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** Up:** REFLECTION AND TRANSMISSION COEFFICIENTS
** Previous:** Plane wave solutions

At a horizontal interface, we assume a displacement-stress vector whose
variables are continuous across the interface , where is
the velocity, and represents the
vertical component of the stress tensor. This vector can be divided
into

| |
(10) |

where the elements of *F* are
| |
(11) |

and the elements of the vector are a function of the wave
amplitudes, as follows:
| |
(12) |

To calculate the amplitude partitioning at an interface between two
layers we equate the displacement-stress vector across the interface,
thus:

| |
(13) |

Translating the coordinate frame so that the interface is at *z*=0,
the exponential terms in *w* are the same in both layers, and we can
write equation (13) as
| |
(14) |

giving a general relation between the up-going and down-going wave
systems in the two media. If we partition so that
is a vector of the amplitudes of
down-going waves and of up-going
waves, we can write the block-matrix equation as

| |
(15) |

In order to calculate the up-going reflected wavefield and the
down-going transmitted wavefield for a downward propagating wavefield
incident on the boundary from above, we need to solve the system

| |
(16) |

After some manipulation, we obtain Nichols (1991)
| |
(17) |

| |

where
| |
(18) |

| |

The matrices *R*_{D} and *T*_{D} convert the vector of
down-going wave amplitudes in layer 1 into a vector of up-going
reflected wave amplitudes in layer 1 and a vector of down-going
transmitted amplitudes in layer 2. The next section studies the PP
wave reflection amplitudes given by the first column, first row
element in matrix *R*_{D}.

** Next:** MODELING REFLECTIONS
** Up:** REFLECTION AND TRANSMISSION COEFFICIENTS
** Previous:** Plane wave solutions
Stanford Exploration Project

11/12/1997