the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
A twostep method to derive combined Fourierwavelet spectra from spacetime data for studying planetaryscale waves, and its Matlab and Python software (cfw v1.0)
Abstract. The combined Fourierwavelet (CFW) transform is a useful technique to characterize planetaryscale waves, such as tides and traveling planetary waves in the Earth's atmosphere. A CFW spectrum, presented in a time versus period diagram, can be used to identify wave activity that is localized in time, similar to a wavelet spectrum. A CFW spectrum can be obtained for each of eastward and westwardpropagating wave components with different zonal wavenumbers. This paper introduces an easytoimplement method to derive CFW spectra in two steps. In the first step, the Fourier transform is performed in space (longitude), and time series of the space Fourier coefficients are derived. In the second step, the wavelet transform is performed on these time series, and wavelet coefficients are derived. It is shown that the CFW transform can be easily derived from these wavelet coefficients. The results suggest that existing Fourier and wavelet software can be utilized to derive CFW spectra. Matlab and Python scripts are created and made available at https://igit.iapkborn.de/yy01/cfw that compute CFW spectra using the wavelet software provided by Torrence and Compo (1998). Some application examples are presented using longitudetime data from atmospheric and geomagneticfield models.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.

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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
 Preprint
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 Final revised paper
Journal article(s) based on this preprint
Yosuke Yamazaki
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221275', JunIchi Yano, 30 Dec 2022
This is an interesting piece of work, potentially worthwhile for a publication: there are already extensive literature performing the continuous wavelet transform to time series in atmospheric science. However, this work is new: the continuous wavelet transform is applied to time series, that itself is nothing new, but in the wavenumber space. The author shows that this methodology can characterized certain wave activities rather well, as shown in Figs. 24.
*
The main problem with the present manuscript, as it stands for now, is to present it with a rather sensational fanfare, calling it specifically a "combined Fourierwavelet spectrum". As far as I can follow, it is nothing other than just performing a continuous wavelet transform in time and a Fourier transform in space (longitude): these are two independent linear operations, that can be performed in any order. There is nothing to "combine", but just to perform two independent things in sequence. In other words, the adjective "combined" is nothing other than just a misnomer.
With an attempt of the author convincing the readers that this is "revolutionary", the author provides rather lengthy technical details of the socalled Hayashi's method (that itself is nothing other than just performing the Fourier transform in time and longitude) and the continuous wavelet in the last half of the introduction section. However, I do not see those technical details to be any importance for this method. As I just said above, what is done in practice is very simple: perform FFT in longitude, and the continuous wavelet analysis in time. We just need few more extra words on specific.
On the other hand, the presentation of the proposed methodology in Sec. 2 is rather "muddled", and difficult to get this very simple point straight.
*
A simple methodology is always a beauty. However, if the author intends to present the present manuscript as a proposal paper of a new methodology, a more careful review of the existing methodologies is required.
First of all, planetary waves can be extracted in a straightforward manner, at least in principle, by the normalmode decomposition. A full description of this methodology is provided in Zager et al. (2015, GMD, https://gmd.copernicus.org/articles/8/1169/2015/) with a software publicly available to apply this methodology. Also please refer to a workshop report for further backgrounds: Zagar et al. (2016, BAMS, https://journals.ametsoc.org/view/journals/bams/97/6/bamsd1500325.1.xml).
In this respect, the introduction is slightly confused as it stands for now: its first half reviews previous works detecting "linear" planetary waves. Then, suddenly, at L55, the author decides to talk about the stratospheric sudden warming: this is clearly a nonlinear process that cannot be described by a single wave. The authors further begins to remark that the observed waves are rather "intermittent" (in own wording), and they can emerge even like bursts: that is all fine with me: these observed waves are not perfectly linear, and they are often generated by forcings as well as instabilities, and those evolution can be very nonlinear. However, after said all those (though the author does not comment on them), if one wishes to understand those phenomena as a part of the wave dynamics, an obvious way to go is to perform the normalmode decompositions so that one can see explicitly which modes are involved in processes in which manner, etc. Those are very basic backgrounds of the atmosphericwave dynamics, that should remind the readers.
