## Abstract

The velocity of cloud droplets has a significant effect on the investigation of the turbulence-cloud microphysics interaction mechanism. The paper proposes an in-line digital holographic interferometry (DHI) technique based on depth expansion and self-fusion algorithm to simultaneously extract particle velocity from eight holograms. In comparison to the two-frame exposure method, the extraction efficiency of velocity is raised by threefold, and the number of reference particles used for particle registration is increased to eight. The experimental results obtained in the cloud chamber show that the velocity of cloud droplets increases fourfold from the stabilization phase to the dissipation phase. The measurement deviations of two phases are 1.138 and 1.153 mm/s, respectively. Additionally, this method provides a rapid solution for three-dimensional particle velocimetry investigation of turbulent field stacking and cloud droplets collisions.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

Investigation of cloud microphysics is of great significance for deepening the understanding of the radiation energy budget, water cycle process, and precipitation mechanism [1–4]. Especially, the turbulence in cloud accelerates the collision of cloud droplets through vortex superposition except shear and inertia forces. Because the velocity distribution of cloud droplets is directly affected by turbulence, their collision efficiency is enhanced. So, turbulence causes the condensation and collision growth of small particles in the cloud droplets spectra, which further complicates cloud microphysical characteristics. During the last few years, more research on the influence of turbulence disturbance on cloud droplets spectra has been carried out for better understanding cloud microphysical properties. Lu [5] found the correlation between relative dispersion and volume-mean radius was from positive to negative as volume-mean radius increased. Falkovich [6] concluded that air turbulence can substantially accelerate the appearance of large droplets that trigger rain. Sardina [7] showed that the variance of the droplet size distribution increases in time as *t*^{1/2}. Kumar [8] presented direct numerical simulations of turbulent mixing followed by droplet evaporation at the cloud-clear air interface in a meter-sized volume. Wojciech [9] delineated the multiscale nature of turbulent cloud microphysical processes and open research issues. Barekzai's [10] simulations suggested that the addition of radiative cooling can lead to a doubling of the droplet size standard deviation. Thomas [11] used direct numerical simulation to study the impact of isotropic homogeneous turbulence on the condensational growth of cloud droplets. However, the above researches are limited to theoretical analysis and model simulation. Therefore, the experimental measurement methods of turbulence in cloud and cloud microphysical parameters have received increasing attention from researchers. The current particle velocimetry method is mainly the particle tracking velocimetry (PTV) and the particle image velocimetry (PIV). Hoyer [12] presented an experimental setup and data processing schemes for three-dimensional scanning particle tracking velocimetry (SPTV), which expanded on the classical three-dimensional particle tracking velocimetry. Buchmann [13] demonstrated the three-dimensional velocity measurements of micron-sized particles suspended in a supersonic impinging jet flow. Novara [14] adapted the Shake-The-Box (STB) particle tracking technique to evaluate the dense velocity fields. Qu [15] presented a synthetic aperture particle image velocimetry (SAPIV) method, which measured the flow field with large particle intensities from the same direction by 16 cameras. PIV and PTV methods are based on the localization and multi-angle observation of tracer particles. Since the tracer particles are added to the turbulent flow field by the jet, the turbulent distribution and cloud droplets velocity distribution are disturbed. Simultaneously, PIV and PTV are mainly used for particle velocity extraction in unidirectional motion to ensure enough reference particles, while the moving directions of cloud droplets in the turbulent field are random. Two methods are difficult to apply to cloud droplets velocity observation.

Therefore, a real-time and uninterrupted three-dimensional measurement method is sought to observe the cloud droplets velocity, such as in-line DHI. Based on the advantages of fast, real-time, non-destructive and full-field optical measurement [16–19], DHI is regarded as a reliable technology for observing dynamic particle field. Meng [20] introduced three experimental setups for holographic particle image velocimetry. Two examples of holographic particle image velocimetry (HPIV) systems and measurement results were given. Sheng [21] introduced that a digital holographic microscope is used to simultaneously measure the instantaneous three-dimensional flow structure in the inner part of a turbulent boundary layer over a smooth wall. Jong [22] presented the measurements of relative velocity statistics of inertial particles in a homogeneous isotropic turbulent flow with three-dimensional holographic particle image velocimetry. Beals [23] used digital holography to measure the cloud droplets spectra during entrainment and mixing with clear air. Toloui [24] presented the latest developments on digital in-line holography, which was suitable for wall-bounded turbulent flow measurements. Gao [25] introduced the application of microscopic dual-view tomographic holography to measure the three-dimensional position and motion of micro-particles located in dense suspensions. In this paper, in-line DHI is used to measure the three-dimensional velocity of cloud droplets. Through the depth expansion and self-fusion process, the particle information of the eight holograms is aggregated into a single image. The extraction process of cloud droplet velocity is performed only once, while the traditional two-frame exposure method is repeated 7 times to match the cloud droplet. Therefore, the registration and velocity extraction efficiency of cloud droplets are greatly improved. The three-dimensional velocities of cloud droplets from 8 holograms are simultaneously extracted by the proposed method. The velocity relationship between cloud droplet and flow field is explained by the Basset-Boussinesq-Oseen (BBO) equation [26–27]. Based on cloud droplet velocity, the results of DHI can be used to study the fluctuation characteristics of the turbulent field. In addition, this method can also provide a three-dimensional particle velocimetry solution for studying fluid dynamics.

