  The most accurate version of the unscented Kalman filter (UKF) involves the
construction of two ensembles. To reduce computational cost, however, UKF is
often implemented without the second ensemble. This simplification comes at a
price, however, since, for linear systems, the one-step variation of the
two-step UKF does not specialize to the classical Kalman filter, with an
associated loss of accuracy. This paper remedies this drawback by developing a
modified one-step UKF that recovers the classical Kalman filter for linear
systems. Numerical examples show that the modified one-step UKF also recovers
the accuracy of the two-step UKF in nonlinear systems with linear outputs.
