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Systematic errors induced by distortions in the pump pulse of conventional Brillouin distributed fiber sensors are thoroughly investigated. Experimental results, supported by a theoretical analysis, demonstrate that the two probe sidebands in standard Brillouin optical time-domain analyzers provide a non-zero net gain on the pump pulse, inducing severe distortions of the pump when scanning the pump-probe frequency offset, especially at high probe power levels. Compared to the impact of non-local effects reported in the state-of-the-art, measurements here indicate that for probe powers in the mW range (below the onset of amplified spontaneous Brillouin scattering), the obtained gain and loss spectra show two strong side-lobes that lead to significant strain/temperature errors. This phenomenon is not related to the well-known spectral hole burning resulting from pump depletion, but it is strictly related to the temporal and spectral distortions that the pump pulse experiences when scanning the Brillouin gain/loss spectrum. As a solution to this problem, a novel scanning scheme for Brillouin sensing is proposed. The method relies on a fixed frequency separation between the two probe sidebands, so that a flat zero net gain is achieved on the pump pulse when scanning the pump-probe frequency offset. The proposed technique is experimentally validated, demonstrating its ability to completely cancel out non-local effects up to a probe power ultimately limited by the onset of amplified spontaneous Brillouin scattering. The method allows for one order of magnitude improvement in the figure-of-merit of optimized long-range Brillouin distributed fiber sensors, enabling measurements along a 100 km-long sensing fiber with 2 m spatial resolution and with no need of added features for performance enhancement.