Assessment of the influence of astronomical cyclicity on sedimentation processes in Eastern Paratethys based on paleomagnetic measurements using Discrete Mathematical Analysis

Dzeboev, Boris; Odintsova, Anastasia; Rybkina, Alena

doi:https://doi.org/10.5194/gi-2021-13

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https://doi.org/10.5194/gi-2021-13

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12 Jul 2021

| 12 Jul 2021

Status: this preprint has been withdrawn by the authors.

Assessment of the influence of astronomical cyclicity on sedimentation processes in Eastern Paratethys based on paleomagnetic measurements using Discrete Mathematical Analysis

Boris Dzeboev, Anastasia Odintsova, and Alena Rybkina

Abstract. The main objective of this study is to apply Discrete Mathematical Analysis (DMA) to the development of the methodology of cyclostratigraphy. This aim is supported by exploring the magnetic properties of rocks, the lithology of sediments and obtained geochronological reference definitions. The analysis was based on measurements of the variability of the magnetic susceptibility of rocks, which reflects climate variations. Astronomical cycles are global; this makes it possible to carry out a correlation analysis over a large area and on different facial types of sediments, considering their lithology and other sedimentary features. The introduction of modern methods of mathematical processing of geological data is one of the prospective areas for investigation and development in geoscience. Astronomical cycles can be revealed from measurements of scalar magnetic parameters of rocks (magnetic susceptibility as presented by the authors). Specific software developed by the authors allows the processing of measurement data and assessment of the presence of stable oscillation cycles based on the obtained measurement base. The present study attempts to apply mathematical methods to magnetic data using the existing PAST program, which allows spectral analysis of primary data with the construction of Lomb-Scargle and REDFIT periodograms. We interpret the spectral analysis data based on paleomagnetic determinations, considering the available dates for the boundaries of direct and reverse polarity chrons on a general stratigraphic scale.

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Received: 05 May 2021 – Discussion started: 12 Jul 2021

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Boris Dzeboev, Anastasia Odintsova, and Alena Rybkina

Interactive discussion

Status: closed

RC1:
'Comment on gi-2021-13', Anonymous Referee #1, 20 Jul 2021
The presented work concerns adaptation of a new DMA method for solving problems in cyclostratigraphy. Authors apply a smart mathematical tool that rests on the analysis of spatial periods and try to detect global astronomical cycles. Despite a clear presentation and an original approach, the article requires a major revision.

The main issues are as follows:

The work describes the validity of frequency maxima for Fourier and Lomb periodograms, but there is not a single word about the validity of DMA periods. The validity criterion of the new method should be described in detail, especially as concerns the detection of several characteristic periods (line 280), not a single one ('global' minima, line 276).

The article should show advances of the new method over the widely used Fourier transform (uniform data) and Lomb-Scargle periodogram (un-uniform data). It seems, resting only on the information presented in the article, that the DMA method loses to the above-mentioned transformations.

Authors use both frequencies and periods to present the results. This complicates the perception. A single physical quantity should be used. Also a uniform presentation should be used in figures for convenience.

A more detailed explanation is required to the transformation of spatial periods of DMA method into time estimations of cyclicity.

A detailed description of the generally used discrete Fourier transform (lines 109-122) is worth to be reduced, especially taking into account the thing that a mathematical algorithm is described in words.

With the above issues the article can be recommended for publication after a major revision.
Citation: https://doi.org/10.5194/gi-2021-13-RC1
RC2: 'Comment on gi-2021-13', Anonymous Referee #2, 05 Sep 2021

I recommend that this paper be returned to the authors so that they may

1) consider estimation of uncertainty / significance for consistency analysis

2) consider estimation of uncertainty / significance for periodicity matching

3) clearly state assumptions required to relate external forcing to sediment deposition rates

4) remove material (eg. DMA) that is not directly required for analysis.

