3.1 Basic cloud microphysical and optical properties

As outlined and summarized by Schmidt et al. (2013, 2014) and Donovan et al. (2015), the basic properties characterizing a liquid-water cloud layer are the cloud droplet number concentration N_d, the cloud droplet effective radius R_e, the cloud droplet (single scattering) light-extinction coefficient α, and the liquid-water content w_l. The liquid-water content of droplets in a given volume is defined as follows:

with the total droplet number concentration $N_{\mathrm{d}}={\int }_{\mathrm{0}}^{\mathrm{\infty }}n\left(r\right)\mathrm{d}r$, the volume mean droplet radius R_v of a given droplet size distribution n(r), and the liquid-water density ρ_w. The droplet number concentration n(r) is described by a modified gamma size distribution (see Eq. 2 in Schmidt et al., 2014).

The light-extinction coefficient of the cloud layer can be approximated by

in the case that the droplets are large in comparison to the laser wavelength. R_s denotes the surface mean droplet radius. Besides the cloud extinction coefficient, the droplet effective radius

is used to characterize an observed liquid-water cloud layer. By combining Eqs. (1), (2), and (3) we can write the following for the liquid-water content:

Based on in situ measurements in warm stratified clouds Martin et al. (1994) found that the cubic power of the measured effective radius and the cubic power of the volume mean droplet radius follow a linear relationship, defining the parameter k:

This linear relationship suggests that, in most cases, a modified gamma function (Eq. 2 in Schmidt et al., 2014, Eq. 6 in Donovan et al., 2015) can describe the droplet size distribution. Lu and Seinfeld (2006) compiled a list of k values for stratiform clouds based on a literature review. The k range of 0.75 ± 0.15 represents well the values found for continental air masses. For marine stratocumulus k was slightly larger (around 0.8).

From Eqs. (2), (3), and (5) an expression for the cloud droplet number concentration can be obtained:

Equations (4) and (6) permit the calculation of the liquid water content w_l and the droplet number concentration N_d from lidar measurements of the cloud extinction coefficient α and the droplet effective radius R_e as already outlined by Schmidt et al. (2013, 2014). In the next sections, we evaluate the possibilities of retrieving information about these two cloud parameters from lidar measurements of depolarization ratios caused by multiple scattering. The investigation is based on simulations with an analytical model (introduced in Sect. 3.3) which can compute the co- and cross-polarized lidar returns in multiple scattering regimes of pure liquid-water clouds.

3.2 Relationship between light depolarization and multiple scattering

It is well known that the polarization state of photons scattered in the strictly backward direction remain invariant in the case of spherical particles. In dense water clouds (multiple scattering regime), however, one or more forward scattering events take place, so the backscatter process that allows the return of laser photons to the receiver telescope within the lidar FOV occurs at a scattering angle close to but different from 180^∘. In this case, multiple scattering causes depolarization of the incident linearly polarized laser light (Sassen and Petrilla, 1986; Zege and Chaikovskaya, 1996).

To provide an easy-to-follow overview of the polarimetric behavior in lidar-relevant multiple scattering regimes, we consider first a simple case of double scattering, as illustrated in Fig. 1, consisting of forward scattering of laser photons by one droplet at height z_f at a small scattering angle θ_f followed by backward scattering by another droplet at height z_b at a large angle ${\mathit{\theta }}_{\mathrm{b}}=\mathit{\pi }-{\mathit{\theta }}_{\mathrm{f}}$ (around 180^∘).

Figure 1Scattering geometry for one forward and one backward scattering event.

Download

The Stokes vector describing the resulting polarization state with respect to the initial coordinate system, which we relate to the laser beam polarization state, can be obtained as follows:

A_lin denotes the Stokes vector for the 100 % linearly polarized laser pulses, associated with the initial laser polarization plane (e_∥cc, e_⟂cc, e_z). The transformation matrix B^cc→rc(ϕ) enables the transition from the Cartesian coordinate system (cc, e_∥cc, e_⟂cc, e_z coordinates in Fig. 1) to the ϕ-rotated system (rc, e_∥rc, e_⟂rc, e_z coordinates) followed by the scattering of the incident wave front. P represents the single scattering matrix defined for an isotropic media (Zege and Chaikovskaya, 2000). The matrices P(θ_f) and P(θ_b) denote the forward and backward scattering. The transformation matrix B^sc→cc finally enables the transition from the scattering-coordinate system (sc, e_∥sc, e_⟂sc, e_r coordinates in Fig. 1) to the original Cartesian system (cc) (Wandinger, 1994).

