SECOND-ORDER REFLECTION TRAVELTIME DERIVATIVES

In this appendix I derive formulas connecting second-order partial derivatives of the reflection traveltime with the geometric properties of the reflector in a constant velocity medium. These formulas are used in the main text of the paper 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  

$$\tau(s,r)=\tau_1\left(s,x(s,r)\right)+\tau_2\left(x(s,r),r\right)\;,$$ (59)

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

$${\partial \tau \over \partial s} = {\partial \tau_1 \over \... ...rtial \tau \over \partial x}\, {\partial x \over \partial s}\;,$$ (60)

 

$${\partial \tau \over \partial r} = {\partial \tau_2 \over \... ...rtial \tau \over \partial x}\, {\partial x \over \partial r}\;.$$ (61)

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

$${\partial \tau \over \partial x} \equiv 0$$ (62)

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

$${\partial^2 \tau \over \partial s^2} = {\partial^2 \tau_1 \over \partial s^2} + B_1\, {\partial x \over \partial s}\;,$$ (63)

 

$${\partial^2 \tau \over \partial r^2} = {\partial^2 \tau_2 \over \partial r^2} + B_2\, {\partial x \over \partial r}\;,$$ (64)

 

$${\partial^2 \tau \over \partial s \partial r} = B_1\, {\partial x \over \partial r}\;= B_2\, {\partial x \over \partial s}\;,$$ (65)

where Differentiating (B-4) gives us the additional pair of equations  

$$C\,{\partial x \over \partial s}+B_1 = 0\;,$$ (66)

 

$$C\,{\partial x \over \partial r}+B_2 = 0\;,$$ (67)

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

$${\partial^2 \tau \over \partial s^2} = {\partial^2 \tau_1 \over \partial s^2} - C^{-1}\,B_1^2\;;$$ (68)

 

$${\partial^2 \tau \over \partial r^2} = {\partial^2 \tau_2 \over \partial r^2} - C^{-1}\,B_2\;;$$ (69)

 

$${\partial^2 \tau \over \partial s \partial r} = - C^{-1}\,B_1\,B_2\;.$$ (70)

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

$$\tau_1(y,x)=\tau_2(x,y)={\sqrt{(x-y)^2+z^2(x)}\over v}\;.$$ (71)

Differentiating (B-13) and taking into account the trigonometric relationships for the incident and reflected rays (Figure 1), one can evaluate all the quantities in (B-10) to (B-12) explicitly. After some heavy algebra, the resultant expressions for the traveltime derivatives take the form  

$${\partial \tau \over \partial s} = {\partial \tau_1 \over \... ...{\partial \tau_2 \over \partial r} = {\sin{\alpha_2}\over v}\;;$$ (72)

 

$${\partial \tau_1 \over \partial x} = {\sin{\gamma}\over v \... ...u_2 \over \partial x} = - {\sin{\gamma}\over v \cos{\alpha}}\;;$$ (73)

 

$$B_1 = {\partial^2 \tau_1 \over \partial s\,\partial x} = {\... ...ft(-1-{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_1}\right)\;;$$ (74)

 

$$B_2 = {\partial^2 \tau_2 \over \partial r\,\partial x} = {\... ...ft(-1+{\sin{\gamma}\over\cos{\alpha}}\,\sin{\alpha_2}\right)\;;$$ (75)

 

$$B_1\,B_2 = {\cos^6{\gamma}\over v^2\,D^2\,a^4}\;;\; B_1+B_2 = -2\,{\cos^3{\gamma}\over v\,D\,a^2}\,\left(2\,a^2-1\right)\;;$$ (76)

 

$${\partial^2 \tau_1 \over \partial x^2} = {{\cos^2{\gamma}+D\... ...^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\,\cos{\alpha_2}\;;$$ (77)

 

$$C={\partial^2 \tau_1 \over \partial x^2}+{\partial^2 \tau_2 ... ...{\gamma}\,{{\cos^2{\gamma}+D\,K}\over{v\,D\,\cos^3{\alpha}}}\;.$$ (78)

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 , and a is the nondimensional function of $\alpha$ and $\gamma$ defined in (35).

The formulas derived in this appendix were used to get the formula  

$$\tau_n\,\left({\partial^2 \tau_n \over \partial y^2}- {\part... ...}\,\left({\sin^2{\alpha}+DK}\over {\cos^2{\gamma}+DK}\right)\;,$$ (79)

which coincides with (38) in the main text.