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 $\,\!\mathbf {\Psi } _{a}$ or forecast $\,\!\mathbf {\Psi } _{f}$ can be efficiently computed using the adjoint model which yields information about the gradients of $\,\!\jmath (\mathbf {\Psi } _{a})$ and $\,\!\jmath (\mathbf {\Psi } _{f})$. We can extent the concept of the adjoint sensitivity to compute the sensitivity of the IS4DVAR cost function, $\,\!J$,

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

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

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:

$\mathbf {\Psi } _{a}=\mathbf {\Psi } _{b}-\mathbf {Q} _{k}\mathbf {T} _{k}^{-1}\mathbf {Q} _{k}^{T}\mathbf {G} ^{T}\mathbf {O} ^{-1}\mathbf {d}$

where $\,\!\mathbf {Q} _{k}=(\mathbf {q} _{i})$ is the matrix of k orthogonal Lanczos vectors, and $\,\!\mathbf {T} _{k}$ is a known tridiagonal matrix. Each of the k-iterations of IS4DVAR employed in finding the minimum of $\,\!J$ yields one column $\,\!\mathbf {q} _{i}$ of $\,\!\mathbf {Q} _{k}$. We can identify the Kalman gain matrix as $\mathbf {K} =-\mathbf {Q} _{k}\mathbf {T} _{k}^{-1}\mathbf {Q} _{k}^{T}\mathbf {G} ^{T}\mathbf {O} ^{-1}$ in which case $\mathbf {K} ^{T}=-\mathbf {O} ^{-1}\mathbf {G} \mathbf {Q} _{k}\mathbf {T} _{k}^{-1}\mathbf {Q} _{k}^{T}$ represents the adjoint of the entire IS4DVAR system. The $\,\!\mathbf {K} ^{T}$ on the vector $\,\!\mathbf {G} ^{T}\mathbf {O} ^{-1}d$ above can be readily computed since the Lanczos vectors $\,\!\mathbf {q} _{i}$ and the matrix $\,\!\mathbf {T} _{k}$ are available at the end of each IS4DVAR assimilation cycle.

Therefore, the basic observation sensitivity algorithm is as follows:

1. Force the adjoint model with $\,\!\mathbf {O} ^{-1}\mathbf {d}$ to yield $\,\!\psi ^{\dagger }=\mathbf {G} ^{\mathbf {T} }\mathbf {O} ^{-1}\mathbf {d} \propto \mathbf {q} _{i}$.
2. Operate $\,\!\psi ^{\dagger }(t_{0})$ with $\,\!\mathbf {Q} _{k}\mathbf {T} _{k}^{-1}\mathbf {Q} _{k}^{T}$ which is equivalent to a rank-k approximation of the Hessian matrix.
3. Integrate the results of (2) forward in time using the tangent linear model and save the solution $\,\!\mathbf {H} ^{'}\psi (t)$, that is, solution at observation locations.
4. Multiply $\,\!\mathbf {H} ^{'}\psi (t)$ by $\,\!-\mathbf {O} ^{-1}$ to yield $\,\!{\partial J}/{\partial \mathbf {y} }$.

Consider now a forecast $\,\!\mathbf {\Psi } _{f}(t)$ fort the interval $\,\!t=[t_{0}+\tau ,\;t_{0}+\tau +t_{f}]$ initialized from $\,\!\mathbf {\Psi } _{a}(t_{0}+\tau )$ obtained from an assimilation cycle over the interval $\,\!t=[t_{0}+\tau ,\;t_{0}+\tau ]$, where $\,\!t_{f}$ is the forecast lead time. In addition, consider $\,\!\jmath (\mathbf {\Psi } _{f}(t))$ that depends on the forecast $\,\!\mathbf {\Psi } _{f}(t)$, and which characterizes some future aspect of the forecast circulation. According to the chain rule, the sensitivity $\,\!{\partial \jmath }/{\partial y}$ of $\,\!\jmath$ to the observations $\,\!\mathbf {y}$ collected during the assimilation cycle $\,\!t=[t_{0}+\tau ,\;t_{0}+\tau ]$ is given by:

${\frac {\partial \jmath }{\partial \mathbf {y} }}={\frac {\partial \mathbf {\Psi } _{a}}{\partial \mathbf {y} }}{\frac {\partial \mathbf {\Psi } _{f}}{\partial \mathbf {\Psi } _{a}}}{\frac {\partial \jmath }{\partial \mathbf {\Psi } _{f}}}=-\mathbf {O} ^{-1}\mathbf {H} \mathbf {R} \mathbf {Q} _{k}\mathbf {T} _{k}^{-1}\mathbf {Q} _{k}^{T}\mathbf {R} ^{T}{\frac {\partial \jmath }{\partial \mathbf {\Psi } _{f}}}$

which again can be readily evaluated using the adjoint model denoted by $\,\!\mathbf {R} ^{T}(t,\;t_{0}+\tau )$ and the adjoint of IS4DVAR, $\,\!\mathbf {K} ^{T}$.