Atmospheric visibility plays a crucial role in aerospace, transportation, and environmental monitoring, directly affecting traffic safety and transportation efficiency. In automatic observation, visibility is represented by the meteorological optical range (MOR), defined as the path length through which the luminous flux of a parallel beam of light emitted by an incandescent lamp with a color temperature of 2700 K is attenuated to 5% of its initial value in the atmosphere. In photon counting mode, lidar captures return energy by receiving backscattered photon signals, which consist of both single-scattered and multiple-scattered photons. Single-scattered photons follow definite paths and directly convey target information, while multiple-scattered photons have complex trajectories and carry significant non-target information, increasing uncertainty and noise. To obtain the true atmospheric visibility, we need to calculate the actual extinction coefficient considering only single-scattered photon returns. However, since lidar cannot distinguish between single-scattered and multiple-scattered photons, the extinction coefficient directly derived from lidar return signals is the apparent extinction coefficient, which is influenced by multiple scattering. Therefore, in-depth research on lidar-based apparent extinction coefficient inversion and the multiple-scattering processes in the atmosphere is crucial for accurately calculating the actual extinction coefficient.

We derive the relationships among the actual extinction coefficient, apparent extinction coefficient, and multiple-scattering factor based on the lidar equation and the parameterized lidar equation, laying a theoretical foundation for subsequent analysis. Then, we generate 10000 sets of simulated signals under typical weather conditions and construct a comprehensive dataset by combining these simulated signals with real measurement data. This ensures that our model has broad adaptability and accurately reflects the complexities in practical applications. After that, we perform data preprocessing to enhance the linear correlation between features and labels and train the temporal convolutional network (TCN). This neural network model can accurately estimate the atmospheric apparent extinction coefficient by analyzing lidar return signals. Based on the apparent extinction coefficient estimated by the TCN, we determine an initial scattering free path and conduct multiple-scattering simulations to obtain an initial multiple-scattering factor and simulated photon number. We then calculate the relative error between the simulated photon number and the theoretical photon number derived from lidar return signals. If the relative error exceeds a predetermined threshold, we correct the scattering free path using the multiple-scattering factor and repeat the multiple-scattering simulation. This iterative process continues until the error falls below the set threshold, yielding the final multiple-scattering factor. Finally, we substitute the estimated apparent extinction coefficient and final multiple-scattering factor into the derived relationships to accurately calculate the actual extinction coefficient.

Under low visibility conditions, the visibility calculated by our proposed method shows significant differences compared to the results without considering multiple-scattering effects. Simulation results show that at a visibility of 100 m, the initial accuracy of the multiple-scattering simulation is relatively low, with substantial deviations between the simulated photon number/multiple-scattering factor and theoretical value. However, as the number of iterations increases, the accuracy of the multiple-scattering simulation gradually improves, with the simulated photon number and multiple-scattering factors converging towards the theoretical values (Figs. 8 and 9). The mean actual extinction coefficient calculated by our proposed method is 29.45 km^-1, with a relative error of 1.70% compared to the theoretical value. In contrast, the mean actual extinction coefficient obtained using the Klett algorithm is 25.88 km^-1, with a relative error of 13.61%. Correspondingly, the average visibility calculated by our proposed method is 101.73 m, while the Klett algorithm yields a visibility of 115.76 m. The root mean square errors of the actual extinction coefficients calculated by our proposed method and the Klett algorithm relative to the theoretical values are 10.49 and 18.63, respectively. The calculated results by our proposed method is closer to the theoretical values, and the calculated visibility is more accurate (Fig. 10). At visibility of 800 m, the multiple-scattering photon number is relatively low, and the effect of multiple scattering on the return signal is smaller (Fig. 11). Using our proposed method, only one iteration is required, and the return signal is nearly identical to the theoretical value, while the multiple scattering factor approaches the theoretical value (Figs. 13 and 14). The mean actual extinction coefficient calculated by our proposed method is 3.73 km^-1, while that of the Klett algorithm is 3.68 km^-1. Compared to the theoretical value, the relative errors are 0.26% and 1.60%, respectively, with root mean square errors of 0.27 and 0.30. The average visibility calculated by our proposed method is 803.21 m, while the Klett algorithm gives 814.13 m. For the experimental signals, we select two actual lidar return signals, A and B, with measured visibilities of 1.41 and 4.44 km, respectively (Fig. 16). The visibility calculated by our proposed method for these signals is 1.23 and 5.93 km, while the Klett algorithm calculates 1.26 and 6.06 km. Compared to the measured values, the relative error for signal A is 2.44% for the Klett algorithm and 0.81% for our proposed method, with a reduction of 1.63 percentage point. For signal B, the relative errors are 2.19% for the Klett algorithm and 1.01% for our proposed method, with a reduction of 1.18 percentage point (Figs. 17 and 18). These results demonstrate that our proposed algorithm, which accounts for multiple-scattering effects, offers significant advantages over the Klett method which does not consider multiple scattering, thus validating its effectiveness under multiple-scattering conditions.

We propose an iterative solution algorithm for atmospheric visibility based on the TCN and multiple-scattering simulation. This algorithm is used to solve the actual extinction coefficient and improve the accuracy of visibility inversion under multiple-scattering conditions. It estimates the apparent extinction coefficient through a TCN, then repeats multiple-scattering simulations to obtain multiple-scattering factors, and finally solves for the actual extinction coefficient to obtain atmospheric visibility. The simulation results show that under visibility conditions of 100 and 800 m, the algorithm we propose significantly improves the accuracy of visibility calculation compared to the case without considering multiple scattering. The comparison of measured signals and calculation results also confirms this conclusion. The advantage of our algorithm is that in estimating the apparent extinction coefficient, the TCN effectively avoids the cumulative error caused by the calculation error of the boundary value of the apparent extinction coefficient. In multiple-scattering simulations, iterative correction of the scattering free path is used to gradually improve simulation accuracy, ultimately enabling accurate calculation of multiple-scattering factors and actual extinction coefficients.