  I present the mean metallicity distribution of stars in the Milky Way Galaxy
based on photometry from the Sloan Digital Sky Survey. I utilize an empirically
calibrated set of stellar isochrones developed in previous work to estimate the
metallicities of individual stars to a precision of $0.2$ dex for reasonably
bright stars across the survey area. I also obtain more precise metallicity
estimates using priors from the $Gaia$ parallaxes for relatively nearby stars.
Close to the Galactic mid-plane ($|Z|<2$ kpc), a mean metallicity map reveals
deviations from the mirror symmetry between the northern and southern
hemispheres, displaying wave-like oscillations. The observed metallicity
asymmetry structure is almost parallel to the Galactic mid-plane, and coincides
with the previously known asymmetry in the stellar number density distribution.
This result reinforces the previous notion of the plane-parallel vertical waves
propagating through the disk, in which a local metallicity perturbation from
the mean vertical metallicity gradient is induced by the phase-space wrapping
of stars in the $Z$-$V_Z$ plane. The maximum amplitude of the metallicity
asymmetry ($\Delta$[Fe/H]$\sim0.05$) implies that these stars have been pulled
away from the Galactic mid-plane by an order of $\Delta|Z|\sim80$ pc as a
massive halo substructure such as the Sagittarius dwarf galaxy plunged through
the Milky Way. This work provides evidence that the $Gaia$ phase-space spiral
may continue out to $|Z|\sim1.5$ kpc.