## 2. Theory

When in-line DHI is used to measure particle field, the twin image is close to a uniform background with increased recording distance. Furthermore, the influence of twin image can be reduced in the reconstruction and identification processes of original particles. When a plane wave is used to irradiate particle, the diffracted light of the particle (as the object light) interferes with the undisturbed plane light (as the reference light), and is recorded as a digital hologram by a Complementary Metal-Oxide-Semiconductor Transistor (CMOS). During the recording process, the target surface of CMOS is the *u*-*v* plane. The coordinate of each pixel on the digital hologram is (*u*, *v*). During the numerical reconstruction process, the negative direction along the beam propagation is the positive direction of Z-axis. The pixel coordinate of the reconstructed hologram at the reconstruction distance *z _{r}* is (

*x*,

*y*,

*z*). In the numerical reconstruction, the reconstruction distance

_{r}*z*is divided to obtain 18 reconstructed holograms. The complex amplitude distribution

_{r}*U*(

_{Ri}*x*,

*y*) of reconstructed hologram with

*z*can be expressed as [28]

_{r}*λ*is the wavelength,

*R*(

*u*,

*v*) is the reference light,

*I*(

_{H}*u*,

*v*) is the intensity of interference fringes on the recording medium,

*k*is the wave number and 0 <

*i*< 19. The depth expansion method is used to aggregate all particles within the 18 reconstructed holograms. Each pixel in the reconstructed hologram is subjected to a maximum grayscale extraction across entire depth. The principle of extraction is as follows where,

*G*

_{0}(

*x*,

*y*) is the depth expansion image, and |

*U*(

_{Ri}*x*,

*y*)|

^{2}is the intensity distribution of reconstructed hologram. Since the diffraction images of partially defocused particles are superimposed on the depth extension, the third-order Laplacian fusion algorithm [29] is used to suppress diffraction images. The first step is to construct a Gaussian pyramid.

*G*

_{0}is set as the 0th layer of the Gaussian pyramid. Then low-pass filtering and down-sampling are used on the 0th layer to obtain the first layer of the Gaussian pyramid. The above process is repeated to form the Gaussian pyramid. The

*l*-th layer image

*G*of the Gaussian pyramid can be expressed as

_{l}When the Laplace Pyramid is constructed, the *G _{l}* interpolation method is used to obtain the enlarged image

*G**, which can be expressed as

_{l}*x*and

*y*are the number of rows and columns of the

*l*-th layer of the Gaussian pyramid;

**w**(

*m*,

*n*) is a two-dimensional 3×3 window function, and the Laplacian decomposition is as follows where, 0 ≤

*l*< 4,

*LP*is the image of the

_{l}*l*-th layer decomposed by the Laplace pyramid.

*LPA*and

_{l}*LPB*are set as the

_{l}*l*-th layer image after the input images

*A*and

*B*are decomposed by the Laplace pyramid. The regional energy of each layer is calculated by

On the basis of regional energy, the corrected image *LP _{l}* can be obtained by

By making *LP _{l}* into Eq. (4) and recursion three times, the self-fusion image

*LP*is obtained, which can be expressed as

By binarization, corrosion, and expansion algorithms, *LP* is processed as identified particle image. The gradation of particle area is set to 1. By labeling the particle identification image, the X-axis and Y-axis coordinate (*x _{p}*,

*y*) of the particle can be obtained. In order to obtain the Z-axis coordinate

_{p}*z*, the boundary gradient variance method of particle is used. Based on a series of three-dimensional coordinates (

_{p}*x*,

_{p}*y*,

_{p}*z*) of particles at different times, the three-dimensional velocity

_{p}*v*of particle can be expressed as

_{p}*p*< 8,

*t*is the time interval for taking two consecutive holograms,

*s*is the displacement of particle during the time interval.

After the particle velocity *v _{p}* is acquired, the BBO equation can be used to research the motion relationship between particle and flow field. It can be expressed as

*ρ*is the density of particle,

_{p}*d*is the particle diameter,

_{p}*µ*is the dynamic viscosity of flow field,

*v*is the velocity of flow field,

_{f}*τ*

_{0}is the particle relaxation time, and

*t*is the particle movement time. The particle velocity

*v*and the flow field velocity

_{p}*v*are expressed as the integral of the complex sine signal, and the equation is as follows

_{f}*η*(

*ω*) is the ratio of the amplitude of the particle velocity

*v*to the fluid velocity

_{p}*v*, and is a function of frequency

_{f}*ω*. The

*φ*(

*ω*) is the difference of phase angle between the particle velocity

*v*and the fluid velocity

_{p}*v*. By combining Eqs. (10) and (11), the following equation can be obtained

_{f}From Eq. (12), the solution of BBO equation can be obtained as

*η*can be expressed as the quantitative relationship between the particle velocity and the turbulence velocity, which is

*η*= |

*v*|/|

_{p}*v*|.

