Mathematical Geophysics

The course introduces the student to mathematical aspects of wave theory as used in geophysics. Focus is on seismic forward models and inversion. Theory for seismic modelling of wave propagation in horizontally layered media is developed in detail. It is shown that several seismic data processing methods like deterministic multiple removal and de-ghosting can be deduced from the modelling theory. Green's functions for the wave and Helmholz equation are given. The reciprocity theorems for acoustic and elastic media are developed. Several seismic data processing schemes which follow by using the principle of reciprocity are outlined. First, it is shown how seismic data can be injected as multiple point sources in finite-difference modelling to separate data into their upgoing and downgoing constituents, or estimate the wavefield emitted by the seismic source. Second, multidimensional deconvolution is derived to eliminate the water-layer effect from seismic data. Finally, waveform inversion of seismograms is treated for the one-dimensional wave equation.

Ingress: The students should learn basic mathematical models of seismic wave theory with the focus on the solution of some specific fundamental problems.

Knowledge: The students should understand how seismic modeling and inversion is applied in petroleum exploration and production.

Skills: The students should be able to develop and apply mathematical models for seismic data processing. They should know various techniques for seismic modeling, processing and inversion.

General competence: The students learn important mathematical aspects of geophysical wave theories. They understand the main principles that are applied in petroleum seismic research. They are able to apply the learning to develop on their own models for seismic data processing and analysis. Critical self-learning is emphasized.

Lectures and exercises.