Seismic tomography is a methodology for estimating the Earth's properties. In the seismology community, seismic tomography is just a part of seismic imaging, and usually has a more specific purpose to estimate properties such as propagating velocities of compressional waves (P-wave) and shear waves (S-wave). It can also be used to recover the attenuation factor Q. Another branch of seismic imaging is seismic migration in which the properties to be estimated include the reflection coefficient or reflectivity.
In another way, we define tomography as a technique whereby a 3-dimensional images are derived from the processing of integrated properties of the medium that rays encounter along their paths through it. Seismic tomography refers to the derivation of the 3-dimensional velocity structure of earth from seismic waves.
The simplest case of seismic tomography is to estimate P-wave velocity. Several methods have been developed for this purpose, e.g., refraction traveltime tomography, finite-frequency traveltime tomography, reflection traveltime tomography, waveform tomography.
Seismic tomography is usually formulated as an inverse problem. In refraction traveltime tomography, the observed data are the first-arrival traveltimes t and the model parameters are the slowness s. The forward problem can be formulated as
t = Ls
where L is the forward operator which, in this case, is the raypath matrix. Refraction traveltime tomography is computationally efficient but can only provide a low-resolution image of the subsurface.
To obtain a higher-resolution image one has to abandon the infinite-frequency approximations of ray theory that are applicable to the time of the wave 'onset' and instead measure travel times (or amplitudes) over a time window of some length using cross-correlation. Finite-frequency tomography takes the effects of wave diffraction into account, which makes the imaging of smaller objects or anomalies possible. The raypaths are replaced by volumetric sensitivity kernels, often named 'banana-doughnut' kernels in global tomography, because their shape may resemble a banana, whereas their cross-section looks like a doughnut, with, at least for direct P and S waves, zero sensitivity of the travel time on the geometrical ray path. In finite-frequency tomography, travel time and amplitude anomalies are frequency-dependent, which leads to an increase in resolution.
To exploit the information in a seismogram to the fullest, one uses waveform tomography. In this case, the seismograms are the observed data. In seismic exploration, the forward model is usually governed by the acoustic wave equation. This is an approximation to the elastic wave propagation. Elastic waveform tomography is much more difficult than acoustic waveform tomography. The acoustic wave equation is numerically solved by some numerical schemes such as finite-difference and finite-element methods. Seismic waveform tomography can be efficiently solved by adjoint methods.
* Stewart, R. R., Exploration Seismic Tomography: Fundamentals, Society of Exploration Geophysicists, 1991