_{f}*φ*is the phase difference between particle and fluid. In the calculation process, the value of

*ω*is taken as the maximum frequency of turbulence. According to the solution

*η*and measured particle velocity

*v*, the flow field velocity

_{p}*v*can be solved. Therefore, the three-dimensional velocity (

_{f}*v*,

_{x}*v*,

_{y}*v*) of a flow field can be obtained.

_{z}## 3. Experimental setup

The experimental setup for cloud droplets velocity measurement based on in-line DHI is shown in Fig. 1. To minimize the influence of the light source in a turbulent environment, an experimental setup employs a continuous laser with a wavelength of 532 nm and a maximum power of 100 mw as the light source. The laser beam is expanded by lenses L1 and L2 and condensed by lenses L3 and L4 into a parallel beam with a diameter of 30 mm. The object beam is composed of the diffracted light generated by the beam impinging on the particles. And the reference beam is composed of the parallel beam that does not pass through the particles. On the focal plane of the microscope objective, the two beams form an interference hologram. A 4.1^{×} magnification microscope is used to increase the accuracy of the X and Y axes in the optical path and to decrease the depth of focus in the Z-axis. Since the parallel light wave is converted into spherical wave by the microscope objective, the actual magnification is affected by the distance between the CMOS and the microscope objective. In the experiment, the positions of the CMOS and the microscope objective are fixed to ensure constant magnification. The 4.1^{×} magnification is the result of actual measurement and calibration. The CMOS (500 frames per second, 1696W × 1710H pixels, 8 µm × 8 µm pixel size) is used to record the amplified interference fringes in order to create a digital hologram. The measurement area of DHI is a 3.3 mm × 3.3 mm × 9.0 mm cuboid. The smallest detectable particle, limited by magnification and pixel size, is 3.9 µm. The aperture P is used to filter out stray light that is reflected internally to the microscope group.

The holographic experiment setup is positioned at the bottom of the cloud chamber on the experimental platform. The experiment is conducted at the temperature of 288.15 ± 2 K, the relative humidity of 99%, and the pressure of 650 hPa. The entire process of cloud formation by expansion takes 5 minutes, while the growth and dissipation processes of cloud droplets take 1.5 minutes. The 3.5 minutes cloud process in the center is relatively stable. In the experiment, the hologram is recorded at 500 frames per second, and the exposure time is set to 10 µs. The resolution of DHI in the XY plane is primarily determined by the pixel size (8 µm), the CMOS chip size (13.56 mm × 13.68 mm), and the optical magnification (4.1^{×}). Theoretically, the particle diameter error is 1.95 µm (the equivalent pixel size), and the X-axis and Y-axis positioning errors are 1.39 µm (the square root of the equivalent pixel size).

In the Z-axis direction, combined with the grayscale gradient variance method [30] of the fusion hologram, the Z-axis coordinate of the particles can be determined. At the boundary of a focused particle in a reconstruction hologram, the light-dark transition zone is narrower than other positions, indicating that the grayscale scale of the particle boundary changes more drastically. The grayscale gradient can be used to characterize the change, and the variance can be used to evaluate the severity of the change, then the grayscale gradient variance *k* is expressed as

*G*is the grayscale gradient of each point in the particle area; `

*G*is the average grayscale gradient of all pixels in the particle area, and

*M*is the total number of pixels in each particle area of a hologram. After binarization, the circular area with the gray value of 1 is the particle area, while the background area is marked as zero. The reconstruction distance of 30 mm is evenly divided into 100 reconstruction intervals with equal intervals for the purpose of determining the particle focus on the Z-axis, and the value of

*k*is determined by Eq. (14). The reconstructed distance corresponding to the maximum value is taken as the particle's Z-coordinate. The reconstruction distance is proportional to the actual distance by a factor of 4.1

^{2}, and the Z-axis coordinate value is divided by 16.81 to eliminate the effect of system magnification on the reconstruction algorithm. The corrected Z-axis coordinate is identical to the actual particle position, the actual distance is 1.78 mm, and each reconstruction interval is 17.8 µm. According to the green curve in Fig. 2, the error of 17.8 µm to 53.4 µm is related to multiple peaks of the curve in a high-concentration particle environment. The multiple peaks on the focus judgment curve are caused by the interference of out-of-focus image. The Gaussian fit is employed in order to maintain the unimodality of the judgment curve. Furthermore, the influence of noise caused by out-of-focus image is reduced. In Figs. 2(a), 2(b), and 2(c), the judgment values are 1.353 mm, 1.193 mm, and 1.388 mm, respectively, while the actual positions are 1.35 mm, 1.20 mm, and 1.40 mm. Because the errors of the three focusing curves are all within a single reconstruction interval, the Z-axis positioning error is 17.8 µm. When the Z-axis coordinate error is less than 4 times the average diameter, the positioning coordinate is high-precision. The average diameter of cloud droplets in the cloud chamber is 8.0 µm. The Z-axis positioning coordinate can be evaluated as high precision.

