Temperature measurements from the HIRDLS instrument on NASA's Aura satellite are analysed for the purposes of detecting and studying internal gravity waves in the terrestrial stratosphere. A detailed description of the methodology used to obtain these data is given, including details of the instrument correction processes used to compensate for errors introduced by a blockage in the instrument optics. A short precis of the relevant theoretical considerations related to atmospheric gravity waves is then outlined. The thesis then discusses the use of the Stockwell (time-frequency) Transform for the detection of gravity waves in HIRDLS data, together with a detailed analysis of the limitations of this method, and the results obtained from this analysis are analysed by comparison to other instruments and climatology. It is concluded that the Stockwell Transform is an appropriate method for the analysis of the HIRDLS dataset, and that the results obtained are robust. We apply these results to analyse stratospheric gravity wave activity during the 2005/06 Arctic sudden stratospheric warming. By comparing the magnitude and form of the gravity wave results to local wind data obtained from ECMWF operational analyses, we conclude that a heavily deformed stratopause observed during this period by other instruments was most probably due to wind-based filtering of the gravity wave spectrum during this period.