dispersion function of early-galaxies are not varying with redshift. The population of strong lenses is dominated by galaxies with velocity dispersion of $\sigma_{ap}=210\pm50$ km/s, while the lens redshift distribution is well approximated by a Gaussian with mean 0.40. Although discovering strong lenses in these surveys will require the development of new methods and algorithms [@Gavazzi14], which is beyond the scope of this paper, we are confident that the simulated population of lenses is a good representation of what the future LSST survey might yield [@Sonnenfeld13; @Gavazzi14].

The scatter plot of the simulated lensing systems is shown in Fig. 1, from which one can see the LSST lenses result in a fair coverage of lenses and sources redshifts.

![The solid red curve in plot indicates the reconstructed $\mathcal{D}_{\rm obs}$ function using Gaussian processes, for the lens redshift ranges $0.30\sim0.32$. The light blue represents the $1\sigma$ confidence region.](fig3.eps){width="0.9\linewidth"}

Gaussian Processes
==================

In order to reconstruct the evolution of angular diameter distances from simulated SGL data sets, we should find a model-independent method to reconstruct $D=D_{ls}/D_s$. Although there are several methods such as principle component analysis [@Huterer03] and Gaussian smoothing [@Shafieloo06], in this paper we will reconstruct $D_{ls}/D_s$ more precisely by using the 