<figure><img src="image_1.png" /></figure>

# Droplet concentration in water spray using short-range elastic backscatter lidar

CHLOÉ LEDENT, CLAIRE LAURENT, BAPTISTE DÉJEAN, OLIVIER ROUZAUD, ANDRES BEDOYA-VELÁSQUEZ, AND ROMAIN CEOLATO*

ONERA, Université de Toulouse, 31000, Toulouse, France

*romain.ceolato@onera.fr

**Abstract:** To our knowledge, a novel experimental method is proposed to remotely measure the droplet concentration in a water spray using a short-range elastic backscatter lidar. A specific calibration technique is proposed to determine the lidar radiometric constant, enabling the conversion of lidar signals into attenuated backscatter signals and, ultimately, into aerosol properties, including average volume concentration. To our knowledge, a new formulation for the lidar constant is proposed, using the total spray transmittance and the particle size distribution of the water droplets, measured using the established laser diffraction technique. The lidar constant value is then compared and discussed in relation to an alternative method based on a Lambertian surface. Ultimately, the attenuated backscatter signals, calculated using the calibration constant, enable the estimation of the average volume concentrations of droplets in the spray under various injection conditions of a full-cone pneumatic atomizer. The retrieved concentrations range between 10⁸ and 10⁹ droplets per cubic meter, which are comparable to those obtained from laser diffraction measurements, especially for sprays with large droplets, i.e., with a Sauter mean diameter greater than 50 µm.

© 2025 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

We are interested in the characterization of droplets spray generated by the atomization process, and more specifically in droplet concentration. To achieve this, the dispersed phase is generally characterized in terms of droplet size and velocity using optical measurement techniques. The techniques used are based on various principles: refractometry, laser diffraction or the Doppler effect. A quantity of interest would be the droplet concentration in the flow since it can provide additional information on atomization quality and therefore on injection efficiency.

Remote sensing is a widely used method for detecting and measuring airborne particles. Numerous "Light Detection and Ranging" (lidar) systems have been developed, from the past years to now, to characterize the optical properties of these particles, to determine their concentration and to classify them according to their composition or their size. The particles measurements can be used to assess the impact of climate change, in particular by analyzing the biomass-burning plumes from the forest fires [1,2], the pollutants responsible for acid rains [3] or the condensation trails that appear at the exit of aircraft engines or wingtips [4,5]. Thereby, this technique could be used to complement the usual optical measurements on a dispersed phase. Depending on the type of interaction between the laser radiation and the studied particles, there are different lidar instruments: the "Differential-Absorption lidar" (DIAL) [6], the Doppler [7] or the elastic backscatter lidar [8]. This study focuses on a short-range elastic backscatter lidar, lidar Colibri [9], developed at ONERA Toulouse. This instrument allows measurements from several meters to a few kilometers. It has been used for quantitative solid aerosols profiling, especially black carbon aerosols produced by incomplete combustion or those remained after pyrolysis [10,11].
Since our study focuses on droplet concentration determined by a short-range elastic backscatter lidar, our primary objective is to experimentally determine the calibration constant of the instrument. To do so, we locally characterize a liquid spray with a laser diffraction particle size analyzer (Malvern Spraytec). This instrument has good accuracy and repeatability, as studied by Dodge et al. [12]. It has been used for several decades in various fields such as medicine [13] or agriculture [14], and it can determine the transmittance and the particle size distributions of droplets. Therefore, in the present study, these parameters are known at a distance y = 20 cm from the outlet of the atomizer. At this distance, the spray is sufficiently dense to be detectable by laser diffraction and to be considered as a surface by the lidar. Then, lidar backscattering measurements are carried out on this spray, at the point where the spray characteristics are known. Different injection conditions are tested by varying independently the air and liquid flow pressures of the atomizer. The lidar technique is thus assessed on liquid particles from a controlled source, which is not the usual application. Finally, a new expression is developed for the calibration constant by taking into account the optical and size parameters of the spray. This new method of calibration, using laser diffraction, is validated by comparing the calibration constant with the one determined by the usual surface target method [15–17]. The average volume concentrations N of the droplets of the spray are finally estimated from the lidar signals.

In this paper, the experimental setup, the principles and the complementarity of laser diffraction and lidar backscattering techniques are presented in § 2. Then, the computation method of the new calibration constant is described in § 3. Finally, § 4 presents the transmittances and the particle size distributions of the sprays from laser diffraction measurements with a Malvern Spraytec instrument, and also the average volume concentrations of the spray determined from attenuated backscatter signals of the lidar Colibri instrument. The results lead to a discussion on the instruments comparison.

