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The building blocks

The field variable transformation
\begin{displaymath}
\pmatrix{ 
 v \cr
 \sigma 
 }
 =
 \pmatrix{ 
 1 & 1 \cr
 \surd(\rho/s) & - \surd(\rho/s)
 }
 \pmatrix{ 
 v^{+} \cr
 v^{-}
 }\end{displaymath} (1)
and its inverse
\begin{displaymath}
\pmatrix{ 
 v^{+} \cr
 v^{-} 
 }
 = 
 \frac{1}{2} 
 \pmatrix...
 .../\rho) \cr
 1 & -\surd(s/\rho)
 }
 \pmatrix{ 
 v \cr
 \sigma
 }\end{displaymath} (2)
and the translation equation
\begin{displaymath}
\pmatrix{ 
 v^{+}(z) \cr
 v^{-}(z) \cr 
 }
 = 
 \pmatrix{ 
 ...
 ...ega z \surd(\rho s)}
 }
 \pmatrix{ 
 v^{+}(0) \cr
 v^{-}(0) 
 }\end{displaymath} (3)
concatenate to form a basic building block  
 \begin{displaymath}
\pmatrix{
 v(z) \cr
 \sigma(z) \cr
 }
 =
 \pmatrix{
 \cos(\o...
 ...s(\omega z \surd(\rho s))
 }
 \pmatrix{
 v(0) \cr
 \sigma(0)
 }\end{displaymath} (4)
which may also be written
\begin{displaymath}
\pmatrix{
 v(z) \cr
 \sigma(z) \cr
 }
 =
 \pmatrix{
 \cos(\o...
 ...&
 \cos(\omega z \lambda)
 }
 \pmatrix{
 v(0) \cr
 \sigma(0)
 }\end{displaymath} (5)
and in case slowness is a constant, then we can re-write this equation in a $\cal Z$-transform notation
\begin{displaymath}
\pmatrix{
 v(z) \cr
 \sigma(z) \cr
 }
 =
 \frac{1}{2 {\cal Z...
 ...\cal Z}) & (1 + {\cal Z})
 }
 \pmatrix{
 v(0) \cr
 \sigma(0)
 }\end{displaymath} (6)
If we denote the RHS operator above by ${\cal P}({\cal Z})$, note the property
\begin{displaymath}
{\cal P}^{2}({\cal Z}) = {\cal P}({\cal Z}^{2})\end{displaymath} (7)

previous up next print clean
Next: Taylor series expansion Up: DEVELOPMENT Previous: DEVELOPMENT
Stanford Exploration Project
11/17/1997