the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Limitations in Wavelet Analysis of NonStationary Atmospheric Gravity Wave Signatures in Temperature Profiles
Abstract. Continuous Wavelet Transform (CWT) is a commonly used mathematical tool when it comes to the timefrequency (or distancewavenumber) analysis of nonstationary signals and is used in a variety of research areas. In this work we use the CWT to investigate signatures of atmospheric internal gravity waves (GW) as observed in vertical temperature profiles obtained for instance by lidar. The focus is laid on the determination of vertical wavelengths of dominant GWs. According to linear GW theory these wavelengths are a function of horizontal wind speed and hence, vertical wind shear causes shifts in the evolution of the vertical wavelength. The resulting signal fulfills the criteria of a chirp. Using complex Morlet wavelets, we apply CWT to test mountain wave signals modeling wind shears of up to 5 ms^{−1} km^{−1} and investigate the capabilities and limitations. We find that the sensitivity of the CWT decreases for large chirp rates, i.e. strong wind shear. For a 4th order Morlet wavelet, edge effects become dominant at a vertical wind shear of 3.4 ms^{−1} km^{−1}. For higherorder wavelets, edge effects dominate at even smaller values. In addition, we investigate the effect of GW amplitudes growing exponentially with altitude on the determination of vertical wavelengths. It becomes evident that, in case of conservative amplitude growth, spectral leakage leads to artificially enhanced spectral power at lower altitudes. Therefore, we recommend to normalize the GW signal before the wavelet analysis and the determination of vertical wavelengths. Finally, the cascading of receiver channels which is typical for middle atmosphere lidar measurements results in an exponential sawtoothlike pattern of measurement uncertainties as function of altitude. With the help of Monte Carlo simulations we compute a wavelet noise spectrum and determine significance levels, which enables the reliable determination of vertical wavelengths. Finally, the insights obtained from the analysis of artificial chirps are used to analyse and interpret real GW measurements from the Compact Rayleigh Autonomous Lidar in April 2018 at Río Grande, Argentina. The comparison of results of commonly used and our suggested wavelet analysis demonstrates improvements in the accuracy of determined wavelengths. For future analyses, we suggest the usage of a 4th order Morlet wavelet, the normalization of GW amplitudes before wavelet analysis, and the significance level computation based on measurement uncertainties.
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RC1: 'Comment on amt2023240', Anonymous Referee #1, 31 Jan 2024
This paper investigates the sensitivity of the wavelet transform of atmospheric vertical profiles to the altitudinal variations of the wavelengths and amplitudes of internal gravity waves. The choice of the order of the Morlet mother wavelet is also discussed. The authors recommend normalizing gravity wave signals before applying wavelet analysis. The analyses presented are helpful for researchers using wavelet transform for atmospheric studies. The paper will become acceptable for publication after answering the following issues.
 Given that the sensitivity of the WPS to the order of the Morlet wavelet is investigated, plots of the wavelet for m0 = 4, 6, 8 should be given.
 Eq.(7): It should be mentioned based on this relation that the COE is wider for larger m0 so that the readers can understand the cause of the different COE for differebt m0 in the subsequent WPSs.
 Different units are connected without spaces, like "ms" and "Kkm", in many places. Insert spaces like "m s" and "K km."
 Figure 3: Why are there many experiments for each combination of the wind shear and wavelet order? There is a description "no additive noise", so there seems to be no Monte Carlo processes. Which parameter is different between those experiments?
 l.177: "Equ,4" might be better as "Eq. 4" or "Equation 4. "
 l.194: How the "growth rates" are applied is unclear. I guess the formulation "a(z) = 1.0 K + z K km1" given in l.139 is modified as "a(z) = 1.0 K + 0.1*z K km1", "a(z) = 1.0 K + 1*z K km1", and "a(z) = 1.0 K + 10*z K km1". More explanations are needed.
 Figure 5: Similar to Fig.3, I do not understand why many experiments exist for each combination of the normalization, wind shear, and growth rate.
 Figure 7: Panel d is not mentioned in the text. Moreover, how this vertical wavelength was obtained is not explained.
 l.260: "(Fig. 7b)": Is this a typo for "(Fig. 9b)"?
 l.262: "due to the smaller COE": Is this smaller COE due to the smaller m0? If so, mention it.
 l.272: An additional example of wavelet analysis for conditions where multiple waves overlap:
Mori, R., Imamura, T., Ando, H., Häusler, B., Pätzold, M., & Tellmann, S. (2021). Gravity wave packets in the Venusian atmosphere observed by radio occultation experiments: Comparison with saturation theory. Journal of Geophysical Research, 126, e2021JE006912. https://doi.org/10.1029/2021JE006912 l.277: For the vertical wavelength to be proportional to the horizontal wind speed, the wave must be a topographically generated gravity wave (mountain wave). This should be mentioned. If the paper focuses on mountain waves, the title of the paper is not appropriate since mountain waves are stationary.
Citation: https://doi.org/10.5194/amt2023240RC1 
AC1: 'Reply on RC1', Robert Reichert, 01 Feb 2024
We thank the first referee for the comments. In order to address all comments properly, we have copied them in here and penned answers in bold below each comment.
 Given that the sensitivity of the WPS to the order of the Morlet wavelet is investigated, plots of the wavelet for m0 = 4, 6, 8 should be given.
Even though it seems a bit redundant to show another illustration of a Morlet wavelet, the suggestion is certainly easy to implement and will not change the manuscript for the worse.
 Eq.(7): It should be mentioned based on this relation that the COI is wider for larger m0 so that the readers can understand the cause of the different COI for different m0 in the subsequent WPSs.
We thank the referee for that comment and will include the following statement: "Larger orders result in a larger extend of the Morlet wavelet and hence to a more extended COI in the WPS."
 Different units are connected without spaces, like "ms" and "Kkm", in many places. Insert spaces like "m s" and "K km."
We thank the referee for that comment and will correct for that.
 Figure 3: Why are there many experiments for each combination of the wind shear and wavelet order? There is a description "no additive noise", so there seems to be no Monte Carlo processes. Which parameter is different between those experiments?
