Kalman - Data assimilation with external model
$M$ 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.