Theory and Techniques of Data Assimilation
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.