Boundary conditions at a horizontal interface

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, $( v_1, v_2, v_3, \sigma_{31}, \sigma_{32}, \sigma_{33} )$ is conserved across a horizontal interface.

This vector can be written as

$$\pmatrix{ {\bf v}\cr {\bf \sigma_N} \cr} = E \cdot f ,$$

where the elements of E are,

$$E_{jn} = a_j(n) \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ j=1,2,3\ \ \ n=1,6$$

$$E_{jn} = - c_{(j-3)3lm} [ a_m(n) p_l(n) + a_l(n) p_m(n) ]/2 \ \ \ \ \ \ \ j=4,5,6\ \ \ n=1,6$$

and the elements of f are a function of the wave amplitudes,

$$f_n = f(n) i \omega e^{ i\omega(t-{\bf x} \cdot {\bf p}(n))}$$

To calculate the amplitude partitioning at an interface between two layers we need to solve the equation,

$$E^1 \cdot f^1 = E^2 \cdot f^2.$$

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 f_n are the same in both layers and we can write,

$$E^2 \cdot f^2(n) = \cdot E^1 \cdot f^1(n).$$

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 $3\times3$ reflection and transmission matrices. If we partition f(n) so that ${\bf d} = ( f(1),f(2),f(3) )$ is a vector of the amplitudes of downgoing waves and ${\bf u} = (f(4),f(5),f(6))$ is a vector if the amplitudes of upgoing waves we can write the following equation,

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

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,

$$\pmatrix{E^1_{11}&E^1_{12}\cr E^1_{21}&E^1_{22}\cr } \pma... ...r E^2_{21}&E^2_{22}\cr } \pmatrix{{\bf d}^2 \cr {\bf 0} \cr}$$

After some manipulation I obtain, where,

The $3\times3$ 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,

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

The two operators that I require are, These are the reflection and transmission operators for upgoing waves.

These operators allow me to calculate the reflected and transmitted wavefields at any horizontal boundary.