Multi-horizon flight trajectory prediction enabled by time-frequency wavelet transform

Domain: physics
Published: 2025-12-11
Editorial outcome: minor_revision

Abstract
Flight trajectory prediction is a fundamental task in air traffic control. While our previous WTFTP framework leveraged wavelet-based time-frequency analysis for trajectory prediction, its iterative single-horizon paradigm suffers from error accumulation in long-horizon forecasts. Here, the WTFTP+ framework is proposed to enhance multi-horizon prediction performance while further exploring the potential of time-frequency analysis in flight trajectory prediction tasks. An encoder-decoder neural architecture is designed to generate wavelet components of predicted trajectories, where a direct multi-horizon prediction paradigm is employed to mitigate the cumulative errors of the WTFTP. Additionally, a time-frequency bridging mechanism is proposed to explore intrinsic correlations among multi-scale flight patterns to enhance the learning ability of wavelet components. Experiments on real-world datasets demonstrate that WTFTP+ not only retains the superior single-horizon prediction performance of WTFTP but also significantly improves multi-horizon prediction accuracy, achieving over 40% mean deviation error reduction at the 5-minute horizon compared with WTFTP.

Introduction
The aviation industry has undergone significant growth over the past few decades, with global passenger traffic reaching 8.7 billion in 2023. The increasing flights are providing heightened complexity to current air traffic control (ATC) systems1. To ensure operational safety and efficiency in congested airspace, trajectory-based operations (TBO) have emerged as a promising solution in future ATC frameworks, exemplified by initiatives such as the next generation air transportation system (NextGen) in the United States and the Single European Sky ATM Research (SESAR) in Europe2. Unlike existing ATC systems that rely on pre-assigned static flight routes, TBO focuses on dynamic, real-time trajectory planning via precise trajectory prediction and sharing to adapt to evolving airspace conditions. As a fundamental technique of the TBO, flight trajectory prediction (FTP) has attracted significant attention from both academic researchers and industry practitioners3–5.

Accurate FTP plays a pivotal role in ATC systems by enhancing coordination, optimizing airspace management, and improving decision-making in real-time ATC operations6–10. In recent years, FTP methods were implemented by well-designed technical frameworks, from traditional physical modeling to state-of-the-art deep learning techniques. Specifically, early FTP methods generally depended on physics-based models with aerodynamics and kinematic equations to capture the motion dynamics of aircraft11–13. Subsequently, filter-based FTP methods were proposed to refine predictions through iterative updates based on observed positions by particle filters14, Kalman filters15,16, etc. Although these methods achieved the desired performance improvements, they still struggle to capture transition patterns due to the high dependency on dynamic models and are prone to error accumulation over extended future horizons, thereby limiting their multi-horizon prediction capability. With the development of artificial intelligence techniques17,18, data-driven methods have emerged as primary approaches for FTP modeling, which learns complex transition patterns and temporal-spatial dependencies by fitting large-scale historical trajectory data19, and finally enable more accurate and adaptive trajectory predictions in various conditions20–24. Nevertheless, most existing data-driven FTP methods primarily focus on advancing temporal-spatial modeling techniques, while often overlooking the exploration of the frequency domain. The inability to model time-frequency patterns inevitably impairs performance in relatively high maneuver scenarios.

In our previous work22, a wavelet transform-based flight trajectory prediction (WTFTP) framework is proposed to tackle the single-horizon FTP tasks, which leverages time-frequency analysis to capture global flight trends and local motion details represented by wavelet components (WTCs). The descriptions of the time-frequency analysis and WTCs can be found in Supplementary Information (Section “Multi-resolution analysis”). Thankstothetime-frequencyperspectiveforthetransitionmodelingofflighttrajectory,theWTFTPachievedsignificantperformanceimprovementsinvariousATCscenarios,especiallyduringthecriticalmaneuverflightphase.Asanalyzedinthediscussion,theWTFTPstillsuffersfromaperformancedegradationinmulti-horizonpredictionduetotheiterativepredictionparadigm.Itisbelievedthatthefollowingtwoissuesarethekeytofurtherimprovingtheperformanceandapplicabilityoftime-frequencyanalysisintheFTPtasks:

• As a single-horizon prediction framework, the performance of the WTFTP is limited by larger error accumulation in multi-horizon prediction, providing only inferior applicability in real-world ATC operations. Specifically, the WTFTP performs the multi-horizon prediction in an iterative manner, where the previous predictions are regarded as pseudo observations to forecast trajectory sequences over an extended time horizon. The pseudo observations inevitably introduce substantial cumulative errors with horizon-wise propagations and further impact the prediction of high-frequency wavelet components, thereby degrading the accuracy of multi-horizon predictions.
• In general, the WTFTP utilizes an encoder-decoder architecture to estimate multi-scale wavelet coefficients to support the FTP task, in which multiple decoders are dedicatedly designed to perform the learning process corresponding to multiple wavelet scales. Considering the orthogonality of the approximation subspace of wavelet components, each decoder performs an independent prediction process without desired feature interaction. A detailed description of the orthogonal space formulation under multi-scale analysis is provided in Supplementary Information (Section “Multi-resolution analysis”). However, in real-world scenarios, flight transition patterns are tightly coupled with inherent dependencies (e.g., driven by a turning maneuver, the trajectory may present a flight trend generally shifting from one direction to another from a macro perspective). It is believed that rigorous mathematical assumptions of orthogonality may hinder performance in the FTP tasks, since the lack of interaction of multi-scale wavelet components ignores the essential inherent dependencies within flight patterns across different time-frequency scales.

