Smoothing data series by means of cubic splines: quality of approximation and introduction of a repeating spline approach

Wüst, Sabine; Wendt, Verena; Linz, Ricarda; Bittner, Michael

doi:https://doi.org/10.5194/amt-10-3453-2017

Articles | Volume 10, issue 9

https://doi.org/10.5194/amt-10-3453-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

https://doi.org/10.5194/amt-10-3453-2017

© Author(s) 2017. This work is distributed under
the Creative Commons Attribution 3.0 License.

Articles | Volume 10, issue 9

Research article

|

21 Sep 2017

Research article |

| 21 Sep 2017

Smoothing data series by means of cubic splines: quality of approximation and introduction of a repeating spline approach

Sabine Wüst, Verena Wendt, Ricarda Linz, and Michael Bittner

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Final revised paper (published on 21 Sep 2017)
Preprint (discussion started on 08 Feb 2017)

Interactive discussion

Status: closed

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

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RC1: 'Referee Comment', Anonymous Referee #2, 07 Mar 2017
- AC1: 'Answer to comments of anonymous referee #2', Sabine Wüst, 28 Apr 2017
RC2: 'Reviewer Comment', Anonymous Referee #1, 21 Mar 2017
- AC2: 'Answer to comments of anonymous referee #1', Sabine Wüst, 28 Apr 2017

Peer-review completion

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

AR by Sabine Wüst on behalf of the Authors (24 May 2017) Author's response Manuscript

ED: Referee Nomination & Report Request started (24 May 2017) by Gerd Baumgarten

RR by Anonymous Referee #1 (08 Jun 2017)

Suggestions for revision or reasons for rejection

The revised version of the Wüst et al. manuscript is strongly improved compared to the original version. The authors clarified the scope of their method and adapted the description according to both reviewers’ comments. I am happy that the scope is much clearer now. Using the spline method for the description of the background and taking the residuals as wave disturbance is a widely used approach in data analyses. Therefore, it has been useful, e.g., to change the term “approximation error” into “sum of squared residuals”. Nevertheless I am still concerned about the structure of the manuscript. Furthermore, a few answers to my previous review stimulated some new comments.

General comments:
The authors made clear that the description of (gravity) waves from the residuals of the spline fit is the major topic of this paper. In detail, the presumably artificial oscillation in averaged SABER GW activity is the main motivation to develop improved methods of spline fitting. On the other hand they only shortly speculate about the true origin of this oscillation. Instead, they state in the Discussion that the analysis “is beyond the scope of this manuscript”. I suggest to at least try to examine the true reason for this oscillation. Is it visible in every profile or in monthly averages? Is it only a result of averaging? Does it also appear if some random GW are added to a CIRA profile? Does it also appear if the sampling point distance is set to, e.g., 15 km? See also below my addition to the old comment #9.
From my point of view it should be made clearer that the difference between normal and repeating spline fitting is largest if the sampling point distance is close to the half wavelength. In case of the constant background the differences between both methods vanish for distances larger than 2 km or smaller than about 1.3 km. In case of the CIRA background, differences are always only about 20% of the sum of squared residuals. Here it would be very interesting to see, which of both methods compares better with the true background profile. This comparison can only be done with CIRA because here the background is exactly known.
From my point of view, the essential information from this paper is twofold:
i) The spline method may produce some artefacts in separating background and waves if the data set contains waves with wavelengths of about double the sampling point distance. This can at least partly be overcome with the repeating spline.
ii) The description of the background by a too coarse spline may produce artificial oscillations in the residuals which should not be confused with GW. This interesting result from the answer to Reviewer 1 should also be presented in the paper, because it demonstrates the limitations of (non-repeating) splines in general, if the background is unknown (as usual).
Further concerns about the structure of the manuscript are described in the specific comments below.

