Course content

The course provides an advanced theoretical and numerical framework for solving geophysical inverse problems, where physical models are adjusted to fit geophysical observations.It covers the mathematical and physical foundations of inversion, regularization, and uncertainty quantification, with applications across seismic, electromagnetic, gravity, and magnetic methods.

Topics include:

• Modeling of electromagnetic and seismic wave propagation in 1D-3D isotropic and anisotropic media.
• The definition and classification of inverse problems after Hadamard.
• Deterministic and statistical (Bayesian) inversion theory.
• Linear and nonlinear least-squares methods and iterative solution strategies.
• Regularization methods and model constraints (smoothness, structural, petrophysical).
• Joint inversion of multi-physics data and cross-gradient coupling.
• Integration of machine learning in geophysical inversion and model parameter estimation.
• Evaluation of model uncertainty, resolution, and non-uniqueness.

The course combines theoretical analysis with numerical implementation and critical discussion of industrial and research applications.

Learning outcome

Competence

After completing the course, the candidate will be able to:

• Formulate and evaluate geophysical inverse problems and justify the choice of deterministic or statistical inversion approaches.
• Critically assess the mathematical well-posedness of an inverse problem and apply appropriate regularization or constraint strategies.
• Integrate physical modeling, numerical algorithms, and data constraints into inversion workflows for seismic, electromagnetic, gravity, and magnetic data.
• Contribute independently to the development of new inversion methodologies and their application in geophysical research or industry.
• Communicate complex inversion concepts and results clearly to both expert and interdisciplinary audiences.

Knowledge and skills

After completing the course, the candidate will be able to:

• Explain the mathematical foundations of inversion theory, including functional analysis, least-squares estimation, and optimization.
• Define and classify inverse problems according to Hadamard’s criteria for existence, uniqueness, and stability.
• Derive and implement solution methods for linear and nonlinear inversion, including iterative and regularized approaches.
• Apply deterministic and Bayesian estimation methods for parameter inference and uncertainty quantification.
• Describe the main inversion strategies for seismic and electromagnetic depth imaging (Kirchhoff, wave-equation migration, full-waveform inversion).
• Evaluate the principles of joint inversion across geophysical datasets with structural or petrophysical constraints.
• Apply and critically assess the use of machine learning for model parameterization and inversion acceleration.
• Develop and document theoretical or numerical inversion studies as part of independent PhD research.
• Discuss the limitations, uncertainty, and interpretational challenges of geophysical inversion in scientific and industrial contexts.

Learning methods and activities

The course is taught by arrangement with the lecturer. The course is given if at least 3 students attend. Lecture, colloquia and self-study.

Specific conditions

Required previous knowledge

Requires admission to the PhD programme in Engineering, specialization within geophysics, or approval from the person with course responsibility. The course requires good mathematical skills.

Course materials

Selected papers from journals.