Application of Lévy processes in modelling (geodetic) time series with mixed spectra

Montillet, Jean-Philippe; He, Xiaoxing; Yu, Kegen; Xiong, Changliang

doi:https://doi.org/10.5194/npg-28-121-2021

Articles | Volume 28, issue 1

https://doi.org/10.5194/npg-28-121-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

https://doi.org/10.5194/npg-28-121-2021

© Author(s) 2021. This work is distributed under
the Creative Commons Attribution 4.0 License.

Articles | Volume 28, issue 1

Research article

|

15 Feb 2021

Research article |

| 15 Feb 2021

Application of Lévy processes in modelling (geodetic) time series with mixed spectra

Jean-Philippe Montillet, Xiaoxing He, Kegen Yu, and Changliang Xiong

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Final revised paper (published on 15 Feb 2021)
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Preprint (discussion started on 24 Jun 2020)

Interactive discussion

Status: closed

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

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- Supplement

RC1: 'Comments', Anonymous Referee #1, 23 Jul 2020
- SC1: 'Application of Levy Processes in Modelling (Geodetic) Time Series With Mixed Spectra - Response to RC1', jean-philippe montillet, 03 Aug 2020
RC2: 'Comments on "Application of Lévy Processes..." by Montillet et al.', Reik Donner, 20 Sep 2020
- AC1: 'Application of LévyProcesses in Modelling (Geodetic) Time SeriesWith Mixed Spectra – Discussions on RC2', Jean-Philippe Montillet, 28 Sep 2020

Peer-review completion

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

AR by Jean-Philippe Montillet on behalf of the Authors (28 Oct 2020) Manuscript

ED: Referee Nomination & Report Request started (01 Nov 2020) by Stéphane Vannitsem

RR by Anonymous Referee #2 (13 Dec 2020)

Suggestions for revision or reasons for rejection

The authors have addressed most of the concerns raised by the two reviewers. While I am generally satisfied with the contents of the revised version, I have to request one necessary scientific correction, along with some improvements regarding the internal referencing of the material as well as moderate revision of language-related issues.

Major comment: For the evaluation of their method’s performance for simulated and real-world data, the authors use (among others) correlations between empirical and fitted probability distribution functions supplemented by an Anderson-Darling test. In this context, the authors should be aware that correlation coefficients generally do not provide a good measure for comparing PDFs. On the one hand, coarse-grained PDFs constitute compositional data due to the inherent normalized sum constraint, which is not properly accounted for by the common Pearson correlation coefficient that has supposedly been used here. Yet, there are possibilities to define correlations for compositional data, too. On the other hand, there are much more natural measures for comparing PDFs, e.g., the Kullback-Leibler divergence or related statistical concepts, which appear more appropriate in the present context. In my opinion, it is neither necessary for the present manuscript to add such additional (more appropriate) measures for the performed statistical model validation, nor do I request removing the results for the correlation coefficient despite the aforementioned concerns. What I however should request is adding some cautionary remarks regarding the feasibility of using correlations for the purpose for which they are used here.
As an attempt to address the aforementioned concerns, the authors have already included results on the Anderson-Darling test for comparing two PDFs, which is well chosen in the present context since the AD test is particularly sensitive to differences in the distribution tails, which is precisely where the studied models might be expected to differ most. However, in discussing the outcomes of this test, the authors frequently speak about an “accepted null hypothesis”, which is a heavily flawed phrasing. The nature of any statistical hypothesis test is attempting to reject a null hypothesis, while the absence of proper evidence for a rejection does NOT(!) imply accepting it. So I have to urge the authors to avoid any statement speaking about the acceptance of a null hypothesis (captions of Tab. 3, 4 and A1, text in ll. 279, 287, 329 – I hope I did not miss any…). Please revise those statements: “accepted” => “not rejected”, etc. This is crucial for the acceptance of this manuscript.

Minor comments:
1. Separating main text, appendices and supplementary information has greatly improved the readability of the manuscript. Given that a concise paper is most appealing for readers, I would even suggest moving the additional Tab. A1 and Figs. A1-A4 to the supplementary material, since they do not add essential information to the manuscript and in fact just blow up its size (thereby also increasing the page charges to be calculated by the publisher). My recommendation would be keeping only Appendix A in the main document, possibly not even using a numbering then and referring to its content in the main body simply to as “see Appendix” or similar.
2. In order to improve the internal referencing of the material, I strongly recommend the authors to revise references to the supplementary material by emphasizing explicitly the part of that document that is referred to, e.g., “see Supplementary Material Sect. A” or “Supplementary Material Fig. S1/Tab. S1” if following my suggestions under the previous point 1.
3. Please use consistently the term “Gaussian Lévy” (or “Lévy Gaussian”, but I would tend to prefer the former) instead of mixing both terms.
4. The block of equations on pp.7-8 misses a few equation numbers, including such that are later referred to in the manuscript (e.g., Eq. (7)). This should be clarified.
5. I experienced some personal confusion with the authors’ use of the term “epoch”. Would you mind explaining its meaning in the context of this work precisely along with its first occurrence (p.3, below Eq. (1))?
6. Ll.19-20: As it is written now, one may think that Flicker noise and white noise are examples of band-pass noise, which I would clearly object. Please rephrase.
7. L.177: “nonstationary signal (around the mean)” – do you mean a signal with a non-stationary mean or (as I suppose) a signal with a stationary (zero?) mean and nonstationary variations about this mean?
8. L.215: “see also the discussion on the Hector Software (Bos et al., 2013) in the Supplementary Material Sect. A).
9. L.231: p and q are NOT lags, but the AR and MA model orders of the considered model.
10. L.372: “uncorrelated assumption” => “assumption of uncorrelated components”
11. Eq. (A.2), first line: the meaning of the brackets is not clear If placed as now, they are not necessary. In Eq. (A.3), the brackets however appear necessary, but you should use \left( and \right) to enforce brackets of a proper size for the enclosed content.
12. Supplementary Material Section A: The text in ll.16-22 is a literal repetition of ll.84-90 of the main manuscript. Just shorten this to one sentence with reference to the corresponding Eq. (3).

