Direct solution of the backscatter equation for ABS instruments through a Riccati reformulation

Measurements of suspended sediment concentration profiles in aquatic environments may be made by remote sensing acoustic backscatter (ABS) instruments. These typically operate at MHz frequencies and narrow beamwidths (1–2°). The backscatter equation for these instruments is not in a closed form. It may be solved by iteration. A direct general solution has previously been presented by recasting into the form of a Bernoulli equation [Thosteson, E.D., Hanes, D.M., 1998. Journal of the Acoustical Society of America 104 (2), 820–830 (Pt. 1)]. It is shown that the direct solution is also enabled by recasting as a Riccati equation. The direct solution obtained by the Riccati formulation has the same form as given previously, but its derivation clarifies inconsistencies in previous work which have disguised its potential usefulness.