EnSR is a variant of Kalman filter like EnKF. There are two fundamental problems associated with the use of EnKF. First is that the ensemble size is limited by the computational cost of applying the forecast model to each ensemble member. The second one is that small ensembles have few degrees of freedom available to represent errors and suffer from sampling errors that will further degrade the forecast error covariance representation. Sampling errors lead to loss of accuracy and underestimation of error covariances. This problem can progressively worsen, resulting in filter divergence.
In ensemble square-root filters (EnSR), the analysis step is done deterministically without generating any observation noise realization (see Tippett,et.al,2003; Evensen, 2004). Since no random sample is generated, this extra source of sampling error is eliminated. Therefore, this method is expected to perform better than the ones with perturbed observations for a certain type of applications.