Introduction. The surface layer of the atmosphere is characterized by polydisperse and nonspherical particles of various natures (including those of anthropogenic origin) with a wide variety of physical and optical parameters [1]. This limits the possibilities of using elastic scattering lidars to determine the microstructure of this layer. Elastic scattering lidars enable to remotely measure the basic coefficients (backscattering and extinction coefficients) [2] and angular transformation of a beam. A lidar with two receiving channels is required to determine the angular transformation of a beam [3]. The main task of interpreting the basic coefficients measured by lidar is development of methods for transition from these coefficients to the microstructure of the scattering layer. A similar problem is solved using nephelometers in the process of local measurements. When interpreting the results of local measurements using nephelometers that use light scattering on individual particles, methods of solving an inverse problem are usually applied, using a set of assumptions about the physical and optical properties of particles [4]. In the atmospheric surface layers, interpretation of measurements of radiation scattered by particles using nephelometers is also associated with considerable difficulties due to the abovementioned diversity of particles. In some cases, particles are presented in the form of prolate and oblate spheroids [5], which is not always applicable. Accordingly, it was proposed to use an equivalent medium consisting of monodisperse particles to interpret the microstructure of the atmospheric surface layer [6]. Simultaneously, methods should be developed for comparing the parameters of real and model scattering mediums. The equivalent model medium can be compared with the medium under study because they have the same scattering parameters (basic lidar coefficients and beam angular transformation). The transition from the studied layer microstructure to the microstructure of the equivalent medium is unambiguous, in contrast to the reverse transition. Previously, methods of measuring the equivalent cross section were substantiated using digital images, beam angular distortions, and three-dimensional (3D) screens [6]. Knowing the equivalent cross section of the particles, the concentration of equivalent particles in the layer can be determined from the transmittance of the scattering layer being probed (atmospheric surface layer). In such a situation, the concentration of equivalent particles is used as a secondary parameter, which is determined from the equivalent cross section of the particle.

This study presents methods of measuring the concentrations of equivalent particles, in which equivalent cross section is used as a secondary parameter measured by the equivalent concentration and the extinction coefficient. This approach does not require using methods of solving an ill-conditioned inverse problem because the measurements are direct at all stages.

This study aims to present a comparative analysis of two methods of determining the concentrations of equivalent particles in a model scattering (surface) layer of the atmosphere by using an elastic scattering lidar and a nephelometer.

Volumetric model of the scattering (surface) layer of the atmosphere. In the considered wavelength range of sounding waves (0.5–0.9 μm) for surface aerosol particles of various natures, whose dimensions are larger than the wavelength, the scattering efficiency factor is in the range from two to four geometric sections of the particle. As the particle size increases, the efficiency factor tends to two and weakly depends on refractive index. The aerosol components of the extinction coefficient α and backscattering coefficient β can be expressed in terms of the geometric characteristics of conducting (reflecting) particles [7]:

$$\left.\begin{array}{c}\mathrm{\alpha }=2\uppi \sum_{k=1}^{n}{R}_{k}^{2};\\\upbeta =\frac{1}{4}\sum_{k=1}^{n}{r}_{k}^{2}\left(\uppi \right),\end{array}\right\}$$
(1)

where Rk represents the radius of the particle section with the number k, nk the concentration of particles with a cross section of radius Rk, rk(π) the radius of the front surface of the particle, and n the total number of particles per unit volume.

For the backscattering coefficient along the radii of the front surface of the particles, both the shape of the front surface of the particles and the single scattering albedo are modeled. Equation (1) allows for further simplification of the model of equivalent monodisperse particles:

$$\left.\begin{array}{c}\mathrm{\alpha }=2\uppi {R}_{\mathrm{eq}}^{2}{n}_{\mathrm{eq}}\left(0\right);\\\upbeta ={r}_{\mathrm{eq}}^{2}\left(\uppi \right){n}_{\mathrm{eq}} \left(\uppi \right)/4,\end{array}\right\}$$
(2)

where Req represents the radius of the section of the equivalent particle; req(π) the radius of curvature of the front surface of the equivalent particle; and neq(0) and neq(π) the concentrations of equivalent particles measured from the direct (at a small angle to the probing beam) and backscattering signals, respectively.

