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Abstract
The study of the optical properties of cirrus clouds is necessary to improve the accuracy of interpreting data from space lidars and ground-based lidar networks. Existing databases of backscattering properties do not include data on hollow columns. In this paper, the backscattering properties of randomly oriented hollow column ice crystal particles in cirrus at wavelengths of 355 nm, 532 nm, and 1064 nm have been investigated. The backscattering cross section (M11), depolarization ratio (δ), lidar ratio (S), and color ratio (χ) of randomly oriented hollow columns with sizes ranging from 10–316.23 µm are calculated within the framework of the physical optical approximation (PO). For the first time, we introduce the concept of modal hollow columns (MHC) suitable for mid-latitude regions, which simplifies the description of hollow ice crystals in nature. It is found that when the mixing proportion of MHC and solid columns (SC) reaches 50%, the mixing ratio can be distinguished by the lidar ratio for a wavelength of 1064 nm and by the χ(1064,532)-δ(532) relation.
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1. Introduction
Cirrus clouds affect the radiation budget of the Earth-atmosphere system and play a crucial role in weather and climate [1]. Understanding the optical properties of cirrus clouds requires addressing the scattering problem of non-spherical ice crystal particles [2]. Scholars have been studying this subject since the 1970s, using rigorous and approximation methods for simulation [3]. Over time, various computational models have been developed, each with its own strengths and weaknesses. For example, the finite-difference time domain method (FDTD) [4–6] and T-Matrix method [7–9] can accurately calculate electromagnetic wave solutions, but FDTD can only be applied to small crystals, while T-Matrix cannot represent the shape of large ice crystals. However, the size of the ice crystals, typically ranging from 30 µm to 300 µm, is much larger than the conventional lidar wavelengths. Consequently, the FDTD or T-Matrix requires extremely large computational resources. On the other hand, the traditional geometric optics approximation (CGOM) could address this problem [10–12], but this method, which uses the Dirac function to approximate Fraunhofer diffraction, has a singularity in the backward scattering direction due to the corner-reflection effect [13]. To overcome this problem, an approximate method of the physical optics, which is based on the beam-splitting technique (PO), has been proposed [14–16]. The method of physical optics has a long history of development. It was independently developed by several scientific groups: in Italy by Guasta et al. [17] in 2001, in the USA by Yang et al. [18] in 1996 and by Bi et al. [19] in 2011, in Japan by Masuda et al. [20] in 2012, in the UK by Hesse et al. [21] in 2018, and in China by Sun et al. [22] in 2022.
Lidar observations are used to study the optical properties of cirrus ice crystal particles. In the past, many papers have used this physical optical (PO) approximation method to simulate the backscattering Mueller matrix of non-spherical particles, trying to combine theoretical calculations with actual observations to improve the accuracy of retrieving ice crystal parameters from lidar observations. Borovoi et al. [23] used the PO method to simulate observations of quasi-horizontally oriented hexagonal ice crystals with the Cloud-Aerosol Lidar with Orthogonal Polarization (CALIOP) onboard the Cloud-Aerosol Lidar and Infrared Pathfinder Satellite Observation (CALIPSO) at two wavelengths and two angles of inclination (initial and present). They reported that the backscattering Mueller matrix is mainly affected by specular and corner-reflection terms. Simulation results also revealed that randomly oriented ice crystal particles have fine angular structure in the backscattering direction, and the approximate equation of the differential scattering cross-section was obtained [24]. The quasi-horizontally oriented ice crystals in cirrus were observed using a dual-wavelength polarization lidar, confirming the validity of the theoretical calculations within the framework of physical optics [25]. Okamoto et al. [26,27] applied the PO method and the modified GOIE method [20] to simulate optics properties of five different shapes of ice crystals: Voronoi aggregates, droxtals, bullets, hexagonal columns, and hexagonal plates, observed by CALIOP onboard CALIPSO and the Atmospheric Lidar (ATLID) on the Earth Clouds and Radiation Explorer (EarthCARE) satellite. They established a lookup table relationship between the depolarization ratio, the lidar ratio, and the color ratio. The results showed a correspondence between the lidar ratio and depolarization ratio, which can be used to detect the habits of quasi-horizontally oriented particles. These works provide theoretical support and the possibility of using multiple channels to observe cirrus particles in the future.
