# Laser Beam Atmospheric Propagation Modelling for Aerospace LIDAR Applications

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## Abstract

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## 1. Introduction

#### 1.1. Aerospace Laser Applications

#### 1.2. Structure of the Article

- Section 2 describes the effects on laser beam performance because of atmospheric attenuation. Covering the Beer–Lambert law governing transmittance, this section introduces the atmosphere composition, the dominant linear propagation effects of absorption and scattering, to then describe the non-linear effects concerning turbulence and thermodynamic propagative effects and the subsequent empirical models and theoretical backgrounds individually.
- Section 3 introduces empirical modelling to collectively combine the propagative effects in terms of laser performance. The benefits of empirical modelling in comparison to atmospheric radiative transfer codes are highlighted, and the approaches are subsequently reflected in practical radiometric measurement techniques for atmospheric extinction.
- Section 4 reviews the main atmospheric radiative transfer codes and emphasizes the underpinning methodology of the line-by-line analysis, the inherent assumptions and applications of each model and identifies trends in model development including more extensive use of absorption and scattering models.

## 2. Atmospheric Extinction and Transmittance

#### 2.1. Atmospheric Properties

_{2}, water, and ozone, as shown in Figure 6.

#### 2.2. Molecular Line Absorption

#### 2.2.1. Absorption Line Profile

#### 2.2.2. Continuum Absorption

#### 2.2.3. Transmittance Attenuated by Molecular Line Absorption

#### 2.3. Atmospheric Scattering

#### 2.3.1. Aerosols

#### 2.3.2. Rayleigh Scattering

#### 2.3.3. Mie Scattering

#### 2.4. Nonlinear Propagation Effects

#### 2.4.1. Thermal Blooming

#### 2.4.2. Kinetic Cooling

_{2}lasers, a temperature drop occurs [51]. As a result, the refractive index and gas density increase, focusing the laser.

#### 2.4.3. Bleaching

#### 2.4.4. Aerodynamic Effects

#### 2.5. Propagation through Haze, Fog and Rain

#### 2.6. Propagation through Atmospheric Turbulence

#### 2.6.1. Refractive Index Structure Coefficient

^{−2/3}[66,67].

**Figure 18.**Height-based refractive index structure coefficient: Fried’s, Brookner’s, Tatarski’s, and Hufnagel’s models [74].

#### 2.6.2. Turbulence Effects

#### 2.6.3. Astronomical Refraction

## 3. Combined and Empirical Propagation Models

#### 3.1. Laser Range Equation

#### 3.2. Signal to Noise Ratio

#### 3.3. Laser Beam Transmittance along a Slant Path

_{2}and O

_{2}, resulting in the broadening pressure in Equation (46), being simplified as the total atmospheric pressure at a certain altitude. The absorption cross section is then expressed as a function of frequency and altitude,

#### 3.4. Particle Retrieval

#### 3.5. Elder–Strong–Langer Model for Absorption

#### 3.6. Elder–Strong–Langer Model for Scattering

#### 3.7. Combined ESLM Model

#### 3.8. Radiometric Measurements of Atmosphere Extinction

#### 3.9. Application of Machine Learning in Laser Propagation

## 4. Atmospheric Radiative Transfer Models

#### 4.1. 4A/OP

#### 4.2. ARTS

#### 4.3. LIDORT/VLIDORT

#### 4.4. 6S/6SV1

#### 4.5. MODTRAN

^{−1}and coupled with absorption and scattering codes [7,125]. The analysis presented in previous work [7] highlights some of the limitations in the area of remote sensing where it is difficult to calculate the top of radiances and the code is computationally expensive due to the aerosol scattering and absorption modelling. For such cases, faster but less accurate models such as 6S are often substituted [124]. The application of MODTRAN is evident in the works in atmospheric correction [127], sensor simulation, calibration, and simulating realistic scenarios.

#### 4.6. LBLRTM

#### 4.7. COART

#### 4.8. DISORT

#### 4.9. MOSART

#### 4.10. RTTOV

#### 4.11. Summary

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

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**Figure 1.**Airborne laser system scanning and georeferenced. Adapted from [13].

**Figure 2.**Direct detection laser radar block diagram from [13].

**Figure 3.**Coherent detection LIDAR schematic from [13].

**Figure 4.**Free space optical communication block diagram. Adapted from [22].

**Figure 6.**Atmospheric transmittance across the spectral range at sea-level along an 1820 m path. Adapted from [28].

**Figure 7.**Absorption line strength of selected molecules from HITRAN 2012 database [29].

**Figure 8.**Absorption line profiles, Doppler, Voigt, and Lorentz. From [32], reproduced with permission.

**Figure 10.**Secondary marine aerosols as a result of biological activity, impacts atmospheric particle composition [39].