There is no Monte Carlo process involved at this place. Line154155: “Figure 3 illustrates the distributions of wavelength ratios derived from the lowermost 50 km of the simulated altitude range […]” With a vertical discretization of dz=0.1km, we obtain 500 wavelength ratios that are plotted as histograms for each wind shear and wavelet order.
 l.177: "Equ,4" might be better as "Eq. 4" or "Equation 4. "
Will be changed as suggested.
 l.194: How the "growth rates" are applied is unclear. I guess the formulation "a(z) = 1.0 K + z K km1" given in l.139 is modified as "a(z) = 1.0 K + 0.1*z K km1", "a(z) = 1.0 K + 1*z K km1", and "a(z) = 1.0 K + 10*z K km1". More explanations are needed.
That is absolutely correct. We apologize for that oversight and will correct the according expressions in line 139 and table 2.
 Figure 5: Similar to Fig.3, I do not understand why many experiments exist for each combination of the normalization, wind shear, and growth rate.
We apologize for the misunderstanding. In fact, we have not mentioned that the distributions are only considering the lowermost 50km as it is the case in figure 3.
 Figure 7: Panel d is not mentioned in the text. Moreover, how this vertical wavelength was obtained is not explained.
We thank the referee for this comment. We will change the following lines as following:
Line 230231: “In addition, we compute the theoretically expected upper limit of vertical wavelengths according to Lambda_z=2*pi*u/N using the Brunt Väisälä frequency determined from the derived temperature background and horizontal winds from ERA5 reanalysis (Fig. 7d).
Line 238: “The ERA5 profile of absolute winds (Fig. 7c) shows […]”
Line 240241: “[…] we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude, i.e. between 20 km and 60 km (see Fig. 7d).”
 l.260: "(Fig. 7b)": Is this a typo for "(Fig. 9b)"?
Thank you. That is a typo indeed. It should be (Fig. 9b).
 l.262: "due to the smaller COI": Is this smaller COI due to the smaller m0? If so, mention it.
We will add the following to line 262: […] which is due to the usage of a smaller order (m0=4).
 l.272: An additional example of wavelet analysis for conditions where multiple waves overlap:
Mori, R., Imamura, T., Ando, H., Häusler, B., Pätzold, M., & Tellmann, S. (2021). Gravity wave packets in the Venusian atmosphere observed by radio occultation experiments: Comparison with saturation theory. Journal of Geophysical Research, 126, e2021JE006912. https://doi.org/10.1029/2021JE006912We will include the reference.
 l.277: For the vertical wavelength to be proportional to the horizontal wind speed, the wave must be a topographically generated gravity wave (mountain wave). This should be mentioned. If the paper focuses on mountain waves, the title of the paper is not appropriate since mountain waves are stationary.
We have mentioned mountain waves at various places in the manuscript:
Line 7: “[…] we apply CWT to test mountain wave signals […]”
Line 40: “The vertical wavelength of stationary mountain waves (MW) […] is approximately given as […]”
Line 224: “The temperature profile in Figure 7a shows significant temperature variability which can be attributed to MWs.”
Line 239241: “Assuming that the signal […] was caused by a MW propagating against the background wind, we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude […].”
Line 283: “[…] consider a MW signal […]”
We want to clarify the title. The manuscript is about nonstationary signatures of atmospheric gravity waves and not about signatures of nonstationary atmospheric gravity waves. The term nonstationary refers to a change of statistical properties such as variance and autocorrelation of the data under consideration over time/length. Even though linear mountain waves are a stationary physical feature in the atmosphere, their signal in the vertical is clearly nonstationary in the statistical sense.
Citation: https://doi.org/10.5194/amt2023240AC1

AC1: 'Reply on RC1', Robert Reichert, 01 Feb 2024

RC2: 'Comment on amt2023240', Anonymous Referee #2, 28 Mar 2024
The manuscript "Limitations in Wavelet Analysis of NonStationary Atmospheric Gravity Wave Signatures in Temperature Profiles" by Reichert et al. is a nice discussion of the limitations of the Continuous Wavelet Transform (CWT) that is often used to analyze nonstationary signals. First, idealized linear chirp signals are used as test signals for optimizing the method. Then the method is applied to a temperature profile observed by the CORAL lidar in South America. Recommendations are made regarding the order of the Morlet wavelet. Further, the authors suggest to normalize observed temperature profiles before performing a wavelet analysis.
Overall, the manuscript is well written and fits into the scope of AMT.
Therefore the manuscript is recommended for publication after addressing my minor but important comments as detailed below.My main comment is that the authors should point out more clearly that the cases discussed in the paper are focused on a very specific meteorological situation (midlatitude winter conditions where wind reversals do not occur at altitudes below 80km), while conditions can be much more complicated in the general case.
SPECIFIC COMMENTS:
(1) l.5358:
As in your paper you are investigating cases of wind shear, you should mention more explicitly that in the presence of wind shear parts of the gravity wave spectrum will dissipate which can lead to rapid decrease of gravity wave amplitudes with altitude. Therefore exponential growth often occurs, but does not hold in every situation.You should also mention that another method to compensate for the growth of gravity wave amplitudes would be the investigation of gravity wave potential energy per unit mass. In this way, not only density variations but also variations of the buoyancy frequency N and of the temperature with altitude are accounted for (Strelnikova et al., 2021).
Strelnikova, I., Almowafy, M., Baumgarten, G., Baumgarten, K., Ern, M., Gerding, M., and Lubken, F.J.:
Seasonal cycle of gravity wave potential energy densities from lidar and satellite observations at 54N and 69N,
J. Atmos. Sci., 78, 13591386, https://doi.org/10.1175/JASD200247.1, 2021.(2) l.6568:
It is important to mention here that close to a critical wind level gravity wave amplitude behavior can be very complicated, and the assumption of exponentially growing amplitudes due to density decrease may no longer hold: On the one hand, close to a critical level, as long as the gravity wave is not saturated, the gravity wave amplitude will grow stronger than suggested by the density decrease in order to compensate for the reduction in vertical wavelength caused by the Doppler shift. On the other hand, even closer to the critical level, the gravity wave amplitude will decrease because the wave saturates which limits the wave amplitude.For a discussion see, for example, Ern et al., 2014, their Section 3.1
Ern, M., F. Ploeger, P. Preusse, J. C. Gille, L. J. Gray, S. Kalisch, M. G. Mlynczak, J. M. Russell III, and M. Riese (2014), Interaction of gravity waves with the QBO: A satellite perspective, J. Geophys. Res. Atmos., 119, 23292355, doi:10.1002/2013JD020731.
