon as described in the paper. Physically the three characteristic celestial motions "beat" on the solar irradiance, which causes it to be distributed over the earth's surface in the form of the insolation.
The point of the paper is to describe the insolation as a timedependent wave resulting from the interference between the PIwave and the obliquity (O) wave.
Figure 12 suggests a match between the precession component of the variations in the metric, and the duration of glacialinterglacial cycles. The match is admittedly intriguing (if not perfect) but the article does not provide a physical interpretation. A second result (Figures 13 and 14) suggests a (less clear) match between the obliquity component of the metric and the amplitude of interglacial maxima. Timings match at the cost of a 10,000 year temporal shift, which the author attributes to timing inaccuracies.
Again, the reference to the precession contradicts the paper. It's the PI contribution that is plotted. The suggested "perfect match" by the reviewer of Figure 12 is not perfect because the plot only involves the PIwave contribution. However, the "approximate alignment" of the PIwavepackets is no accident. The PI contribution depends upon the quasiperiodic behaviors of the precession and eccentricity, which is important for accounting for the approximate recurrence of the prominent features in the paleoclimate data. Figure 12 represents the comparison of time series data with the time series prediction of the deconvolution model.
The purpose of Figures 12 and 13 is to highlight a phase difference between the PI and Owaves. That is described in the text. The 10,000 year shift in the Owave is offset by the PIwave through interference to reproduce the beat structure in Figure 3. It has nothing to do with timing discrepancies as suggested by the reviewer.
The insolation undergoes 74 transitions between maxima and minima over 800,000 years with significantly fewer prominent temperature excursions in the paleoclimate proxy data. Why? Are there special insolation changes of a certain magnitude that are reflected in prominent features in the paleoclimate proxy data. Or could there be trends in the insolation that cause interglacial inceptions and terminations. Examining the percentage change between insolation maxima and minima could identify those special insolation transitions that are dominant. Also, the interpolation between the percentage changes provides trends in the insolation.
The discovery in this paper is that by deconvolving the PI and O contributions to the percentage change between insolation maxima and minima, the dominant insolation effects are identified, and the trends are revealed. The inception and termination of interglacial inceptions and terminations are primarily due to trends in the PIwave contribution that is enhanced or reduced by the Owave through interference. This approach provides a novel interpretation of Milankovitch Theory based on the physics of insolation as a wave. The reviewer appears to missed this point.
The author then comments on the timing of the next glacial inception.
# Comments## Position with respect to the state of the art
Quite a number of phenomenological models connecting astronomical forcing have been reasonably convincing at predicting the timing of glacialinterglacial transitions (many of these models take the form a dynamical system) but it is fair that the duration of interglacials, and the interglacial/glacial transitions may have received a bit less attention than the timing of terminations. The argument about "pattern recognition" brought about by the author is not obvious to me because, precisely, such rules are derived by examining all interglacials, see, e.g.,
Tzedakis P. C., J. E. T. Channell, D. A. Hodell, H. F. Kleiven and L. C. Skinner (2012), Determining the natural length of the current~interglacial, Nature Geoscience, (5) 138141 doi:10.1038/ngeo1358 and Tzedakis P. C., E. W. Wolff, L. C. Skinner, V. Brovkin, D. A. Hodell, J. F. McManus and D. Raynaud (2012), Can we predict the duration of an interglacial?, Climate of the Past, (8) 1473–1485 doi:10.5194/cp814732012
The above comment confounds several issues. My paper's single reference to pattern recognition on page 2 has to do with several papers that examine the similar behaviors of the eccentricity, precession, and obliquity to propose other MIS analogs to MIS 1. One of those papers suggests that the MIS 1 termination will be like MIS 19c. As I discuss later in my paper, unlike that paper, my paper takes a comprehensive approach over 800,000 years using the deconvolution model to interglacial inceptions, terminations, durations, the timing of their recurrences, etc. It demonstrates that there are consistent and recurring insolation conditions that account for these phenomena through a dynamic model.
My paper is simply looking at the insolation at 65N during June across 800,000 years based on the Milankovitch hypothesis. That hypothesis alleges that the changes in the insolation at northern latitudes during the summer solstice is the cause of ice ages. Many people have described this hypothesis. My paper explores it through two assumptions.
The first involves the assumption that the percentage change between successive meandailyinsolation maxima and minima at 65 degrees northern (65N) latitude during the summer solstice (June) over the last 800,000 years, substantially influenced the prominent features in paleoclimate data. Those features are interglacial inceptions, terminations, durations, the timing of their recurrence, their classification into two types based on wave interference, etc.
The second involves the deconvolution model, which is a simple physical model. The model is based on the comparatively small role the obliquity plays in contributing to the insolation, which essentially shifts the sun's rays north and south by 2.4 degrees. It changes the angle of the sun's rays with respect to the vertical and the insolation distribution over the earth's surface. That is demonstrated by the derivation of its contribution in Appendix A. The insolation amplitude is primarily influenced by the precession index because of changes in eccentricity and the direction the earth's axis points during the summer and winter solstices. The timescale of obliquity changes is approximately twice that of the precession index. Therefore, in first approximation the obliquity is constant. The model simply multiplies these contributions to obtain an approximation to the insolation. Its magnitude does not matter because the deconvolution model calculates the percentage change. The justification for this assumption is born out in the results both in terms of the numerical precision compared to Laskar et al and the qualitative and quantitative behaviors of the PI and Owave contributions to the insolation.
Is the reviewer asking that I compare my model with all other models? That would seem to be unreasonable given the extent of the current paper and its novel assumptions. It could be a comprehensive followon paper that would require reconciling different models.
## No physical modelling
It should be made clear that the current contribution focuses entirely on the one hand, on insolation at 65N and astronomical components, and, on the other hand, the EPICA Dome C Deuterium curve. There are no explicitly physical assumption linking both. When the author uses the word 'model', it is to name the "deconvolution model" that decomposes the metric into its obliquity and precession components. One decomposition is based on arithmetic developments (eqs 1 to 10) and compared with a numerical deconvolution.
The deconvolution model and the EPICA Dome C (EDC) data are connected through Milankovitch hypothesis. The deconvolution model explores the validity of the that hypothesis through a comparison of the model and the paleoclimate data. As I describe above, the assumptions of the paper are two and the motivation for them is reasonably clear. If the reviewer would like more elaboration in the paper as I indicated above, I am happy to do that.
I do not understand the remark about equations (110) being arithmetic developments. Those equations follow from the physical description earlier in the paper concerning the roles of the precession index and obliquity with respect to the insolation. The calculations contain the relevant celestial parameters, and the results reveal the dependency of the percentage change between insolation extrema at 65N during June on these physical parameters. There are no arbitrary parameters, and the celestial parameters are specified at a given time by the precise calculations of Laskar et al. The model results for the percentage change between meandaily insolation extrema at 65N during June are compared to the same quantity as Laskar et al showing very good agreement Figure 7.
Furthermore, the estimated PIwave and Owave results are consistent with the assumptions of the model and the approximation made to neglect higher order terms in equation (3).
The EDC is a temperature reconstruction of paleoclimate proxy data. It represents Marine Isotope Stages some of which have been identified with interglacial inceptions and terminations. All paleoclimate data are model dependent; however, there is remarkable consistency amongst the data sets, although there are timing discrepancies among them. For example, there are slight timing discrepancies between the EDC and Vostok data sets. As Parennin et al point out there are timing discrepancies between EDC and LR04 data sets, which I identify in the paper.
I could not read a justification for using 'percent change between two successive maxima' as a reference metric. There is no argument about its physical relevance, and there is no empirical evidence (here or in the literature) that using this metric provides superior results to other, more standard, metrics. For example, the obliquity imprint on the EDC curve would be more straightforwardly explained as an influence of annual mean insolation (controlled by obliquity) on Southern Ocean seasurface temperatures.
## Lack of method description
Numerical deconvolution and estimates of instantaneous amplitude and frequencies have become fairly standard in astronomical theory of palaeoclimates and cyclostratigraphy. The author does not give much detail about the numerical methods used here. He explains that he uses "Laskar's tool", which is a reference to the IMCCE website. As far as I could tell the website generates insolation for usersupplied latitudes and seasons but (again as far as I could tell) does not make demodulation.