However, these studies have not been able to accurately match numerical simulations with actual observations. One of the main reasons for this discrepancy is the variety of shapes that ice crystal particles in cirrus clouds can take. There is a distinct difference between the shapes of cirrus ice particles formed in situ and those in anvils generated near convection [28]. Most of the time, we only consider simple shapes like columns and plates. However, when a regular hexagonal column particle is distorted, the backscattering Mueller matrix parameters will oscillate as the distortion angle increases. This leads to an increase in lidar ratio, depolarization ratio, and color ratio, while the backscattering cross-section decreases [29]. Some studies have reported on the backscatter properties of irregularly shaped particles [30–32]. Additionally, a novel method has been proposed in recent years to solve the scattering problem of non-spherical particles. It involves replacing non-spherical ice crystals such as columns, plates, and bullets with hyperspheroids [33,34]. However, all of these methods assume that ice crystals in the atmosphere are solid, which is not entirely true. Schmitt et al. [35] detected cavities in bullet and column ice crystals using balloon-borne Formvar replicators, indicating that hollow particles are common in cirrus clouds. They found out that about 50-80% of these ice crystals have cavities. Takano et al. [36] used the Monte Carlo geometric ray tracing method to calculate the scattering of ice crystals with various irregular structures. They discovered that hexagonal columns have completely different scattering characteristics when they are hollow instead of solid. In cloud chamber experiments, the hollow hexagonal column forms different cavity structures at different ambient temperatures, and these different cavity structures have varying scattering effects [37].
To improve the detection quality of cirrus using lidars, this paper focuses on studying the backscattering properties of hollow hexagonal columns. The physical-optics method, based on the beam-splitting technique (PO) [14–16], taking into account the interference and diffraction, allows us to calculate the backscattering properties relatively accurately and with high speed. In this study, we utilize the improved version [38] called modified beam-splitting 1 (MBS-1) algorithm to investigate the backscattering Mueller matrix of randomly oriented hollow hexagonal columns and the corresponding lidar parameters at wavelengths of 355 nm, 532 nm, and 1064 nm. Future studies will also explore the case of quasi-horizontal orientation.
2. Mueller matrix for randomly oriented hollow hexagonal column particles
In order to establish a hollow hexagonal column ice crystal model that is close to the actual atmosphere, the column is assumed to form symmetrical pyramidal cavities on either side. The cone tapers from the sides to the center, which resembles an hourglass shape [39]. We define a parameter µ to describe the degree of hollowness, which is defined as µ = 2 h/L (h is the height of the cavity cone) ranging from 0 to 1. The µ = 0 represents a solid column, while 1 represents cavities reaching the center of the column. Figure 1 illustrates the model.
For ice crystals in cirrus, the lengths L and widths D are not independent. In previous studies, there are mainly three empirical formulas to describe this relationship of hollow columns [40–42], as shown in Fig. 2. When L < 200 µm, the results of Mitchell 1994 and Auer 1970 are almost identical. In the radiative transfer ice crystal scattering database calculated later, the model by Mitchell 1994 was adopted [43]. Considering that the lengths L in this study are all less than 350 µm, so it is reasonable to adopt the model by Mitchell 1994 in this case. The relationship of L and D obeys Eq. (1) and Eq. (2) [40].
The Mueller matrix describes the depolarization and wave intensity, which can effectively characterize the backscattering properties of particles [2]. Generally, we only consider the zenith angle (t) orientation of the particle, assuming that the azimuth angle (α) and its main axis angle (γ) are uniformly distributed (here the α, t, and γ are the Euler angles according to “zyz-notation” or “y-convention”). However, when crystals have a random orientation, t is also uniformly distributed from 0 to $\pi $. The probability density function is shown in Eq. (3).
After averaging over α and γ angles, the backscattering Mueller matrix of randomly oriented ice crystal particles can be expressed as Eq. (4) [23,44]
If the crystal shapes have a plane of symmetry, the M14 = 0. Both the solid and hollow columns have the plane of symmetry, so the M14 element can be ignored. The M11 element represents the backscattering cross section, while the M22 element represents the depolarization properties. Directly observing the Mueller matrix with lidar is challenging, and an accurate Mueller matrix can only be obtained through numerical simulation. Even lidars that aim to measure all elements of the Mueller matrix require complex transformations of multiple independent measurements. However, the lidar parameters such as backscattering coefficient β, depolarization ratio δ, color ratio χ, and lidar ratio S can be calculated from the simulated Mueller matrix:
Here, the parameter c is a constant (representing the concentration of particles), β is proportional to M11. The quantities ${\sigma _\textrm{||}}$ and ${\sigma _ \bot }$ characterize the parallel and perpendicular components of the backscattered signal. The element ${\sigma _{ext}}$ is the extinction cross section, which proved to be equal to twice the crystal projected area (s) [45]. In the process of simulating the lidar signal from randomly oriented ice crystal particles, s can be considered as ¼ of the surface area of the solid crystals. In fact, it has the same value for a hollow crystal as well [46].
