Assessment of the influence of astronomical cyclicity on sedimentation processes in Eastern Paratethys based on paleomagnetic measurements using Discrete Mathematical Analysis
 ^{1}Geophysical Center of the Russian Academy of Sciences (GC RAS), Moscow 119296, Russia
 ^{2}Geophysical Institute  the Affiliate of Vladikavkaz Scientific Centre of the Russian Academy of Sciences (VSC RAS), Vladikavkaz 362002, Russia
 ^{3}Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences (IPE RAS), Moscow 123995, Russia
 ^{1}Geophysical Center of the Russian Academy of Sciences (GC RAS), Moscow 119296, Russia
 ^{2}Geophysical Institute  the Affiliate of Vladikavkaz Scientific Centre of the Russian Academy of Sciences (VSC RAS), Vladikavkaz 362002, Russia
 ^{3}Schmidt Institute of Physics of the Earth of the Russian Academy of Sciences (IPE RAS), Moscow 123995, Russia
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 LombScargle 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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Boris Dzeboev et al.
Interactive discussion
Status: closed

RC1: 'Comment on gi202113', 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 LombScargle periodogram (ununiform data). It seems, resting only on the information presented in the article, that the DMA method loses to the abovementioned 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 109122) 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.

RC2: 'Comment on gi202113', 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 ratesÂ
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)Â
to the development of the methodology of cyclostratigraphy."Âwith a concise summary of cyclostratigraphy provided subsequently in lines [90105].
In this study, three Fourier analysis methods are used to identify variations inÂ
magnetic susceptibility of geological sediments. Â Fourier results are compared toÂ
periodicity estimates obtained with a different method: "constancy" analysis.The first method described in lines [106119] is essentially standard FourierÂ
analysis that is well suited for detecting harmonics in evenly sampled data. ÂThe second method is the LombScargle periodogram which was designed to deal with unevenly
spaced or gappy data.The third method (REDFIT) is an extension of the LombScargle method to include theÂ
presence of background red noise.Some results are shown in section 3 for a reference data set:
[189190] "This series contains the last 400,000 years of the record in 1,000yearÂ
increments (Figure 1)."Some discussion of the selection of this data set would be nice:
[190191] "This series was chosen for demonstrating the application of spectral methods asÂ
it has been comprehensively studied."
Does this mean that periodicity in this data set is well understood? Â Citation(s) would seem appropriate.
The Fourier and LombScargle power spectra are similar but not identical.
In the limit of regular sampling the results should be the same. Â 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 [155184] that does not address uncertainties
or significance estimates.The largest minimum in Figure 3 occurs at 115,000 which is within 10% of theÂ
largest Fourier result. Â Other minima are not particularly noteworthyÂ
("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 largestÂ
amplitude periodic variation, but is otherwise not clearly superior to the FourierÂ
methods.
The analysis is then applied to two sediment layers (41.2 and 30m respectively) at 20cm spacing. Â
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. Â As before, the LombScargle
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 weakÂ
minimum at 15.2m; the second layer has a decent constancy minimum at 7m.The authors assert that:
[299301] "These signals correspond to the precession and obliquity cycles.Â
The 3.6 m peak corresponds to the precession periodicity (19â€“24 thousand years). The 7 m peakÂ
corresponds to the periods of changes in the angle of inclination of the Earthâ€™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, 493â€“510]
"In practice, sedimentation rate rarely changes in a uniform way.Â
Moreover, the response of the climatic system can change the amplitude ratioÂ
among the astronomical components of the original insolation forcing signal."Different deposition rates might be in effect for each of the two layers,Â
which might help explain the different observations. In fact, there could in principle be significantÂ
variations which could map astronomical periodicities to sediment variations with significantlyÂ
different power spectra. Â If the mapping is sufficiently nonlinear as to change
the periodicities between layers then it could also cause distortion within layers, whichÂ
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 146154:
"The proposed technique is based on DMA [Agayan et al., 2018]. The new method is an originalÂ
technique for analysing discrete data, developed at the GC RAS. DMA is a series of algorithms unitedÂ
by a common formal basis: fuzzy comparisons of numbers, a measure of proximity in discrete spaces,Â
and a discrete limit. DMA was developed to create discrete equivalents of the concepts of classicalÂ
mathematical analysis: for example, limit, continuity, smoothness, connectivity, monotonicity, andÂ
extremum. DMA methods and algorithms have proven to be useful in numerous studies related to theÂ
processing and analysis of various [...] data."Similarly bold statements occur near the end of the paper:
[311315] "It is necessary to say in conclusion a few words, about the mathematical apparatus of DMAÂ
[Agayan et al., 2018]. It is being developed at the Geophysical Center of the Russian Academy ofÂ
Sciences and forms the basis of the methodology for searching periods in the series of cyclostratigraphicÂ
data presented in the article. It is important to emphasize that DMA is a direction of modern appliedÂ
systems analysis [Zgurovsky and Pankratova, 2007]."[316320] "DMA has all the necessary tools to generate mining algorithms for geological and geophysicalÂ
data, including searching for hidden periods / cycles. Based on fuzzy sets and fuzzy logic, DMA has theÂ
ability to convey expert ideas about the structure, morphology, monotony, and other of studied dataÂ
series. Thus, DMA enables a systematic approach to the analysis of complex data series of EarthÂ
sciences." ÂWhile this may be true, none of it is supported by the results of this paper. Â
Â
Interactive discussion
Status: closed

RC1: 'Comment on gi202113', 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 LombScargle periodogram (ununiform data). It seems, resting only on the information presented in the article, that the DMA method loses to the abovementioned 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 109122) 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.