I don't understand these statements. Every number contained in this paper follows from Laskar's tool on the IMCCE website. That tool is based on extensive work done by Laskar et al over many years. The references to the relevant papers are on the website. As an example, a user can generate the climate precession (also known as the precession index) and divide it by the eccentricity to produce the precession.
## Questions about equations equation (1) featuring a product between obliquity and precession is not justified. Insolation anomalies would have come much more naturally as as sum of obliquity and precession components, not a product.
 I suspect indices a and p are swapped in equations (4) and (6)Again, the reviewer's terminology is inconsistent with the paper. It's not the precession but rather the precession index. As argued above, the obliquity contribution is slowly varying, so to obtain an approximation to the insolation it's the product of the precession index contribution and the obliquity that is relevant. All the analysis that follows is demonstrated to be consistent with equation (1). Regarding the reviewer's additivity remark, note equation (3). All the results that follow from equation (1) are justified numerically as evident from Figure 8. Note also, the socalled arithmetic reveals approximations that are quite insightful given the complexity of celestial mechanical calculations of the insolation.
The reason for the swapping of indices is physics. There are two types of transitions between maxima and minima, one from perihelion to aphelion and the reverse.
## Lose semantics
 the author says that between maxima the curve "should not be trusted for numerical precision", but given that the percent change in successive maxima is only defined at maximum points, the meaning to be given to "numerical precision" is not clear
 some sentences are clumsy. E.g. l. 136: "According to the Milankovitch hypothesis, their determination [I understand, of the behaviours of astronomical parameters] provides a consistent temporal calibration that should correlate insolation changes with features of palaeoclimate data. Things could have been stated more clearly. The discussion about "ameliorating timing differences" through "constructive and destructive interference" is somewhat obscure. l. 214: "While the time scale of the precession index contribution to insolation is affected by eccentricity, its shortterm halfcycle is primarily due to precession" remained impenetrable I am afraid. l. 547: "both MIS 18d and 13c represent deep ice cores". With some good will one can see what the author is referring to, but semantics are inaccurate.
 aperiodic and quasiperiodic seem to be used interchangeably, while they should not.
Conclusions about the "catastrophic consequences to future of civilization from another ice age" really need to be recast and put in context given that the catastrophe we are all facing is definitely not that of a coming ice age.Relevance of Appendix A is arguable and seem to be mostly standard material about insolation but I would leave it to the editor.
I do not understand these remarks. The PIwave and Owave are computed at a sparce set of points. The accuracy of those points in time is set by Laskar's tool. The rest is interpolation. The paper is simply pointing out that the numerical accuracy between the extrema points is not precise. In fact, the paper simply uses the interpolation qualitatively to interpret the paleoclimate data. In all honesty, it's quite remarkable how well the model results consistently interpret the paleoclimate data over 800,000 years. All interglacial terminations occur in the same manner based on the PIwave and Owave behaviors. The only timing discrepancies occur with two interglacial inceptions, namely, MIS 18d and MIS 13c. These by the nature of ice cores are deep, so I do not understand the semantic issue. Depth in ice cores is related to time; however, the association of time at ice core depth is model dependent because of physical effects. As the paper points out, timing discrepancies are to be expected since paleoclimate data is model dependent. To support this statement, the paper identifies the work of Parrenin et al that demonstrates such a discrepancy between two sets of paleoclimate data that occur during the inceptions of MIS 18d and MIS 13c. Given the consistency of the deconvolution model with the paleoclimate data over 800,000 years, the paper suggests that the data be reexamined.
It appears that the reviewer may be confused about the precession index. It is the modulation of the precession by the eccentricity. There are two different timescales involved here. However, the insolation transitions from maxima to minima approximately coincide with the precession halfcycles. The paper demonstrates that the PIwave is dominant, and the transitions are primarily due to eccentricity changes over the precession halfcycles.
Regarding aperiodic versus quasiperiodic, the latter is an example of aperiodic behavior. If the reviewer prefers, a simple word change can be accomplished by replacing aperiodic with quasiperiodic throughout the paper.
The paper constrains the conclusions by specifying that the forcing is only celestial mechanical. The paper also points only provides recurring insolation conditions associated with terminations over the last 800,000 years. The earth's climate response to this is unknown. The catastrophe has happened many times before based on celestial mechanical forcing. By inductive reasoning, it could happen again, and the PI wave packets in Figure 12 make that case very clearly.
The paper speculates that the instability of the polar vortex may be the earth's climate response but that is currently an active area of research. We do not understand the earth's climate well enough to predict what that mechanism is. Basically, there is no complete theory of the earth's climate. There are models which are not equivalent to a theory.
Regarding Appendix A, I would appreciate a reference that goes through the computation with all the assumptions made.
# Conclusion
A lot of work has been put in this article and I feel sorry to come with a recommendation that sounds negative. Given the countless possibilities given by playing with insolation curves and data, it seems to me that new metrics have to be introduced with care, and the possible link with the palaeoclimate proxy for climate, which they are supposed to explain, have to be physically justified. This is not the case here.
The linkage between the behavior of the insolation and paleoclimate data is the Milankovitch
hypothesis, which many other authors have explored. The deconvolution model results support that hypothesis using a wave formulation of the insolation resulting in a novel interpretation of the paleoclimate data having remarkable consistency over the last 800,000 years. There are many new results among them are interglacial classifications, the relationship between recurring PI wave packets and recurring features in the paleoclimate data, estimations of interglacial terminations, an estimation of the MIS 1 termination, etc. That latter is a lowresolution estimate of 500 years based on the model described in the paper.
The paper is consistent and the results unexpected through a novel interpretation of paleoclimate data using the Milankovitch hypothesis.
Citation: https://doi.org/10.5194/egusphere2022569AC1

AC1: 'Reply on RC1', John Parmentola, 22 Aug 2022

RC2: 'Comment on egusphere2022569', Anonymous Referee #2, 15 Aug 2022
The first issue here is terminological, but is indicative of major problems throughout the paper:In mathematics and physics including geophysics, generally, "convolution" and "deconvolution" refer to a specific mathematical operation, equivalent to multiplying and dividing fourier transforms. The author seems to think a "deconvolution" model means an "explanatory" model, but if there is any deconvolution or convolution going on here, I've completely missed it. The "model" is a quasiinstantaneous linear forcing represented through the simple Eq. 3 with no reference to the physics or chemistry of the climate system.More generally, this manuscript seems to represent a big step backwards: among otherissues, the author implies, without ever saying so, that the climate system on these timescales is linear. But even the original Hayes et al. paper referred to Milankovich "pacing" ofglacial intervals, recognizing that the relationship between forcing and response was notthat of a linear system. In the intervening time interval, a number of papers (see for example,Tziperman et al., 2006, Paleoceanography, or various papers by Peter Ditlevsen) have dealt explicitly with the anticipated results of forcing a nonlinear system with quasiperiodic deterministic driving. None of this is even mentioned.The paper is really a collection of wiggly lines. But a huge literature exists on the response of linear and nonlinear systems to deterministic narrowband (in the Fourier sense) forcing. (Studies of ordinary ocean tides are one much discussed, analogous, application dating back to the 19th Century.) The author repeatedly invokes "correlation" between two wiggly linesall based on some perceived visual resemblance. But the human eye is notoriously poor at separating apparent patterns from real ones (a branch of psychology known as pareidolia), and for that reason, specific statistical tools have been developed to obtain objective tests. It is a truism of data analysis that even two unrelated records will show a correlation, both numerical and visual, and hence a level of no significance is *always* necessary. In most cases here, one wonders what would have led the author to reject his own hypothesis of apparent correlation? The model being used is fully deterministic and the distribution e.g., of zerocrossing times, is also deterministic unless the system is nonlinear and chaotic. Deviations from claimed correlations are explained away by armwaving stories (lines 517+ are one example). The use of halfcycles, rather than the usual use of frequencies, here would imply some kind of obscure rectification mechanism at work. Furthermore, the paper takes no account of any kind of noise in the ice core records, including issues of dating.Smaller issues:the 65degree N dependence is invoked without any discussion of its relevance, particularly to an individual ice core.No error bars are placed on any of the ice core timings (e.g, Fig. 1)Several figures refer to curve fits to data points (e.g., line 150), but with no specificationof what constituted those curves.Why don't the author's hypotheses operate in the interval before 1MY BP, when theglacialinterglacial intervals are widely believed to have occurred only at ~40KY?Was the precession index ever defined? Is A truly timeindependent?Line 97 and elsewhere, what the author means by aperiodic would be the simple beating of two or more nearby frequencieseach individually periodic.Line 299. Why is it surprising?Line 413. What is meant by "judiciously"?Citation: https://doi.org/
10.5194/egusphere2022569RC2 
AC2: 'Reply on RC2', John Parmentola, 30 Aug 2022
The first issue here is terminological, but is indicative of major problems throughout the paper:
In mathematics and physics including geophysics, generally, "convolution" and "deconvolution" refer to a specific mathematical operation, equivalent to multiplying and dividing fourier transforms. The author seems to think a "deconvolution" model means an "explanatory" model, but if there is any deconvolution or convolution going on here, I've completely missed it. The "model" is a quasiinstantaneous linear forcing represented through the simple Eq. 3 with no reference to the physics or chemistry of the climate system.