We adopt the MBS-1 algorithm to obtain the Mueller matrix within the physical optical approximation, which uses the beam-splitting algorithm to calculate the geometric optical near-zone field and then obtains the far field by using the Kirchhoff diffraction [14–16,38]. We consider 12 reflection/refraction events during the beam-splitting process. The calculations were carried out for three wavelengths: 355 nm, 532 nm, and 1064 nm, with the polarization taken into account. In the case of randomly oriented particles, the lidar tilt angle is not significant; therefore, we only calculate the case where the lidar angle is 0°.
3. Backscattering properties of typical randomly oriented hexagonal hollow columns
The size of ice crystals in cirrus usually ranges from 5 µm up to 4000 µm, which exceeds the lidar wavelengths. In this section of the paper, we calculate several typical lengths L of the crystals, which are shown in Fig. 3 and Fig. 4. The dimension parameter D is defined as the width of the column, which complies with Eq. (1) and Eq. (2). There are two main reasons for studying these typical crystals. One is that they can represent the size state of the actual cirrus crystals in most cases, and the other is that they can be easily translated into pairwise coordinates for subsequent studies. The main difference between the scattering properties of hollow ice crystal particles and solid particles is that the cavity structure eliminates the corner reflection and increases the complexity of the beam splitting process. The light beam from the cavity has a certain chance to re-enter the crystal. Therefore, in the backscatter direction, the scattering characteristics are not only affected by the crystal size but also sensitive to the degree of hollowness.
Fig. 3. The backscattered signals with wavelength of 355 nm, 532 nm and 1064 nm vary with the hollowness of randomly oriented hexagonal columns (µ).
We analyzed the backscattering characteristics of ice crystals ranging in size from 10 µm to 316 µm to study the combined influence of crystal size and hollowness degree µ. The physical optics method has a lower limit of applicability for particle sizes of 10 µm. As a result, calculations were only conducted for particles larger than 10 µm. However, the computational complexity of the problem increases significantly with particle size, so we restricted the particle size to 316 µm. The crystal sizes were selected to ensure that the interval between them is equal on a logarithmic scale. When averaging over orientations, it is important to choose a sufficient number of Euler angle steps to achieve convergence. Fortunately, averaging over one of the Euler angles can be done analytically, so it is necessary to perform calculations only for two angles, for which the crystal symmetry can be taken into account. According to our earlier estimation [47], it requires approximately 13,600 orientations for a column of 10 µm, 1,360,000 for a column of 100 µm, and 13,600,000 for a column of 316 µm for a wavelength of 0.532 µm. We choose the refractive index according to Ref. [48]: n = 1.3249 for 0.355 µm, n = 1.3116 for 0.532 µm, and n = 1.3004 + 0.0000019i for 1.064 µm.
Since the fitting relationship between length and width changes when the crystal length is 100 µm, we discuss the backscattering characteristics separately for crystal lengths less than 100 µm and the one greater than 100 µm. In Fig. 3(a), (d), (g), for the three wavelengths simulation, when the crystal length is less than 100 µm, the backscattering cross section M11 has a significant peak as the hollowness µ approaches 0.1 and 0.6, and oscillates between 0.1 and 0.3. For crystals of different sizes, the variation trend of M11 with hollowness is basically the same, but with the decrease of crystal size and the increase of wavelength, this trend gradually smooths out. For example, in Fig. 3(g) crystal with L of 10 µm has almost no such characteristics. This phenomenon indicates that the backscattering of ice crystals of these sizes is mainly affected by the diffraction effects because the crystal surface integration areas are reduced due to the reduction of crystal size. And the longer wavelengths cause the scattering angle to increase. That's why we can see the value of M11 is proportional to the size of the crystal for each hollowness. And for crystals of different sizes, the maximum M11 is achieved when the hollowness is 0 (solid) due to the corner reflection, which means if we ignore the existence of the hollow ice crystal particles in the actual cirrus observation, the accuracy of our inversion will be greatly affected. The depolarization ratio (δ) and lidar ratio (S) are also affected by diffraction. In Fig. 3(b), (c), (e), (f), (h), (i), when the crystal size is larger and the wavelength is shorter, the width of the peaks and valleys in the variation trend are narrower and the shape is sharper. When the hollowness degree is 0–0.3, the oscillation frequency is higher and the intensity is larger. It is worth noting that the trend characteristics of δ and S are roughly opposite. For example, when μ is approximately 0.45, the δ reaches a minimum, while the S reaches a maximum there. The same phenomenon applies when µ is equal to 0.6. And if µ is in the range of 0.6–0.9, the δ shows an obvious “W” shape, while the S shows an “M” shape. By comparing Fig. 3(b), (e), (h) and Fig. 3(c), (f), (i), it can also be found that in the backscattering direction of hollow columns, the depolarization ratio is more sensitive than the lidar ratio to wavelength. For example, in Fig. 3(h) when the wavelength is equal to 1064 nm and µ is between 0.3–0.6, the trough of the δ of small ice crystals decreases significantly, while the peak of the S is still obvious in Fig. 3(i).
