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Abstract
This paper introduces a novel back propagation (BP) neural network method to accurately characterize optical properties of liquid cloud droplets, including black carbon. The model establishes relationships between black carbon volume fraction, wavelength, cloud effective radius, and optical properties. Evaluated on a test set, the value of the root mean square error (RMSE) of the asymmetry factor, extinction coefficient, single-scattering albedo, and the first 4 moments of the Legendre expansion of the phase function are less than 0.003, with the maximum mean relative error (MRE) reaching 0.2%, which are all better than the traditional method that only uses polynomials to fit the relationship between the effective radius and optical properties. Notably, the BP neural network significantly compresses the optical property database size by 37,800 times. Radiative transfer simulations indicate that mixing black carbon particles in water clouds reduces the top-of-atmosphere (TOA) reflectance and heats the atmosphere. However, if the volume fraction of black carbon is less than 10−6, the black carbon mixed in the water cloud has a tiny effect on the simulated TOA reflectance.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Radiative forcing from tropospheric aerosols caused by human activities significantly impacts climate change. Sulfate and carbonaceous aerosols have been found to be of particular importance in climate research [1–3]. Black carbon (BC) aerosols, mainly resulting from the incomplete combustion of fossil fuels and biofuels, are present throughout the Earth system [4–6]. BC strongly absorbs solar energy, affects cloud microphysical processes, and plays an important role in the Earth system [7–11]. Mixing BC particles in water clouds has several profound results. On the one hand, BC absorbs solar radiation from ultraviolet to visible wavelengths, heats the atmosphere, and cools the surface, directly affecting radiative forcing [12]. On the other hand, BC particles also serve as cloud condensation nuclei (CCN), influencing the microphysical characteristics of clouds, cloud reflection, and emissivity [13]. The formation of clouds and precipitation can be promoted by BC particles, which have a cooling effect on the surface, the resulting atmospheric radiation change is called indirect radiative forcing [14,15]. In addition, BC embedded in clouds heats them, causing evaporation, and this aerosol effect on clouds is called the semi-direct effect [16]. Climate simulation and weather forecasting require the precise parameterization of the optical characteristics of water clouds. Consequently, simulating optical properties in water clouds containing BC becomes crucial, offering essential support for forward radiative transfer computations and remote sensing inversion techniques.
Initially emitted, BC aerosols exhibit primarily hydrophobic characteristics; however, they gradually become hydrophilic over time due to physicochemical processes in the atmosphere. These hydrophilic BC particles can serve as effective CCN [17–19] leading to the incorporation of BC into cloud droplets.
Early studies on the influence of BC mixed into clouds on radiative transfer and climate mainly considered the effect of BC on the single-scattering albedo ($\omega$) of cloud droplets, and $\omega$ of cloud droplets mixed with black carbon was calculated through empirical formulas. These studies have found that BC in cloud droplets reduces the reflection of solar radiation by clouds, resulting in positive radiative forcing at the top-of-atmosphere (TOA) [12,20]. Nevertheless, these empirical formulas lack physical explanations and may not be universally applicable. Other studies have used multiterm rational functions to model the relationship between cloud effective radius and optical properties [21–25] including asymmetry factor, single-scattering albedo, and extinction coefficient. Among them, Li et al. [25] have proposed a method to deal with the BC impact on cloud optical properties. They discovered that the alterations in cloud optical properties due to a mixture of black carbon can be treated as a perturbation to existing cloud optical property parameterizations in climate models, such that the proposed scheme is that current cloud optical property parameterizations used in climate models can be kept. Li et al.’s scheme is built upon band mean for two types of specific band structures.