## 2. Materials and methods

The experimental setup of the two optical measurement instruments that allows to determine the sprays properties using the laser diffraction (Malvern Spraytec) and the backscattering (lidar Colibri) techniques is presented in § 2.1. This section also presents the atomization of the water liquid sprays. Then, § 2.2 and § 2.3 describe the operation of each instrument.

### 2.1. Experimental setup

The atomizer was oriented parallel to the ground floor and located at $r_s = 7.1$ m from the lidar source. A calibrated Lambertian surface target was placed behind the spray, at $r_t = 9.6$ m from the lidar source, so as to stop the laser beam. Both laser diffraction and lidar backscattering measurements were carried out on a continuous water spray, at a distance y = 20 cm from the outlet of the atomizer. The transmitter and receiver modules of Malvern Spraytec, separated by 0.5 m, are aligned and located on either side of the spray. Since the parameters provided by laser diffraction are stable over time, each measurement lasts 30 s. This has been verified with a 5-minute measurement on a continuous spray. In addition, some measurements were repeated to ensure repeatability. Thus, the transmittances and the particle size distributions of the droplets are known locally for the 25 injection conditions.

Figure 1 shows the top view of the experimental setup of Malvern Spraytec and lidar Colibri on a controlled spray. For the sake of a simple representation, the two measurement techniques are shown side by side, although their measurements were carried out at the same distance, y = 20 cm, from the outlet of the atomizer, as aforementioned. At this position, the theoretical spray width is approximately 7 cm. This value was verified and confirmed using backlighting measurements. The uncertainty is about ± 0.40 cm.

The atomizer used to generate a continuous spray of liquid water droplets is a full-cone pneumatic atomizing nozzle (Lechler, 136.115.1Y.A2) with a 0.5 mm diameter outlet orifice.
<figure><img src="image_2.png" /><figcaption>Fig. 1. Top view scheme of the experimental setup of Malvern Spraytec and lidar Colibri to characterize a controlled spray from a pneumatic atomizer.</figcaption></figure>

In this case, the atomizer operates according to an internal mixing principle, whereby gas and liquid are mixed within the nozzle [18]. Distilled water has been used since it enables sprays to be characterized under standard conditions. The atomizer has been used for different liquid and air injection conditions: from 0.661 g s⁻¹ to 4.320 g s⁻¹ for the liquid flow rate, and from 2.18 bar to 6.59 bar for the air pressure, for a total of 25 injection conditions. The different flow speeds of gas and liquid generate pressure waves, breaking up the liquid into fine droplets.

Usually, such sprays are characterized by mean diameters that are computed from a notation standardized by Mugele and Evans [19]:

$$D_{p,q} = \left( \frac{\sum_{i=1}^{n} N_i \cdot D_i^p}{\sum_{i=1}^{n} N_i \cdot D_i^q} \right)^{\frac{1}{p-q}}, \quad (1)$$

where i is the index of a size class, Nᵢ is the number of droplets in the class i, Dᵢ is the diameter of the center of class i, and p and q are coefficients that can take different values depending on the chosen mean diameter. The Malvern Spraytec directly provides the mean diameters D₁₀, D₃₂ and D₄₃ of the droplets.

## 2.2. Laser diffraction technique

The Malvern Spraytec is a laser diffraction particle size analyzer that can determine a particle size distribution by using the Fraunhofer and Mie theories [20]. When a droplet crosses the laser beam, the light is scattered and the sensors of the Malvern Spraytec detect the angular variation of the light. Assuming spherical droplets [21,22], a lens of 300 mm allows to measure droplets with sizes ranging from 0.1 to 900 µm. The laser wavelength is 632.8 nm and its diameter is about 1 cm, which is ten times the one of the lidar Colibri beam.

Moreover, the transmittance T, which characterizes the optical density of a spray, is a parameter that is defined as the transmitted-to-incident light intensity ratio in a spray. The three mean diameters provided by the Malvern Spraytec are the most common to compare sprays and to characterize mass transfers, especially for combustion phenomena. While measuring these diameters, the instrument can also provide a particle density function (PDF) that follows a log-normal distribution for the droplets diameters [18].
Using the Beer-Lambert law, we can highlight that $T$ and the extinction coefficient $\alpha$ [$m^{-1}$] are linked by the following formula:

$$T = \exp\left(-\int_{\Delta_0}^{\Delta_r} \alpha(r')dr'\right), \qquad (2)$$

where $\Delta_0$ [m] and $\Delta_r$ [m] are, respectively, the beginning and the end of the spray, such as the spray width is $\delta_{spray} = \Delta_r - \Delta_0$ [m]. The initial position $\Delta_0$ corresponds to the position just before the laser beam goes into the spray.

