## 3 Results and Discussion
### Time-Based Markov Chain Multiple Scattering Inversion  
The time-based model can utilize temporal resolved transmitted photons as reconstruction inputs, which is more desirable compared with finding the scattering order (i.e., how many times a specific photon has been scattered before transmission) of the transmitted photons experimentally. Furthermore, for the backscattered photons, there is a maximum penetrating distance during a specific time period. For instance, within t = 0-20\(\Delta z/c\), the furthest distance a back-transmitted photon can reach is 10\(\Delta z\), which requires this photon to propagate perpendicular to the entrance plane (\(\theta=0^{\circ}\)), scattered at \(z=10\Delta z\), then return to the entrance plane with a transmitted angle of \(\theta=180^{\circ}\). Therefore, the optical properties  
Figure 4: Time-based Markov chain model predictions compared with the Monte Carlo simulation results. (**a–h**) Photons transmitted within different ranges of time. within \(z=0\)-\(10\Delta z\) can be inferred by the transmitted angular distribution obtained during \(t=0\)-\(20\Delta z/c\). In a similar approach, the optical properties within \(z=10\)-\(20\Delta z\) can be found by the transmitted angular distribution obtained during \(t=20\)-\(40\Delta z/c\), together with the known properties in \(z=0\)-\(10\Delta z\). In an iterative manner, the spatially resolved optical properties of the turbid slab can be reconstructed without brute force trial and error of all possible combinations. The schematic of such a method has been depicted in Figure 5. The transition matrix to be solved was initially a zero matrix. With the angular distribution data during the first time period (Iter 1), the optical properties of the first set of layers (10 in this work) could be found by minimizing the difference of the simulated angular distribution and the measured angular distribution, assuming the optical properties in the first set of layers were uniform. The entries at other locations (for instance, in Iter 2) had no contribution to the measurements, so the entries in these locations could be set to 0. After fixing the optical properties in the first set of layers (in this work, the droplet diameter), the corresponding entry values (denoted by \(x\) in Figure 5) were updated and fixed in the iterations that followed. Next, for Iter 2, we performed the same procedures and updated the corresponding values (denoted by \(y\)), and so on so forth. Eventually, we solved for the final set of layers and updated the rest of the entries (denoted by \(z\)). By using this scheme and backscattered photons (backscattered photons were more readily accessible in the experiments), we successfully decoupled the connections between different layers through multiple scattering and substantially reduced the computational cost compared with using Monte Carlo simulation or brute force calculations. To examine such a capacity of the time-based model, we established a synthetic slab with a customized water diameter distribution for reconstruction. The numerical setup of this reconstruction effort is depicted in Figure 6. The synthetic phantom was in agreement with the one in our previous work [18] (i.e., the total \(OD\) of the turbid slab was 5.0). The slab was discretized into 10 layers with the same \(OD_{\Delta z}\) and \(OD\). The diameter of the water droplets in layers 1-3 and 8-10 was 1.3 \(\upmu\)m, and for layers 4-7 it was 1.6 \(\upmu\)m. In this simulation, we had 11 candidate droplet diameter sizes (1.0-2.0 \(\upmu\)m, with a step length of 0.1 \(\upmu\)m). Therefore, there were 1110 different possible diameter spatial distributions. Solving for the true diameter distributions by brute force could be impossible in the sense of computational cost, even with the Markov chain models.