Specific Comments to the Author Response:
Old Comment 7: In the Summary (p. 9, l. 8) the “ability of a spline to approximate oscillations” is mentioned, not the ability to reproduce a background state (and wave-induced residuals).
Answer: We hope that we have clarified all confusing points mentioned above and leave the formulation in the summary as it was since this meets the purpose of the manuscript.
My comment: At different positions of the manuscript you write that the motivation is to remove the oscillations in the wave activity (Fig. 1). Obviously this can only be done by improved description of the background by the spline, not by optimal approximation of the oscillations.

Old Comment 9. (i) P. 8, l. 17-18: i) It is still surprising that the annual mean residuals (~500 profiles) show such a pronounced oscillation. Please comment on this.
Answer: See page 10, first paragraph for possible explanations. The answer to the major comment, number 3 of the other reviewer helps to better understand the information which we added to this paragraph.
My comment: As written above, the new figure provided in the answer to Reviewer 1 would also improve the manuscript. It shows limitations of the standard spline approximation by an impressive example. Furthermore, it gives some hints on the question why oscillations with a wavelength similar to the sampling point distance appear in the average data of Fig. 1, but may not be visible in a single GW profile.

Specific Comments to the manuscript (line numbers refer to the manuscript without tracked changes):
p.1, l. 14: Maybe I am too strict on this, but I suggest writing “approximation of the background state”
Section 3, case study with constant background: Obviously in this section it is intended to show a method for a good approximation of the oscillations (disturbed profile), see, e.g., p. 5, l. 27, “sinusoidal approximation is better adapted”. This is somewhat outside of the main motivation. Therefore it should be made clear that, e.g., this section helps to understand the general behavior of splines if the data set contains waves with a wavelength of double the sampling point distance – which may happen in the general case of an unknown mixture of waves.
Section 3, case study with CIRA: Now it seems that a good approximation of the original CIRA profile is intended. Therefore, it would be very helpful to show also the original profile in Fig. 7. This helps the reader to estimate whether the conventional spline or the repeating spline removes the waves better. Additionally, also the residuals between both retrieved profiles and the original CIRA should be calculated.
p. 7, l. 1: Again, I thought that the motivation is not to reproduce oscillations by the spline, but to reproduce the background and to calculated the GW from the residuals.
p. 8, l. 8-13: The calculation is straightforward. I suggest shortening this part and giving the numbers in dB and K^2 without further explanation.
p. 8, l. 14-16: I think it is hard to say from the comparison with Shuai et al., which method is better. Their numbers are comparable to both, Fig. 1 and Fig. 8, especially if potential longitudinal variations are taken into account. I suggest adding an appropriate statement. Unfortunately I do not have access to the Shuai et al. paper. If they show profiles, it would be helpful to learn whether these are also showing the “surprising” oscillations as Fig. 1.
p. 9, l. 3: From my understanding, the motivation of this paper is to remove these oscillations (see Introduction and other places). Therefore, the analysis should not be “beyond the scope”. See above.
p. 10, l. 8: I suggest (again) rephrasing “ability of the spline to approximate oscillations”. See above.

Typos:
p. 9, l. 1: “not very much” should read “not vary much”
Fig. 6: “reisduals” should read “residuals”

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RR by Anonymous Referee #2 (13 Jun 2017)

ED: Publish subject to minor revisions (Editor review) (19 Jun 2017) by Gerd Baumgarten

AR by Sabine Wüst on behalf of the Authors (29 Jun 2017)

ED: Publish as is (13 Jul 2017) by Gerd Baumgarten

AR by Sabine Wüst on behalf of the Authors (18 Jul 2017)

Short summary

Cubic splines with equidistant spline sampling points are a common method in atmospheric science for the approximation of background conditions by means of filtering superimposed fluctuations from a data series. However, splines can generate considerable artificial oscillations in the background and the residuals. We introduce a repeating spline approach which is able to significantly reduce this phenomenon and to apply it to TIMED-SABER vertical temperature profiles from 2010 to 2014.