To ensure the reliability of three-dimensional velocity of particle, the high-precision three-dimensional position and identified particle area are required. For multi-layer object reconstruction in the measured field, the reconstruction in the second plane may be interfered with the image in the first plane. The identified area of particle is larger due to the increased noise caused by out-of-focus image. In order to reduce the error of particle area, the gray threshold is used to binarize the self-fusion image in the experiment. A standard diffraction plate with 10 µm spot array is used to determine the gray threshold 0.41 as a constraint on particle boundary. Figure 3 shows four labeled images taken from the upper left, upper right, lower left, and lower right regions of the dot array diffraction plate, each with 121 particles. Among them, eight particles in Figs. 3(a) and 3(d) are unidentified, while the remaining particles in Figs. 3(b) and 3(c) are all identified. The recognition rate of particles is determined to be 96.7 percent. The main reason for the unidentified particles is the dust adhering to the lens, and the images of these “particles” are also picked up by the camera. The diffraction images of dust in two adjacent holograms are highly similar when acquired by a CMOS at the rate of 500 fps. These diffraction images and background are removed by subtracting two adjacent holograms. However, the diffraction rings of some particles are damaged by subtraction. The grayscale values of these particles are significantly reduced in numerical reconstruction. Eventually, some particles with incomplete diffraction rings are unable to be identified, which is one of the reasons for the identification rate calibration. After the diameters of the particles in the four regions are summarized, the uncertainty is 10.6 ± 0.8 µm, and the particle size error of DHI is corrected to 0.8 µm. The particle spacing on the diffraction plate is 100 µm, and the identified spacings are generally distributed at 101.3 ± 3.7 µm. Hence, the X and Y axes positioning accuracy is corrected to 3.7 µm.

## 4. Analysis and discussion

As illustrated in Fig. 4, the velocity extraction process for digital holograms is divided into several phases: numerical reconstruction, depth extension [31–33], image self-fusion, particle identification, particle marker, focus judgment on the Z-axis and three-dimensional velocity calculation. Among them, depth expansion is critical for rapidly extracting particle velocities. By depth extension, the focused particles of 18 reconstructed holograms are extracted into an extended image. Image Self-Fusion Algorithm (ISFA) is used to suppress out-of-focus particle noise in this expanded image. In the whole hologram processing, five times of ISFA are adopted to obtain the final self-fusion image. The self-fusion image is used for subsequent particle identification, particle marker, and focus judgment on the Z-axis. During the particle identification, algorithms such as binarization, erosion and dilation are used to obtain particle area. The labeling algorithm based on the connected region is used to mark the center coordinate (*x _{p}*,

*y*) of each particle. Through the focus judgment on the Z-axis, the Z-axis coordinate

_{p}*z*is solved. The three-dimensional velocity

_{p}*v*of particle is obtained by combining the particle coordinate (

_{p}*x*,

_{p}*y*,

_{p}*z*) and Eq. (9).

_{p}To evaluate the efficiency of velocity extraction, the proposed method is compared with the traditional two-frame exposure method based on the time consumed. The two-frame exposure method is less efficient at extracting velocity than the proposed method for processing multiple holograms. Particle velocity extraction from 8 holograms is used as an example. The time consumption of the proposed method for numerical reconstruction, depth expansion, image self-fusion, particle identification and marker, and focus judgment is 157.76 s, 0.19 s, 26.04 s, 1.94 s, and 516.23 s, respectively. The total time is 702.16 s. In the two-frame exposure method, the three processes are repeated seven times including numerical reconstruction, particle identification and marker, and focus judgment. The time consumed is 278.13 s, 14.85 s and 1925.69 s, respectively. The total time is 2218.67 s. As a result, the extraction efficiency of the proposed method is about 3 times that of the two-frame exposure method.” have been added in the revision.

#### 4.1 Simulation result

Particle velocimetry experiment in water is designed and performed to assess the algorithm’s reliability from numerical reconstruction to particle marker. A magnetic rotor spinning at a constant speed (700 r/min) is used to drive the water flow in a cubic container with a side length of 50 mm to create the vortex field. To simulate cloud droplets, 35 µm standard particles are added to the container, and DHI is used to observe the particles movement. Figure 5(a) illustrates the results of internal flow field simulation using flow field simulation. The experiment recorded 3000 holograms, with the 1101st and 1102nd holograms being used to detect the entire process. The obtained particle velocity field is shown in Fig. 5(b). Following the velocity comparison, the measured particle velocity amplitudes are consistent, with a particle velocity range of 0.0347 m/s ∼ 0.0878 m/s in the experiment. Due to the high viscosity of water, the particle velocity can be approximated by the velocity of the flow field, and the simulated particles velocity is assumed to be 0.033 m/s ∼ 0.089 m/s. Thus, the simulation results agree well with the experimental results, demonstrating the reliability of the DHI velocity measurement.