Technical comments (mostly regarding language, but some of the comments also include stylistic suggestions):
• L.8: “The fractional Lévy process…”
• L.9: “The stable process…”
• L.10: “Therefore, it implies…” – what does “it” refer to?
• L.15: “phenomena”
• L.15: “earthquakes”
• Ll.16-17: “These time series provide the estimated daily positions of…”
• L.19: “describe”
• L.20: “cyclic”
• L.20: “referred to”
• Ll.27-28: “To name a few, previous choices include the…”
• L.39: “distributions”
• L.40: “from a Gaussian distribution”
• L.47: “to the residual…”
• L.77: “a multivariate continuous-time stochastic process”
• L.80: “transposition operator”
• L.81: “$K\in [0,2]$”
• L.88: “represents”
• L.89: “model for x”
• L.93: “correctly” might better read “properly” (there is not ground truth…)
• L.98: “bursts”
• L.107: “a random walk”
• L.109: use \citet instead of \citep for reference Eke et al.(2002)
• L.114: “based on different”
• L.123: close bracket not after “2018”, but after “1.6”
• L.127: “where sign(.)…”
• L.128: “sets”
• L.132: use \citet instead of \citep for reference Weron et al. (2005)
• L.133: “with $H$ being”
• L.140: “similar approach as used”
• L.144: “Gaussian Lévy”
• L.146: “This is the special case”
• L.152: “ratio between the amplitudes of the coloured and white noise components determines…”
• L.153: “most suitable among the FARIMA…”
• L.157: “modelled”
• L.167: “as a random walk”
• L.220: “Gaussian Lévy”
• L.233: “selection (ARMA, FARIMA)”
• L.234: “(e.g., Panahi (2016))”
• L.238: “than that of the white noise… should be oreferred de facto....”
• L.245: “power-law noise”
• L.250: “between 1 and 3”
• Ll.254-256: don’t put physical units in math mode (no italics)
• Ll.257, 269: “in scenario C”
• L.272: “the fits”
• L.273: “over 50 simulations”
• L.274: “the average correlation”
• L.275: I suggest not to capitalize “normal” (also in what follows from here)
• L.282: do not capitalized “intermediate”
• L.282: “However, scenario C….”
• L.283: “introduce”
• L.284: “performs”
• L.286: “displayed”
• L.287: “…in Table 2 give the probability…”
• L.297: “of the three GNSS stations DRAO,…” (note that I would also suggest not to put the station abbreviations in italics, but this is up to the authors…)
• L.309: “dependence on the…”
• L.320: “explained by…”
• L.321: use \citep instead of \citet for reference to He et al. (2019)
• L.323: “series of ASCO”
• L.326: “of the noise that is much larger…”
• L.327: “is approximately the same”
• Ll.332,333: “Up coordinate”
• L.341: “develop closed-form expressions for the mean….”
• L.344: “equal to 1”
• Fig. 3, caption: “function of the residual” (wo times, also in the captions of the other similar figures currently placed in the appendix)
• L.351: “intersect” => “intercept”?
• L.356: remove “Note that” (not necessary) and “if” (grammatically useless)
• L.358: the inline equation seems to miss brackets around the content that is summed up
• L.368: “where Cross…”
• L.371: “with the residual signals”
• L.392: remove “exhibiting”
• L.393: “to using”
• L.398: “an N-step”
• L.417: “in the presence of remaining offsets can be written as”
• L.431: remove “note that”
• SI, l.3: “from the GNSS…”
• SI, l.8: what do you mean by “nominated time”?
• SI, l.14: remove “Furthermore”
• SI, l.33: remove “Note that”
• SI, l.53: “aggregation of fractional…”
• SI, l.69: remove “Note that”
• SI, l.73: “The representation of fLsm in Eq. (C.7) is similar…”
• SI, ll.101-102: The sum… can be written as…” – this sentence is incomplete.

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ED: Publish subject to minor revisions (review by editor) (13 Dec 2020) by Stéphane Vannitsem

AR by Jean-Philippe Montillet on behalf of the Authors (17 Dec 2020) Author's response Author's tracked changes Manuscript

ED: Publish as is (23 Dec 2020) by Stéphane Vannitsem

AR by Jean-Philippe Montillet on behalf of the Authors (23 Dec 2020) Manuscript

Short summary

Recently, various models have been developed, including the fractional Brownian motion (fBm), to analyse the stochastic properties of geodetic time series, together with the estimated geophysical signals. The noise spectrum of these time series is generally modelled as a mixed spectrum, with a sum of white and coloured noise. Here, we are interested in modelling the residual time series after deterministically subtracting geophysical signals from the observations with the Lévy processes.