From the Stokes vector $\mathit{A}({\mathit{\theta }}_{\mathrm{f}},{\mathit{\theta }}_{\mathrm{b}},\mathit{\varphi })$ we can extract the co- and cross-polarized lidar signal components S_∥ and S_⟂. In Fig. 2b, the computed azimuthal patterns (in the backscatter plane orthogonal to the z axis in Fig. 1) of the co- and cross-polarized signal components for scattering angles from 170 to 180^∘ are shown for four different droplet sizes. Those azimuthal patterns can be observed with imaging polarization lidars, which are supplied with a charged coupled device matrix as a photo-receiving element (Roy et al., 2004). But a common lidar receiver collects the scattered light over the entire azimuthal range and stores it as one signal. However, by selecting a certain receiver FOV, we define the range of scattering angles θ_f and θ_b that a lidar can detect in the multiple scattering regime, and by measuring lidar return signals at different FOVs and thus for different ranges of θ_f and θ_b, a way is opened to derive information about the droplet sizes as emphasized in Fig. 2a–d. This is the basic idea of combining lidar measurements at different FOVs to retrieve the effective radius of the droplets and, in the next step, further cloud properties as will be described in Sect. 4.

Figure 2(a) Normalized scattering matrix element P₁₁ (normalized to the maximum at 0^∘ scattering angle) as a function of forward scattering angle θ_f for four droplet effective radii (given as numbers), (b) azimuthal patterns (computed with the MS model for the entire range of azimuthal angles from 0 to 2π; see Fig. 1) of the co-polarized ∥ and the cross-polarized ⟂ signal components at scattering angles θ_b between 170 and 189.5^∘ for the different droplet diameters, (c) droplet linear depolarization ratio $\mathit{\delta }=S_{⟂}/S_{\parallel }$ with the lidar signal components S_⟂ and S_∥ (obtained from azimuthal integration over the range from ϕ=0–2π in b) as a function of the backscattering angle θ_b from 174^∘ to 180^∘ for the four droplet sizes, and (d) scattering matrix element P₁₁(θ) at θ_f (in a) multiplied by the depolarization ratio at ${\mathit{\theta }}_{\mathrm{b}}=\mathit{\pi }-{\mathit{\theta }}_{\mathrm{f}}$ (in c).

Download

From the two observed lidar signal components, S_⟂ and S_∥ backscattered at height z_b, the so-called volume or, in the case of dense water clouds, droplet linear depolarization ratio defined as

is obtained. After forward scattering, the laser photons are backscattered at a certain backscatter angle θ_b. The dependence of the depolarization ratio on the backscatter angle θ_b is shown in Fig. 2c for the four droplet effective radii. It can be seen that the depolarization ratio increases to considerable values when the scattering angle deviates from 180^∘. This sensitivity of the non-180^∘ backscattering angle θ_b on light depolarization and the strong forward scattering peak in Fig. 2a are the features used in the dual-FOV polarization lidar technique to retrieve the basic cloud microphysical properties. Figure 2a, c, and d provide an impression of the sensitive impact of cloud droplet size on measurable lidar quantities and thus suggest again that polarization lidars operated at two FOVs have the potential to derive R_e and subsequently also the cloud extinction coefficient α. Both R_e and α are closely linked to the cloud droplet number concentration N_d (see Eq. 6). The relationship between MS-induced light depolarization measured at several FOVs and the cloud droplet size characteristics has already been illuminated and discussed in previous studies (Veselovskii et al., 2006; Roy et al., 2016). In the next sections, we will show that a dual-FOV polarization lidar can already provide trustworthy information about the size and extinction coefficient in the cloud base region of liquid water clouds.

To emphasize the dominating impact of the receiver FOV on the measured multiple scattering effects let us, at the end of this subsection, compare the influence of the laser beam width and divergence, receiver telescope area, and the receiver field of view on the observable cloud volume. In the case of a receiver FOV of 1 mrad, the lidar sees or observes a geometrical cross section (circular area in the horizontal plane at cloud base height z_bot) of about 0.8, 7, and 20 m² for a cloud with base height at 1, 3, and 5 km, respectively. The observable cross sections increase to about 3, 28, and 80 m² when using a 2 mrad FOV. In contrast, in the case of a 30 cm receiver telescope (and a theoretical FOV of 0 mrad), the monitored circular cloud area at the cloud base is less than 0.1 m². Also, the divergence of the laser beam (0.1 to 0.2 mrad) has only a minor impact on the amount of backscattered photons (and MS effects). The illuminated cloud cross section at cloud base is always < 1 m² for a cloud base height of < 5 km. So, the FOV clearly determines the cloud volume (geometrical cross section at cloud base times 50–100 m laser beam penetration depth into the cloud) available for MS cloud studies with lidar.