To address the mentioned limitations, in this work, an FTP framework, called WTFTP+, is proposed to improve the previous WTFTP framework to enhance multi-horizon FTP performance. The WTFTP+ retains the core concept of time-frequency analysis from the WTFTP framework, i.e., generating the optimal wavelet coefficients based on the observations and constructing the future trajectory sequence using inverse discrete wavelet transform (IDWT) operation. In this paradigm, the WTFTP+ framework not only can fully exploit the advantages of the time-frequency modeling technique but also is able to mitigate the error accumulation problem in multi-horizon prediction scenarios.

As depicted in Fig. 1, to mitigate the problem of error accumulation of the iterative multi-horizon prediction, the WTFTP+ is designed in a direct multi-horizon prediction paradigm with a generalized encoder-decoder architecture. The encoder is applied to extract the temporal-spatial features from the observed trajectory sequence, while the decoder is used to generate the optimal wavelet coefficients for the predicted horizons. The improved Transformer-style neural network serves as the backbone of the WTFTP+ framework to facilitate the time-frequency domain interaction.

Unlike the multiple decoders in the WTFTP framework, the WTFTP+ is designed with a single-decoder architecture, where the scale-aware wavelet component decoder (SAWCD) is proposed to generate predictions by considering the underlying correlations among the multi-scale flight patterns. Specifically, to obtain the required feature interaction, as well as retain the component orthogonality, a time-frequency bridging mechanism is proposed to support the learning of the multi-scale wavelet coefficients, in which wavelet-domain-to-time-domain (W2T) and time-domain-to-wavelet-domain (T2W) blocks with the learnable transform matrices are proposed to bridge the domain gap. In addition, a cross-domain attention (CDA) mechanism is designed to establish the underlying correlations between observations and the predictions. The above two mechanisms are expected to simultaneously learn the flight transition patterns from both the time and frequency domains, thereby improving the overall performance of FTP tasks.

To validate the effectiveness of the proposed WTFTP+ framework, experiments are conducted on a real-world flight trajectory dataset collected from an industrial ATC system. Furthermore, several competitive baselines are also selected to provide a comprehensive evaluation and comparison. The experimental results demonstrate that the proposed framework outperforms the baselines across nearly all prediction horizons and evaluation metrics. Notably, WTFTP+ achieves approximately a 40% relative reduction in mean deviation error (MDE) for the 5-min prediction horizon compared to the previous WTFTP framework. All the proposed technique improvements are confirmed by extensive ablation studies. Most importantly, intuitive visualizations and in-depth analyses are conducted to investigate the interpretability and generalizability of the proposed WTFTP+ framework. It is believed that the proposed WTFTP+ framework not only broadens the scope of the time-frequency analysis for FTP modeling but also provides a powerful and practical tool for the modern ATC systems and downstream applications. In summary, the proposed WTFTP+ framework contributes to the FTP research in the following ways:

• Based on the previous WTFTP work, an enhanced framework called WTFTP+ is proposed to achieve the multi-horizon FTP task by employing a direct prediction paradigm, which effectively addresses the error accumulation in long-horizon predictions.
• A Transformer-based encoder-decoder neural architecture is proposed to obtain expected wavelet coefficients, in which a SAWCD is designed to perform feature extraction towards intra-scale (i.e., within a certain scale) flight patterns while capturing the inherent correlations of inter-scale (i.e., among different scales) flight patterns.
• The time-frequency bridging mechanism is designed in the SAWCD to enable bidirectional transforms of multi-scale representations across wavelet and time domains, as well as capture inter-scale intrinsic dependencies through the time domain interaction.
• Extensive experiments conducted on a real-world trajectory dataset demonstrate that the proposed WTFTP+ framework outperforms competitive baselines. The results further validate the effectiveness of the proposed direct multi-horizon prediction paradigm and the time-frequency bridging mechanism in enhancing FTP performance.

Results
Problem definition
In general, the short-term FTP tasks can be formulated as temporal-spatial sequential modeling problems, which aim to predict the flight status over a few minutes based on observations. Let the trajectory sequence {P_i} = N-M, N-M+1, ..., N-1 represent the past M observations, where each trajectory point P_i consists of six attributes:

P_i = [Lon_i, Lat_i, Alt_i, Vx_i, Vy_i, Vz_i]^T (1)

where Lon, Lat, and Alt correspond to the longitude, latitude, and altitude, respectively, while Vx, Vy, and Vz denote the velocities along the longitudinal, latitudinal, and vertical directions.

The single-horizon FTP methods aim to estimate the state of the trajectory point at the next time step, which can be described as:

^P_N = f_s(P_{N-M:N-1}) (2)

P_{N-M:N-1} = [P_{N-M}, P_{N-M+1}, ..., P_{N-1}] (3)

where f_s(⋅) denotes the single-horizon FTP model, and ^P_N represents the predicted trajectory point at the next time step. For H-step horizon prediction tasks, existing single-horizon methods typically employ an iterative approach, where previous predicted values are recursively used as pseudo observations for subsequent predictions, which can be formulated as follows:

^P_N = f_s(P_{N-M:N-1})
^P_{N+1} = f_s([P_{N-M+1:N-1}, ^P_N]) (4)
...
^P_{N+H-1} = f_s([P_{N-M+H-1:N-1}, ^P_{N:N+H-2}])

It is clear that the single-horizon FTP methods suffer from error accumulations and low efficiency for multi-horizon predictions scenarios. Therefore, the WTFTP+ is designed as a direct multi-horizon FTP model to further improve both prediction accuracy and inference efficiency. To be specific, the direct multi-horizon FTP method outputs the trajectory states over the entire H-step prediction horizon in a single inference process, which can be described as:

^P_{N:N+H-1} = f_m(P_{N-M:N-1}) (5)

where the f_m(⋅) represents the direct multi-horizon FTP model.

