The Nth-root stack; theory, applications, and examples

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.