ODIN: Improved Narrowband Ly$\alpha$ Emitter Selection Techniques for $z$ = 2.4, 3.1, and 4.5

Lyman-Alpha Emitting galaxies (LAEs) are typically young, low-mass, star-forming galaxies with little extinction from interstellar dust. Their low dust attenuation allows their Ly$\alpha$ emission to shine brightly in spectroscopic and photometric observations, providing an observational window into the high-redshift universe. Narrowband surveys reveal large, uniform samples of LAEs at specific redshifts that probe large scale structure and the temporal evolution of galaxy properties. The One-hundred-deg$^2$ DECam Imaging in Narrowbands (ODIN) utilizes three custom-made narrowband filters on the Dark Energy Camera (DECam) to discover LAEs at three equally spaced periods in cosmological history. In this paper, we introduce the hybrid-weighted double-broadband continuum estimation technique, which yields improved estimation of Ly$\alpha$ equivalent widths. Using this method, we discover 6032, 5691, and 4066 LAE candidates at $z =$ 2.4, 3.1, and 4.5 in the extended COSMOS field ($\sim$9 deg$^2$). We find that [O II] emitters are a minimal contaminant in our LAE samples, but that interloping Green Pea-like [O III] emitters are important for our redshift 4.5 sample. We introduce an innovative method for identifying [O II] and [O III] emitters via a combination of narrowband excess and galaxy colors, enabling their study as separate classes of objects. We present scaled median stacked SEDs for each galaxy sample, revealing the overall success of our selection methods. We also calculate rest-frame Ly$\alpha$ equivalent widths for our LAE samples and find that the EW distributions are best fit by exponential functions with scale lengths of $w_0$ = 53 $\pm$ 1, 65 $\pm$ 1, and 59 $\pm$ 1 Angstroms, respectively.