With the development of machine learning, neural network algorithms have been applied to solve numerous nonlinear problems, such as estimating wind speed, precipitation, radiative transfer, etc. [26–32]. Among them, Yu et al. [28] have proposed a neural network to fit the optical characteristics of non-spherical particles, which reduced the size of the optical property database while ensuring high accuracy, and promoted the application of non-spherical particle models in atmospheric correction and radiative transfer models. Further, Ukkonen et al. [31] have utilized a neural network algorithm to replace the calculation of gas absorption parameters in radiative transfer and accelerate the solution process of radiative transfer. All of the above demonstrates that neural network algorithms can be effective in solving problems with clear physical relationships.
We introduced a neural network algorithm aimed at precisely simulating the optical characteristics of water clouds containing diverse volume fractions of black carbon across various wavelengths. Our scheme not only can be used in climate models but also can be used in remote sensing since the result of individual wavelength is available. Chapter 2 detailedly describes the construction of the database of optical properties used for the proposed method. Water cloud droplets mixed with black carbon are assumed to be spherical droplets, the Maxwell-Garnett (MG) mixing rule [33,34] is used to calculate the refractive index of cloud-black carbon mixtures. Afterwards, these values are fed into the Lorenz-Mie theory [35] to compute the optical characteristics of the spherical droplets. In Chapter 3, we present the Back Propagation (BP) neural network algorithm, along with the experimental data and network configurations employed in the experiment. Subsequently, we assess the fitting accuracy of the optical property results derived from the BP model. In Chapter 4, we conduct a comparative analysis to assess the simulation proficiency of the proposed model in comparison to the traditional approach. This analysis focuses specifically on the optical properties of cloud-black carbon mixtures. In Chapter 5, the application and performance analysis of the proposed algorithm in radiative transfer mode are introduced. Finally, the conclusions are presented in Chapter 6.
2. Theory and dataset
2.1 Cloud optical properties
Cloud droplet size distribution is a core characteristic of atmospheric cloud microstructure, which is generally assumed to be a gamma distribution and can be expressed as:
whereThe simulation of atmospheric radiative transfer involves at least four kinds of cloud optical parameters: asymmetry factor, extinction coefficient (optical thickness), single-scattering albedo, and the phase function. Water cloud droplets are typically assumed to be spherical particles, and the optical properties of water cloud droplets are calculated by Lorenz-Mie theory based on the complex refractive index of water. Then, the optical properties of water cloud clusters (extinction coefficient, absorption coefficient, etc.) under any effective radius of the water cloud are calculated according to the size distribution of water cloud droplets.
The extinction coefficient and absorption coefficient at wavelength $\lambda$ are given by:
In the discrete ordinate radiative transfer [36] (DISORT) method, the phase function is given by:
where $C_{n}^{\lambda }$ is the dimensionless Legendre coefficient, which defined as $C_{n}^{\lambda } = {\frac {2n+1}{2}}{\int _{-1}^{1}f(x,\lambda )P_n(x)dx}$. Four-stream radiative transfer calculations can be performed by using the first four moments of Legendre polynomials [37–39]. $f(x,\lambda )$ is the phase function at wavelength $\lambda$ for different scattering angles. $x$ is the cosine of the scattering angle. $P_n(x)$ is the Legendre polynomial of order $n$. The asymmetry factor at wavelength $\lambda$ is calculated as:2.2 Refractive index of the cloud-black carbon mixture
The dielectric constant of the hybrid particles can be calculated using MG theory, and the formula is as follows:
Figure 1 shows the variation of the imaginary component of the refractive index with wavelength for water clouds mixed with different volume fractions of black carbon. It can be seen that in the visible range, the larger the black carbon volume fraction, the larger the imaginary component of the refractive index, resulting in stronger particle absorption. The imaginary component of the refractive index of the cloud-black carbon mixture mainly changes in the solar wavelength range, particularly when the wavelength is less than 1.0 $\mu m$. Similar conclusions were also presented in [12,25,40].
Fig. 1. The imaginary part of refractive index for water cloud containing different volume fractions of black carbon.