## 2.3. Short-range aerosol lidar technique

The lidar Colibri (CAL-120) has been developed to remotely measure aerosol concentrations in real-time for various needs and measurement scenarios. The CAL-120 is a small, compact, and lightweight version of the forward-looking short-range elastic backscatter lidar (SR-EBL), featuring a bi-static and multi-axial architecture for remote aerosol measurements. It offers a range resolution of 10 cm and a time resolution of 1 ms. The emitter unit consists of a compact, air-cooled Nd:YAG laser that emits ns pulses with a pulse energy of 25 mJ, wavelength of $\lambda = 532.8$ nm, 20 Hz, and a beam divergence of 0.5 mrad. A pair of lenses expands and collimates the laser beam before directing it toward the particulate matter, here the droplets of the liquid water spray. The receiver unit, equipped with a Cassegrain telescope with a 90 mm effective diameter and a field-stop (FS) to regulate the field of view, captures the backscattered light. A series of optical components, including achromatic doublet (AD) lenses, a neutral-band filter (NBF), and an interference filter (IF), are mounted on a translation stage for precise adjustment of the focal plane. This capability is essential for addressing focusing issues encountered at short ranges. Additionally, the bi-static angle — the angle between the transmitter and receiver units — is a critical feature for short-range measurements, as it allows complete control over the overlap function. The sensing unit includes an ultrafast photomultiplier tube (PMT), whose signal is digitized by a 12-bit analog-to-digital converter (ADC) at a 1 GS/s sampling rate and then processed by a field-programmable gate array (FPGA) to extract the distance-resolved backscatter signal. The data is then transmitted to a computer for analysis and processing to determine the range-resolved backscatter signal using the Time-of-Flight (TOF) principle, which depends on the number and mass concentration.

Our method is based on lidar backscattering measurements under the single-scattering approximation, as described by the elastic lidar equation [23]:

$$P(r) = K_o \cdot O(r) \cdot U(r) \cdot r^{-2}, \qquad (3)$$

where $P(r)$ [V] is the lidar backscattered power converted to the detector voltage at a given range $r$ from the laser source, $K_o$ [V m$^3$ sr] is the radiometric lidar constant, $O(r)$ is the overlap function that corrects the incomplete light beam collection between the transmitter and the receiver [24], and $U(r)$ [m$^{-1}$ sr$^{-1}$] is the attenuated backscatter signal that depends on the extinction and backscattering coefficients, $\alpha$ and $\beta$ [m$^{-1}$ sr$^{-1}$], at a given range $r$.

Aerosol profiling and optical properties are derived from the elastic lidar equation, and since there is only one equation to solve that depends on both extinction coefficient $\alpha$ and backscattering coefficient $\beta$, it is necessary to use inverse methods, such as those of Klett [25,26] and Fernald [27]. However, the inversion requires knowledge of three quantities:

(i) the radiometric lidar constant $K_o$, which includes losses in the transmitting and receiving optics. This constant is of paramount importance since it permits to convert the raw measured signals into attenuated backscatter signals. Generally, it is determined with a measure into a "clean" atmosphere, i.e., an atmosphere without particles, far from the laser source. Several recent studies have proposed different methods to calibrate the lidar
instrument, especially by using a surface method with a calibrated Lambertian target [15–17]. Since the lidar calibration constant governs the attenuated backscatter signals and therefore all of the parameters derived from its analysis, it is important to determine it correctly. That is why one of the aims of this study is to propose a new calibration method, based on previous works [9], by using a liquid spray of which characteristics — transmittance and size distributions — are determined by laser diffraction ;

(ii) the lidar ratio LR [sr], which is the extinction-to-backscattering coefficients ratio, is assumed to be constant over the measurement distance [28], particularly because the optical properties of the studied particles are a priori unknown ;

(iii) the molecular particles contribution, α$_{mol}$ and β$_{mol}$ , are known and assessed with pressure and temperature models. The contribution is important for atmospheric measurements and the molecular lidar ratio LR$_{mol}$ is generally of 8 π /3 sr [27].

