Next: MODELING REFLECTIONS Up: REFLECTION AND TRANSMISSION COEFFICIENTS Previous: Plane wave solutions

## Boundary conditions at a horizontal interface

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 RD and TD 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 RD.

Next: MODELING REFLECTIONS Up: REFLECTION AND TRANSMISSION COEFFICIENTS Previous: Plane wave solutions
Stanford Exploration Project
11/12/1997