2.3 Dataset
The primary objective of this study is to retrieve the optical characteristics of water clouds mixed with BC utilizing volume fraction, wavelength, and the effective radius of droplets. The water cloud droplets mixed with black carbon are regarded as spherical particles, and the optical properties of the cloud are calculated using Lorenz-Mie theory. The refractive index of water is derived from Hale et al. [41] and the refractive index of black carbon is derived from Chang et al. [42]. Firstly, the refractive index of mixture droplets with the volume fraction of black carbon ranging from $10^{-2}$ to $10^{-10}$ can be calculated using the method described above. Secondly, the optical properties of mixed droplets are simulated, with the effective radius of the droplets taken from 2-60 microns and the wavelength taken from 0.2-4 microns.
Ideally, a neural network training set should comprise extensive, realistic, and large amounts of data. In our experiments, 100 interpolation points were taken for each input parameter range, containing black carbon volume fraction, wavelength, and cloud effective radius (Table 1). The volume fraction values were taken as logarithmically spaced sampling, and the other parameters were equidistant sampling. There are a total of 1,000,000 samples. Among them, 70% of the data is allocated for the training set, 10% for validation, and the remaining 20% for the test set. The test set is kept unseen throughout the training process. The BP neural network is trained to obtain the asymmetry factor ($g$), single-scattering albedo ($\omega$), extinction coefficient ($\beta _{e}$), and the first 4 moments of the Legendre expansion of the phase function ($C_{1}, C_{2}, C_{3} , C_{4}$) for the mixed BC cloud.
Table 1. The input parameters (volume fractions, effective radii, and wavelengths) and output optical properties of the BP neural network ($g$, $\omega$, $\beta _{e}$, $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$). The value range of each input parameter is given.
While obtaining the optical properties of BC-mixed clouds is efficient from the available dataset, expanding parameters and reducing intervals exponentially increase the database size. The parameters in Table 1 alone occupy approximately 17GB of storage. In addition, if the required volume fraction, wavelength, and effective radius are not included in the database, the corresponding optical properties need to be obtained by interpolation.
3. Methodology and results
3.1 Back propagation neural network
The BP algorithm is a typical artificial intelligence algorithm [43]. The algorithms earliest and most classic application involves approximating highly intricate nonlinear function relationships via training. Its training employs gradient descent, minimizing the output-target error to update network weights and biases.
The BP neural network consists of two main stages: the feed-forward stage and the backward transmission stage. In the forward pass, the training set propagates the output result to the output layer through the hidden layer, compares the output value with the truth value of the actual target, and calculates the error between the output value and the true value. Then, the gradient of the error is calculated by the back propagation algorithm for the weight and bias of each layer, and the gradient descent method is used to update the values of the weight and bias, continuously reduce the error and improve the accuracy of the network.
Since both the model size and the accuracy of the simulation are considered simultaneously. After several experiments, we obtain the optimal structure of the hidden neurons for each optical property simulation network while ensuring that the coefficient of determination ($R^2$) is larger than 0.999, as shown in Table 2. The first layer of the model serves as the input layer, with the input parameters consisting of $\eta$ , $\lambda$, $r$, $1/r$, respectively. The last layer outputs the optical properties of the simulation. The model simultaneously simulates $C_{1}, C_{2}, C_{3} , C_{4}$ and then outputs the results. Finally, a total of four neural network models are used to simulate $g$, $\omega$, $\beta _{e}$, and $C_{1}$ to $C_{4}$, respectively.
Table 2. Network architectures that are developed for the prediction of optical properties.
Figure 2 shows the neural network structure used by the proposed method. The network fitting Legendre coefficients has three hidden layers, and the hidden layers simulating other optical properties are two layers. Further, we employ Adaptive Moment Estimation (Adam) as the gradient descent optimization algorithm and the Mean Squared Error (MSE) function as the loss function in the model. Table 2 lists the sizes of models trained for different optical properties. Compared with the 17GB dataset mentioned in section 2.3, the total size of disk memory occupied by the four models is reduced by about 37,800 times.
