  X-ray photons scattered by the interstellar medium carry information about
dust distribution, dust grain model, scattering cross section, and the distance
of the source; they also take longer time than unscattered photons to reach the
observer. Using a cross-correlation method, we study the light curves of the
X-ray dust scattering halo of Cyg X-1, observed with the \textit{Chandra X-ray
Observatory}. Significant time lags are found between the light curves of the
point source and its halo. This time lag increases with the angular distance
from Cyg X-1, implying a dust concentration at a distance along the line of
sight (LOS) of 2.0 kpc $\times$ (0.876 $\pm$ 0.002) from the Earth. By fitting
the observed light curves of the halo at different radii with simulated light
curves, we obtain a width of $\mathit{\Delta L}=33_{-13}^{+18}$ pc of this dust
concentration. The origin of this dust concentration is still not clearly
known. The advantage of our method is that we need no assumption of scattering
cross section, dust grain model, or dust distribution along the LOS. Combining
the derived dust distribution from the cross-correlation study with the surface
brightness distribution of the halo, we conclude that the two commonly accepted
models of dust grain size distribution need to be modified significantly.
