A high-energy atmospheric physics phenomenon, referred to as a terrestrial gamma ray flash (TGF), is associated with lightning and produces large bursts of energetic photon radiation. TGFs will be investigated using a suite of gamma-ray instruments designed and constructed to fly on ten United States Air Force (USAF) WC-130J Hurricane Hunter aircraft as part of an aircrew ionization study led by the Air Force Institute of Technology (AFIT) and the United States Air Force School of Aerospace Medicine (USAFSAM), in cooperation with the 53rd Weather Reconnaissance Squadron (WRS). Each instrument consists of one NaI and one plastic detector, a GPS timing device, and an instrument computer that performs data acquisition. High Elevation Radiation Array (HERA) detectors will be employed to maximize the chances of observing TGFs near their source and to gain a better understanding of their origin, mechanism, ubiquity, and to assess potential hazards posed to military and commercial aircrew and passengers. The HERA program, deployed on 10 separate Air Force aircraft over a multi-year campaign, will result in thousands of observational flight hours and be the largest concerted effort to date to observe TGFs in situ through aircraft observations. In this paper, we give an overview of the scientific goals of this campaign and how the HERA instruments have been designed to meet those goals. We include a detailed description of the HERA instrument, along with mass model and signal processing simulations.