Analytical path-summation imaging of seismic diffractions

Appendix B: Double path-summation migration

Double path-integral formulation was proposed by Schleicher and Costa (2009), Santos et al. (2016). In unweighted path-summation migration framework it can be described by the following equation:

$$I_{DPI}(t,x,v_a,v_b) = \int^{v_b}_{v_a} v\ P(t,x,v)\,dv\ .$$ (10)

Velocity distribution can be acquired through a smooth division of the double path-summation integral by the path-summation integral:

$$v(t,x) = I_{DPI}(t,x,v_a,v_b)/I_{PI}(t,x,v_a,v_b)\ .$$ (11)

We propose to evaluate the double path-summation integral analytically as well:

$$I_{DPI}(\Omega,k,v_a,v_b) = \int^{v_b}_{v_a} v\ \hat{P}_0(k,\Omega)\ e^{-\frac{i k^2 v^2}{16\Omega}}dv\$$

$$= \hat{P}_0(\Omega,k)\ \int^{v_b}_{v_a} v\ e^{-\frac{i k^2 v^2}{16\Omega}}dv$$

$$= \hat{P}_0(\Omega,k) \cdot F_{DPI}(\Omega,k,v_a,v_b),$$ (12)

where the filter $F_{DPI}(\Omega,k,v_a,v_b)$ has the following analytical expression:

$$F_{DPI}(\Omega,k,v_a,v_b) = \frac {8 i \Omega}{k^2} e^{-\frac{ik^2v^2}{16\Omega}}\ \bigg\vert _{v_a}^{v_b}.$$ (13)

To extract migration velocity as described by the equation (A-5) we need to evaluate both double path-summation integral and path-summation integral. These evaluations can be done analytically as described by equations 5 and A-6. A smooth division procedure (Fomel, 2007) is applied in equation A-5 to acquire a regularized time migration velocity estimate, which then can be applied to image the subsurface using any available time imaging algorithm.

Analytical path-summation imaging of seismic diffractions