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.