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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

\begin{displaymath}
{ \partial t \over \partial x }   \eq \
{ \sin   \theta \over v } \eq  p
\end{displaymath} (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
\begin{displaymath}
{ \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 $f$:
$\displaystyle { \partial U \over \partial z }     $ $\textstyle =$ $\displaystyle     - \
{ \partial t \over \partial z } \
{ \partial U \over \partial t }$ (4)
$\displaystyle U     $ $\textstyle =$ $\displaystyle    \
f \left(  t  - \int_0^z  {\partial t \over \partial z}  dz \right)$ (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
\begin{displaymath}
{ \partial U \over \partial z } \quad = \quad \sqrt{ {1 \ove...
...\partial x}  \right)^2
 } { \partial U
\over \partial t }
\end{displaymath} (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
\begin{displaymath}
{ \partial U \over \partial z }  \eq  -   i \omega \
\sqrt{
{1 \over v^2 }  - { k_x^2 \over \omega^2}  }  U
\end{displaymath} (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 $v(x)$.
next up previous [pdf]

Next: The DSR equation in Up: SURVEY SINKING WITH THE Previous: The survey-sinking concept

2009-03-16