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A Monte-Carlo Method for Making SDSS $u$-Band Magnitude more accurate

We develop a new Monte-Carlo-based method to convert the SDSS (Sloan Digital Sky Survey) $u$-band magnitude to the SCUSS (South Galactic Cap of $u$-band Sky Survey) $u$-band magnitude. Due to more accuracy of SCUSS $u$-band measurements, the converted $u$-band magnitude becomes more accurate comparing with the original SDSS $u$-band magnitude, in particular at the faint end. The average $u$ (both SDSS and SCUSS) magnitude error of numerous main-sequence stars with $0.2<g-r<0.8$ increase as $g$-band magnitude becomes fainter. When $g=19.5$, the average magnitude error of SDSS $u$ is 0.11. When $g=20.5$, the average SDSS $u$ error is up to 0.22. However, at this magnitude, the average magnitude error of SCUSS $u$ is just half as much as that of SDSS $u$. The SDSS $u$-band magnitudes of main-sequence stars with $0.2<g-r<0.8$ and $18.5<g<20.5$ are converted, therefore the maximum average error of converted $u$-band magnitudes is 0.11. The potential application of this conversion is to derive more accurate photometric metallicity calibration from SDSS observation, especially for those distant stars. Thus, we can explore stellar metallicity distributions either in the Galactic halo or some stream stars.