Christian Melsheimer
IUP, University of Bremen

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

# Prologue

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 * = (complex conjugate, transpose).

# Basic Idea

## Forward Model: ( ) =   Fi(x1,...xn) = yi where is a complicated, non-linear, nasty function (the Model''); is the model input (often the retrieval variable); is the model output (simulated measurement).

## Sensitivity Analysis: x  y = ?

where x is a small disturbance;

or, the other way round: y  x = ?

where y is a small disturbance;

or even: J  J = ?

where J( ( )) is, e.g., a cost function, a measure for some error of ( ) with respect to the measurement . and are the gradients with respect to and , respectively.

## Tangent-Linear Model: yi =   xk   = K .  where K is the Jacobian of !

Here we have a linear mapping from y to x that involves the first derivative (something like a tangent) of the forward model, hence the name tangent-linear''.     (chain rule)  J( ) = KT J( )

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''.

If: ( ) = (...( ( ( ))..)    (a sequence of operations)

then:

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

and:

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

# So What?

• Adjoint models are widely used in meteorology, particularly for assimilation schemes
• ECMWF, MétéoFrance and others have adopted official coding conventions for adjoint models
• There are adjoint model compilers (at least in FORTRAN77) to automatically generate the code for the adjoint model from the code for the forward model, e.g. TAMC (Tangent Linear and Adjoint Model Compiler) by Ralf Giering (see below)
BUT:
• How to apply adjoint models to the actual retrieval in atmospheric sounding (OEM, Levenberg-Marquardt etc.)?
• Would it really make things easier (we still need the Jacobian ...)?

[...]

# Sources of Information, References

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