A 4.3 dB stimulated Brillouin scattering (SBS) threshold suppression is measured in a passive Al-doped acoustically anti-guiding single mode optical fiber relative to that of a Ge-doped acoustically guiding single mode optical fiber. Stimulated scattering is generated by the electrostrictive acoustic wave generated in the fiber core. This acoustic excitation has a decay length L(d) related to the sound absorption decay length L(abs) and the acoustic waveguide decay length L(wg) by: L(d)(-1)= L(abs)(-1)+ L(wg)(-1). The acoustic waveguide decay length L(wg) is associated with the diffraction, refraction and reflection of the acoustic wave in the elastically inhomogeneous optical fiber cores. The SBS gain is proportional to the net acoustic decay length L(d) and the relative SBS suppression is proportional to the ratio of the net decay lengths of the Al and Ge doped cores (L(Al)/ L(Ge)). An acoustic beam propagation model is used to calculate the evolution of the complex acoustic excitations in the optical cores and determine the acoustic wave decay lengths L(wg). Model predictions for the relative SBS suppression for the two fibers are in good agreement with experimental values obtained from Stokes power and optical heterodyne linewidth measurements.