The particle size distribution (PSD) of aerosol particles is regarded as an important indicator in fields such as industrial emissions monitoring, global warming research, and medical research. In most aerosol measurement techniques, the dynamic light scattering (DLS) method has advantages under certain measurement conditions because the refractive index of aerosol particles does not need to be determined. When using the autocorrelation function (ACF) of the measured particle intensity to invert the PSD in flowing aerosol DLS, flow velocity is a necessary parameter. However, the measurement error of flow velocity leads to errors in the coefficient matrix of the inversion equation. The Tikhonov regularization (LS-Tik) inversion method based on the classical least squares model only considers errors in the measurement correlation function and does not consider those in the coefficient matrix. Meanwhile, the increase in the flow velocity term exacerbates the ill posedness of the particle size inversion equation, leading to an increase in PSD sensitivity to noise and poorer anti-interference performance, resulting in lower solution accuracy. To address this issue, a Tikhonov?total variation (TV) hybrid regularization (TLS-Tik-TV) inversion method based on total least squares (TLS) is proposed, which considers both the coefficient matrix and measurement correlation function errors, to reduce the ill posedness of the inversion equation, minimize noise sensitivity, and improve inversion accuracy.

First, we established a TLS model that considered both coefficient matrix and measurement correlation function errors. Subsequently, based on the TLS model, the traditional LS-Tik was combined with the noise-resistant TV regularization to establish the TLS-Tik-TV inversion algorithm. To verify the performance of the algorithm, Johnson’s SB function was used to generate simulation data. Different flow rates were selected at noise levels of 10^-2 and 10^-3, and the LS-Tik and TLS-Tik-TV algorithms were used to simulate unimodal and bimodal particles. Two performance indicators, peak error (E_p) and distribution error (E_r), were introduced to evaluate the accuracy of PSD inversion, and the conclusions of the simulation experiment were verified by inverting the measured data.

Compared with the LS-Tik algorithm, the established TLS-Tik-TV algorithm (Fig. 1) has smaller peak and distribution errors, stronger bimodal resolution, and higher inversion accuracy. When simulating particles with a unimodal distribution, at the same particle size, the peak and distribution errors of both the LS-Tik and TLS-Tik-TV methods gradually increase as the flow rate increases, manifesting as a left shift in the peak position and PSD broadening. However, the amplitude of the left shift in the peak position and PSD broadening of TLS-Tik-TV is smaller than those of LS-Tik (Figs. 2?4). At the same particle size and flow rate, TLS-Tik-TV exhibits smaller peak and distribution errors compared to LS-Tik (Table 2). For example, for 601 nm particles, TLS-Tik-TV can reduce the peak and distribution errors by up to 0.033 (Table 2). For particles with a simulated bimodal distribution, under the same set of bimodal distributions, as the flow velocity increases, both methods result in a left shift in the peak position and a decrease in the peak height in the PSD obtained by inversion. However, at a selected flow rate of 0?1.5 m/s, for the 143/584 nm and 234/700 nm particle systems, the TLS-Tik-TV peak position shifts to the left less and the peak height decreases less compared to those of LS-Tik (Figs. 5 and 6). At the same particle size and flow rate, the peak and distribution errors of TLS-Tik-TV are generally lower than those of LS-Tik (Table 3). The inversion results of the measured particles show that TLS-Tik-TV exhibits smaller peak and distribution errors and stronger bimodal resolution (Figs. 8 and 9, Table 4), revealing the same trend as the inversion results of simulated data.

A TLS-Tik-TV inversion algorithm is proposed. This algorithm uses the TLS model to balance the coefficient matrix and correlation function errors and combines Tikhonov and TV regularization to reduce the ill posedness of the inversion equation two-fold and improve the inversion accuracy. Under different flow rates and noise levels, the traditional LS-Tik and TLS-Tik-TV methods were used to invert simulated particles. The results show that compared with LS-Tik, TLS-Tik-TV exhibits smaller peak and distribution errors, stronger bimodal resolution, and stronger noise resistance. The inversion results of 584 nm unimodal and 243/825 nm bimodal measured particles show that compared to LS-Tik, TLS-Tik-TV exhibits smaller peak and distribution errors, stronger bimodal resolution, and can reduce peak errors by up to 0.027 and 0.123/0.091, as well as distribution errors by 0.032 and 0.038, respectively. Therefore, TLS-Tik-TV is superior to LS-Tik, and the inversion of the measured particles also verifies the conclusions of simulated data.