3.2 Performance of BP neural network
Two intuitive error metrics, mean relative error (MRE) and root mean square error (RMSE), are used to evaluate the precision of the fitting of optical properties. They are calculated as follows:
Table 3 lists the errors of the model in predicting parameters $g$, $\omega$, $\beta _{e}$, $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ based on 200,000 samples of the test set, respectively. Among them, the RMSE of the seven parameters is all less than 0.003, highlighting the BP model’s strong prediction capability. The mean relative errors are 0.0477%, 0.0341%, 0.1883%, 0.0428%, 0.0270%, 0.0287%, and 0.0267%. The error of the simulated results of the test data shows that the generalization of the model is good.
Table 3. Evaluation score of the BP neural network for seven optical properties based on the independent test set.
Figure 3 gives the average simulated values of the model simulation for different effective radii ($C_{1}$ is three times as much as $g$) , as a function of volume fraction and wavelength. And the corresponding mean absolute error (MAE) between the simulated values and the true values. Ten effective radii were sampled equally spaced between 2 and 60 $\mu m$. Figures 3(a)–3(c), 3(g)–3(i) highlights that the fluctuation of six optical properties with the change in the volume fraction of black carbon is relatively low. Only when the volume fraction is greater than $10^{-3}$, did the optical properties alter slightly with the increase of the volume fraction. Simultaneously, it is clear that six optical properties will oscillate around the wavelength of about 3.0 $\mu m$, except that $\beta _{e}$ does not fluctuate significantly with the wavelength. A minimum of $\omega$ occurs at a wavelength equal to about 3.0 $\mu m$, which indicates that the absorption coefficient of the water cloud containing BC at 3.0 $\mu m$ rises sharply and reaches a maximum value. This is consistent with the change in the imaginary component of the complex refraction index in Fig. 1. At this wavelength, the imaginary of the complex refraction index has a maximum value, and the droplet absorption is the strongest. As shown in Fig. 3(a), the maximum value of $g$ appears at a wavelength of about 3.0 $\mu m$, which indicates that the asymmetry between the backward and forward scattering of the cloud is the strongest at this time. Figures 3(d)–3(f), 3(j)–3(l), highlights the good simulation ability of the BP model through the upper and lower limits of the error and the distribution of errors. The mean absolute error of $g, \omega, \beta _{e}$ does not exceed $\pm$0.0015, and the mean absolute error of the first 2 to 4 moments of the Legendre expansion of the phase function are mainly between −0.003 and 0.003.
Fig. 3. Average optical properties at ten effective radii of liquid cloud mixed with black carbon under different volume fraction and wavelength: (a)-(c) asymmetry factor, single-scattering albedo, extinction coefficient, (g-i)$C_{2}$, $C_{3}$, and $C_{4}$, based on the BP neural network model. And their mean absolute errors (d)-(f) ,(j)-(l) compared with the Lorenz-Mie theory calculation.
Figure 4 demonstrates the changes in average optical properties with volume fractions and effective radii when ten wavelength points are sampled equally spaced from 0.2 to 4$\mu m$. The corresponding MAE between the prediction values and the Lorenz-Mie theory calculation results are also analyzed. As can be seen from Figs. 4(a)–4(c) and 4(g)–4(i), when the effective radius is fixed and the volume fraction of black carbon is greater than $10^{-3}$, $g$, $\omega$, $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$ monotonically increases or decreases with the increase of volume fraction. Nevertheless, this only occurred at an effective radius greater than 4.0 $\mu m$, since the six optical properties exhibited almost no change with an increasing volume fraction of BC and an effective radius of about 4.0 $\mu m$. Furthermore, $\beta _{e}$ does not change with the change in the volume fraction, which is consistent with the result shown in Fig. 3(c). Figures 4(d)–4(f) and 4(j)–4(l) highlights the prediction errors of the BP model for six optical parameters. The absolute errors of $g$, $\omega$, and $\beta _{e}$ do not exceed $\pm$0.0015, and the absolute errors of the first 2 to 4 moments of the Legendre expansion of the phase function are mainly between −0.002 and 0.002.
