## 2 Mathematical Formation of the Time-Based Markov Model  
The mathematical grounds of both the scattering-based Markov chain model and the time-based Markov chain model are similar. In the model, a transition matrix is built, and photon transmission results are given by matrix manipulations, such as matrix multiplication and inversion. The major difference between the two models is the definition of what an event is in the Markov formation. Such a difference in definition is depicted in Figure 1. Figure 1a demonstrates the light propagation procedure of the scattering-based model, and Figure 1b shows the same procedure for the time-based model, respectively. In scattering-based models, each photon scattering is considered an event. Therefore, the photon shown in the figure experienced two events before being transmitted. In this formation, the corresponding transition matrix is usually nonzero, since there is always a nonzero probability for a photon to propagate from layer \(i\) to layer \(j\) without scattering during the propagation. In comparison, the time-based model considers that a new event takes place once a certain amount of time has elapsed. For instance, if we define the time during which the photon propagates for one layer, then the photon will experience approximately \(n\) events before transmission if there are \(n\) layers in total. It is also worth noting that the propagation angle will affect the time for a photon to pass through one layer; we just used this illustration for demonstration purposes. In this model, the transition matrix can be established by mostly zero entries. For instance, if the set time interval \(t\) corresponds to the time that a photon propagates for a distance of the thickness of one layer, the photon cannot propagate for more than one layer during this time period, which means only those representing probabilities from the first layer to the second layer are not zero in the transition matrix. With such characteristics, it is possible to reconstruct the optical properties of the turbid slab using linear algebraic methods. Although the time-based model is intrinsically similar to the scattering-based model, two assumptions are made to simplify the establishment of the time-based model. The first assumption is that the photon will be scattered at most once during one time period, and the scattering happens at the end of this time period. The error induced by this assumption can be reduced by finer discretization so that the likelihood of multiple scattering within one time period can be minimized. Assuming the \(OD\) in a layer is \(OD_{\Delta z}\), the probability of multiple scattering can be readily found by Equation (1):  
\[P_{err}=1-\exp(-OD_{\Delta z})-\int_{0}^{OD_{\Delta z}}\exp(-x)\exp(x-OD_{ \Delta z})dx=1-\exp(-OD_{\Delta z})-OD_{\Delta z}\exp(-OD_{\Delta z}) \tag{1}\]  
In this expression, the second term on the right-hand side represents the probability that the photon transmitted a specific layer with an \(OD\) (with the value of \(OD_{\Delta z}\)) without being scattered, and the third term represents the likelihood that this photon is scattered only once in this layer. It can be readily found that the error introduced by this assumption is 1.97%, with an \(OD_{\Delta z}\) = 0.2, and 9%, with an \(OD_{\Delta z}\) = 0.5. Since 100 layers are used in this current model, this error is acceptable, but as the \(OD\) in each layer grows, this error also increases and accumulates with each scattering, which should be taken into account in actual applications. This error can be reduced by using finer spatial meshes or taking a higher scattering order into consideration, which will make the algorithm more complex. The second assumption is that in each layer, the photon starts its propagation at a random location \(z\) within this layer rather than a preset location. This assumption is also utilized in the scattering-based Markov chain model.