Dataset and evaluation metrics
To validate the effectiveness of the proposed framework, a real-world dataset19 collected from an industrial ATC system in China is applied to train and evaluate the proposed WTFTP+ framework. The dataset covers the period from February 19 to February 27, 2021, with 20-second intervals, and the regions of interest are [94.616°, 113.689°], [19.305°, 37.275°], [0, 12500] meters for longitude, latitude, and altitude, respectively. The first 7 days of data serve as the training set, while the data from the 8th and 9th days are for the validation and test. It is noted that the mentioned data splits are applied to all experiments to ensure experimental fairness.

During the model evaluation, 4 metrics are applied to measure the FTP performance, including mean absolute error (MAE), mean absolute percentage error (MAPE), root of mean square error (RMSE), and MDE. Specifically, the MAE, MAPE, and RMSE are standard metrics to evaluate the performance for each trajectory attribute separately. The MDE metric is introduced to measure the distance in nautical mile (NM) between predicted and ground truth trajectory points in three-dimensional (3D) earth space. Moreover, considering the real-time requirements of practical ATC systems, the mean time cost (MTC) metric is introduced to evaluate the computational efficiency of models in multi-horizon prediction scenarios. The detailed definitions of these metrics can be found in our previous work23,25.

Comparison methods
In this work, a total of 7 competitive baselines with diverse technical frameworks are selected to validate the effectiveness of the proposed WTFTP+ framework, as listed below:
• Kalman filter: a classical recursive state estimation algorithm that predicts and corrects flight states by integrating historical observations with system dynamics models15.
• CNN-LSTM: a recurrent neural network-based FTP model, in which one-dimensional convolution kernel is employed to extract the trajectory point embeddings, and long short-term memory (LSTM) network is applied to capture the temporal dependencies, implemented by referring to 26.
• FlightBERT: an FTP framework that proposes the binary encoding to represent the trajectory attributes and converts the FTP task into a multi-binary classification problem4.
• WTFTP: a time-frequency analysis based FTP framework proposed in our previous work, which validates the effectiveness of wavelet analysis in the FTP tasks22.
• Attn-LSTM: a sequence-to-sequence (Seq2Seq) LSTM network that utilizes an attention mechanism to predict multiple future timesteps directly27.
• TD-Transformer: a Transformer network that directly predicts the multi-horizon trajectory in an autoregressive manner. In addition, a trend decomposition (TD) strategy28 is applied to enhance the model performance.
• FlightBERT++: an enhanced FTP framework of the FlightBERT, which addresses the limitations of high-bit misclassifications errors23.

Based on the inference style of the multi-horizon prediction, the baselines are further categorized into iterative prediction models (Kalman filter, CNN-LSTM, FlightBERT, WTFTP) and direct prediction models (Attn-LSTM, TD-Transformer, and FlightBERT++). Iterative prediction models perform one-step prediction in an inference procedure, where the predicted results are used as pseudo observations to iteratively obtain multi-horizon predictions. In contrast, the direct prediction models generate multi-horizon prediction results in a single inference pass.

Experimental configuration
The trajectory encoder consists of 4 Transformer encoder layers, while 2 improved Transformer decoder (ITD) layers are applied to construct the SAWCD. The hidden state dimension is set to 128 for both the trajectory encoder and the SAWCD. The multi-head attention mechanism is utilized for the trajectory encoder and the SAWCD to enhance the feature extraction ability of the model, in which the number of attention heads is set to 4. The level of wavelet analysis is 4, using the Haar wavelet basis with a filter length of 2. The symmetric extension is selected to ensure continuity at the boundaries of the trajectory. The L2 regularization weight in the loss function is set to 1×10−4 to ensure consistency with the magnitude of the wavelet component loss.

For the FlightBERT and FlightBERT++ baselines, experimental configurations follow the original works4,23. As to remaining baselines and the proposed WTFTP+ framework, the z-score normalization is applied to longitude and latitude attributes to address their sparse distributions, while other attributes are normalized to the [0, 1] range using the min-max method. In the training process, the batch size is set to 1024, and the number of training epochs is 150. The Adam optimizer with an initial learning rate of 10−3 is used to train all deep learning models.

All experiments are implemented using the open-source deep learning framework PyTorch 1.9.0. The models are trained on a server with an Ubuntu 16.04 operating system, 8 NVIDIA GeForce RTX 2080 GPUs, an Intel(R) Core(TM) i7-7820X@3.6GHz CPU, and 128GB of memory.

