Inversion of seismic wavefields and travel times

Ludek Klimes

NGEO051, summer term, 30 hours
(free extension of lectures Theoretical foundations of ray methods and Modelling seismic waves in 3-D)

Forward modelling. Structural inversion. Seismic source inversion.

Elastodynamic equation. Attenuation, viscoelastodynamic equation, constitutive equations (global, local, without memory, with memory). Representation theorem. Born approximation. High-frequency approximation: amplitude and travel time, Taylor expansions, paraxial approximation.

Stochastic description of geological structures.
Geological structures. Medium correlation functions. Velocity model of the medium. Covariances.

Sensitivity of seismic waves to geological structures.
Structural Gabor functions. Sensitivity Gaussian packets.

Velocity models.
Velocity model (macro model) and structural images (images of the gradients of elastic impedances). Velocity model for ray methods. Ray chaos, Lyapunov exponents and rotational numbers. Refraction and reflection seismic methods.

Nonlinear inverse theory, probability density functions.
A priori p.d.f., data p.d.f., theoretical p.d.f., a posteriori p.d.f., marginal p.d.f.. Standard distribution (Gaussian). Linearization.

Kinematic hypocentre determination (linearized and non-linear).

Stochastic travel-time tomography.
Data, model parameters, medium and model covariance functions. Linearized stochastic travel-time tomography. Model smoothness, Sobolev scalar products. Linearized travel-time tomography constrained by Sobolev scalar products. Resolution and covariance matrices, standard deviations. Transfer of errors in seismic modelling.

Seismic migrations.
Prestack depth migration. General algorithm: extrapolation, decomposition, imaging. Imaging functionals. Kirchhoff migrations.

Resolution of seismic migrations.
Born approximation. Back-propagated Green tensor evaluated using the representation theorem. Back-propagated wavefield. Reflectivity function. Analysis of migrated image. Example.

Ray characteristic functions.
Travel-time characteristic function, amplitude characteristic function. Density of rays, overlapping.

Seismic wavefield tomography.
Representation theorems, Born scattering, many local minima of the objective function, frequency-dependent information contained within wide-angle seismic data, Gabor transform, seismic wavefield tomography.

Seismic modelling.
Software, portability of source codes and data files, programming language standards, graphics standards, software packages developed and used at the Department of Geophysics.