I do not understand what the reviewer means by an explanatory model. Deconvolution is used in the sense of partitioning two effects associated with insolation, namely the contribution from the obliquity and the other from the precession index (the precession modulated by the eccentricity). It is not immediately obvious that such a partitioning is quantitatively valid, but systematic computations presented in the paper bear this out as represented by Figures 7 and 8.
This partitioning is performed in temporal space unlike the typical frequency space analyses. There are physical reasons for this choice. The first has to do with the EDC data, which is represented as a time series. As I discuss in the introduction section of the paper, typical frequency analyses of the data and insolation reveal characteristic frequencies, and these are associated with the average cycles of the three celestial parameters. My paper treats the behavior of the three celestial parameters as they are in nature. They are quasiperiodic functions of time with wide ranging halfcycles as demonstrated by the numerous graphs in the paper. A temporal comparison between the EDC time series data and the deconvolution model requires that the obliquity and precession index contributions to the insolation be treated as quasiperiodic functions of time. This approach is born out in Figure 12 where I show that the precession index contribution to the percentage change in insolation between maxima and minima correlate with the temperature trends and interglacialglacial periods quite well. This result is no accident.
The deconvolution model is an approximate physical description of the insolation. The model is based on the comparatively small role the obliquity plays in contributing to the insolation. It essentially shifts the sun’s rays north and south by 2.4 degrees. It changes the angle of the sun’s rays with respect to the vertical and primarily the angular distribution of the insolation over the earth’s surface. That is demonstrated by the derivation of its contribution in Appendix A. The insolation amplitude is primarily influenced by the precession index because of changes in eccentricity and the direction the earth’s axis points during the summer solstice. The timescale of obliquity changes is approximately twice that of the precession index. Therefore, in first approximation the obliquity is constant. The model simply multiplies these contributions to obtain an approximation to the insolation. The insolation magnitude does not matter because the percentage change in the insolation between extrema is calculated and used to compare with the EDC data. The justification for this approximation is born out in the results both in terms of the numerical precision compared to Laskar et al and the qualitative and quantitative behaviors of the PI and Owave contributions to the insolation and their comparison with the data.
The paper demonstrates that the precession index dominates the insolation contrary to many papers in the field of Milankovitch Theory. Even Milankovitch believed the obliquity was the dominant effect in the cause of ice ages.
I have no idea what the reviewer means by “a quasiinstantaneous linear forcing represented through the simple Eq. 3.” Linear in what parameter? Equation (3) is nonlinear in the obliquity and precession index contributions to the percentage change in the mean daily insolation between maxima and minima. It’s clearly not linear in time or the three celestial parameters as born out in the subsequent equations to equation (3). What does the reviewer mean by quasiinstantaneous? The estimates of Laskar et al for the insolation and the three celestial parameters have a resolution of 100 years as I state in the paper. Is this what the reviewer means by quasiinstantaneous? What is the point of this comment?
Is the reviewer objecting to the simplicity of the model? What matters are the physical assumptions, the translation of those assumptions into mathematics, and the comparison with the data. That is the scientific method. The paper makes two key assumptions.
The first involves the assumption that the percentage change between successive meandailyinsolation maxima and minima at 65 degrees northern (65N) latitude during the summer solstice (June) over the last 800,000 years, substantially influenced the prominent features in paleoclimate data. Those features are interglacial inceptions, terminations, and durations, the timing of their recurrence, their classification into two types based on wave interference, etc.
The second involves the deconvolution model, which is a simple physical model as described above. The rest of the paper describes the consequences of these assumptions and semiquantitative comparisons to the EDC time series data.
The paper is not a climate model. As discussed in the abstract and the introduction section, the paper explores the Milankovitch hypothesis, which focuses on the mean daily insolation at 65 N degrees latitude during the summer solstice. The issue is what does the behavior of the insolation tell us about the occurrence of interglacial and glacial periods over the last 800,000 years. Are there quantitative and qualitative behaviors of the insolation that account for prominent features in the paleoclimate data such as the timing of interglacial inceptions and terminations, interglacial durations, the recurrence of interglacialglacial periods, etc. The issue of how the earth’s climate responds to insolation changes is not the purpose of this paper. That is a totally different matter. The specific climate mechanism that causes an interglacial termination is unknown. We do not understand the earth’s climate well enough to predict how it responds to insolation changes and there is no calibration of insolation to temperature. The paper merely is establishing recurring patterns in the behavior of the insolation that correlate with the prominent features in the EDC time series data.
There maybe an important point that has been missed by the reviewer. The insolation is inherently a wave phenomenon, which is emphasized in the paper. It is an amplitude modulated wave in time. This wave description involves the interference between the obliquity and precession index waves as defined in the paper that accounts for both quantitative and qualitative features in the paleoclimate data. It appears to be a useful language to describe the prominent features in the paleoclimate data based on the Milankovitch hypothesis. It’s a new and novel contribution to the field.
More generally, this manuscript seems to represent a big step backwards: among other
issues, the author implies, without ever saying so, that the climate system on these time
scales is linear. But even the original Hayes et al. paper referred to Milankovich "pacing" of
glacial intervals, recognizing that the relationship between forcing and response was not
that of a linear system. In the intervening time interval, a number of papers (see for example,
Tziperman et al., 2006, Paleoceanography, or various papers by Peter Ditlevsen) have dealt explicitly with the anticipated results of forcing a nonlinear system with quasiperiodic deterministic driving. None of this is even mentioned.
I have no idea what the reviewer means by this paper implicitly assumes that the climate system on these time scales is linear. Linear in what parameter? The reviewer seems to miss the point that the paper is not a climate model. The paper is simply exploring the Milankovitch hypothesis through a model that partitions the obliquity and precession index contributions to the percentage change in the mean daily insolation at 65N latitude during the summer solstice. The model provides strong support for the Milankovitch hypothesis. If there is a valid theory or model of the earth’s climate that incorporates a response to insolation force according to the patterns established in this paper, then that would be a major advance in understanding the cause of ice ages. I know of no such validated theory or model. If the reviewer knows of one, I would appreciate a reference.
The paper is really a collection of wiggly lines. But a huge literature exists on the response of linear and nonlinear systems to deterministic narrowband (in the Fourier sense) forcing. (Studies of ordinary ocean tides are one much discussed, analogous, application dating back to the 19th Century.) The author repeatedly invokes "correlation" between two wiggly linesall based on some perceived visual resemblance. But the human eye is notoriously poor at separating apparent patterns from real ones (a branch of psychology known as pareidolia), and for that reason, specific statistical tools have been developed to obtain objective tests. It is a truism of data analysis that even two unrelated records will show a correlation, both numerical and visual, and hence a level of no significance is *always* necessary. In most cases here, one wonders what would have led the author to reject his own hypothesis of apparent correlation? The model being used is fully deterministic and the distribution e.g., of zerocrossing times, is also deterministic unless the system is nonlinear and chaotic. Deviations from claimed correlations are explained away by armwaving stories (lines 517+ are one example). The use of halfcycles, rather than the usual use of frequencies, here would imply some kind of obscure rectification mechanism at work. Furthermore, the paper takes no account of any kind of noise in the ice core records, including issues of dating.
