Underground gas storage facilities are an important element of the natural gas supply system. They compensate for seasonal fluctuations in natural gas consumption. Their expected lifetime is in tens of years. Continuous monitoring of underground gas storage is therefore very important to ensure its longevity. Periodic injection and withdrawal of natural gas can cause, among other things, vertical movements of the terrain surface. Radar interferometry is a commonly used method for tracking changes in the terrain height. It can register even relatively small height changes (mm/year). The primary aim of our research was to verify whether terrain behavior above a relatively deep underground gas storage can be monitored by this method and to assess the possibility of detecting the occurrence of anomalous terrain behavior in an underground gas storage area such as reactivation of faults in the area. The results show a high correlation between periodic injection and withdrawal of natural gas into/from the underground reservoir and periodic changes in terrain height above it (the amplitude of the height changes is in centimeters), which may allow the detection of anomalous phenomena. We documented special behavior of storage structures in the Vienna Basin: the areas adjacent to the underground gas storages show exactly the opposite phase of vertical movements, i.e., while the terrain above the underground reservoirs rises as natural gas is injected, the adjacent areas subside, and vice versa. Based on the analysis of geological conditions, we tend to conclude that this behavior is conditioned by the tectonic fault structure of the studied area.