We present a thorough investigation into the accuracy and reliability of gravity wave (GW) spectral estimation methods when dealing with observational gaps. GWs have a significant impact on atmospheric dynamics, exerting influence over weather and climate patterns. However, empirical atmospheric measurements often suffer from data gaps caused by various factors, leading to biased estimations of the spectral power-law exponent (slope)

Gravity waves (GWs) are ubiquitous phenomena that play a crucial role in the dynamics of the Earth's atmosphere, where they impact weather and climate patterns

On that note, VanZandt first introduced the concept of a “universal atmospheric GW spectrum”

This GW spectrum exhibits a power law scaled by an exponent (or slope)

Determining

In this paper, we systematically quantify the advantages and limitations of estimation methods of GW spectra in handling these error sources. We also propose a procedure for selecting unambiguously suitable methods for

Previous studies have investigated these spectral methods and others for estimating power-law spectra and compared their performance using synthetic and observed data. For instance,

The rest of the paper is organized as follows: in Sect.

FFT is the most commonly used method for estimating frequency spectra of evenly sampled data

The GLS periodogram developed by

The LS method has often been used to seek dominant periodic frequencies or cycles

HSF is a mathematical tool used in conducting scaling analysis of signals

A power spectrum which follows a power-law

In this section, we present the simulation procedures used to generate time series similar to actual GWs measurements. In measurements, GWs can exhibit various behaviours, ranging from superposed waves within wave packets with multiple frequencies, amplitudes, and phases to more coherent quasi-monochromatic waves

By analysing both simulations, we can determine the accuracy of each of the methods at different levels of signal complexity and identify potential limitations and sources of error in the analysis of GWs spectra. Random gaps are then introduced to resemble observational gaps for both simulations. The units and values of the variables used in this simulation have been chosen to represent average values or ranges characteristic of typical GW time series.

Quasi-monochromatic GWs can be observed under specific conditions where a single frequency dominates other components

The amplitude of the time series is equivalent to the estimated height of the main peak in the spectrum. It is computed from the FFT coefficients (Eq.

As reported before (see Table

The PSD is obtained by FFT using the relation

After creating a time series with the desired spectral properties, gaps are introduced by randomly removing data points (except both endpoints), assuming that all data points are equally likely to be removed (i.e. a uniform distribution). Based on the simulated GP

Before applying spectral methods to the generated time series from the simulations in Sect.

The time series is first interpolated using the original time step of 5 min; this is only necessary for FFT.

The mean of the signal

The frequency range spans from

The frequency spacing is given by

This approach is more appropriate for GLS and HSF than applying a “pseudo-Nyquist” limit based on an average or a minimum value of

A maximum likelihood estimator (MLE) is employed to determine the fit parameter

The MLE fit involves minimizing the negative log-likelihood function

The MLE fit is recommended over least-squares regression because the latter assumes a Gaussian distribution of periodogram residuals, leading to a biased estimate of

First, we show an example of the time series generated by the simulation described in Sect.

Time series of a 0.5 h wave in a 6 h observation time generated according to Sect.

When comparing average relative period bias for different simulation periods, Fig. 2a shows that GLS demonstrates no period bias below 80 % GP and a negligible bias beyond (within

Comparison of relative period bias (Eq.

Similarly, FFT's amplitude bias experiences clear dependency on the frequency of the signal, while GLS demonstrates a negligible amplitude bias at GPs below 80 %; see Fig.

Comparison of relative amplitude bias (Eq.

Overall, GLS provides a more robust estimation of the period and amplitude of gapped time series, while FFT's performance is simultaneously dependent on the GP and frequency of the signal.

A time-series example is shown in Fig.

A time-series example (upper left and right) generated by the spectral power-law simulation (according to Sect.

