Article,
The Nth-root stack: Theory, applications, and examples
Geophysics, 51 (10): 1879--1892 (Oct 1, 1986)
DOI: 10.1190/1.1442045
Abstract
Multichannel geophysical data are usually stacked by calculating the
average of the observations on all channels. In the Nth-root stack,
the average of the Nth root of each observation is raised to the
Nth power, with the signs of the observations and average maintained.
When N = 1, the process is identical to conventional linear stacking
or averaging. Nth-root stacking has been applied in the processing
of seismic refraction and teleseismic array data. In some experiments
and certain applications it is inferior to linear stacking, but in
others it is superior. Although the variance for an Nth-root stack
is typically less than for a linear stack, the mean square error
is larger, because of signal attenuation. The fractional amount by
which the signal is attenuated depends in a complicated way on the
number of data channels, the order (N) of the stack, the signal-to-noise
ratio, and the noise distribution. Because the signal-to-noise ratio
varies across a wavelet, peaking where the signal is greatest and
approaching zero at the zero-crossing points, the attenuation of
the signal varies across a wavelet, thereby producing signal distortion.
The main visual effect of the distortion is a sharpening of the legs
of the wavelet. However, the attenuation of the signal is accompanied
by a much greater attenuation of the background noise, leading to
a significant contrast enhancement. It is this sharpening of the
signal, accompanied by the contrast enhancement, that makes the technique
powerful in beam-steering applications of array data. For large values
of N, the attenuation of the signal with low signal-to-noise ratios
ultimately leads to its destruction. Nth-root stacking is therefore
particularly powerful in applications where signal sharpening and
contrast enhancement are important but signal distortion is not.