When the crystal length exceeds 100 µm, the width is obtained by the exponential relation of Eq. (2), which results in the size change no longer being linear. The trend characteristics of Fig. 3 will also change. Figure 4 describes this fact. As shown in Fig. 4(a), (d), (g), the variation trend of the backscattering cross section M11 is similar to that in Fig. 3(a), (d), (g), except that the crest and trough move forward with the increase of particle size. This shows that when the lengths of the crystals are greater than 100 µm, the widths of the crystals increase exponentially with the lengths, which leads to an increase in the difference in the transmission of the beams in the crystals. Similar changes in backscattering cross section waveforms due to changes in the optical path in solid ice crystals have been discussed in the past [49,50]. For δ and S, there is also oscillation when µ is between 0.1 and 0.3 in Fig. 4. However, when µ is greater than 0.3, it becomes worse regularly. It can be determined that when µ is less than 0.15, the variation trend of the S and δ of large-size crystals is basically consistent with that of ice crystals with L equal to 100 µm, and then gradually deviates with the increase of the degree of hollowing.
In Fig. 5, the color ratio has similar characteristics to Fig. 4 when µ is 0.1–0.3. It is also affected by the relationship between the lengths and widths of the crystal. When µ is greater than 0.3, the trend of color ratio change for crystal lengths greater than 100 µm disappears. In this figure, we also notice that when the hollowness is relatively small, the color ratio is easily greater than 1, which may be a typical feature of hollow columns compared to solid ones. In particular, when the hollowing degree is between 0.1 and 0.3, the color ratio reaches a maximum there and is accompanied by a sharp oscillation. This indicates that the refractive index of these wavelengths in this interval has a significant influence on the final position of the beam on the crystal surface and the shape formed. However, we cannot discuss in detail here the effect of the reflection/refraction beam inside the crystal on the backscatter, as it requires a more rigorous argumentation process, and we need to discuss the contribution of each characteristic light path [24,49,50].
Fig. 5. The color ratio of 1064/532 nm and 532/355 nm vary with the hollowness of randomly oriented hexagonal columns (µ).
It should be noted that for particles larger than 100 µm, the aspect ratio starts to change, resulting in different angles of hollowness for the same value of µ. Since the position of the peaks of element M11 depends on the angles of hollowness rather than the parameter µ, we observe a shift in the maximum of the peaks in Fig. 4 as the crystal size increases.
4. Backscattering properties for an ensemble of columns with different values of hollowness
In Ref. [35], it shows the relationship between the lengths of hollow columns L and the central values of hollowness µ. When L is larger than 100 µm, µ ranges from 0.7 to 0.9 with a central value of about 0.8. When L is small, there is a dependence, as shown in Fig. 6(a) (black line). The parameter µ will oscillate around the central value by about ±0.1. We found a quadratic function to fit it if L is less than 85 µm and then as L increases, it becomes equal to 0.8. We found a Gaussian distribution to describe it, like Eq. (6). The parameter m is equal to the central value µ, which depends on the lengths of hollow columns. For the element s, we assume that it is 0.05. The reason is to make the peak width of the probability density function close to 0.2. Figure 6(b) shows the case when L > 85 µm.
