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.