Can Stacking Absolute Values Give Better Signal-to-noise Ratio than Raw Signal Stacking?

It is common knowledge that stacking suppresses uncorrelated noise increasing signal-to-noise ratio (SNR). A coherent signal is summed constructively but random noise cancels out. The SNR on the stack increases with the square root of the number of the stacked traces. Stacking of incoherent signal is less effective and it can reduce the SNR. Incoherency is common for microseismic events due to polarity reversals. In such cases stacking of the absolute values of the traces is used. But the stacking of absolute values is not as effective in noise suppression as stacking positive and negative values. Although it is true in most of the practical cases of low SNR values on individual traces, it is not true in general. We present a synthetic case study, and theoretical analysis, which shows that for some definitions of the SNR this phenomenon appears.