Mueller matrices of crystals of every size will be averaged using the following equation: $\delta = \mathop \smallint \nolimits_0^1 M(\mu ) p({\mu ,s = 0.05,m} )d\mu $ . We call crystals that are averaged like this as “modal hollow columns”. In an experiment of lidar to observe cirrus, the laser has a divergence angle such that the signal received is the sum of the backscattered contributions of a large number of ice crystals. Therefore, the modal hollow columns is close to the real. In this section, we calculate the modal hollow columns of crystals sized about 10–316 µm to analyze the backscattering properties that vary with the lengths of hollow columns L.
Fig. 6. (a) The relationship of hollowness and crystal length. (b) Gaussian distribution of µ = 0.8, s = 0.05.
Figure 7 shows the results. An intuitive feeling is that the backscattering characteristics of modal hollow columns (MHC) and solid columns (SC) are very different. Therefore, if the existence of hollow ice crystals is ignored and the hollow ice crystals cannot be estimated correctly in the calculation of radiative transfer mode, the final calculation result will be affected. From the first column of Fig. 7, we can see that the backscattering cross section of MHC is lower than that of the same size of SC. From the second and third columns, we can find that δ and S of the three wavelengths of MHC are much larger than that of SC. Like SC, the backscattering cross section of HMC also increases monotonically with increasing crystal size, with the difference that the rate of change is not stable. For SC, the depolarization ratio of the three laser wavelengths is less than 0.3, and the lidar ratio is less than 50. For MHC, the depolarization ratio is almost above 0.4, and the lidar ratio is almost above 100 except for the 355 nm wavelength. For δ and S of MHC, there are roughly opposite trends which are similar to the cases before in Fig. 4 when δ and S vary with hollowness. This should be caused by the dependence of hollow degree and crystal size in HMC model. Also, from Fig. 3, we can see that the M11 element varies less with increasing particle size compared to δ and S. Therefore, we observe less oscillation of M11 after averaging.
Fig. 7. µ obeys Gaussian distribution, the backscattered signals with wavelength of 355 nm, 532 nm and 1064 nm vary with the lengths of randomly oriented hexagonal columns.
As shown in Fig. 8, the color ratio of MHC varies widely and is always greater than 0.6, and sometimes even greater than 1. In contrast, SC is stable, with the value of χ(1064, 532) ranging from 0.4-0.6 and floating around 0.6 for χ(532, 355).
Fig. 8. µ obeys Gaussian distribution, the color ratio of 1064/532 nm and 532/355 nm vary with the lengths of randomly oriented hexagonal columns.
We believe that this is the most universal model to describe hollow ice crystals, as summarized in the previous observations. Of course, it may not be perfect, because the observed ice crystals are mainly from the middle latitudes [35]. At different latitudes, there may be some differences in the relationship between the degree of hollowness and the size of the columns, which needs further study and discussion. Because we have noticed that the frequency of hollow ice crystals in the high latitudes is slightly different from that in the middle latitudes [51]. This dependence may also be related to the season because the formation of the cavity is closely related to the environmental temperature [52]. The biggest benefit of this MHC model idea is that it simplifies the description of hollow ice crystals in ice clouds. However, we can learn from Fig. 7 that in the real atmosphere, MHC does not appear alone. Usually, the MHC and SC should be mixed in a cloud, which is why the lidar ratio of atmospheric ice clouds above 100 is unrealistic.
5. Backscattering properties of the modal hollow columns with the size averaging
Previously, we discussed randomly oriented hollow columns of various sizes and degrees of hollowness, which represent the backscattering properties of single crystals. This will help us gain a better understanding of how changes in crystal parameters affect backscattering properties. It can serve as a reference point for studying the backscattering characteristics of hollow ice crystals. However, due to its specificity, it cannot be used for real lidar observation experiments. In the actual atmosphere, when the lidar detects cirrus clouds, there is an ensemble of crystals with different sizes, similar to the concept of averaging over hollowness mentioned earlier. We suggest that the crystal size in cirrus follows the Gamma distribution, which has been used by many researchers [27,44,53].
The ${N_0}$ is the number concentration of ice crystals per unit volume, the rm is the modal size, and p is the dispersion of the distribution. Based on previous studies [54,55], it is reasonable to set p to 2. Thus, the Gamma function $\mathrm{\Gamma }(p )= 1$. The req is the volume-equivalent spherical radius which is defined as ${\left( {\frac{{3v}}{{4\pi }}} \right)^{\frac{1}{3}}}$. The final probability density function can be written as Eq. (8).
Using Eq. (7) we can get the effective radius reff by Eq. (9).
The averaging method is the same as in section 4. Therefore, in this section, we consider not only the joint contribution of the coupling relationship between the hollow degree µ and crystal length L in the backscattering direction but also the spectral distribution of crystal size in cirrus.
