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
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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
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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
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