*
Second is lack of a proper review of the wavelet method. The most important question, in this context, for me is a choice between the continuous redundant wavelet and the discrete orthogonal wavelet. Here, the author chooses the former, but without explanation. The choice is just puzzling for me considering a very fact that the latter is much more robust, with much more potential applicabilities, as my series of work suggest: see a list of reference below.
Performing a continuous wavelet analysis is like a decomposition of a finite domain data into continuous wavenumbers, when only the discrete integer wavenumbers have a meaning.
*
Finally, if detection of a planetarywave packet is the main issue, a discrete set of wavelets can be constructed based on normal modes, in an equivalent manner as the Meyer wavelet is constructed based on the Fourier modes. Though I do not think that the author has to try this possibility in the present work, all those potential possibilities must be clearly mentioned in the manuscript.
It is obvious that the author is only taking a small first step forwards for exploring all those wider possibilities.
*
Specifics:
L7980, the standard wavelet technique is not directly applicable to longitudetime data: false. 2D wavelet transform can easily performed in analogy with the 2D Fourier transform: refer to my publications below.
L90, parameter > variable
Eqs. (5): the frequency, omega, must be discrete, as the case for the wavenumbers. please comment on this
L122: state explicitly that continuous wavelet is applied in time
Eq. (16): the actual data set only has a finite length in time. comment on this
Eq. (21): if not, this expression is puzzling: since continuous wavelet is applied here, obtained coefficients must also be continuous: why we suddenly get a discrete expression?
Sec. 2: as far as I can follow, the longitudinal dependence does not play any role in the presentation, though Eqs. (29) and (30) retain it. at least a word would be required for a clarification: otherwise, in my own reading, Sec. 2 is essentially just repeating Sec. 1.3. If not, what is a difference, except for a longitudinal dependence added in Eqs. (29) and (30)?
Eqs. (29), (30): the exponent, t^2/2 must be replaced by t^2/2s? if not, I do not know how to connect this expansion with (17), as invoked after Eq. (34).
Eqs. (29), (30): the given decomposition modes are only localized in time, thus it appears to me that the author essentially fails to address a question of the propagation of a wave packet, that should happen both in time and space.
Eqs. (35) and (36): Psi* here must depend on both s and omega: how do you specify them?
Fig. 5b, c: they should be better presented by standard Fourier transforms: the plots do not take any advantage of wavelet, either.
References:
J.I. Yano, M. W. Moncrieff, X. Wu, and M. Yamada, 2001 : Wavelet Analysis of Simulated Tropical Convective Cloud Systems Part I: Basic Analysis. J. Atmos. Sci., 58, 850867.
J.I. Yano, M. W. Moncrieff, and X. Wu, 2001 : Wavelet Analysis of Simulated Tropical Convective Cloud Systems Part II: Decomposition of Convectivescale and Mesoscale structures. J. Atmos. Sci., 58, 868876.
J.I. Yano, R. Blender, Chidon Zhang, and K. Fraedrich, 2004: 1/f Noise and Pulselike Events in the Tropical Atmospheric Surface Variabilities. Quator. J. Roy. Meteor. Soc, 300, 16971721.
J.I. Yano, P. Bechtold, J.L. Redelsperger, and F. Guichard, 2004: WaveletCompressed Representation of deep moist convection. Mon Wea Rev, 132, 14721486.
J.I. Yano, and B. Jakubiak, 2016: WaveletBased Verification of the Quantitative Precipitation Forecast. DAO, 74, 1429
Å½agar, N., A. Kasahara, K. Terasaki, J. Tribbia and H. Tanaka, 2015: Normalmode function representation of global 3D data sets: openaccess software for the atmospheric research community. Geosci. Model Dev., 8, 1169–1195, doi:10.5194/gmd811692015.