Fig. 4. Similar to Fig. 3 but the average optical properties at ten different wavelengths and their mean absolute errors vary with the volume fraction and effective radius.
In order to further understand the accuracy of the model, the points with large errors (dark areas) in Figs. 3-4 are selected, and the BP model prediction and Lorenz-Mie theory calculation results are compared and analyzed. We picked the point in Fig. 3 with a wavelength equal to 3.5 $\mu m$ and a volume fraction of black carbon equal to $10^{-8}$ as Case 1, and we selected the point in Fig. 4 with an effective radius of 5 $\mu m$ and a volume fraction of black carbon equal to $10^{-6}$ as Case 2.
The results for Case 1 are depicted in Fig. 5. There is excellent agreement between the BP model and the Lorenz-Mie theory. Furthermore, the change of optical parameters with the monotonic rise or fall of the effective radius can be simulated.
Fig. 5. Comparison of asymmetry factor, single-scattering albedo, extinction coefficient, Legendre coefficient $C_{2}$, Legendre coefficient $C_{3}$ and Legendre coefficient $C_{4}$ predicted by the BP neural network against the Lorenz-Mie theory calculation. The wavelength and volume fraction are set to 3.5 $\mu m$ and $10^{-8}$, respectively.
Figure 6 illustrates the result for Case 2. In this calculation, the variation of the BP model predicted optical parameters with wavelength. As presented in Figs. 3 and 6, the optical properties oscillate significantly with an increase in wavelength, especially around 3 $\mu m$. Figure 6 shows that there is a good agreement between the optical properties simulated by the BP model and the Lorenz-Mie theory calculation results, and the trend changes of the six optical properties with wavelength are roughly simulated.
Fig. 6. Similar to Fig. 5 but effective radius and volume fraction set to 5$\mu m$ and $10^{-6}$, respectively.
4. Comparison with traditional method
In this section, in order to prove the reliability and superiority of our model, we compare the performance of the BP neural network and the traditional multiterm rational functions (given by Eqs. (12)–(14)) using the same dataset.
where $a_{n},b_{n},c_{n}$ are the fitting coefficients at each wavelength and each volume fraction. The result of the Lorenz-Mie theory calculation is taken as the true value. The effective radius is taken at 2 to 60 $\mu m$, and 100 points are taken at equal intervals. The wavelengths are 0.55, 1.0, 2.0, and 3.0 $\mu m$, and the volume fractions are $10^{-8}$, $10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$. The optical properties of the water cloud mixed with BC at different wavelengths and volume fractions are fitted as per the multiterm rational functions approach, and 26 different equations are obtained for each optical property.Table 4 compares the MRE corresponding to the asymmetry parameter, single-scattering albedo, and extinction coefficient fitted by these two methods, respectively. The fitting errors of the BP method in 26 cases are smaller than those of the traditional method for the asymmetry factor. When the wavelengths are 0.55, 1.0, and 2.0 $\mu m$, the relative error of the single-scattering albedo calculated by the BP neural network is slightly higher than the traditional method. In addition, the fitting error is smaller than that of traditional method in the remaining cases. For the extinction coefficient, at wavelengths of 0.55 and 1.0 $\mu m$, the relative error of the simulated $\beta _{e}$ of the BP method is slightly higher than that of the traditional method, and the fitting error is smaller than that of the traditional method in the remaining cases.
Table 4. MRE of the asymmetry factor, single-scattering albedo, and extinction coefficient using different simulation methods: the BP neural network and the traditional method. Wavelengths are set to 0.55, 1.0, 2.0 and 3.0$\mu m$, and volume fractions set to $10^{-8}$, $10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$. Our error values in the Table 4 omit the percent mark.
