# Lidar and Radar Signal Simulation: Stability Assessment of the Aerosol–Cloud Interaction Index

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## Abstract

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## 1. Introduction

## 2. Methodology

#### 2.1. Lidar and Radar Signal Simulator (LARSS)

- The PSD and DSD are described by a log-normal size distribution.
- The aerosol number concentration is constant with height, presenting the same PSD except for hygroscopic growth.
- The minimum $SS$ to start the droplet formation is established by the PSD bin with the largest radius.
- The LWP is constant, as it is a requirement for retrieving the ACI indexes.
- The updraft velocity is constant with height.
- The droplets are only allowed to grow through condensation, making the maximum radius of a droplet at $r=20$ $\mathsf{\mu}$m (the limit where coalescence growth starts to present a considerable contribution).

#### 2.2. ACI Indices Estimation by LARSS

#### 2.2.1. ACI Uncertainty Based on Monte Carlo Technique

- 1.
- Table 1 shows the 13 initial parameters required to initialize one simulation with LARSS (${P}_{1}$, ${P}_{2}$,…, ${P}_{13}$). This set of parameters is noted as S.
- 2.
- The uncertainty associated to each parameter ${P}_{i}$ is represented by its relative error $\Delta {P}_{i}$.
- 3.
- A Gaussian distribution is associated to the uncertainty of each parameter ${P}_{i}$ where its standard, ${\sigma}_{P}i$, is derived from $\Delta {P}_{i}$.
- 4.
- From each Gaussian distribution, h random values are selected (e.g., ${P}_{1,1}$, ${P}_{1,2}$, …, ${P}_{1,h}$).
- 5.
- Random values are grouped in h sets (e.g., ${S}_{1}$, ${S}_{2}$, …, ${S}_{h}$). For example, the set ${S}_{1}$ is given by ${P}_{1,1}$, ${P}_{2,1}$, …, ${P}_{13,1}$.
- 6.
- h ACI indexes are retrieved with the generated sets.
- 7.
- The ACI index uncertainty is the standard deviation of the h ACI indexes.

## 3. LARSS Evaluation and ACI Assessment for Simulated Data

#### 3.1. LARSS Evaluation against Experimental Data

#### 3.2. Analysis of the ACI Indices for Different Aerosol Types

## 4. Results: ACI Index Sensitivity to Atmospheric Conditions

## 5. Proposal of ACI Index Based on Remote-Sensing Measurements (ACI_{Rs})

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

Symbol | Description | Units |

${A}_{1}$ | $\frac{g}{RT}\left(\right)open="("\; close=")">\frac{{L}_{w}R}{{c}_{p}{R}_{v}T}-1$ | ${\mathrm{m}}^{-1}$ |

${A}_{2}$ | $\frac{1}{{q}_{v}}+\frac{{L}_{w}^{2}}{{c}_{p}{R}_{v}{T}^{2}}$ | - |

$ACI$ | Aerosol cloud interaction | - |

$AC{I}_{AOD}$ | Aerosol cloud interaction index based on the backscatter coefficient | - |

$AC{I}_{reff}$ | Aerosol cloud interaction based on the aerosol optical depth | - |

$AF$ | Activation fraction | - |

$AOD$ | Aerosol optical depth | - |

C | Initial supersaturation | - |

$CCN$ | Cloud condensation nuclei | $\#/{\mathrm{m}}^{3}$ |

${c}_{p}$ | Specific heat capacity of most air at constant pressure | $\mathrm{J}/\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

${D}_{d}$ | Droplet diameter | $\mathsf{\mu}\mathrm{m}$ |

${D}_{reff}$ | Droplet effective radius | $\mathsf{\mu}\mathrm{m}$ |

${D}_{rmod}$ | Droplet modal radius | $\mathsf{\mu}\mathrm{m}$ |

DSD | Droplet number size distribution | - |

g | Mean gravitational acceleration | ${\mathrm{m}/\mathrm{s}}^{2}$ |

${q}_{v}$ | water vapor mixing ratio (mass of water per 1 kg of air) | - |

${q}_{w}$ | Liquid water mixing ratio (mass of liquid water per 1 kg of dry air) | - |

