The accurate estimation of spectral power-law exponents (β) in atmospheric gravity wave (GW) spectra plays a crucial role in understanding energy transfer mechanisms within the atmosphere. However, dealing with observational gaps in atmospheric measurements poses a significant challenge for obtaining reliable estimations. Using simulated data of varying complexity, we rigorously evaluate the performance of these estimation methods and aim to offer valuable guidance in selecting the most appropriate approach for accurately determining β from observational datasets with gaps. Our findings are of paramount importance for researchers, as they can significantly impact our understanding of energy propagation and dissipation in the atmosphere.To further extend the scope and applicability of our investigation, we have embarked on analyzing lidar measurements of temperatures and winds in the stratosphere and mesosphere above Kühlungsborn (54°N, 12°E) and ALOMAR (69°N, 16°E). By estimating the spectral power-law exponents of atmospheric GWs in these distinct regions, we intend to shed light on the energy transfer processes occurring during complex atmospheric dynamics in the vicinity of the polar vortex.This research aims to bridge the gap between theoretical predictions and empirical findings in atmospheric science by providing robust methodologies for accurate spectral estimation, even in the presence of sparse observational data.

An advanced hodograph technique was applied to extract quasi-monochromatic, linear gravity wave (GW) packets from simultaneous wind and temperature measurements by a ground-based Doppler-Rayleigh-Mie-Raman lidar located in Kühlungsborn (54\,\textdegree N, 12\,\textdegree E). During the night of 11 to 12 February 2022, the stratospheric polar vortex was slightly elongated towards Central Europe. The polar night jet’s (PNJ) core was located nearly above the lidar. This unique meteorological situation allowed horizontal wind and temperature measurements in a stratospheric high-wind speed regime. GW refraction in the strong vertical gradient of the horizontal winds associated with the PNJ was observed. In addition, the subsequent low-pass filtering of upward and downward propagating GWs in the strong winds of the PNJ was demonstrated. A pair of upward and downward propagating GWs at the top and bottom edge of the PNJ’s core could be the result of shear excitation in the PNJ. A statistical analysis of intrinsic GW parameters is provided for all resolved linear GWs. Approximately 35\,\% of the resolved GWs propagate downward, and there is stronger filtering of the downward waves compared to the upward waves by the PNJ core. ECMWF-IFS horizontal winds and temperatures had a lower variability than the lidar and a poorer overall agreement above 50\,km. Stratospheric temperatures in ECMWF show a cool bias of greater than 2\,K.