#### 4.2 Self-fusion algorithm

When processing holograms including cloud droplets, it is necessary to suppress or filter out diffraction images of particle edges based on a fixed binarization threshold and identify more effective particles. As a result, the conventional Laplacian fusion algorithm is modified to recognize the contours of tiny particles. The *LP*_{0}, *LP*_{1}, and *LP*_{2} obtained from Eq. (7) are multiplied by 0.8, 0.4, and 0.1, respectively. The resulting modified algorithm is ISFA. According to Eq. (7), the premise for obtaining *LP _{l}* is to input two distinct images. The depth expansion image

*G*

_{0}is multiplied by 0.90 and 0.99 to create image

*A*and

*B*for normal fusion execution. The effect of particle fusion after processing with the ISFA is analyzed in Fig. (6). The particles depicted in Figs. 6(b) and 6(d) have a higher mean grayscale value and more defined edges than the particles depicted in Figs. 6(a) and 6(c). To better describe the changes in particle edges, the grayscale value distributions of particles 1 and 2 along the red dotted line in Fig. 6 are extracted, as shown in Figs. 6(e) and 6(f). Typically, the edge rise distance (between 10% and 90% of the maximum grayscale value) is used to describe the sharpness of particles. The processed edge rise distance decreased from 6.92 µm to 4.25 µm based on the grayscale curve of particle 1. The edge rise distance of particle 2 is reduced from 2.58 µm to 1.42 µm, and the sharpness of both particles is improved. For particle identification, the increase in particle center gray value and edge sharpness directly improves the accuracy of size identification and X-axis and Y-axis positioning.

#### 4.3 Depth expansion

The depth extension results for 2, 4, 6, and 8 groups of reconstructed images are shown in Fig. 7. One or two of the eight trajectory images of some particles are not identified, and the method of adjacent coordinate interpolation is used to fill in the gaps. Then, for the entire depth range (0 mm ∼ 9 mm), the particle trajectories are extracted as described previously. The extended eight-frame hologram performs well in particle identification. Although overlapping particles and some particles with incomplete trajectories are removed, over 83 percent of the particle is retained, which means that the velocity of 71 ∼ 82 particles can be obtained simultaneously on a single image.

#### 4.4 Cloud droplets velocity extraction

Figure 8 shows the extended fusion images, particle identification images, and particle marker images of 0 mm ∼ 3.6 mm, 3.6 mm ∼ 6 mm, and 6 mm ∼ 9 mm in the Z-axis direction, respectively. When a high-speed camera is used to capture images, the motion of particles between two adjacent frames is considered uniform motion. As a result, the difference in three-dimensional coordinate between the two frames represents the particle displacement, and the ratio of the displacement to the time interval represents the current velocity of particle. Figures 9(a), 9(b), and 9(c) depict the particle velocity fields obtained from the three marker images.

In conjunction with the labeling results in Fig. 8, the velocities of 232 particles in the range of 0 mm ∼ 9 mm are extracted. The complete eight-frame particle three-dimensional velocity field is shown in Fig. 9(d). The particle velocity measurements range of 7.641 mm/s ∼ 32.365 mm/s, with an average of 18.032 mm/s. The movement direction of particle swarm is changed between 2.4 mm and 5.2 mm along the Z-axis, revealing a more concentrated spatial distribution and a chaotic movement trajectory. This demonstrates that in the sampling space (98 mm^{3}), the random turbulence clearly affects the three-dimensional space motion of particles, and it is also an important direction of particle velocity research. The size distribution of cloud droplets is depicted in Figs. 9(e) and 9(f) in terms of both spatial distribution and numerical properties of cloud droplets size. The results indicate that cloud droplets have a relatively uniform spatial distribution. Simultaneously, the size distribution is generally between 6 µm and 10 µm. The data is primarily used for cloud droplets follow-up investigation.

Statistical analysis of cloud droplets velocity reveals that cloud droplets velocities of 7 mm/s and 40 mm/s coexist in small cloud droplets groups. Thus, as illustrated in Fig. 10(a), the fluctuation of velocity and cloud droplets diameter are considered to be intrinsically linked. With increasing diameter, the fluctuation range of cloud droplets velocity gradually decreases. This trend is particularly pronounced in the 6.5 µm ∼ 7 µm region, which has a high density of data points. As a result, it can be considered that as the size of cloud droplets increases, the influence of flow field fluctuations on their movement gradually diminishes. This means that the ability of cloud droplets to follow the fluid is diminished. As illustrated in Fig. 10(b), the magnitude distribution of cloud droplets velocity is closer to the Gamma distribution than the size distribution, demonstrating the data stability of DHI velocity measurement system. The effect of the flow field on the cloud droplets velocity distribution is also worth examining in an expanding cloud chamber.

