Many of the measurements available in the atmospheric or oceanographic systems are only available within a limited interval of actual variation of the quantity due to the limitation of gauge or data retrieval techniques i.e, observations with detection limit. For e.g., SMOS retrieved sea-ice thickness. These observations with detection limits contains hard data (quantitative) and soft data (qualitative). The current work focuses on the development and application of the data assimilation scheme for the observations with detection limit.

Travelling waves appear naturally in many various domains in physics such as fluid mechanics,
electromagnetic theory, etc. Intuitively, a travelling wave is a recognizable shape (of energy for
instance) which is transferred from one part of the medium to another part with a constant
speed of propagation. A broad question tackled in
geophysics is whether small errors in initial conditions can bring huge errors when

Hydrological models are used extensively to monitor and manage water resources, and provide flood forecasts. These complex, physically based models are inherently uncertain due to imperfect parameterization, meteorological forcing data, initial conditions, and model discretization. Data assimilation offers a means to incorporate information from measurements to both correct model forecasts and, importantly, provides quantitative uncertainty estimates useful for decision makers.

In recent years, there has been a growing interest in applying data
assimilation (DA) methods, originally designed for state estimation, to
the model selection problem. Along this stream of efforts, a contextual
formulation of model evidence (CME) has recently been introduced in
Carrassi et al. (2017) and it has been shown that it can be efficiently
computed using a hierarchy of ensemble-based DA procedures. While the DA
methods analyzed in Carrassi et al. (2017) cover those operationally
used worldwide for atmosphere or ocean prediction, they were not studied