I cannot make logical sense of these confounding comments. Are the assumptions of the paper wrong and what specifically is wrong in terms of physics? Is the translation of the assumptions into mathematics and the model wrong and what is specifically wrong? The reviewer seems to be stuck on frequencies as the only approach to understanding paleoclimate data. My paper takes a time series approach because of the quasiperiodic nature of the three celestial parameters and the fact that there is more information in such an approach to compare with data. What is obscure about this?
The paper points out the very limited number of points that represent predictions of the model. There are not enough points to perform a statistical comparison with the data. Furthermore, the history of science is fundamentally a search for patterns and the development of theories to explain the patterns. Would the reviewer consider the experiments of Thomas Young concerning the interference patterns of light as a wave a statistical aberration? Explaining those patterns took about a century through the development of quantum theory that could fundamentally account for those patterns quantitatively and qualitatively. Richard Feynman has hand waving lectures that describe all of this without using a single formula. There is more to a good physical argument than most people think.
My paper is based on two assumptions as clearly described above and in the paper. Celestial behaviors, i.e., physics, are used to translate these assumptions into mathematics to define the model. The approximations are justified quantitatively in the paper based on wellestablished benchmark computations of Laskar et al. The results are used both quantitatively and qualitatively to identify the prominent features in the EDC time series data. Wave interference, a novel aspect of the paper, is used to interpret the data. The model accounts approximately for the recurrence of interglacialglacial periods over the last 800,000 years, temperature trends, the timing of interglacial inceptions and terminations, the duration of interglacial periods, the classification of interglacial periods into two types, and quantitative estimates of interglacial terminations. All interglacial terminations consistently occur vin the same manner over the last 800,000 years.
Smaller issues:
the 65degree N dependence is invoked without any discussion of its relevance, particularly to an individual ice core.
Many (A. Berger et al) have explored the relationship between insolation predictions at 65N during the summer solstice from celestial mechanics and their correlation with paleoclimate data. This paper simply uses EDC data, which is a temperature model rendering of ice core data. There is remarkable consistency amongst the various paleoclimate data sets, however there are timing issues as I point out in the paper.
No error bars are placed on any of the ice core timings (e.g, Fig. 1)
Several figures refer to curve fits to data points (e.g., line 150), but with no specification
of what constituted those curves.
As pointed out in the paper, the EDC ice core data is a temperature rendering of the paleoclimate data published on the NOAA site as referenced. The timing issues with these data are well known as I indicate in the paper. The issue is how well can a model account for the prominent features in the data. Expecting precision from this data is inconsistent with the methods used to infer it. At best, a model comparison with the data can be semiquantitative. That is the approach taken in the paper and the results obtained worthy of publication.
The caption under Figure 3 and other Figures explains the origin of the graphs. There is a reference, which directs the reader to a wellestablished computational tool that is used to create the graph based on the simple definition of what is computed.
Why don't the author's hypotheses operate in the interval before 1MY BP, when the
glacialinterglacial intervals are widely believed to have occurred only at ~40KY?
The earth was undergoing a major transition likely due to the Isthmus of Panama (about 2.3 million years ago), which changed the dynamic mixing of the oceans. The average global temperature steadily dropped, and the temperature excursions steadily increased likely due to the increasing accumulation of snow and ice at the poles. Eventually, the earth stabilized into the more recent pattern of ice ages over the last 800,000 years. The model required to describe the earlier period would be different because the physical circumstances were different.
Was the precession index ever defined? Is A truly timeindependent?
It’s defined in the abstract and it’s clearly not time independent because it depends on the eccentricity and precession. Anyone familiar with the subject of this paper and celestial mechanics would know this.
Line 97 and elsewhere, what the author means by aperiodic would be the simple beating of two or more nearby frequencieseach individually periodic.
The paper demonstrates that the three celestial parameters are aperiodic. What two or more nearby frequencies that are periodic is the reviewer talking about?
Line 299. Why is it surprising?
Given the derivation in Appendix A, it is surprising that the obliquity contribution to the insolation is well approximated by the formula provided.
Line 413. What is meant by "judiciously"?
The vertical lines are chosen based on the recurring pattern in the PIwave.
Citation: https://doi.org/10.5194/egusphere2022569AC2

AC2: 'Reply on RC2', John Parmentola, 30 Aug 2022
Status: closed

RC1: 'Comment on egusphere2022569', Anonymous Referee #1, 09 Aug 2022
Review of "Celestial Mechanics and Estimating the Termination of the Holocene"
# Summary
The article claims (in the abstract) to address several issues regarding the Milankovitch theory and "its relationship to palaeoclimate data over the last 800,000 years". The author observes that a substantial number of papers have connected astronomical parameters with "palaeoclimate data" but still fail to completely account for the interglacial and glacial durations, and the "timing" of prominent excursion in palaeoclimate data. He also remarks that some form of "pattern recognition" has enabled "quantitative estimates of the Holocene termination" but implies that this pattern recognition has not been successfully been applied to other terminations.
The originality of the approach presented here is to examine the percentage change between successive meandailyinsolation maxima and minima at 65N. This metric is, in principle defined only at successive insolation maxima, but it is being interpolated and then partitioned into its precession and obliquity components, which themselves are characterised by cycle durations (e.g. his Figures 4, 5, 6).
Figure 12 suggests a match between the precession component of the variations in the metric, and the duration of glacialinterglacial cycles. The match is admittedly intriguing (if not perfect) but the article does not provide a physical interpretation. A second result (Figures 13 and 14) suggests a (less clear) match between the obliquity component of the metric and the amplitude of interglacial maxima. Timings match at the cost of a 10,000 year temporal shift, which the author attributes to timing inaccuracies.
The author then comments on the timing of the next glacial inception.
# Comments## Position with respect to the state of the art
Quite a number of phenomenological models connecting astronomical forcing have been reasonably convincing at predicting the timing of glacialinterglacial transitions (many of these models take the form a dynamical system) but it is fair that the duration of interglacials, and the interglacial/glacial transitions may have received a bit less attention than the timing of terminations. The argument about "pattern recognition" brought about by the author is not obvious to me because, precisely, such rules are derived by examining all interglacials, see, e.g.,
Tzedakis P. C., J. E. T. Channell, D. A. Hodell, H. F. Kleiven and L. C. Skinner (2012), Determining the natural length of the current~interglacial, Nature Geoscience, (5) 138141 doi:10.1038/ngeo1358 and Tzedakis P. C., E. W. Wolff, L. C. Skinner, V. Brovkin, D. A. Hodell, J. F. McManus and D. Raynaud (2012), Can we predict the duration of an interglacial?, Climate of the Past, (8) 1473–1485 doi:10.5194/cp814732012
## No physical modelling
It should be made clear that the current contribution focuses entirely on the one hand, on insolation at 65N and astronomical components, and, on the other hand, the EPICA Dome C Deuterium curve. There are no explicitly physical assumption linking both. When the author uses the word 'model', it is to name the "deconvolution model" that decomposes the metric into its obliquity and precession components. One decomposition is based on arithmetic developments (eqs 1 to 10) and compared with a numerical deconvolution.
I could not read a justification for using 'percent change between two successive maxima' as a reference metric. There is no argument about its physical relevance, and there is no empirical evidence (here or in the literature) that using this metric provides superior results to other, more standard, metrics. For example, the obliquity imprint on the EDC curve would be more straightforwardly explained as an influence of annual mean insolation (controlled by obliquity) on Southern Ocean seasurface temperatures.
## Lack of method description
Numerical deconvolution and estimates of instantaneous amplitude and frequencies have become fairly standard in astronomical theory of palaeoclimates and cyclostratigraphy. The author does not give much detail about the numerical methods used here. He explains that he uses "Laskar's tool", which is a reference to the IMCCE website. As far as I could tell the website generates insolation for usersupplied latitudes and seasons but (again as far as I could tell) does not make demodulation.