Therefore, the fundamental lidar equation of U ( r ) that has to be solved is:

$$
U(r) = \frac{RCS(r)}{K_o \cdot O(r)} = \beta(r) \cdot \exp \left( -2 \int_{0}^{r} \alpha(r') dr' \right), \quad (4)
$$

where RCS ( r ) [V m $^{2}$] is the range-corrected signal calculated with the backscattered power and the square range at which the measurement is performed. Besides, for lidar systems, we introduce the two-way transmittance T $^{2}$( r ) due to the back-and-forth movement of the beam in the spray:

$$
T^2(r) = \exp \left( -2 \int_0^r \alpha(r') dr' \right). \qquad (5)
$$

Additionally, it is important to present the extinction-to-backscattering ratio also known as the lidar ratio LR :

$$
LR = \frac{\alpha(r)}{\beta(r)} = \frac{C_{\text{ext}}(R)}{dC_{\text{back}}(R)}, \qquad (6)
$$

where C$_{ext}$ ( R ) and C$_{back}$ ( R ) [m$^{2}$] are, respectively, the extinction and backscattering cross-sections of a particle of radius R , and dC$_{back}$ ( R ) := dC$_{back}$ ( R ) / d Ω [m$^{2}$ sr$^{-1}$] is the differential backscattering cross-section with d Ω an element of the solid angle. Usually computed with the extinction and backscattering coefficients, it is however possible to approximate the lidar ratio LR from the extinction and differential backscattering cross-sections, C$_{ext}$ and dC$_{back}$ [8,9,29]. This ratio strongly depends on several particle properties like size, shape, composition and complex refractive index. The wavelength of the incident light should also be taken into account because some particles absorb more in the UV [30]. This is the reason why some studies are carried out with multiple-wavelength lidars [2,30,31]. Typical values for LR are ranging from 42 to 80 sr for desert dusts [30,32], from 26 to 87 sr for biomass-burning plumes [2,33], and from 22 to 32 sr for sea spray aerosols [34,35], at 532 nm. Outside the spray, there is only the molecular contribution.

The laser diffraction and backscattering measurements complement each other as they are non-intrusive techniques to determine, on the one hand, the particles sizes of a spray and, on the other hand, the average particles concentrations.

3. Lidar calibration

3.1. Usual calibration method using a Lambertian surface

Lidar backscattering measurements were carried out on a configuration that was previously studied in the literature [15–17]. This configuration requires a calibrated Lambertian surface target with a known Bidirectional Reflectance Distribution Function (BRDF) f$_{r}$ ( θ$_{i}$ ) [sr$^{-1}$] defined
for an incident angle $\theta_i$. The signal measured, without any particles, at the target allows to compute the lidar calibration constant $K_{\text{target}}$ [V m$^3$ sr] [9,17]:

$$K_{\text{target}} = K_o \cdot O(r = r_s) = \text{RCS}(r = r_s) \frac{c\tau \pi}{2 \rho}, \qquad (7)$$

where $c$ [m s$^{-1}$] is the light celerity, $\tau$ [s] is the laser pulse width and $\rho$ is the reflectance of the surface target. It is important to note that $K_{\text{target}}$ is named "constant" whereas it varies with the range since it depends on the overlap function $O(r)$, which has been nearly optimized (close to unity) in the experiment. In the present study, $O(r)$ is not specifically determined for each $r$ since all of the measurements — on the Lambertian surface and on the sprays — have been carried out at the same distance from the lidar source, $r_s = 7.1$ m.

### 3.2. New calibration method using a water spray

In the following, we introduce a novel lidar calibration method that employs a water spray, which is itself calibrated using laser diffraction. In this approach, the lidar radiometric constant $K_o$ is determined using the two-way transmittance $T^2$, measured by laser diffraction, and the lidar ratio $LR$, estimated from particle size distributions. Both the transmittance and size distribution are derived from laser diffraction measurements (Malvern Spraytec). Let us consider the derivative of the two-way transmittance:

$$\frac{d}{dr}(T^2(r)) = -2 \cdot \alpha(r) \cdot T^2(r) = -2 \cdot LR \frac{\text{RCS}(r)}{K_o \cdot O(r)}, \qquad (8)$$

where the derivative has been determined considering the lidar ratio $LR$ and the Eqs. (4) and (5). This equation is then integrated over [0, $r$] and the integral term of $U$ is averaged over the spray width $\delta_{\text{spray}} = \Delta_r - \Delta_0$. Indeed, despite a high lidar spatial resolution of 10 cm, $\delta_{\text{spray}}$ is narrower than it, so this leads to consider the spray as an unique point of measurement of the backscatter signal. Initial conditions can be written for $T^2$, $U$ and $\beta$ in the absence of particles because, in this case, there is no extinction and so the intensity of the light is maximum outside the spray:

$$T^2(r) = 1 \quad \text{and} \quad U(r) = \beta(r), \quad \text{for} \quad 0<r<\Delta_0. \qquad (9)$$