$|{K}_{p}|$ | ${\left(\right)}^{\frac{{m}_{r}-1}{{m}_{r}+2}}$ | - |

${m}_{r}$ | Refractive index of water | - |

$n\left({D}_{d}\right)$ | Droplet number concentration with a diameter d | $\#/{\mathrm{m}}^{3}$ |

${N}_{a}$ | Aerosol number concentration | $\#{/\mathrm{m}}^{3}$ |

${N}_{d}$ | Droplet number concentration | $\#/{\mathrm{m}}^{3}$ |

PSD | Particle number size distribution | - |

R | Specific gas constant of most gases | $\mathrm{J}/\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

${R}^{2}$ | Correlation coeficient | - |

RH | Relative humidity | % |

${r}_{m,dry}$ | Dry aerosol-particle modal radius | $\mathsf{\mu}\mathrm{m}$ |

${r}_{m,wet}$ | Wet aerosol-particle modal radius | $\mathsf{\mu}\mathrm{m}$ |

${R}_{v}$ | Specific gas constant of water vapor | $\mathrm{J}\mathrm{/}\mathrm{k}\mathrm{g}\phantom{\rule{0.166667em}{0ex}}\mathrm{K}$ |

$SS$ | Supersaturation | - |

$S{S}_{max}$ | Maximum supersaturation | - |

T | Temperature | K |

${T}_{o}$ | Reference temperature | K |

w | Updraft velocity | $\mathrm{m}/\mathrm{s}$ |

z | Height above ground level | m |

${Z}_{e}$ | Radar reflectivity | $\mathrm{d}\mathrm{B}$ |

${\alpha}_{ext}$ | Extinction coefficient | ${\mathrm{m}}^{-1}$ |

$\beta $ | Backscatter coefficient | $1/\mathrm{M}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\mathrm{r}$ |

${\beta}_{att}$ | Attenuated backscatter coefficient | $1/\mathrm{M}\mathrm{m}\phantom{\rule{0.166667em}{0ex}}\mathrm{s}\mathrm{r}$ |

$\gamma $ | aerosol proxy | − |

$\Delta $ ACI | Variability between the ACI index with and without fluctuations | % |

$\Delta {r}_{m,dry}$ | Fluctuation of the dry aerosol-particle modal radius | % |

$\Delta {r}_{m,wet}$ | Fluctuation of the wet aerosol-particle modal radius | % |

