Non-hyperbolic common reflection surface

Appendix B: Circular reflector

In the case of a circular (cylindrical or spherical) reflector in a homogeneous velocity model, the closed-form analytical solution is complicated, because it involves a solution of a high-order polynomial equation (Landa et al., 2010). However, the traveltime surface can be easily described analytically by parametric relationships (Glaeser, 1999).

crefl
Figure 5. Reflection from a circular reflector in a homogeneous velocity model (a scheme).

Consider the reflection geometry shown in Figure B-1. According to the trigonometry of the reflection triangles, the source and receiver positions can be expressed as

$$s = R\,\sin{\gamma} + (L - R\,\cos{\gamma})\,\tan{(\gamma-\theta)}\;,$$ (32)

$$r = R\,\sin{\gamma} + (L - R\,\cos{\gamma})\,\tan{(\gamma+\theta)}\;,$$ (33)

where $R$ is the reflector radius, $D$ is the minimum reflector depth, $L=D+R$ , $\gamma$ is the reflector dip angle at the reflection point, and $\theta$ is the reflection angle. Correspondingly, the midpoint and half-offset coordinates are expressed as

$$m = R\,\sin{\gamma} + (L - R\,\cos{\gamma})\,\frac{\cos{\gamma}\,\sin{\gamma}}{\cos^2{\theta} - \sin^2{\gamma}}\;,$$

$$h = (L - R\,\cos{\gamma})\,\frac{\cos{\theta}\,\sin{\theta}}{\cos^2{\theta} - \sin^2{\gamma}}\;,$$ (34)

and the reflection traveltime is expressed as

$$t = \frac{L - R\,\cos{\gamma}}{V}\,\left[\frac{1}{\cos{(\gamma-\theta)}} + \frac{1}{\cos{(\gamma+\theta)}}\right]$$

$$= 2\,\frac{L - R\,\cos{\gamma}}{V}\,\frac{\cos{\gamma}\,\cos{\theta}}{\cos^2{\theta} - \sin^2{\gamma}}\;,$$ (35)

where $V$ is the medium velocity. Equations (B-3-B-4) define the reflection traveltime surface $t(m,h)$ parametrically via the dependence $\{m(\gamma,\theta),h(\gamma,\theta),t(\gamma,\theta)\}$ . The connection with the multifocusing parameters is given by

$$t_0 = \frac{2\,\left(\sqrt{m_0^2 + L^2}-R\right)}{V}$$ (36)

$$K_{NIP} = \frac{1}{\sqrt{m_0^2 + L^2}-R}\;,$$ (37)

$$K_N = \frac{1}{\sqrt{m_0^2 + L^2}}\;,$$ (38)

$$\sin{\beta} = \frac{m_0}{\sqrt{m_0^2 + L^2}}\;,$$ (39)

and $V_0=V$ .

Non-hyperbolic common reflection surface