This more complex situation in the general case would be a justification for the method you propose later in lines 186192 which allows for both increasing and decreasing amplitudes.
(3) l.277: This is not correct. "Proportionality" is given only for orographic gravity waves, and even in this case proportionality applies only to the wind component parallel to the gravity wave horizontal wave vector. In case of midfrequency gravity waves vertical wavelength is proportional to intrinsic phase speed, and there is a "linear" relationship between vertical wavelength and the wind speed parallel to the gravity wave horizontal wave vector.
TECHNICAL COMMENTS:l.50: Please state that m_0 in Table 1 is the order of the Morlet wavelet.
This is defined later in the paper, but needs to be introduced already here.
l.141: (Fig.1cd). > (Fig.1c, d).
Caption of Fig.1 and elsewhere: same problem, please separate labels of the different panels by commas
l.385, reference ChaneMing et al.: Journal title is sufficient, publisher (Springer) can be omitted
l.393: Please check title of reference Ehard et al.
l.405, reference Ge: Journal title is sufficient, publisher (Copernicus) can be omittedCitation: https://doi.org/10.5194/amt2023240RC2 
AC3: 'Reply on RC2', Robert Reichert, 16 May 2024
Dear Referee #2,
Thank you very much for your comments! We have copied them here and answer each comment below in bold.
The manuscript "Limitations in Wavelet Analysis of NonStationary Atmospheric Gravity Wave Signatures in Temperature Profiles" by Reichert et al. is a nice discussion of the limitations of the Continuous Wavelet Transform (CWT) that is often used to analyze nonstationary signals. First, idealized linear chirp signals are used as test signals for optimizing the method. Then the method is applied to a temperature profile observed by the CORAL lidar in South America. Recommendations are made regarding the order of the Morlet wavelet. Further, the authors suggest to normalize observed temperature profiles before performing a wavelet analysis.
Overall, the manuscript is well written and fits into the scope of AMT.
Therefore the manuscript is recommended for publication after addressing my minor but important comments as detailed below.My main comment is that the authors should point out more clearly that the cases discussed in the paper are focused on a very specific meteorological situation (midlatitude winter conditions where wind reversals do not occur at altitudes below 80km), while conditions can be much more complicated in the general case.
SPECIFIC COMMENTS:
(1) l.5358:
As in your paper you are investigating cases of wind shear, you should mention more explicitly that in the presence of wind shear parts of the gravity wave spectrum will dissipate which can lead to rapid decrease of gravity wave amplitudes with altitude. Therefore, exponential growth often occurs, but does not hold in every situation.That is correct and we thank the reviewer for that comment. We have altered l.5357 in the following way:
“Secondly, according to linear GW theory, not only can vertical wavelengths change rapidly, but the amplitudes of GWs also vary with altitude. Generally, amplitudes increase exponentially with altitude enforced by conservation of energy and decreasing air density. However, when thermal or dynamical instability is reached, wave dissipation occurs, causing GW amplitudes to decrease above the breaking altitude. This variation in GW amplitude may lead to an undesirable shift in the localization of the wavelet during the computation of the CWT.”
You should also mention that another method to compensate for the growth of gravity wave amplitudes would be the investigation of gravity wave potential energy per unit mass. In this way, not only density variations but also variations of the buoyancy frequency N and of the temperature with altitude are accounted for (Strelnikova et al., 2021).
Strelnikova, I., Almowafy, M., Baumgarten, G., Baumgarten, K., Ern, M., Gerding, M., and Lubken, F.J.:
Seasonal cycle of gravity wave potential energy densities from lidar and satellite observations at 54N and 69N,
J. Atmos. Sci., 78, 13591386, https://doi.org/10.1175/JASD200247.1, 2021.We disagree with this statement. It is correct, that the GW potential energy per unit mass is a function not only of density but also of ambient temperature and thermal stability. However, the mere investigation of GW potential energies is not an adequate method to compensate for the growth of GW amplitudes.
(2) l.6568:
It is important to mention here that close to a critical wind level gravity wave amplitude behavior can be very complicated, and the assumption of exponentially growing amplitudes due to density decrease may no longer hold: On the one hand, close to a critical level, as long as the gravity wave is not saturated, the gravity wave amplitude will grow stronger than suggested by the density decrease in order to compensate for the reduction in vertical wavelength caused by the Doppler shift. On the other hand, even closer to the critical level, the gravity wave amplitude will decrease because the wave saturates which limits the wave amplitude.For a discussion see, for example, Ern et al., 2014, their Section 3.1
Ern, M., F. Ploeger, P. Preusse, J. C. Gille, L. J. Gray, S. Kalisch, M. G. Mlynczak, J. M. Russell III, and M. Riese (2014), Interaction of gravity waves with the QBO: A satellite perspective, J. Geophys. Res. Atmos., 119, 23292355, doi:10.1002/2013JD020731.
This more complex situation in the general case would be a justification for the method you propose later in lines 186192 which allows for both increasing and decreasing amplitudes.
We wrote the paragraph in l.6572 to highlight the importance to determine vertical wavelengths of GWs and not their amplitudes. The approximation of amplitudes is just a necessity in order to determine proper vertical wavelengths. We agree, that the behavior of GW amplitudes close to a critical wind level can be very complicated, but since we use a 4^{th} order polynomial to approximate GW amplitude variation with altitude, the proposed method in l.190195 results obviously only in a rough estimation of GW amplitudes. To put it in a nutshell, since we are not interested in the precise determination of GW amplitudes, we will not lead the reader at this point in the manuscript down the wrong path.
(3) l.277: This is not correct. "Proportionality" is given only for orographic gravity waves, and even in this case proportionality applies only to the wind component parallel to the gravity wave horizontal wave vector. In case of midfrequency gravity waves vertical wavelength is proportional to intrinsic phase speed, and there is a "linear" relationship between vertical wavelength and the wind speed parallel to the gravity wave horizontal wave vector.We thank the reviewer for pointing this out. We have altered l.280 in the following way:
“Linear GW theory shows that vertical wavelengths of GWs are a function of horizontal wind speed.”