This leads to a new expression of the two-way transmittance across the spray region:

$$T^2(r) = 1 - 2 \cdot LR \frac{\text{RCS}(r)}{K_o \cdot O(r)} \delta_{\text{spray}}, \qquad (10)$$

and finally, the lidar calibration constant $K_{\text{spray}}$ using liquid spray measurements, at $r_s = 7.1$ m, is given by:

$$K_{\text{spray}} = K_o \cdot O(r = r_s) = \frac{2 \cdot \text{RCS}(r = r_s) \cdot LR \cdot \delta_{\text{spray}}}{1 - T^2}. \qquad (11)$$

It is crucial to note, at this stage, that the transmittance $T^2$, used for the computation, is measured by laser diffraction, as we are relying on the previously mentioned condition (Eq. (9)). Besides, the signal intensity $\text{RCS}(r = r_s)$ corresponds to the peak value of the lidar signal at the spray position since the signal other than this peak point is negligibly small, $\text{RCS} \sim 0$, as inferred from the conditions in the Eq. (9).

The lidar ratio $LR$ is estimated from Eq. (6) with the extinction and differential backscattering cross-sections, $C_{\text{ext}}$ and $dC_{\text{back}}$, but also with the probability density function (PDF) of droplets radii $f(R)$ obtained from Malvern Spraytec measurements, to analyze the whole spray according
to:

$$C_{\text{ext}}(R) = \sum_{i=1}^{N_d} C_{\text{ext}}(R_i) \cdot f(R_i) = \sum_{i=1}^{N_d} \pi R_i^2 \cdot Q_{\text{ext}}(\lambda, R_i) \cdot f(R_i), \quad (12)$$

$$dC_{\text{back}}(R) = \frac{1}{4\pi} C_{\text{back}}(R) = \frac{1}{4\pi} \sum_{i=1}^{N_d} C_{\text{back}}(R_i) \cdot f(R_i) = \frac{1}{4\pi} \sum_{i=1}^{N_d} \pi R_i^2 \cdot Q_{\text{back}}(\lambda, R_i) \cdot f(R_i), \quad (13)$$

where $\lambda$ is the beam wavelength taken at 532 nm, $Q_{\text{ext}}$ and $Q_{\text{back}}$ are, respectively, the extinction and the backscattering efficiencies, $\pi R^2$ is the geometrical cross-section for a spherical particle of radius $R$ and $N_d$ is the number of droplet size classes. The efficiencies are estimated for each radius of the distribution using the Lorenz-Mie theory calculated by Bohren and Huffman's code [36]. It is important to note that since each injection condition leads to a specific range-corrected signal RCS and a specific LR, each test has its own calibration constant $K_{\text{spray}}$. Nevertheless, if one wants to determine the aerosol concentration distributions in other configurations, it is necessary to determine an unique lidar calibration constant. Consequently, the general calibration constant is obtained with a least squares analysis on the attenuated backscatter signals of the 25 injection conditions. The general calibration constant $K_{\text{opt}}$ is the value of $K$ that minimizes the following relation:

$$\min \left( \sum_{n=1}^{25} \left( \frac{\text{RCS}(r)}{K_{\text{spray}}^{(n)}} - \frac{\text{RCS}(r)}{K} \right)^2 \right), \quad (14)$$

where $K_{\text{spray}}^{(n)}$ is the calibration constant of the test $n$.

# 4. Results and discussion

The particle size distributions and the transmittance factor, which have been determined from the laser diffraction technique, provide information about the physical and optical structure of the spray. Their use in the new lidar constant allows to calculate the attenuated backscatter signals and, ultimately, the average volume concentrations.

## 4.1. Water spray characterization

Since the 25 injection conditions show a widespread distribution of the transmittance, it is convenient to identify each measurement with its Sauter mean diameter $D_{32}$. This diameter defines the volume-to-surface ratio of the droplets. The injection conditions, from 0.661 g s⁻¹ to 4.320 g s⁻¹ for the liquid flow rate and from 2.18 bar to 6.59 bar for the air pressure, lead to different attenuations of the laser beam as shown in Fig. 2. The higher the air pressure, the lower the transmittance since the spray attenuates the light considerably. The lowest transmittance, 43.0 ± 0.3 %, is obtained for an air pressure of 6.59 bar and a liquid flow rate of 4.320 g s⁻¹. Note that at a constant air injection pressure, the transmittance varies. This is due to the change in spray characteristics because each liquid flow rate creates a spray with a specific particle density function. The coefficients of variation, which are the ratios of the standard deviation to the mean, of the parameters $T$ and $D_{32}$ are, respectively, lower than 1.21 % and 4.52 %, considering all tests except the injection condition of 3.37 bar and 0.661 g s⁻¹. Indeed, for this condition, we have measured $T = 66.88 \pm 6.07 \%$ and $D_{32} = 22.75 \pm 5.15 \mu m$.

