Generalized nonhyperbolic moveout approximation

Appendix B: LINEAR SLOTH MODEL

The linear sloth model is defined by

$${\frac{1}{V^2(z)}} = {\frac{1}{V_0^2}}\,(1+G\,z)\;.$$ (45)

where $G$ is the sloth gradient and $V_0$ is velocity at zero depth.

The equation for traveltime can be computed analytically, as follows (Cervený, 2001):

$$t^2(x) = t_0^2 + \frac{x^2}{v^2} - \frac{x^4\,(r^2-1)^2\,(2\,Q+1)}{144\,Q^3\,H^2\,v^2\,(Q+1)^2}\;,$$ (46)

where $H$ is the depth of the reflector, $r=V(H)/V_0$ is the ratio of velocity at the bottom and the top of the model and

$$Q = \sqrt{1-\frac{x^2\,(r^2-1)^2}{16\,r^2\,H^2}}\;.$$

The main traveltime parameters are given by

$$t_0 = \frac{4\,H}{3\,V_0}\,\frac{1+r+r^2}{r\,(r+1)}\;,$$ (47)

$$v^2 = V_0^2\,\frac{3\,r^2}{1+r+r^2}\;,$$ (48)

$$A = -\frac{(r-1)^2}{6\,r}\;.$$ (49)

The maximum offset and traveltime are defined by

$$X = \frac{4\,H}{\sqrt{r^2-1}}\;,$$ (50)

$$T = \frac{4\,H}{3\,V_0}\,\frac{r^2+2}{r\,\sqrt{r^2-1}}\;.$$ (51)

Substituting equations B-6 and B-7 into equations 22-23 and using the expressions for traveltime parameters B-3, B-4, and B-5 results in the following analytical expressions for additional parameters $B$ and $C$:

$$B = - \frac{(r-1)^2\,(1+r+r^2)}{2\,r\,(r+2)\,(2\,r+1)}\;,$$ (52)

$$C = - \frac{(r-1)^4\,(1+r+r^2)^2}{3\,r\,(r+2)\,(2\,r+1)^2}\;.$$ (53)

Generalized nonhyperbolic moveout approximation