TECHNICAL COMMENTS:
l.50: Please state that m_0 in Table 1 is the order of the Morlet wavelet.
This is defined later in the paper, but needs to be introduced already here.We add “order” in front of m_0 in the headline of table 1.
l.141: (Fig.1cd). > (Fig.1c, d).We change that as suggested.
Caption of Fig.1 and elsewhere: same problem, please separate labels of the different panels by commasWe change that as suggested.
l.385, reference ChaneMing et al.: Journal title is sufficient, publisher (Springer) can be omittedWe change that as suggested.
l.393: Please check title of reference Ehard et al.We thank the reviewer for pointing that out. We change that accordingly.
l.405, reference Ge: Journal title is sufficient, publisher (Copernicus) can be omittedWe change that as suggested.
Kind Regards,
Robert Reichert
Citation: https://doi.org/10.5194/amt2023240AC3

AC3: 'Reply on RC2', Robert Reichert, 16 May 2024

EC1: 'Comment on amt2023240', Robin Wing, 26 Apr 2024
Dear Authours,
In addition to the excellent comments provided by Reviewer 1 and Reviewer 2 it may be worth considering the following four points:
1) The angle of propagation between the gravity wave and horizontal wind can change (i.e. theta=48 degrees).
2) The change in vertical wavenumber can depend on background wind, intrinsic frequency, buoyancy frequency, and horizontal wavenumber. You could start from the TaylorGoldstein or Raytracing equations to confirm all of your assumptions.
3) There could be two (or more) families of gravity waves in your lidar observation.
4) It is not always easy to separate the geophysical from the technical. Simulating a test wave which resembles a realistic wave seen in lidar could allow you more control over all the technical variables without having to worry about the complex geophysics happening in the background.
Cheers,
Robin WingCitation: https://doi.org/10.5194/amt2023240EC1 
AC2: 'Reply on EC1', Robert Reichert, 16 May 2024
Dear Editor,
We have revised our manuscript and considered the points you raised above.
Since it was suggested by referee #1 we added a figure (now Figure 1) to the manuscript which illustrates three Morlet wavelets with the same scale but different orders. In addition, we show their representation in the distancewavelengthspace and Fourier space in order to highlight the localization in each space as function of the order.
We have revised the text at the following lines:
L100101: “Three Morlet wavelets with orders of 4, 6, and 8 are illustrated in Figure 1.”
L126: “A scale of s=5km is equivalent to a wavelength of 3.9km for an order of 4 and 7.6km for an order of 8 (see Fig. 1).”
L223: “[…] we analyze a temperature profile obtained at Río Grande (53.7°S, 67.7°W), Argentina on the night of 21 May 2018 00UTC.”
L232: We change the cutoff from 22km to 20km.
L235: “[…] and juxtapose the measured vertical wavelength using our best practice (Fig. 8d).”
L235236: “Reanalysis data is spectrally truncated at wavenumber T21 in order to define a synopticscale background (e.g. Reichert et al., 2021).”
L238: We change the maximum growth rate from 0.21K/km at 47km to 0.82K/km at 36km.
L241: We change the maximum from 10K at 75km to 20K at 55km.
We erased the following:
L242245: “The ERA5 profile of absolute winds shows a stratospheric wind minimum around 2025km and a maximum at 60km. The vertical shear of horizontal wind exceeds 3.4m/s/km between 29km and 35km. Assuming that the signal shown in Figure 8b was caused by a MW propagating against the background wind, we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude, i.e. between 20km and 60km.”
And replaced it with:
L242245: “The ERA5 profiles of zonal and meridional wind show a rather steady increase of wind speeds between 20km and 50km. The vertical shear of horizontal wind exceeds 3.4m/s/km between 32km and 37km. At this altitude, we find a discontinuity in the profile of measured vertical wavelengths. Computed and measured vertical wavelengths agree quite well below 35km but differ by up to a factor of two above 35km.”
We have rewritten subsection 3.4.2
“We now create a WPS in the conventional way in which the amplitudes of the temperature disturbance are not normalized, the order of the wavelet is set to m_0=6, and no significance levels are determined (Fig. 10a). Furthermore, we create a WPS based on our bestpractice procedure where the amplitudes of the temperature perturbation are normalized, the order of the wavelet is set to m_0=4, and significance levels are calculated (Fig. 10b).
In the conventional WPS we find only little variation in the vertical wavelength with values ranging from 12.7km to 16.4km. Values decrease from the upper and lower edge of the profiles towards 65km altitude. Due to the COI, determined Lz can be considered reliable only between 35km and 75km altitude. The conventional WPS shows no other interesting features.
Let us now turn to the bestpractice WPS (Fig. 10b).
Similar to the conservative case, we find an extended altitude region from 30km to 80km with only little variation in the vertical wavelength with values of 11.1km to 15.0km which are smaller than the values found in the conservative case. This agrees with the results from our sensitivity study (Section 3.1). In contrast to the conventional case we are able to identify maxima in the WPS now at vertical wavelengths from 4.7km at 20km altitude to 9.8km at 30km altitude and vertical wavelengths in the order of 6km to 7km above 80km altitude.”
L297: We changed “Lz=7km” to “Lz=10km”
We added:
L301304: “ERA5 temperature perturbation fields and colocated OHairglow imagery provide more evidence that the MW observed by CORAL propagates steeply within the lidar’s field of view. On the other hand, the difference between computed and measured vertical wavelength (Fig. 8d) could be an indication for an obliquely propagating MW. After all, this work is of methodological nature and the geophysical interpretation of the results is not in our focus.”
L325: “[…] revealing an increase of vertical wavelength from Lz=10km to Lz=15km at approximately 32km altitude.”
L352: “[…] midfrequency MW”
To your 1):
We addressed the angle between wind direction and wave propagation direction by pointing out that we see a difference in the measured and computed vertical wavelength which could be a hint on lateral propagation.