**Figure 14.**Illustrative example of Mie scattering. Adapted from [28].

**Figure 15.**Thermal blooming with a transverse wind. Adapted from [32].

**Figure 17.**Attenuation due to fog, rain, and drizzle. From [54], reproduced with permission.

**Figure 20.**Beam wander as a result of laser beam deflection by turbulent cells greater than the beam diameter. Adapted from [47].

**Figure 21.**Beam intensity profile is distorted by turbulent cells smaller than the beam diameter. Adapted from [47].

**Figure 22.**Receiver system for lasers. Adapted from [28].

**Figure 23.**Geometry of a laser beam propagation along a slant path adapted from [28].

**Figure 24.**Regularization approach to the inverse problem from [28].

**Figure 25.**Extinction measurement using laser spot energy to determine the intensity as a result of extinction [13].

**Figure 26.**Laser transmission measurement technique [28].

**Figure 27.**Line-by-line approach of atmospheric models for radiative transfer, considering atmospheric layers from the top of the atmosphere. Adapted from [110].

**Figure 28.**Simulation process of the radiative transfer model. Adapted from [115], reproduced with permission.

**Table 1.**Types of LIDAR [13].

Laser Type | Carrier Wavelength | |
---|---|---|

CO_{2} | 9.2–11.2 µm | |

Er:YAG | 2 µm | |

Raman Shifted Nd:YAG | 1.54 µm | |

Nd:YAG | 1.06 µm | |

GaAlAs | 0.8–0.904 µm | |

HeNe | 0.63 µm | |

Frequency Doubled Nd:YAG | 0.53 µm | |

Detection Technique | Interferometer Type | Modulation Technique |

Direct Detection | Not applicable | Pulsed |

Amplitude Modulation (AM) | ||

Coherent Detection | Heterodyne | Pulsed |

Homodyne | Amplitude Modulation (AM) | |

Offset Homodyne | Frequency Modulation (FM) | |

Hybrid (AM/FM, Pulse Burst) None (CW) | ||

Functions | Measurements | |

Tracking | Reflectance (Amplitude) | |

Moving Target Indication | Range (Time Delay) | |

Machine Vision | Velocity (Differential Range or Doppler Shift) | |

Velocimetry | Angular Position | |

Wind Shear Detection | Vibration | |

Target Identification | ||

Imaging | ||

Vibration Sensing |

**Table 2.**Atmospheric molecular composition. Adapted from [27], reproduced with permission.

Permanent Constituents | Variable Constituents | ||
---|---|---|---|

% by volume | % by volume | ||

Nitrogen (N_{2}) | 78.084 | Water Vapor (H_{2}O) | 0–0.04 |

Oxygen (O_{2}) | 20.948 | Ozone (O_{3}) | 0–12 × 10^{−4} |

Argon (Ar) | 0.934 | Sulfur Dioxide (SO_{2}) | 0.001 × 10^{−4} |

Carbon Dioxide (CO_{2}) | 0.036 | Nitrogen Dioxide (NO_{2}) | 0.001 × 10^{−4} |

Neon (Ne) | 18.18 × 10^{−4} | Ammonia (NH_{3}) | 0.004 × 10^{−4} |

Helium (He) | 5.24 × 10^{−4} | Nitric Oxide (NO) | 0.0005 × 10^{−4} |

Krypton (Kr) | 1.14 × 10^{−4} | Hydrogen Sulfide (H_{2}S) | 0.00005 × 10^{−4} |

Xenon (Xe) | 0.089 × 10^{−4} | Nitric acid vapor (HNO_{3}) | Trace |

Hydrogen (H_{2}) | 0.5 × 10^{−4} | Chlorofluorocarbons | Trace |

Methane (CH_{4}) | 1.7 × 10^{−4} | ||

Nitrous Oxide (N_{2}O) | 0.3 × 10^{−4} | ||

Carbon Monoxide (CO) | 0.08 × 10^{−4} |

Window Number | Window Boundaries (µm) |
---|---|

I | 0.72–0.94 |

II | 0.94–1.13 |

III | 1.13–1.38 |

IV | 1.38–1.90 |

V | 1.90–2.70 |

VI | 2.70–4.30 |

VII | 4.30–6.00 |

Type of Scattering | Size of Scatterer |
