  Recently \cite{Ramos2017a} presented a subspace system identification
algorithm for 2-D purely stochastic state space models in the general Roesser
form. However, since the exact problem requires an oblique projection of
$Y_f^h$ projected onto $W_p^h$ along $\widehat{X}_f^{vh}$, where $W_p^h=
\begin{bmatrix}\widehat{X}_p^{vh} \\ Y_p^h \end{bmatrix}$, this presents a
problem since $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$ are unknown. In the
above mentioned paper, the authors found that by doing an orthogonal projection
$Y_f^h/Y_p^h$, one can identify the future horizontal state matrix
$\widehat{X}_f^{h}$ with a small bias due to the initial conditions that depend
on $\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. Nevertheless, the results on
modeling 2-D images were very good despite lack of knowledge of
$\{\widehat{X}_p^{vh},\widehat{X}_f^{vh}\}$. In this note we delve into the
bias term and prove that it is insignificant, provided $i$ is chosen large
enough and the vertical and horizontal states are uncorrelated. That is, the
cross covariance of the state estimates $x_{r,s}^{h}$ and $x_{r,s}^{v}$ is
zero, or $P_{hv}=0_{n_x\times n_x}$ and $P_{vh}=0_{n_x\times n_x}$. Our
simulations use $i=30$. We also present a second iteration to improve the state
estimates by including the vertical states computed from a vertical data
processing step, i.e., by doing an orthogonal projection $Y_f^v/Y_p^v$. In this
revised algorithm we include a step to compute the initial states. This new
portion, in addition to the algorithm presented in \cite{Ramos2017a}, forms a
complete 2-D stochastic subspace system identification algorithm.
