The helical coordinate

The helical coordinate

Jon Claerbout

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 multidimensional space. The helical coordinate system makes a giant step toward overcoming this difficulty.

Many geophysical map estimation applications appear to be multidimensional, but in reality they are one-dimensional. To see the tip of the iceberg, consider this example: On a 2-dimensional Cartesian mesh, the function $$\begin{array}{\vert r\vert r\vert r\vert r\vert} \hline 0 ... ... 0 & 1 & 1 &0 \\ \hline 0 & 0 & 0 &0 \\ \hline \end{array}$$

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 1-dimensional cartesian mesh,

the function $$\begin{array}{\vert r\vert r\vert r\vert r\vert r\vert r\ver... ... r\vert} \hline 1&1&0&0& \cdots& 0& 1&1 \\ \hline \end{array}$$

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

Observe the numbers in the 1-dimensional world are identical with the numbers in the 2-dimensional world. This correspondence is no accident.