---|---|

Rayleigh Scattering | Electron Size of Scatterer $\mathsf{\lambda}$ |

Mie Scattering | Size of Scatterer ≈ $\mathsf{\lambda}$ |

Non-selective Scattering | Size of Scatterer $\mathsf{\lambda}$ |

Source | Amount, Tg/yr [10^{6} Metric Tons/yr] | |
---|---|---|

Range | Best Estimate | |

Natural | ||

Soil Dust | 1000–3000 | 1500 |

Sea Salt | 1000–10,000 | 1300 |

Botanical Debris | 26–80 | 50 |

Volcanic Dust | 4–10,000 | 30 |

Forest Fires | 3–1500 | 20 |

Gas-to-particle conversion (total) | 100–260 | 180 |

Sulphate from H_{2}S | 130–200 | |

Ammonium salts from NH_{3} | 80–270 | |

Nitrate from NO_{x} | 60–430 | |

Hydrocarbons from plant exudations | 75–200 | |

Photochemical | 40–200 | 60 |

Subtotal | 2200–24,000 | 3100 |

Anthropogenic | ||

Direct Emissions | 50–160 | 120 |

Gas-to-particle conversion (total) | 260–460 | 330 |

Sulphate from SO_{2} | 130–200 | |

Nitrate from NO_{x} | 30–35 | |

Hydrocarbons | 15–90 | |

Photochemical | 5–25 | 10 |

Subtotal | 320–640 | 460 |

**Table 6.**Laser transmittance through rainfall [47].

Rainfall Rate (cm/h) | Transmittance, τ, for 1800 m Path |
---|---|

0.25 | 0.88 |

1.25 | 0.74 |

2.5 | 0.65 |

10.0 | 0.38 |

**Table 7.**Rain intensity characterization [47].

Rain Intensity | Rainfall (mm/h) |
---|---|

Mist | 0.025 |

Drizzle | 0.25 |

Light | 1.0 |

Moderate | 4.0 |

Heavy | 16 |

Thundershower | 40 |

Cloudburst | 100 |

**Table 8.**Typical refractive index structure coefficient values for turbulence adapted from [47].

Turbulence Strength | Refractive Index Structure Coefficient, ${\mathit{C}}_{\mathit{n}}^{2}$ |
---|---|

Strong | ${C}_{n}^{2}=5\times {10}^{-14}$ [m^{−2/3}] |

Intermediate | ${C}_{n}^{2}=4\times {10}^{-16}$ [m^{−2/3}] |

Weak | ${C}_{n}^{2}=8\times {10}^{-18}$ [m^{−2/3}] |

$\mathit{b}$ | ${\mathit{h}}^{\prime}$ | ${\mathit{C}}_{\mathit{n}0}^{2}$ | Reference | |
---|---|---|---|---|

Fried’s Model | 1/3 | 3200 m | $4.22\times {10}^{-14}{\mathrm{m}}^{-1/3}$ | [70] |

Brookner’s Model | 5/6 | 320 m | $3.6\times {10}^{-13}{\mathrm{m}}^{-1/6}$ | [71] |

Tatarski’s Model | 4/3 | ∞ | $4.16\times {10}^{-13}{\mathrm{m}}^{-2/3}$ | [72] |

Hufnagel Condition I | −10 | 1000 m | $5.94\times {10}^{-53}{\mathrm{m}}^{-2/3}$ | [73] |

Hufnagel Condition II | 0 | 1500 m | $2.7\times {10}^{-16}{\mathrm{m}}^{-2/3}$ | [73] |

${\mathit{C}}_{\mathit{n}}^{2}$ | Equation | Ref. | |
---|---|---|---|

Fried’s Model | ${C}_{n}^{2}\left(h\right)=4.22\times {10}^{-14}{h}^{-1/3}{e}^{\left(-\frac{h}{3200}\right)}{\mathrm{m}}^{-2/3}$ | (22) | [70] |

Brookner’s Model | ${C}_{n}^{2}\left(h\right)=3.6\times {10}^{-13}{h}^{-5/6}{e}^{\left(-\frac{h}{320}\right)}{\mathrm{m}}^{-2/3}$ | (23) | [71] |

Tatarski’s Model | ${C}_{n}^{2}\left(h\right)=4.16\times {10}^{-14}{h}^{-4/3}{\mathrm{m}}^{-2/3}$ | (24) | [72] |

Hufnagel Model | ${C}_{n}^{2}\left(h\right)=2.7\times {10}^{-16}\left[2.2\times {10}^{-37}{h}^{10}{\left(\raisebox{1ex}{$w$}\!\left/ \!\raisebox{-1ex}{$27$}\right.\right)}^{2}\times {e}^{\left(\raisebox{1ex}{$-h$}\!\left/ \!\raisebox{-1ex}{$1000$}\right.\right)}+{e}^{\left(\raisebox{1ex}{$-h$}\!\left/ \!\raisebox{-1ex}{$1500$}\right.\right)}\right]$ | (25) | [73] |

Altitude (km) | ${\mathit{C}}_{\mathit{n}}\left({\mathbf{m}}^{-1/3}\right)\times {10}^{8}$ |
---|---|

