Derivation of the PS-DMO operator

The total traveltime (t) for a reflector at depth z in a medium with constant P-velocity (v_p) and S-velocity (v_s) is:  

$$t=\frac{\sqrt{z^2 + (h+x)^2}}{v_p} + \frac{\sqrt{z^2 + (h-x)^2}}{v_s}.$$ (10)

where h is the half-offset and x is the surface midpoint-location. By introducing the ratio, $\gamma=\frac{v_p}{v_s}$, and following Huub Den Rooijen's 1991 derivations, as well as work by Xu et al. (2001), I have the following standard form of elliptical equation:

 

$$z^2+\left ( 1-\frac{4\gamma h^2}{v_p^2 t^2} \right ) (x+D)^2 = \frac{v_p t_n^2}{(1+\gamma)^2}$$ (11)

This equation introduces two more terms that were absent in equation , namely t_n, the NMO-corrected time; and, D, this term represents the lateral shift responsible for the CMP to CCP/CRP correction. The main difference among the existing PS-DMO operators is in the computation and correction from CMP to CCP (D). Throughout this thesis I use the definition presented by Xu et al. (2001):

$$D=\left[ 1 + \frac{4\gamma h^2}{v_p^2 t_n^2 + 2\gamma(1-\gamma)h^2} \right] \frac{1-\gamma}{1+\gamma}h.$$ (12)

The PS-DMO smile equation is presented as follows:

 

$$\frac{t_0^2}{t_n^2} + \frac{y^2}{H^2} = 1,$$ (13)

where

$$\begin{eqnarray} y & = & x+D, \nonumber \ H & = & \frac{2\sqrt{\gamma}}{1+\gamm... ... t_n^2 + 2\gamma(1-\gamma) h^2}\right] \frac{1-\gamma}{1+\gamma}h.\end{eqnarray}$$ (14) (15)