To your 2):
The focus of this work is to determine the limitations of the continuous wavelet transform. One of the aspects was to investigate which chirp rates, i.e. frequency shifts can be resolved using the CWT. In order to learn more about the implications of this study, we relate the linear chirps we define in Section 2.2 to horizontal wind shear using a) the midfrequency approximation, b) assuming we are dealing with mountain waves, and c) assuming that wind and wave propagation direction are constant.
To your 3):
Indeed, additional ERA5 analysis has shown that the nightly mean profile we used previously from the 1718 April 2018 showed very likely a superposition of two mountain waves that propagated obliquely through CORAL’s field of view. Therefore, we decided to show another temperature profile from 21 May 2018 (see Figure 8a, b in the manuscript). ERA5 analysis shows that mountain waves are excited close by the lidar location and propagate steeply (see supplemented figure c, d).
To your 4):
We refrain from performing idealized numerical simulations in order to improve the interpretation of the results from section 3.4. As stated before, the geophysical interpretation is not the focus of this work.
Kind regards,
Robert Reichert

AC2: 'Reply on EC1', Robert Reichert, 16 May 2024
Status: closed

RC1: 'Comment on amt2023240', Anonymous Referee #1, 31 Jan 2024
This paper investigates the sensitivity of the wavelet transform of atmospheric vertical profiles to the altitudinal variations of the wavelengths and amplitudes of internal gravity waves. The choice of the order of the Morlet mother wavelet is also discussed. The authors recommend normalizing gravity wave signals before applying wavelet analysis. The analyses presented are helpful for researchers using wavelet transform for atmospheric studies. The paper will become acceptable for publication after answering the following issues.
 Given that the sensitivity of the WPS to the order of the Morlet wavelet is investigated, plots of the wavelet for m0 = 4, 6, 8 should be given.
 Eq.(7): It should be mentioned based on this relation that the COE is wider for larger m0 so that the readers can understand the cause of the different COE for differebt m0 in the subsequent WPSs.
 Different units are connected without spaces, like "ms" and "Kkm", in many places. Insert spaces like "m s" and "K km."
 Figure 3: Why are there many experiments for each combination of the wind shear and wavelet order? There is a description "no additive noise", so there seems to be no Monte Carlo processes. Which parameter is different between those experiments?
 l.177: "Equ,4" might be better as "Eq. 4" or "Equation 4. "
 l.194: How the "growth rates" are applied is unclear. I guess the formulation "a(z) = 1.0 K + z K km1" given in l.139 is modified as "a(z) = 1.0 K + 0.1*z K km1", "a(z) = 1.0 K + 1*z K km1", and "a(z) = 1.0 K + 10*z K km1". More explanations are needed.
 Figure 5: Similar to Fig.3, I do not understand why many experiments exist for each combination of the normalization, wind shear, and growth rate.
 Figure 7: Panel d is not mentioned in the text. Moreover, how this vertical wavelength was obtained is not explained.
 l.260: "(Fig. 7b)": Is this a typo for "(Fig. 9b)"?
 l.262: "due to the smaller COE": Is this smaller COE due to the smaller m0? If so, mention it.
 l.272: An additional example of wavelet analysis for conditions where multiple waves overlap:
Mori, R., Imamura, T., Ando, H., Häusler, B., Pätzold, M., & Tellmann, S. (2021). Gravity wave packets in the Venusian atmosphere observed by radio occultation experiments: Comparison with saturation theory. Journal of Geophysical Research, 126, e2021JE006912. https://doi.org/10.1029/2021JE006912 l.277: For the vertical wavelength to be proportional to the horizontal wind speed, the wave must be a topographically generated gravity wave (mountain wave). This should be mentioned. If the paper focuses on mountain waves, the title of the paper is not appropriate since mountain waves are stationary.
Citation: https://doi.org/10.5194/amt2023240RC1 
AC1: 'Reply on RC1', Robert Reichert, 01 Feb 2024
We thank the first referee for the comments. In order to address all comments properly, we have copied them in here and penned answers in bold below each comment.
 Given that the sensitivity of the WPS to the order of the Morlet wavelet is investigated, plots of the wavelet for m0 = 4, 6, 8 should be given.
Even though it seems a bit redundant to show another illustration of a Morlet wavelet, the suggestion is certainly easy to implement and will not change the manuscript for the worse.
 Eq.(7): It should be mentioned based on this relation that the COI is wider for larger m0 so that the readers can understand the cause of the different COI for different m0 in the subsequent WPSs.
We thank the referee for that comment and will include the following statement: "Larger orders result in a larger extend of the Morlet wavelet and hence to a more extended COI in the WPS."
 Different units are connected without spaces, like "ms" and "Kkm", in many places. Insert spaces like "m s" and "K km."
We thank the referee for that comment and will correct for that.
 Figure 3: Why are there many experiments for each combination of the wind shear and wavelet order? There is a description "no additive noise", so there seems to be no Monte Carlo processes. Which parameter is different between those experiments?
There is no Monte Carlo process involved at this place. Line154155: “Figure 3 illustrates the distributions of wavelength ratios derived from the lowermost 50 km of the simulated altitude range […]” With a vertical discretization of dz=0.1km, we obtain 500 wavelength ratios that are plotted as histograms for each wind shear and wavelet order.
 l.177: "Equ,4" might be better as "Eq. 4" or "Equation 4. "
Will be changed as suggested.
 l.194: How the "growth rates" are applied is unclear. I guess the formulation "a(z) = 1.0 K + z K km1" given in l.139 is modified as "a(z) = 1.0 K + 0.1*z K km1", "a(z) = 1.0 K + 1*z K km1", and "a(z) = 1.0 K + 10*z K km1". More explanations are needed.
That is absolutely correct. We apologize for that oversight and will correct the according expressions in line 139 and table 2.
 Figure 5: Similar to Fig.3, I do not understand why many experiments exist for each combination of the normalization, wind shear, and growth rate.
We apologize for the misunderstanding. In fact, we have not mentioned that the distributions are only considering the lowermost 50km as it is the case in figure 3.
 Figure 7: Panel d is not mentioned in the text. Moreover, how this vertical wavelength was obtained is not explained.