$\Delta w$ | Fluctuation of the updraft velocity | % |

$\kappa $ | Hygroscopicity parameter | - |

${\lambda}_{r}$ | Radar wavelength | $\mathrm{m}\mathrm{m}$ |

$\sigma $ | Standard deviation | - |

${\sigma}_{b}$ | Droplet effective cross-section | ${\mathrm{m}}^{2}$ |

## Appendix A. Additional LARSS Simulations

#### Appendix A.1. Simulation of the Twomey Effect by LARSS

**Figure A1.**Twomey effect of a cloud formed from the activation of ammonium sulfate particles. Supersaturation (SS) (

**a**), droplet number concentration (${N}_{d}$) (

**b**), droplet effective radius (${D}_{reff}$) (

**c**), and radar reflectivity (${Z}_{e}$) (

**d**) profiles inside the cloud.

#### Appendix A.2. Variations in ACI Related to the Presence of a Second (Coarse) Mode in the PSD

#### Appendix A.3. Range-Dependence of Cloud Microphysics

**Figure A3.**Early development of cloud properties: (

**a**) supersaturation (SS), (

**b**) droplet number concentration (${N}_{d}$), (

**c**) droplet effective radius (${D}_{reff}$), (

**d**) standard deviation ($\sigma $), (

**e**) liquid water content (LWC) in air ascending at constant velocity of 2 $\mathrm{m}/\mathrm{s}$.

## Appendix B. Additional Figures

#### Appendix B.1. ACI_{Rs} Sensitivity to r_{m,dry}, r_{m,wet} and w

**Figure A4.**${R}^{2}$ (left axis) and $\Delta ACI$ (right axis) with Respect to the ${r}_{m,dry}$ Uncertainty.

**Figure A5.**${R}^{2}$ (left axis) and $\Delta ACI$ (right axis) with respect to the ${r}_{m,wet}$ uncertainty. ${R}^{2}{N}_{d}-AOD$ and $\Delta ACIreff-AOD$ (not shown) are identical to ${R}^{2}{N}_{d}-\beta $ and $\Delta ACIreff-\beta $ because the uncertainty of ${r}_{m,wet}$ in the vecinity of the CBH does not affect the dry aerosol properties.

**Figure A6.**${R}^{2}$ (left axis) and $\Delta ACI$ (right axis) with respect to the w uncertainty. ${R}^{2}{N}_{d}-AOD$ and $\Delta ACIreff-AOD$ (not shown) are identical to ${R}^{2}{N}_{d}-\beta $ and $\Delta ACIreff-\beta $ because the uncertainty of w only affect within the cloud.

#### Appendix B.2. ACI_{Nd} and ACI_{reff} for All the Cases Used in LARSS

#### Appendix B.3. ACI_{Rs} for All the Cases Used in LARSS

#### Appendix B.4. ACI_{Rs} to ACI_{reff} Relation for All the Cases Used in LARSS

**Figure A9.**Relation between the $AC{I}_{reff}$ with the $AC{I}_{Rs}$ for different cases according to the legend.

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**Figure 1.**Simulation scheme (left to right columns): The ascent of the dry air parcel, cloud formation, atmospheric process and theories, retrieval of aerosol and cloud optical properties from the PSD and DSD, respectively; and generation of lidar (aerosol and cloud) and radar (cloud) signals.

**Figure 2.**(

**a**) Relative humidity (RH), (

**b**) effective radius and attenuated backscatter coefficient (${\beta}_{att}$) (

**c**) profiles retrieved using LARSS with standard atmospheric conditions at surface (298.15 K, 101.325 kPa, and $RH=30\%$).

**Figure 3.**(

**a**) Supersaturation (SS), (

**b**) droplet effective radius (${D}_{reff}$) and (

**c**) backscatter attenuated (${\beta}_{att}$) profiles retrieved from LARSS within the cloud.

**Figure 4.**Dependence between $log\left({N}_{a}\right)$ and $log\left({N}_{D}\right)$ (

**left**), between $log\left({N}_{a}\right)$ and $log\left({r}_{eff}\right)$ (

**center**), and between $log\left({Z}_{e}\right)$ and $log\left(\beta \right)$ (

**right**) at three different heights varying the initial number of aerosol for the accumulation mode ammonium sulfate using ten iterations. Input parameters of Table 1.

**Figure 5.**Monte Carlo technique scheme showing the process to derive ACI uncertainty from input parameter uncertainties $({P}_{i}\pm \Delta {P}_{i})$.

**Figure 6.**$\Delta AC{I}_{reff-\beta}$ and $\Delta AC{I}_{reff-AOD}$ with respect to the number of simulations (N).

**Figure 8.**$AC{I}_{Nd}$, $AC{I}_{reff}$ (based in $\beta $) and radar reflectivity factor (${Z}_{e}$) of ammonium sulfate, where the red dashed line marks the height at which the minimum radar sensitivity is reached.

**Figure 9.**Aerosol cloud interaction index ($AC{I}_{Nd}$ and $AC{I}_{reff}$) for different aerosol types, as a function of height, according to the legend.

**Figure 10.**Aerosol cloud interaction index ($AC{I}_{Nd}$ and $AC{I}_{reff}$) for different mixtures of two aerosol types, as a function of height, according to the legend.

**Figure 11.**$AC{I}_{Nd}$, $AC{I}_{reff}$ for different mixtures of three and four aerosol types according to the legend.

**Figure 12.**Correlation coefficients (${R}^{2}ACI$) and variability ($\Delta $ACI) for $AC{I}_{Nd-\beta}$ and $AC{I}_{reff-\beta}$ as a function of the fluctuation of the aerosol modal radius at surface (${r}_{m,dry}$) using LARSS.

