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Abstract
In this paper, we propose a new approach to solve the radiative transfer equation (RTE) and determine the path loss for line-of-sight (LOS) propagation with laser diode sources in underwater wireless optical channels, which severely suffers from attenuation due to inevitable absorption and scattering. The scheme is based on an effective combination of Monte-Carlo (MC) simulation employed for dataset generation and a partially pruned deep neural network (PPDNN) utilized to predict the received optical power. First, a parallel MC algorithm is newly introduced and applied to speed up the dataset-generation process. Compared with the conventional single-step MC, the dataset-generation time of the parallel MC can be reduced by at least 95%. Meanwhile, a deep neural network (DNN) is partially pruned to acquire a compact structure and adopted to predict the path loss in three typical water types. The simulation results yield that the mean square errors (MSEs) between the predictive and the reference ones are all lower than 0.2, while the sparsity of the original DNN’s weights can be appropriately increased to 0.9, 0.7, and 0.5 for clear water, coastal water, and harbor water, respectively. Finally, the occupied storage space of the original DNN can be dramatically compressed by at least 40% with a small performance penalty. In view of this, the received optical power under certain parameters could be instantly obtained by employing the proposed PPDNN, which can effectively help design underwater wireless optical communication systems in future work.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the global economy flourishing nowadays, land resources have gradually been unable to satisfy the increasing demands of human life. Considering the abundant natural resources that the ocean contains (e.g., fishery, mineral, oil, and gas), it has become a second home for humans and is playing an indispensable role in human activities. Meanwhile, aiming to carry out the investigation, research, and exploitation of oceans effectively and flexibly, underwater wireless communication (UWC) has already become an essential and widely used technical means over the past decades [1].
As an emerging UWC technology, underwater wireless optical communication (UWOC) has developed rapidly and attracted lots of attention in the last few years [2]. Compared with the conventional UWC methods, including underwater acoustic communication and underwater radio-frequency communication, UWOC has been increasingly regarded as an appropriate alternative for the underwater wireless sensor networks, uncrewed marine vehicles, ocean resource exploration, and seabed environmental monitoring, which is owing to its advantages of tremendous bandwidth over hundreds of GHz, achievable transmission distance up to hundreds of meters, low latency, low cost as well as high security [3–6]. Given this situation, a number of studies have been reported to analyze and validate the feasibility of UWOC systems via diverse experiments and numerical simulations [7].
Since the two major physical mechanisms, namely absorption and scattering, strongly corrupt the light propagation in marine medium, the transmission performance of UWOC systems is significantly governed by the underwater channel characteristics. The two mentioned inherent optical properties (IOPs), arising from water molecules, particulates, and dissolved matters, will affect the amplitude, phase, arrival angle, spot size, and energy distribution of light beam at the receiving end, which can eventually result in temporal dispersion and optical power reduction [8–9]. However, such channel characteristics and corresponding effects are more difficult to be accurately described than those of free space optical (FSO) communication and visible light communication (VLC), due to the difficulties of conducting field experiments in highly dynamic and complex waters [10]. As a substitute, analytical and numerical approaches have been developed and gradually become the alternative methods to mathematically study and analyze the underwater channel characteristics of UWOC links [11]. In this case, one of the main targets is to evaluate the overall optical path loss, which can be quite helpful for further link budget calculations to some extent.
Within the past few decades, many approaches have been reported in the literature to evaluate the underwater optical path loss. Among these, Beer-Lambert’s law is the most widely used model in the line-of-sight (LOS) UWOC systems because of its simplicity [12]. But considering its rough assumption that the scattered photons are annihilated and cannot be captured by the detector after multiple scattering events, Beer-Lambert’s law will severely underestimate the total received optical power, especially in the turbid water. Besides, the radiative transfer equation (RTE), which takes both absorption and scattering effects into account, has been employed as a deterministic solution to fully describe underwater light propagation. However, it should be pointed out that since the RTE is a complicated integrodifferential equation containing plenty of independent variables, it cannot be analytically solved for underwater conditions. Therefore, several numerical approaches have been proposed to obtain an approximate solution of the RTE [13]. Table 1 outlines four typical numerical RTE methods that have been studied in recent years [13–17], and the partially pruned deep neural network method proposed by us, namely PPDNN, is also presented here for comparison, which will be comprehensively discussed in the following sections. The assessment items used here are from Ref. [13].
