Anomalous signed passive fathometer impulse response when using adaptive beam forming

The impulse response of the seabed can be extracted from sea surface ambient noise by cross-correlating the time series from an upward and a downward steered beam. When the steering for each beam is standard minimum variance adaptive beam forming it has been found that the impulse response for significant echoes appears to have the same amplitude but opposite sign. A mathematical explanation is offered for this strange phenomenon. Crucial contributing factors are that the cross-spectral density matrix for the vertical array typically consists of the sum of a Toeplitz matrix and a much weaker Hankel matrix and that it is ill-conditioned.