Examples: Determine sources and sinks of carbon dioxide by means of inverse modelling of the atmospheric transport

Numerical simulation models of  atmospheric transport can be used to interpret spatio-temporal differences in the observed concentrations of an atmospheric constituent such as carbon dioxide in terms of its sources and sinks. Inverse modelling of the atmospheric transport is the systematic search for both a source(/sink) field that yields an optimal match between modelled and observed concentrations and, equally importantly,  the uncertainties in this inferred source field. Fortunately, for a time span of a few years, feedbacks of a change in the source field onto a change in the atmospheric circulation can be neglected. Furthermore, for a constituent such as carbon dioxide, the spatio-temporal differences in the observed concentration are linear in the source field. Hence, all necessary information about atmospheric transport can be condensed into a Jacobian matrix of the transport model, which maps changes in the source field onto changes into the simulated concentration differences.

Typically, the atmospheric observations are collected at a sparse observational network, but the source field is unknown at each point in space and time. Due to this imbalanced ratio of dimensions, the inverse problem is underdetermined, i.e. there are many source configurations yielding the same modelled concentration at the observational sites. But this is the nature of the problem, the challenge is to pick those configurations that at the same time achieve consistency with the observations and reliable additional (prior) information about the source fields.

On the other hand, this imbalance between the large number of unknowns and the small number of observations allows a computational trick: A Jacobian matrix representing a model with few output variables but many input variables can be most efficiently computed by means of the (vector valued) adjoint of the model, i.e. using the reverse mode of automatic differentiation. During his work at the Max-Planck-Institute in Hamburg, Thomas was the first to exploit this opportunity and to compute the Jacobian (see  Kaminski et al. 1999, see an animation) for the full source resolution of the transport model TM2 (Heimann, 1995). Of course, to generate the adjoint of the transport model, he used the Tangent linear and Adjoint Model Compiler (TAMC) and profited enormously from Ralf's experience in automatic differentiation. Since then the Jacobian has been the key ingredient of several inversion studies (see  references).

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