A sequential Bayesian approach for the estimation of the age–depth relationship of the Dome Fuji ice core

Nakano, Shin'ya; Suzuki, Kazue; Kawamura, Kenji; Parrenin, Frédéric; Higuchi, Tomoyuki

doi:https://doi.org/10.5194/npg-23-31-2016

Articles | Volume 23, issue 1

https://doi.org/10.5194/npg-23-31-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/npg-23-31-2016

© Author(s) 2016. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 23, issue 1

Research article

|

29 Feb 2016

Research article |

| 29 Feb 2016

A sequential Bayesian approach for the estimation of the age–depth relationship of the Dome Fuji ice core

Shin'ya Nakano, Kazue Suzuki, Kenji Kawamura, Frédéric Parrenin, and Tomoyuki Higuchi

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Final revised paper (published on 29 Feb 2016)
Preprint (discussion started on 26 Jun 2015)

Interactive discussion

Status: closed

AC: Author comment | RC: Referee comment | SC: Short comment | EC: Editor comment

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- Supplement

RC C293: 'review of sequential Bayesian approach for the estimation of the age-depth relationship', Anonymous Referee #1, 27 Jul 2015
- AC C548: 'Reply to the comments by Referee #1', Shinya Nakano, 16 Nov 2015
RC C377: 'Review of ”A sequential Bayesian approach for the estimation of the age-depth relationship of Dome Fuji ice core” by S. Nakano et al', Anonymous Referee #2, 05 Sep 2015
- AC C549: 'Reply to the comments by Referee #2', Shinya Nakano, 16 Nov 2015
RC C418: 'Referee comment', Anonymous Referee #3, 14 Sep 2015
- AC C550: 'Reply to the comments by Referee #3', Shinya Nakano, 16 Nov 2015

Peer-review completion

AR: Author's response | RR: Referee report | ED: Editor decision

AR by Shinya Nakano on behalf of the Authors (21 Dec 2015) Author's response Manuscript

ED: Reconsider after major revisions (further review by Editor and Referees) (22 Dec 2015) by Ana M. Mancho

AR by Shinya Nakano on behalf of the Authors (04 Jan 2016) Author's response Manuscript

ED: Referee Nomination & Report Request started (05 Jan 2016) by Ana M. Mancho

RR by Anonymous Referee #1 (15 Jan 2016)

Suggestions for revision or reasons for rejection

I believe that the new version is much better. It contains more details and interesting comments, while being generally clearer than the previous version.

Minor remarks:
- page 5, line 135 and below. As in my previous review, I point out that there
is nothing Bayesian about a state space model. The Bayesian aspect comes in
later, when priors are put on parameters theta.
- Likewise, the distribution p(x_{0:z} | y_{1:z}, theta) is referred to as the
posterior distribution (e.g. page 7 line 188), but it is customary to call it
the filtering distribution (on paths), or the smoothing distribution. The
term "conditional distribution", used page 8 line 224, is neutral and
might be preferred.
- Likewise, "sequential Bayesian model" are usually called state space models
(e.g. around Eq 23).
- page 7, line 210. I believe you
should indicate that you consider the uniform distribution on the real line
for each parameter, also referred to as the flat prior (as opposed to the uniform
distribution on an interval). Thus you use what is called an improper prior.
- page 10 line 284 "can accordingly obtained" + be
- page 11 line 320: note that if you needed to store the whole trajectories,
there would be efficient ways to do so (Jacob, P. E., Murray, L. M., &
Rubenthaler, S. (2013). Path storage in the particle filter. Statistics and
Computing, 25(2), 487-496.).
- page 13 line 363: "is sufficiently converged" should be rewritten...
perhaps "is considered reliable" ?
- I believe the language is very understandable, I didn't see many spelling
mistakes, but some sentences are a bit clumsy. For instance, I believe that
most of the articles "the" could be safely deleted ("the MCMC", "the SMC",
...).

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RR by Anonymous Referee #2 (28 Jan 2016)

Suggestions for revision or reasons for rejection

Review of “A sequential Bayesian approach for the estimation of the age-depth relationship of Dome Fuji ice core” by Nakano et al

In the revised version of the manuscript, the authors have corrected some errors in the flow model equations, and have provided several improved explanations. Overall, the manuscript is significantly improved and I recommend its publication with some minor changes, most of these concern the readability of the paper and a clarification of the underlying model assumptions.

The manuscript would benefit from being written more to-the-point. Several places – especially in the newly added sections – the text contain redundancies. I have noted some of these under “technical comments”, but they exist throughout the text. I encourage the authors to carefully go through the manuscript and remove such redundancies.

