Generalized nonhyperbolic moveout approximation

Appendix C: REFLECTION FROM A HYPERBOLIC REFLECTOR IN A HOMOGENEOUS VELOCITY MODEL

In this appendix, we derive an analytical expression for reflection traveltime from a hyperbolic reflector in a homogeneous velocity model (Figure C-1). Similar derivations apply to an elliptic reflector and were used previously in the theory of dip moveout, offset continuation, and non-hyperbolic common-reflection surface (Fomel and Kazinnik, 2009; Stovas and Fomel, 1996; Fomel, 2003).

hyper
Figure 9. Reflection from a hyperbolic reflector in a homogeneous velocity model (a scheme).

Consider the source point $x_s$ and the receiver point $x_r$ at the surface $z=0$ above a 2-D constant-velocity medium and a hyperbolic reflector defined by the equation

$$z(x) = \sqrt{h^2 + x^2\,\tan^2{\alpha}}\;.$$ (54)

The reflection traveltime as a function of the reflection point location $y$ is

$$t = \frac{\sqrt{(x_s-y)^2 + z^2(y)} + \sqrt{(x_r-y)^2+z^2(y)}}{V}\;.$$ (55)

According to Fermat's principle, the traveltime should be stationary with respect to the reflection point $y$:

$$0 = \frac{\partial t}{\partial y} = \frac{y-x_s + y\,\tan^2{... ...2{\alpha}}{V\,\sqrt{(x_r-y)^2 + h^2 + y^2\,\tan^2{\alpha}}}\;.$$ (56)

Putting two terms in equation C-3 on different sides of the equation, squaring them, and reducing their difference to a common denominator, we arrive at the equation

$$0 = \left[\frac{y}{\cos^2{\alpha}} - x_s\right]^2\,\left[(x_r-y)^2 + h^2 + y^2\,\tan^2{\alpha}\right]$$

$$- \left[\frac{y}{\cos^2{\alpha}} - x_r\right]^2\,\left[(x_s-y)^2 + h^2 + y^2\,\tan^2{\alpha}\right]$$ (57)

which simplifies to the following quadratic equation with respect to $y$:

$$y^2\,(x_s+x_r)\,\tan^2{\alpha} - 2\,y\,\left(x_s\,x_r\,\s... ...{\alpha} - h^2\right) - h^2\,(x_s+x_r)\,\cos^2{\alpha} = 0\;.$$ (58)

The discriminant is

$$D = \left(x_s\,x_r\,\sin^2{\alpha}-h^2\right)^2 + h^2\,(x... ... = (h^2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})\;.$$ (59)

Only one of the two branches of the solution

$$y = \frac{x_s\,x_r\,\sin^2{\alpha}-h^2 + \sqrt{(h^2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})}}{(x_s+x_r)\,\tan^2{\alpha}}$$

$$= \frac{h^2\,(x_s+x_r)\,\cos^2{\alpha}}{h^2 - x_s\,x_r\,\sin^2{\alpha} + \sqrt{(h^2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})}}$$ (60)

has physical meaning. Substituting equation C-7 into equation C-2, we obtain, after a number of algebraic simplifications,

$$t = \frac{\sqrt{2 h^2 + x_s^2 + x_r^2 - 2\,x_s\,x_r\,\cos... ...2+x_s^2\,\sin^2{\alpha})\,(h^2+x_r^2\,\sin^2{\alpha})}}}{V}\;.$$ (61)

Making the variable change in equation C-8 from $x_s$ and $x_r$ to the midpoint and offset coordinates $m$ and $x$ according to $x_s=m-x/2$, $x_r=m+x/2$, we notice that this equation is exactly equivalent to equation 1 with the following definition of parameters:

$$t_0 = \frac{2\,\sqrt{h^2 + m^2\,\sin^2{\alpha}}}{V}\;,$$ (62)

$$a = \frac{2-\sin^2{\alpha}}{V^2}\;,$$ (63)

$$b = \frac{\sin^2{\alpha}}{V^2}\,\frac{h^2 - m^2\,\sin^2{\alpha}}{h^2 + m^2\,\sin^2{\alpha}}\;,$$ (64)

$$c = \frac{\sin^4{\alpha}}{V^4}\;,$$ (65)

$$\xi = \frac{1}{2}\;.$$ (66)

Generalized nonhyperbolic moveout approximation