These concentrations of equivalent particles generally differ from each other because neq(π) depends on the shape of the front surface of the particles, whereas neq(0) practically depends only on the cross section of the particles. In such a model, the relationship between the geometric characteristics of an equivalent particle and the particles of the medium under study can be presented as a system:

$$\left.\begin{array}{c}{R}_{\mathrm{eq}}^{2}=\frac{1}{{n}_{\mathrm{eq}}\left(0\right)}\sum_{k=1}^{n}{R}_{k}^{2};\\ {r}_{\mathrm{eq}}^{2}\left(\uppi \right)=\frac{1}{{n}_{\mathrm{eq}}\left(\uppi \right)}\sum_{r=1}^{n}{r}_{k}^{2}\left(\uppi \right).\end{array}\right\}$$
(3)

Equation (3) represents a transition to the microphysical characteristics of equivalent particles, as for a real medium, in addition to the basic coefficients, one must determine the concentration of equivalent particles from forward scattering neq(0) and backscattering neq(π). Equations (2) and (3) enable to reveal a method of determining the geometric characteristics of an equivalent scattering particle using Req and neq(π).

Table 1 presents the parameters of various 3D models of a reflecting equivalent particle; for all particles, the extinction coefficient is α = 2πR2neq.

Table 1 Parameters of Various Scattering Objects
Full size table

From Table 1, it can be seen that the real and imaginary parts of the refractive index can be disregarded in the reflective model of a scattering object. Using the factor 4π gives a clear physical meaning for the ratio between the basic coefficients α and β (the last column of Table 1). In this case, a ratio of the basic coefficients, α/(4πβ), is expressed in terms of twice the ratio of the square of the equivalent particle radius to the square of the radius of curvature of the front surface of the particle. Such angular backscattering patterns correspond to scattering by particles with a spherical reflecting surface of a certain curvature. A disk with a Lambert scattering diagram can be replaced by a spherical segment with radius r(π) = 2R, and a disk with an isotropic scattering diagram can be replaced with a spherical segment with radius r(π) = \(R\sqrt{2}\). The model of a molecule as a scattering particle can be represented as a spherical segment with radius r(π) = \(R\sqrt{3}\). As a reference, a conductive spherical surface with radius r(π) = R can be selected. Using the factor 4π for the ratio between the basic coefficients is based on the interpretation of backscattering in radar problems, where this factor is used to determine radar cross section [8].

Considering the factor 4π, the estimate of the total geometric cross section of particles per unit volume is 4πβ, and that of the corresponding backscattering cross section is 2 πβ. In this case, we write

$$\frac{\mathrm{\alpha }}{4\pi\upbeta }=\frac{\upsigma \left(R\right)}{\upsigma \left(r\right)}=\frac{2{R}_{\mathrm{eq}}^{2}{n}_{\mathrm{eq}}\left(0\right)}{{r}_{\mathrm{eq}}^{2}{n}_{\mathrm{eq}}\left(\pi \right)},$$
(4)

where σ(R) and σ(r) represent the extinction and backscattering cross sections, respectively.

From Eq. (4), it can be seen that for reflecting spheres larger than the wavelength, σ(R) is twice the geometric cross section of the particle, and σ(r) is twice less than the geometric cross section of scattering on the sphere. In this case, the concentrations of equivalent spherical particles measured from the forward and backward scattering signals are equal. In other cases, the relationship between the basic coefficients can be interpreted using a truncated cone and a spherical segment.

A backscatter lidar with an optimal coaxial design [7] can be calibrated to measure the backscattering coefficient. The convenience of such a calibration is due to many factors:

  • the backscattering signal is concentrated on relatively small paths up to 200 m. On such paths, a scattering surface with a given backscattering angular pattern can be installed;

  • both scattering surfaces and reflecting spheres can be used for calibration. This is because for the optimal coaxial circuit, the geometric form factor of the receiving and transmitting channels is 0.25;

  • using perforated screens, the instrumental function (path dependence of a lidar signal from a homogeneous atmosphere without attenuation) can be measured at one point of the path to calibrate the lidar along the entire path. These aspects enable to measure the probed volume practically at any point of the route.

Thus, the optimal coaxial circuit can be used to measure the basic coefficients for some scattering layer.

The coefficient β measured from the backscattering signal can be decomposed into molecular and aerosol components. The molecular component is determined from the tabular values of the spectral dependence of the backscattering coefficient for a purely molecular atmosphere based on the thermodynamic and optical model of the atmosphere [2]. The molecular component determined from the tables should be subtracted from the value of β measured by the lidar for direct measurements of the aerosol component. The extinction coefficient is measured on paths with a fixed range. Further interpretation of the aerosol component of the basic coefficients is related to determination of the concentration of equivalent particles of the probed object by backward and forward scattering.

Statistical approach to describing the particle concentration of the investigated medium. One must measure the radiation scattered by individual particles for the microphysical interpretation of the basic coefficients. Let us use an equivalent model for the microstructure of the medium under study [6]. This representation is based on direct methods associated with the registration of signals generated by light radiation scattering by individual particles inside the studied scattering layer (scattering signals). The microphysical properties of the equivalent medium are expressed in terms of nonnormalized, first- and second-order moments for the registered signals of scattering particles.

