The use of acoustic measurement systems in prior art downhole applications, such as logging while drilling (LWD) and wireline logging applications is well known.  In one application, acoustic signals may be generated at one or more transmitters deployed in the borehole.  These acoustic signals may then be received at an array of longitudinally spaced receivers deployed in the borehole.  Acoustic logging in this manner provides an important set of borehole data and is commonly used in both LWD and wireline applications to determine compressional and shear wave velocities (also referred in the art in terms of their slowness values) of a formation.  These velocities may then be used to compute certain elastic constants of the formation. 
Many subterranean formations exhibit elastic anisotropy that may be intrinsic, for example, owing to orientation and layering of sediments (e.g., in shales) or may be induced by the presence of fractures or the stress sensitivity of the formation.  Such formations are commonly modeled as being transversely isotropic media having five elastic constants.  However, conventional axial acoustic measurements only enable three (or occasionally four) of these five elastic constants to be determined.  The remaining constant(s) are sometimes inferred from various assumptions about the geophysics of the formation; however, such inferences can be problematic.  Therefore, there is a need in the art for a downhole acoustic measurement system and methodology for directly determining the five elastic constants of a transversely isotropic medium.