## Questions about equations equation (1) featuring a product between obliquity and precession is not justified. Insolation anomalies would have come much more naturally as as sum of obliquity and precession components, not a product.
 I suspect indices a and p are swapped in equations (4) and (6)## Lose semantics
 the author says that between maxima the curve "should not be trusted for numerical precision", but given that the percent change in successive maxima is only defined at maximum points, the meaning to be given to "numerical precision" is not clear
 some sentences are clumsy. E.g. l. 136: "According to the Milankovitch hypothesis, their determination [I understand, of the behaviours of astronomical parameters] provides a consistent temporal calibration that should correlate insolation changes with features of palaeoclimate data. Things could have been stated more clearly. The discussion about "ameliorating timing differences" through "constructive and destructive interference" is somewhat obscure. l. 214: "While the time scale of the precession index contribution to insolation is affected by eccentricity, its shortterm halfcycle is primarily due to precession" remained impenetrable I am afraid. l. 547: "both MIS 18d and 13c represent deep ice cores". With some good will one can see what the author is referring to, but semantics are inaccurate.
 aperiodic and quasiperiodic seem to be used interchangeably, while they should not.
Conclusions about the "catastrophic consequences to future of civilization from another ice age" really need to be recast and put in context given that the catastrophe we are all facing is definitely not that of a coming ice age.Relevance of Appendix A is arguable and seem to be mostly standard material about insolation but I would leave it to the editor.
# Conclusion
A lot of work has been put in this article and I feel sorry to come with a recommendation that sounds negative. Given the countless possibilities given by playing with insolation curves and data, it seems to me that new metrics have to be introduced with care, and the possible link with the palaeoclimate proxy for climate, which they are supposed to explain, have to be physically justified. This is not the case here.
Citation: https://doi.org/10.5194/egusphere2022569RC1 
AC1: 'Reply on RC1', John Parmentola, 22 Aug 2022
Review of "Celestial Mechanics and Estimating the Termination of the Holocene"
# Summary
The article claims (in the abstract) to address several issues regarding the Milankovitch theory and "its relationship to palaeoclimate data over the last 800,000 years". The author observes that a substantial number of papers have connected astronomical parameters with "palaeoclimate data" but still fail to completely account for the interglacial and glacial durations, and the "timing" of prominent excursion in palaeoclimate data. He also remarks that some form of "pattern recognition" has enabled "quantitative estimates of the Holocene termination" but implies that this pattern recognition has not been successfully been applied to other terminations.
The originality of the approach presented here is to examine the percentage change between successive meandailyinsolation maxima and minima at 65N. This metric is, in principle defined only at successive insolation maxima, but it is being interpolated and then partitioned into its precession and obliquity components, which themselves are characterised by cycle durations (e.g. his Figures 4, 5, 6).
The statements in the above paragraph are inconsistent with the paper. The deconvolution (partitioning) is not between the precession and obliquity components. It's between the precession index (PI), sometimes referred to as the climate precession, and the obliquity. There is a significant difference. The PI is the precession modulated by the eccentricity, which is shown in the paper to dominate the insolation. There are many papers that conclude the obliquity dominates. There is important physics in the PI because it provides the characteristic timescale relevant to the behaviors observed in the paleoclimate data and makes the largest amplitude contribution to the insolation primarily due to changes in the eccentricity over a precession halfcycle, which is not constant as the paper demonstrates.
Also, Figures 4,5, and 6 represent the halfcycle behaviors of the three celestial parameters, the eccentricity, precession, and obliquity, and not their cyclical behaviors. The reason as stated in the paper for this is to make clear that these celestial parameters are quasiperiodic, and the quasiperiodicity is important for making a time series comparison between the deconvolution model predictions and the paleoclimate proxy data.
I believe a major point has been missed here. The insolation is inherently a wave phenomenon as described in the paper. Physically the three characteristic celestial motions "beat" on the solar irradiance, which causes it to be distributed over the earth's surface in the form of the insolation.
The point of the paper is to describe the insolation as a timedependent wave resulting from the interference between the PIwave and the obliquity (O) wave.
Figure 12 suggests a match between the precession component of the variations in the metric, and the duration of glacialinterglacial cycles. The match is admittedly intriguing (if not perfect) but the article does not provide a physical interpretation. A second result (Figures 13 and 14) suggests a (less clear) match between the obliquity component of the metric and the amplitude of interglacial maxima. Timings match at the cost of a 10,000 year temporal shift, which the author attributes to timing inaccuracies.
Again, the reference to the precession contradicts the paper. It's the PI contribution that is plotted. The suggested "perfect match" by the reviewer of Figure 12 is not perfect because the plot only involves the PIwave contribution. However, the "approximate alignment" of the PIwavepackets is no accident. The PI contribution depends upon the quasiperiodic behaviors of the precession and eccentricity, which is important for accounting for the approximate recurrence of the prominent features in the paleoclimate data. Figure 12 represents the comparison of time series data with the time series prediction of the deconvolution model.
The purpose of Figures 12 and 13 is to highlight a phase difference between the PI and Owaves. That is described in the text. The 10,000 year shift in the Owave is offset by the PIwave through interference to reproduce the beat structure in Figure 3. It has nothing to do with timing discrepancies as suggested by the reviewer.
The insolation undergoes 74 transitions between maxima and minima over 800,000 years with significantly fewer prominent temperature excursions in the paleoclimate proxy data. Why? Are there special insolation changes of a certain magnitude that are reflected in prominent features in the paleoclimate proxy data. Or could there be trends in the insolation that cause interglacial inceptions and terminations. Examining the percentage change between insolation maxima and minima could identify those special insolation transitions that are dominant. Also, the interpolation between the percentage changes provides trends in the insolation.
The discovery in this paper is that by deconvolving the PI and O contributions to the percentage change between insolation maxima and minima, the dominant insolation effects are identified, and the trends are revealed. The inception and termination of interglacial inceptions and terminations are primarily due to trends in the PIwave contribution that is enhanced or reduced by the Owave through interference. This approach provides a novel interpretation of Milankovitch Theory based on the physics of insolation as a wave. The reviewer appears to missed this point.
The author then comments on the timing of the next glacial inception.
# Comments## Position with respect to the state of the art
Quite a number of phenomenological models connecting astronomical forcing have been reasonably convincing at predicting the timing of glacialinterglacial transitions (many of these models take the form a dynamical system) but it is fair that the duration of interglacials, and the interglacial/glacial transitions may have received a bit less attention than the timing of terminations. The argument about "pattern recognition" brought about by the author is not obvious to me because, precisely, such rules are derived by examining all interglacials, see, e.g.,
Tzedakis P. C., J. E. T. Channell, D. A. Hodell, H. F. Kleiven and L. C. Skinner (2012), Determining the natural length of the current~interglacial, Nature Geoscience, (5) 138141 doi:10.1038/ngeo1358 and Tzedakis P. C., E. W. Wolff, L. C. Skinner, V. Brovkin, D. A. Hodell, J. F. McManus and D. Raynaud (2012), Can we predict the duration of an interglacial?, Climate of the Past, (8) 1473–1485 doi:10.5194/cp814732012
The above comment confounds several issues. My paper's single reference to pattern recognition on page 2 has to do with several papers that examine the similar behaviors of the eccentricity, precession, and obliquity to propose other MIS analogs to MIS 1. One of those papers suggests that the MIS 1 termination will be like MIS 19c. As I discuss later in my paper, unlike that paper, my paper takes a comprehensive approach over 800,000 years using the deconvolution model to interglacial inceptions, terminations, durations, the timing of their recurrences, etc. It demonstrates that there are consistent and recurring insolation conditions that account for these phenomena through a dynamic model.
My paper is simply looking at the insolation at 65N during June across 800,000 years based on the Milankovitch hypothesis. That hypothesis alleges that the changes in the insolation at northern latitudes during the summer solstice is the cause of ice ages. Many people have described this hypothesis. My paper explores it through two assumptions.
The first involves the assumption that the percentage change between successive meandailyinsolation maxima and minima at 65 degrees northern (65N) latitude during the summer solstice (June) over the last 800,000 years, substantially influenced the prominent features in paleoclimate data. Those features are interglacial inceptions, terminations, durations, the timing of their recurrence, their classification into two types based on wave interference, etc.
