Reconstruction of the frequency-wavenumber spectrum of water waves with an airborne acoustic Doppler array for non-contact river monitoring

This work presents a novel method to reconstruct the frequency-wavenumber spectrum of water waves based on the complex acoustic Doppler spectra of scattered sound measured with an array of microphones. The reconstruction is based on a first-order small-roughness-amplitude expansion of the acoustic wave scattering equation, which is discretized and inverted by means of a singular value decomposition. An analogy of this approach to the first-order Bragg scattering problem is demonstrated by means of a stationary phase expansion. The approach enables the reconstruction of the dispersion relation of water waves when the ratio between roughness height and acoustic wavelength is less than 0.1, and when the surface wavelength is larger than 1/2 of the acoustic wavelength. The method is validated against synthetic data and data from laboratory and field experiments, to demonstrate its applicability to two- and three-dimensional complex patterns of water waves, and specifically to the surface deformations that arise naturally in a turbulent open-channel flow. Fitting the reconstructed data with the analytical dispersion relation enables the non-contact estimate of the underlying flow velocity for hydraulic conditions where the coexistence of different types of turbulence-forced and freely propagating water waves would limit the accuracy of standard non-contact Doppler velocimetry approaches, paving the way for robust and accurate non-contact river monitoring using acoustics.