Masterclass: Numerical methods for forward and inverse problems in geophysics

In the last twenty years, many improvements have been made in earth imaging at different scales using different technologies such as active/passive seismics, electromagnetism, potentials (gravity, magnetism, electric potentials),….

The wide variety of data to be inverted to retrieve the earth's properties needs to develop or use different data inversion methods at different scales in time and space. Those methods can be also combined to take advantage of their respective potentialities.

The inversion methods can be based on local/global optimisation approaches (generally gradient like approaches) or stochastic approaches (simulated annealing, genetic algorithms, neighbourhood methods). The advantages and disadvantages will be discussed and some simple/theoretical or realistic examples in electrical capacitance tomography or seismics will be shown.

It is also extremely important to have a good forward problem solver able to approximate the data as accurately as possible. Those techniques can be based on finite differences on different grids at different orders in space (staggered, compact, collocated), finite volumes or finite elements. Some stability and dispersion criteria will be also provided.

Dr Martin will first show, in the case of the wave propagation equation, the different schemes that are commonly used, their advantages and drawbacks.

The boundary conditions used in the direct problem are also important and will be treated such as the paraxial conditions and the perfectly matched layers approaches. This is crucial for many applications in seismic imaging, for instance, where solutions should not introduce spurious modes from the outer boundaries into the computational domain that could deteriorate the solutions during the inverse problem.

Dr. Roland Martin senior research scientist at the National Centre for Scientific Research, Universit̩ Paul Sabatier РToulouse 3, France and has been working for many years in France where he obtained his PhD in Geophysics (1998). He has been a researcher in Mexico City (1999-2004) before integrating the French CNRS (equivalent to the Australian CSIRO) in 2005 at Pau University and GET laboratory in Toulouse. His main interests are the numerical modelling in geophysics at different scales using different numerical techniques for the forward and inverse problems. He is developing and applying those techniques to the modelling and imaging the Earth at different scales: from the near subsurface or laboratory scale to the Earth crust scale with some specific sites of study like the well monitored Pyrenees chain located between Spain and France. Seismic and gravity dense measurements are mainly used to obtain more information on both seismic wave velocities and densities in the Earth crust and to couple the structures to the surface using not only high resolution numerical tools but also more complex physics in solid-fluid mechanical systems. In 2017, Roland was awarded an Institute of Advanced Studies Robert and Maude Gledden Visiting Senior Fellowship.