FEATURES OF 1-D THAT APPLY TO MANY DIMENSIONS

Time-series analysis is rich with concepts that the helix now allows us to apply to many dimensions. First is the notion of an impulse function. Observe that an impulse function on the 2-D surface of the helical cylinder maps to an impulse function on the 1-D line of the unwound coil. An autocorrelation function that is an impulse corresponds both to a white (constant) spectrum in 1-D and to a white (constant) spectrum in 2-D. A causal filter in one dimension has a curious shape on the two-dimensional helix. In one dimension, the causal filter has zeros before the ``1.0'' and various values after it. Say the nonzero filter coefficients on the cylinder lie within a short distance (two lags) from the ``1.0''. Extract the little 2-D patch (which is the end of the 1-D filter). I display it reversed on both axes so the reader can envision it as crosscorrelation, first moving down the first seismogram and then down the next.  

$$\begin{array} {ccccc} \begin{array} {ccc} u & g & b \\ s ... ... \cdot &\cdot & 0 \\ \cdot &\cdot & 0 \end{array}\end{array}$$ (6)

where a,b,c,...,u are adjustable coefficients. Thus we conclude that the 2-D analog of a 1-D causal filter has its abrupt beginning along the side of the 2-D filter.

A special causal filter that unites many well established concepts in time-series analysis is the prediction-error-filter (PEF). A 2-D PEF, like a 1-D PEF, is a causal filter with adjustable coefficients as in the array (6), that are adjusted to minimize the filter's output energy (for a particular input signal). That the 2-D PEF should have its beginning along a side (instead of at a corner) is an abstract idea that I have always found difficult to teach clearly, until I fell upon the helix explanation.

Here is a brief summary of important ideas in time-series analysis that the helix makes applicable in higher dimensions:

• The filter (a, b, c, ..., u) is the negative of the prediction filter. The filter (1, a, b, c, ..., u) is the prediction-error filter.
• The method of least-squares is used to find the prediction filter. This is also called ``autoregression''.
• Textbooks Claerbout (1998b) show that the spectrum of the output of the PEF tends towards whiteness as the filter length increases. Thus the spectrum of the PEF itself tends to the inverse of that of its input. (Noncausal filters do not have white outputs and cannot be used recursively.)
• A time series can be decomposed into random impulses (white spectrum) convolved with a natural wavelet that is the inverse of the PEF.
• For any power spectrum, there is a causal wavelet (with that spectrum) that can be found by ``spectral factorization''. In the frequency domain this is known as the Kolmogoroff 1939 method.
• The PEF has the property of ``minimum phase'' which means that both it and its convolutional inverse are causal. Thus, we can design stable multidimensional recursive filters as we do in one dimension.
• Stable filters can be modeled as layered media where waves resonate among reflection coefficients bounded in absolute value by unity. Such models help in PEF estimation by the Burg 1975 method.

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. The many features of 1-D theory outlined above are now awaiting multidimensional application.