Additionally, I strongly suggest the authors to improve the readability of the paper by starting each section with an introductory paragraph explaining “what is the underlying purpose of the equations that we’re going to develop now”. While such explanations do exist in the paper, many of these are currently located towards the end of the section, after the equations have been developed. E.g. the idea behind introducing error functions is described almost parenthetically in line 130-133, after the ice dynamic equations have been developed. Another example of this is e.g. on page 8, line 245.

On a similar note, it would improve readability if variables were introduced in the text before, rather than after, the equation in which they are used.

To clarify how the PMCMC method is used, I further suggest the authors to explicitly provide the sequence of steps taken in the algorithm, either in the text as a series of numbered steps, or in a figure as e.g. a flow diagram.

Apart from these more general comments, the paper lacks a thorough discussion of the following two points:

The paper lacks a paragraph stating explicitly which glaciological parameters are assumed to be (approximately) constant in time, and which parameters are allowed to vary. Currently, this important information is not obvious from reading the manuscript. This is especially important to clarify since the age model equations are developed under the assumption of steady state, an assumption which is subsequently relaxed with e.g. the accumulation rate allowed to change over time. It appears that the authors assume the horizontal velocity profile (described by variables p, s) to stay constant over time, as well as the thickness of the ice sheet, H. Also µ is held constant in time, and with that also the basal melt rates (please correct me if I’m wrong). Consequently, the entire thinning profile is held constant in time, although this is not consistent with changes in accumulation. I agree that these are good starting approximations, but all this should be clear from the text. I encourage the authors to explicitly state, in a single paragraph, all the assumptions that go into the development of the age-model equations.

A second issue concerns the employed tie points. The authors have now included some information about the tie points, but I’m still missing some discussion about the nature of these, and, not least, their associated uncertainties.

Most of the employed tie points are age estimates derived from O2/N2 cycles. These tie points have an absolute uncertainty. However, many tie points commonly used for providing age constraints for ice core timescales are not based on N2O2 measurements. Indeed, probably the most common age constraint for ice cores is based on volcanic tie points; volcanic acidity horizons that can be correlated to sulfate horizons in other ice cores that are more accurately dated. Such more accurate ice core timescales may e.g. have been developed by counting annual layers in the ice core data. In this case, however, the uncertainty of the age of a tie point will no longer be absolute. Instead, one would have an estimate of the uncertainty of the time interval (i.e. the number of layers wrongly counted/not counted) between two age markers. Would it be possible to include such age markers in the current version of the proposed model? I don’t think so, since as they would imply a relations between the ages of the ice core at different depths, in contradiction with the discussion at P. 9, line 270-272. I hence suggest the authors to include in the paper a discussion on which type of tie points can be used in the model (and to mention the origin of the uppermost two tie points).

I have below outlined some smaller comments, and pointed out sections where the text need to be tightened up to improve the readability of the paper.

Specific comments:

P.1, line 7: Explain what the d18O data is used for (this is not trivial information)

P. 1, line 4, and other places: The accumulation process (i.e. “how does the snow fall”) is not important here - what we don’t know is how the accumulation rate has changed over time. This should be made clear, here as well as in the rest of the paper.

P. 3, line 66-68, and other places: While I understand what the authors mean by A(z), this should be more carefully described: A(z) is the accumulation rate at the time when the ice now at depth z was deposited. Similarly, in the rest of the paper, the authors still on several occasions mention “accumulation at the surface, A0” (e.g. P. 4, line 90, P. 2, line 337). Accumulation always take place at the surface! Please use correct wording; A0 is present accumulation rate, and A(z) could e.g. be described as “accumulation rate corresponding to the ice at depth z”. Also, note that the proper term for A is not accumulation, but accumulation rate.

P. 3, line 88: The Lliboutry equation is put forward without any justification

P. 4, line 107: What kind of age markers are the two upper tiepoints? This should be mentioned, as it is important for the way their uncertainties are incorporated in the model. Additionally, this information could be included (for all tie points) in table 2.

P. 8, line 220: I do not understand the sentence “the posterior .. d18O variation”. The sentence seems to imply that the discrepancy between actual accumulation and that predicted from isotopes is caused by something that has to do with the isotope data. But the differences are because the relationship between isotopes and accumulation is only approximate. Please revise the sentence to reflect this.

P. 10, line 296-289: Why is the value of lambda chosen as 1/5? Are there 5 times as many isotope data points to age markers? How does the resulting age model depend on the chosen value of lambda? Is it robust?

P. 10, line 290-292: This is the kind of information that should go into an introductory paragraph of the section.

P. 11, line 299: Are also the variance parameters assumed to be belong to zero-mean Gaussian distributions? If so, the variances may become negative.

P. 11, line 303: I assume that these variances were selected to produce reasonable acceptance rates during the MC sampling. If so, this should be stated. In that case, the way that these are selected should not influence the results (but would affect the burn-in period and the rapidity that the algorithm is moving around in parameter space). If absolutely required, I suggest these numbers to be placed in a table instead.