The BBO equation is used to characterize the strength of cloud droplets followability in order to construct the influence mechanism between cloud droplets velocity and flow field velocity. Equation (13) is solved for obtaining the ratio of fluid velocity to cloud droplets velocity. Table 1 summarizes the pertinent environmental parameters. As shown in Table 2, both cloud droplets size and turbulence frequency increase result in a decrease in cloud droplets followability. The ratio of 6 to 10 µm cloud droplets ranges between 0.9728 and 0.8472, indicating that cloud droplets exhibit strong flow field following characteristics, which is an inherent advantage of small particle measurement. The *η* (0.9251) corresponding to the median diameter (8 µm) is usually used to represent the overall followability of the cloud droplets swarm. Then, since the velocity distribution of cloud droplets is in the range of 7.641 mm/s ∼ 32.365 mm/s, the flow field velocity range is estimated to be 8.259 mm/s ∼ 34.985 mm/s.

After the three-dimensional field of cloud droplets velocity has been constructed, it is required to evaluate the velocity error. The in-line DHI velocity measuring system has a Z-axis positioning accuracy of 17.8 µm, a resolution of 3.9 µm, and a sample time interval of 2 ms. In the extreme instance, the displacement of the cloud droplets in the Z-axis results in an error of 17.8 µm every eight times intervals. The cloud droplets spacing measurement of the spot array diffraction plate indicates a positioning error of 3.7 µm in both the X-axis and Y-axis directions, as shown in Fig. 3. It can be considered that the displacement error is centered primarily along the z-axis. Therefore, the synthesized three-dimensional displacement error *δ _{s}* is 18.2 µm within 0.016 seconds. The time error associated with image storage within 1 second is typically a time interval (2 ms), and the time error

*δ*associated with cloud droplets collection within 0.016 seconds is 3.2 × 10

_{t}^{−5}s. According to the error synthesis equation, the velocity error

*δ*can be expressed as

_{v}*t*is 0.016 s, the average velocity of cloud droplets is 18.032 mm/s, and

*s*is 0.289 mm. Solving Eq. (15),

*δ*is 1.138 mm/s. The relative error of the recorded cloud droplets velocity is 6.311 percent when

_{v}*δ*is divided by the average velocity.

_{v}Cloud droplets have a higher velocity and a lower number concentration during the cloud dissipation phase. As a result, even after expanding and fusing the eight holograms at a depth of 9 mm in the Z-axis, the particle distribution remains sparse. Furthermore, some particles move out of view, resulting in incomplete motion trajectories. Figure 11 shows the velocity extraction process of sparse cloud droplets, from image fusion to image labeling, which is consistent with the cloud stabilization phase. From the particle labeling information in Fig. 11(c), the position and size of cloud droplets are obtained, which are transformed into a three-dimensional velocity field to define the particle motion.

Figure 12(a) depicts the three-dimensional velocity field, with the beginning positions of 32 cloud droplets represented by scaled-up red spheres. Two voids are observed in the spatial distribution of cloud droplets in Fig. 12(b), which are marked by the red dashed circle. From Fig. 12(c), the step-like variation of cloud droplets size distribution matches well with the discrete Gamma distribution. The median diameter of size distribution is 7.4 µm. During the dissipation phase, the velocity of the cloud droplets in the Z-axis is 2 to 5 times that of the other directions. Meanwhile, the velocity distribution of cloud droplets is 54.104 mm/s ∼ 162.530 mm/s, with an average velocity of 94.763 mm/s. The median size is 7.4 µm, and the turbulent frequency is 2 × 10^{3} Hz. According to the BBO equation, the velocity ratio *η* is 0.9427. Therefore, the velocity of the flow field is 54.104 mm/s ∼ 172.409 mm/s. Combined with the average velocity of 94.763 mm/s, *s* is taken as 1.516 mm. Solving Eq. (15), the standard deviation *δ _{v}* of velocity is 1.153 mm/s, and the relative error of cloud droplets velocity is 1.22 percent. The velocity relative error is minor in comparison to the cloud stabilization phase. The reason is that the standard deviation increases only by 0.013 and the average speed increases by 4.255.

Given the cloud chamber volume of 70 m^{3}, the vortex field formed by internal depressurization is larger than 1 m in diameter. When the large-scale vortex field is moving, the local cloud droplets group typically moves in a uniform direction. This is evident in observations of the cloud stabilization and dissipation phases, with cloud droplets constantly traveling in the positive direction of the Z-axis. From Fig. 13(a), in the cloud dissipation phase, the size distribution range of cloud droplets is reduced to 6 µm ∼ 9.5 µm, and the cloud droplets velocity is increased to 4 times that of the stable phase. However, the dissipation and stabilisation phases have similar velocity distribution characteristics. The velocities of 6.5 µm ∼ 7.5 µm cloud droplets vary widely, and the cloud droplets velocity follows the Gamma distribution. As showed in Fig. 13(b), even with the data of 38 cloud droplets, the fitting effect of the velocity amplitude distribution is still great. This further demonstrates that the DHI velocity system accurately represents the velocity characteristics of cloud droplets.

