Objective Due to the cavitation of the propeller and the breaking waves against the hull, a bubble curtain is generated during the ship's navigation. The presence of this bubblecurtain creates optical differences between the ship’s wake area and the surrounding bubble-free seawater. The distribution of wake bubbles generated by ships with ship displacement and sailing speed, which provides important basis for the identification and tracking of ships. The light scattering method has become an effective research tool in this field due to its advantages of high inversion accuracy, wide measurement range, and non-interference with the bubble structure. To further improve the ill posedness and noise sensitivity in light scattering inversion,the traditional Chahine algorithm was improved. The enhanced algorithm demonstrated significant improvements in inversion accuracy and noise resistance.

Methods An improved Chahine algorithm based on adaptive regularization operator is proposed, which constructs an adaptive regularization matrix, preserves larger singular values and corrects smaller singular values to enhance the stability and noise resistance of bubble distribution inversion. A set of underwater laser diffraction devices for bubble distribution inversion was built, and the experiment used a bubble machine to simulate the wake bubble curtain of a ship. Combining simulation and experimental verification to validate the effectiveness of the method.

Results and Discussions Both simulation and experimental results have confirmed that the improved algorithm has high inversion accuracy. Under 10% high noise conditions, the inversion errors of the improved algorithm for standard normal and Rosin–Rammler distribution bubble groups are 7.95% and 9.70%, respectively, which are significantly lower than the traditional Chahine algorithm (78.81% and 52.46%) and the regularized Chahine algorithm (12.14% and 15.80%) (Tab.1). In addition, the improved algorithm effectively suppressed the pseudo peaks and oscillations caused by high noise (Fig.6). Compared with other algorithms, under the same noise level, the improved algorithm has inversion errors of 9.78% and 7.52% for the two distributions, respectively, while the errors of the regularized Landweber and Projection algorithms both exceed 17% (Tab.2). In the experimental verification, the standard particle GBW(E)120030 was used for quantitative testing, and the peak inversion error of the improved algorithm was only 2.39%, and the inversion distribution was highly consistent with the actual distribution of particles; The experimental results under different airflow conditions are also consistent with expectations (Fig.12).

Conclusions This paper proposes a Chahine iterative algorithm based on adaptive regularization operator to invert and simulate bubbles that conform to standard normal distribution and R-R distribution under different noises. Based on the algorithm proposed in this paper, an underwater laser device for bubble distribution measurement is built to measure the distribution of bubbles generated in the experiment. Simulation and experimental results show that the improved algorithm in this paper can obtain more accurate inversion results. Under high noise levels, the relative root mean square errors of inversion for normal distribution and R-R distribution bubbles under 10% noise interference are 7.95% and 9.70%, respectively. The inversion error is relatively small and the inversion effect is better than traditional Chahine and Tikhonov Chahine algorithms. Compared with Tikhonov Landweber and Projection algorithms in independent inversion algorithms under high noise levels, the results show that our algorithm has better noise resistance and stability. The measurement system built through experiments was used to invert standard particles, and the inversion error of standard particles was 2.39%, which verified the effectiveness of the underwater laser device. Finally, the inversion of bubbles was carried out to verify the practicality and effectiveness of the device proposed in this paper for the inversion of bubble distribution, providing an effective means for the inversion of bubble distribution in ship wake.