RC2: 'Comment on gi202113', 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 ratesÂ
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)Â
to the development of the methodology of cyclostratigraphy."Âwith a concise summary of cyclostratigraphy provided subsequently in lines [90105].
In this study, three Fourier analysis methods are used to identify variations inÂ
magnetic susceptibility of geological sediments. Â Fourier results are compared toÂ
periodicity estimates obtained with a different method: "constancy" analysis.The first method described in lines [106119] is essentially standard FourierÂ
analysis that is well suited for detecting harmonics in evenly sampled data. ÂThe second method is the LombScargle periodogram which was designed to deal with unevenly
spaced or gappy data.The third method (REDFIT) is an extension of the LombScargle method to include theÂ
presence of background red noise.Some results are shown in section 3 for a reference data set:
[189190] "This series contains the last 400,000 years of the record in 1,000yearÂ
increments (Figure 1)."Some discussion of the selection of this data set would be nice:
[190191] "This series was chosen for demonstrating the application of spectral methods asÂ
it has been comprehensively studied."
Does this mean that periodicity in this data set is well understood? Â Citation(s) would seem appropriate.
The Fourier and LombScargle power spectra are similar but not identical.
In the limit of regular sampling the results should be the same. Â 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 [155184] that does not address uncertainties
or significance estimates.The largest minimum in Figure 3 occurs at 115,000 which is within 10% of theÂ
largest Fourier result. Â Other minima are not particularly noteworthyÂ
("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 largestÂ
amplitude periodic variation, but is otherwise not clearly superior to the FourierÂ
methods.
The analysis is then applied to two sediment layers (41.2 and 30m respectively) at 20cm spacing. Â
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. Â As before, the LombScargle
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 weakÂ
minimum at 15.2m; the second layer has a decent constancy minimum at 7m.The authors assert that:
[299301] "These signals correspond to the precession and obliquity cycles.Â
The 3.6 m peak corresponds to the precession periodicity (19â€“24 thousand years). The 7 m peakÂ
corresponds to the periods of changes in the angle of inclination of the Earthâ€™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, 493â€“510]
"In practice, sedimentation rate rarely changes in a uniform way.Â
Moreover, the response of the climatic system can change the amplitude ratioÂ
among the astronomical components of the original insolation forcing signal."Different deposition rates might be in effect for each of the two layers,Â
which might help explain the different observations. In fact, there could in principle be significantÂ
variations which could map astronomical periodicities to sediment variations with significantlyÂ
different power spectra. Â If the mapping is sufficiently nonlinear as to change
the periodicities between layers then it could also cause distortion within layers, whichÂ
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 146154:
"The proposed technique is based on DMA [Agayan et al., 2018]. The new method is an originalÂ
technique for analysing discrete data, developed at the GC RAS. DMA is a series of algorithms unitedÂ
by a common formal basis: fuzzy comparisons of numbers, a measure of proximity in discrete spaces,Â
and a discrete limit. DMA was developed to create discrete equivalents of the concepts of classicalÂ
mathematical analysis: for example, limit, continuity, smoothness, connectivity, monotonicity, andÂ
extremum. DMA methods and algorithms have proven to be useful in numerous studies related to theÂ
processing and analysis of various [...] data."Similarly bold statements occur near the end of the paper:
[311315] "It is necessary to say in conclusion a few words, about the mathematical apparatus of DMAÂ
[Agayan et al., 2018]. It is being developed at the Geophysical Center of the Russian Academy ofÂ
Sciences and forms the basis of the methodology for searching periods in the series of cyclostratigraphicÂ
data presented in the article. It is important to emphasize that DMA is a direction of modern appliedÂ
systems analysis [Zgurovsky and Pankratova, 2007]."[316320] "DMA has all the necessary tools to generate mining algorithms for geological and geophysicalÂ
data, including searching for hidden periods / cycles. Based on fuzzy sets and fuzzy logic, DMA has theÂ
ability to convey expert ideas about the structure, morphology, monotony, and other of studied dataÂ
series. Thus, DMA enables a systematic approach to the analysis of complex data series of EarthÂ
sciences." ÂWhile this may be true, none of it is supported by the results of this paper. Â
Â
Boris Dzeboev et al.
Boris Dzeboev et al.
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