In this paper a technique based on spectral factorization for restoring the phase of incoherent bottom sediment reflection coefficient measurements is presented, so that by Fourier transformation one can then obtain the minimum phase impulse response at each grazing angle. The method is developed and discussed 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). 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. For instance, in both cases one can see clear, matching arrivals from each layer boundary at angles above critical. Finally the method is demonstrated on experimental reflection loss inferred from ambient noise measurements at three moored VLA sites and one VLA drift track in the Mediterranean Sea. Convincing angular variation of the impulse response is shown for the moored sites. Sub-bottom profiles (impulse response vs position) are shown for the drift track demonstrating that one can survey with just a single directional receiver.