We analyze the prospects of employing a distributed global network of precision measurement devices as a dark matter and exotic physics observatory. In particular, we consider the atomic clocks of the Global Positioning System (GPS), consisting of a constellation of 32 medium-Earth orbit satellites equipped with either Cs or Rb microwave clocks and a number of Earth-based receiver stations, some of which employ highly-stable H-maser atomic clocks. High-accuracy timing data is available for almost two decades. By analyzing the satellite and terrestrial atomic clock data, it is possible to search for transient signatures of exotic physics, such as "clumpy" dark matter and dark energy, effectively transforming the GPS constellation into a 50,000 km aperture sensor array. Here we characterize the noise of the GPS satellite atomic clocks, describe the search method based on Bayesian statistics, and test the method using simulated clock data. Our method is not limited in scope to GPS or atomic clock networks, and can also be applied to other networks of precision measurement devices.