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Until now the proof was pure physics. But because in seismology the
depth axis is ``special,'' we will change notations. Axis x3 will be
denoted by z, and the will denote the surface position vector
with the components (x1, x2). Also, the Laplacian will be the
operator,
| |
(23) |
We will use the Fourier transform of the pressure field along the time
axis:
| |
(24) |
The following property of the Fourier transform will be needed:
| |
(25) |
The spatial frequency is defined as:
| |
(26) |
Let be a spatial average of v in the medium, a known constant
that does not depend on or z. We also define
| |
(27) |
and the function
| |
(28) |
The index notation for derivatives will be used from now on. The
symbol will denote the phrase ``for all.''
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Stanford Exploration Project
6/8/2002