Zagar, N., J. Boyd, E. Kallen, A. Kasahara, J. Tribbia, H. Tanaka, J.I. Yano, 2016: Normal modes of atmospheric variability in observations, numerical weather prediction and climate models. BAMS, 97, ES125–ES128, doi.org/10.1175/BAMSD1500325.1
Citation: https://doi.org/10.5194/egusphere20221275RC1 
RC2: 'Comment on egusphere20221275', Anonymous Referee #2, 15 May 2023
Review comments for “A twostep method to derive combined Fourierwavelet spectra from spacetime data for studying planetaryscale waves, and its Matlab and Python software (cfw v1.0)” by Yosuke Yamazaki (2022) discussed in EGUsphere.
This paper introduces a simple method to perform combined Fourierwavelet (CFW) transform to extract planetaryscale waves in the gridded 2D longitudetime atmospheric data set. Although the concept of this method is not new, or the idea is not difficult to understand, the implementation of this method has always been a difficulty for the space physics or mesosphere and lower thermosphere community. After testing the programs provided by the authors, I think this method has the following advantages:
 The calculation speed is fast since this method only reads the data once. This advantage is meaningful, especially to the extraction of tides, which usually requires data with a high temporal resolution and is a challenge for the memory and computing resources of a personal computer;
 The output is directly the real amplitudes of the waves, which is similar to the Stransform, but this method obtains the amplitudes in the 2D data set;
 In fact, there has been no good way to accurately extract the planetary wave activity during sudden stratospheric warmings in the longitudetime satellite observations or simulations. However, the results in this manuscript show that this method does a relatively good job on this issue, which is important for related research.
Overall, the authors implemented the CFW transform in a relatively simple way with MATLAB and Python, which is very worthy of recognition and will bring a lot of convenience to the middle and upper atmosphere community. I only have two minor comments on the current manuscript:
 The authors may consider comparing the results obtained by different methods (e.g., CFW transform, 2D fast Fourier transform, and least squares fitting) to better demonstrate the superiority of the CFW transform in studying the temporal variations of planetaryscale waves.
 The authors can appropriately show the results on the planetary waves or tides in the ionospheric parameters (e.g., total electron content) extracted by the CFW transform.
Citation: https://doi.org/10.5194/egusphere20221275RC2  AC1: 'Comment on egusphere20221275', Yosuke Yamazaki, 13 Jun 2023
Interactive discussion
Status: closed

RC1: 'Comment on egusphere20221275', JunIchi Yano, 30 Dec 2022
This is an interesting piece of work, potentially worthwhile for a publication: there are already extensive literature performing the continuous wavelet transform to time series in atmospheric science. However, this work is new: the continuous wavelet transform is applied to time series, that itself is nothing new, but in the wavenumber space. The author shows that this methodology can characterized certain wave activities rather well, as shown in Figs. 24.
*
The main problem with the present manuscript, as it stands for now, is to present it with a rather sensational fanfare, calling it specifically a "combined Fourierwavelet spectrum". As far as I can follow, it is nothing other than just performing a continuous wavelet transform in time and a Fourier transform in space (longitude): these are two independent linear operations, that can be performed in any order. There is nothing to "combine", but just to perform two independent things in sequence. In other words, the adjective "combined" is nothing other than just a misnomer.
With an attempt of the author convincing the readers that this is "revolutionary", the author provides rather lengthy technical details of the socalled Hayashi's method (that itself is nothing other than just performing the Fourier transform in time and longitude) and the continuous wavelet in the last half of the introduction section. However, I do not see those technical details to be any importance for this method. As I just said above, what is done in practice is very simple: perform FFT in longitude, and the continuous wavelet analysis in time. We just need few more extra words on specific.
On the other hand, the presentation of the proposed methodology in Sec. 2 is rather "muddled", and difficult to get this very simple point straight.
*
A simple methodology is always a beauty. However, if the author intends to present the present manuscript as a proposal paper of a new methodology, a more careful review of the existing methodologies is required.
