**Christian Melsheimer
IUP, University of Bremen**

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

Then the adjoint operator *L*^{ * } defined by

< *u*, *L*^{ * }*v* > = < *Lu*, *v* >

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

() =

- is a complicated, non-linear, nasty function (``the
Model'');

- is the model input (often the retrieval
variable);

- is the model output (simulated measurement).

or, the other way round:

or even:

= *K*^{ . }

where
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)

If:

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

then:

- 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)

- 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 ...)?

[...]

- Errico, R.M.: What is an adjoint model? Bull. Am. Met. Soc., vol. 78, p. 2577, 1997.
- Rodgers, C.D.: Inverse Methods for Atmospheric Sounding: Theory and Practice, 238 pp., World Scientific, Singapore, 2000. ISBN 981-02-2740-X
- Giering, R.: Automatic Adjoint and Tangent Linear Code
Generation,

http://www.npaci.edu/Research/ESS/seminars/giering.htm - Kornblueh, L., Working group on Model evaluation and data assimilation, Technical Explanations, http://www.mpimet.mpg.de/working_groups/wg3/node3.html

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- ...
operator
^{1} - The usual conditions apply, such as
*u*vector space*U*and*v*vector space*V*etc.