Proof of kinematic equivalence

In order to prove the validity of equation (5), it is convenient to transform it to the coordinates of the initial shot gathers: s=y-h, r=y+h, and $\tau = \sqrt{\tau_n^2+{{4h^2} \over {v^2}}}$. The transformed equation takes the form  

$$\left( \tau^2 + {{(r-s)^2} \over {v^2}} \right) \left( {\par... ...er \partial r} {\partial \tau \over \partial s} \right) \,\,\,.$$ (7)

Now the goal is to prove that any reflection traveltime function $\tau(r,s)$ in a constant velocity medium satisfies equation (7).

Let S and R be the source and the reflection locations, and O be a reflection point for that pair. Note that the incident ray SO and the reflected ray OR form a triangle with the basis on the offset SR (l=|SR|=r-s). Let $\alpha_1$ be the angle of SO from the vertical axis, and $\alpha_2$ be the analogous angle of RO (Figure 1). Elementary trigonometry (the law of sines) gives us the following explicit relationships between the sides and the angles of the triangle SOR:

$$\begin{eqnarray} \vert SO\vert\,=\,\vert SR\vert\, {\cos{\alpha_1} \over \sin{\l... ...\cos{\alpha_2} \over \sin{\left(\alpha_2-\alpha_1\right)}} \,\,\,.\end{eqnarray}$$ (8) (9)

Hence, the total length of the reflected ray is  

$$v \tau = \vert SO\vert+\vert RO\vert=\vert SR\vert\, {{\cos{... ...a_1\right)}} = (r-s)\,{\cos{\alpha} \over \sin{\gamma}} \,\,\,.$$ (10)

Here $\gamma$ is the reflection angle ($\gamma = (\alpha_2 - \alpha_1)/2$), and $\alpha$ is the central ray angle ($\alpha = (\alpha_2 + \alpha_1)/2$) coincident with the local dip angle of the reflector at the reflection point. Recalling the well-known relationships between the ray angles and the first-order traveltime derivatives

$$\begin{eqnarray} {{\partial \tau} \over {\partial s}} \,=\,{ {\sin{\alpha_1}} \o... ...au} \over {\partial r}} \,=\, {{\sin{\alpha_2}} \over {v}} \,\,\,,\end{eqnarray}$$ (11) (12)

we can substitute (10), (11), and (12) into (7), which leads to the simple trigonometric equality  

$$\cos^2{\left( {\alpha_1 + \alpha_2} \over 2 \right)} + \sin... ...ver 2 \right)}\, = \, 1 - \sin{\alpha_1} \sin{\alpha_2} \,\,\,.$$ (13)

It is now easy to prove that equality (13) is true for any $\alpha_1$ and $\alpha_2$.

 

ocoray Figure 1 Reflection rays in a constant velocity medium (a scheme).

Thus we have proved that equation (7), equivalent to (5), is valid in constant velocity media independently of the reflector geometry and the offset. This means that high-frequency asymptotic components of the waves, described by the OC equation, are located on the true reflection traveltime curves.

The theory of characteristics can provide other ways to prove the kinematic validity of equation (5), as described in Fomel (1994); Goldin (1994).