At a horizontal interface the conserved quantities are the displacements, or velocities, and the tractions on the interface. I choose to work in terms of velocities as this simplifies some later expressions. It is convenient to write these quantities as a function of the amplitudes of the six wavetypes in the layer.
The vector, is conserved across a horizontal interface.
This vector can be written as
where the elements of E are,
and the elements of f are a function of the wave amplitudes,
To calculate the amplitude partitioning at an interface between two layers we need to solve the equation,
The superscript denotes the layer.
The coordinate frame can be translated so that the interface is at x3=0. If we do this the exponential terms in fn are the same in both layers and we can write,
This gives the relationship between the amplitudes of the six wavetypes propagating in the layers. The quantities that are required for the modeling program are the four reflection and transmission matrices. If we partition f(n) so that is a vector of the amplitudes of downgoing waves and is a vector if the amplitudes of upgoing waves we can write the following equation,
This is the block matrix form of the layer matrix operator discussed in the text.
For a downward propagating wavefield incident on the boundary from above I wish to calculate an operator that gives the upgoing reflected wavefield and one that gives the downgoing transmitted wavefield. The equation I need to solve is,
After some manipulation I obtain,
The matrices downrefl and downtrans convert the 3 vector of downgoing wave amplitudes in layer 1 into a three vector of upgoing reflected wave amplitudes in layer 1 and a three vector of downgoing transmitted amplitudes in later 2.
For an upward propagating wavefield incident on the boundary from below I wish to calculate an operator that gives the downgoing reflected wavefield and one that gives the upgoing transmitted wavefield. On solving,
The two operators that I require are,
These operators allow me to calculate the reflected and transmitted wavefields at any horizontal boundary.