3.3 Multiple scattering model

After presenting the principle relationship between the measured linear depolarization ratio, forward scattering, and droplet size, next we introduce the multiple scattering model used to develop our retrieval method presented in Sect. 4. The simulation model allows us to simulate realistic cloud scenarios with varying cloud height, droplet number concentration, cloud extinction coefficient, and droplet size distribution and the resulting co- and cross–polarized lidar signal components S_∥ and S_⟂ for given lidar configuration parameters such as laser beam divergence and receiver FOV.

In several articles, the radiative transfer problem of polarized light undergoing multiple scattering in an optically dense medium has been analytically addressed, and several solutions have been proposed and tested (Zege et al., 1995; Zege and Chaikovskaya, 1999, 2000). The so-called small-angle approximation is used. This solution is justified in the case of a narrow and pronounced forward scattering peak of the droplet scattering phase function which in turn is the case when the droplet size (on the order of 5–20 µm) is large compared to the laser wavelength (532 nm). Such an elongated forward-phase-function medium allows a simplification of Green's matrix. The vector equation can thus be split into a system of scalar-like equations which are simpler and include less integral terms than the original ones and can thus be solved by using well developed radiative-transfer-equation techniques.

The Stokes vector A has the general form $\mathit{A}=(I,Q,U,V{)}^{\mathrm{T}}=(S_{\parallel }+S_{⟂},S_{\parallel }-S_{⟂},U,V{)}^{\mathrm{T}}$ and the Stokes vector is ${\mathit{A}}_{\mathrm{lin}}=(\mathrm{1},\mathrm{1},\mathrm{0},\mathrm{0}{)}^{\mathrm{T}}$ in the case of linearly polarized laser pulses (in x direction in Fig. 1) in Eq. (7).

The first element of the Stokes vector is the light intensity I which satisfies the radiative transfer equation using the single scattering matrix element P₁₁(θ) (shown after normalization in Fig. 2a). The Q component of the Stokes vector describes the linear polarization and satisfies the same equation but with the “modified” angular scattering function that equals $(P_{\mathrm{22}}\pm P_{\mathrm{33}})/\mathrm{2}$ with the sign “+” for the forward and “−” for backward scattering. P₂₂ and P₃₃ are also elements of the scattering matrix P. In this way, the Stokes vector components can be solved separately as a scalar radiative transfer problem (Zege and Chaikovskaya, 2000). The more elongated the phase function, the more accurate the solution. This solution is not restricted to single and double scattering events. It simulates multiple forward scattering processes and one backscattering process. The approach offers high accuracy for optical depths up to 5 together with high computing efficiency (Chaikovskaya, 2008). The authors emphasized the potential of the model for developing new retrieval techniques.

The modeled components I and Q enable the calculations of the cross- and co-polarized returns S_⟂(z_b) and S_∥(z_b) for backscatter height z_b within a liquid-water cloud layer,

The geometrical vector $\mathit{G}({\mathrm{\Theta }}_{\mathrm{ldiv}},{\mathrm{\Theta }}_{\mathrm{fov}},d_{\mathrm{lb}},d_{\mathrm{m}\mathrm{1}},d_{\mathrm{m}\mathrm{2}\mathrm{sd}})$ required to solve Eqs. (9) and (10) provides all necessary information about the lidar configuration, such as the full divergence angle of the laser beam Θ_ldiv, the beam diameter d_lb, the FOV full divergence angle of the receiver Θ_fov, the diameter of the primary receiver telescope d_m1 and its respective secondary mirror shadow d_m2sd. The atmospheric state vector $\mathit{X}(z_{\mathrm{b}},\mathit{\alpha }(z_{\mathrm{b}}),R_{\mathrm{e}}(z_{\mathrm{b}}\left)\right)$ provides the cloud information, i.e., cloud extinction coefficient α(z_b) (assumed as the scattering coefficient) and effective radius R_e at height z_b.

