  Continuous data assimilation (CDA) is a method that continuously integrates
observational data into a dynamical system to improve model accuracy in
real-time. The AOT algorithm is one of the most widely used methods in CDA due
to its efficiency in incorporating observational data to enhance model
accuracy. However, no research to date has evaluated the performance of the AOT
algorithm compared to the most widely used DA method, the ensemble Kalman
filter (EnKF). Hence, in this paper, we conduct an extensive numerical
examination to evaluate and compare these two algorithms for CDA problems with
measurement error, addressing this gap. By analyzing the one-dimensional
Kuramoto-Sivashinsky equation and the two-dimensional Navier-Stokes equation,
which are central to many applications and representative in CDA problems, we
found a significant computational advantage of the AOT algorithm.
