Uncertainty Quantification of Radio Wave Propagation over Irregular Terrains Using Adaptive Polynomial Chaos Expansion

TL;DR

This paper presents the first uncertainty quantification (UQ) study of realistic antenna configurations for irregular-terrain propagation using an adaptive polynomial chaos expansion method and coupled with a two-way parabolic wave equation (PWE) method to address this problem efficiently.

Abstract

Accurate modeling of radio wave propagation over irregular terrains is crucial for designing reliable wireless communication systems in such environments, yet uncertainties in the antenna configuration are not quantified within deterministic models. In this paper, we present, to the best of our knowledge, the first uncertainty quantification (UQ) study of realistic antenna configurations for irregular-terrain propagation. An adaptive polynomial chaos expansion (APCE) method is improved and coupled with a two-way parabolic wave equation (PWE) method to address this problem efficiently. The polynomial basis is extended according to variance contributions and terminated by a composite criterion combining validation error and sample-to-basis ratio, enabling stable coefficient estimations via least-square regression without additional regularization. Convergence analysis shows a monotonic error decay with increasing training samples, producing compact, low-interaction models and improved accuracy and robustness over the previous APCE methods. For two realistic terrain profiles, the proposed method accurately predicts the mean and the 5th-95th percentile range of the path loss, matching Monte Carlo (MC) references using only 30 PWE simulations. Using a fixed sampling budget, APCE outperforms standard and sparse PCE, with the largest gains observed for the 5th and 95th percentile estimates; as the sample size increases, APCE maintains low errors with reduced trial-to-trial variability.

Figures (6)

• Figure 1: Two-way PWE method over an irregular terrain. The forward (blue) and backward (red) propagating waves are shown; the light-green domain represents the true terrain profile, and the dark-green denotes its discretized approximation. For clarity, the discretization grid is exaggerated in this illustration; in the actual simulations, a much finer terrain discretization is employed.
• Figure 2: Jerslev and Hjorringvej terrain profiles used to validate the proposed uncertainty quantification framework.
• Figure 3: Geometry of the antenna configuration and definition of the uncertain geometric parameters: transmitter height, elevation angle, and receiver height.
• Figure 4: Validation error $\varepsilon_v$ [defined in \ref{['eq: l2_error']}] of the path-loss surrogate versus the number of training samples $N_s$ for the Jerslev and Hjorringvej terrain profiles. The solid lines represent the mean validation error, while the shaded bands (bounded by dashed lines) indicate the minimum and maximum errors across the independent trials.
• Figure 5: Mean and reliability interval (5%--95% quantiles) of the path loss versus range for the Jerslev and Hjorringvej terrain profiles. The proposed APCE method with $30$ PWE simulations is validated against the Monte Carlo (MC) reference using $10^5$ simulations.