We thank the referee for this comment. We will change the following lines as following:
Line 230231: “In addition, we compute the theoretically expected upper limit of vertical wavelengths according to Lambda_z=2*pi*u/N using the Brunt Väisälä frequency determined from the derived temperature background and horizontal winds from ERA5 reanalysis (Fig. 7d).
Line 238: “The ERA5 profile of absolute winds (Fig. 7c) shows […]”
Line 240241: “[…] we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude, i.e. between 20 km and 60 km (see Fig. 7d).”
 l.260: "(Fig. 7b)": Is this a typo for "(Fig. 9b)"?
Thank you. That is a typo indeed. It should be (Fig. 9b).
 l.262: "due to the smaller COI": Is this smaller COI due to the smaller m0? If so, mention it.
We will add the following to line 262: […] which is due to the usage of a smaller order (m0=4).
 l.272: An additional example of wavelet analysis for conditions where multiple waves overlap:
Mori, R., Imamura, T., Ando, H., Häusler, B., Pätzold, M., & Tellmann, S. (2021). Gravity wave packets in the Venusian atmosphere observed by radio occultation experiments: Comparison with saturation theory. Journal of Geophysical Research, 126, e2021JE006912. https://doi.org/10.1029/2021JE006912We will include the reference.
 l.277: For the vertical wavelength to be proportional to the horizontal wind speed, the wave must be a topographically generated gravity wave (mountain wave). This should be mentioned. If the paper focuses on mountain waves, the title of the paper is not appropriate since mountain waves are stationary.
We have mentioned mountain waves at various places in the manuscript:
Line 7: “[…] we apply CWT to test mountain wave signals […]”
Line 40: “The vertical wavelength of stationary mountain waves (MW) […] is approximately given as […]”
Line 224: “The temperature profile in Figure 7a shows significant temperature variability which can be attributed to MWs.”
Line 239241: “Assuming that the signal […] was caused by a MW propagating against the background wind, we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude […].”
Line 283: “[…] consider a MW signal […]”
We want to clarify the title. The manuscript is about nonstationary signatures of atmospheric gravity waves and not about signatures of nonstationary atmospheric gravity waves. The term nonstationary refers to a change of statistical properties such as variance and autocorrelation of the data under consideration over time/length. Even though linear mountain waves are a stationary physical feature in the atmosphere, their signal in the vertical is clearly nonstationary in the statistical sense.
Citation: https://doi.org/10.5194/amt2023240AC1

AC1: 'Reply on RC1', Robert Reichert, 01 Feb 2024

RC2: 'Comment on amt2023240', Anonymous Referee #2, 28 Mar 2024
The manuscript "Limitations in Wavelet Analysis of NonStationary Atmospheric Gravity Wave Signatures in Temperature Profiles" by Reichert et al. is a nice discussion of the limitations of the Continuous Wavelet Transform (CWT) that is often used to analyze nonstationary signals. First, idealized linear chirp signals are used as test signals for optimizing the method. Then the method is applied to a temperature profile observed by the CORAL lidar in South America. Recommendations are made regarding the order of the Morlet wavelet. Further, the authors suggest to normalize observed temperature profiles before performing a wavelet analysis.
Overall, the manuscript is well written and fits into the scope of AMT.
Therefore the manuscript is recommended for publication after addressing my minor but important comments as detailed below.My main comment is that the authors should point out more clearly that the cases discussed in the paper are focused on a very specific meteorological situation (midlatitude winter conditions where wind reversals do not occur at altitudes below 80km), while conditions can be much more complicated in the general case.
SPECIFIC COMMENTS:
(1) l.5358:
As in your paper you are investigating cases of wind shear, you should mention more explicitly that in the presence of wind shear parts of the gravity wave spectrum will dissipate which can lead to rapid decrease of gravity wave amplitudes with altitude. Therefore exponential growth often occurs, but does not hold in every situation.You should also mention that another method to compensate for the growth of gravity wave amplitudes would be the investigation of gravity wave potential energy per unit mass. In this way, not only density variations but also variations of the buoyancy frequency N and of the temperature with altitude are accounted for (Strelnikova et al., 2021).
Strelnikova, I., Almowafy, M., Baumgarten, G., Baumgarten, K., Ern, M., Gerding, M., and Lubken, F.J.:
Seasonal cycle of gravity wave potential energy densities from lidar and satellite observations at 54N and 69N,
J. Atmos. Sci., 78, 13591386, https://doi.org/10.1175/JASD200247.1, 2021.(2) l.6568:
It is important to mention here that close to a critical wind level gravity wave amplitude behavior can be very complicated, and the assumption of exponentially growing amplitudes due to density decrease may no longer hold: On the one hand, close to a critical level, as long as the gravity wave is not saturated, the gravity wave amplitude will grow stronger than suggested by the density decrease in order to compensate for the reduction in vertical wavelength caused by the Doppler shift. On the other hand, even closer to the critical level, the gravity wave amplitude will decrease because the wave saturates which limits the wave amplitude.For a discussion see, for example, Ern et al., 2014, their Section 3.1
Ern, M., F. Ploeger, P. Preusse, J. C. Gille, L. J. Gray, S. Kalisch, M. G. Mlynczak, J. M. Russell III, and M. Riese (2014), Interaction of gravity waves with the QBO: A satellite perspective, J. Geophys. Res. Atmos., 119, 23292355, doi:10.1002/2013JD020731.
This more complex situation in the general case would be a justification for the method you propose later in lines 186192 which allows for both increasing and decreasing amplitudes.
(3) l.277: This is not correct. "Proportionality" is given only for orographic gravity waves, and even in this case proportionality applies only to the wind component parallel to the gravity wave horizontal wave vector. In case of midfrequency gravity waves vertical wavelength is proportional to intrinsic phase speed, and there is a "linear" relationship between vertical wavelength and the wind speed parallel to the gravity wave horizontal wave vector.
TECHNICAL COMMENTS:l.50: Please state that m_0 in Table 1 is the order of the Morlet wavelet.
This is defined later in the paper, but needs to be introduced already here.
l.141: (Fig.1cd). > (Fig.1c, d).