The second involves the deconvolution model, which is a simple physical model. The model is based on the comparatively small role the obliquity plays in contributing to the insolation, which essentially shifts the sun's rays north and south by 2.4 degrees. It changes the angle of the sun's rays with respect to the vertical and the insolation distribution over the earth's surface. That is demonstrated by the derivation of its contribution in Appendix A. The insolation amplitude is primarily influenced by the precession index because of changes in eccentricity and the direction the earth's axis points during the summer and winter solstices. The timescale of obliquity changes is approximately twice that of the precession index. Therefore, in first approximation the obliquity is constant. The model simply multiplies these contributions to obtain an approximation to the insolation. Its magnitude does not matter because the deconvolution model calculates the percentage change. The justification for this assumption is born out in the results both in terms of the numerical precision compared to Laskar et al and the qualitative and quantitative behaviors of the PI and Owave contributions to the insolation.
Is the reviewer asking that I compare my model with all other models? That would seem to be unreasonable given the extent of the current paper and its novel assumptions. It could be a comprehensive followon paper that would require reconciling different models.
## No physical modelling
It should be made clear that the current contribution focuses entirely on the one hand, on insolation at 65N and astronomical components, and, on the other hand, the EPICA Dome C Deuterium curve. There are no explicitly physical assumption linking both. When the author uses the word 'model', it is to name the "deconvolution model" that decomposes the metric into its obliquity and precession components. One decomposition is based on arithmetic developments (eqs 1 to 10) and compared with a numerical deconvolution.
The deconvolution model and the EPICA Dome C (EDC) data are connected through Milankovitch hypothesis. The deconvolution model explores the validity of the that hypothesis through a comparison of the model and the paleoclimate data. As I describe above, the assumptions of the paper are two and the motivation for them is reasonably clear. If the reviewer would like more elaboration in the paper as I indicated above, I am happy to do that.
I do not understand the remark about equations (110) being arithmetic developments. Those equations follow from the physical description earlier in the paper concerning the roles of the precession index and obliquity with respect to the insolation. The calculations contain the relevant celestial parameters, and the results reveal the dependency of the percentage change between insolation extrema at 65N during June on these physical parameters. There are no arbitrary parameters, and the celestial parameters are specified at a given time by the precise calculations of Laskar et al. The model results for the percentage change between meandaily insolation extrema at 65N during June are compared to the same quantity as Laskar et al showing very good agreement Figure 7.
Furthermore, the estimated PIwave and Owave results are consistent with the assumptions of the model and the approximation made to neglect higher order terms in equation (3).
The EDC is a temperature reconstruction of paleoclimate proxy data. It represents Marine Isotope Stages some of which have been identified with interglacial inceptions and terminations. All paleoclimate data are model dependent; however, there is remarkable consistency amongst the data sets, although there are timing discrepancies among them. For example, there are slight timing discrepancies between the EDC and Vostok data sets. As Parennin et al point out there are timing discrepancies between EDC and LR04 data sets, which I identify in the paper.
I could not read a justification for using 'percent change between two successive maxima' as a reference metric. There is no argument about its physical relevance, and there is no empirical evidence (here or in the literature) that using this metric provides superior results to other, more standard, metrics. For example, the obliquity imprint on the EDC curve would be more straightforwardly explained as an influence of annual mean insolation (controlled by obliquity) on Southern Ocean seasurface temperatures.
## Lack of method description
Numerical deconvolution and estimates of instantaneous amplitude and frequencies have become fairly standard in astronomical theory of palaeoclimates and cyclostratigraphy. The author does not give much detail about the numerical methods used here. He explains that he uses "Laskar's tool", which is a reference to the IMCCE website. As far as I could tell the website generates insolation for usersupplied latitudes and seasons but (again as far as I could tell) does not make demodulation.
I don't understand these statements. Every number contained in this paper follows from Laskar's tool on the IMCCE website. That tool is based on extensive work done by Laskar et al over many years. The references to the relevant papers are on the website. As an example, a user can generate the climate precession (also known as the precession index) and divide it by the eccentricity to produce the precession.
## Questions about equations equation (1) featuring a product between obliquity and precession is not justified. Insolation anomalies would have come much more naturally as as sum of obliquity and precession components, not a product.
 I suspect indices a and p are swapped in equations (4) and (6)Again, the reviewer's terminology is inconsistent with the paper. It's not the precession but rather the precession index. As argued above, the obliquity contribution is slowly varying, so to obtain an approximation to the insolation it's the product of the precession index contribution and the obliquity that is relevant. All the analysis that follows is demonstrated to be consistent with equation (1). Regarding the reviewer's additivity remark, note equation (3). All the results that follow from equation (1) are justified numerically as evident from Figure 8. Note also, the socalled arithmetic reveals approximations that are quite insightful given the complexity of celestial mechanical calculations of the insolation.
The reason for the swapping of indices is physics. There are two types of transitions between maxima and minima, one from perihelion to aphelion and the reverse.
## Lose semantics
 the author says that between maxima the curve "should not be trusted for numerical precision", but given that the percent change in successive maxima is only defined at maximum points, the meaning to be given to "numerical precision" is not clear
 some sentences are clumsy. E.g. l. 136: "According to the Milankovitch hypothesis, their determination [I understand, of the behaviours of astronomical parameters] provides a consistent temporal calibration that should correlate insolation changes with features of palaeoclimate data. Things could have been stated more clearly. The discussion about "ameliorating timing differences" through "constructive and destructive interference" is somewhat obscure. l. 214: "While the time scale of the precession index contribution to insolation is affected by eccentricity, its shortterm halfcycle is primarily due to precession" remained impenetrable I am afraid. l. 547: "both MIS 18d and 13c represent deep ice cores". With some good will one can see what the author is referring to, but semantics are inaccurate.
 aperiodic and quasiperiodic seem to be used interchangeably, while they should not.
Conclusions about the "catastrophic consequences to future of civilization from another ice age" really need to be recast and put in context given that the catastrophe we are all facing is definitely not that of a coming ice age.Relevance of Appendix A is arguable and seem to be mostly standard material about insolation but I would leave it to the editor.
I do not understand these remarks. The PIwave and Owave are computed at a sparce set of points. The accuracy of those points in time is set by Laskar's tool. The rest is interpolation. The paper is simply pointing out that the numerical accuracy between the extrema points is not precise. In fact, the paper simply uses the interpolation qualitatively to interpret the paleoclimate data. In all honesty, it's quite remarkable how well the model results consistently interpret the paleoclimate data over 800,000 years. All interglacial terminations occur in the same manner based on the PIwave and Owave behaviors. The only timing discrepancies occur with two interglacial inceptions, namely, MIS 18d and MIS 13c. These by the nature of ice cores are deep, so I do not understand the semantic issue. Depth in ice cores is related to time; however, the association of time at ice core depth is model dependent because of physical effects. As the paper points out, timing discrepancies are to be expected since paleoclimate data is model dependent. To support this statement, the paper identifies the work of Parrenin et al that demonstrates such a discrepancy between two sets of paleoclimate data that occur during the inceptions of MIS 18d and MIS 13c. Given the consistency of the deconvolution model with the paleoclimate data over 800,000 years, the paper suggests that the data be reexamined.
It appears that the reviewer may be confused about the precession index. It is the modulation of the precession by the eccentricity. There are two different timescales involved here. However, the insolation transitions from maxima to minima approximately coincide with the precession halfcycles. The paper demonstrates that the PIwave is dominant, and the transitions are primarily due to eccentricity changes over the precession halfcycles.
Regarding aperiodic versus quasiperiodic, the latter is an example of aperiodic behavior. If the reviewer prefers, a simple word change can be accomplished by replacing aperiodic with quasiperiodic throughout the paper.
The paper constrains the conclusions by specifying that the forcing is only celestial mechanical. The paper also points only provides recurring insolation conditions associated with terminations over the last 800,000 years. The earth's climate response to this is unknown. The catastrophe has happened many times before based on celestial mechanical forcing. By inductive reasoning, it could happen again, and the PI wave packets in Figure 12 make that case very clearly.