Nonnormalized moments EΣ(uk) for the abovementioned signals of scattering of individual particles diff er from the usual moments without normalization and are defined as

$${E}^{\Sigma }\left({u}^{k}\right)=\sum_{i}{u}_{i}^{k},i=1, 2, 3, \cdots , N,$$

where k represents the order of the nonnormalized moment, ui the scattering signal of the ith particle, and N the number of particles.

We define these moments as nonnormalized. The signals ui of individual particles can be pulses of current, voltage, etc. The physical meaning of the nonnormalized, first-order moment EΣ(u) is the sum of signals of individual particles. In terms of nonnormalized moments, the number of equivalent particles, Neq, and the signal of the equivalent particle, ueq, can be expressed as follows:

$$\left.\begin{array}{c}{N}_{\mathrm{eq}}\left(u\right)={E}^{\Sigma }{\left(u\right)}^{2}/{E}^{\Sigma }\left({u}^{2}\right);\\ {u}_{\mathrm{eq}}={u}_{21}={E}^{\Sigma }{\left(u\right)}^{2}/{E}^{\Sigma }\left(u\right),\end{array}\right\}$$
(5)

where ueq is proportional to the equivalent cross section, and Neq represents the number of equivalent signals in terms of which the nonnormalized moment EΣ(u) can be expressed.

Equation (5) includes nonnormalized moments of the first and second orders. Thus, from the nonnormalized moments of the first and second orders, the concentration of equivalent particles can be determined [see Eq. (2)].

The ratio of the number of equivalent particles Neq to volume provides the volume concentration of equivalent particles, neq. The following is the equation for the concentration of equivalent particles in an explicit form:

$${n}_{\mathrm{eq}}\left(u\right)={N}_{\mathrm{eq}}\left(u\right){V}^{-1},$$

where

$${N}_{\mathrm{eq}}\left(u\right)={\left[\sum_{m=1}^{N}{u}_{m}\right]}^{2}{\left[\sum_{m=1}^{N}{u}_{m}^{2}\right]}^{-1}.$$

The probed volume V is set either according to the characteristics of the optimal coaxial system, or with the use of perforated screens [6]. If the backscattering coefficient and the concentration of equivalent cross sections, neq, are determined for the layer under study, the ratio β/neq gives the diff erential cross section for backscattering by an equivalent particle.

The abovementioned method of measuring the concentration of particles by using basic coefficients provides sufficient accuracy and is applicable for a wide range of transverse particle sizes. The concentration measurement error is determined by the statistical characteristics of the backscattering signal. The measurement error of the return signal used by us for a prototype miniature backscatter lidar was 10–20%, depending on the sounding conditions (weather conditions, aerosol sources, dust, steam, among others). Calibration accuracy is due to the error of beam angular distortions when using calibrated screens, uneven location of holes on the calibrated screen, uneven angular scattering diagram of the standard screen limiting the path, and changes in the geometric form factor of the receiving and transmitting channels.

Determination of the concentration of equivalent particles from forward and backward scattering signals. The concentration of equivalent particles is one of the main parameters of the microstructure of the scattering layer. Let us consider the schemes typical for lidars for measuring the signals formed by the scattering of light by individual particles. The atmospheric surface layer is characterized by polydisperse and nonspherical particles with diff erent physical and optical properties.

Let us assume that the signal (intensity) of backscattering of the mth particle is represented by Im(π). Some volume V comprises N such particles. The value of the concentration of these particles can then be determined from the ratio of nonnormalized moments of the first and second orders for the inverse signals Im(π) of individual particles:

$${n}_{\mathrm{eq}}\left(\pi \right)={N}_{\mathrm{eq}}\left(\pi \right){V}^{-1},$$

where

$${N}_{\mathrm{eq}}\left(\pi \right)={\left[\sum_{m=1}^{N}{I}_{m}\right]}^{2}{\left[\sum_{m=1}^{N}{I}_{m}^{2}\right]}^{-1},$$
(6)

Neq(π) represents the number of equivalent particles measured by direct scattering in volume V.

In Eq. (6), other values that correspond to Im(π), for example, the amplitude of a current or voltage pulse, can be used. This indicates the relative nature of measurements of concentration of equivalent particles. For monodisperse, spherical particles, Neq(π) = N; for polydisperse and nonspherical particles, Neq(π) < N.