++++ Summary +++++

The abstract states that:

"the main objective of this study is to apply Discrete Mathematical Analysis (DMA)

to the development of the methodology of cyclostratigraphy."

with a concise summary of cyclostratigraphy provided subsequently in lines [90-105].

In this study, three Fourier analysis methods are used to identify variations in

magnetic susceptibility of geological sediments. Fourier results are compared to

periodicity estimates obtained with a different method: "constancy" analysis.

The first method described in lines [106-119] is essentially standard Fourier

analysis that is well suited for detecting harmonics in evenly sampled data.

The second method is the Lomb-Scargle periodogram which was designed to deal with unevenly

spaced or gappy data.

The third method (REDFIT) is an extension of the Lomb-Scargle method to include the

presence of background red noise.

Some results are shown in section 3 for a reference data set:

[189-190] "This series contains the last 400,000 years of the record in 1,000-year

increments (Figure 1)."

Some discussion of the selection of this data set would be nice:

[190-191] "This series was chosen for demonstrating the application of spectral methods as

it has been comprehensively studied."

Does this mean that periodicity in this data set is well understood? Citation(s) would seem appropriate.

The Fourier and Lomb-Scargle power spectra are similar but not identical.

In the limit of regular sampling the results should be the same. This

indicates that there must be some data gaps or uneven spacing, which is not

consistent with the description of the data set.

Periodicity detection is also carried out with "constancy" analysis as described

in a fairly abstract presentation [155-184] that does not address uncertainties

or significance estimates.

The largest minimum in Figure 3 occurs at 115,000 which is within 10% of the

largest Fourier result. Other minima are not particularly noteworthy

("strong" [182]), and overall there is not an unambiguous 1:1 match with all of the Fourier results.

For this data set constancy analysis seem reasonably effective for the largest

amplitude periodic variation, but is otherwise not clearly superior to the Fourier

methods.

The analysis is then applied to two sediment layers (41.2 and 30m respectively) at 20cm spacing.

Results appear much noisier than first example. All Fourier methods are generally in agreement

with spatial periodicity of 3.97m and 22.52m for first, 7m for second. As before, the Lomb-Scargle

results should be the same as classic Fourier analysis for evenly spaced data but is not.

Constancy analysis for the first layer has a fairly clear minimum around 3.6m and a very weak

minimum at 15.2m; the second layer has a decent constancy minimum at 7m.

The authors assert that:

[299-301] "These signals correspond to the precession and obliquity cycles.

The 3.6 m peak corresponds to the precession periodicity (19–24 thousand years). The 7 m peak

corresponds to the periods of changes in the angle of inclination of the Earth’s axis (41,000 years)."

However, it is first necessary to address the question of why both layers do not exhibit exactly the same periodicities.

As noted by [Sinnesael et al 2018 Paleoceanography and Paleoclimatology, 33, 493–510]

"In practice, sedimentation rate rarely changes in a uniform way.

Moreover, the response of the climatic system can change the amplitude ratio

among the astronomical components of the original insolation forcing signal."

Different deposition rates might be in effect for each of the two layers,

which might help explain the different observations. In fact, there could in principle be significant

variations which could map astronomical periodicities to sediment variations with significantly

different power spectra. If the mapping is sufficiently non-linear as to change

the periodicities between layers then it could also cause distortion within layers, which

could invalidate any hope of matching sedimentary periods with Milankovitch cycles.

Finally, DMA is not clearly defined in this report or in what is supposed to be a

canonical reference [Agayan et al 2018].

DMA is introduced in lines 146-154:

"The proposed technique is based on DMA [Agayan et al., 2018]. The new method is an original

technique for analysing discrete data, developed at the GC RAS. DMA is a series of algorithms united

by a common formal basis: fuzzy comparisons of numbers, a measure of proximity in discrete spaces,

and a discrete limit. DMA was developed to create discrete equivalents of the concepts of classical

mathematical analysis: for example, limit, continuity, smoothness, connectivity, monotonicity, and

extremum. DMA methods and algorithms have proven to be useful in numerous studies related to the

processing and analysis of various [...] data."