Caption of Fig.1 and elsewhere: same problem, please separate labels of the different panels by commas
l.385, reference ChaneMing et al.: Journal title is sufficient, publisher (Springer) can be omitted
l.393: Please check title of reference Ehard et al.
l.405, reference Ge: Journal title is sufficient, publisher (Copernicus) can be omittedCitation: https://doi.org/10.5194/amt2023240RC2 
AC3: 'Reply on RC2', Robert Reichert, 16 May 2024
Dear Referee #2,
Thank you very much for your comments! We have copied them here and answer each comment below in bold.
The manuscript "Limitations in Wavelet Analysis of NonStationary Atmospheric Gravity Wave Signatures in Temperature Profiles" by Reichert et al. is a nice discussion of the limitations of the Continuous Wavelet Transform (CWT) that is often used to analyze nonstationary signals. First, idealized linear chirp signals are used as test signals for optimizing the method. Then the method is applied to a temperature profile observed by the CORAL lidar in South America. Recommendations are made regarding the order of the Morlet wavelet. Further, the authors suggest to normalize observed temperature profiles before performing a wavelet analysis.
Overall, the manuscript is well written and fits into the scope of AMT.
Therefore the manuscript is recommended for publication after addressing my minor but important comments as detailed below.My main comment is that the authors should point out more clearly that the cases discussed in the paper are focused on a very specific meteorological situation (midlatitude winter conditions where wind reversals do not occur at altitudes below 80km), while conditions can be much more complicated in the general case.
SPECIFIC COMMENTS:
(1) l.5358:
As in your paper you are investigating cases of wind shear, you should mention more explicitly that in the presence of wind shear parts of the gravity wave spectrum will dissipate which can lead to rapid decrease of gravity wave amplitudes with altitude. Therefore, exponential growth often occurs, but does not hold in every situation.That is correct and we thank the reviewer for that comment. We have altered l.5357 in the following way:
“Secondly, according to linear GW theory, not only can vertical wavelengths change rapidly, but the amplitudes of GWs also vary with altitude. Generally, amplitudes increase exponentially with altitude enforced by conservation of energy and decreasing air density. However, when thermal or dynamical instability is reached, wave dissipation occurs, causing GW amplitudes to decrease above the breaking altitude. This variation in GW amplitude may lead to an undesirable shift in the localization of the wavelet during the computation of the CWT.”
You should also mention that another method to compensate for the growth of gravity wave amplitudes would be the investigation of gravity wave potential energy per unit mass. In this way, not only density variations but also variations of the buoyancy frequency N and of the temperature with altitude are accounted for (Strelnikova et al., 2021).
Strelnikova, I., Almowafy, M., Baumgarten, G., Baumgarten, K., Ern, M., Gerding, M., and Lubken, F.J.:
Seasonal cycle of gravity wave potential energy densities from lidar and satellite observations at 54N and 69N,
J. Atmos. Sci., 78, 13591386, https://doi.org/10.1175/JASD200247.1, 2021.We disagree with this statement. It is correct, that the GW potential energy per unit mass is a function not only of density but also of ambient temperature and thermal stability. However, the mere investigation of GW potential energies is not an adequate method to compensate for the growth of GW amplitudes.
(2) l.6568:
It is important to mention here that close to a critical wind level gravity wave amplitude behavior can be very complicated, and the assumption of exponentially growing amplitudes due to density decrease may no longer hold: On the one hand, close to a critical level, as long as the gravity wave is not saturated, the gravity wave amplitude will grow stronger than suggested by the density decrease in order to compensate for the reduction in vertical wavelength caused by the Doppler shift. On the other hand, even closer to the critical level, the gravity wave amplitude will decrease because the wave saturates which limits the wave amplitude.For a discussion see, for example, Ern et al., 2014, their Section 3.1
Ern, M., F. Ploeger, P. Preusse, J. C. Gille, L. J. Gray, S. Kalisch, M. G. Mlynczak, J. M. Russell III, and M. Riese (2014), Interaction of gravity waves with the QBO: A satellite perspective, J. Geophys. Res. Atmos., 119, 23292355, doi:10.1002/2013JD020731.
This more complex situation in the general case would be a justification for the method you propose later in lines 186192 which allows for both increasing and decreasing amplitudes.
We wrote the paragraph in l.6572 to highlight the importance to determine vertical wavelengths of GWs and not their amplitudes. The approximation of amplitudes is just a necessity in order to determine proper vertical wavelengths. We agree, that the behavior of GW amplitudes close to a critical wind level can be very complicated, but since we use a 4^{th} order polynomial to approximate GW amplitude variation with altitude, the proposed method in l.190195 results obviously only in a rough estimation of GW amplitudes. To put it in a nutshell, since we are not interested in the precise determination of GW amplitudes, we will not lead the reader at this point in the manuscript down the wrong path.
(3) l.277: This is not correct. "Proportionality" is given only for orographic gravity waves, and even in this case proportionality applies only to the wind component parallel to the gravity wave horizontal wave vector. In case of midfrequency gravity waves vertical wavelength is proportional to intrinsic phase speed, and there is a "linear" relationship between vertical wavelength and the wind speed parallel to the gravity wave horizontal wave vector.We thank the reviewer for pointing this out. We have altered l.280 in the following way:
“Linear GW theory shows that vertical wavelengths of GWs are a function of horizontal wind speed.”
TECHNICAL COMMENTS:
l.50: Please state that m_0 in Table 1 is the order of the Morlet wavelet.
This is defined later in the paper, but needs to be introduced already here.We add “order” in front of m_0 in the headline of table 1.
l.141: (Fig.1cd). > (Fig.1c, d).We change that as suggested.
Caption of Fig.1 and elsewhere: same problem, please separate labels of the different panels by commasWe change that as suggested.
l.385, reference ChaneMing et al.: Journal title is sufficient, publisher (Springer) can be omittedWe change that as suggested.
l.393: Please check title of reference Ehard et al.We thank the reviewer for pointing that out. We change that accordingly.
l.405, reference Ge: Journal title is sufficient, publisher (Copernicus) can be omittedWe change that as suggested.
