Total time required for seismic wave to travel through the rock towards the seismic receiver:
| T_x = \int_0^{L_x} \frac{dl}{V_p(l)} |
where
\{ x, \, y, \, z \} is cartesian coordinates in 3D space with x-axis aligned between seismic source and seismic sensor, y-axis is traversal to x-axis and z-axis is oriented towards Earth centre,
x is a lateral offset between the seismic source and seismic receiver
l(x,y,z) – trajectory of reflection wave from seismic source @ (x = 0, \, y = 0, \, z = 0) and seismic receiver @ (x, \, y = 0, \, z = 0)
dl = \sqrt{dx^2 + dy^2 + dz^3} is differential element of the distance along the reflection travel trajectory,
V_p(l) is p-wave velocity of rocks found at travel point l.
In relatively simple geological structures the travel time can be approximated by a Dix equation:
| (1) | T^2_x = T^2_0 + \frac{4 x^2}{V^2_{rms}} |
where T^2_0 is reflection time at zero offset (which means the normal incident wave reflection):
| T_0 = 2 \cdot \int_0^H \ \frac{\delta z}{V_p(z)} |
where H is the depth of the reflecting boundary,
V_{rms} – average p-wave velocity through the reflecting travel distance between the seismic source and seismic receiver:
| V^2_{rms} = \frac{\sum_i^N V_p^2(t_i) \, \delta t_i}{\sum_i^N \delta t_i}= \frac{\sum_i^N V_p(t_i) \, \delta h_i}{\sum_i^N \frac{\delta h_i}{V_p(t_i)}} |
where
V_p(t_i) is p-wave velocity of rocks found at travel time t_i,
\delta t_i is travel time through the rock element of thickness \delta h_i in tghe rock element found at travel time t_i.