Adjoint Models

Christian Melsheimer
IUP, University of Bremen

4th International Radiative Transfer Modeling Workshop, Bredbeck, 9 July, 2002


Let < u, v > be some scalar product, L some linear operator1.

Then the adjoint operator L * defined by

< u, L * v > = < Lu, v >

If L is a real matrix, then L * = LT (transpose); if L is a complex matrix, then L * = $ \bar{{L}}^{T}_{}$ (complex conjugate, transpose).

Basic Idea

Forward Model:

$\displaystyle \vec{{F}} $($\displaystyle \vec{{x}} $) = $\displaystyle \vec{{y}} $

$\displaystyle \Big($     $\displaystyle \Leftrightarrow$ Fi(x1,...xn) = yi    $\displaystyle \Big)$

$ \vec{{F}} $
is a complicated, non-linear, nasty function (``the Model'');
$ \vec{{x}} $
is the model input (often the retrieval variable);
$ \vec{{y}} $
is the model output (simulated measurement).

Sensitivity Analysis:

$\displaystyle \delta$x $\displaystyle \Rightarrow$ $\displaystyle \delta$y = ?

where $ \delta$x is a small disturbance;

or, the other way round:

$\displaystyle \delta$y $\displaystyle \Rightarrow$ $\displaystyle \delta$x = ?

where $ \delta$y is a small disturbance;

or even:

$\displaystyle \nabla_{{\!y}}^{}$J $\displaystyle \Rightarrow$ $\displaystyle \nabla_{{\!x}}^{}$J = ?

where J($ \vec{{y}} $($ \vec{{x}} $)) is, e.g., a cost function, a measure for some error of $ \vec{{y}} $($ \vec{{x}} $) with respect to the measurement $ \vec{{y}}_{m}^{}$. $ \nabla_{{\!y}}^{}$ and $ \nabla_{{\!y}}^{}$ are the gradients with respect to $ \vec{{x}} $ and $ \vec{{y}} $, respectively.

Tangent-Linear Model:

$\displaystyle \delta$yi = $\displaystyle \sum_{k}^{}$$\displaystyle {\frac{{\partial F_i}}{{\partial x_k}}}$$\displaystyle \delta$xk

$\displaystyle \;\stackrel{{\frac{\partial F_i}{\partial x_k} \equiv K_{ik}}}{{\Longleftrightarrow}}\;$$\displaystyle \delta$$\displaystyle \vec{{y}} $ = K . $\displaystyle \delta$$\displaystyle \vec{{x}} $

where K is the Jacobian of $ \vec{{F}} $!

Here we have a linear mapping from $ \delta$y to $ \delta$x that involves the first derivative (something like a tangent) of the forward model, hence the name ``tangent-linear''.

Adjoint (Tangent-Linear) Model:

$\displaystyle {\frac{{\partial J(\vec{y}(\vec{x}))}}{{\partial x_i}}}$$\displaystyle \;\stackrel{{\vec{y}=\vec{F}(\vec{x})}}{{=}}\;$$\displaystyle \sum_{k}^{}$$\displaystyle {\frac{{\partial F_i}}{{\partial x_k}}}$$\displaystyle {\frac{{\partial J}}{{\partial y_i}}}$         (chain rule)

$\displaystyle \equiv$ $\displaystyle \nabla_{{\!x}}^{}$J($\displaystyle \vec{{y}} $) = KT$\displaystyle \nabla_{{\!y}}^{}$J($\displaystyle \vec{{y}} $)

where KT is the adjoint of K, i.e. the adjoint of the tangent linear model, hence the name. It is usually just called ``adjoint model'', although the correct name is ``adjoint tangent-linear model''.


$\displaystyle \vec{{F}} $($\displaystyle \vec{{x}} $) = $\displaystyle \vec{{F}}^{{(N)}}_{}$(...($\displaystyle \vec{{F}}^{{(1)}}_{}$($\displaystyle \vec{{F}}^{{(0)}}_{}$($\displaystyle \vec{{x}} $))..)    (a sequence of operations)


K = K(N) ... K(1)K(0)


KT = K(0)TK(1)T ... K(N)T

So What?



Sources of Information, References

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Adjoint Models

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... operator1
The usual conditions apply, such as u $ \in$ vector space U and v $ \in$ vector space V etc.

Christian Melsheimer