The linear depolarization ratio according to Eqs. (8)–(10) is then given by

3.4 Model quality check: comparison with ECSIM Monte Carlo simulations and CALIPSO multiple scattering observations

We investigated to what extent the MS model used is able to simulate real-world polarization lidar observations and thus can be used to develop new lidar analysis methods with a focus on clouds. We compared our simulations with results obtained with the Monte Carlo simulation model ECSIM (EarthCARE Simulator) (Donovan et al., 2015, 2010) and observations with the CALIPSO (Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observation) lidar (Hu et al., 2007). EarthCARE (Earth Clouds, Aerosols and Radiation Explorer) is a planned spaceborne lidar and radar mission, designed within a co-operation of the European Space Agency (ESA) and the Japan Aerospace Exploration Agency (JAXA) (Illingworth et al., 2015). Based on the 4×4 α–R_e combinations (4 α(z_b) values in the range from 5 to 26 km⁻¹, 4 R_e(z_b) values in the range from 3 to 15 µm), we performed more than 200 different simulations for these 16 cloud scenarios by considering cloud penetration depths from 10 to 70 m (with step width of 10 m), two different FOVs of 0.5 and 2.0 mrad, and assuming a liquid-cloud layer with a cloud base height at 3000 m. We compared the obtained volume depolarization ratio with respective values simulated with the Monte Carlo simulation model ECSIM in Fig. 3a. As can be seen, our simulations (δ(our model)) are in good agreement with results of the sophisticated Monte Carlo model. Both models agree for most of the depolarization ratio, except for the largest penetration depth exhibiting values close to 0.1. On average the depolarization ratios obtained with ECSIM are larger than our values by 0.016. The small differences indicate that our method delivers a realistic picture of multiple scattering in liquid-water clouds. The growing disagreement for depolarization ratios >0.05 is caused by different assumptions and implementations regarding the considered narrow ranges of the small-angle forward scattering processes and the one wide-angle backscattering process in the different models.

Figure 3(a) Comparison of the volume linear depolarization ratios δ(z) computed with our analytical MS model and computed with ESA's Monte Carlo model ECSIM (more details in the text, 1:1 line is given as solid diagonal line) and (b) comparison of our computations of the relationship between the single-scattering-to-total-scattering-attenuated-backscatter ratio ${\stackrel{\mathrm{‾}}{\mathit{\gamma }}}_{\mathrm{ss}}/\stackrel{\mathrm{‾}}{\mathit{\gamma }}$ and the cloud-integrated depolarization ratio $\stackrel{\mathrm{‾}}{\mathit{\delta }}$ (red and black circles) with the respective values for this relationship as retrieved from CALIPSO multiple scattering observations (solid black line). For the two different FOVs (0.5 mrad in blue, 2.0 mrad in red) 4×4 R_e(z_ref) – α(z_ref) combinations are considered together with different cloud penetration depths Δz_ref from 10 to 70 m (with 10 m step width). All in all more than 200 simulations are included in each of the panels (a) and (b).

Download

In a second approach, we compared our simulations with CALIPSO polarization lidar observations. Hu et al. (2007) investigated the relationship between the ratio of the total, cloud-integrated lidar return signal $\stackrel{\mathrm{‾}}{\mathit{\gamma }}$ (from cloud top to base in the case of the CALIPSO lidar) to the one caused by single scattering (${\stackrel{\mathrm{‾}}{\mathit{\gamma }}}_{\mathrm{ss}}$ caused by one backscattering process) and the respective cloud-integrated linear depolarization ratio $\stackrel{\mathrm{‾}}{\mathit{\delta }}$. This study was based on observations with ground-based and spaceborne lidars supported by sophisticated Monte Carlo simulations of the multiple scattering impact on the observed cloud lidar returns. By performing a polynomial regression analysis to all observations they found the following best matching relationship:

The measured cloud-integrated CALIPSO lidar signal $\stackrel{\mathrm{‾}}{\mathit{\gamma }}$ results from single plus multiple scattering events and corresponds to the respective cloud-integrated depolarization ratio $\stackrel{\mathrm{‾}}{\mathit{\delta }}$ for a given receiver FOV full angle Θ.

In Fig. 3b, the relationship presented by Hu et al. (2007) is shown as a solid black line. As can be seen, our individual simulations for the two FOVs (red and black circles) are in good agreement with Eq. (12) (black solid line in Fig. 3b), which again corroborates that our model describes well the link between cloud multiple scattering and light depolarization.

6.1 Lidar-derived aerosol properties

For completeness of the theoretical Part 1, we briefly introduce the aerosol parameters needed for the ACI studies. Examples of aerosol observations with the multiwavelength polarization Raman lidar Polly (portable lidar system) (Engelmann et al., 2016) used in Part 2 and upgraded to a dual-FOV polarization lidar can be found in Baars et al. (2016) and Hofer et al. (2017, 2020). The sketch in Fig. 4 illustrates our overall concept of lidar-based ACI studies. The aerosol parameters are measured with the lidar at the smaller FOV (FOV_in) several hundreds of meters below the cloud base. The cloud- and ACI-relevant aerosol proxies are the particle extinction coefficient α_par(z) and the cloud condensation nucleus concentration N_CCN(z). The methodology to derive N_CCN profiles from measurements of particle optical properties is outlined in Mamouri and Ansmann (2016). A brief summary of the method, denoted as the POLIPHON (Polarization Lidar Photometer Networking) method, is given here.

