An advanced hodograph-based analysis technique to derive gravity-wave (GW) parameters from observations of temperature and winds is developed and presented as a step-by-step recipe with justification for every step in such an analysis. As the most adequate background removal technique the 2-D FFT is suggested. For an unbiased analysis of fluctuation whose amplitude grows with height exponentially, we propose applying a scaling function of the form

It is generally accepted that atmospheric gravity waves (GWs) produce global effects on the atmospheric circulation from the surface up to the mesosphere and lower thermosphere (MLT) region

Our knowledge about gravity-wave parameters can be improved by means of high-resolution measurements of atmospheric GWs. Ideally, the measurement range should cover the entire path of the waves, starting from their sources in the troposphere to the level of their dissipation that is up to the MLT region. This type of measurement ultimately faces high experimental challenges, which explains why we still do not have satisfactory and conclusive observational data on these processes.

In the altitude range of the mesosphere only a few observation techniques exist.
In the last decades the only source of high-resolution GW observations based on both temperatures and winds in the stratosphere and mesosphere region were rocket soundings

Satellite-borne remote-sensing techniques can provide excellent global coverage; their observations deliver unique horizontal information about GWs

Ground-based radar systems are able to measure winds at heights of 0–30 and 60–100 km. From the altitudes between 30 and 60 km radars do not receive sufficient backscatter and, therefore, cannot provide wind measurements in this region. While the vertical wave structure can be resolved from rocket profiles, the long and irregular time intervals between successive launches prevent the study of temporal gravity-wave fluctuations over a larger time span

Recent developments in lidar technology give us new possibilities to study GWs experimentally on a more or less regular basis and resolve spatial sales of 150 m on vertical and temporal scales of 5 min

All those quantities, i.e. winds and temperature, when measured with high temporal and spatial resolution, reveal structuring at scales down to minutes and hundreds of meters. In our analysis technique we focus solely on such fluctuations which are generated by GWs. By applying a proper data analysis technique, one can extract several important parameters of GWs from the advanced lidar measurements.

In this paper we describe a newly developed analysis technique which allows for derivation of GW parameters such as vertical wavelength, the direction of propagation, phase speed, kinetic and potential energy, and momentum flux from the advanced lidar measurements.
We aim at presenting a step-by-step recipe with justification of every step in such an analysis. Every single step, if considered independently, is in general well known. The strength and novelty of our work is their combination and some justification on their importance and how they affect analysis results.
The paper is structured as follows. In the next section a short description of the lidar measurement technique is given. The theoretical basis used by the data analysis technique is shortly summarized in Sect.

The ALOMAR Rayleigh–Mie–Raman lidar in northern Norway (69.3

A GW field consists of various waves with different characteristics. An attempt to describe this system as a whole is made, for example, by Stokes analysis

For this analysis we use the assumption that a wave packet at a fixed time point and in a limited altitude range can be considered to be QMGWs, i.e., dispersion within one wave packet is neglected. Also we assume that all the observed parameters (

Equation (

Following

This theoretical basis allows us to describe the main GW parameters and to derive them from observations. However, in practice, noisy data and/or insufficient resolution of measurements may lead to large uncertainties when applying these equations directly to the measured time series. Therefore, the most common technique, based on linear theory of gravity waves to derive the propagation direction, intrinsic frequency and phase velocity of GWs from ground-based observations, is the hodograph method

For mid- and low-frequency GWs the velocity perturbations in the propagation direction and perpendicular to this direction are related by the polarization relation (e.g.

If we assume that

To summarize, the basic theory, briefly described in this section, allows us to derive the main GW parameters: intrinsic frequency, amplitude and the direction of propagation. From these parameters one can derive a more extended set of GW parameters: horizontal and vertical phase speed, group velocity, kinetic and potential energy, and vertical flux of horizontal momentum, as summarized in Appendix

In this section we describe the procedure for deriving wave parameters from the measured lidar data. For our analysis we need simultaneously measured wind and temperature profiles. Technically we can extract wave parameters from a single measurement – that is, using two wind and one temperature profile. However, for a robust estimation of the atmospheric background we need an observational data set that is several hours long.

Schematics of the method.

The first step is to remove the background from the measured data. The background removal procedure plays a key role in GW analysis techniques and may even lead to strongly biased results. This is because most analysis techniques rely on fluctuation's amplitudes remaining after subtraction of the background to infer wave energy

We define the background as fluctuations with periods and vertical wavelengths longer that typical GW parameters. This means that tidal fluctuations and planetary waves are attributed to the background. Tidal periods are integer fractions of a solar day. Semidiurnal tides have period of 12 h, and the Coriolis period (

Thus, we define the background as wind or temperature fluctuations with periods longer than 12 h and vertical wavelengths larger than 15 km. Fluctuations with periods shorter than 12 h, which have any vertical wavelength (also greater than 15 km), are attributed to GWs and are the subject of further analysis.

Temperature observations.

To extract such a background from measurements, we apply a low-pass filter to the altitude vs. time data. Specifically, we use the two-dimensional fast Fourier transform (

The same as Fig.