The paper speculates that the instability of the polar vortex may be the earth's climate response but that is currently an active area of research. We do not understand the earth's climate well enough to predict what that mechanism is. Basically, there is no complete theory of the earth's climate. There are models which are not equivalent to a theory.
Regarding Appendix A, I would appreciate a reference that goes through the computation with all the assumptions made.
# Conclusion
A lot of work has been put in this article and I feel sorry to come with a recommendation that sounds negative. Given the countless possibilities given by playing with insolation curves and data, it seems to me that new metrics have to be introduced with care, and the possible link with the palaeoclimate proxy for climate, which they are supposed to explain, have to be physically justified. This is not the case here.
The linkage between the behavior of the insolation and paleoclimate data is the Milankovitch
hypothesis, which many other authors have explored. The deconvolution model results support that hypothesis using a wave formulation of the insolation resulting in a novel interpretation of the paleoclimate data having remarkable consistency over the last 800,000 years. There are many new results among them are interglacial classifications, the relationship between recurring PI wave packets and recurring features in the paleoclimate data, estimations of interglacial terminations, an estimation of the MIS 1 termination, etc. That latter is a lowresolution estimate of 500 years based on the model described in the paper.
The paper is consistent and the results unexpected through a novel interpretation of paleoclimate data using the Milankovitch hypothesis.
Citation: https://doi.org/10.5194/egusphere2022569AC1

AC1: 'Reply on RC1', John Parmentola, 22 Aug 2022

RC2: 'Comment on egusphere2022569', Anonymous Referee #2, 15 Aug 2022
The first issue here is terminological, but is indicative of major problems throughout the paper:In mathematics and physics including geophysics, generally, "convolution" and "deconvolution" refer to a specific mathematical operation, equivalent to multiplying and dividing fourier transforms. The author seems to think a "deconvolution" model means an "explanatory" model, but if there is any deconvolution or convolution going on here, I've completely missed it. The "model" is a quasiinstantaneous linear forcing represented through the simple Eq. 3 with no reference to the physics or chemistry of the climate system.More generally, this manuscript seems to represent a big step backwards: among otherissues, the author implies, without ever saying so, that the climate system on these timescales is linear. But even the original Hayes et al. paper referred to Milankovich "pacing" ofglacial intervals, recognizing that the relationship between forcing and response was notthat of a linear system. In the intervening time interval, a number of papers (see for example,Tziperman et al., 2006, Paleoceanography, or various papers by Peter Ditlevsen) have dealt explicitly with the anticipated results of forcing a nonlinear system with quasiperiodic deterministic driving. None of this is even mentioned.The paper is really a collection of wiggly lines. But a huge literature exists on the response of linear and nonlinear systems to deterministic narrowband (in the Fourier sense) forcing. (Studies of ordinary ocean tides are one much discussed, analogous, application dating back to the 19th Century.) The author repeatedly invokes "correlation" between two wiggly linesall based on some perceived visual resemblance. But the human eye is notoriously poor at separating apparent patterns from real ones (a branch of psychology known as pareidolia), and for that reason, specific statistical tools have been developed to obtain objective tests. It is a truism of data analysis that even two unrelated records will show a correlation, both numerical and visual, and hence a level of no significance is *always* necessary. In most cases here, one wonders what would have led the author to reject his own hypothesis of apparent correlation? The model being used is fully deterministic and the distribution e.g., of zerocrossing times, is also deterministic unless the system is nonlinear and chaotic. Deviations from claimed correlations are explained away by armwaving stories (lines 517+ are one example). The use of halfcycles, rather than the usual use of frequencies, here would imply some kind of obscure rectification mechanism at work. Furthermore, the paper takes no account of any kind of noise in the ice core records, including issues of dating.Smaller issues:the 65degree N dependence is invoked without any discussion of its relevance, particularly to an individual ice core.No error bars are placed on any of the ice core timings (e.g, Fig. 1)Several figures refer to curve fits to data points (e.g., line 150), but with no specificationof what constituted those curves.Why don't the author's hypotheses operate in the interval before 1MY BP, when theglacialinterglacial intervals are widely believed to have occurred only at ~40KY?Was the precession index ever defined? Is A truly timeindependent?Line 97 and elsewhere, what the author means by aperiodic would be the simple beating of two or more nearby frequencieseach individually periodic.Line 299. Why is it surprising?Line 413. What is meant by "judiciously"?Citation: https://doi.org/
10.5194/egusphere2022569RC2 
AC2: 'Reply on RC2', John Parmentola, 30 Aug 2022
The first issue here is terminological, but is indicative of major problems throughout the paper:
In mathematics and physics including geophysics, generally, "convolution" and "deconvolution" refer to a specific mathematical operation, equivalent to multiplying and dividing fourier transforms. The author seems to think a "deconvolution" model means an "explanatory" model, but if there is any deconvolution or convolution going on here, I've completely missed it. The "model" is a quasiinstantaneous linear forcing represented through the simple Eq. 3 with no reference to the physics or chemistry of the climate system.
I do not understand what the reviewer means by an explanatory model. Deconvolution is used in the sense of partitioning two effects associated with insolation, namely the contribution from the obliquity and the other from the precession index (the precession modulated by the eccentricity). It is not immediately obvious that such a partitioning is quantitatively valid, but systematic computations presented in the paper bear this out as represented by Figures 7 and 8.
This partitioning is performed in temporal space unlike the typical frequency space analyses. There are physical reasons for this choice. The first has to do with the EDC data, which is represented as a time series. As I discuss in the introduction section of the paper, typical frequency analyses of the data and insolation reveal characteristic frequencies, and these are associated with the average cycles of the three celestial parameters. My paper treats the behavior of the three celestial parameters as they are in nature. They are quasiperiodic functions of time with wide ranging halfcycles as demonstrated by the numerous graphs in the paper. A temporal comparison between the EDC time series data and the deconvolution model requires that the obliquity and precession index contributions to the insolation be treated as quasiperiodic functions of time. This approach is born out in Figure 12 where I show that the precession index contribution to the percentage change in insolation between maxima and minima correlate with the temperature trends and interglacialglacial periods quite well. This result is no accident.
The deconvolution model is an approximate physical description of the insolation. The model is based on the comparatively small role the obliquity plays in contributing to the insolation. It essentially shifts the sun’s rays north and south by 2.4 degrees. It changes the angle of the sun’s rays with respect to the vertical and primarily the angular distribution of the insolation over the earth’s surface. That is demonstrated by the derivation of its contribution in Appendix A. The insolation amplitude is primarily influenced by the precession index because of changes in eccentricity and the direction the earth’s axis points during the summer solstice. The timescale of obliquity changes is approximately twice that of the precession index. Therefore, in first approximation the obliquity is constant. The model simply multiplies these contributions to obtain an approximation to the insolation. The insolation magnitude does not matter because the percentage change in the insolation between extrema is calculated and used to compare with the EDC data. The justification for this approximation is born out in the results both in terms of the numerical precision compared to Laskar et al and the qualitative and quantitative behaviors of the PI and Owave contributions to the insolation and their comparison with the data.
The paper demonstrates that the precession index dominates the insolation contrary to many papers in the field of Milankovitch Theory. Even Milankovitch believed the obliquity was the dominant effect in the cause of ice ages.
I have no idea what the reviewer means by “a quasiinstantaneous linear forcing represented through the simple Eq. 3.” Linear in what parameter? Equation (3) is nonlinear in the obliquity and precession index contributions to the percentage change in the mean daily insolation between maxima and minima. It’s clearly not linear in time or the three celestial parameters as born out in the subsequent equations to equation (3). What does the reviewer mean by quasiinstantaneous? The estimates of Laskar et al for the insolation and the three celestial parameters have a resolution of 100 years as I state in the paper. Is this what the reviewer means by quasiinstantaneous? What is the point of this comment?
Is the reviewer objecting to the simplicity of the model? What matters are the physical assumptions, the translation of those assumptions into mathematics, and the comparison with the data. That is the scientific method. The paper makes two key assumptions.
The first involves the assumption that the percentage change between successive meandailyinsolation maxima and minima at 65 degrees northern (65N) latitude during the summer solstice (June) over the last 800,000 years, substantially influenced the prominent features in paleoclimate data. Those features are interglacial inceptions, terminations, and durations, the timing of their recurrence, their classification into two types based on wave interference, etc.
