## 3 Results and Discussion  
### Numerical Verification against Monte Carlo Simulations  
In this section, we compare the performance of the time-based Markov chain model against the Monte Carlo simulation model. Although it would be more desirable to directly compare the results from the time-based model to the experiments, matching all the parameters in such experimental efforts can be challenging. Therefore, we used the Monte Carlo algorithm in [4] as the benchmark in this work, which was validated against a few experiments under multiple scattering conditions. In the Monte Carlo model, the trajectory and scattering events of each photon were determined by randomly generated numbers and then tracked and monitored by the algorithm. Under each condition, 100 million photons were sent. Results from the scattering-based model will also be presented, but the scattering-based Markov model cannot estimate temporally resolved angular distributions for transmitted photons, so we only compared the results of the time-based Markov chain model and the Monte Carlo model when applicable. Figure 2 compares the total angular distributions under different _OD_s when using different algorithms. The phase functions used were in agreement with those in our previous work [18]. The external light wavelength used in this simulation was set 532 nm, which is usually seen from Nd:YAG laser sources for diagnostics purposes. The scattering medium was water, and the ambient medium was air. The water droplet diameter ranged from 1.0 \(\upmu\)m to 2.0 \(\upmu\)m (a diameter resolution of 0.1\(\upmu\)m was used). The Mie scattering theorem was adopted to calculate multiple scattering characteristics. In Figure 2, and in the rest of the simulation, a uniform _OD_ distribution was used, and the phase function from the 1.0 \(\upmu\)m water droplets was adopted. The turbid slab was assumed to be infinitely large, and the optical properties of the slab were uniform across the plane perpendicular to the thickness direction (2). All photons entered the slab from the same point and initially propagated in the \(z\)-direction. The turbid slab was discretized into 100 layers in the \(z\)-direction, and the propagation angle \(\theta\) was discretized into 180 bins. Therefore, the dimensions of the transition matrix were 18,000 \(\times\) 18,000, which was the same as in the previous work of the authors. For more detailed interpretation of the results shown in Figure 2, each data point represents the probability that the photon is transmitted in the propagation angle \(\theta\). For instance, the value at \(\theta\) = 45\({}^{\circ}\) reflects the percentage of photons transmitted with 45\({}^{\circ}\)\(<\)\(\theta\)\(<\) 46\({}^{\circ}\). Ballistic photons were excluded (photons without scattering) in the figure, and the total percentage plotted from \(\theta\) = 0\({}^{\circ}\) to \(\theta\) = 180\({}^{\circ}\) (including ballistic photons) equaled 1.0. In the Monte Carlo simulations, such probabilities could be obtained by counting individual photons statistically and then dividing them by the total number of photons simulated by the Monte Carlo model. Although it was not the purpose of this work to compare the computational cost and accuracy, since the time-based model was mainly designed for reconstruction purposes, it is worth noting that the computational cost of the time-based model was around 5 min, and it took about 60 min for the Monte Carlo algorithm to simulate 100 million photons under  
Figure 2: Total angular distribution of transmitted photons under different total optical depths (ODs). (**a**) Time-based model results compared with the scattering-based model results; (**b**) Time-based model results compared with the Monte Carlo simulation results. \(OD=1.0\) conditions.