Augmented orthogonal arrays (AOAs) were introduced by Stinson, who showed the equivalence between ideal ramp schemes and augmented orthogonal arrays (Discrete Math. 341 (2018), 299-307). In this paper, we show that there is an AOA$(s,t,k,v)$ if and only if there is an OA$(t,k,v)$ which can be partitioned into $v^{t-s}$ subarrays, each being an OA$(s,k,v)$, and that there is a linear AOA$(s,t,k,q)$ if and only if there is a linear maximum distance separable (MDS) code of length $k$ and dimension $t$ over $\mathbb{F}_q$ which contains a linear MDS subcode of length $k$ and dimension $s$ over $\mathbb{F}_q$. Some constructions for AOAs and some new infinite classes of AOAs are also given.