We also performed a robustness test to check how different background removals influence our advanced hodograph-based method. To derive the background (for both wind and temperature data) we additionally made use of (a) a running mean with different smoothing window lengths, (b) different splines and (c) constant values in time. It turned out that our analysis results were near identical for all these different backgrounds. The new technique is not sensitive to the background derivation schemes and may even allow us to skip this step from the analysis or to apply simple methods. A more in-depth analysis showed that the robustness to the background removal is a consequence of the analysis approach. We only search for waves which are prominent simultaneously in temperature and both wind components. Even though we are confident in the robustness of our GW analysis technique to the various background derivation methods, we need a well-defined and well-behaved (i.e. continuous and smooth) background (1) to derive the basic parameters of atmosphere like buoyancy frequency and wind shear and (2) to find out how the background wind and temperature fields affect (or at least correlate with) the GW field. Thus, we consider the 2-D-FFT-based approach to be the one most adequate for this purpose.

The same as Fig.

Under the assumptions of conservative propagation (i.e., without wave breaking and dissipations) in the isothermal atmosphere without background winds, the amplitude of fluctuations increases with altitude as

Wavelet transform of zonal and meridional wind and temperature fluctuations at 02:07 UT on 10 January 2016.

Starting from this point we only analyze the altitude profiles at every time step. At every time step we have measured profiles of wind and temperature, which are split into fluctuations and background profiles.

First, we search for dominant waves in both altitude and wave-number domains. For this purpose we apply the continuous wavelet transform (CWT) to every profile of the extracted fluctuations. We use a Morlet wavelet of the sixth order

An example of the resulting scalograms for one time step is shown in Fig.

Combined wavelet transform of profiles shown in Fig.

In this step we fit a wave function to all three measured profiles, i.e.,

The fit can be performed using a least-squares regression algorithm implemented in numerous routines. The data (measurements) to which the wave function is to be fitted are the three profiles

Thus, to derive the first set of initial parameters

Thus, the updated values for this demonstration case are

We recall that the vertical extend of wave packet

A similar way of deriving initial guess parameters was implemented by

Generally speaking, intrinsic frequency and the propagation direction can be estimated from the obtained fitting results by applying Eqs. (

According to the theory

To extract the essential parameters of the wave packet found in the previous steps we apply the hodograph analysis around the center of the wave packet

In order to minimize an error in the hodograph analysis due to the presence of other waves

The vertical propagation direction of the wave is unambiguously determined by the rotation direction of the zonal wind versus meridional wind hodograph. In the Northern Hemisphere the (anti-clockwise) clockwise rotation of the hodograph indicates a (downward-propagating) upward-propagating wave.

The rotation direction of the hodograph is defined as a phase angle change of either

An additional hodograph of the parallel wind fluctuations versus temperature fluctuations is used to resolve an ambiguity in the horizontal propagation direction that arises from the orientation of the ellipse in Fig.

If, in the previous step, the rotation of the hodograph does not make a full 360

The ratio of the major and minor ellipse axes is further used to derive the intrinsic frequency of GWs (Appendix

Knowing all these wave parameters and applying the linear wave theory we derive further wave characteristics as summarized in Sect.

Figure

After the first QMGW is identified in all three profiles, it is subtracted from those data. We repeat the procedure described above for all of the maxima seen in the combined spectrogram (Fig.

Total number of waves obtained per altitude profile for the entire data set.

Finally, this algorithm for a single point in time is subsequently applied to all time points of the entire data set shown in Figs.

Examples of hodograph results from 10 January 2016 at 02:07:30 UT.

In this section we demonstrate, on a real data set, how our analysis works, and results are summarized in the form of different statistics.

The data used in this study were obtained from 9 to 12 January 2016. During this time period a strong jet with wind speeds of more than 100 m s

After applying the new analysis technique to the

First, a short discussion about the exponential scaling factor

Reconstructed temperature fluctuations of upward-propagating GWs

The number of detected waves per altitude profile is summarized in the histogram in Fig.

Figure

It is interesting to compare these results with the mean background wind shown in Fig.

Color-coded bars show the intrinsic period of upward-propagating

The existence of the downward-propagating waves was reported earlier from observations by different methods

To investigate the time and altitude dependence of the GWs detected by our hodograph technique, we reconstructed the temperature and the wind fluctuation fields from the derived waves parameters using Eq. (

We use similar representations to investigate the temporal variability in any of the other derived GW properties. For example, Fig.

Histograms of different GW properties separated for upward- and downward-propagating waves.

Fourier power spectra of measured temperature fluctuations (blue) and of the reconstructed GWs (orange).

Kinetic

In Fig.

Polar histograms of the direction of the background wind

The distribution of phase velocities in Fig.

Another way to check the consistency of our technique is to look at the spectrum of fluctuations before and after analysis. As an example, Fig.

Histogram of the absolute value of the angle between the group velocity vector and the horizon, separated for upward- and downward-propagating waves.

