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Compulsory: Inverse Modelling (IM)
Identification number


180 h
Term of studying
1st or 2nd semester
Frequency of

Summer term

1 semester
1 Type of lessons
a) Lectures
b) Tutorials
Contact times
30 h
30 h
Self-study times
60 h
60 h
Intended group size

Aims of the module and acquired skills

Understanding inverse modelling methods for the determination of meteorological and geophysical parameters from measurements, gaining knowledge in major spatial-temporal data assimilation methods.

Acquired skills are the mathematical foundation of linear and non-linear inverse problems, formulation of inverse problems, assessment of statistical prerequisites and numerical complexity, assessment of inverse solutions, practical limitation of current assimilation methods, critical judgment of model simulations and capacity of model development.


Contents of the module

  • Basics: Inverse problems in geophysics and data assimilation in meteorology, overview of methods and definitions
  • Deterministic approaches: linear problems, general formulation, least-squares method, normal equations, Jacobian matrix, generalised matrix inverse, adjoint and tangent-linear models, SVD decomposition, data and model gain matrices, data and model covariance matrices (data error and model assessment), nonlinear problems, Jacobian matrix, iterative conjugate gradient and Gauss- Newton methods, regularisation (Occam, Levenberg-Marquardt)
  • Stochastic approaches, general formulation, Bayes theorem, optimal estimation, information content, error assessment
  • data assimilation, optimum interpolation, 3d-var, Kalman filtering and 4d-var
  • Applications: geoelectric and electromagnetic methods, gravity, magnetics, remote sensing of the atmosphere (humidity and temperature), weather forecasting
4 Teaching/Learning methods

Lectures and tutorials (compulsory attendance in the tutorials)
5 Requirements for participation

Formal: None.

Regarding content: Basics of mathematics and physics

Type of module examinations

Written Examination (graded)

7 Requisites for the allocation of credits

Successful participation in the tutorials (50 % of the possible points have to be obtained) and passing a final examination.

At the end of the semester or to the beginning of the following semester a possibility to repeat the examination is offered. A failed examination may be repeated twice. Additional possibilities to repeat an examination exist according to the examination regulations (§ 20 section 1).

Assessments which have been passed are not allowed to be taken again. There is an exception: If at the end of a module which consists of a lecture and tutorial classes, the student takes the assessment at the first available date after having received admission to the module exam, he/she is then allowed to take the assessment again at the next available date for the purpose of improving the grade, even if he/she passed the assessment the first time – in this case, the better of the two grades will count towards the final degree grade (§ 20 section 9).

The module mark is the grade obtained in the assessment. In the case of two passed assessments the module mark is the better grade.
8 Compatibility with other Curricula

9 Significance of the module mark for the overall grade

10 Module coordinator

B. Tezkan and H. Elbern
11 Additional information

Recommended Literature:

Aster, R.C., B. Borchers, C.H. Thurber, Parameter estimation and inverse problems, Elsevier, 2005.
Bennet, A. F., 2005. Inverse Modeling of the Ocean and Atmosphere. Cambridge University Press, ISBN: 9780521021579.
Evensen, G., 2009. Data Assimilation: the Ensemble Kalman Filter. Springer, SBN 978-3-642-03711-5
Kalnay, E., 2003. Atmospheric Modelling, data assimilation and predictability, Cambridge Univ. Press, 342 pp.
Meju, M.A., 1994. Geophysical data analysis: Understanding inverse problems, Theory and practice, Society of Exploration Geophysicists.
Rodgers, C. D., 2000. Inverse methods for atmospheric sounding: Theory and practice. World Scientific, 238 pp.
Menke, W., 2012. Geophysical Data Analysis: Discrete Inverse Theory – 3rd Ed., Elsevier.
Oliver et al., 2008, Inverse Theory for Petroleum Reservoir Characterization and History Matching, Cambridge Univ. Press.
Tarantola, A., 2005. Inverse Problem Theory and Methods for Model Parameter Estimation. SIAM. ISBN 978-0-89871-572-9.