0.001 | 30 |

0.003 | 20 |

0.01 | 15 |

0.03 | 10 |

0.1 | 6 |

0.3 | 4 |

1.0 | 1 |

3.0 | 1 |

**Table 12.**Constant values used in Equations (58) and (59) adapted from [28].

Constants | ${\mathit{A}}_{\mathit{i}}$ | ${\mathit{k}}_{\mathit{i}}$ | ${\mathit{\beta}}_{\mathit{i}}$ | ${\mathit{w}}_{\mathit{i}}$ |
---|---|---|---|---|

Window | ||||

I | 0.0305 | 0.800 | 0.112 | 54 |

II | 0.0363 | 0.765 | 0.134 | 54 |

III | 0.1303 | 0.830 | 0.093 | 2.0 |

IV | 0.211 | 0.802 | 0.111 | 1.1 |

V | 0.350 | 0.814 | 0.1035 | 0.35 |

VI | 0.373 | 0.827 | 0.095 | 0.26 |

VII | 0.598 | 0.784 | 0.122 | 0.165 |

**Table 13.**Transmittance equations for collocated transmitter and receiver [28].

Case | Condition | Equations | |
---|---|---|---|

A | $V\ge 6\mathit{km}$ $w<{w}_{i}$ | ${\tau}_{atm}={k}_{i}{\left(\frac{{w}_{i}}{w}\right)}^{{\beta}_{i}}{e}^{-z\frac{3.91}{V}{\left(\frac{{\mathsf{\lambda}}_{i}}{0.55}\right)}^{-\left(0.0057V+1.025\right)}}$ | (69) |

B | $V\ge 6\mathit{km}$ $w>{w}_{i}$ | ${\tau}_{atm}={e}^{-z\left[{A}_{i}\sqrt{w}+\frac{3.91}{V}{\left(\frac{{\mathsf{\lambda}}_{i}}{0.55}\right)}^{-\left(0.0057V+1.025\right)}\right]}$ | (70) |

C | $V<6\mathit{km}$ $w<{w}_{i}$ | ${\tau}_{atm}={e}^{-z\left[{A}_{i}\sqrt{w}+\frac{3.91}{V}{\left(\frac{{\mathsf{\lambda}}_{i}}{0.55}\right)}^{-0.585\sqrt[3]{V}}\right]}$ | (71) |

D | $V<6\mathit{km}$ $w>{w}_{i}$ | ${\tau}_{atm}={k}_{i}{\left(\frac{{w}_{i}}{w}\right)}^{{\beta}_{i}}{e}^{-z\frac{3.91}{V}{\left(\frac{{\mathsf{\lambda}}_{i}}{0.55}\right)}^{-0.585\sqrt[3]{V}}}$ | (72) |

R_{1} | Rain $w<{w}_{i}$ | ${\tau}_{atm}={e}^{-{A}_{i}\sqrt{w}}{e}^{-z\left[0.365{\left(\frac{\mathsf{\Delta}x}{\mathsf{\Delta}t}\right)}^{0.63}\right]}$ | (73) |

R_{2} | Rain $w>{w}_{i}$ | ${\tau}_{atm}={k}_{i}{\left(\frac{{w}_{i}}{w}\right)}^{{\beta}_{i}}{e}^{-z\left[0.365{\left(\frac{\mathsf{\Delta}x}{\mathsf{\Delta}t}\right)}^{0.63}\right]}$ | (74) |

Areas of Contribution | ML Algorithms | References | |
---|---|---|---|

Algorithms | Their Functions | ||

SNR improvement | GPML | Filtering | [105] |

SVM, random forest, decision tree, gradient boosting tree | Supervised learning | [106] | |

CNN | Supervised learning | [107] | |

Smoke detection | t-SNE, density-based spatial clustering | Clustering | [106] |

Cirrus cloud detection | CNN | Supervised learning | [107] |

Improving the prediction accuracy | RNN, CNN, RNN-CNN | Supervised learning in sensor fusion | [108] |

Dealing with atmospheric turbulence | ANN | Supervised learning | [109] |

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**MDPI and ACS Style**

Fahey, T.; Islam, M.; Gardi, A.; Sabatini, R.
Laser Beam Atmospheric Propagation Modelling for Aerospace LIDAR Applications. *Atmosphere* **2021**, *12*, 918.
https://doi.org/10.3390/atmos12070918

**AMA Style**

Fahey T, Islam M, Gardi A, Sabatini R.
Laser Beam Atmospheric Propagation Modelling for Aerospace LIDAR Applications. *Atmosphere*. 2021; 12(7):918.
https://doi.org/10.3390/atmos12070918

**Chicago/Turabian Style**

Fahey, Thomas, Maidul Islam, Alessandro Gardi, and Roberto Sabatini.
2021. "Laser Beam Atmospheric Propagation Modelling for Aerospace LIDAR Applications" *Atmosphere* 12, no. 7: 918.
https://doi.org/10.3390/atmos12070918