⚡ electrical engineering

A Three-Dimensional Path Loss Model for THz Band Aerial Communications

This paper proposes a new analytical three-dimensional path loss model for Terahertz band aerial communications that accounts for arbitrary transceiver geometries and frequency-selective absorption, validated against atmospheric propagation data across various drone-to-drone and high-altitude link scenarios.

Original authors: Sina Jorjani, Caglar Tunc, Ozgur Gurbuz, Akhtar Saeed

Published 2026-04-01

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CC BY 4.0

Original authors: Sina Jorjani, Caglar Tunc, Ozgur Gurbuz, Akhtar Saeed

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to send a high-speed message from one drone to another, or from a drone to a satellite, using a super-fast "invisible laser" of data called Terahertz (THz) waves. This is the technology that will power the internet of the future (6G), capable of downloading entire movies in a split second.

However, there's a big problem: The air gets in the way.

Just like trying to shout across a foggy field, these high-speed waves get absorbed by water vapor and oxygen in the atmosphere. The higher you go, the thinner the air, and the clearer the path. But calculating exactly how much signal is lost is incredibly hard because:

The drones can be at any height.
They can be flying in any direction (not just side-by-side or one directly above the other).
The air behaves differently depending on the specific "color" (frequency) of the wave.

The Problem with Old Maps

Previous attempts to map this signal loss were like using a flat, 2D map to navigate a mountainous terrain. They only worked if the drones were flying in a straight line horizontally or a straight line vertically. If the drones were flying in a complex, 3D zig-zag pattern, the old maps failed.

The New Solution: A "Smart 3D GPS" for Signals

This paper introduces a new, 3D mathematical model that acts like a smart GPS for signal loss. It doesn't just look at distance; it understands the shape of the journey through the sky.

Here is how the authors built it, using a simple analogy:

1. The "Two-Lane Highway" Concept

Imagine the signal traveling from Drone A to Drone B. The authors realized they could split the journey into two parts:

The Horizontal Lane: How far it travels sideways.
The Vertical Lane: How far it travels up or down.

They treated the signal loss in these two lanes separately, like calculating the fuel cost for driving across a state and then the fuel cost for climbing a mountain, and then adding them together.

2. The "Recipe" (The Regression Pipeline)

To create their model, the researchers didn't just guess. They used a computer tool (called am) to simulate millions of different flight paths and weather conditions. Then, they used a three-step "recipe" to turn this massive data into a simple formula:

Step 1 (The Distance Check): They figured out how much signal is lost just by how far the drones are apart.
Step 2 (The Altitude Check): They noticed that the "steepness" of the loss changes depending on how high up the drones are. Higher up = less loss. They created a rule to predict this based on altitude.
Step 3 (The Frequency Check): Finally, they realized that different "colors" of THz waves behave differently. They created a polynomial (a fancy math curve) to predict how the loss changes as you tune the frequency.

By chaining these three steps together, they created a single, elegant formula that can predict signal loss for any 3D position, any altitude, and any frequency between 0.1 and 1 THz.

Two Ways to Use the Recipe

The paper offers two ways to use this new model, depending on how much computing power you have:

The "One-Size-Fits-All" Approach (θ-agnostic):
- Analogy: Like using a single, universal map for the whole world.
- Pros: It's fast and simple. You don't need to know the exact angle of the flight path.
- Cons: It's slightly less accurate when the drones are low to the ground where the air is thick and foggy.
The "Custom-Made" Approach (θ-adaptive):
- Analogy: Like hiring a local guide for every specific mountain pass.
- Pros: It is extremely accurate, especially for low-altitude flights where the air is tricky.
- Cons: It requires more computer power because it calculates a specific rule for every possible angle.

The Results: Why It Matters

The researchers tested their model against the computer simulations and found:

Low Altitude (Drone-to-Drone): The "Custom-Made" approach was a winner, predicting the signal loss with near-perfect accuracy.
High Altitude (Satellite-to-Satellite): The air is so thin up there that the signal barely gets absorbed. In this case, the simpler "One-Size-Fits-All" approach was almost as good as the complex one, saving valuable computing time.

