A new technique for Brillouin scattering-based, distributed fiber-optic measurements of temperature and strain is proposed, analyzed, simulated, and demonstrated. Broadband Brillouin pump and signal waves are drawn from the filtered amplified spontaneous emission of an erbium-doped fiber amplifier, providing high spatial resolution. The reconstruction of the position-dependent Brillouin gain spectra along 5 cm of a silica single-mode fiber under test, with a spatial resolution of 4 mm, is experimentally demonstrated using a 25 GHz-wide amplified spontaneous emission source. A 4 mm-long localized hot spot is identified by the measurements. The uncertainty in the reconstruction of the local Brillouin frequency shift is ± 1.5 MHz. The single correlation peak between the pump and signal is scanned along a fiber under test using a mechanical variable delay line. The analysis of the expected spatial resolution and the measurement signal-to-noise ratio is provided. The measurement principle is supported by numerical simulations of the stimulated acoustic field as a function of position and time. Unlike most other Brillouin optical correlation domain analysis configurations, the proposed scheme is not restricted by the bandwidth of available electro-optic modulators, microwave synthesizers, or pattern generators. Resolution is scalable to less than one millimeter in highly nonlinear media.