Using Fraunhofer and Mie theories, Malvern Spraytec is able to provide a PDF of the droplets radii. This PDF is given in volume, $v(R)$, but it is possible to convert it in number $f(R)$ or in surface $s(R)$, as presented in Fig. 3(a). It can be noticed that the volume PDF follows a log-normal distribution and, for instance, the parameters are N(105.73, 3.76) [µm] for 3.37 bar and 4.320 g s⁻¹. Furthermore, the small droplets, below 5 µm in radius, are majoritary in number but their volume is negligible compared with large droplets; only one mode is observed on the
<figure><img src="image_3.png" /><figcaption>**Fig. 2.** Measured laser-diffraction transmittance *T*, for all tests identified by their Sauter mean diameter *D*<sub>32</sub>. For each pressure, the results are shown for the five liquid flow rates between 0.661 and 4.320 g s<sup>-1</sup>.</figcaption></figure>

number and volume distributions whereas two modes are detected for the surface distribution. Figure 3(b) presents the PDF in volume and the cumulative volume $C_v$ [%] for 6.59 bar and 4.320 g s<sup>-1</sup>.

<figure><img src="image_4.png" /><figcaption>**Fig. 3.** (a) Number (red), surface (green) and volume (blue) PDF for 3.37 bar and 4.320 g s<sup>-1</sup>, and (b) volume PDF as well as cumulative volume $C_v$ (%) for 6.59 bar and 4.320 g s<sup>-1</sup>.</figcaption></figure>

## 4.2. Assessment of the lidar radiometric constant

Figure 4(a) shows the time-average range-corrected signal $\overline{RCS}$ for a test with 3.37 bar and 4.320 g s<sup>-1</sup>. The peak at $r_s = 7.1$ m corresponds to the interaction between the laser beam (532 nm) and the liquid spray. The maximum value of this signal is used to compute the maximum range-corrected signal presented in Fig. 4(b). The stability of the backscatter measurements over 30 s has been assessed for 6 injection conditions. The coefficient of variation (ratio of the standard deviation to the mean) of the maximum $\overline{RCS}$ is lower than 5 %. The maximum $\overline{RCS}$ increases with the air pressure of the atomizer. This corresponds to lower transmittances which
involve a similar evolution of the coefficients $\alpha$ and $\beta$. The predominance of one over the other is not obvious because the lidar ratios $LR$ calculated with $Q_{\text{ext}}$, $Q_{\text{back}}$ — at 532 nm — and $f(R)$ are rather constant. Indeed, for all tests, $18.31 \text{ sr} < LR < 22.50 \text{ sr}$. This is consistent with the $LR$ of pure or salt liquid water aerosols [34,35], even though the studied droplets are micrometric in size and generated by a controlled atomizer. Concerning this limited range of $LR$, the lowest values are associated to the highest Sauter mean diameters.

<figure><img src="image_5.png" /><figcaption>**Fig. 4.** Average backscatter range-corrected signal $\overline{\text{RCS}}$ for (a) 3.37 bar and 4.320 g s⁻¹ (—), and (b) maximum RCS for all tests at the measurement distance $r_s = 7.1 \text{ m}$.</figcaption></figure>

The calibration constants $K_{\text{spray}}$ of each of the 25 tests, under various injection conditions, have been computed. By minimizing the differences between computed attenuated backscatter signals, we may determine the lidar calibration constant, denoted $K_{\text{opt}}$. The value is compared to the one obtained with a surface target measurement, without any droplets. The comparison of the lidar calibration constants leads to a relative error of 6.31 %. Indeed, $K_{\text{target}} = 134.47 \text{ V m}^3 \text{ sr}$ and $K_{\text{opt}} = 125.99 \text{ V m}^3 \text{ sr}$. Figure 5 shows the maximum attenuated backscatter signals calculated with the Eq. (4), the constant $K_{\text{target}}$ from the calibrated surface target and the constants $K_{\text{spray}}^{(n)}$ relative to each test $n$ (between 1 and 25). The latter range from 113.30 to 160.57 $\text{V m}^3 \text{ sr}$. Experimental values, for the two configurations, are close which justifies the relative error of 6.31 %.