Overall results and quantitative analysis
Table 1 presents the overall prediction performance of the WTFTP+ framework and other baseline models, with the predictions for 1-, 3-, 9-, and 15-step horizons (corresponding to 20s, and 1, 3, 5 min) reported in terms of the four mentioned FTP metrics. In general, predictions for 1- and 3-step horizons reflect short-horizon prediction accuracy, while those for 9- and 15-step horizons demonstrate the ability to capture long-range dependencies. From the proposed evaluation metrics, the WTFTP+ framework outperforms the baseline models across almost all prediction horizons. Specifically, the WTFTP+ shows significant improvements in the MDE metric, obtaining approximately 26%, 22%, and 13% reduction relative to the best baseline (FlightBERT++) for 1-, 3-, and 9-step horizons, respectively, and achieving comparable performance for 15-step horizon. The following key conclusions can be drawn from the results:

(1) For the multi-horizon FTP task, direct prediction methods significantly outperform their iterative counterparts. Although the Attn-LSTM model shows inferior performance compared to other direct prediction methods for 9- and 15-step horizons, it still effectively captures long-range dependencies in flight trajectories. Compared to the best iterative prediction method (FlightBERT), Attn-LSTM achieves approximately 14% and 25% MDE reduction for 9- and 15-step horizons, respectively, also outperforming in other metrics. In contrast, the iterative prediction methods suffer from substantial noise by using the prediction results as pseudo observations29,30, leading to considerable error accumulation as the prediction horizon increases. By considering the previous prediction as pseudo observations, due to the fixed input length, more collected trajectory points will be removed from the input sequence, thereby diminishing prior knowledge and further degrading performance. Therefore, the WTFTP+ framework is implemented in a direct multi-horizon prediction paradigm and achieves an expected performance improvement. Most importantly, the WTFTP+ addresses the limitations of the WTFTP framework, which supports our motivation in this work and provides a promising time-frequency analysis solution for the FTP task.

(2) For iterative prediction methods, the time-frequency analysis-based model (WTFTP) achieves superior short-horizon prediction accuracy compared to other models, while the FlightBERT obtains the best performance for 9- and 15-step horizons. Specifically, WTFTP focuses on capturing multi-scale transition patterns from observed trajectories and predicting short-horizon trajectory transition, which validates the effectiveness and efficiency of the time-frequency analysis in the FTP task, as outlined in previous work22. However, the pseudo observations during the iterative prediction impact the ability of WTFTP to predict high-frequency wavelet components, leading to the performance degradation over longer prediction horizons. Benefiting from the capability of the Transformer in capturing long-range dependencies as well as binary encoding, FlightBERT achieves the best performance at longer predictions. However, FlightBERT fails to capture overall trends and fine-grained local details, which limits its performance in short-horizon predictions. In addition, the Kalman filter-based FTP method only obtains inferior performance, especially facing longer prediction horizon scenarios, because it relies solely on pseudo observations without real collected measurement updates during the iterative prediction process.

(3) As to direct prediction methods, WTFTP+ not only achieves a higher performance in short horizons but also outperforms other baseline models at long horizons. TD-Transformer generally outperforms the Attn-LSTM in terms of almost all evaluation metrics, which confirms that the Transformer-based models are more effective in capturing long-range dependencies than the LSTM architecture. Benefiting from the differential prediction mechanism and non-autoregressive inference architecture, FlightBERT++ mitigates high-bit misclassifications and achieves substantial performance improvements over FlightBERT. Although obtaining comparable performance to WTFTP+ for 15-step horizon, FlightBERT++ still exhibits an obvious performance gap for 1-, 3-, 9-, and 15-step horizons, demonstrating the effectiveness of the proposed WTFTP+ framework. WTFTP+ exhibits slightly higher RMSE in the altitude dimension, primarily due to the limited adaptability of unified time-frequency modeling to the distinct variation patterns of altitude compared with longitude and latitude (see Discussion). In addition, FlightBERT++ achieves marginally better MAPE at the 15-step horizon, which can be attributed to its tendency to focus on long-term trend modeling. In contrast, WTFTP+ consistently achieves lower absolute errors and spatial deviations across most horizons and demonstrates a balanced capability in both short- and long-term prediction, which aligns with our motivation of leveraging wavelet analysis to capture both local details and global trends.

(4) Based on the time-frequency analysis capabilities derived from the WTFTP framework, WTFTP+ significantly enhances the ability to capture multi-scale flight patterns in long-range scenario, and finally reaches a state-of-the-art performance for the FTP task in this dataset. As illustrated in Table 1, WTFTP+ achieves relative error reductions of approximately 40% and 34% for 1- and 15-step horizons, respectively. It is believed that the performance improvements can be attributed that: On the one hand, WTFTP+ utilizes the direct multi-horizon prediction paradigm to address the error accumulation of the WTFTP. On the other hand, the proposed time-frequency bridging mechanism promotes the feature extraction and interaction towards multi-scale flight patterns and enables inherent correlations to enhance the trajectory prediction. By combining the mentioned two improvements, the WTFTP+ harvests the higher prediction performance across multiple prediction horizons in the proposed metrics. In summary, the experimental results demonstrate that time-frequency analysis not only facilitates higher prediction performance in single-horizon scenarios but also enables effective multi-horizon FTP tasks. As an enhancement of the WTFTP framework, WTFTP+ addresses error accumulation and orthogonality challenges by incorporating a direct prediction paradigm and a time-frequency bridging mechanism, which enable the WTFTP+ framework to better leverage time-frequency analysis to improve the FTP performance.

Evaluation across different flight phases
In ATC, flight trajectories are typically divided into different phases: takeoff/climb, cruise, and approach/descent. In general, the FTP task for the takeoff/climb and approach/descent phases are more challenging due to external factors such as control instructions and airspace congestion. Furthermore, according to the Global Flight Accident Report31 issued by the Federal Aviation Administration, takeoff/climb and approach/descent phases account for 21% and 46% of total accidents, respectively, whereas the cruise phase accounts for only 9%. Therefore, it is crucial to evaluate the prediction performance across different flight phases to validate the reliability of the proposed WTFTP+ framework for real-world applications.

