  The Ensemble Kalman filter assumes the observations to be Gaussian random
variables with a pre-specified mean and variance. In practice, observations may
also have detection limits, for instance when a gauge has a minimum or maximum
value. In such cases most data assimilation schemes discard out-of-range
values, treating them as "not a number", at a loss of possibly useful
qualitative information.
  The current work focuses on the development of a data assimilation scheme
that tackles observations with a detection limit. We present the Ensemble
Kalman Filter Semi-Qualitative (EnKF-SQ) and test its performance against the
Partial Deterministic Ensemble Kalman Filter (PDEnKF) of Borup et al. (2015).
Both are designed to explicitly assimilate the out-of-range observations: the
out-of-range values are qualitative by nature (inequalities), but one can
postulate a probability distribution for them and then update the ensemble
members accordingly. The EnKF-SQ is tested within the framework of twin
experiments, using both linear and non-linear toy models. Different sensitivity
experiments are conducted to assess the influence of the ensemble size,
observation detection limit and a number of observations on the performance of
the filter. Our numerical results show that assimilating qualitative
observations using the proposed scheme improves the overall forecast mean,
making it viable for testing on more realistic applications such as sea-ice
models.
