The authors have addressed several of my concerns regarding the methodology and improved their manuscript significantly. However, there remain a number of points in the methodology that in my opinion require further consideration:
1. I disagree with the authors’ statement that they can investigate more change-points than the number defined in the model (which is 6+2). Each realization of the MCMC fits the data with n change-points. Hence, the PDF is also only true for the defined n. In their reply to my previous comment, the authors state that they generate one continuous PDF for the likelihood of a given x being a change-point, and that this PDF can have multiple (more than n) peaks. It is of course true, that there can be more than n peaks in the PDF, but some of them will be modes of the same change-point and thus, mutually exclusive. It would thus be more informative to generate one PDF per change-point. The way MCMC works, I am surprised that this posterior distribution for each change-point is not standard output of the method.
This would affect the error estimates on the change-points and thus, the inferred delays. Furthermore, it would affect the significance test. Currently, the absolute probability threshold does not measure significance but precision. We can easily imagine how few iterations placing a change-point in the same location create a high and narrow peak in the PDF. On the other hand, many iterations placing a change-point into a slightly more variable location will result in a wider and thus, lower peak in the PDF. The cumulative probability is however higher in the latter example. Eventually, it is not a question of how many significant change-points there are: By defining n change-points, the answer is n. The question would then be whether a model with more/less change-points leads to a significantly improved fit to the data.
Ideally, I would like the authors to address this and show posterior distribution per change-point. However, given the tests with more (SI) and less (SI of previous version of the manuscript) change-points I can also see how treating the change-points separately would probably not change the results drastically. Thus, I would appreciate if the authors could include both figures (more and less change-points, the latter also updated with the AR-model of this version of the manuscript) in the supplementary materials and add a short section in the main text, where they discuss the differences with respect to change-points and the resulting delays. I would appreciate if the authors could be more explicit/quantitative in their discussion of the differences than in this version of the manuscript (P8, L5-6: “probability distributions with 8 and 10 points are rather similar.”) since the choice of number of change-points is ultimately arbitrary and represents an uncertainty in the method.
2. The AR-model should also be applied to the filtered data.
In the manuscript the authors state (P 10, L11-12): “In the two filtered series, the sub-millenial scale AR(1) noise present in the original series should be essentially removed. As such, fitting change points to these two series, assuming the residuals to be uncorrelated, provides a second form of verification of the appropriateness of the covariance matrix we use to fit the raw data.”
This is not true. Savitsky-Golay filters are smoothing filters that do not remove, but introduce autocorrelation. It is obvious that data filtered with a 500yr cutoff but supplied at 200yr resolution cannot be white noise. Furthermore, the method itself estimates autocorrelation (equation 2 and 3) and thus, if “a” in eq. 2 and 3 was indeed zero, then the method should detect it, set “a” to zero and hence, treat it as white noise automatically.
Please, redo the analysis and figure 4 accordingly.
3. The authors point out correctly, that the PDFs of the change-points are multimodal and skewed and should hence not be reported as μ±σ. Looking at figure 6 I believe the same is true for the lead/lag PDFs. Please, throughout the manuscript, provide the most likely values (probability peak) and the 68.2/95.4% probability intervals.
4. I apologize for not commenting on this earlier: Could you rearrange the figures so that the time-series and the histograms do not overlap with each other, but are connected with X-gridlines? As it is now, some features discussed in the manuscript are impossible to see in the figures.
Specific Comments:
P1, L7: Delete “abrupt”. The method makes no estimate of the abruptness of changes.
P1, L11: replace “nearly synchronously” with a more precise “within xx years [68.2% probability]” – the uncertainty is quite big for this change.
P1, L11: “250 years”: Figure 6 reads -188±154? Please correct (also according to main comment #3)
P1, L12: “after the ACR onset in the temperature record”: I think one of the interesting results of this paper is, that this change point is actually only present in East Antarctica (EDC and TD), which could be worthwhile mentioning in the abstract?
P2, L3: enter spaces before the two en dash uses.
P4, L23 and P11, L. 24: Bintanja et al. 2013 is cited as a reference for the correction of water isotope records for source isotopic variations. Is this the correct reference? They do not study isotopes, and focus on the last 30 years? Please check.
Page 5, figure caption 2: “ratio of the age difference”. I still don’t entirely understand this formulation. Do you mean the ratio of the age difference between neighbouring tie-points on WDC2014 and the same tie-points on EDC (on AICC12?). I.e., if the ratio is 1.01, then the duration between tie-points differs by 1% between the two independent timescales? Please clarify.
