# Theory and Techniques of Data Assimilation

#### grandy

##### Member
Suppose we have a dynamical system for a vector x = (u,v,p)^T where u,v,p are scalar quantities. Let the dynamical system be represented by the equations
u_(k+1) = u_k +v_k +2p_k
v_(k+1) = 2u_k +v_k +2p_k
p_(k+1) = 3u_k +3v_k + p_k
where k indicates the time index. we wish to apply a four- dimensional data assimilation scheme to determine the vector x_0 at time t_0.
Suppose that we take observations of both u and p together at the two times t_0 and t_1. Determine whether we have enough information to reconstruct the vector x_0 uniquely.

= we can write the dynamical system as
x_(k+1) =F * x_(k)

[1 1 2]
F= [ 2 1 2]
[3 3 1]

Where F is matrix
For observations of u and p together the observation operator is
H = [1 0 0]
[0 0 1]
The observability matrix is
P = [H]
[HF]
HF = [1 1 2]
[3 3 1]
Hence F = [1 0 0]
[0 0 1]
[1 1 2]
[3 3 1]
we can see this is not full rank. hence we do not have sufficient information to reconstruct x_0.

(This is what I have try)