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The chain rule for partial differentiation says that
| ![\begin{displaymath}
\left[ \
\matrix { \partial_t \cr \partial_x \cr \partial_z...
...tial_{ t' } \cr \partial_{{x}' }
\cr \partial_{{z}' } }
\right]\end{displaymath}](img83.gif) |
(35) |
In Fourier space, the top two rows of the above matrix may be interpreted as
| ![\begin{eqnarray}
-\,i\,\omega \ \ \ &=&\ \ \ -\,i\,\omega'
\\ i\, k_x \ \ \ &=&\ \ \ +\,p\,\omega' \ \ +\ \ i\,
{{ k' }_{ \kern-0.25em x }}\end{eqnarray}](img84.gif) |
(36) |
| (37) |
Of particular interest is the energy that is flat
after linear moveout (constant with x').
For such energy
.Combining (36) and (37) gives the familiar equation
| ![\begin{displaymath}
p\ \eq \ {k \over \omega }\end{displaymath}](img86.gif) |
(38) |
EXERCISES:
-
Explain the choice of sign of the s-axis in Figure 11.
-
Equations (30), (31) and (32) are for
upgoing
Snell waves.
What coordinate system would be appropriate for
downgoing
Snell waves?
-
Express the scalar wave equation in the coordinate system
(30), (31) and (32).
Neglect first derivatives.
-
Express the dispersion relation of the scalar wave equation in terms of the
Fourier variables
.
Next: INTERVAL VELOCITY BY LINEAR
Up: SNELL WAVES AND SKEWED
Previous: Snell wave coordinates
Stanford Exploration Project
10/31/1997