I4DVAR ANA SENSITIVITY

Technical Description

The mathematical development presented here closely parallels that of Zhu and Gelaro (2008). The sensitivity of any functional, $\,\!\jmath$, of the analysis $\,\!\psi _{a}$ or forecast $\,\!\psi _{f}$ can be efficiently computed using the adjoint model which yields information about the gradients of $\,\!\jmath (\psi _{a})$ and $\,\!\jmath (\psi _{f})$. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the IS4DVAR cost function, $\,\!J$,

$J(\psi )={\frac {1}{2}}\,\psi ^{T}B^{-1}\psi +{\frac {1}{2}}\,(G\psi -d)^{T}O^{-1}(G\psi -d)$

and any other function $\,\!\jmath$ of the forecast $\,\!\psi _{f}$ to the observations, $\,\!y$. Here, $\,\!\psi$ is the ocean state vector, $\,\!O$ is the observation error and error of representativeness matrix, $\,\!B$ represents the background error covariance, $\,\!d$ is the innovation vector that represents the difference between the nonlinear background solution and the observations, $\,\!d_{i}=y_{i}-H_{i}(\psi _{b}(t))$, $\,\!H_{i}$ is an operator that samples the nonlinear model at the observation location, and $\,\!G=H_{i}^{'}R(t_{0},t_{i})$.

In the current IS4DVAR/LANCZOS data assimilation algorithm, the above cost function is identified using the Lanczos method (Golub and Van Loan, 1989), in which case:

$\psi _{a}=\psi _{b}-Q_{k}T_{k}^{-1}Q_{k}^{T}G^{T}O^{-1}d$

where $\,\!Q_{k}=(q_{i})$ is the matrix of k orthogonal Lanczos vectors, and $\,\!T_{k}$ is a known tridiagonal matrix. Each of the k-iterations of IS4DVAR employed in finding the minimum of $\,\!J$ yields one column $\,\!q_{i}$ of $\,\!Q_{k}$.