  This note examines the influence of covariance inflation on the distance
between the measured observation and the simulated (or predicted) observation
with respect to the state estimate. In order for the aforementioned distance to
be bounded in a certain interval, some sufficient conditions are derived,
indicating that the covariance inflation factor should be bounded in a certain
interval, and that the inflation bounds are related to the maximum and minimum
eigenvalues of certain matrices. Implications of these analytic results are
discussed, and a numerical experiment is presented to verify the validity of
our analysis.