In this section, the flight trajectories in the test set are further split into trajectory segments according to the above three flight phases, and the best baseline FlightBERT++ in the direct multi-horizon prediction methods serves as the comparison. Table 2 presents the prediction performance of two methods across three flight phases. The results indicate that WTFTP+ demonstrates superior performance by outperforming FlightBERT++ in the MDE metric across almost all prediction horizons. Specifically, both WTFTP+ and FlightBERT++ exhibit a higher prediction accuracy during the cruise phase compared to the maneuvering phases (takeoff/climb and descent/approach). It can be attributed that the complex transition patterns in both horizontal and vertical dimensions during maneuvering phases that result in additional challenges for the FTP task. Fortunately, WTFTP+ achieves notable performance improvements during these maneuvering phases over FlightBERT++, benefiting from time-frequency analysis in modeling multi-scale flight patterns. By leveraging time-frequency analysis, the WTFTP+ framework captures global and local transition patterns in different scales. In addition, the time-frequency bridging mechanism enhances inherent dependencies of inter-scale patterns, and the direct multi-horizon prediction paradigm further contributes to the prediction performance in multi-horizon scenarios. Therefore, WTFTP+ is able to learn trajectory transition during maneuvering phases and providing high-confidence predictions for real-world ATC operations. In summary, through an analysis of performance across different flight phases, it can be concluded that the WTFTP+ framework not only retains the single-horizon prediction advantages of the WTFTP framework but also harvests superior performance in multi-horizon scenarios by integrating a time-frequency bridging mechanism and a direct multi-horizon prediction paradigm.

Visualization and qualitative analysis
To intuitively demonstrate the prediction performance of the WTFTP+ framework and other selective baselines, as shown in Fig. 2, a total of 4 representative flight trajectories are selected from the test set to perform the visualization of the predicted trajectories in both 3D space and the two-dimensional (2D) plane (i.e., latitude-longitude horizon). The selected trajectories concern common flight phases (such as climb, descent, and turns) and complex maneuvering scenarios (such as the coupling of descent and turns). In the 3D visualizations, the altitude is measured in 10 meters. Based on the analysis of these visualizations, the following conclusions can be drawn:

(1) The WTFTP+ framework achieves higher visualized predictions in both common and complex flight scenarios. In climb and descent scenarios, the WTFTP+ provides excellent prediction accuracy for latitude and longitude, as well as effectively capture transition trends in the altitude dimension. For instance, as the common descent and climb trajectory with relatively simple transition patterns in Fig. 2a, b, e, f, the WTFTP+ and most baseline methods can achieve the desired performance (overlapped to the ground truth) due to the simple motion patterns, but prediction deviations from the ground truth can be gradually observed for longer prediction horizons in certain baselines. The FlightBERT++ (best baseline) still suffers from similar issues, i.e., short-horizon deviations directly affect subsequent trajectory predictions, even if the designed differential binary mechanism can mitigate high-bit misclassification errors. Similar to the quantitative analysis, the Kalman filter-based method exhibits larger error accumulation at longer horizons without real-time observations. In contrast, for more complex flight scenarios, such as the coupled descent and right-turn case during the approach phase shown in Fig. 2c, d, it can be seen that the WTFTP+ framework significantly outperforms other baselines in the visualization results. Except for the CNN-LSTM, Kalman filter, and WTFTP affected by error accumulation, other baseline models can only capture long-term and turn-right trends but fail to accurately predict the latitude-longitude transition during the descent phase. Although the deviations occurred in the last 6-step prediction horizons, the WTFTP+ provides the best performance, i.e., superior latitude-longitude prediction capabilities and comparable altitude prediction accuracy, which also validates the motivation of the time-frequency analysis in this work.

(2) As the prediction horizon increases, direct multi-horizon prediction methods demonstrate higher learning capabilities for long-range dependency compared to iterative multi-horizon prediction methods, as that of the quantitative results in Table 2. For instance, the iterative WTFTP and CNN-LSTM frameworks show significant prediction errors from the ground truth at longer horizons, as illustrated in step 4 of Fig. 2b, step 7 of Fig. 2d, and step 5 of Fig. 2f. Even the FlightBERT, which achieves the best multi-horizon performance among iterative methods, still suffers from considerable prediction deviations in long-horizon predictions, such as step 5 of Fig. 2h. It can be also observed that the performance limitations of iterative multi-horizon methods are deteriorated in altitude prediction. In contrast, direct multi-horizon prediction methods obtain higher prediction accuracy, which can be attributed that its unique prediction paradigm to mitigate error accumulation and sufficient modeling towards the entire historical prior information to enhance prediction robustness. It is also noted that the direct multi-horizon prediction methods only obtain marginal performance improvement in short-horizon prediction accuracy, particularly within 3 steps, whereas the WTFTP+ framework achieves both desired multi-horizon prediction performance and reliable short-horizon forecasting confidence, as concluded in the quantitative analysis of Section “Overall results and quantitative analysis”.

Computational performance
To further evaluate the practical applicability of the proposed WTFTP+ framework, Fig. 3 presents the model parameters and computational efficiency of WTFTP+ and selected data-driven baseline models. The vertical axis is log-scaled to improve readability and distinguishability. To ensure the comparison fairness, the batch size and prediction horizons is fixed to 1 and 15 for all models, respectively.