The Bottom Line

This paper gives engineers a universal toolkit to design the future of aerial internet. Instead of running slow, complex simulations every time they want to move a drone, they can now use this formula to instantly calculate: "If I move my drone here, and fly at this speed, will I still have a strong connection?"

It turns the chaotic, 3D sky into a predictable highway, ensuring that our future high-speed aerial networks don't crash into the atmosphere.

1. Problem Statement

The emergence of 6G communication systems necessitates the use of Terahertz (THz) and sub-THz bands (0.1–10 THz) to support ultra-high data rates. However, signal propagation in these bands is severely limited by free-space path loss (FSPL) and atmospheric absorption loss (primarily due to water vapor and oxygen).

Existing path loss models for aerial networks suffer from significant limitations:

Geometric Constraints: Most current models assume strictly horizontal or vertical transceiver alignments. They fail to account for the random 3D geometry inherent in aerial networks where drones and aircraft operate at varying altitudes and relative positions.
Scenario Specificity: Many models are tailored to specific scenarios (e.g., ground-to-air or drone-to-drone) and do not offer a unified framework for diverse aerial links.
Frequency Selectivity: Atmospheric absorption is highly frequency-selective, varying drastically across the THz spectrum, yet many simplified models do not capture this nuance across a broad range of sub-bands.

The paper aims to fill this gap by developing a unified, analytical path loss model that handles arbitrary 3D transceiver geometries and frequency-selective absorption across the 0.1–1 THz range.

2. Methodology

The authors propose a data-driven, analytical modeling framework based on a cascaded regression approach.

A. System Model

The model considers two transceivers ( $N_1, N_2$ ) in a 3D Cartesian coordinate system.

Total Path Loss ( $L_{total}$ ): Composed of Free-Space Path Loss ( $L_{fspl}$ ) and Absorption Loss ( $L_{abs}$ ).
Geometry: The distance is decomposed into horizontal ( $d_h$ ) and vertical ( $d_v$ ) components based on the zenith angle ( $\theta$ ).
Transmittance ( $\tau$ ): The absorption loss is modeled via transmittance, where $L_{abs} = -10 \log_{10}(\tau)$ .

B. Data Generation

Reference data was generated using the Atmospheric Model (am) tool under the U.S. Standard Atmosphere 1976 profile.

Scenarios: Three distinct aerial scenarios were simulated:
1. Dr2Dr: Low-altitude Drone-to-Drone (0–0.5 km).
2. MAAC: Medium-Altitude Aerial Communication (1–15 km).
3. U2U: High-Altitude UAV-to-UAV (15–50 km).
Frequency Range: 0.1–1 THz, covering 10 representative sub-bands (excluding sharp molecular absorption peaks).
Variables: Altitude, distance, zenith angle ( $0^\circ$ to $90^\circ$ ), and frequency.

C. Modeling Approach: Cascaded Regression

The authors model transmittance $\tau$ as a function of altitude ( $l$ ), horizontal distance ( $d_h$ ), vertical distance ( $d_v$ ), and frequency ( $f$ ) using a three-step regression pipeline:

Step 1 (Distance): Model transmittance as an exponential function of $d_h$ $d_{h}$ and $d_v$ $d_{v}$ for fixed altitude and frequency.
- $\tau \approx \exp(\hat{b}_{1,h} d_h + \hat{b}_{1,v} d_v)$
Step 2 (Altitude): Model the distance coefficients ( $\hat{b}_{1,h}, \hat{b}_{1,v}$ $\hat{b}_{1, h}, \hat{b}_{1, v}$ ) as exponential functions of altitude ( $l$ $l$ ).
- $\hat{b}_{1,\xi} \approx \hat{a}_{2,\xi} e^{\hat{b}_{2,\xi} l}$
Step 3 (Frequency): Model the altitude coefficients ( $\hat{a}_{2,\xi}$ $\overset{a}{^}_{2, ξ}$ ) as polynomial functions of frequency ( $f$ $f$ ).
- $\hat{a}_{2,\xi} \approx \sum \lambda f^p$

