Gravity gradiometry inversion can provide important knowledge about a salt body and can assist in subsalt imaging. However, such inversions are faced with difficulties associated with the lack of response from the nil zone in which the salt density is nearly identical to that of the background sediments and weak signals from the deeper portion of the salt. It is well understood that such difficulties could be alleviated by incorporating prior information, such as the top of salt from seismic imaging and petrophysical data, into the inversions. How to effectively incorporate such prior information is still a challenge, and what level of increased knowledge such constrained inversions can provide remains to be understood. We have investigated and compared the additional knowledge provided by incorporating different forms of prior information, such as a top-of-salt surface and an expected density contrast model. These different types of information are incorporated through different strategies of constrained inversion, including an inversion with bound constraints on the density contrast, inversion after a reduction-to-binary process, and discrete-valued inversion. We apply these strategies first to synthetic gravity gradiometry data calculated from the SEG/EAGE salt body and evaluate the improvements to the recovered salt provided from successive imposition of increased prior information. We further apply the strategies to a set of marine gravity gradiometry data collected in the Gulf of Mexico and examine the additional knowledge gained from the imaging of the salt in the region. We find that much more valuable knowledge about the salt can be obtained with the right prior information imposed through an effective strategy, and we determine that such gravity gradiometry data contain information about the salt body at depths much greater than previously recognized.