Although the Kalman filter-based FTP method achieves superior MTC performance (0.64 ms), its prediction error rapidly propagates to reduce the performance as the prediction horizon increases (as shown in Table 1), which demonstrates its inapplicability for long-horizon prediction scenarios. Except for FlightBERT and FlightBERT++, the number of model parameters of all models is maintained within the order of millions. Although FlightBERT++ adopts a non-autoregressive prediction mechanism and obtains a faster inference speed, it still requires additional computational overhead for the binary encoding process in practice. Prior works have reported that the average visual reaction time of air traffic controllers can reach up to 170ms32. Therefore, all FTP models except FlightBERT can meet real-time operational requirements. The slower performance of FlightBERT is primarily due to its base transformation of trajectory attributes, which further burdens computational overhead during iterative multi-horizon prediction. Compared with the WTFTP framework, WTFTP+ removes the multi-decoder structure but increases the latent feature dimensionality and incorporates a Transformer-style backbone, thereby resulting in larger trainable parameters than that of the WTFTP framework. Fortunately, the proposed WTFTP+ follows a direct multi-horizon prediction paradigm, achieving 15-step predictions within 24ms even under four-level wavelet decomposition. In summary, considering both the model parameters and MTC measurement, WTFTP+ achieves a favorable balance between model complexity and computational efficiency, while maintaining superior prediction accuracy and robustness across multi-horizon prediction scenarios. The results indicate that the WTFTP+ is a promising solution for the real-world operational ATC systems, further supporting real-time downstream applications, such as conflict detection and decision-making.

Case study
Unlike the previous WTFTP, the WTCs generated by the WTFTP+ are applied to obtain only the predicted trajectory sequence without reconstructing the observed trajectory sequence. Considering the multi-scale feature analysis described in our previous work22, a flight trajectory with climbing and right-turning is selected to analyze flight trends and motion details represented by WTCs and intuitively correlate multi-scale flight patterns. Since the length of the low-frequency component is one in the 4th level of wavelet analysis, only the 1–3 levels of wavelet components are presented in Fig. 4. Figure 4k displays the flight profile of the WTFTP+ and selected baselines. Figure 4a–j show horizon-plane flight patterns derived from different WTCs. Specifically, y-axis label indicates the longitude (Lon) or latitude (Lat) transition pattern at certain wavelet scales. fHi represents the case of motion details at the i-th level of wavelet analysis, while Lo_i, Lo_0 are flight trend and time-domain trajectory, respectively.

As illustrated in Fig. 4k, a predominant northeastward flight trend (i.e., increasing latitude and longitude) is presented within 3 steps of prediction horizon, while steps 4–6 capture the local motion details of a right turn. The subsequent 6 steps maintain an approximate southeastward flight trend. From step 13, the aircraft performs a deceleration accompanied by a right turn, resulting in a heading shift toward the southwest. Meanwhile, the altitude of the aircraft is gradually increasing throughout this maneuver process. Based on the above observations, the following conclusions can be drawn:

(1) The multi-scale wavelet components provide an enriched diversity of flight patterns. In previous work22, it was established that low- and high-frequency WTCs represent global flight trends and local motion details, respectively. Considering the multi-level wavelet analysis, Fig. 4c, g capture the flight patterns corresponding to low-frequency WTCs. Compared to level-1 and level-2 wavelet components, the level-3 exhibits higher frequency resolution and lower time resolution, showing an overall upward trend in longitude and a downward trend in latitude. However, they fail to capture the fine-grained details, such as the longitude initially decelerated, increasing, and the latitude slightly increasing before decreasing. Figure 4d–f and h-j illustrate high-frequency components in the form of oscillations, with time resolution gradually increasing as the analysis level decreases and manifested by a higher discriminability on the occurrence time of extrema. The multi-scale high-frequency WTCs demonstrate more intricate flight patterns, i.e., their oscillation amplitudes present local maneuverability, with the slope of latitude and longitude in the time domain proportional to the oscillation amplitude in the wavelet domain, for example, as shown between steps 1–4 and steps 6–9 in Fig. 4a, f. On the other hand, the phase changes indicate shifts in local transition direction, for example as observed in steps 12–14 in Fig. 4a, f.

From the prediction results of the WTFTP+, the absence of the prior intent information fail to capture the phase changes of high-frequency components near step 13, resulting in a deviation at the end of the predicted trajectory. Nevertheless, the proposed model accurately captures the trend and details in earlier-step horizons, demonstrating expected performance advantages over other selective baseline models.

(2) For the same flight pattern, the orthogonal multi-scale components can provide shared and complementary clues. For instance, the latitude component initially increases and then decreases in steps 1–6 (reach an extremum at step 4). Although the Hi_1 and Hi_2 components of latitude distribute in orthogonal approximation spaces, Fig. 4i, j both clearly reveal the phase changes in step 1–6 horizons, indicating a shift in the local transition direction within this time window. Figure 4i identifies that the maneuver occurs in around steps 1–6, while Fig. 4j further refines it to steps 4–5, benefiting from higher time resolution. A similar pattern can be observed in the longitude component in steps 12–14, where the transition direction changes. In addition, the comparison between Fig. 4e, f as well as Fig. 4i, j, shows a correspondence for the transition of wavelet-domain oscillation amplitudes, i.e., with the rate of longitude decreasing and the rate of latitude increasing. In fact, even flight trends and motion details with distinct dynamic characteristics also have indicative relationships. The end of the observed trajectory demonstrates a general northeastward flight trend, with Fig. 4c, g showing an increase in longitude and a decrease in latitude from a macro perspective, suggesting that the future trajectory will evolve from northeast to southeast. Meanwhile, Fig. 4i, j reveal a phase change in the latitude within steps 1–6, while Fig. 4e and f show no phase change until step 12. It is indicated that the trajectory trend will shift from northward to southward motion, with the eastward transition trend reflecting a deceleration, further supporting the prediction that the future trajectory will generally evolve from northeast to southeast.

