Second-order reflection traveltime derivatives

 In this appendix, I derive equations connecting second-order partial derivatives of the reflection traveltime with the geometric properties of the reflector in a constant velocity medium. These equations are used in the main text of Chapter  for the amplitude behavior description. Let $\tau(s,r)$ be the reflection traveltime from the source s to the receiver r. Consider a formal equality  

$$\tau(s,r)=\tau_1\left(s,x(s,r)\right)+\tau_2\left(x(s,r),r\right)\;,$$ (266)

where x is the reflection point parameter, $\tau_1$ corresponds to the incident ray, and $\tau_2$ corresponds to the reflected ray. Differentiating () with respect to s and r yields

$$\begin{eqnarray} {\partial \tau \over \partial s} & = & {\partial \tau_1 \over ... ...\partial \tau \over \partial x}\, {\partial x \over \partial r}\;.\end{eqnarray}$$ (267) (268)

According to Fermat's principle, the two-point reflection ray path must correspond to the traveltime stationary point. Therefore  

$${\partial \tau \over \partial x} \equiv 0$$ (269)

for any s and r. Taking into account () while differentiating () and (), we get

$$\begin{eqnarray} {\partial^2 \tau \over \partial s^2} & = & {\partial^2 \tau_1 ... ...tial x \over \partial r}\;= B_2\, {\partial x \over \partial s}\;,\end{eqnarray}$$ (270) (271) (272)

where

$$B_1={\partial^2 \tau_1 \over \partial s \partial x}\;;\; B_2={\partial^2 \tau_2 \over \partial r \partial x}\;.$$

Differentiating equation () gives us the additional pair of equations

$$\begin{eqnarray} C\,{\partial x \over \partial s}+B_1 & = & 0\;, \\ C\,{\partial x \over \partial r}+B_2 & = & 0\;,\end{eqnarray}$$ (273) (274)

where

$$C={\partial^2 \tau \over \partial x^2}= {\partial^2 \tau_1 \over \partial x^2}+ {\partial^2 \tau_2 \over \partial x^2}\;.$$

Solving the system () - () for $\partial x \over \partial s$ and $\partial x \over \partial r$ and substituting the result into () - () produces the following set of expressions:

$$\begin{eqnarray} {\partial^2 \tau \over \partial s^2} & = & {\partial^2 \tau_1 ... ...l^2 \tau \over \partial s \partial r} & = & - C^{-1}\,B_1\,B_2\;.\end{eqnarray}$$ (275) (276) (277)

In the case of a constant velocity medium, expressions () to () can be applied directly to the explicit equation for the two-point eikonal  

$$\tau_1(y,x)=\tau_2(x,y)={\sqrt{(x-y)^2+z^2(x)}\over v}\;.$$ (278)

Differentiating () and taking into account the trigonometric relationships for the incident and reflected rays (Figure ), one can evaluate all the quantities in () to () explicitly. After some heavy algebra, the resultant expressions for the traveltime derivatives take the form

$$\begin{eqnarray} {\partial \tau \over \partial s} = {\partial \tau_1 \over \par... ...\tau_2 \over \partial x} = - {\sin{\gamma}\over v \cos{\alpha}}\;;\end{eqnarray}$$ (279) (280)

$$\begin{eqnarray} B_1 & = & {\partial^2 \tau_1 \over \partial s\,\partial x} = {... ...\left(-1+{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_2}\right)\;;\end{eqnarray}$$ (281) (282)

 

$$B_1\,B_2 = {\cos^6{\gamma}\over v^2\,D^2\,a^4}\;;\; B_1+B_2 = -2\,{\cos^3{\gamma}\over v\,D\,a^2}\,\left(2\,a^2-1\right)\;;$$ (283)

 

$${\partial^2 \tau_1 \over \partial x^2} = {{\cos^2{\gamma}+D\... ...^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\,\cos{\alpha_2}\;;$$ (284)

 

$$C={\partial^2 \tau_1 \over \partial x^2}+{\partial^2 \tau_2 ... ...{\gamma}\,{{\cos^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\;.$$ (285)

Here D is the length of the normal (central) ray, $\alpha$ is its dip angle ($\alpha={{\alpha_1+\alpha_2}\over 2}$, $\tan{\alpha}=z'(x)$), $\gamma$ is the reflection angle $\left(\gamma={{\alpha_2-\alpha_1}\over 2}\right)$, K is the reflector curvature at the reflection point $\left(K=z''(x)\,\cos^3{\alpha}\right)$, and a is the dimensionless function of $\alpha$ and $\gamma$ defined in ().

The equations derived in this appendix were used to get the equation  

$$\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}- {\part... ...}\,\left({\sin^2{\alpha}+DK}\over {\cos^2{\gamma}+DK}\right)\;,$$ (286)

which coincides with () in the main text.