Table 1. Comparison of numerical RTE methods
The previously reported methods mentioned in Table 1 can be generally divided into two categories: probabilistic and deterministic methods [15]. Among these, the Monte-Carlo (MC) simulation is the most popular probabilistic approach widely applied to numerically solve the RTE owing to its simplicity, intuitiveness, and understandability. Nevertheless, its main drawback lies in that it is very time-consuming, especially when accurately simulating a long-distance or strong-scattering channel since millions of photons are required to assure sufficient received photons. On the other hand, the deterministic methods, such as the conventional discrete ordinate method [14], the direct RTE solver [15], and Markov chains [16] mentioned in Table 1, will partly suffer from the disadvantages of mathematical complexity and poor universality. Besides, they are more difficult to be programmed and only specific to some given parameters such as a certain volume scattering function (VSF), whereas the MC simulation can directly use the look-up table method without any probability recalculations for a different VSF [13]. Meanwhile, even though the deterministic ones can obtain the path loss more quickly, yet the final accuracy highly depends on the granularity of the parameter discretization process, and the performance will gradually become worse as the transmission distance increases [17–18].
Inspired by the deep learning methods applied to solve the Navier-Stoke equation [19–20], the deep neural network (DNN) method is first introduced here to effectively solve the RTE and determine the path loss for UWOC systems. Specifically, a partially pruned DNN model is proposed to predict the received optical power in underwater wireless optical channels, and the required model datasets are generated by the parallel MC simulation. It is worth noting that the deterministic methods are not suitable for datasets generation due to the aforementioned drawbacks. Once trained, the proposed PPDNN can instantly provide a predicted value of the optical path loss without additional bulky computation or iterative operation. The major contributions of this paper are summarized as follows:
- (1) A parallel algorithm is newly employed to speed up the MC simulation via vectorization operation and graphics processing unit (GPU) acceleration. Additionally, a comprehensive MC simulation with a detailed scattering angle generation process is introduced.
- (2) A DNN model is trained and employed to learn how to solve the RTE and predict the path loss under different transmission distances, fields of view (FOV), and detector radiuses at the receiving end, with an interesting finding that the mean square errors (MSEs) between the predictive and the reference ones are significantly lower than those obtained by other methods for three typical water types.
- (3) A pruning strategy is utilized to compress the size of the original DNN model [21], and the simulation results indicate that the MSEs are all lower than 0.2 while the sparsity of weights can be appropriately increased to 0.9, 0.7, and 0.5 for clear water, coastal water, and harbor water, respectively. Finally, the occupied storage space of the proposed PPDNN can be reduced by at least 40% of the original size.
2. Parallel MC algorithm
As noted in Ref. [22], there are four typical MC methods proposed for photon migration simulations, namely albedo-weight (AW), albedo-rejection (AR), absorption-scattering path length rejection, and microscopic Beer-Lambert law, respectively. In our simulations, the most favored AW method was chosen since there is no wasted computation [22]. To provide a better pre-understanding before introducing the parallel MC algorithm, a simplified flowchart generalizing the process of a typical single-step MC simulation and some corresponding sketches are presented in Fig. 1. Firstly, a laser diode (LD) light source with a Gaussian beam distribution is commonly employed to propagate along the z-axis in the x-y plane with radius W and divergence angle θ. The initial 3D emission coordinates (x0, y0, z0) and direction cosines (ux0, uy0, uz0) of each photon shown in Fig. 1 can be expressed as:
Fig. 1. (a) Flowchart of the single-step MC algorithm; (b) Initial 3D emission coordinate and direction cosines of a single photon; (c) Trajectory of a single photon.
Since it has been assumed that the irradiance distribution of the light beam strictly conforms to the ideal 2D Gaussian distribution, x0 and y0 will independently decrease smoothly from the center outwards in the form of the Gaussian function ${e^{ - 2{r^2}/{W^2}}}$. Therefore, r0 is subject to Rayleigh distribution, and its probability distribution function (PDF) and cumulative distribution function (CDF) can be given as:
After the photon initialization, the random path length or step size of the photon can be given as [24]:
where ξs ∼ U (0,1), and c is the attenuation coefficient of light underwater, which is equal to the sum of the absorption coefficient a and the scattering coefficient b. After the photon travels a certain distance s, the new weight w1 and coordinates (x1, y1, z1) can be updated as:
Fig. 2. (a) FF phase function (solid red line) compared with the normalized histogram of 109 scattering angles generated by the tabulated method; (b) Time-consuming comparison of parallel and single-step MC simulation with 105 received photons needed.