This results in a closed-form expression for transmittance:
$\hat{\tau} = \exp\left( \hat{\Lambda}_h(f) e^{\hat{b}_{2,h}l} d_h + \hat{\Lambda}_v(f) e^{\hat{b}_{2,v}l} d_v \right)$

D. Implementation Approaches

Two variations of the model were implemented and compared:

$\theta$ -agnostic Approach: Fits the model to the entire dataset regardless of the zenith angle. It separates horizontal and vertical components.
- Pros: Lower complexity, unified coefficients.
- Cons: May lose accuracy at extreme angles.
$\theta$ -adaptive Approach: Fits the model independently for each discrete zenith angle $\theta$ $θ$ .
- Pros: Higher accuracy, especially for low-altitude links.
- Cons: Higher computational complexity (requires $N_\theta$ sets of coefficients).

3. Key Contributions

3D Generalization: The first analytical model to handle arbitrary 3D transceiver geometries (random combinations of altitude and horizontal distance) rather than being restricted to purely horizontal or vertical links.
Unified Frequency Modeling: Covers a wide spectrum (0.1–1 THz) with detailed analysis across 10 representative sub-bands, capturing frequency-dependent absorption characteristics.
Dual-Approach Framework: Proposes and compares two implementation strategies ( $\theta$ -agnostic vs. $\theta$ -adaptive), offering a trade-off between computational complexity and modeling accuracy.
Comprehensive Validation: Validated against high-fidelity atmospheric simulation data across three distinct aerial scenarios (Dr2Dr, MAAC, U2U).

4. Results

The models were evaluated using Normalized Root Mean Square Error (NRMSE) against the reference am tool data.

Accuracy vs. Altitude:
- Low Altitude (Dr2Dr): The $\theta$ -adaptive approach significantly outperformed the $\theta$ -agnostic approach (nearly an order of magnitude lower NRMSE). In low altitudes, atmospheric absorption is dominant and highly sensitive to geometry.
- Medium Altitude (MAAC): The performance gap narrowed as absorption decreased, though $\theta$ -adaptive remained slightly more accurate.
- High Altitude (U2U): At high altitudes (15–50 km), the atmosphere is thin, making absorption negligible compared to FSPL. Consequently, both models converged, and the simpler $\theta$ -agnostic approach performed nearly as well as the complex $\theta$ -adaptive model, closely matching the FSPL-only baseline.
Distance and Angle Sensitivity:
- NRMSE generally increased with distance.
- The $\theta$ -agnostic model showed degradation at extreme zenith angles, whereas the $\theta$ -adaptive model maintained stable accuracy across all angles.
Complexity: The $\theta$ -agnostic approach has lower computational overhead and fewer coefficients, making it suitable for high-altitude scenarios where absorption is minimal.

5. Significance and Impact

Unified Framework: Provides a single, closed-form analytical tool for link budget analysis and coverage planning in future 6G aerial networks, eliminating the need for repeated, computationally expensive numerical simulations.
Design Guidance: Helps engineers decide which modeling approach to use based on the operational altitude:
- Use $\theta$ -adaptive for low-altitude, absorption-dominated scenarios (e.g., drone swarms).
- Use $\theta$ -agnostic for high-altitude scenarios (e.g., HAPS, satellites) to save computational resources without sacrificing accuracy.
Future Applications: The model supports dynamic applications such as space-air-ground integrated networks, mobility-aware path tracking, and adaptive beamforming where transceiver geometry changes rapidly.
Scalability: The framework is designed to be extended to varying weather conditions and validated with experimental data as it becomes available.