Flight trajectory prediction enabled by time-frequency wavelet transform

The study addresses short-term flight trajectory prediction (FTP), a core capability for trajectory-based operations in air traffic management. Rising traffic volumes and airspace complexity demand accurate forecasting of 4D trajectories to support downstream tasks such as ETA estimation, conflict detection, and traffic flow prediction. FTP is commonly framed as multivariate time-series forecasting over position and velocity attributes; however, existing physics-based and state-estimation approaches struggle with generalization and environmental uncertainty, and machine learning/deep learning methods focusing solely on time-domain temporal modeling often miss subtle maneuver patterns, especially during climb/descent. The authors hypothesize that incorporating time-frequency analysis via wavelet transforms can better capture both global flight trends (low-frequency) and local maneuver details (high-frequency), thereby improving short-term prediction accuracy and robustness. The work proposes a wavelet-transform-based framework to test this hypothesis and demonstrates its practical benefits on real-world surveillance data.

Prior FTP methods fall into four categories: (1) Kinetics-and-aerodynamics models using physical rules and handcrafted mathematical formulations, which often lack generalization and are sensitive to environmental uncertainties; (2) State-estimation methods (e.g., Kalman filtering, state-space models) that treat flight as a state transition process but still face uncertainty impacts; (3) Machine-learning models that learn hidden motion features from data yet struggle with complex maneuver controls; and (4) Deep-learning approaches (RNNs, CNN-LSTMs, Transformers) that have improved fitting capacity but primarily operate in the time domain, limiting their ability to capture fine-grained maneuver dynamics. Frequency-domain and time-frequency analyses have shown promise in time series forecasting, yet remain underexplored in FTP. The gap identified is the lack of explicit multi-resolution time-frequency modeling (global trends vs. local details) for flight trajectories.

Proposed framework (WTFTP): a wavelet-transform-based time-frequency approach for short-term FTP. Core idea: predict wavelet coefficients (WTCs) of trajectory attributes and reconstruct both historical sequence and the next-step state via inverse discrete wavelet transform (IDWT).

• Problem setup: Given past M multivariate points P_{N−M:N−1} with attributes [Lon, Lat, Alt, Vx, Vy, Vz], predict next state P_N.
• Wavelet analysis: Use discrete wavelet transform (DWT) to decompose each attribute into multi-scale components. Low-frequency components capture global flight trends; high-frequency components capture local maneuver details. Predicted WTCs are fed to an IDWT module to reconstruct the historical trajectory and predict the next point.
• Neural architecture: Encoder–multiple-decoders design. • Input embedding: Two-layer FC with ReLU maps each low-dimensional trajectory point to a D-dimensional feature (I ∈ R^{M×D}). • Encoder: LSTM processes embedded sequence to produce trajectory embeddings H capturing temporal dynamics; final hidden/cell states provide memory. • Decoders: One decoder per WTC scale (for L-level wavelet analysis, decoders i=0..L). Each decoder includes a Wavelet Attention (WAtt) module followed by an LSTM and projection layer. WAtt computes attention scores over historical steps to enhance scale-oriented features and applies 1D convolutions to align enhanced embeddings to the target WTC temporal length h_i (accounting for downsampling and symmetric boundary extension). Decoder LSTM initializes cell states with encoder memory to strengthen scale-specific learning, then outputs WTC Q_i via LayerNorm+FC. • IDWT module: Stacks predicted WTCs per attribute and performs iterative reconstruction with predefined filters to obtain a time-domain sequence of length M+1 (first M reconstructed history + 1 predicted next step), trimming boundary artifacts.
• Training objective: Wavelet loss equals the sum of MSE across all attributes and all WTC scales between predicted and ground-truth wavelet coefficients; supervision uses DWT of the real historical sequence plus next-step ground truth.
• Dataset and preprocessing: Real-world surveillance data from multi-source Secondary Surveillance Radar (SSR) and ADS-B in China over ~45 days. Steps: data check, parsing, registering, alignment, filtering, and fusion. Attributes include timestamp, positions, and speeds (3D Earth space). Update interval: 20 s. Normalization: min–max (Lon/Lat in degrees; Alt in 10 m; speed in km/h). Split: first 40 days for training, following 1 day for hyperparameter tuning, last 4 days for testing.
• Evaluation metrics: MAE, RMSE, MRE (per-attribute), and Mean Deviation Error (MDE) computed as Euclidean distance in ECEF coordinates (km) to assess overall 3D positional deviation.
• Baselines: Vanilla LSTM, TCN, CNN-LSTM, Transformer, FlightBERT (Transformer with binary encoding and attribute-correlation attention).
• Training/config: PyTorch (wavelet ops, DWT/IDWT), Adam optimizer (lr=0.001, decay 0.5 every 10 epochs). Model configs tuned on small/validation sets (details in Supplementary).

