Frequency dependence in phonon surface scattering is a debated topic in fundamental phonon physics. Recent experiments and theory suggest such a phenomenon, but an independent agreement between the two remains elusive. We report low-temperature dependence of thermal conductivity in silicon nanowires fabricated using a two-step, metal-assisted chemical etch. By reducing etch rates down to 0.5 nm/s from the typical >100 nm/s, we report controllable roughening of nanowire surfaces and selectively focus on moderate roughness scales rather than the extreme scales investigated previously. This critically enables direct comparison with perturbation-based spectral scattering theory. Using experimentally characterized surface roughness, we show that a multiple scattering theory provides excellent agreement and explanation of the observed low-temperature dependence of rough surface nanowires. The theory does not employ any fitting parameters. A 5-10 nm roughness correlation length is typical in metal-assisted chemical etching and resonantly scatters dominant phonons in silicon, leading to the observed ~T(1.6-2.4) behavior. Our work provides fundamental and quantitative insight into spectral phonon scattering from rough surfaces. This advances applications of nanowires in thermoelectric energy conversion.