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Basics of Seismic Traveltime Tomography

I present the theory of seismic traveltime migration. Denote
the earth's slowness distribution by the vector m, the
recorded traveltime data (picked first-arrival or reflection
traveltimes) by the vector d, and the traveltime forward
modeling operator (either raypath integral or Rytov equa-
tion) by L. In this case, d=Lm represents an (high-fre-
quency or weak scattering) approximation to traveltime mod-
eling so that traveltime tomography is defined to be m_est =
[L^T L]^{-1} L^Td. I show several interpretations of trav-
eltime tomography, including "raypath slowness tomogram is
obtained by smearing+summing traveltime residuals along ray-
paths", and "wavepath slowness tomogram is obtained by
smearing+summing traveltime residuals along fat raypaths,
i.e, wavepaths". I show how the resolution of a tomogram
image is obtained using ideas borrowed from the generalized
Radon transform. Finally, examples from fill waveform
tomography are shown.