A specific problem in ACI studies is the retrieval of the particle backscatter and extinction profiles below extended liquid-water cloud layers in the first step. The required calibration of the lidar profiles in clear air (in pure Rayleigh scattering conditions), i.e., in the upper troposphere and lower stratosphere, is then not possible. In these cases with aerosol backscatter signals up to cloud base height z_bot only, the so-called lidar constant is required in the retrieval of aerosol properties. The determination of the lidar constant (considering all instrumental constants, such as laser pulse energy and receiver telescope area, in the basic lidar equation) following the procedure of Wiegner and Geiß (2012) is performed during cloud-free situations before or after the passage of the cloud fields or during periods with cloud holes so that clear air layers (Rayleigh scattering regime) are available for the lidar calibration. Subsequently, the determined lidar constant is used during the cloudy periods in the data analysis to retrieve the backscatter coefficient profiles up to the base of the optically dense water clouds again following the procedure of Wiegner and Geiß (2012).

By means of height profiles of the aerosol particle depolarization ratio and the particle backscatter coefficient, the POLIPHON data analysis separates particle backscatter and extinction contribution of the three basic aerosol types (marine aerosol, mineral dust, anthropogenic haze). The aerosol-type-dependent 532 nm extinction coefficients below the cloud base z_bot are then converted into particle number concentrations and respective CCN concentrations (for a water supersaturation level of 0.2 % or relative humidity over water of 100.2 %) as described by Mamouri and Ansmann (2016).

For pure marine conditions, we obtain N_CCN from α_par by using the following conversion:

with N_CCN per cubic centimeter (cm⁻³) and α_par in units of inverse megameter (Mm⁻¹). For urban haze conditions (central European pollution conditions), we apply

and for desert dust

The N_CCN values assume that all dry particles with radius > 50 nm (marine, urban) and > 100 nm are potential cloud condensation nuclei. The parameterization holds for an ambient relative humidity of 60 % relative humidity for continental fine-mode aerosol and 80 % relative humidity in the case of marine particles. Respective water-uptake effects by aerosol particles are considered and corrected in Eqs. (36) and (37). In the case of hydrophobic dust particles, no water uptake effect is considered and corrected.

The uncertainty in the basic aerosol-type-dependent extinction coefficients and in the retrieved N_CCN values is on the order of 20 % and 50 %–100 %, respectively. However, aircraft comparisons (Düsing et al., 2018; Haarig et al., 2019) and long-term field studies at a central European background station (Schmale et al., 2018) revealed that the uncertainty is typically on the order of 50 % for lidar-derived CCN estimates. It should be emphasized at the end that the Raman lidar Polly permits the retrieval of profiles of the water-vapor mixing ratio and relative humidity (RH) (Dai et al., 2018) so that, in principle, actual RH measurements are available for the required aerosol water uptake effects in the N_CCN conversion procedure as described by Mamouri and Ansmann (2016)

6.2 Aerosol–cloud-interaction (ACI) parameter

The study of the influence of aerosol particles on liquid-water cloud evolution and cloud microphysical properties is based on two ACI parameters defined as follows (Feingold et al., 2001; McComiskey et al., 2009; Schmidt et al., 2013, 2014):

and

The so-called nucleation-efficiency parameter $E_{\mathrm{ACI},{\mathit{\alpha }}_{\mathrm{par}}}$ describes the relative change of the cloud droplet number concentration N_d with a relative change in the particle extinction coefficient α_par. Correspondingly, $E_{\mathrm{ACI},{\mathrm{N}}_{\mathrm{CCN}}}$ characterizes the relative increase in N_d with a relative increase in the cloud condensation nucleus concentration N_CCN. The higher the ACI value, the stronger the impact of the observed aerosol conditions on the cloud microphysical properties.

Table 3Overview of the cloud and aerosol retrieval procedure (step-by-step data analysis). The data analysis starts with a precise determination of the cloud base height z_bot. The cloud products are given at the reference height z_ref, 75 m above the cloud base height z_bot. In the estimation of the ACI efficiency, particle extinction and cloud condensation nucleus concentration at z_aer, usually several hundreds of meters below the cloud base, are considered.

Download Print Version | Download XLSX