Page 5, figure caption 2: The 20% uncertainty is assumed to be 1 sigma? If so please change to: “…which is determined as 20% (1σ) of the distance to the nearest tie point, …”
P6, L6: “…and can improve the balance of precision and accuracy of the fits.” What do you mean? Delete?
P6, L7: “related with” replace “related to”
P6, L9: “first source of uncertainty” replace with “measurement uncertainty”
P6, L15-17: “These challenges can be circumvented…” That would only be true if we knew that the resolution was lower than the decorrelation time. Is this the case? Looking at figure 4 in Parrenin et al. also there the residuals are autocorrelated. Delete?
P7, L32-33: replace “confirm” with “test” and “is accurate” with “cannot be rejected”.
P7, L33: “(Supplement)” what does this refer to? On that note: in the supplementary P1, L15, it reads that “a” is set to “2”? How do these two sections relate to each other? Is “a” estimated as described in eq. 2-3 or is it set to a constant value? Why would it be constant? I cannot find the motivation to set “a” to 2 in Goodman and Weare 2010.
P7, L37: “(IQR(R)/1.349)^2”: Please explain IQR (Inter-Quartile-Range?) and where the 1.349 is coming from.
P8, L4-6: See main comment #1. Yes, there can be more modes, but eventually there can only be as many change-points as allowed by the user (6+2). Please rephrase, and provide a short section (here or elsewhere in the manuscript) discussing differences arising from the choice of allowed change-points.
P8, L8: “second derivative of the fits”: I don’t understand. The fits are straight lines, so the second derivative is always 0?
P8, L8-16: I do not understand what is done here. 1000 change points at once? Or 1000x1 random change-point? Or 1000 x (6+2) random change-points? How do the histograms reflect the approximate slope? Please clarify this section.
P9, Figure caption 3: Line 3, typo “Rrobability”; Line 5: “the sum” - linear? Quadrature?
P9, L36: “one at the ACR onset”: see main comment #4 – this is impossible to see.
P10, L44-45: see main comment #4 – impossible to see.
P10, L48-49: “roughly concurrent with the downward CO2 change point” replace with “concurrent with the second mode of the downward CO2 change point” Also, this statement is at odds with P12, L16, where you state, that you do not identify a CO2 change-point. Please check for consistency.
P10, L1 (50?, line-numbering is off): “No corresponding change point is detected for CO2”. This is briefly discussed in section 3.2 but maybe it is already here worthwhile mentioning that this change point is only present in East Antarctic temperature records?
P10, L4: “The Holocene onset is well defined”. I suppose the way the Holocene is defined it cannot be seen in Antarctic isotope records. Maybe replace with “The end of the deglacial warming is well-defined..”
P10, L10-13: See main comment #2.
P11, L24: See earlier – is Bintanja et al. 2013 the correct reference?
P11, L30: “rather broad, non-significant”. That is by design. See comment #1. Maybe change to: “rather broad and hence, non-significant.”
P12, L41: “…with temperature change at WD appearing to slightly precede temperature in the East Antarctic records,…”. Omit the second use of “temperature” or change to “temperature change”
Figure 6: The PDFs in the top-left and bottom-right panels are the same plot. Please check/replace.
Figure 6 and throughout this section: See main comment #3.
P13, L4-5: Mention again, that interestingly this is not the case for Dome Fuji?
P13, L12: Mention again that the 14k change-point is not present everywhere?
P14, L15-16: “was most likely synchronous” change to: “occurred within xx years of the CO2 change”
P14, L16-17: “Further, we do not identify an analog in CO2 of the marked temperature decrease…”. But figure 3 and 4 do show downward histograms there? As mentioned in main comment #1, I think this is related to how significance is defined. There seems to be a downward change-point that is just not very well constrained in time, and hence, yields a low and wide probability distribution. See also earlier comment – check for consistency with P10, L48-49.
P14, L37: Within error also the ACR onset “allows” for the CO2-ATS phasing to be reversed. And for both events, phasing is zero within error.
P14, L38: “But the phasings are opposite in direction and different in magnitude”: This sentence is unclear. Do you mean that CO2 and CH4 change in opposite directions? Why is it relevant that the magnitude of CO2 and CH4 change differs, and how do you define the magnitude for 2 gases that have very different background concentrations?
P14, L42: “None of the five isotopic records show significant probability in this region” change to: “We do not detect a significant change-point for any of the isotope records during this period.”
Figure 7: Please also add a dashed vertical line to the oldest change-point/Mt. Takahe eruption that extends through all panels.
P15, L3: “only roughly consistent” Can you please be more precise?
P15, L11, “ice-air shift” Please replace with “delta-age”
P16, L21. Also add, that the end of deglacial warming occurs later in DF? |