## 5. Conclusion

Combined with the in-line DHI method, observation of cloud droplets velocity in warm stratiform cloud can be achieved by using a pulsed laser with sub-microsecond pulse width and a high-speed sampling camera. Three-dimensional velocity is mainly obtained using a three-dimensional ultrasonic anemometer. Since the propagation speed of ultrasonic wave is affected by the liquid water in cloud, there is the measurement error of wind velocity. The wind velocity can be inverted from the cloud droplets velocity measured by the proposed method. However, the cloud droplets velocity is limited by the frame rate of the camera. High frame rate and large pixel count are a pair of inherent contradictions in the commercial camera. Due to higher wind velocity in convective cloud, the proposed method should be optimized in subsequent studies. The research results can provide data support for the entrainment, collision and spatiotemporal evolution of turbulence in cloud.

## Funding

National Natural Science Foundation of China (41875034, 41975045, 52127802).

## Acknowledgments

Statistical support was provided by Jingjing Liu and Qing Yan. Writing assistance was provided by Pan Gao. Optical Design support was provided by Jun Wang and Jiabin Tang. Detection experiment was provided by Dengxin Hua and Yangzi Gao.

## 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.

## References

**1. **Y. N. Shi, F. Zhang, K. L. Chan, T. Trautmann, and J. Li, “Multi-layer solar radiative transfer considering the vertical variation of inherent microphysical properties of clouds,” Opt. Express **27**(20), A1569–A1590 (2019). [CrossRef]

**2. **Y. P. Zhang, C. F. Zhao, K. Zhang, J. Ke, H. C. Che, X. Shen, Z. F. Zheng, and D. Liu, “Retrieving the microphysical properties of opaque liquid water clouds from CALIOP measurements,” Opt. Express **27**(23), 34126–34140 (2019). [CrossRef]

**3. **J. M. Li, Q. Y. Lv, M. Zhang, T. H. Wang, K. Kawamoto, S. Y. Chen, and B. D. Zhang, “Effects of atmospheric dynamics and aerosols on the fraction of supercooled water clouds,” Atmos. Chem. Phys. **17**(3), 1847–1863 (2017). [CrossRef]

**4. **S. N. Gao, C. S. Lu, Y. G. Liu, F. Mei, J. Wang, L. Zhu, and S. Q. Yan, “Contrasting Scale Dependence of Entrainment-Mixing Mechanisms in Stratocumulus Clouds,” Geophys. Res. Lett. **47**(9), 086970 (2020). [CrossRef]

**5. **C. S. Lu, Y. G. Liu, S. S. Yum, J. Y. Chen, L. Zhu, S. N. Gao, Y. Yin, X. C. Jia, and Y. Wang, “Reconciling Contrasting Relationships Between Relative Dispersion and Volume-Mean Radius of Cloud Droplet Size Distributions,” J. Geophys. Res. Atmos. **125**(9), 031868 (2020). [CrossRef]

**6. **G. Falkovich, A. Fouxon, and M. G. Stepanov, “Acceleration of rain initiation by cloud turbulence,” Nature **419**(6903), 151–154 (2002). [CrossRef]

**7. **G. Sardina, F. Picano, L. Brandt, and R. Caballero, “Continuous Growth of Droplet Size Variance due to Condensation in Turbulent Clouds,” Phys. Rev. Lett. **115**(18), 184501 (2015). [CrossRef]

**8. **B. Kumar, P. Götzfried, N. Suresh, J. Schumacher, and R. A. Shaw, “Scale Dependence of Cloud Microphysical Response to Turbulent Entrainment and Mixing,” J. Adv. Model. Earth Syst. **10**(11), 2777–2785 (2018). [CrossRef]

**9. **W. W. Grabowski and L. P. Wang, “Growth of Cloud Droplets in a Turbulent Environment,” Annu. Rev. Fluid Mech. **45**(1), 293–324 (2013). [CrossRef]

**10. **M. Barekzai and B. Mayer, “Broadening of the Cloud Droplet Size Distribution due to thermal radiative cooling: Turbulent Parcel Simulations,” J Atmos. Sci. **77**(6), 1993–2010 (2020). [CrossRef]

**11. **L. Thomas, W. W. Grabowski, and B. Kumar, “Diffusional growth of cloud droplets in homogeneous isotropic turbulence: DNS, scaled-up DNS, and stochastic model,” Atmos. Chem. Phys. **20**(14), 9087–9100 (2020). [CrossRef]

**12. **K. Hoyer, M. Holzner, B. Lüthi, M. Guala, A. Liberzon, and W. Kinzelbach, “3D scanning particle tracking velocimetry,” Exp. Fluids **39**(5), 923–934 (2005). [CrossRef]

**13. **N. A. Buchmann, C. Cierpka, C. J. Kähler, and J. Soria, “Ultra-high-speed 3D astigmatic particle tracking velocimetry: application to particle-laden supersonic impinging jets,” Exp. Fluids **55**(11), 1842 (2014). [CrossRef]

**14. **M. Novara, D. Schanz, N. Reuther, C. J. Kähler, and A. Schröder, “Lagrangian 3D particle tracking in high-speed flows: Shake-The-Box for multi-pulse systems,” Exp. Fluids **57**(8), 128 (2016). [CrossRef]