The equivalent concentration can be determined from the forward scattering signals of individual particles by using a nephelometer:

$${n}_{\mathrm{eq}}\left(0\right)={N}_{\mathrm{eq}}\left(0\right){V}^{-1},$$

where

$${N}_{\mathrm{eq}}\left(0\right)={\left[\sum_{m=1}^{N}{I}_{m}\left(0\right)\right]}^{2}{\left[\sum_{m=1}^{N}{I}_{m}^{2}\left(0\right)\right]}^{-1}.$$
(7)

Here, the signal Im(0) is measured at a small angle to the direction of light propagation. Equations (6) and (7), in the general case, illustrate two characteristics of the concentration of equivalent particles measured from forward and backward scattering signals. The concentration measured from the backscattering coefficient depends on single scattering albedo, whereas that measured from the extinction coefficient is practically independent of albedo. This characteristic results in possible diff erences between the measured concentrations of equivalent particles according to the forward and backward scattering signals in Eq. (2). Hence, we must determine the coefficient relating these concentrations. The concentration of equivalent particles, measured from forward scattering signals and practically independent of particle albedo, can be selected as the main characteristic of the equivalent medium.

The presented approach to determining the concentration of equivalent particles is implemented as follows:

  • determination of the number of equivalent particles on the basis of the results of primary processing of forward scattering signals measured by a nephelometer;

  • determination of the number of equivalent particles measured from backscattering signals by using a nephelometer.

The implementation of these measurement schemes requires simultaneous use of a lidar and a nephelometer. However, one cannot always install a nephelometer inside the scattering layer under study. In this case, the conditions for the spatial homogeneity of the scattering medium in the studied layer must be satisfied.

There are characteristics typical for lidars with two receiving channels in measuring the concentration of equivalent particles by the basic coefficients and the angular size of the halo in the case of forward scattering [3]. In this interpretation, angular size of the halo can be employed to estimate the transverse size of an equivalent scattering particle. Then, from the extinction coefficient and the scattering cross section of the probing radiation by an individual particle, the concentration of equivalent particles in the probed layer can be determined; from this concentration and the backscattering coefficient, the diff erential cross section for backscattering of the probing radiation by an individual equivalent particle can be estimated.

Knowing the transmission coefficient for a given layer and the equivalent concentration neq(0) enables to determine the equivalent particle scattering cross section inside the probed object using Eq. (2).

Let us consider the limitations of the possibility of using direct measurements of the microphysical characteristics of a medium only by lidar methods:

  • the optimal condition for using lidar methods is not always observed, as there is a topographic object behind the probed layer (a path with a fixed range);

  • particles with sufficiently large cross sections should prevail inside the layer;

  • the accuracy of determining the angular size of the scattering halo is limited due to the relatively small transformation of the probing beam power into the angular size of the halo.

These considerations entail a conclusion regarding possible implementations of methods of measuring the characteristics of scattering cross sections and the concentration of equivalent particles without employing methods of solving an ill-conditioned problem [9, 10]. For example, if the concentrations of equivalent particles measured in the abovementioned two directions in a certain layer and the basic coefficients are known, then these data are sufficient to determine the diff erential backscattering cross section and the scattering cross section for an individual equivalent particle [see Eq. (2)].

In the atmospheric surface layer, contact measurement methods characteristic of nephelometers can be used. For these measurements, the volume of air pumped is usually known. For monodisperse, spherical particles, the number of equivalent particles measured from forward and backward scattering signals using a nephelometer is the same. In this case, the measured number of equivalent particles is equal to the number of registered particles. For spherical particles with sizes distributed according to the lognormal law, the number of equivalent particles measured from forward and backward scattering signals will be less than the number of particles registered by a nephelometer.

The results of the abovementioned analysis on the ratio of the basic coefficients, and microscopic parameters (diff erential backscattering cross section and extinction cross section), can be used to characterize the scattering medium and solve the lidar equation.

Conclusion. Within the model of an equivalent medium (equivalent to atmospheric layer microstructure) with monodisperse particles, a microphysical interpretation of the basic coefficients, the main measured quantities for an elastic scattering lidar, could be performed. A lidar with two receiving channels could be used to measure the basic coefficients and the angular size of the halo in the case of forward scattering. These data are sufficient to determine the concentration of equivalent particles in the layer. However, the lidar method has a limited applicability because the angular size of the halo, related to the size of the particles and their location, must be comparable with the angular size of the beam.

The concentration of equivalent particles can also be determined from the measurements results of the forward and backward scattering signals of individual particles by using a nephelometer, taking into account the given ratios for the transition from the concentration of real particles measured using the nephelometer to that of equivalent particles. This method is more versatile because it is applicable to particles with a wide range of transverse sizes.

The proposed methods allow for unambiguous comparison of measured backscattering signal with the concentration of equivalent particles. In future, this is expected to enable to create remote systems for primary monitoring of the atmospheric surface layer, as well as to identify spatial and temporal dynamics and level of aerosol pollution.

Conflict of Interest. The author declares no conflict of interest.