average proper motions can be used to estimate the expected positions of *Gaia* stars at the epoch of each SDSS scan. $$\hat\alpha_{\rm SDSS} = \alpha_{Gaia} - \langle \mu_{\alpha}(\mathrm{hpx},g-i,i)  \rangle \delta T$$ where $\delta T$ is the timespan between *Gaia* and SDSS observation of a given star, hpx is the HEALPix pixel number of the star and $g-i$, and $i$ are colors and magnitudes of the star. With those positions $(\hat{\alpha}_{\rm SDSS}, \hat{\delta}_{\rm SDSS})$ computed for all the stars with both SDSS and *Gaia* measurements we redetermine the astrometric mapping in SDSS between $(x',y')$ pixel coordinates and on the sky great circle $(\mu,\nu)$ coordinates by using a flexible spline model. There are many more stars available in *Gaia* DR1 compared to the UCAC catalog, so in the model we are able to much better describe the anomalous refraction along the SDSS scans and, therefore, noticeably reduce the systematic uncertainties of the astrometric calibration. Furthermore, as a final step of the calibration, we also utilise the galaxies observed by Gaia and SDSS to remove any residual large scale astrometric offsets in the calibrated SDSS astrometry. With the SDSS astrometry recalibrated, the SDSS-[*Gaia*]{} proper motions are then simply obtained from the *Gaia* positions 