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Time and depth derivatives--the bilinear transform

You might be inclined to think a second derivative is a second derivative and that there is no mathematical reason to do time derivatives differently than space derivatives. This is not the case. A hint of disparity between t and x derivatives comes from boundary conditions. With time derivatives (and often with the depth z derivative) we must consider causality--which means the future is determined solely from the present and past. Appropriate boundary conditions on the time axis are initial conditions--the function (and perhaps some derivatives) is specified at one point, the initial point in time. For depth z that special point is the earth's surface at z=0. But lateral space derivatives are different: they require boundary conditions at two widely separated points, usually at the left and right sides of the volume.

The differential equation  
{dq \over dz } \eq \,i\,{ {k }_z ( \omega , k_x ) } \,q\end{displaymath} (42)
is associated with the very definition of kz . The analogous difference equation will define $ \hat k_z $: 
{ q_{{z} + \Delta z} \ -\ q_z \over \Delta z }
 \eq \,i\, {\hat k}_z \ 
{ q_{{z} + \Delta z} \ +\ q_z \over 2 }\end{displaymath} (43)
Inserting the solution of (42) $q = q_0 \exp ( i {k_z } z )$ into (43) gives us the relation between the desired kz and the actual $ \hat k_z $. 
\,i\, {\hat k}_z \, \Delta z \eq 
2 \ 
{ e^{ i {k_z } \, \De...
 ...{k_z } \, \Delta z /2 } \ +\ e^{{-} i {k_z } \, \Delta z /2 } }\end{displaymath} (44)
This equation is known as the bilinear transform.
\,i\, {\hat k}_z \, \Delta z \ \ \ &=&\ \ \ 
2 \, i \ { \sin \,...
 ... z \over 2 }\ \ \ &=&\ \ \ 
\tan \, { {k_z } \, \Delta z \over 2 }\end{eqnarray} (45)

Equation (46) gives the accuracy of first derivatives obtained using the Crank-Nicolson method. Recall the migration differencing schemes in chapter [*]. We did the time differencing in the same way that we did the depth differencing. So the same accuracy limitation must apply, namely,  
{ \hat \omega \, \Delta t \over 2 } \eq 
 \tan \, { \omega \, \Delta t \over 2 }\end{displaymath} (47)
Series expansion shows that $ \hat \omega$ goes to $\omega$ as $ \Delta t $ goes to zero. Relative errors in $\omega$ at (4, 10, and 20) points per wavelength are (30%, 3%, and 1%). These errors are quite large, calling for either a choice of small $ \Delta t $ or a more accurate method than (46).

The bad news is that there does not seem to exist a representation of causal differentiation that is any more accurate than the Crank-Nicolson representation. There is nothing like the 1/6 trick. Thus the sample intervals of $ \Delta z $ and $ \Delta t $ must be reduced considerably from the Nyquist criterion. The practical picture may not be as bleak as the one I am painting. Many people are pleased with both the speed and accuracy of time-domain migrations at $\Delta t = 4\ $ milliseconds.

Stolt's classic paper [1978] besides introducing the fast Fourier transform migration method, points out that more accuracy can be achieved when the requirement of causality is dropped. Stolt shows how dropping causality at the known depth level while retaining it at the next level allows stable finite differencing. With the depth z-axis we are stuck with causal derivatives, although Fourier methods could be used for discrete layers. The depth axis is not so troublesome as the x- and t-axes, however, because it affects computer time only, not data storage.

Finite difference solutions don't just approximate the frequency--what they really do is to approximate $\exp i k_z \Delta z$.Solve (43) for the unknown.  
{q_{{z} + \Delta z} }
{1 \ +\ i\, {\hat k}_z { ...
 ...\over 1 \ -\ i\, {\hat k}_z { \Delta z } / 2 }
\ {q_z }\end{displaymath} (48)
So for Nz layers in depth $z = N \Delta z$ we have the approximation  
e^{ i k_z N \Delta z }
\ \ \ \approx \ \ \ 
{1 \ +\ i...
 ...} / 2 
\over 1 \ -\ i\, {\hat k}_z { \Delta z } / 2 }
\right)^N\end{displaymath} (49)
which will be of later use for Fourier domain simulations of finite difference programs. Such simulations enable us to compare the accuracy of various migration methods.

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