The second involves the deconvolution model, which is a simple physical model as described above. The rest of the paper describes the consequences of these assumptions and semiquantitative comparisons to the EDC time series data.
The paper is not a climate model. As discussed in the abstract and the introduction section, the paper explores the Milankovitch hypothesis, which focuses on the mean daily insolation at 65 N degrees latitude during the summer solstice. The issue is what does the behavior of the insolation tell us about the occurrence of interglacial and glacial periods over the last 800,000 years. Are there quantitative and qualitative behaviors of the insolation that account for prominent features in the paleoclimate data such as the timing of interglacial inceptions and terminations, interglacial durations, the recurrence of interglacialglacial periods, etc. The issue of how the earth’s climate responds to insolation changes is not the purpose of this paper. That is a totally different matter. The specific climate mechanism that causes an interglacial termination is unknown. We do not understand the earth’s climate well enough to predict how it responds to insolation changes and there is no calibration of insolation to temperature. The paper merely is establishing recurring patterns in the behavior of the insolation that correlate with the prominent features in the EDC time series data.
There maybe an important point that has been missed by the reviewer. The insolation is inherently a wave phenomenon, which is emphasized in the paper. It is an amplitude modulated wave in time. This wave description involves the interference between the obliquity and precession index waves as defined in the paper that accounts for both quantitative and qualitative features in the paleoclimate data. It appears to be a useful language to describe the prominent features in the paleoclimate data based on the Milankovitch hypothesis. It’s a new and novel contribution to the field.
More generally, this manuscript seems to represent a big step backwards: among other
issues, the author implies, without ever saying so, that the climate system on these time
scales is linear. But even the original Hayes et al. paper referred to Milankovich "pacing" of
glacial intervals, recognizing that the relationship between forcing and response was not
that of a linear system. In the intervening time interval, a number of papers (see for example,
Tziperman et al., 2006, Paleoceanography, or various papers by Peter Ditlevsen) have dealt explicitly with the anticipated results of forcing a nonlinear system with quasiperiodic deterministic driving. None of this is even mentioned.
I have no idea what the reviewer means by this paper implicitly assumes that the climate system on these time scales is linear. Linear in what parameter? The reviewer seems to miss the point that the paper is not a climate model. The paper is simply exploring the Milankovitch hypothesis through a model that partitions the obliquity and precession index contributions to the percentage change in the mean daily insolation at 65N latitude during the summer solstice. The model provides strong support for the Milankovitch hypothesis. If there is a valid theory or model of the earth’s climate that incorporates a response to insolation force according to the patterns established in this paper, then that would be a major advance in understanding the cause of ice ages. I know of no such validated theory or model. If the reviewer knows of one, I would appreciate a reference.
The paper is really a collection of wiggly lines. But a huge literature exists on the response of linear and nonlinear systems to deterministic narrowband (in the Fourier sense) forcing. (Studies of ordinary ocean tides are one much discussed, analogous, application dating back to the 19th Century.) The author repeatedly invokes "correlation" between two wiggly linesall based on some perceived visual resemblance. But the human eye is notoriously poor at separating apparent patterns from real ones (a branch of psychology known as pareidolia), and for that reason, specific statistical tools have been developed to obtain objective tests. It is a truism of data analysis that even two unrelated records will show a correlation, both numerical and visual, and hence a level of no significance is *always* necessary. In most cases here, one wonders what would have led the author to reject his own hypothesis of apparent correlation? The model being used is fully deterministic and the distribution e.g., of zerocrossing times, is also deterministic unless the system is nonlinear and chaotic. Deviations from claimed correlations are explained away by armwaving stories (lines 517+ are one example). The use of halfcycles, rather than the usual use of frequencies, here would imply some kind of obscure rectification mechanism at work. Furthermore, the paper takes no account of any kind of noise in the ice core records, including issues of dating.
I cannot make logical sense of these confounding comments. Are the assumptions of the paper wrong and what specifically is wrong in terms of physics? Is the translation of the assumptions into mathematics and the model wrong and what is specifically wrong? The reviewer seems to be stuck on frequencies as the only approach to understanding paleoclimate data. My paper takes a time series approach because of the quasiperiodic nature of the three celestial parameters and the fact that there is more information in such an approach to compare with data. What is obscure about this?
The paper points out the very limited number of points that represent predictions of the model. There are not enough points to perform a statistical comparison with the data. Furthermore, the history of science is fundamentally a search for patterns and the development of theories to explain the patterns. Would the reviewer consider the experiments of Thomas Young concerning the interference patterns of light as a wave a statistical aberration? Explaining those patterns took about a century through the development of quantum theory that could fundamentally account for those patterns quantitatively and qualitatively. Richard Feynman has hand waving lectures that describe all of this without using a single formula. There is more to a good physical argument than most people think.
My paper is based on two assumptions as clearly described above and in the paper. Celestial behaviors, i.e., physics, are used to translate these assumptions into mathematics to define the model. The approximations are justified quantitatively in the paper based on wellestablished benchmark computations of Laskar et al. The results are used both quantitatively and qualitatively to identify the prominent features in the EDC time series data. Wave interference, a novel aspect of the paper, is used to interpret the data. The model accounts approximately for the recurrence of interglacialglacial periods over the last 800,000 years, temperature trends, the timing of interglacial inceptions and terminations, the duration of interglacial periods, the classification of interglacial periods into two types, and quantitative estimates of interglacial terminations. All interglacial terminations consistently occur vin the same manner over the last 800,000 years.
Smaller issues:
the 65degree N dependence is invoked without any discussion of its relevance, particularly to an individual ice core.
Many (A. Berger et al) have explored the relationship between insolation predictions at 65N during the summer solstice from celestial mechanics and their correlation with paleoclimate data. This paper simply uses EDC data, which is a temperature model rendering of ice core data. There is remarkable consistency amongst the various paleoclimate data sets, however there are timing issues as I point out in the paper.
No error bars are placed on any of the ice core timings (e.g, Fig. 1)
Several figures refer to curve fits to data points (e.g., line 150), but with no specification
of what constituted those curves.
As pointed out in the paper, the EDC ice core data is a temperature rendering of the paleoclimate data published on the NOAA site as referenced. The timing issues with these data are well known as I indicate in the paper. The issue is how well can a model account for the prominent features in the data. Expecting precision from this data is inconsistent with the methods used to infer it. At best, a model comparison with the data can be semiquantitative. That is the approach taken in the paper and the results obtained worthy of publication.
The caption under Figure 3 and other Figures explains the origin of the graphs. There is a reference, which directs the reader to a wellestablished computational tool that is used to create the graph based on the simple definition of what is computed.
Why don't the author's hypotheses operate in the interval before 1MY BP, when the
glacialinterglacial intervals are widely believed to have occurred only at ~40KY?
The earth was undergoing a major transition likely due to the Isthmus of Panama (about 2.3 million years ago), which changed the dynamic mixing of the oceans. The average global temperature steadily dropped, and the temperature excursions steadily increased likely due to the increasing accumulation of snow and ice at the poles. Eventually, the earth stabilized into the more recent pattern of ice ages over the last 800,000 years. The model required to describe the earlier period would be different because the physical circumstances were different.
Was the precession index ever defined? Is A truly timeindependent?
It’s defined in the abstract and it’s clearly not time independent because it depends on the eccentricity and precession. Anyone familiar with the subject of this paper and celestial mechanics would know this.
Line 97 and elsewhere, what the author means by aperiodic would be the simple beating of two or more nearby frequencieseach individually periodic.
The paper demonstrates that the three celestial parameters are aperiodic. What two or more nearby frequencies that are periodic is the reviewer talking about?
Line 299. Why is it surprising?
Given the derivation in Appendix A, it is surprising that the obliquity contribution to the insolation is well approximated by the formula provided.
Line 413. What is meant by "judiciously"?
The vertical lines are chosen based on the recurring pattern in the PIwave.
Citation: https://doi.org/10.5194/egusphere2022569AC2

AC2: 'Reply on RC2', John Parmentola, 30 Aug 2022
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