Similarly bold statements occur near the end of the paper:

[311-315] "It is necessary to say in conclusion a few words, about the mathematical apparatus of DMA

[Agayan et al., 2018]. It is being developed at the Geophysical Center of the Russian Academy of

Sciences and forms the basis of the methodology for searching periods in the series of cyclostratigraphic

data presented in the article. It is important to emphasize that DMA is a direction of modern applied

systems analysis [Zgurovsky and Pankratova, 2007]."

[316-320] "DMA has all the necessary tools to generate mining algorithms for geological and geophysical

data, including searching for hidden periods / cycles. Based on fuzzy sets and fuzzy logic, DMA has the

ability to convey expert ideas about the structure, morphology, monotony, and other of studied data

series. Thus, DMA enables a systematic approach to the analysis of complex data series of Earth

sciences."

While this may be true, none of it is supported by the results of this paper.

Citation: https://doi.org/10.5194/gi-2021-13-RC2

Interactive discussion

Status: closed

RC1:
'Comment on gi-2021-13', Anonymous Referee #1, 20 Jul 2021
The presented work concerns adaptation of a new DMA method for solving problems in cyclostratigraphy. Authors apply a smart mathematical tool that rests on the analysis of spatial periods and try to detect global astronomical cycles. Despite a clear presentation and an original approach, the article requires a major revision.

The main issues are as follows:

The work describes the validity of frequency maxima for Fourier and Lomb periodograms, but there is not a single word about the validity of DMA periods. The validity criterion of the new method should be described in detail, especially as concerns the detection of several characteristic periods (line 280), not a single one ('global' minima, line 276).

The article should show advances of the new method over the widely used Fourier transform (uniform data) and Lomb-Scargle periodogram (un-uniform data). It seems, resting only on the information presented in the article, that the DMA method loses to the above-mentioned transformations.

Authors use both frequencies and periods to present the results. This complicates the perception. A single physical quantity should be used. Also a uniform presentation should be used in figures for convenience.

A more detailed explanation is required to the transformation of spatial periods of DMA method into time estimations of cyclicity.

A detailed description of the generally used discrete Fourier transform (lines 109-122) is worth to be reduced, especially taking into account the thing that a mathematical algorithm is described in words.

With the above issues the article can be recommended for publication after a major revision.
Citation: https://doi.org/10.5194/gi-2021-13-RC1
RC2: 'Comment on gi-2021-13', Anonymous Referee #2, 05 Sep 2021

I recommend that this paper be returned to the authors so that they may

1) consider estimation of uncertainty / significance for consistency analysis

2) consider estimation of uncertainty / significance for periodicity matching

3) clearly state assumptions required to relate external forcing to sediment deposition rates

4) remove material (eg. DMA) that is not directly required for analysis.

++++ Summary +++++

The abstract states that:

"the main objective of this study is to apply Discrete Mathematical Analysis (DMA)

to the development of the methodology of cyclostratigraphy."

with a concise summary of cyclostratigraphy provided subsequently in lines [90-105].

In this study, three Fourier analysis methods are used to identify variations in

magnetic susceptibility of geological sediments. Fourier results are compared to

periodicity estimates obtained with a different method: "constancy" analysis.

The first method described in lines [106-119] is essentially standard Fourier

analysis that is well suited for detecting harmonics in evenly sampled data.

The second method is the Lomb-Scargle periodogram which was designed to deal with unevenly

spaced or gappy data.

The third method (REDFIT) is an extension of the Lomb-Scargle method to include the

presence of background red noise.

Some results are shown in section 3 for a reference data set:

[189-190] "This series contains the last 400,000 years of the record in 1,000-year

increments (Figure 1)."

Some discussion of the selection of this data set would be nice:

[190-191] "This series was chosen for demonstrating the application of spectral methods as

it has been comprehensively studied."

Does this mean that periodicity in this data set is well understood? Citation(s) would seem appropriate.

