Imaging in shot-geophone space

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 } \$$ (3)

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

$${ \partial U \over \partial z } = - \ { \partial t \over \partial z } \ { \partial U \over \partial t }$$ (4)

$$U = \ f \left( t - \int_0^z {\partial t \over \partial z} dz \right)$$ (5)

The partial derivatives in equation (9.4) are taken to be at constant $x$, just as is equation (9.3). After inserting (9.3) into (9.4) 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, (9.6) 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 (9.7) 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 (9.7) 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 $v(x)$.

Imaging in shot-geophone space