As mentioned before, the common single-step MC simulation is extremely time-consuming due to tremendous repetitions. To overcome this shortcoming, we propose a parallel MC algorithm that employs vectorization operation and GPU acceleration to reduce the running time, as shown in Algorithm 1. In general, the parallel operation packs a mass of photons and emits them simultaneously, and then the states of all photons, such as coordinates, direction cosines, step sizes, and weights, will be updated together at each iteration no matter these photons are eliminated, received, or ongoing. Meanwhile, the GPU is used to accelerate the whole process. Compared with the conventional single-step MC, parallel MC considers the independence of photons, thus making better use of computing resources via vectorization operations. As shown in Fig. 2(b), the running time between single-step and parallel MC simulation under three typical water types reported by Petzold is compared [30]. As the single-step MC requires such a long running time for turbid water and long-distance transmission, a comparison is made for a maximum distance of 20 m here. It can be observed that the parallel MC simulation can effectively reduce the running time by at least 95% or acquire a time gain over 20, and the trend becomes more significant as the water becomes more turbid or the transmission distance increases. In view of this, the parallel MC method will be used to generate adequate datasets fast in our work.
3. Partially pruned DNN model
After generating training and testing datasets by the above parallel MC simulation, the DNN model is developed for predicting the received optical power in underwater wireless optical channels. The DNN model before pruned can be theoretically expressed as:
To reduce the size of the original DNN model, the pruning algorithm has been applied [21]. To be more specific, the DNN model is first trained to obtain a relatively high accuracy through the mini-batch method. Afterwards, the weights of each layer are sorted in descending order. According to the preset sparse value, some low weights are set to 0 by comparing absolute values, namely pruning. Then the DNN model will be retrained to further fine-tune the remained weights. In this way, the training MSE performance can converge to a relatively low value again. Instead of merely pruning partial weights in one step to achieve the desired sparsity, we use the automated gradual pruning algorithm proposed in [33]:
4. Simulation results and discussions
The parameters of the parallel MC simulation used for training data generation are given in Table 2, and the values in parentheses represent the step intervals. During the testing process, two datasets are properly chosen with the parameters (R, FOV) set as (0.15 m, 25°) and (0.75 m, 180°), respectively, which represents two different conditions for a certain water type. Meanwhile, the maximum testing value of parameter L is set as 132 m, 96 m, and 60 m for three water types, respectively. Note that the parameter L of testing datasets is further increased to be distinguished from the training datasets.
Table 2. Parallel MC simulation parameters for training data generation
We first compared the prediction performance between the original unpruned DNN model and three typical models, including linear regression (LR), support vector regression (SVR) [35], and XGBoost [36], and the two aforementioned testing datasets are used here. The MSE is properly employed as an effective indicator of prediction performance. For instance, Fig. 4 indicates the differences between the predicted and reference values with different methods in harbor water. Besides, the coefficient of determination R2 is also given for a better understanding of the goodness of fit. It can be found that the DNN model has the best agreement as its testing MSE value is the lowest and R2 is closest to 1, while LR, SVR, and XGBoost cannot yield satisfactory results with the optimal parameters after traversal. More specifically, the LR model based on the least square method is not suitable for such a nonlinear relationship. Meanwhile, the SVR and XGBoost models with optimized parameters are no longer feasible as the transmission distance exceeds the range of training datasets. In such a case, no matter how the other two parameters (R, FOV) are selected, the fitting performance of SVR and XGBoost will be poor. However, since the DNN model can establish a potential correlation between the parameters (L, R, FOV) and the optical power Pp more accurately, it can lead to better fit performance. In Table 3, the MSE values of three water types under different methods are indicated. Compared to other common models, the original DNN model obtains minimum MSE values and has a good predictive ability for the received optical power under different water types.