Although the traditional method can fit the optical properties by using multiterm rational functions, it only considers the fitting relationship with the effective radius. Different wavelengths and volume fractions need different formulas to fit the optical properties. And interpolation is required to compensate for the volume fraction and wavelength that are not considered by the formula. Figure 7(a)(c)(e) evaluate the relative errors between the results calculated by the Lorenz-Mie theory and the results calculated by the multiterm rational functions and linear interpolation, respectively. Relative errors at wavelengths equal to 0.55, 1.0, 2.0, and 3.0 $\mu m$ are obtained by multiterm rational functions (blue dots), and relative errors at 0.8, 1.5, and 2.5 $\mu m$ are obtained by interpolation (red stars). Figure 7(b)(d)(f) show the relative errors at 0.55, 0.8, 1.0, 1.5, 2.0, 2.5, and 3.0 $\mu m$ calculated by the BP neural network. With the testing, the volume fractions are $10^{-8}$, $10^{-7}$, $10^{-6}$, $10^{-5}$, $10^{-4}$, and $10^{-3}$.
Fig. 7. Relative errors of asymmetry factor, single-scattering albedo, and extinction coefficient of the interpolation between two wavelength points (red stars). Relative errors of multiterm rational function and BP neural network(blue dots). (a)(c)(e) multiterm rational function method. (b)(d)(f) BP neural network method.
Figure 7(a) shows that the interpolation method causes a sudden increase in the error of the result, as $g$ fluctuateds with the wavelength (Fig. 3(a)). Nevertheless, the relative error of $g$ predicted by the BP neural network method does not increase suddenly, and is generally small(Fig. 7(b)). Figure 7(c) presents the relative error of $\omega$ between the Lorenz-Mie theory calculation results and those calculated by the multiterm rational functions method and the linear interpolation, respectively. As indicated in Fig. 7(c), the linear interpolation method will cause the error of $\omega$ to increase suddenly, especially at wavelengths around 2.5 $\mu m$, where $\omega$ changes drastically with the increase of the wavelength. However, the error in $\omega$ predicted by the BP neural network does not exhibit sudden increases and remains below 0.1%. Figure 7(e) shows that the relative error caused by interpolation reaches 4% (red star), while the relative error of $\beta _{e}$ simulated by the BP neural network is less than 0.4%. Errors of $\beta _{e}$ simulated by the linear interpolation method exhibit a sudden increase at a wavelength of around 2.5 $\mu m$.
The results demonstrate the selected neural network scheme’s capability to predict patterns based on interpolated parameter values. Additionally, the error size remained relatively consistent without sudden fluctuations. Under various conditions, the same model yield minor relative errors for $g$, $\omega$, and $\beta _{e}$ all below 0.4%. For the multiterm rational function method, distinct functions were necessary for different wavelengths and volume fractions to fit optical properties. Despite some results exhibiting smaller errors compared to BP neural networks, linear interpolation led to a high 23.5% error.
5. Model application
In the previous section, we developed a BP neural network model and evaluated its accuracy. We demonstrated that the BP neural network can reproduce the database of the optical properties of the water cloud mixed with BC computed by Lorenz-Mie theory and predict the optical properties corresponding to the unknown volume fraction, wavelength, and effective radius. The small storage size of the model makes it easy to embed into various modes, such as atmospheric radiative transfer models, global climate simulations, or remote sensing retrieval algorithms.
In this section, we further evaluate the model by investigating the effect of water clouds containing black carbon on the reflectance at the TOA. We utilized $\delta$-64 DISORT for the radiative transfer calculation, and the gas absorption coefficients are calculated using the line-by-line radiative transfer model [44]. The mid-latitude summer (MLS) atmospheric profile is employed, and the water cloud(liquid water content=0.22$gm^{-3}$) is set in the region from 1–2km [25,45]. For the calculation of the optical properties of the water cloud, the volume fraction of black carbon ranges from $10^{-10}$ to $10^{-2}$, and the effective radius ranges from 2 to 60 $\mu m$, while the wavelength ranges from 0.2 to 4 $\mu m$. 20 points were taken at equal intervals and the cumulative data quantity within the set is 8000.
