Liquid water clouds are an important radiative constituent of the Earth–atmosphere system, but microphysical processes related to clouds remain highly uncertain and poorly represented in climate models [1]. Remote sensing observations are crucial to improving our knowledge on cloud processes, but traditional methods are often biased especially in cases of broken or small clouds [2]. Furthermore, ignoring the temperature dependence of the refractive index may lead to biases in cloud remote sensing products [3]. The polarimetric cloudbow observations from NASA’s Plankton, Aerosol, Cloud, ocean Ecosystem (PACE) mission will enable retrievals of cloud-top effective radius, variance, and droplet number concentration [4–6]. In addition, lidar observations contain information on cloud optical and microphysical properties [7]. NASA’s future Atmosphere Observing System (AOS) mission will likely include an imaging polarimeter and high spectral resolution lidar that will make collocated measurements further advancing cloud remote sensing capabilities. In this Letter we present a new cloud look-up table (LUT) that enables fast and precise retrievals of the microphysical properties of liquid water clouds from existing and next-generation remote sensing instruments. The LUT can also be used to compute the optical properties of clouds from airborne in situ droplet size distribution data. Our LUT supports a wide range of size distributions of spherical liquid water cloud droplets and derives the corresponding optical properties across the ultraviolet, visible, near-infrared, and shortwave infrared spectra.

The purpose of our LUT is to provide a fast and precise way to compute the liquid cloud inherent optical properties (IOPs) for a single-scattering case. These cases are characterized by a low value of depolarization measured by lidar and can be seen near the cloud edges [8]. The list of IOPs covered by the cloud LUT is the same as for our aerosol LUT and includes the normalized scattering matrix, the asymmetry parameter, the absorption, backscatter, extinction, and scattering coefficients [9]. Both LUTs use the same principles of work and similar unit test framework to assess the precision of estimations. Both LUTs can be used with the advanced single-instrument and combined microphysical retrievals as a part of radiative transfer model for an arbitrary lidar, spectrometer, or polarimeter wavelength. Our cloud LUT enables the polarimeter retrievals of aerosols above clouds and other new remote sensing techniques in development. Furthermore, the single-scattering IOPs can be used as input for the multiple-scattering radiative transfer simulations [10,11].

For radiative transfer calculations the normalized scattering matrix that relates incident and scattered Stokes parameters is needed. In the standard Lorenz–Mie theory of light scattering by homogeneous spheres this matrix can be represented as [10,11]

The elements of the matrix $\mathbf{P}\left (\Theta,m,\lambda \right )$ can be computed as

The scattering coefficient $k_\mathrm{sca}\left (m,\lambda \right )$ that appears in Eq. (2) and the extinction coefficient $k_\mathrm{ext}\left (m,\lambda \right )$ can be computed as

We compute the lidar backscatter coefficient as

Computation of the elements of the normalized scattering matrix $\mathbf{P}$ [see Eqs. (1) and (2)] and other single-scattering properties [see Eqs. (3)–(7)] takes hours if high precision is required and drizzle droplets are covered. For the purpose of fast retrievals of cloud microphysical properties, we set the goal to compute all these IOPs to within $\pm 1\%$ precision using a precomputed LUT.

The most efficient way to organize the LUT for the case of Lorenz–Mie scattering calculations is to use the scale invariance rule [9,12,13]. The scale invariance rule (SIR) exploits the fact that all the Lorenz–Mie computations are done using the size parameter $x=2\pi r/\lambda$ that relates radius and wavelength [10,11]. For a given CRI, the Lorenz–Mie scattering properties for a specified radius and wavelength are the same as those at another wavelength after adjusting the radius. If the CRI is fixed then we can establish a direct connection between the efficiencies [see Eqs. (2) and (3)] at wavelengths $\lambda$ and $\lambda _\mathrm{r}$ using a simple scaling in the radius domain given by

Equation (8) allows us to calculate the integral values [see Eqs. (2) and (3)] using the corresponding values of integrated cross sections $C_p$ precomputed at the reference wavelength $\lambda _\mathrm{r}$ on the discretized grids (quadratures) of radius, scattering angle, CRI, and stored on the hard drive as a LUT file [9].

The quadratures of radius, scattering angle, and CRI define the precision of LUT and the amount of information stored on the hard drive and in random-access memory. It is inefficient to store redundant information but its reduction can have a negative effect on the precision. The SIR-C LUT represents balance between the two conflicting criteria of precision and size.

Based on numerical simulations and earlier studies [9], we use the radius quadrature consisting of 700 log-equidistant grid bins to cover the full size range of liquid water cloud particles from $r_\mathrm{min}=10^{-3}$ to $r_\mathrm{max}=500\,\mathrm{\mu}$m. The SIR-C LUT coefficients are computed over the integration intervals formed by the neighboring radius grid bins [9]. We used $10^{4}$ points over each integration interval by setting the Mishchenko et al. Lorenz–Mie program’s parameters $\mathrm{N}=100$ and $\mathrm{NK}=100$ [11,14].