Table 3. Testing MSE values of three water types with different methods
Since the original DNN model is redundant, the pruning algorithm mentioned in section 3 is employed to compress its size. In Fig. 5, the MSE results in the training process with different NN models and pruning strategies are compared, where the harbor water is chosen as an example, and the final sparsity is properly set as 0.5. The blue curve represents the training process of the original unpruned DNN model, and the dotted red curve shows the sparsity curve in which the pruning process starts at the 600th epoch and stops at the 1500th epoch with a pruning interval of 100 epochs. As the black curve shows, the artificial neural network (ANN) model containing only one hidden layer of 100 neurons is more unstable than other DNN models due to its lack of deep-network fitting ability. The red curve represents the partial pruning strategy, in which the weights of hidden layers connecting the input layers and output layers are not pruned. Besides, it can be observed that the red curve has the smoothest training convergence process by employing the pre-trained weights obtained from the originally unpruned DNN model. However, as the green curve shows, the training process without the pre-trained weights would be slower and rougher. That is, the pre-trained weights instead of random initial weights can further accelerate the convergence process since the pruning process can be regarded as a fine-tuning process for the weights in some way. Additionally, compared with other strategies, the purple curve indicates that the full pruning strategy, in which the weights of all layers are pruned, will lead to a significant degradation of MSE performance, which proves the selectivity of pruning is necessary.
Fig. 5. MSE performance of training process for different NN and pruning strategies in harbor water.
The final training MSE values of different strategies are presented in Table 4. It can be found that the PPDNN with pre-trained weights can reduce the number of weights from 701 to 401. Even though the original DNN model is compressed to 57.2%, the final training MSE value increases only by 1.36%. At the same time, although the ANN model with only one hidden layer is simpler, its final MSE value is a little higher than the PPDNN. Using the same pruning algorithm, the final MSE results for three typical water types with different sparsity are obtained. In Fig. 6, it indicates that the lower sparsity is needed as the water quality deteriorates. However, even if the sparsity increases to 0.9, the final testing MSE performance will hardly increase for clear water. As well, the MSE values are also almost unchanged when the sparsity increases to 0.7 and 0.5 for coastal water and harbor water, respectively. But as the sparsity is further improved, the MSE performance of harbor water will be further degraded. That is because the scattering and absorption effects are more obvious in turbid water. In this case, any changes in the three variables (L, FOV, R) of the input layer will significantly affect the final optical power loss so that the PPDNN needs more weights to fit the entire process.
Fig. 6. Impact of different sparsity on the MSE performance of testing datasets for three typical water types.
Table 4. Number of parameters and final training MSE values of different methods
Furthermore, more detailed results are shown in Fig. 7. The corresponding MC simulation results are used as a reference, and “0.15 m/20°” in figures represents the values of (R, FOV). It can be found that the predicted values are in good agreement with the reference values, and all testing MSE values are lower than 0.2. In general, the obtained results have proved the feasibility of the pruning method for different water types.
To further understand the state of weight pruning, for instance, the heatmaps of pruned DNN model for clear water with sparsity 0.9 are shown in Fig. 8. The x-axis and y-axis indicate the neurons for the corresponding layers, and the color block represents the weights between the corresponding two neurons. It can be observed that lots of blocks have zero values owing to the large sparsity, leading to a significant size compression and relatively low space complexity. Meanwhile, it is worth noting that some biases are also trained to zero since lots of weights are deleted, further decreasing the number of parameters needed for the DNN model. Table 5 shows the final occupied storage space of the PPDNN for three typical water types. According to Table 5, it can be observed that more turbid water whose sparsity cannot be too large requires more parameters, resulting in a larger storage space correspondingly. Nevertheless, the compression ratios of the three water types are all less than 40%, which can be used to decrease the occupied storage space effectively.
Table 5. Occupied storage space of DNN models for different water types
5. Conclusions
In this paper, a partial pruned DNN model coupled with parallel MC simulation has been proposed to predict the path loss in underwater wireless optical channels. The parallel algorithm is employed to speed up the MC simulation and generate essential datasets for the DNN model. Compared with LR, SVR, and XGBoost methods, the DNN model has the best MSE performance. As well, the partial pruned DNN model is trained and employed to predict the path loss in various water types, resulting in a relatively low MSE value below 0.2 and a compression ratio of storage space less than 40%. The proposed PPDNN can be employed to instantly provide a reference value for the design of UWOC systems without intensive time consumption. The proposed path loss prediction scheme can be deployed on devices owing to the relatively small required storage space. In the future, more channel parameters and corresponding datasets will be included in the proposed PPDNN, and it is well encouraged to consider more conditions like bubbles and turbulence to build a universal prediction model in our follow-up work.
Funding
National Natural Science Foundation of China (61971378); Strategic Priority Research Program of the Chinese Academy of Sciences (XDA22030208); Zhoushan-Zhejiang University Joint Research Project (2019C81081).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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