Figure 9 shows the results of mixing the MHC and SC in a certain proportion. In Schmitt et al. [35], in the mid-latitudes, the proportion of hollow ice crystals is more than 50% and sometimes as high as 80%. After mixing, we noticed that the lidar ratio is significantly reduced, especially at the wavelength of 355 nm, it is below 50. This is close to the actual observation [56]. According to the numerical simulation results, although the backscattering characteristics of MHC and SC are very different, once the two are mixed, they will approximate the characteristics of SC. According to the backscattering cross-section in Fig. 9, the backscattering energy of MHC and SC mixed in different proportions is significantly different. With the increase of the effective radius of the crystals, δ of MHC and SC with different mixing ratios may tend to converge, which needs to be proved by further calculation of larger sizes. Therefore, the δ may not be suitable for directly distinguishing the mixture ratio of MHC and SC. However, when the proportion of MHC is higher than 50%, S has a better differentiation, especially when the wavelength is longer, such as 1064 nm.
Fig. 9. $\mu $ obeys Gaussian distribution, D obeys Gamma distribution, the backscattered signals with wavelength of 355 nm, 532 nm and 1064 nm vary with the reff of randomly oriented hexagonal columns.
Because long wavelengths are more sensitive to mixtures of MHC and SC, we can see in Fig. 10 that χ(1064, 532) has a better distinction than χ(532, 355). For SC, the χ remains almost constant as the effective radius increases. However, for the mixture of MHC and SC, it is sensitive, and the χ decreases with the increase of the effective radius.
Fig. 10. µ obeys Gaussian distribution, D obeys Gamma distribution, the color ratio of 1064/532 nm and 532/355 nm vary with the lengths of randomly oriented hexagonal columns.
In some of the previous studies, the relationship of χ(1064,532)-δ(532) can be used to identify particle types. And the S(355)-δ(355) relationship can further distinguish the habit and orientation of particles [26,27]. In this paper, we made a similar relationship, like for solid particles. As shown in Fig. 11, we find that when the proportion of MHC is more than 50%, it can be distinguished by using this relationship, especially in Fig. 11(c), where we can see that the graph of χ(1064,532)-δ(532) shows an obvious linear progressive relationship as the proportion of MHC increases. Of course, there are large gaps between the patterns of 100% MHC and 95% MHC. These gaps are caused by the nature of SC, as seen in Fig. 9 and Fig. 10 above. Only when the SC proportion is infinitely small, these gaps can be filled.
Fig. 11. The relationship between lidar parameters of different ice crystals. (a) the relationship of χ(532, 355)-δ(355). (b) the relationship of S(355)-δ(355). (c) the relationship of χ(1064,532)-δ(532). (d) the relationship of S(532)-δ(532).
6. Conclusions
We conducted a detailed study on the backscattering characteristics of randomly oriented hollow hexagonal prisms of various sizes, considering the variation in hollowness. It was observed that when L exceeds 100 µm, the characteristic trend of backscattering property with µ becomes indiscernible due to the dependence on crystal lengths and widths. The variation of δ and S exhibits an approximately opposite trend with µ. On the basis of previous studies, we propose the concept of modal hollow columns (MHC), which simplifies the description of hollow ice crystals, particularly their random orientation. Using this model, we investigate the mixture of hollow columns and solid columns in specific proportions. Our findings suggest that the lidar ratio at a wavelength of 1064 nm and χ(1064,532)-δ(532) are effective in distinguishing the mixing ratio when the MHC proportion reaches 50%. It is important to note that the MHC model is a novel concept and may not be perfect. We plan to refine it in the future by incorporating observational data.
Funding
National Natural Science Foundation of China (41975038, 42375122); Russian Science Foundation (21-77-10089); CAS Project for Young Scientists in Basic Research (YSBR-066); Key Research and Development Program of Anhui Province (2022h11020008); Natural Science Foundation of Anhui Province (2008085J33, 2208085UQ01); Hefei Institutes of Physical Science, Chinese Academy of Sciences Director's Fund (2021YZGH01).
Acknowledgments
The authors thank the Chinese Academy of Sciences (CAS-PIFI, 2021VTA0009 and CAS-YIPA, Y2021113) for their support in this work. V. Shishko and A. Konoshonkin acknowledge the support of the Tomsk State University Development Program (Priority-2030). The authors also thank all reviewers and editors.
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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