Kalman - Data assimilation with external model

Question

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.

  

Answer 1

$M$ 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.

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• 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.