Spectral factorization is shown to restore the phase of an incoherent layered sediment reflection coefficient so that its Fourier transform is the minimum phase impulse response at each angle. The method requires the reflection coefficient to be known over a range of frequencies and the grazing angles in question to be above critical. It is developed here in the context of another recently established technique for extracting the seabed’s plane wave reflection coefficient from ambient noise data measured on a moored or drifting vertical array (VLA). Thus it offers the possibility of sub-bottom profiling from a single platform with no sound source. Limitations of the phase restoration method are discussed and, using modeled data, comparisons are made between the “true” impulse response derived from the known complex reflection coefficient and the result of applying spectral factorization to the absolute value of the reflection coefficient. The method is also demonstrated on experimental reflection loss inferred from ambient noise measurements at three moored VLA sites and one VLA drift track in the Mediterranean Sea. Sub-bottom profiles (impulse response versus position) are shown for the drift track demonstrating that one can indeed survey with only a single directional receiver. The technique appears to perform well when compared with other profiling techniques and published results.