Improving LSTM-based inversion of ultrasonic particle size distribution

In hydraulic engineering, the particle size distribution (PSD) of suspended solids is a key parameter with important implications for both research and practical applications. Conventional approaches for estimating PSD typically rely on solving first-kind Fredholm integral equations. This study presents a PSD prediction method that integrates acoustic physical mechanisms with deep learning. Based on the ECAH theory of sound propagation in multiphase media, we constructed a feature library of ultrasonic attenuation responses and developed a PSD prediction framework using an SSA-optimized LSTM (SSA-LSTM). Experimental samples were classified by particle size through sieving, yielding 15 standard particle-size intervals. A total of 45 experimental samples were then prepared and measured. The dataset was deliberately designed to cover four benchmark distribution types normal, uniform, random, and specially designed non-continuous distributions to enable a comprehensive evaluation of model performance. Ultrasonic attenuation experiments provided both the physical parameters of particles in the suspended sediment solutions and the corresponding attenuation parameters. The SSA-LSTM model was trained using selected feature and label parameters, allowing it to predict PSDs from the input acoustic features. Four suspended-sediment samples with different distributions were then evaluated using the model, and the predictions were compared against PSDs obtained from sieving, random forest, and a standard RNN approach. The coefficient of determination exceeded 0.80 for all samples, and the median particle-size errors were 6.25%, 6.09%, 3.09%, and 2.07%, respectively. These results demonstrate that the proposed method provides a practical and effective alternative for estimating PSD from ultrasonic attenuation.