Bispectra of climate cycles show how ice ages are fuelled

Liebrand, Diederik; de Bakker, Anouk T. M.

doi:https://doi.org/10.5194/cp-15-1959-2019

Articles | Volume 15, issue 6

https://doi.org/10.5194/cp-15-1959-2019

© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/cp-15-1959-2019

© Author(s) 2019. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 15, issue 6

Research article

|

22 Nov 2019

Research article |

| 22 Nov 2019

Bispectra of climate cycles show how ice ages are fuelled

Diederik Liebrand and Anouk T. M. de Bakker

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Final revised paper (published on 22 Nov 2019)
Supplement to the final revised paper
Preprint (discussion started on 24 Apr 2019)
Supplement to the preprint

Interactive discussion

Status: closed

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

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- Supplement

RC1: 'Review', Michel Crucifix, 14 May 2019
- AC1: 'Rebuttal to M. Crucifix', Diederik Liebrand, 11 Aug 2019
RC2: 'Interactive comment on “Bispectra of climate cycles show how ice ages are fuelled” by Diederik Liebrand and Anouk T. M. de Bakker', Mathieu Martinez, 12 Jul 2019
- AC2: 'Rebuttal to M. Martinez', Diederik Liebrand, 11 Aug 2019

Peer-review completion

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

ED: Reconsider after major revisions (19 Aug 2019) by Qiuzhen Yin

AR by Diederik Liebrand on behalf of the Authors (25 Aug 2019)

ED: Referee Nomination & Report Request started (27 Aug 2019) by Qiuzhen Yin

RR by Anonymous Referee #1 (11 Sep 2019)

RR by Mathieu Martinez (29 Sep 2019)

Suggestions for revision or reasons for rejection

In their revised manuscript, Drs. Liebrand and de Bakker made substantial modifications which clarified the method and the interpretations in terms of climate dynamics. Clarification is still possible by defining terms when they are used for the first time in the manuscript. Nonetheless, the manuscript is a great contribution tothe understandng of Plio-Pleistocene climatic changes and will be suitable for publication after minor revisions would be done. Below are my detailed comments:

1) A possible mistake I saw in the revised version, is the definition of the kurtosis. Kurtosis, which, in the absence of outlier, measures how flat or peaked is the top of a distribution of values, ranges between 1 and the infinite. For a normal distribution, the kurtosis value is 3. For a flat-top distribution, the kurtosis is 1. Then, themore peaked is a distribution, the higher is the kurtosis. Subtracting 3 to the kurtosis value means the normal distribution is centred to 0. In that case, what is measured is called the excess kurtosis which ranges from -2 to infinite. See for instance Collis et al. (1998). If the authors decide to keep this formulation, they should replace "kurtosis" by "excess kurtosis" throughout the text and in Figure 2.

2) In Fig. 4c and in section 3.2., how there can be in the same bispectrum the following triad interactions: 𝐵Imp(40↑, ∞↑, 40↓) (Line 20) and BImp(40↓, ∞↓, 40↑) (Line 24)? ow can a triad interaction be positive and negative in the same time?

3) Page 7, lines 24: section 4.3. does not exist. Do the authors refer to secton 3.4. instead?

4) In the method section, I think that defining terms which are not reorganizing ideas or paragraphs would make the text muh straightfoward to read. For instance, „conservative energy exchange“ is an important concept in the manuscript. This is mentioned for the time in page 4 but only defined in page 13. In my opinion, the concepts the authors use should be defined when they are mentioned for the time in the manuscript. Below, I list possible rephrasing and additional explanations in the method section. These additional explanation may be more inclusive for readers who are interested in Pleistocene climate changes but feel not confident in higher order statistics. The authors can feel free to account them or modify them if misunderstanding appeared to come from me:

2.2. Quantifying geometries using central moments

The geometry of a distribution of values can be quantified by a infinity of statistical moments. Generally, only the first four moments are used. These four statistical moments are respectively the average, the variance, the skewness (or asymmetry) and the kurtosis. Here, the cycles are assimilated to a distribution of values through time. The skewness is here defined as a disymmetry of a cycle relatively to a horizontal axis (Fig. 2b). The asymetry is here the dissymetry of a cycle through time (Fig. 2c). The kurtosis quantifies the flatness of the extrema of a cycle. Flat-top (flat-bottom) cycles have low-kurtosis values, while sharp-top (sharp-bottom) cycles have a high kurtosis value. Kurtosis ranges from 1 to infinite. A Gaussian curve has a kurtosis value of 3. Thus, the deviation of a curve to the Gaussian shape can be calculate by the "excess kurtosis" which is defined as follows: excess kurtosis = kurtosis – 3. Using third-moment quantities, skewness is determined by Eq. (1)...

2.3.1. The bispectrum

The Fourier Transform calculates a spectrum showing the distribution of variance (being related to power and energy, as power is the energy per time unit) with frequency. The Fourier Transforms is however unable to document higher order statistical moments of the signal considered and, for this, higher order spectral analyses must be done. Bispectral analyses describe the distribution of nonsinusoidality with frequency, in both the real and imaginary parts (King, 1996). The skewness of a cycle geometry is related to the real part of the bispectrum while the asymetry is related to its imaginary part (Fig. 2). The bispectrum shows nonconservative, relative energy exchanges among frequencies of a single time series. Conservative energy exchanges are here defined as exchanges of energy and relative energy exchanges as … (I think this is important to introduce the terms here otherwise, many readers may be lost depending on their own usage of the terminology). Energy transfers can occur ...

2.3.2. Interpreting the bispectrum

Page 5, Line 30: „The former are the so-called difference frequencies, while the latter is referred to as a sum frequency“: using „former“ and „latter“ can be confusing beause it is not always to what they refer. I usually tend to avoid these terms. I would write instead: f1 and f2 are so-called difference frequencies, while f3 is referred to as sum frequency“.

I find the subsection „2.3.2 Interpreting the bispectrum“ very practical and useful. Nonetheless, in this subsection, the notions of energy gains and losses are mentioned, while they are defined further in subsection 2.3.4. In my opinion. It would be more logical to define the bispectrum, the calculation of geometries from the bispectrum, the energy exchanges, and then show how to interpret the bispectrum. Again, this organisation should ensure that terms and concepts are defined when they are mentioned for the first time in the manuscript, which is important.

Reference:

Collis, W.B., White, P.R., Hammon, J.K., 1998. Higher-order spectra: the bispectrum and the trispectrum. Mechanical Systems and Signal Processing 12(3), 375-394.

Referee Report: PDF

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ED: Publish subject to minor revisions (review by editor) (04 Oct 2019) by Qiuzhen Yin

AR by Diederik Liebrand on behalf of the Authors (15 Oct 2019) Author's response Manuscript

ED: Publish as is (17 Oct 2019) by Qiuzhen Yin

AR by Diederik Liebrand on behalf of the Authors (17 Oct 2019) Author's response Manuscript

Short summary

We present a new analysis and interpretation of a well-established climate record that spans the past 5 million years. We describe how the energy the Earth receives from the Sun is transferred among climate cycles with different duration. This analysis offers new insights into the complex evolution of the global climate system and land-ice volumes during this time. Furthermore, it provides a more complete solution to the long-standing 40 000- and ~100 000-year problems of the ice ages.