<p>$M$ is the forward model you are using. It is the dynamical model used to solve whatever discrete equation evolves the field ($d?/dt=$). </p>

<p>The best way to start with data assimilation and understand its nomenclature is to read <a href="http://twister.caps.ou.edu/OBAN2004/IdeEtal_1997_Notation.pdf" rel="nofollow noreferrer">Ide et al. (1997)</a>. In their work, they explain it as: </p>

<blockquote>
  <p>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 </p>
  
  <p>$x^f(t_{i+1})=M_i[x^f(t_{i})]$,          (1)</p>
  
  <p>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. </p>
</blockquote>

<hr>

<ul>
<li>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.</li>
</ul>

CC BY-SA 3.0
2016-11-02T14:18:07.847Z
I have changed the link. Let me know if it works for you
arkaia
CC BY-SA 3.0
2016-08-11T02:04:02.910Z
2016-11-02T14:17:20.303Z
2016-11-02T14:17:20.303Z
arkaia
<p>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 : </p>

<ul>
<li><p>I implemented equation (2a) and (2b), but I have no idea how to implement the model operator <em>M</em> quoted in equation (3)... If you have any idea or document that describes carefully this operator please provide it to me.</p>

<p><a href="https://i.sstatic.net/NdDC8.png" rel="noreferrer"><img src="https://i.sstatic.net/NdDC8.png" alt="enter image description here"></a></p></li>
</ul>

CC BY-SA 3.0
2016-08-09T13:32:31.180Z
$M$ is the forward model you are using. It is the dynamical model that you are using to solve whatever discrete equation evolves your field (d?/dt=).
arkaia
CC BY-SA 3.0
2016-08-10T22:50:27.620Z
@aretxabaleta why not put that down as an answer?
BarocliniCplusplus
<models><climate-models><data-analysis>
Kalman - Data assimilation with external model
Mohamed wassim Baba
# Kalman - Data assimilation with external model

(Asked by: username_0 on 2016-08-09)

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.

  [![enter image description here](https://i.sstatic.net/NdDC8.png)](https://i.sstatic.net/NdDC8.png)

**Tags:** models, climate-models, data-analysis

> **Comments:**
> **username_1**: $M$ is the forward model you are using. It is the dynamical model that you are using to solve whatever discrete equation evolves your field (d?/dt=).  
> **username_2**: @aretxabaleta why not put that down as an answer?  

---

## 1 answers

### Answer by username_1 on 2016-08-11. Score: 8 | Accepted answer

$M$ 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)](http://twister.caps.ou.edu/OBAN2004/IdeEtal_1997_Notation.pdf). 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.

> **Comments:**
> **username_1**: I have changed the link. Let me know if it works for you  

---

