Multi-frequency acoustic backscatter inversion for measuring multi-class sediment suspensions

Multi-frequency acoustic backscatter is widely used to estimate suspended sediment concentration and particle size, but its application to heterogeneous suspensions is complicated by the nonlinear and noise-sensitive nature of sonar equation inversion. We revisit the sonar equation for suspensions composed of multiple sediment classes insonified at several frequencies, reformulate it in matrix form, and derive an integral expression that enables a direct inversion for sediment concentration. A nonnegative least squares solver is applied to recover the concentration of each sediment class while minimizing the residual norm also allows the estimation of sediment sizes when the number of frequencies exceeds the number of classes. By analyzing the spectra of the backscattered signal, we separate genuine concentration fluctuations from statistical noise associated with changes in particle configuration within the insonified volume. This distinction enables a simple temporal filtering prior to inversion, improving stability and providing direct control over information loss and interpretability. Validation against independent high-quality laboratory experiments demonstrates accurate recovery of both single- and dual-class concentration profiles. The proposed approach offers a practical framework for studying suspended sediment transport and cohesive-noncohesive interactions in fluvial, estuarine, and coastal environments.