This question is related to MA processes
Suppose $(X_t) \sim MA(1)$: $$X_t = \epsilon_t + \alpha \epsilon_{t-1}, \quad \epsilon_t \overset{iid}{\sim}N(0,\sigma_1).$$ Now let $\eta_t \overset{iid}{\sim}N(0,\sigma_2)$. Define $Z_t := X_t + \eta_t$ \begin{equation}\label{I}\tag{I} Z_t = X_t + \eta_t = \epsilon_t + \alpha \epsilon_{t-1} + \eta_t \end{equation}
It is well known that $Z_t \sim MA(1)$, i.e.: \begin{equation}\label{II}\tag{II} Z_t = \nu_t + \theta\nu_{t-1} \end{equation} Is it possible to determine $\nu_t$ and $\theta$ in (\ref{II}) from (\ref{I})?
If not, what can we learn about $\nu_t$ and $\theta$?
In (I) we have:
$$ \text{Var}(Z_t)=(1+\alpha^2)\sigma_1 + \sigma_2 \\ \gamma(1) = \alpha \, \sigma_{1} \,, \gamma(k) =0 \quad \forall k\ge2 $$
In (II) we have, with $\nu_t \sim \mathcal{N}(o,\sigma_3)$:
$$ \text{Var}(Z_t)=(1+\theta^2)\sigma_3\\ \gamma(1) = \theta \, \sigma_{3} \,, \gamma(k) =0 \quad \forall k\ge2 $$
From the ACF we see that the processes are indeed $\text{MA}(1)$ and equating both the variances and ACF we get a system of equations in the unknowns $\theta$ and $\sigma_3$ which we can solve to get the parameters in terms of the old ones.
$$ \begin{cases} (1+\theta^2)\sigma_3 = (1+\alpha^2)\sigma_1 + \sigma_2 \\ \theta \, \sigma_{3} = \alpha \, \sigma_{1} \end{cases} $$