**15. **X. Qu, Y. Song, Y. Jin, Z. H. Li, X. Z. Wang, Z. Y. Guo, Y. J. Ji, and A. Z. He, “3D SAPIV particle field reconstruction method based on adaptive threshold,” Appl. Opt. **57**(7), 1622–1633 (2018). [CrossRef]

**16. **J. Wang, J. L. Zhao, C. Qin, J. L. Di, A. Rauf, and H. Z. Jiang, “Digital holographic interferometry based on wavelength and angular multiplexing for measuring the ternary diffusion,” Opt. Lett. **37**(7), 1211–1213 (2012). [CrossRef]

**17. **P. Su, D. Sun, J. S. Ma, Z. P. Luo, H. Zhang, S. L. Feng, and L. C. Cao, “Axial resolution analysis in compressive digital holographic microscopy,” Opt. Express **29**(2), 1275–1288 (2021). [CrossRef]

**18. **Y. Y. Zhang, J. L. Zhao, J. L. Di, H. Z. Jiang, Q. Wang, J. Wang, Y. Z. Guo, and D. C. Yin, “Real-time monitoring of the solution concentration variation during the crystallization process of protein-lysozyme by using digital holographic interferometry,” Opt. Express **20**(16), 18415–18421 (2012). [CrossRef]

**19. **S. Shao, K. Mallery, S. S. Kumar, and J. Hong, “Machine learning holography for 3D particle field imaging,” Opt. Express **28**(3), 2987–2999 (2020). [CrossRef]

**20. **H. Meng, G. Pan, Y. Pu, and S. H. Woodward, “Holographic particle image velocimetry: From film to digital recording,” Meas. Sci. Technol. **15**(4), 673–685 (2004). [CrossRef]

**21. **J. Sheng, E. Malkiel, and J. Katz, “Using digital holographic microscopy for simultaneous measurements of 3D near wall velocity and wall shear stress in a turbulent boundary layer,” Exp Fluids **45**(6), 1023–1035 (2008). [CrossRef]

**22. **J. De Jong, J. P. L. C. Salazar, S. H. Woodward, L. R. Collins, and H. Meng, “Measurement of inertial particle clustering and relative velocity statistics in isotropic turbulence using holographic imaging,” International Journal of Multiphase Flow **36**(4), 324–332 (2010). [CrossRef]

**23. **M. A. Beals, J. P. Fugal, R. A. Shaw, J. Lu, S. M. Spuler, and J. L. Stith, “Holographic measurements of inhomogeneous cloud mixing at the centimeter scale,” Science **350**(6256), 87–90 (2015). [CrossRef]

**24. **M. Toloui, K. Mallery, and J. Hong, “Improvements on digital inline holographic PTV for 3D wall-bounded turbulent flow measurements,” Meas. Sci. Technol. **28**(4), 044009 (2017). [CrossRef]

**25. **J. Gao and J. Katz, “Self-calibrated microscopic dual-view tomographic holography for 3D flow measurements,” Opt. Express **26**(13), 16708–16725 (2018). [CrossRef]

**26. **G. D. Catalano, “A prediction of particle behavior via the Basset-Boussinesq-Oseen equation,” AIAA J. **23**(10), 1627–1628 (1985). [CrossRef]

**27. **M. Parmar, A. Haselbacher, and S. Balachandar, “Generalized Basset-Boussinesq-Oseen equation for unsteady forces on a sphere in a compressible flow,” Phys. Rev. Lett. **106**(8), 084501 (2011). [CrossRef]

**28. **L. C. Yao, J. Chen, P. E. Sojka, X. C. Wu, and K. Cen, “Three-Dimensional Dynamic Measurement of Irregular Stringy Objects Via Digital Holography,” Opt. Lett. **43**(6), 1283–1286 (2018). [CrossRef]

**29. **J. Du, W. S. Li, B. Xiao, and Q. Nawaz, “Union Laplacian pyramid with multiple features for medical image fusion,” Neurocomputing **194**, 326–339 (2016). [CrossRef]

**30. **H. A. Ilhan, M. Doğar, and M. Özcan, “Digital holographic microscopy and focusing methods based on image sharpness,” J. Microsc. **255**(3), 138–149 (2014). [CrossRef]

**31. **Y. C. Wu, X. C. Wu, J. Yang, Z. H. Wang, X. Gao, B. W. Zhou, L. H. Chen, K. Z. Qiu, G. Gérard, and K. F. Cen, “Wavelet-based depth-of-field extension, accurate autofocusing, and particle pairing for digital inline particle holography,” Appl. Opt. **53**(4), 556–564 (2014). [CrossRef]

**32. **D. K. Singh and P. K. Panigrahi, “Improved digital holographic reconstruction algorithm for depth error reduction and elimination of out-of-focus particles,” Opt. Express **18**(3), 2426–2448 (2010). [CrossRef]

**33. **J. Gao, D. R. Guildenbecher, P. L. Reu, and J. Chen, “Uncertainty characterization of particle depth measurement using digital in-line holography and the hybrid method,” Opt. Express **21**(22), 26432–26449 (2013). [CrossRef]