Overall performance (Table 1): WTFTP achieves best results across most metrics and substantially improves 3D deviation.

• Aggregate metrics (test set): • MAE: Lon 0.0025 deg, Lat 0.0022 deg, Alt 1.14 (10 m units). • MRE: Lon 0.0023%, Lat 0.0078%, Alt 0.23%. • RMSE: Lon 0.0148 deg, Lat 0.0125 deg, Alt 8.91 (10 m units). • MDE: 0.3855 km (<400 m), ≈35% reduction vs. best baseline (FlightBERT 0.5910 km).
• Relative gains vs. best baseline (FlightBERT): ≥30% reduction in MAE and ≥20% reduction in MRE on longitude/latitude; strong MDE improvement indicating practical accuracy for ATC use.
• Flight phases (Table 2): WTFTP maintains robust performance across cruise, climb, descent. • MDE: Cruise 0.3405 km; Climb 0.4846 km; Descent 0.3753 km. Baselines show larger degradation during maneuvering (climb/descent). • WTFTP excels especially under maneuver control despite slightly higher RMSE on altitude in some cases.
• Complex approach/traffic control case study: In approach with a flying circle (intention changes), WTFTP yields MDE 0.5003 km and DTW 15.00 km, outperforming baselines (best FlightBERT: MDE 0.7873 km; DTW 23.45 km). Over 36% relative reduction in both MDE and DTW vs. FlightBERT.
• Ablation studies (Table 3): • Wavelet levels: 2-level (B1) slightly improves over 1-level WTFTP (MDE 0.3727 km vs. 0.3855 km); 3-level (B2) degrades due to overly short WTC sequences hampering temporal modeling. • Wavelet Attention (WAtt): Removing WAtt increases errors at all levels (e.g., C1 MDE 0.5090 km). WAtt improves scale-oriented feature learning and prediction of WTCs.
• Interpretability: Attention visualizations show low-frequency WTCs attend more to recent history (trend), while high-frequency WTCs focus on short, localized segments reflecting maneuver details. Reconstructing with only WTCo recovers global trends; only WTC1 captures local changes; combining all WTCs closely matches ground truth and predicts the next step accurately, including during descending/turning.
• Practicality: Achieves sub-400 m 3D deviation on real-world data with 20 s update interval, indicating readiness for operational support of trajectory-based operations.

The findings validate the central hypothesis that explicit time-frequency modeling via wavelet decomposition enhances short-term FTP by jointly capturing global trends and local maneuver details. WTFTP’s encoder–multi-decoder architecture, guided by supervision in the wavelet domain and augmented with a wavelet attention mechanism, delivers consistent accuracy gains over strong deep-learning baselines, particularly in maneuver-intensive phases (climb, descent, approach). The time-frequency perspective improves robustness to intention changes and complex airspace scenarios, translating into substantially lower 3D positional deviations (MDE). While altitude RMSE can be affected by high-frequency estimation noise during cruise, the overall MAE/MRE on altitude remain competitive or better, and the comprehensive MDE demonstrates operational relevance. The ablation analyses further underscore that an appropriate wavelet level (e.g., 1–2) and the WAtt module are crucial to balancing detail capture with temporal context. Collectively, the results suggest that wavelet-based multi-resolution representations are a powerful inductive bias for FTP, complementing temporal models and overcoming limitations of purely time-domain approaches.

This work introduces WTFTP, a wavelet-transform-based framework for short-term flight trajectory prediction that predicts wavelet coefficients and reconstructs trajectories via IDWT. The encoder–multi-decoder design with a wavelet attention module effectively learns scale-oriented features, enabling accurate modeling of both global trends and local maneuvers. On real-world SSR/ADS-B data, WTFTP significantly outperforms state-of-the-art baselines across MAE, MRE, RMSE (most dimensions), and notably reduces MDE to under 0.4 km. Case studies confirm robustness in complex, maneuver-rich airspace. Future directions include: (1) improving altitude prediction during cruise by controlling high-frequency noise and component convergence (e.g., loss design); and (2) extending to non-autoregressive multi-step prediction to mitigate error accumulation over longer horizons.

• Altitude during cruise: The model may over-emphasize fast dynamics on altitude in the largely steady cruise phase, introducing estimation noise; altitude RMSE can be higher than some baselines despite strong MAE/MRE.
• Multi-step horizons: Beyond ~80 s, WTFTP’s iterative predictions become sensitive to accumulated historical deviations and may underperform FlightBERT for longer horizons; a non-autoregressive strategy is needed.
• Wavelet level sensitivity: Excessively high decomposition levels shorten WTC sequences, degrading temporal modeling and prediction stability.
• Data availability: Full real-world trajectory data cannot be publicly released due to safety constraints, which may limit external replication; only processed examples are shared.