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 (Pwave) and shear waves (Swave). 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 3dimensional 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 3dimensional velocity structure of earth from seismic waves. The simplest case of seismic tomography is to estimate Pwave velocity. Several methods have been developed for this purpose, e.g., refraction traveltime tomography, finitefrequency 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 firstarrival 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 lowresolution image of the subsurface. To obtain a higherresolution image one has to abandon the infinitefrequency 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 crosscorrelation. Finitefrequency 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 'bananadoughnut' kernels in global tomography, because their shape may resemble a banana, whereas their crosssection 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 finitefrequency tomography, travel time and amplitude anomalies are frequencydependent, 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 finitedifference and finiteelement methods. Seismic waveform tomography can be efficiently solved by adjoint methods. References * Stewart, R. R., Exploration Seismic Tomography: Fundamentals, Society of Exploration Geophysicists, 1991
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