First of all, planetary waves can be extracted in a straightforward manner, at least in principle, by the normalmode decomposition. A full description of this methodology is provided in Zager et al. (2015, GMD, https://gmd.copernicus.org/articles/8/1169/2015/) with a software publicly available to apply this methodology. Also please refer to a workshop report for further backgrounds: Zagar et al. (2016, BAMS, https://journals.ametsoc.org/view/journals/bams/97/6/bamsd1500325.1.xml).
In this respect, the introduction is slightly confused as it stands for now: its first half reviews previous works detecting "linear" planetary waves. Then, suddenly, at L55, the author decides to talk about the stratospheric sudden warming: this is clearly a nonlinear process that cannot be described by a single wave. The authors further begins to remark that the observed waves are rather "intermittent" (in own wording), and they can emerge even like bursts: that is all fine with me: these observed waves are not perfectly linear, and they are often generated by forcings as well as instabilities, and those evolution can be very nonlinear. However, after said all those (though the author does not comment on them), if one wishes to understand those phenomena as a part of the wave dynamics, an obvious way to go is to perform the normalmode decompositions so that one can see explicitly which modes are involved in processes in which manner, etc. Those are very basic backgrounds of the atmosphericwave dynamics, that should remind the readers.
*
Second is lack of a proper review of the wavelet method. The most important question, in this context, for me is a choice between the continuous redundant wavelet and the discrete orthogonal wavelet. Here, the author chooses the former, but without explanation. The choice is just puzzling for me considering a very fact that the latter is much more robust, with much more potential applicabilities, as my series of work suggest: see a list of reference below.
Performing a continuous wavelet analysis is like a decomposition of a finite domain data into continuous wavenumbers, when only the discrete integer wavenumbers have a meaning.
*
Finally, if detection of a planetarywave packet is the main issue, a discrete set of wavelets can be constructed based on normal modes, in an equivalent manner as the Meyer wavelet is constructed based on the Fourier modes. Though I do not think that the author has to try this possibility in the present work, all those potential possibilities must be clearly mentioned in the manuscript.
It is obvious that the author is only taking a small first step forwards for exploring all those wider possibilities.
*
Specifics:
L7980, the standard wavelet technique is not directly applicable to longitudetime data: false. 2D wavelet transform can easily performed in analogy with the 2D Fourier transform: refer to my publications below.
L90, parameter > variable
Eqs. (5): the frequency, omega, must be discrete, as the case for the wavenumbers. please comment on this
L122: state explicitly that continuous wavelet is applied in time
Eq. (16): the actual data set only has a finite length in time. comment on this
Eq. (21): if not, this expression is puzzling: since continuous wavelet is applied here, obtained coefficients must also be continuous: why we suddenly get a discrete expression?
Sec. 2: as far as I can follow, the longitudinal dependence does not play any role in the presentation, though Eqs. (29) and (30) retain it. at least a word would be required for a clarification: otherwise, in my own reading, Sec. 2 is essentially just repeating Sec. 1.3. If not, what is a difference, except for a longitudinal dependence added in Eqs. (29) and (30)?
Eqs. (29), (30): the exponent, t^2/2 must be replaced by t^2/2s? if not, I do not know how to connect this expansion with (17), as invoked after Eq. (34).
Eqs. (29), (30): the given decomposition modes are only localized in time, thus it appears to me that the author essentially fails to address a question of the propagation of a wave packet, that should happen both in time and space.
Eqs. (35) and (36): Psi* here must depend on both s and omega: how do you specify them?
Fig. 5b, c: they should be better presented by standard Fourier transforms: the plots do not take any advantage of wavelet, either.
References:
J.I. Yano, M. W. Moncrieff, X. Wu, and M. Yamada, 2001 : Wavelet Analysis of Simulated Tropical Convective Cloud Systems Part I: Basic Analysis. J. Atmos. Sci., 58, 850867.
J.I. Yano, M. W. Moncrieff, and X. Wu, 2001 : Wavelet Analysis of Simulated Tropical Convective Cloud Systems Part II: Decomposition of Convectivescale and Mesoscale structures. J. Atmos. Sci., 58, 868876.
