Brillouin amplification of an OTDR signal due to Rayleigh scattering in an optical fiber is demonstrated using two different configurations. In the first technique, which we call internal Brillouin amplified Rayleigh scattering (iBARS) the Rayleigh scattering and amplification occurs simultaneously in the same fiber. In the second technique, which we call external Brillouin amplified Rayleigh scattering (eBARS) the amplification takes place in a second fiber. The differences between the two techniques are delineated. Using the second technique, we demonstrate single-sideband off-resonant Brillouin amplification of the Rayleigh signal. This technique is shown to enhance the SNR of a signal that is due to vibration-induced strain on the fiber. Using eBARS, we demonstrate a sensitivity of 50 nanostrain.