By investigating the flight trends and motion details represented by multi-scale WTC, WTFTP+ achieves higher performance and interpretability for multi-horizon trajectory prediction. In addition, it can be seen that multi-scale WTCs within orthogonal spaces also present certain correlations to provide shared and complementary clues. Actually, this is the key motivation for WTFTP+ to introduce the time-frequency bridging mechanism, which facilitates the interactions of multi-scale flight patterns and further enhances wavelet feature representations. Consequently, the proposed WTFTP+ further strengthens both the prediction performance and the interpretability of time-frequency analysis via the bridging bypass.

Ablation study
In this section, extensive ablation experiments are conducted to evaluate the contribution of the proposed technical improvement of the WTFTP+ framework, focusing on the levels of wavelet analysis and the time-frequency bridging mechanism.

Levels of the wavelet analysis. Based on the 15-step prediction horizon in this work, the maximum decomposition level under the Haar wavelet basis is set to 4. Table 3 presents the prediction performance of WTFTP+ in terms of the levels of wavelet analysis from 1 to 4. The following conclusions can be drawn:

(1) The prediction performance of WTFTP+ is progressively improved with the increasing of the levels of wavelet analysis. In short-horizon predictions, only marginal performance improvements are obtained from additional wavelet levels. As in the comprehensive MDE metric, by employing two or higher level of the wavelet analysis, both 1- and 3-step horizon prediction errors are approximately 0.12 and 0.23 NM, respectively, which indicates that a single-level wavelet analysis is already able to obtain desired short-horizon prediction performance. In addition, it can be seen that a higher level of wavelet analysis can provide increasing performance improvement for longer prediction horizons. By leveraging time-frequency analysis to capture both global flight trends and local motion details, WTFTP+ effectively learns multi-scale transition patterns in long-range trajectory sequences. Furthermore, by the incorporation of direct multi-horizon prediction paradigm and time-frequency bridging, WTFTP+ has the ability to address the limitations of WTFTP, i.e., the impaired modeling of wavelet component dependencies caused by insufficient time resolution in high-level wavelet analysis. The time-frequency bridging mechanism is able to enhance the multi-scale representations and further build interactions across different wavelet scales. Finally, the proposed model significantly enhances the robustness of multi-scale features, even facing wavelet components with critically low time resolution at certain scales.

(2) As can be seen from the results, the WTFTP+ demonstrates improved prediction accuracy for the latitude-longitude plane with increasing levels of wavelet analysis, the performance and robustness exhibit a gradual decline in the altitude prediction. The results can be attributed that the continuous transition patterns of the latitude-longitude plane inherently incorporates abundant flight trends and motion details, which allows the WTFTP+ to fully leverage the time-frequency analysis capabilities to support the FTP task. However, as in RMSE metrics, the stability of the altitude prediction progressively deteriorates with additional wavelet levels. It can be attributed that during the cruise phases (i.e., dominated flight phase in the operation), the altitude component provides limited motion details to facilitate effective multi-scale feature learning of the proposed WTFTP+. Fortunately, the improvements in terms of the MAE and MAPE metrics are still able to validate the altitude prediction in higher levels of wavelet analysis, demonstrating the fundamental capability of WTFTP+ to capture critical transition patterns in the vertical dimension.

Time-frequency bridging mechanism. Table 4 reports the performance of the WTFTP+ framework with or without the time-frequency bridging mechanism. In addition, the prediction results with different levels of wavelet analysis from 1 to 4 are also reported to investigate the performance improvements by incorporating the time-frequency bridging mechanism. The following conclusions can be observed from the ablation experiments:

(1) In general, the time-frequency bridging bypass can enhance the FTP performance of the WTFTP+. By considering the error distributions across different levels between Tables 3 and 4, it is demonstrated that the proposed time-frequency bridging mechanism improves the learning capability to capture multi-scale flight patterns. By extracting inherent time-frequency correlations among multi-scale features, the WTFTP+ framework empowered by 4 levels of wavelet analysis reaches the state-of-the-art performance in this dataset, i.e., under 1.1 NM of MDE benchmark at the 15-step horizon prediction. It can also be seen that the proposed framework with time-frequency bridging mechanism also degrades robustness for the 1 level of wavelet analysis in latitude and longitude prediction (as shown in Tables 3 and 4). Furthermore, the prediction performance for the 1-step horizon in Table 4 only show marginal improvements with increasing levels of wavelet analysis, as the same in Table 3. The results indicate that a higher level of wavelet analysis can significantly benefit long-range predictions, while the resulting improvement on short-horizon prediction remains limited, as short-horizon predictions primarily rely on recent correlated trajectory patterns more than multi-scale feature interactions.