The TOA reflectance is calculated, and the correlation analysis is performed with the TOA reflectance obtained from the optical properties of the Lorenz-Mie theory. Figure 8 illustrates the results of the experiment. $R^{2}$ reaches 0.999 and the RMSE reaches 0.005. The results indicate that the integration of our model into the radiative transfer model has little impact on the precision of radiative transfer calculation, under the premise of the reduction of the disk memory occupied by the data.
Fig. 8. Joint probability distribution of TOA reflectance from the optical properties of water clouds calculated by BP neural network and Lorenz-Mie theory.
The $\delta$-64 DISORT was utilized to calculate the TOA reflectance with the volume fractions of black carbon set to 0, $10^{-5}$, $10^{-4}$, and $10^{-3}$ in the water cloud, respectively. When the volume fraction of black carbon is less than or equal to $10^{-6}$, the simulated TOA reflectance is comparatively consistent with that of pure water clouds, and the black carbon contained in the water cloud has a tiny effect on the radiative forcing of the TOA. So we didn’t show it in the results. The dashed line is the result of the TOA reflectance simulated by the optical properties of the water cloud mixed with BC calculated by the Lorenz-Mie theory, and the solid line is the result of the TOA reflectance simulated based on the optical properties calculated by the BP neural network.
To assess the impact of water clouds with varying black carbon volume fractions and different effective radii on TOA reflectance, we focus on a wavelength of 0.55 $\mu m$ and a cosine of the solar zenith angle (SZA) represented by 0.65. Figure 9(a) demonstrates that under the same atmospheric state, the larger the volume fraction of black carbon, the smaller the TOA reflectance value, because the existence of black carbon in cloud droplets increases the absorption of solar energy. As the effective radius of the cloud increases, the extinction coefficient of the cloud decreased (Fig. 4(c)), more solar energy reaches the Earth’s surface, and the TOA reflectance value decreases. As can be seen from the relative error depicted in Fig. 9(b), the optical properties of mixed BC clouds simulated by the BP neural network have little effect on the error of the TOA reflectance calculation, less than 1%.
Fig. 9. Optical properties calculated by Lorenz-Mie theory are used to simulate TOA reflectance (dashed line). Simulation of TOA reflectance using the optical properties calculated by BP neural network(solid line). Black dashed line is the simulated result of TOA reflectance for the pure liquid cloud. The green, pink, and purple lines correspond to volume fractions of $10^{-5}$, $10^{-4}$, and $10^{-3}$, respectively. (a) TOA reflectance varying with the effective radius of the cloud, (b) the relative error (%) of simulated TOA reflectance between the optical properties calculated by BP neural network and the optical properties calculated by Lorenz-Mie theory.
Figure 10 shows the value of TOA reflectance as a function of SZA for water clouds containing different volume fractions of black carbon. The effective radius of the cloud droplets is 5.89 $\mu m$, and the wavelength is 1.2 $\mu m$. Figure 10(a) shows that the rate of change in TOA reflectance is gradual for a SZA below 70 degrees, but becomes more pronounced as SZA surpasses 70 degrees. This behavior can be attributed to the Earth’s curvature, which impacts TOA reflectance particularly when SZA exceeds 70 degrees. Further, the relative error of TOA reflectance simulated based on the optical properties of the cloud fitted by the BP neural network is less than 2%, and most of the errors are concentrated in 0-1% (Fig. 10(b)).