The quadrature of scattering angle consists of 203 angles $\Theta$ in the range from 0° to 180° (see Table 1). As in our aerosol-focused LUT [9], the quadrature near angles of 0° and 180° has 0.2° and 0.5° spacing because the elements of the normalized scattering matrix $\mathbf{p}$ [see Eq. (1)] can rapidly change there [15]. The rate of change in $\mathbf{p}$ is smaller between the angles of 5° and 175° that allows a coarser 1° spacing. The values of IOPs of interest for the other scattering angles can be estimated using interpolation [9].

Table 1. Scattering Angles Included in the SIR-C LUT

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The quadrature of complex refractive index has 1,400 CRIs: 56 real parts $\left (m_\mathrm{R}\right )$ of the CRI in the range between 1.25 and 1.36 with a step 0.002 [16]; 25 imaginary parts $\left (m_\mathrm{I}\right )$ of the CRI: 0, and 24 log-equidistant values between $10^{-5}$ and $10^{-3}$ [16].

To demonstrate the capabilities of SIR-C LUT, we compute the IOPs [see Eqs. (2)–(7)] using the LUT and compare them with the simulated truth values. The targeted $\pm 1\%$ precision shall be achieved at all wavelengths between 0.35 and 2.3 $\mathrm{\mu}$m that covers the range of channels provided by most existing and planned lidars, spectrometers, and polarimeters including for the NASA PACE and AOS missions [4].

It is an important question how many radius grid bins should be used for integration in Eqs. (2) and (3) to compute the simulated truth values for the case $r_\mathrm{min}=10^{-3}$ and $r_\mathrm{max}=500\,\mathrm{\mu}$m. We can demonstrate that an insufficient number of radius grid bins can lead to imprecise estimates of IOPs. Let us use Simpson’s rule for integration to compute the lidar ratio [see Eq. (7)] at wavelength $\lambda =0.532\,\mathrm{\mu}$m, assuming the CRI to be equal to $m=1.33557-i\cdot 1.825\times 10^{-9}$. The monomodal lognormal PSD $n\left (r\right )$ in our test is defined by its fixed effective variance $\nu _\mathrm{eff}=0.05$ and variable effective radius $r_\mathrm{eff}=1\ldots 100\,\mathrm{\mu}$m [9].

Figure 1 shows the lidar ratios resulting from the direct integration along with the estimations provided by SIR-C LUT. One can see that the insufficient number of $10^{3}$ and $10^{4}$ radius grid bins leads to the lidar ratios that differ by dozens of percent (see dashed and short-dashed curves in Fig. 1). The differences decrease as the number of radius grid bins increase. As an answer to our question, we computed the simulated truth values using $10^{8}$ radius grid bins (see dotted curve in Fig. 1), which sufficiently eliminated integration errors. The SIR-C LUT results (see solid curve in Fig. 1 that almost coincides with the dotted curve) are very close to the estimations with $10^{8}$ grid bins even with the radius quadrature consisting of $7\times 10^{6}$ bins due to the use of Gaussian integration scheme [11,14].

 

Fig. 1. Lidar ratio at wavelength $\lambda =0.532\,\mathrm{\mu}$m as a function of effective radius for the fixed effective variance $\nu _\mathrm{eff}=0.05$ and CRI $m=1.33557-i\cdot 1.825\times 10^{-9}$. Dashed, short-dashed, and dotted curves are correspondingly the results of integration with Simpson’s rule using $10^{3}$, $10^{4}$, and $10^{8}$ radius grid bins. Solid curve is the estimation by SIR-C LUT.

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For the comparisons we conducted $10^{5}$ random $\lambda$–CRI–PSD unit tests. We used monomodal lognormal PSDs $n\left (r\right )$ defined by effective radius $r_\mathrm{eff}$ and effective variance $\nu _\mathrm{eff}$ [9]. A uniform random number generator provided evenly distributed values of wavelength $\lambda$ in the range from 0.35 to $2.3\,\mathrm{\mu}$m, effective radius $r_\mathrm{eff}$ from 0.1 to $100\,\mathrm{\mu}$m, effective variance $\nu _\mathrm{eff}$ from 0.05 to 0.6, real part of the CRI $m_\mathrm{R}$ from 1.25 to 1.36, and imaginary part of the CRI $m_\mathrm{I}$ from 0 to $10^{-3}$. Using these random inputs together with the Bohren and Huffman program [10], we computed the simulated truth values for all the IOPs [see Eqs. (2)–(7)] and compared them with the corresponding SIR-C LUT values.

