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5.1.1 Forward or direct sensitivity

Consider a perturbation to the input variables $ \delta \vec{u}$ (typically a single component $ \delta \vec{u} = \delta u_{i} \, {\vec{e}_{i}}$). Their effect on the output may be obtained via the linear approximation of the model $ {\cal M}$ in terms of its Jacobian matrix $ M $, evaluated in the point $ u^{(0)}$ according to

$\displaystyle \delta \vec{v} \, = \, M \vert _{\vec{u}^{(0)}} \, \delta \vec{u}$ (5.2)

with resulting output perturbation $ \delta \vec{v}$. In components $ M_{j i} \, = \, \partial {\cal M}_{j} / \partial u_{i} $, it reads

$\displaystyle \delta v_{j} \, = \, \sum_{i} \left. \frac{\partial {\cal M}_{j}}{\partial u_{i}} \right\vert _{u^{(0)}} \, \delta u_{i}$ (5.3)

Eq. (5.2) is the tangent linear model (TLM). In contrast to the full nonlinear model $ {\cal M}$, the operator $ M $ is just a matrix which can readily be used to find the forward sensitivity of $ \vec{v}$ to perturbations in $ u$, but if there are very many input variables $ (\gg O(10^{6})$ for large-scale oceanographic application), it quickly becomes prohibitive to proceed directly as in (5.2), if the impact of each component $ {\bf e_{i}} $ is to be assessed.

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Next: 5.1.2 Reverse or adjoint Up: 5.1 Some basic algebra Previous: 5.1 Some basic algebra   Contents
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