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  

$$\tau(s,r)=\tau_1\left(s,x(s,r)\right)+\tau_2\left(x(s,r),r\right)\;,$$ (1)

where x is the reflection point parameter, $\tau_1$ corresponds to the incident ray, and $\tau_2$ corresponds to the reflected ray. Differentiating (A-1) with respect to s and r yields

$$\begin{eqnarray} {\partial \tau \over \partial s} & = & {\partial \tau_1 \over ... ...\partial \tau \over \partial x}\, {\partial x \over \partial r}\;.\end{eqnarray}$$ (2) (3)

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

$${\partial \tau \over \partial x} \equiv 0$$ (4)

for any s and r. Taking into account (A-4) while differentiating (A-2) and (A-3), we get

$$\begin{eqnarray} {\partial^2 \tau \over \partial s^2} & = & {\partial^2 \tau_1 ... ...tial x \over \partial r}\;= B_2\, {\partial x \over \partial s}\;,\end{eqnarray}$$ (5) (6) (7)

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 (A-4) gives us the additional pair of equations

$$\begin{eqnarray} C\,{\partial x \over \partial s}+B_1 & = & 0\;, \\ C\,{\partial x \over \partial r}+B_2 & = & 0\;,\end{eqnarray}$$ (8) (9)

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 (A-8) - (A-9) for $\partial x \over \partial s$ and $\partial x \over \partial r$ and substituting the result into (A-5) - (A-7) produces the following set of expressions:

$$\begin{eqnarray} {\partial^2 \tau \over \partial s^2} & = & {\partial^2 \tau_1 ... ...l^2 \tau \over \partial s \partial r} & = & - C^{-1}\,B_1\,B_2\;.\end{eqnarray}$$ (10) (11) (12)

In the case of a constant velocity medium, expressions (A-10) to (A-12) can be applied directly to the explicit equation for the two-point eikonal  

$$\tau_1(y,x)=\tau_2(x,y)={\sqrt{(x-y)^2+z^2(x)}\over v}\;.$$ (13)

Differentiating (A-13) and taking into account the trigonometric relationships for the incident and reflected rays (Figure ), one can evaluate all the quantities in (A-10) to (A-12) explicitly. After some heavy algebra, the resultant expressions for the traveltime derivatives take the form

$$\begin{eqnarray} {\partial \tau \over \partial s} = {\partial \tau_1 \over \par... ...\tau_2 \over \partial x} = - {\sin{\gamma}\over v \cos{\alpha}}\;;\end{eqnarray}$$ (14) (15)

$$\begin{eqnarray} B_1 & = & {\partial^2 \tau_1 \over \partial s\,\partial x} = {... ...\left(-1+{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_2}\right)\;;\end{eqnarray}$$ (16) (17)

 

$$B_1\,B_2 = {\cos^6{\gamma}\over v^2\,D^2\,a^4}\;;\; B_1+B_2 = -2\,{\cos^3{\gamma}\over v\,D\,a^2}\,\left(2\,a^2-1\right)\;;$$ (18)

 

$${\partial^2 \tau_1 \over \partial x^2} = {{\cos^2{\gamma}+D\... ...^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\,\cos{\alpha_2}\;;$$ (19)

 

$$C={\partial^2 \tau_1 \over \partial x^2}+{\partial^2 \tau_2 ... ...{\gamma}\,{{\cos^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\;.$$ (20)

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  

$$\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}- {\part... ...}\,\left({\sin^2{\alpha}+DK}\over {\cos^2{\gamma}+DK}\right)\;,$$ (21)

which coincides with () in the main text.