We again demonstrate that the $P_{12}\left (\Theta \right )$ element of the normalized scattering matrix is the most difficult IOP to estimate [9]. The $\pm 1\%$ precision was not achieved in 3.3% (or 3,342) of the test cases. Figure 2 shows the distribution of the problematic cases. Approximately 96% (or 3,209) of them have an imaginary part of the CRI $m_\mathrm{I}$ below $10^{-4}$ [see Fig. 2(a)] and approximately 96% (or 3,193) of issues happen for a scattering angle $\Theta$ below 1° [forward peak, see Fig. 2(d)]. The number of problematic cases increases with increasing effective radius [see Fig. 2(b)]. The relative difference [see Fig. 2(e)] is computed as

 

Fig. 2. Distribution of problematic test cases for (a) and (b) microphysical properties, (c) wavelengths, (d) scattering angles, and (e) relative difference for scattering matrix element $P_{12}\left (\Theta \right )$.

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The $\pm 1\%$ precision also was not achieved in 1% (or 993) of the test cases for backscatter coefficient, in 1% (or 1,003) for lidar ratio, and in 0.55% (or 549) for absorption coefficient. The problematic cases for these IOPs exhibit similar distributions of microphysical properties and wavelength as shown in Figs. 2(a)–2(c) and not shown here for brevity. All other IOPs were computed using the SIR-C LUT within the targeted $\pm 1\%$ precision for 100% of the test cases.

It is worth pointing out that the simulated truth values were computed using the Bohren and Huffman program [10], whereas the LUT was created using the Mishchenko et al. program [11,14]. The systematic distribution of problematic cases in Fig. 2(a) suggests that they are not caused by differences between the programs and will disappear if we increase the number of real and especially imaginary parts in the CRI quadrature. We do not increase this number due to the limitations in LUT file size, but the cross-checking of two well-established Lorenz–Mie programs helped us to achieve a higher level of confidence in our results. The reason to expect the reduction of problematic cases is based on our earlier study [9], where we noticed that for the imaginary parts below $10^{-4}$ the efficiencies [see Eqs. (2) and (3)] are generally becoming extremely oscillatory for large radii. In turn, those oscillations affect the quality of SIR-C LUT estimations.

Let us provide an example of practical application for the SIR-C LUT. NASA’s Aerosol Cloud meTeorology Interactions oVer the western ATlantic Experiment (ACTIVATE) field campaign [17] is a rich source of data to investigate relationships between cloud microphysics and lidar ratio. ACTIVATE is a five-year project (January 2019–December 2023) to gather data about marine boundary layer cloud systems and atmospheric aerosols. ACTIVATE employs two aircraft with complementary flight patterns: UC-12 King Air with the NASA Langley second-generation High Spectral Resolution Lidar (HSRL-2) on-board [18]; and HU-25 Falcon with instruments conducting in situ aerosol, cloud, and trace gas measurements.

We will use the results of in situ measurements from the Cloud Droplet Probe (CDP) manufactured by Droplet Measurement Technologies [19]. CDP measures the concentration and size distribution of cloud droplets in the radius range of 1–25 $\mathrm{\mu}$m in 30 size bins. We can assume the CRI to be equal to ${m=1.33557-i\cdot 1.825\times 10^{-9}}$ [20], and then with the help of SIR-C LUT compute the lidar ratio [see Eq. (7)] at wavelength ${\lambda =0.532\,\mathrm{\mu}}$m using the measured size distribution.

Using HSRL-2 data, it is possible to determine the cloud top height and measure the lidar ratio at wavelength $\lambda =0.532\,\mathrm{\mu}$m. For the comparisons with CDP results, we should include only the single-scattering liquid cases near cloud top. Low values of depolarization measured by HSRL-2 can help us to identify such kind of liquid water cloud cases [8].

The UC-12 King Air and HU-25 Falcon aircraft flight patterns are complementary but different. For the comparisons, we use only the HSRL-2 and CDP data that are collocated in space and time. The measurement day of September 22nd 2020 provided us with five cloud measurement points that are collocated within 150 seconds in time, 500 meters horizontally, and 50 meters vertically. Figure 3 shows the distribution of lidar ratios measured by HSRL-2 and derived from CDP measurements using direct integration (see triangles in Fig. 3) and SIR-C LUT (see squares in Fig. 3) for these five points. The results obtained from direct integration match those obtained from the SIR-C LUT within 1%, which reaffirms the high precision for the SIR-C LUT values. But the SIR-C LUT lidar ratio results were obtained a factor of $10^{3}$ faster than the direct integration lidar ratio results (integration from 1 to 25 $\mathrm{\mu}$m in radius), which highlights the core advantage of the SIR-C LUT values. We remark further that the HSRL-2 and CDP lidar ratios are within 0.8 sr from each other (see dashed lines in Fig. 3 showing the 0.8 sr border).

 

Fig. 3. Comparison of lidar ratios at wavelength $\lambda =0.532\,\mathrm{\mu}$m resulting from the HSRL-2 and CDP measurements.

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In conclusion, the single-scattering cloud IOPs derived by using our SIR-C LUT have a number of potential applications. The LUT is to be applied in a range of wavelengths starting at 0.35 and going up to at least 2.3 $\mathrm{\mu}$m. Overall, the precision of liquid cloud IOPs is nearly equivalent to direct integration of the PSD, but makes the calculations up to $10^{4}$ times faster, within a fraction of a second instead of hours if drizzle droplets are covered.