Kalman - Data assimilation with external model I'm working with a distributed snow model which simulates Snow Depth and Snow Cover area. It takes in input some atmospherics variables : Temperature / Precipitation / Wind speed ... 
I want to assimilate the output : Snow Depth retrieving remote sensing data ( I have snow cover indices for each 10 days ) and by using Kalman Filter. But some ambiguities that prevent me to do it : 


I implemented equation (2a) and (2b), but I have no idea how to implement the model operator M quoted in equation (3)... If you have any idea or document that describes carefully this operator please provide it to me.
$M$ is the forward model you are using. It is the dynamical model used to solve whatever discrete equation evolves the field ($d?/dt=$). 

The best way to start with data assimilation and understand its nomenclature is to read Ide et al. (1997). In their work, they explain it as: 


  A discrete model for the evolution of an atmospheric, oceanic, or coupled system from time $t_i$ to time $t_{i+1}$ is governed by an equation 
  
  $x^f(t_{i+1})=M_i[x^f(t_{i})]$,          (1)
  
  where $x$ and $M$ are the model's state vector and its corresponding dynamic operator, respectively. The state vector $x$ has dimension $n$. The dynamics $M$ of the model evolution (1) in a computer simulation or prediction is commonly nonlinear and deterministic, while the true geofluid might differ from (1) by random or systematic errors. 





Ide, K., Courtier, P., Ghil, M., Lorenc, A.C., 1997. Unified notation
for data assimilation: operational, sequential and variational. J.
Met. Soc. Jpn. 75, 181– 189.