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Free surface boundary conditions

At the free surface we have to consider different boundary conditions. If an upcoming wave hits the free surface it will be measured as a particle velocity vector at the surface. Reflected downgoing waves will also be produced. The free surface boundary condition is that there are no normal stresses on the free surface. For a wave hitting the free surface we have,

\begin{displaymath} \pmatrix{ E_{11} & E_{12} \\ E_{21} & E_{22} \\ } \cdot \... ...{{\bf d} \cr {\bf u}\cr} = \pmatrix{{\bf v} \cr {\bf 0}\cr}\end{displaymath}

This gives the solutions,
\begin{eqnarraystar} {\bf d} = surfrefl \cdot {\bf u} \\ {\bf v} = surfdisp \cdot {\bf u}\end{eqnarraystar}

where,
\begin{eqnarraystar} surfrefl = - E_{21}^{-1}E_{22} \\ surfdisp = E_{12} - E_{11}E_{21}^{-1}E_{22}.\end{eqnarraystar}

For a traction applied to the free surface, there is no upgoing wave. To solve for the downgoing waves produced by a source traction, we have

\begin{displaymath} \pmatrix{ E_{11} & E_{12} \\ E_{21} & E_{22} \\ } \cdot \... ... \cr {\bf 0}\cr} = \pmatrix{{\bf v} \cr {\bf \sigma}_N \cr }\end{displaymath}

This gives the solutions,
\begin{eqnarraystar} {\bf d} = tracdown \cdot {\bf \sigma}_N \\ {\bf v} = tracdisp \cdot {\bf \sigma}_N\end{eqnarraystar}
where,
\begin{eqnarraystar} tracdown = E_{21}^{-1} \\ tracdisp = E_{11} E_{21}^{-1}.\end{eqnarraystar}

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Stanford Exploration Project
12/18/1997