Modal beamforming array gain

In a previous article T. C. Yang, J. Acoust. Soc. Am. 82, 1736–1745 (1987), range and depth were successfully estimated by modal beamforming. In this article, the modal beamforming array gain is derived in closed form for a vertical array of N hydrophones. For the case of white noise and a long well-populated vertical array, the array gain nearly equals 10 log N. For the case of colored noise and a finite aperture, finite element vertical array, the array gain degrades from the theoretical by an amount determined by the mode depth amplitudes evaluated at the source and receiver depths and the modal covariance matrix of the noise. Numerical estimates of array gain are given for the FRAM IV vertical array for a colored noise environment in the Arctic sound channel. Approximately 1-dB array gain degradation at 47 Hz is found. Proper array configuration and deployment depth are not only important for optimizing array gain but also critical for maximizing the input signal-to-noise ratio on a vertical array, which together with the array gain determine the array output signal-to-noise ratio.