## 1 Introduction  
There is an increasing need for interpreting light scattering signals from turbid media, since scattering signals can contain crucial information about the optical characteristics of the media. The scattering process is usually complicated due to the occurrence of multiple scattering. Literally, multiple scattering means a photon or ray of light is scattered more than once within the medium before transmitting (i.e., leaving the scattering medium). Such chaotic motions can significantly impact optical diagnosis in, for instance, optical computed tomography techniques. Meanwhile, multiple scattering signals can be used to infer scattering phase function-related information by tracking the transmitting process of the photons. Therefore, it is expected that multiple scattering signals can be used for applications such as tissue diagnosis [1, 2], atmosphere detection [3] and spray droplet size estimation [4], among others. However, the inversion procedure for reconstruction of the turbid media can be complex when considering multiple scattering. When the optical depth (OD) of the medium (representing an expected number of scattering events for a single photon penetrating the medium) is within the range of 2-10, no satisfying approximation can be applied to obtain the solution of the turbid media for the scattering process. The most commonly seen methods to solve the multiple scattering problem are solving the radiative transfer equation (RTE) [5, 6, 7] or using Monte Carlo simulations [8, 4, 9, 10]. As an analytical method, RTE can be solved to obtain the exact solutions, while it is challenging and time-consuming unless strong assumptions are made [11, 12, 13, 14]. The Monte Carlo simulation method is also popular in that the algorithm itself is relatively simple. The scattering event in a Monte Carlo simulation is modeled using a random number generator, which determines whether or not a photon will be scattered at a certain location and the new propagation direction. The disadvantage of Monte Carlo simulations is also notable. The Monte Carlo simulation is a stochastic algorithm in which a great number of photons need to be sent to warrant statistically meaningful results, especially when the signal level is expected to be low. Furthermore, the stochastic nature of this algorithm prevents any analytical inversion or reconstruction approach based on Monte Carlo simulations. In previous works, the authors proposed a Markov chain-based model [15; 16; 17; 18] to calculate the transmitted angular distribution through turbid slabs. In this model, multiple scattering was converted into a matrix (transition matrix), which represented the likelihood of a photon to transit from one state (location and propagation angle) to another. It has been demonstrated that this method can obtain results showing angular distribution (i.e., the probability distribution of transmitted photons as a function of transmitted angles) through infinitely large turbid slabs with the assumption that the optical properties within each parallel layer is uniform. It has been demonstrated that the Markov chain method could incorporate practical scattering phase functions and be executed with a much lower computational cost. In the recent work of the authors, the angular distribution results from photons scattered once and twice were used as inputs, and the optical properties of the turbid slab (droplet diameter) were reconstructed with good accuracy and noise resistance. The use of global optimization algorithms notably increases the computational costs to around 150 minutes, although such a capacity is new and has hardly been demonstrated before. However, the previously introduced Markov chain method is still not perfect, since it relies on obtaining the angular distribution of the photons with an exact scattering order. Although possible, such a requirement is hard to achieve in practical experiments.