**Figure 13.**Correlation coefficients (${R}^{2}ACI$) and variability ($\Delta $ACI) for $AC{I}_{Nd-\beta}$ and $AC{I}_{reff-\beta}$ as a function of the fluctuation of the aerosol modal radius at the CBH (${r}_{m,wet}$) using LARSS.

**Figure 14.**Correlation coefficients (${R}^{2}ACI$) and variability ($\Delta $ACI) for $AC{I}_{Nd-\beta}$ and $AC{I}_{reff-\beta}$ as a function of the fluctuation of the updraft velocity (w) using LARSS.

**Figure 16.**Relation between the $AC{I}_{Nd}$ and $AC{I}_{reff}$ with the $AC{I}_{Rs}$ considering ACI values where ${Z}_{e}$ greater than $-60\mathrm{d}\mathrm{B}$.

**Figure 17.**Relation between the $AC{I}_{reff}$ with the $AC{I}_{Rs}$ for different cases with AE from $0.6$ to $1.4$ (${Z}_{e}\ge -60$ dB).

Atmospheric properties | Specie | Ammonium sulfate | Atmospheric conditions |
Temperature $\left[{T}_{o}\right]$ (K) | 298 |

Number concentration $\left[{N}_{a}\right]$ ($\#{10}^{8}/{\mathrm{m}}^{3}$) | $5.0$ | ||||

Minimun radius ($\mathsf{\mu}\mathrm{m}$) | $0.01$ | Pressure $\left[{e}_{o}\right]$ ($\mathrm{kPa}$) | 101 | ||

Modal radius ($\mathsf{\mu}\mathrm{m}$) | $0.1$ | Updraft $\left[w\right]$ ($\mathrm{m}/\mathrm{s}$) | 2 | ||

Maximun radius ($\mathsf{\mu}\mathrm{m}$) | $0.5$ | Water-vapor ratio $\left[{q}_{v}\right]$ | 8 | ||

PSD standard deviation $\left[\sigma \right]$ | $1.6$ | Instrumental parameters | Lidar wavelength ($\mathrm{n}\mathrm{m}$) | 355 | |

Hygroscopicity parameter $\left[\kappa \right]$ | $0.51$ | ||||

Density (${\mathrm{g}/\mathrm{c}\mathrm{m}}^{3}$) | $1.77$ | Radar wavelength ($\mathrm{m}\mathrm{m}$) | $3.18$ | ||

Refractive index | $1.448+$ $i7.49\times {10}^{-6}$ |

**Table 2.**$AC{I}_{Nd}$, $AC{I}_{reff}$, and $AC{I}_{Rs}$ values for different height according to the slope of Figure 4.

Height (m) | 100 | 120 | 140 |
---|---|---|---|

$AC{I}_{Nd}$ | 0.97 ± 0.04 | 0.92 ± 0.04 | 0.81 ± 0.04 |

Height (m) | 260 | 280 | 300 |

$AC{I}_{reff}$ | 0.17 ± 0.01 | 0.20 ± 0.01 | 0.21 ± 0.01 |

Height (m) | 300 | 320 | 340 |

$AC{I}_{Rs}$ | 0.59 ± 0.03 | 0.61 ± 0.03 | 0.64 ± 0.03 |

Uncertainty | |||
---|---|---|---|

h | $AC{I}_{Rs}$ | $AC{I}_{reff}$ | $AC{I}_{Nd}$ |

10 | $0.05$ | $0.020$ | $0.07$ |

20 | $0.04$ | $0.016$ | $0.05$ |

30 | $0.03$ | $0.011$ | $0.04$ |

40 | $0.03$ | $0.014$ | $0.05$ |

Atmospheric aerosol properties | Measurement location | UGR | SNS |

Number concentration $\left[{N}_{d}\right]$ ($\#{10}^{8}/{\mathrm{m}}^{3}$) | 130 | 27 | |