Next, we analyze and sum up the wave energetics. Figure

The directions of background wind and wave propagation are summarized in Fig.

Figure

To find the vertical angles at which the GWs propagate, we show histograms of the angle between the group velocity vector and the horizon (

The vertical group velocities

Finally, in Fig.

In this paper, a detailed step-by-step description of a new algorithm for derivation of GW parameters with justification for every step is presented. Most of these steps, if considered independently, are well known and validated in numerous experimental works. The advantage and novelty of this work are their combination and some justifications of their importance and how they affect GW analysis results.

Thus the very first action normally performed on the measured time series is background removal. Since most conventional techniques based on smoothing or averaging in time or altitude ultimately introduce artifacts, we justify that application of the 2-D FFT for background removal is the most appropriate. The advantage of this method is that it simultaneously accounts for both variability in space and time.

A specific feature of our algorithm for GW analysis is that it is insensitive to the particular background removal scheme. Therefore, to avoid any degree of arbitrariness, the background removal can be excluded from fluctuation analysis when applying further steps of the analysis technique described in the paper.

As a next step we proposed applying a scaling function of the form

Polar histogram of the upward

The most essential part of the proposed analysis technique consists of fitting cosine waves to simultaneously measured profiles of winds and temperature and subsequent hodograph analysis of these fitted waves. We emphasize that this fit must be applied to all three quantities, i.e., zonal and meridional wind and temperature (

All these advantages are especially important, since modern advanced measurement techniques (e.g., our lidar system described in Sect.

One obvious advantage of the proposed algorithm is that it allows for simultaneous detection of any kind of waves presented in the measurements. This includes not only GWs but also tides. Since the new analysis algorithm allows us to apply the simplest background removal techniques like subtraction of a mean, the necessity of the removal of tidal components a priori, which cannot be done unambiguously, is eliminated. All the detected waves can be sorted out on a statistical basis after the observational database is analyzed by using the proposed algorithm.

Vertical flux of horizontal momentum averaged through all observed hodographs (dashed) and upward-propagating (blue) and downward-propagating (orange) waves.

Another specific feature of our analysis technique is the extension to the linear wave theory introduced in Sect.

By applying this new methodology to real data obtained by lidar during about 60 h of observations in January 2016 we found 4507 single hodographs. In general, 5 to 10 waves were detected from every vertical profile. This allowed identifying and analyzing quasi-monochromatic waves in about

Polar histogram of upward-propagating

The main characteristics of the upward- and downward-propagating GWs were investigated statistically. We find that the downward-propagating GWs reveal shorter intrinsic periods and lower phase speeds than the upward-propagating GWs. Downward waves propagate at steeper angles than the upward-propagating ones. Currently, our analysis does not allow us to distinguish between primary and secondary GWs. The next step will be to look for similar wave characteristics (horizontal, vertical wavelengths and propagation direction) in the upward- and downward-propagating waves. The nearby occurrence of similar waves with an opposite vertical propagation direction is an indication of secondary GWs

A monochromatic gravity-wave (GW) perturbation in Cartesian coordinates (

where

Alternatively, these equations can be rewritten in the form

Finally, we take into account that the quasi-monochromatic gravity wave (QMGW) is limited in space; i.e. it appears in our observations within a limited altitude range:

The most common technique, based on the linear theory of gravity waves to derive the propagation direction, intrinsic frequency and phase velocity of GW from ground-based observations, is the hodograph method

In order to keep the polarization relation as simple as Eq. (

The relationship between fluctuations in new (

Amplitudes of the ellipse in the new coordinate system

Thus,

On the other hand the intrinsic frequency is a function of the buoyancy frequency (

From this equation the horizontal wave number along the propagation direction can be derived

The horizontal–vertical phase speed is the ratio of intrinsic frequency to the horizontal–vertical wave number

The vertical component of the group velocity

The angle between the group velocity vector and the horizon can be estimated from

The kinetic energy density of GWs estimated from observed fluctuations

Thus, the kinetic energy density as a function of fitted amplitudes of the wind hodograph is

The potential energy density of GWs estimated from observed fluctuations is

The vertical flux of horizontal momentum in the wave propagation direction can be written as

The data used in this paper are available upon request.

IS developed the analysis technique algorithm and code and performed the calculations. GB designed experiments and conducted measurements. GB and IS analyzed the data. IS, GB and FJL contributed to the final paper.

The authors declare that they have no conflict of interest.

This study benefited from the excellent support by the dedicated staff at the ALOMAR observatory. The DoRIS project was supported by Deutsche Forschungsgemeinschaft (DFG – German Research Foundation; grant no. BA2834/1-1). This project has received funding from the European Union's Horizon 2020 Research and Innovation program under grant agreement no. 653980 (ARISE2) and was supported by the Deutsche Forschungsgemeinschaft (DFG – German Research Foundation) under project LU1174/8-1 (PACOG), FOR1898 (MS-GWaves).

The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Gerhard Ehret and reviewed by four anonymous referees.