Kind Regards,
Robert Reichert
Citation: https://doi.org/10.5194/amt2023240AC3

AC3: 'Reply on RC2', Robert Reichert, 16 May 2024

EC1: 'Comment on amt2023240', Robin Wing, 26 Apr 2024
Dear Authours,
In addition to the excellent comments provided by Reviewer 1 and Reviewer 2 it may be worth considering the following four points:
1) The angle of propagation between the gravity wave and horizontal wind can change (i.e. theta=48 degrees).
2) The change in vertical wavenumber can depend on background wind, intrinsic frequency, buoyancy frequency, and horizontal wavenumber. You could start from the TaylorGoldstein or Raytracing equations to confirm all of your assumptions.
3) There could be two (or more) families of gravity waves in your lidar observation.
4) It is not always easy to separate the geophysical from the technical. Simulating a test wave which resembles a realistic wave seen in lidar could allow you more control over all the technical variables without having to worry about the complex geophysics happening in the background.
Cheers,
Robin WingCitation: https://doi.org/10.5194/amt2023240EC1 
AC2: 'Reply on EC1', Robert Reichert, 16 May 2024
Dear Editor,
We have revised our manuscript and considered the points you raised above.
Since it was suggested by referee #1 we added a figure (now Figure 1) to the manuscript which illustrates three Morlet wavelets with the same scale but different orders. In addition, we show their representation in the distancewavelengthspace and Fourier space in order to highlight the localization in each space as function of the order.
We have revised the text at the following lines:
L100101: “Three Morlet wavelets with orders of 4, 6, and 8 are illustrated in Figure 1.”
L126: “A scale of s=5km is equivalent to a wavelength of 3.9km for an order of 4 and 7.6km for an order of 8 (see Fig. 1).”
L223: “[…] we analyze a temperature profile obtained at Río Grande (53.7°S, 67.7°W), Argentina on the night of 21 May 2018 00UTC.”
L232: We change the cutoff from 22km to 20km.
L235: “[…] and juxtapose the measured vertical wavelength using our best practice (Fig. 8d).”
L235236: “Reanalysis data is spectrally truncated at wavenumber T21 in order to define a synopticscale background (e.g. Reichert et al., 2021).”
L238: We change the maximum growth rate from 0.21K/km at 47km to 0.82K/km at 36km.
L241: We change the maximum from 10K at 75km to 20K at 55km.
We erased the following:
L242245: “The ERA5 profile of absolute winds shows a stratospheric wind minimum around 2025km and a maximum at 60km. The vertical shear of horizontal wind exceeds 3.4m/s/km between 29km and 35km. Assuming that the signal shown in Figure 8b was caused by a MW propagating against the background wind, we can expect an almost linear chirp in the altitude range where the horizontal wind increases linearly with altitude, i.e. between 20km and 60km.”
And replaced it with:
L242245: “The ERA5 profiles of zonal and meridional wind show a rather steady increase of wind speeds between 20km and 50km. The vertical shear of horizontal wind exceeds 3.4m/s/km between 32km and 37km. At this altitude, we find a discontinuity in the profile of measured vertical wavelengths. Computed and measured vertical wavelengths agree quite well below 35km but differ by up to a factor of two above 35km.”
We have rewritten subsection 3.4.2
“We now create a WPS in the conventional way in which the amplitudes of the temperature disturbance are not normalized, the order of the wavelet is set to m_0=6, and no significance levels are determined (Fig. 10a). Furthermore, we create a WPS based on our bestpractice procedure where the amplitudes of the temperature perturbation are normalized, the order of the wavelet is set to m_0=4, and significance levels are calculated (Fig. 10b).
In the conventional WPS we find only little variation in the vertical wavelength with values ranging from 12.7km to 16.4km. Values decrease from the upper and lower edge of the profiles towards 65km altitude. Due to the COI, determined Lz can be considered reliable only between 35km and 75km altitude. The conventional WPS shows no other interesting features.
Let us now turn to the bestpractice WPS (Fig. 10b).
Similar to the conservative case, we find an extended altitude region from 30km to 80km with only little variation in the vertical wavelength with values of 11.1km to 15.0km which are smaller than the values found in the conservative case. This agrees with the results from our sensitivity study (Section 3.1). In contrast to the conventional case we are able to identify maxima in the WPS now at vertical wavelengths from 4.7km at 20km altitude to 9.8km at 30km altitude and vertical wavelengths in the order of 6km to 7km above 80km altitude.”
L297: We changed “Lz=7km” to “Lz=10km”
We added:
L301304: “ERA5 temperature perturbation fields and colocated OHairglow imagery provide more evidence that the MW observed by CORAL propagates steeply within the lidar’s field of view. On the other hand, the difference between computed and measured vertical wavelength (Fig. 8d) could be an indication for an obliquely propagating MW. After all, this work is of methodological nature and the geophysical interpretation of the results is not in our focus.”
L325: “[…] revealing an increase of vertical wavelength from Lz=10km to Lz=15km at approximately 32km altitude.”
L352: “[…] midfrequency MW”
To your 1):
We addressed the angle between wind direction and wave propagation direction by pointing out that we see a difference in the measured and computed vertical wavelength which could be a hint on lateral propagation.
To your 2):
The focus of this work is to determine the limitations of the continuous wavelet transform. One of the aspects was to investigate which chirp rates, i.e. frequency shifts can be resolved using the CWT. In order to learn more about the implications of this study, we relate the linear chirps we define in Section 2.2 to horizontal wind shear using a) the midfrequency approximation, b) assuming we are dealing with mountain waves, and c) assuming that wind and wave propagation direction are constant.
To your 3):
Indeed, additional ERA5 analysis has shown that the nightly mean profile we used previously from the 1718 April 2018 showed very likely a superposition of two mountain waves that propagated obliquely through CORAL’s field of view. Therefore, we decided to show another temperature profile from 21 May 2018 (see Figure 8a, b in the manuscript). ERA5 analysis shows that mountain waves are excited close by the lidar location and propagate steeply (see supplemented figure c, d).
To your 4):
We refrain from performing idealized numerical simulations in order to improve the interpretation of the results from section 3.4. As stated before, the geophysical interpretation is not the focus of this work.
Kind regards,
Robert Reichert

AC2: 'Reply on EC1', Robert Reichert, 16 May 2024
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