Minimun radius ($\mathsf{\mu}\mathrm{m}$) | 0.012 | 0.012 | |

Modal radius $\left[{D}_{rmod}\right]$ ($\mathsf{\mu}\mathrm{m}$) | 0.045 | 0.062 | |

Maximun radius ($\mathsf{\mu}\mathrm{m}$) | 0.514 | 0.514 | |

Hygroscopicity parameter $\left[\kappa \right]$ | 0.186 | 0.198 | |

Density (${\mathrm{g}/\mathrm{c}\mathrm{m}}^{3}$) | 1.76 | 2.08 | |

Refractive index | 1.51 +0.005i | 1.51 +0.005i | |

Cloud properties | Supersaturation $\left[SS\right]$ (%) | 0.2 | 0.25 |

Activation-related properties | CCN concentration ($\#{10}^{8}/{\mathrm{m}}^{3}$) | 10.06 | 4.13 |

Activation fraction $\left[AF\right]$ | 0.077 | 0.152 |

Case | Aerosol Type | Aerosol Mode |
---|---|---|

${A}_{a}$ | Ammonium sulfate | Accumulation |

${B}_{a}$ | Biomass burning | Accumulation |

${D}_{a}$ | Dust | Accumulation |

${D}_{c}$ | Dust | Coarse |

**Table 6.**Initial parameters used for the different aerosol types. Source: ${}^{\alpha}$ Mészáros [32], ${}^{\beta}$ Tu and Kanapilly [35], ${}^{\gamma}$ D’Almeida et al. [50], ${}^{\delta}$ Remer et al. [47], ${}^{\u03f5}$ Keil and Haywood [48], ${}^{\zeta}$ Dentener et al. [46], ${}^{\eta}$ Curtis et al. [34], ${}^{\theta}$ Denjean et al. [49], ${}^{\iota}$ Hande et al. [33], ${}^{\lambda}$ Zhai et al. [45], and ${}^{\mu}$ Psichoudaki et al. [44].

Atmospheric properties | Specie | Ammonium sulfate [${A}_{a}$] | Burning Biomass [${B}_{a}$] | Dust Accumulation [${D}_{a}$] | Dust Coarse [${D}_{c}$] | Atmospheric conditions | Temperature $\left[{T}_{o}\right]$(K) | 298 |
---|---|---|---|---|---|---|---|---|

Number concentration $\left[{N}_{a}\right]$ ($\#{10}^{8}/{\mathrm{m}}^{3}$) | ${}^{\alpha}$ 5.00 | ${}^{\u03f5}$ 18.90 | ${}^{\zeta}$ 7.00 | ${}^{\zeta}$ 0.35 | ||||

Minimun radius ($\mathsf{\mu}\mathrm{m}$) | ${}^{\alpha}$ 0.01 | ${}^{\delta}$ 0.05 | ${}^{\zeta}$ 0.05 | ${}^{\zeta}$ 0.50 | Pressure $\left[{e}_{o}\right]$ ($\mathrm{kPa}$) | 101 | ||

Modal radius ($\mathsf{\mu}\mathrm{m}$) | ${}^{\beta}$ 0.10 | ${}^{\delta}$ 0.14 | ${}^{\zeta}$ 0.20 | ${}^{\zeta}$ 1.30 | Updraft $\left[w\right]$ ($\mathrm{m}/\mathrm{s}$) | 2 | ||

Maximun radius ($\mathsf{\mu}\mathrm{m}$) | ${}^{\alpha}$ 0.5 | ${}^{\delta}$ 0.5 | ${}^{\zeta}$ 0.5 | ${}^{\zeta}$ 6.0 | Water-vapor ratio $\left[{q}_{v}\right]$ ($\mathrm{g}/\mathrm{k}\mathrm{g}$) | 8 | ||

Standard deviation $\left[\sigma \right]$ | ${}^{\iota}$ 1.6 | ${}^{\gamma}$ 1.3 | ${}^{\iota}$ 1.59 | ${}^{\iota}$ 2.00 | Instrumental parameters | Lidar wavelength ($\mathrm{n}\mathrm{m}$) | 355 | |