P. 12, line 333: It does not seem likely that every 5th sample is independent. Later, the authors (P. 15, line 433) quote an acceptance rate of 26%, meaning that on average every 5th sample might be different - but only slightly - and they will definitely not be independent. As the Metropolis-Hastings sampling requires approximately independent samples, it would be preferable for obtaining proper statistics that samples were selected further apart. This is even more important when the authors test the algorithm when using fewer particles (P. 14-15), in which case they quote an acceptance rate of only 7.6%, making every 5th sample highly dependent (indeed, often every 5th sample would be the same as the previous).

P. 12, line 340: Provide the physical interpretation of the results for these parameters. For instance, with µ being approximately 0, this implies a most likely basal melt rate of zero. This is important glaciological information and should be noted.

P. 12, line 346-348, P. 13, line 384-386: Collect these sentences in a single paragraph about the differences to the model used in Parrenin et al (2007).

P. 14, line 408-410: These differences might also have to do with the large covariance between exactly these three variables.

P. 14, line 426-437: Since this run with fewer particles has not converged, I’m not convinced how much information these results contain. I suggest to significantly reduce this paragraph, as well as remove figure 12.

P. 15, line 464: A better proposal distribution – for what?

P. 15: Part of the text in the conclusion should be moved to the discussion

Technical corrections:

P. 1, line 18-20: Write sentence more to-the-point, remove redundancy

P. 2, line 24-26: same

P. 2, line 33: Klauenberg et al (2011) also produces a timescale - not just “accumulation and some parameters from d18O”. Which are these “other parameters”?

P. 2, line 57: section 6 -> Section 6 and 7

P. 3, line 59: Remove “It is thought that”. It is unquestionable that the age-depth relationship is determined by accumulation and thinning.

P. 3, line 54: Remove “in year”

P. 3, line 64: z is later referred to in units of m, not cm. Which unit is used?

P. 3, line 65: The word “past” refers to time, not depth, hence z cannot be referred to as positive towards the past.

P. 3, line 81, P. 4, line 115: The values used for H and Z are appropriate only to Dome Fuji, and should be mentioned in section about Dome Fuji model results, not here.

P. 4, line 98-104: This paragraph should be written more to-the-point. Further, it should be made abundantly clear that the accumulation here is allowed to change over time, and that this is in contrary to the assumption of steady state, based on which the thinning rate equation is developed.

P.4, line 105: Remove “more reliable age information”. Only the age markers provide age information (d18O provides accumulation information).

P. 4, line 118, line 128: Provide information here on the distributions of ν_z, η_z

P. 5, line 121-123: Correct the grammar, remove redundancies

P. 6, line 156-162: Write more to-the-point, remove redundancies as well as the apparent contradiction (“accumulation and thinning might be affected … accumulation and thinning would not be sensitive”). Indeed, I agree that any depth uncertainty on the tie points could to large extent be compensated by increasing the age uncertainties. The effect of such depth uncertainties would, however, depend on the proximity of the age markers.

P. 6, line 175-181: Remove redundancies

P. 8, line 225: Spell out the name of PMCMC

P. 9, line 255-256: Redundancy

P. 10, approx. line 277: No need to include this middle step of the equation, since it is not used in the following.

P. 11, line 301: What is the difference between theta’ and theta*?

P. 12, line 337, 239: Provide mass balance in the same unit in the two places.

P. 13, line 366-369, figure 4: Since all four lines are on top of each other, it makes no sense to include them all on the figure. Their differences are shown well in figure 5.

Table 1: This table should include a description for all the employed variables, also a, b, σ_v etc.

Figure 7: Units for the accumulation data (top panel) are missing. Blue line is not mentioned in figure text. Label on the axis is likely wrong (according to the text, it shows accumulation differences).

Figure 8: Same

Figure 10: Does not contain any new information, figure can be removed. Instead it would be interesting to see a figure similar to figure 5 for the case where the five lowermost age markers are removed.

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ED: Publish subject to minor revisions (further review by Editor) (28 Jan 2016) by Ana M. Mancho

AR by Shinya Nakano on behalf of the Authors (07 Feb 2016) Author's response Manuscript

ED: Publish as is (12 Feb 2016) by Ana M. Mancho

AR by Shinya Nakano on behalf of the Authors (14 Feb 2016) Manuscript

Short summary

This paper proposes a technique for dating an ice core. The proposed technique employs a hybrid method combining the sequential Monte Carlo method and the Markov chain Monte Carlo method, which is referred to as the particle Markov chain Monte Carlo method. The sequential Monte Carlo method, which is also known as the particle filter, is widely used for nonlinear time-series analysis. This paper demonstrates the usefulness of the approach in time-series analysis for dating an ice core.