Travel-time relations

In a homogeneous medium, the seismic energy generated by a shot propagates along the straight rays that connect the shot position x_s, reflector position (x,z), and receiver position x_r. The total travel-time of this propagation can be formulated as  

$$t={1 \over 2} \left[ \sqrt{t^2_0+{4(x-x_s)^2 \over v^2}}+ \sqrt{t^2_0+{4(x-x_r)^2 \over v^2}} \right],$$ (1)

where v is the velocity of the medium, and

$$t_0={2z \over v}$$

is the vertical two-way travel time. We can solve equation (1) for t₀,  

$$t_0={1 \over t} \sqrt{t^2-{(x_r-x_s)^2 \over v^2}} \sqrt{t^2-{(2x-x_r-x_s)^2 \over v^2}}.$$ (2)

Equations (1) and (2) define the relations between two travel times and are the key equations of kinematic operators for imaging.