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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
| ![\begin{displaymath}
\pmatrix{ \vec{\bf u}\cr \mbox{\boldmath$\tau_N$} \cr} = F \cdot \vec{\bf w}\end{displaymath}](img54.gif) |
(10) |
where the elements of F are
| ![\begin{displaymath}
F_{ij} = \left(
\begin{array}
{cccccc}
v_x^1 & v_x^2 & v_x^3...
...zz}^3 &\tau_{zz}^4 &\tau_{zz}^5 &\tau_{zz}^6 \end{array}\right)\end{displaymath}](img55.gif) |
(11) |
and the elements of the vector
are a function of the wave
amplitudes, as follows:
| ![\begin{displaymath}
w_j(z) = S_j e^{ i\omega(t-\vec{\bf x} \cdot \vec{\bf p}^j)}\end{displaymath}](img57.gif) |
(12) |
To calculate the amplitude partitioning at an interface between two
layers we equate the displacement-stress vector across the interface,
thus:
| ![\begin{displaymath}
F^{top} \cdot \vec{\bf w}^{top} = F^{bottom} \cdot \vec{\bf w}^{bottom}\end{displaymath}](img58.gif) |
(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
| ![\begin{displaymath}
F^{top} \cdot \vec{\bf S}^{top} = F^{bottom} \cdot \vec{\bf S}^{bottom}\end{displaymath}](img59.gif) |
(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
| ![\begin{displaymath}
\pmatrix{F^{top}_{11} & F^{top}_{12}\cr
F^{top}_{21} & E^{t...
...\pmatrix{\vec{\bf S}^{bottom}_D \cr
\vec{\bf S}^{bottom}_U\cr}\end{displaymath}](img63.gif) |
(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
| ![\begin{displaymath}
\pmatrix{F^{top}_{11} & F^{top}_{12}\cr
F^{top}_{21} & E^{t...
...22}\cr }
\pmatrix{\vec{\bf S}^{bottom}_D \cr
\vec{\bf 0} \cr}\end{displaymath}](img64.gif) |
(16) |
After some manipulation, we obtain Nichols (1991)
| ![\begin{eqnarray}
\vec{\bf S}^{top}_U & = & R_D \cdot \vec{\bf S}^{top}_D \\ \vec{\bf S}^{bottom}_D & = & T_D \cdot \vec{\bf S}^{top}_D \nonumber\end{eqnarray}](img65.gif) |
(17) |
| |
where
| ![\begin{eqnarray}
R_D & = & ( F^{bottom}_{21}(F^{bottom}_{11})^{-1}F^{top}_{12}
...
...top}_{11} - F^{top}_{11}(F^{top}_{22})^{-1}F^{top}_{21})
\nonumber\end{eqnarray}](img66.gif) |
(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