  This paper is devoted to the problem of time parallelization of assimilation
methods applying on unbounded time domain. In this way, we present a general
procedure to couple the Luenberger observer with time parallelization
algorithm. Our approach is based on a posteriori error estimates of the latter
and preserves the rate of the non-parallelized observer. We then focus on the
case where the Parareal algorithm is used as time parallelization algorithm,
and derive a bound of the efficiency of our procedure. A variant devoted to the
case a large number of processors is also proposed. We illustrate the
performance of our approach with numerical experiments.