Hygroscopicity parameter $\left[\kappa \right]$ | ${}^{\iota}$ 0.51 | ${}^{\mu}$ 0.22 | ${}^{\iota}$ 0.14 | ${}^{\iota}$ 0.14 | ||||

Density (${\mathrm{g}/\mathrm{c}\mathrm{m}}^{3}$) | ${}^{\iota}$ 1.77 | ${}^{\lambda}$ 1.15 | ${}^{\iota}$ 2.60 | ${}^{\iota}$ 2.60 | Radar wavelength ($\mathrm{m}\mathrm{m}$) | $3.18$ | ||

Refractive index | ${}^{\eta}$ 1.448 + $i7.49\times {10}^{-6}$ | ${}^{\delta}$ 1.520 + $i0.025$ | ${}^{\theta}$ 1.530 + $i0.008$ | ${}^{\theta}$ 1.530 + $i0.008$ |

**Table 7.**Fluctuation range of each parameter to retrieve the ACI index based in the backscatter coefficient ($\beta $) or the aerosol optical depth ($AOD$) with a variability lower than $10\%,20\%$ and $30\%$. Parameters are sorted according to their influence on $AC{I}_{reff}$ uncertainty (Up, higher influence). ${R}^{2}$ remains above 0.6 for all parameters, except for hygroscopicity parameter at CBH (${R}^{2}=0.31$) and the updraft velocity (${R}^{2}=0.34$).

ACI Index Variability | |||||
---|---|---|---|---|---|

ACI Based on | <10% | <20% | <30% | ||

Parameter fluctuation (%) | Refractive index | $\beta $ | 1 | 3 | 5 |

AOD | 15 | 24 | 35 | ||

PSD standard deviation at surface | $\beta $ | 4 | 6 | 10 | |

AOD | 6 | 10 | 16 | ||

Dry modal radius at surface | $\beta $ | 7 | 13 | 16 | |

AOD | 13 | 16 | 18 | ||

Wet modal radius at CBH | $\beta $ | 15 | 17 | 20 | |

AOD | |||||

Dry maximum radius at surface | $\beta $ | 46 | 57 | 63 | |

AOD | 63 | 65 | 67 | ||

Hygroscopicity parameter at CBH | $\beta $ | 56 | 87 | 94 | |

AOD | |||||

Updraft | $\beta $ | 59 | 95 | 119 | |

AOD | |||||

Wet maximum radius at CBH | $\beta $ | 74 | 144 | 178 | |

AOD | |||||

Hygroscopicity parameter at surface | $\beta $ | 82 | 135 | 216 | |

AOD |

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## Share and Cite

**MDPI and ACS Style**

Fajardo-Zambrano, C.M.; Bravo-Aranda, J.A.; Granados-Muñoz, M.J.; Montilla-Rosero, E.; Casquero-Vera, J.A.; Rejano, F.; Castillo, S.; Alados-Arboledas, L.
Lidar and Radar Signal Simulation: Stability Assessment of the Aerosol–Cloud Interaction Index. *Remote Sens.* **2022**, *14*, 1333.
https://doi.org/10.3390/rs14061333

**AMA Style**

Fajardo-Zambrano CM, Bravo-Aranda JA, Granados-Muñoz MJ, Montilla-Rosero E, Casquero-Vera JA, Rejano F, Castillo S, Alados-Arboledas L.
Lidar and Radar Signal Simulation: Stability Assessment of the Aerosol–Cloud Interaction Index. *Remote Sensing*. 2022; 14(6):1333.
https://doi.org/10.3390/rs14061333

**Chicago/Turabian Style**

Fajardo-Zambrano, Carlos Mario, Juan Antonio Bravo-Aranda, María José Granados-Muñoz, Elena Montilla-Rosero, Juan Andrés Casquero-Vera, Fernando Rejano, Sonia Castillo, and Lucas Alados-Arboledas.
2022. "Lidar and Radar Signal Simulation: Stability Assessment of the Aerosol–Cloud Interaction Index" *Remote Sensing* 14, no. 6: 1333.
https://doi.org/10.3390/rs14061333