In the absence of gaps, the true value of

For a statistically significant picture, we show the distributions of estimated

Histograms of the estimated

To further explore the behaviour of the bias under different conditions, we evaluated the effect of changing the simulated value of

Comparison of the bias in the mean

In the instance of gapped time series, our results show that as the GP increases, the biases in the estimated exponent also became more pronounced for all three methods. Similarly to the non-gapped situation, GLS demonstrates an exceptional efficiency in estimating flat spectra where

In similar fashion, both FFT and HSF demonstrate a relatively constant bias for

In light of the aforementioned considerations, it can be argued that the FFT technique demonstrates competence in generating accurate spectral estimations for non-gapped time series. Nevertheless, it encounters challenges as the data incorporate an increasing number of gaps, necessitating interpolation techniques which introduce inherent biases. Meanwhile, HSF is demonstrated to be a particularly reliable approach for analysing GW time series with spectral power-law exponents

Conversely, the GLS method yields similarly favourable outcomes, particularly for time series whose spectra are flat and high-frequency-dominated, so that it even surpasses the accuracy of HSF when

The problem of power leakage from low frequencies into higher ones arises as a result of the constrained frequency range, which itself is limited by the observed time span

In both cases, longer-than-

Averaged temporal spectra of non-gapped time series generated by the spectral power-law simulation with a spectral exponent of

While a weighted fit of the spectra can reduce the bias, it does not fully rectify the problem of leakage; it also requires a smoothed spectrum and may be confounded by other biases from observational noise, gaps, or method inefficiencies. One approach to counteract this leakage from unresolved low-frequency power is based on “prewhitening” the time series and then “postdarkening” the spectra

In Fig. 7c, we present the postdarkened spectra after prewhitening the time series for

The spectral analysis of GW time-series data is a complex task that requires careful consideration of various factors. Based on our simulation results, we propose a flowchart (see Fig.

Recommended procedure for estimating power spectra of gravity wave time series.

The domain of observational analyses, especially in atmospheric physics, is vast and complex. The intrinsic nature of these measurements requires precision, reliability, and adaptability of analytical methods to extract meaningful insights. Observational gaps due to instrument failures or adverse conditions are also common in atmospheric physics. Our research on synthetic atmospheric gravity wave (GW) time series offers a comprehensive overview of different spectral estimation methods, shedding light on their strengths and limitations. The methods compared in this study are the fast Fourier transform (FFT) (which requires interpolation of gaps) and the generalized Lomb–Scargle method (GLS) and the Haar structure function (HSF) (both which can handle unevenly sampled data without interpolation). Their performance is assessed by evaluating whether the output spectra of simulated time series (with known a priori features and gap distributions) match the input parameters of the data, including the frequency, amplitude, and spectral slope. Building on our findings, we propose the following recommendations:

These sequential decisions highlight the importance of tailoring analysis methods to the characteristics of the data at hand. Employing a one-size-fits-all method can result in biased spectra, which are especially critical in GW parameterizations in all atmospheric models. Our recommendations, grounded in rigorous research on synthetic gravity wave time series, aim to guide researchers in making informed decisions, ensuring the accuracy of spectral results and advancing our understanding of the dynamics of the atmosphere.

All codes used in this paper are available under

MM wrote the codes to conduct these simulations, analysed their results, and drafted the manuscript. IS, RW, and GB provided supervision and scientific insight and edited the text of the manuscript.

At least one of the (co-)authors is a member of the editorial board of

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This paper is a contribution to the project W1 (Gravity Wave Parameterization for the Atmosphere) of the Collaborative Research Centre TRR 181 “Energy Transfers in Atmosphere and Ocean” and the project Analyzing the Motion of the Middle Atmosphere Using Nighttime RMR-lidar Observations at the Midlatitude Station Kühlungsborn (AMUN).

This research has been supported by the Deutsche Forschungsgemeinschaft (grant nos. 274762653 and 445400792).The publication of this article was funded by the Open Access Fund of the Leibniz Association.

This paper was edited by Lars Hoffmann and reviewed by two anonymous referees.