The Fourier and Lomb-Scargle power spectra are similar but not identical.

In the limit of regular sampling the results should be the same. This

indicates that there must be some data gaps or uneven spacing, which is not

consistent with the description of the data set.

Periodicity detection is also carried out with "constancy" analysis as described

in a fairly abstract presentation [155-184] that does not address uncertainties

or significance estimates.

The largest minimum in Figure 3 occurs at 115,000 which is within 10% of the

largest Fourier result. Other minima are not particularly noteworthy

("strong" [182]), and overall there is not an unambiguous 1:1 match with all of the Fourier results.

For this data set constancy analysis seem reasonably effective for the largest

amplitude periodic variation, but is otherwise not clearly superior to the Fourier

methods.

The analysis is then applied to two sediment layers (41.2 and 30m respectively) at 20cm spacing.

Results appear much noisier than first example. All Fourier methods are generally in agreement

with spatial periodicity of 3.97m and 22.52m for first, 7m for second. As before, the Lomb-Scargle

results should be the same as classic Fourier analysis for evenly spaced data but is not.

Constancy analysis for the first layer has a fairly clear minimum around 3.6m and a very weak

minimum at 15.2m; the second layer has a decent constancy minimum at 7m.

The authors assert that:

[299-301] "These signals correspond to the precession and obliquity cycles.

The 3.6 m peak corresponds to the precession periodicity (19–24 thousand years). The 7 m peak

corresponds to the periods of changes in the angle of inclination of the Earth’s axis (41,000 years)."

However, it is first necessary to address the question of why both layers do not exhibit exactly the same periodicities.

As noted by [Sinnesael et al 2018 Paleoceanography and Paleoclimatology, 33, 493–510]

"In practice, sedimentation rate rarely changes in a uniform way.

Moreover, the response of the climatic system can change the amplitude ratio

among the astronomical components of the original insolation forcing signal."

Different deposition rates might be in effect for each of the two layers,

which might help explain the different observations. In fact, there could in principle be significant

variations which could map astronomical periodicities to sediment variations with significantly

different power spectra. If the mapping is sufficiently non-linear as to change

the periodicities between layers then it could also cause distortion within layers, which

could invalidate any hope of matching sedimentary periods with Milankovitch cycles.

Finally, DMA is not clearly defined in this report or in what is supposed to be a

canonical reference [Agayan et al 2018].

DMA is introduced in lines 146-154:

"The proposed technique is based on DMA [Agayan et al., 2018]. The new method is an original

technique for analysing discrete data, developed at the GC RAS. DMA is a series of algorithms united

by a common formal basis: fuzzy comparisons of numbers, a measure of proximity in discrete spaces,

and a discrete limit. DMA was developed to create discrete equivalents of the concepts of classical

mathematical analysis: for example, limit, continuity, smoothness, connectivity, monotonicity, and

extremum. DMA methods and algorithms have proven to be useful in numerous studies related to the

processing and analysis of various [...] data."

Similarly bold statements occur near the end of the paper:

[311-315] "It is necessary to say in conclusion a few words, about the mathematical apparatus of DMA

[Agayan et al., 2018]. It is being developed at the Geophysical Center of the Russian Academy of

Sciences and forms the basis of the methodology for searching periods in the series of cyclostratigraphic

data presented in the article. It is important to emphasize that DMA is a direction of modern applied

systems analysis [Zgurovsky and Pankratova, 2007]."

[316-320] "DMA has all the necessary tools to generate mining algorithms for geological and geophysical

data, including searching for hidden periods / cycles. Based on fuzzy sets and fuzzy logic, DMA has the

ability to convey expert ideas about the structure, morphology, monotony, and other of studied data

series. Thus, DMA enables a systematic approach to the analysis of complex data series of Earth

sciences."

While this may be true, none of it is supported by the results of this paper.

Citation: https://doi.org/10.5194/gi-2021-13-RC2

Boris Dzeboev, Anastasia Odintsova, and Alena Rybkina

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