Survey sinking with the double-square-root equation

An equation was derived for paraxial waves. The assumption of a single plane wave means that the arrival time of the wave is given by a single-valued t(x,z). On a plane of constant z, such as the earth's surface, Snell's parameter p is measurable. It is  

$${ \partial t \over \partial x } \ \ \eq \ { \sin \, \theta \over v }\ \eq \ p$$ (2)

In a borehole there is the constraint that measurements must be made at a constant x, where the relevant measurement from an upcoming wave would be  

$${ \partial t \over \partial z } \ \ \eq \ -\ { \cos \, \the... ...} \ -\ \left( {\partial t \over \partial x} \ \right)^2 \ } \\ end{displaymath}$$ (3)

Recall the time-shifting partial-differential equation and its solution U as some arbitrary functional form f:

$$\begin{eqnarray} { \partial U \over \partial z } \ \ \ \ &=&\ \ \ \ - \ { \part... ...ft( \ t \ -\ \int_0^z \ {\partial t \over \partial z} \ dz \right)\end{eqnarray}$$ (4) (5)

The partial derivatives in equation (4) are taken to be at constant x, just as is equation (3). After inserting (3) into (4) we have  

$${ \partial U \over \partial z } \quad = \quad \sqrt{ {1 \ove... ... \partial x} \ \right)^2 \ }\ { \partial U \over \partial t }$$ (6)

Fourier transforming the wavefield over (x,t), we replace $\partial / \partial t$ by $-\,i \omega$.Likewise, for the traveling wave of the Fourier kernel $\exp (-\,i \omega t \ +\ ik_x x )$,constant phase means that ${\partial t}/{\partial x} \,=\, k_x / \omega$.With this, (6) becomes  

$${ \partial U \over \partial z } \ \eq \ - \, i \omega \ \sqrt{ {1 \over v^2 } \ -\ { k_x^2 \over \omega^2} \ } \ U$$ (7)

The solutions to (7) agree with those to the scalar wave equation unless v is a function of z, in which case the scalar wave equation has both upcoming and downgoing solutions, whereas (7) has only upcoming solutions. We go into the lateral space domain by replacing i k_x by $\partial / \partial x$.The resulting equation is useful for superpositions of many local plane waves and for lateral velocity variations v(x).