<figure><img src="image_6.png" /><figcaption>**Fig. 5.** Maximum attenuated backscatter signals $U$ for the target (▼) and spray (▼) measurements, for all tests identified by their Sauter mean diameter $D_{32}$.</figcaption></figure>
By estimating the parameter uncertainties in Eqs. (7) and (11), we obtain a range of acceptable values for the calibration constants. One possible error is the spray width $\delta_{spray}$. With a spray angle of 20°, given by the manufacturer, we can deduce a spray width of 7.05 cm at 20 cm from the atomizer outlet. This value has been verified using backlighting measurements. For instance, the image analysis of a water spray at 3.37 bar and 4.320 g s⁻¹ shows a spray opening angle of 7.09 ± 0.34 cm, but it may be higher for a poorer atomization. Thus, a ± 10% uncertainty of the spray width value leads to a general calibration constant $K_{opt}$ between 113.39 and 138.59 V m³ sr. The uncertainty of this parameter may be due to the spray angle value specified by the manufacturer and to the presence of droplets beyond the theoretical spray width. Regarding the target measurements, the uncertainty of the reflectance is low because the target is calibrated. However, the laser pulse width may be different from the stated manufacturer value of 1.25 ns. A ± 10% uncertainty conducts to a calibration constant $K_{target}$ between 121.01 and 147.90 V m³ sr.

Concerning the order of magnitude of $U$, it is difficult to compare it with reference values because it is generally expressed in km⁻¹ sr⁻¹ for far aerosols measurements. The range of values of the present study may be attributed to the short-range distance measurements and to the density of the spray. This type of spray from a pneumatic atomizer is denser than diluted atmospheric aerosols [37] and it is also very stable in the time as shown in Fig. 6. The standard deviation of $U$ over time allows to visualize this tendency. The plot of the time-average attenuated backscatter profile $\bar{U}$ is similar to the one of the $\overline{RCS}$ because they are linked by the fundamental lidar equation.

<figure><img src="image_7.png" /><figcaption>Fig. 6. Attenuated backscatter profile measured on the liquid spray for 3.37 bar and 4.320 g s⁻¹ with the general lidar constant $K_{opt}$ = 125.99 V m³ sr.</figcaption></figure>

### 4.3. Water spray volume concentrations from lidar

Finally, assuming $U(r) \approx \beta(r)$, it is possible to approximate the average volume concentration $N$ of the spray droplets from:

$$ \beta(r) = dC_{back}(R) \cdot N_{Lidar}(r, R), \qquad (15) $$

$$ N_{Lidar}(r, R) \approx \frac{U(r)}{dC_{back}(R)} = \frac{RCS(r)}{K_{opt} \cdot dC_{back}(R)}, \qquad (16) $$
where $N_{Lidar}$ [droplets m⁻³] denotes the average volume concentration obtained from lidar backscattering measurements in opposition to the average volume concentration obtained from laser diffraction measurements and denoted $N_{Malvern}$ [droplets m⁻³].

The average volume concentrations, in logarithmic scale, evolve with the $D_{32}$ values of the droplets as shown in Fig. 7. The increase in the Sauter mean diameter $D_{32}$ means that the spray is probably constituted of large droplets which are less numerous. Thus, at fixed liquid flow rate, the atomization is more effective, when the air pressure of the atomizer increases because there are more droplets that attenuate the laser intensity to a lesser extent due to their small sizes. For a constant air pressure, increasing the liquid flow rate results in the increase of $D_{32}$ [38]. We can interpolate the lidar experimental data with a least squares regression such as: $\log_{10}(N_{Lidar}) = -9.2 \cdot 10^{-3} \cdot D_{32} + 9.56$ droplets m⁻³, with a coefficient of determination of $R^2 = 0.81$. Due to a logarithmic scale for $N$ and a linear scale for $D_{32}$, the least squares regression is not linear in Fig. 7.

<figure><img src="image_8.png" /><figcaption>Fig. 7. Comparison of the average volume concentrations $N$ obtained from lidar backscattering (▼) and laser diffraction (*) techniques, for all tests identified by their Sauter mean diameter $D_{32}$. The regression (dotted curve) is plotted for the average volume concentrations from lidar backscattering measurements.</figcaption></figure>

Moreover, for this type of atomizer, the average volume concentration seems to be dependent on the diameter $D_{32}$, and its evolution is inversely proportional to this diameter. It is unusual to look for a relation between concentration and diameter. Generally, the Sauter mean diameter $D_{32}$ is related to the Weber number $We$, which characterizes the inertia forces to the forces resulting from the surface tension:

$$We = \frac{\rho_p \cdot V_p^2 \cdot D_p}{\gamma_p}, \qquad (17)$$

where $\rho$ [kg m⁻³], $V$ [m s⁻¹], $D$ [m] and $\gamma$ [N m⁻¹] are, respectively, the density, the velocity, the diameter and the surface tension of a particle $p$. Studies have demonstrated that there exists an inversely proportional relation between these two quantities [39–41], especially with correlations such as $D_{32} \propto We^{-0.6}$ [40]. Increasing the Weber number enhances the atomization process. Thus, smaller droplets are created which coincides with a higher average volume concentration.

