Covariance Functions and Related Statistical Properties of Acoustic Backscattering from a Randomly Rough Air-Water Interface.

An experimental and theoretical study of the scattering of sound from the surface of a freshwater lake was performed. The study was limited to backscattered sound (reverberation) from a pulsed sound source. The experimental study employed nine vertically separated receivers and four horizontally separated receivers in order to examine the spatial and temporal covariance of the backscattered sound for both orientations simultaneously. Approximately 1000 narrowband reverberation returns were generated by repetitively projecting a 100 (mu)s CW pulse centered at 80 kHz at the wind-roughened surface of a freshwater lake at a 10.5 degree grazing angle. The returns were sampled digitally for a 10 ms interval during which surface reverberation was predominant, and were formed into ensembles at each sample time. Various univariate moments and the time-difference component of the covariance were computed experimentally utilizing ensemble averages. The statistical validity of the ensembles was determined by testing each ensemble for randomness and homogeneity. The ensembles were also tested for normality and found to be non-gaussian, in contrast to previous studies. The experimental results were compared to a theoretical model developed by D. Middleton and others, based on point scattering and Poisson statistics. The reverberation is modeled as weak scattering from random point sources representing inhomogeneities at the air-water interface of an otherwise homogeneous medium. In the present study the point source was assumed to be a perfect point reflector distributed uniformly over the surface. The implementation of the model took into account most of the relevant geometrical parameters (spatial distribution of sensors, grazing angle, range) and acoustic parameters (frequency, pulse length, directionality, aperture response, bandwidth, signal spectrum) but did not take into account environmental parameters (surface waveheight, wave spectrum, wind direction). Significant differences were observed between the covariances of the vertical and horizontal arrays. The envelope of the covariance between the vertical receivers maintained a significant level at much larger separations than the horizontal receivers. The phase of the vertical covariance was found to change linearly with time, resulting in a slow oscillation of the covariance with time, while the phase of the horizontal covariance was constant (and non-zero). The vertical covariance was also shown to depend on the location of the elements on the array. The theory, which had been applied previously only to horizontal arrays, was extended to a vertical array. The theory correctly predicted the dependence of the horizontal and vertical covariance on the time of observation and on the time-difference of the observations. In addition, it predicted the dependence of the envelope of the covariance on horizontal separation of the receivers. The change of the phase of the vertical covariance with time was also predicted by theory. However, the theory failed to predict the dependence of the envelope of the covariance on vertical separation and the non-zero phase of the horizontal covariance. The two failures of the theory were shown to most likely be the result of the failure to include the environmental parameters into the implementation of the model. Ways to include these environmental parameters were suggested.