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The helical coordinate

 For many years it has been true that our most powerful signal-analysis techniques are in one-dimensional space, while our most important applications are in multi-dimensional space. The helical coordinate system makes a giant step towards overcoming this difficulty.

Many geophysical map estimation problems appear to be multidimensional, but actually they are not. To see the tip of the iceberg, consider this example: On a two-dimensional cartesian mesh, the function $
\begin{array}
{\vert r\vert r\vert r\vert r\vert}
 \hline
 0 & 0 & 0 &0 \\  \h...
 ... & 1 &0 \\  \hline
 0 & 1 & 1 &0 \\  \hline
 0 & 0 & 0 &0 \\  \hline\end{array}$

has the autocorrelation $
\begin{array}
{\vert r\vert r\vert r\vert} \hline
 1 & 2 & 1\\  \hline
 2 & 4 & 2\\  \hline
 1 & 2 & 1
 \\  \hline\end{array}$.

Likewise, on a one-dimensional cartesian mesh,

the function $
\mathcal b =
\begin{array}
{\vert r\vert r\vert r\vert r\vert r\vert r\vert r\...
 ...rt r\vert r\vert r\vert} \hline
 1&1&0&0& \cdots& 0& 1&1
 \\  \hline\end{array}$

has the autocorrelation $ \mathcal r =
\begin{array}
{\vert r\vert r\vert r\vert r\vert r\vert r\vert r\...
 ...rt r\vert} \hline
 1&2&1&0&\cdots&0&2&4&2&0&\cdots&1&2&1
 \\  \hline\end{array}$.

Observe the numbers in the one-dimensional world are identical with the numbers in the two-dimensional world. This correspondence is no accident.


 
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Stanford Exploration Project
4/27/2004