Linearized inverse dynamic problem for the telegraph equation

Method to find the Minimum 1D Linear Gradient Model for Seismic Tomography

“…However, in this case in the equation (1) there is a complex dependence of ray path ξ on the nonlinear velocity function v(x, y, z), where z is a depth. Therefore the task to determine the slowness field f (x, y, z) can be solved with linearized formulation that has been proposed and investigated by [9,10]. Considering travel times t and t 0 along ray paths ξ and ξ 0 for the two media with velocities v(x, y, z) and v 0 (z) (Figure 1) the problem of a search of f (x, y, z) can be written in the following form:…”

“…Note that the integral ( 2) is taken along the curve ξ 0 (S, R) that corresponds to the source-receiver pair (S, R) and represents the seismic ray trajectory based on the a priori starting 1D model v 0 (z). This statement has been made in [9] under the assumption that the velocity v(x, y, z) can be presented as v(x, y, z) = v 0 (z) + v(x, y, z), where v 0 >> |v 1 |.…”