To evaluate the effect of water clouds containing BC on the radiative transfer at wavelengths ranging from 0.2 to 4 $\mu m$, we take the effective radius of cloud droplets as 5.89 $\mu m$ and the cosine of the SZA as 0.65. The $\delta$-64 DISORT is used to simulate the variation of TOA reflectance for water clouds containing different black carbon volume fractions. When the wavelength fell below 1.4 $\mu m$, the change of TOA reflectance is more susceptibility to the volume fraction of black carbon within the water cloud (Fig. 11(a)), corresponding to Chýlek et al. [40]. Figure 11(a) shows that when the wavelength is 0.2um and the black carbon volume fraction increases from 0 to $10^{-3}$, the maximum change of TOA reflectance reaches 0.6. For a wavelength larger than 1.4 $\mu m$, the black carbon contained in the water cloud has less effect on the radiative forcing at the TOA. The maximum relative error of the TOA reflectance simulated by the cloud optical properties of the BP neural network is less than 6%. Meanwhile, when the wavelength fell below 2.4 $\mu m$, the relative error is typically less than 2%. Furthermore, we are conducting an equidistant sampling of 20 values for effective radii in the range of 2 to 60. Using the BP neural network, we simulate cloud optical properties. TOA reflectance is calculated using $\delta$-64 DISORT for wavelengths ranging from 0.2 to 4 $\mu m$. We compute the average relative error as −0.43% with a value of maximum relative error is 5.20%.
Given the above experiments, it can be concluded that when the volume fraction of black carbon mixed with the water cloud is greater than $10^{-6}$, the presence of black carbon in the water cloud has a more significant impact on the TOA reflectance. When the wavelength is greater than 2.4$\mu m$, the impact of black carbon in water clouds on TOA reflectance is minimal.
6. Conclusion
The inclusion of BC in water clouds results in the absorption of solar radiation and contributes to atmospheric heating. Accurate simulation of optical properties for clouds containing BC is crucial in atmospheric radiative transfer models and remote sensing retrievals. This study focuses on precisely calculating the optical properties of mixed clouds by leveraging MG mixing rules and Lorenz-Mie theory. Diverging from prior approaches that solely parameterized cloud size distribution’s effective radius for optical property simulation, our approach establishes a comprehensive relationship between optical properties, black carbon volume fraction, wavelength, and cloud size distribution’s effective radius. Employing the back propagation neural network enables us to attain precise simulated optical parameters while also showcasing robust model generalization. This methodology holds significant promise for advancing cloud remote sensing and enhancing the modelling of cloud radiation interactions.
The majority of radiative transfer models utilize optical property look-up tables to obtain data. The look-up table occupies a large amount of hard disk storage. Furthermore, look-up tables generally cannot cover all the cloud optical properties. If the interpolation method is used, it will introduce additional errors. The method proposed in this paper can accurately obtain all the optical properties of BC included within water clouds in the wavelength range of 0.2-4 $\mu m$, a volume fraction of BC between $10^{-10}$-$10^{-2}$, and an effective radius of 2-60 $\mu m$. The seven optical parameters ($g$, $\omega$, $\beta _{e}$, $C_{1}$, $C_{2}$, $C_{3}$, and $C_{4}$) of the cloud for the radiative transfer calculation can be obtained by using four back propagation neural networks. Utilizing our model, the optical property database was compressed by a factor of 37,800, which occupies a small storage volume and is convenient. Additionally, it is stably applied to the radiative transfer model, which is conducive to the study of the TOA reflectance change and radiative forcing caused by black carbon contained in cloud droplets.
Integrating the trained BP neural network model into the $\delta$-64 DISORT enables the analysis of atmospheric remote sensing effects. A higher volume of black carbon within water clouds enhances the absorption of solar radiation, leading to reduce reflectance at the TOA and contributing to global warming. When the volume fraction of black carbon exceeds $10^{-6}$, the impact on TOA reflectance becomes notably pronounced, resulting in a positive radiative forcing on the atmosphere. In the wavelength range of 0.2-1.4 $\mu m$, reflectance demonstrates clearer variation with changes in black carbon volume fractions in water clouds, facilitating the identification of black carbon particles within the cloud. In the future, we will apply the BP neural network model to the Dayu model [46].
Funding
National Key Research and Development Program of China (2021YFC3000801); National Natural Science Foundation of China (42222506).
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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