J.I. Yano, R. Blender, Chidon Zhang, and K. Fraedrich, 2004: 1/f Noise and Pulselike Events in the Tropical Atmospheric Surface Variabilities. Quator. J. Roy. Meteor. Soc, 300, 16971721.
J.I. Yano, P. Bechtold, J.L. Redelsperger, and F. Guichard, 2004: WaveletCompressed Representation of deep moist convection. Mon Wea Rev, 132, 14721486.
J.I. Yano, and B. Jakubiak, 2016: WaveletBased Verification of the Quantitative Precipitation Forecast. DAO, 74, 1429
Å½agar, N., A. Kasahara, K. Terasaki, J. Tribbia and H. Tanaka, 2015: Normalmode function representation of global 3D data sets: openaccess software for the atmospheric research community. Geosci. Model Dev., 8, 1169–1195, doi:10.5194/gmd811692015.
Zagar, N., J. Boyd, E. Kallen, A. Kasahara, J. Tribbia, H. Tanaka, J.I. Yano, 2016: Normal modes of atmospheric variability in observations, numerical weather prediction and climate models. BAMS, 97, ES125–ES128, doi.org/10.1175/BAMSD1500325.1
Citation: https://doi.org/10.5194/egusphere20221275RC1 
RC2: 'Comment on egusphere20221275', Anonymous Referee #2, 15 May 2023
Review comments for “A twostep method to derive combined Fourierwavelet spectra from spacetime data for studying planetaryscale waves, and its Matlab and Python software (cfw v1.0)” by Yosuke Yamazaki (2022) discussed in EGUsphere.
This paper introduces a simple method to perform combined Fourierwavelet (CFW) transform to extract planetaryscale waves in the gridded 2D longitudetime atmospheric data set. Although the concept of this method is not new, or the idea is not difficult to understand, the implementation of this method has always been a difficulty for the space physics or mesosphere and lower thermosphere community. After testing the programs provided by the authors, I think this method has the following advantages:
 The calculation speed is fast since this method only reads the data once. This advantage is meaningful, especially to the extraction of tides, which usually requires data with a high temporal resolution and is a challenge for the memory and computing resources of a personal computer;
 The output is directly the real amplitudes of the waves, which is similar to the Stransform, but this method obtains the amplitudes in the 2D data set;
 In fact, there has been no good way to accurately extract the planetary wave activity during sudden stratospheric warmings in the longitudetime satellite observations or simulations. However, the results in this manuscript show that this method does a relatively good job on this issue, which is important for related research.
Overall, the authors implemented the CFW transform in a relatively simple way with MATLAB and Python, which is very worthy of recognition and will bring a lot of convenience to the middle and upper atmosphere community. I only have two minor comments on the current manuscript:
 The authors may consider comparing the results obtained by different methods (e.g., CFW transform, 2D fast Fourier transform, and least squares fitting) to better demonstrate the superiority of the CFW transform in studying the temporal variations of planetaryscale waves.
 The authors can appropriately show the results on the planetary waves or tides in the ionospheric parameters (e.g., total electron content) extracted by the CFW transform.
Citation: https://doi.org/10.5194/egusphere20221275RC2  AC1: 'Comment on egusphere20221275', Yosuke Yamazaki, 13 Jun 2023
Peer review completion
Journal article(s) based on this preprint
Yosuke Yamazaki
Model code and software
CFW Software for Matlab and Python Yosuke Yamazaki https://igit.iapkborn.de/yy01/cfw
Matlab and Python software for computing combined Fourierwavelet spectra: cfw v1.0 Yosuke Yamazaki https://doi.org/10.5281/zenodo.7458051
Matlab and Python software to compute Fourierwavelet spectra (fourierwavelet v1.1) using longitudetime data for studying globalscale atmospheric waves Yosuke Yamazaki https://zenodo.org/record/8033686
Yosuke Yamazaki
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The requested preprint has a corresponding peerreviewed final revised paper. You are encouraged to refer to the final revised version.
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