Geometrical derivation

To derive equation (2) we will change the coordinates system in order to find the 2-D case illustrated in Figure -a) and follow the derivation given in Fomel (1996).

 

geometry Figure 8 a) 2-D geometry described by Sava and Fomel (2000). b) Common-azimuth formulation of the problem. The derivation of equation (1) relies on the definition of the problem into the propagation plane. Defined this way, the 3-D problem becomes equivalent to the 2-D problem.

Consider the source and the receiver rays constrained to a slanted propagation plane as illustrated in Figure -b). The plane has dip angle $\delta$.Within the propagation plane, the source ray has a dip angle $\alpha_S$ and the receiver ray a dip angle $\alpha_R$.We define z' as being a new vertical axis within the propagation plane: the 3-D problem becomes a 2-D problem in this new basis. The aperture angle $\gamma$ and the dipping angle $\alpha_n$ of the normal of the interface at the reflection point can be expressed as a function of $\alpha_S$ and $\alpha_R$:

$$\begin{eqnarray} \gamma &=& \frac{\alpha_R+\alpha_S}{2} \\ \alpha_n &=& \frac{\alpha_R-\alpha_S}{2}\end{eqnarray}$$ (8) (9)

The offset ray parameter in the propagation plane is:

$$\begin{eqnarray} \frac{\partial t}{\partial h_x} = \frac{\sin\alpha_S}{v}+\frac{\sin\alpha_R}{v} = \frac{2}{v} \cos\alpha_n \sin\gamma\end{eqnarray}$$ (10)

We can write a 2-D version of the Double Square Root (DSR) in the propagation plane:

$$\begin{eqnarray} \frac{\partial t}{\partial z'} = \frac{\cos\alpha_S}{v}+\frac{\cos\alpha_R}{v} = \frac{2}{v} \cos\alpha_n \cos\gamma\end{eqnarray}$$ (11)

The change of variable between the pseudo vertical axis z' and the real vertical axis z is made with the relation:  

$$\frac{\partial z'}{\partial z} = \cos \delta$$ (12)

With equations (11) and (12) the DSR expression in the original coordinates system becomes:  

$$\frac{\partial t}{\partial z} = \frac{2}{v} \cos \alpha_n \cos \gamma \cos \delta$$ (13)

The quotient of equations (10) and (13) leads to the expected relation 2:

$$\frac{\partial z}{\partial h}= \frac{k_{h_x}}{k_z} = \frac{\tan \gamma}{\cos \delta}$$ (14)