The order of magnitude of the droplet concentration may be assessed with the laser diffraction method. Indeed, from the Beer-Lambert law of Eq. (2) and similarly to Eqs. (15) and (16), the
average volume concentration $N$ can be expressed as:

$$N_{\text{Malvern}}(R) \approx \frac{-\ln(T)}{\delta_{\text{spray}} \cdot C_{\text{ext}}(R)} \quad (18)$$

The logarithmic average volume concentrations computed from lidar backscattering measurements (Eq. (16)) and from laser diffraction measurements (Eq. (18)) are similar but the relative error in volume concentrations tends to be greater for sprays containing small droplets, i.e., for higher air pressure injection and so $D_{32} < 50 \mu m$. As the lidar Colibri instrument usually measures submicrometers droplets whereas the range of detectability of the Malvern Spraytec is 0.1 to 900 µm, the lidar Colibri could be more sensitive and precise while measuring the small droplets. Furthermore, the difference in wavelengths of the instruments should have no impact because for large water droplets, the Ångström coefficient, that characterizes the ratio of transmittances at two wavelengths, tends towards 0 [42,43]. However, other parameters could strongly influence these concentrations, like the spray width $\delta_{\text{spray}}$, the measured transmittances $T$ or the laser pulse width $\tau$.

The average volume concentrations calculated considering the uncertainty of the parameters are not displayed in Fig. 7. However, the spray width uncertainty leads to a minimal concentration between $6.27 \cdot 10^8$ and $7.67 \cdot 10^8$ droplets m$^{-3}$ and a maximal concentration between $2.64 \cdot 10^9$ and $3.23 \cdot 10^9$ droplets m$^{-3}$ for lidar backscattering measurements. Regarding the laser pulse width uncertainty, we may obtain a minimal concentration between $5.88 \cdot 10^8$ and $7.19 \cdot 10^8$ droplets m$^{-3}$, and a maximal concentration between $2.48 \cdot 10^9$ and $3.03 \cdot 10^9$ droplets m$^{-3}$.

## 5. Conclusion

In this paper, a new calibration method for the backscattering measurements of the lidar Colibri is proposed using laser diffraction measurements on sprays and a Mie theory code. The measured transmittances and PDF derived from the laser diffraction measurements are used to compute $C_{\text{ext}}$, $dC_{\text{back}} := dC_{\text{back}}/d\Omega$, $LR$ and the lidar calibration constant used for the lidar Colibri instrument. Using a least squares method, an optimal constant $K_{\text{opt}}$ is calculated over the 25 injection conditions and compared with a calibration constant $K_{\text{target}}$ determined by the surface calibration method. Note that in the present study, the optimized overlap function is included into the constants since all the spray and target measurements are carried out at the same distance from the lidar source. The constants show good agreement with a relative error of 6.31 %. This low relative error demonstrates the interest of using the Malvern Spraytec to estimate the lidar Colibri constant even though they operate at different wavelengths (632.8 nm for Malvern Spraytec and 532 nm for lidar Colibri). Therefore, lidar backscattering technique can be used on water droplets with low absorption and diameters greater than the atmospheric particles usually considered.

Moreover, assuming at first sight that $U(r) \approx \beta(r)$, the local average volume concentration of the sprays can be estimated. For the 25 injection conditions, these concentrations range from $6.90 \cdot 10^8$ droplets m$^{-3}$ to $2.91 \cdot 10^9$ droplets m$^{-3}$. The order of magnitude has been validated with the computed concentrations from the laser diffraction measurements. It is similar for droplets with $D_{32} > 50 \mu m$ but discrepancies appear below this limit. This can be due to the aforementioned assumption, to the range detectability of the two instruments or to the uncertainties of the used parameters such as the laser pulse width $\tau$, for instance.

Finally, the lidar Colibri can be used on dense sprays from a controlled source. This new application of backscattering technique can involve other configurations on sprays of varying densities for remote sensing of agricultural spray drift or bioaerosols, for instance.

**Funding.** This work has been funded by ONERA.

**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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