Calculation of reflection and transmission operators

The vector of velocities and normal stresses in a layer can be related to the amplitudes of the up- and downgoing waves by a block $3\times3$ operator as shown in the appendix.

$$\pmatrix{E_{11}&E_{12}\cr E_{21}&E_{22}\cr } \pmatrix{{\bf d} \cr {\bf u}\cr} = \pmatrix{ {\bf v}\cr {\bf \sigma_N} \cr}$$

At a horizontal boundary between two layers the vectors ${\bf v}$ and ${\bf \sigma_N}$ are continuous so we can write.

$$\pmatrix{E^1_{11}&E^1_{12}\cr E^1_{21}&E^1_{22}\cr } \pma... ... E^2_{21}&E^2_{22}\cr } \pmatrix{{\bf d}^2 \cr {\bf u}^2 \cr}$$

I wish to calculate four reflection and transmission operators: In the appendix I show how to calculate the four reflection and transmission operators that my algorithm needs for steps three and five.

The operators downrefl and downtrans are the reflection and transmission operators for a downgoing wave incident on an interface. The operators uprefl and uptrans are the reflection and transmission operators for an upgoing wave incident on an interface.