(2) The WTFTP+ framework demonstrates robust prediction performance even without the time-frequency bridging bypass. As to the 1-step horizon prediction, the WTFTP+ inherits the superior performance from the WTFTP, thereby validating the effectiveness of time-frequency analysis for the FTP task. Referred to baseline performance in Table 1, the WTFTP+ has higher performance over other baseline models at 1-, 3-, and 9-step horizons even without leveraging the time-frequency bridging enhancement. The results also demonstrate that the proposed framework achieves both higher accuracy in single-step prediction by leveraging time-frequency analysis and competitive multi-horizon performance by the scale-aware decoder with direct multi-horizon prediction paradigm. Correspondingly, the experimental results also validate that the time-frequency bridging mechanism with the addition of bridging bypass obtains expected performance improvements, especially the lower MDE metric in all prediction horizons compared to that of without bridging bypass.

Integrating ATC instruction into WTFTP+ framework
In real-world ATC operation, the ATC instructions issued by ATC controllers are a critical factor influencing flight trajectory evolution. Existing FTP methods with only historical observations typically suffer from a significant performance degradation during instruction-driven maneuvering phases. In our previous work25, a spoken instruction-aware flight trajectory prediction (SIA-FTP) framework was proposed to incorporate textual ATC instructions into the FTP process through a 3-stage progressive multi-modal learning strategy. The results confirmed that integrating spoken instructions is a key factor in ensuring robust prediction performance during instruction-driven phases. In this section, to validate the scalability and generalizability, the SIA-FTP with the proposed WTFTP+ is also performed by replacing the FlightBERT++ backbone to implement a multi-modal FTP model with instruction-aware ability. For the multi-modal fusion process, the instruction embedding is concatenated with the observed trajectory embeddings along the temporal dimension as shown in Fig. 5a, which serves as an informative prior knowledge for the FTP task. In this context, all other experimental configurations are the same with that of in the original SIA-FTP framework25. During the test phase, these models are evaluated on the test set with only instruction-driven maneuvering trajectories.

The experimental results are illustrated in Fig. 5. Compared to the original WTFTP+ framework, the WTFTP+ model enhanced with ATC instructions demonstrates approximately a 20% improvement in performance at the 9- and 15-step prediction horizons. These results indicate that the WTFTP+ framework exhibits strong generalizability and can serve as a robust foundational model for trajectory prediction, with the flexibility to incorporate additional modalities (such as ATC instructions) to further enhance predictive performance. Compared to the SIA-FTP framework, WTFTP+, integrating ATC instructions, harvests superior performance at short-horizons (step 1-3), which can be attributed that the wavelet-based decomposition enables WTFTP+ to effectively capture fine-grained motion details. Meanwhile, in the longer horizons (step 9-15), the WTFTP+ also achieves comparable performance with the SIA-FTP framework, with only 0.04 NM and 0.14 NM MDE gap, respectively. The performance results from that the relatively small parameter size, i.e., 1.2 M in WTFTP+ vs. 44.13 M in SIA-FTP, may impact the learning capacity to fully exploit the multi-modal information during the fusion process. The detailed experimental results and more exploration of the integration strategies can be found in Supplementary Information (Section “Integrating ATC instruction into WTFTP+ framework”).

Discussion
In this work, to address the performance gap of the previous WTFTP framework, a WTFTP+ framework is proposed to explore the effectiveness of time-frequency analysis techniques for direct multi-horizon FTP tasks. To this end, a Transformer-style encoder-decoder neural network is designed to generate the multi-scale wavelet components of the future trajectory sequence, which advances the time-frequency analysis ability of the WTFTP framework. To capture the underlying correlations among the multi-scale wavelet components, a time-frequency bridging mechanism is designed to enable bidirectional transformations of multi-scale representations across the wavelet and time domains by improving the Transformer decoder layer. Extensive experimental results demonstrate that the proposed WTFTP+ framework is able to generate accurate predictions over multiple prediction horizons with lower error accumulation. Furthermore, the proposed framework exhibits strong interpretability by visualizing the horizon-plane flight patterns derived from multi-scale wavelet components. Most importantly, the extensibility is also validated through the integration of ATC instructions, confirming its adaptability to more complex, instruction-driven air traffic scenarios. It is believed that the WTFTP+ framework not only provides a time-frequency perspective on multi-horizon FTP modeling but also offers a practical and extensible solution for modern ATC systems.

Although the proposed WTFTP+ framework achieves the expected performance improvements for multi-horizon FTP tasks, its prediction performance on the altitude attribute can be further enhanced compared to selected baselines (as shown in Table 1). As described in Section “Ablation Study”, the transition patterns derived from wavelet components in the altitude attribute differ significantly from those in the longitude and latitude attributes. It can be attributed to the fact that altitude typically exhibits noticeable changes only during climbing or descending maneuvers, whereas longitude and latitude undergo continuously changes throughout flight operations. Consequently, the unified time-frequency modeling across longitude, latitude, and altitude attributes may limit the predictive accuracy of the altitude attribute. In future work, the following potential directions can be further explored to address this limitation:

(1) Designing attribute-independent modeling strategies that decouple vertical dynamics from horizontal motion, thereby capturing the sparse yet abrupt transition patterns of altitude more effectively. Meanwhile, it is essential to design an appropriate interaction mechanism to capture the inherent dependencies between vertical and horizontal dynamics. A channel-attention mechanism may then be employed to capture inter-channel correlations, which alleviates the limitations of unified modeling across attributes with distinct dynamic characteristics while preserving their potential dependencies.
(2) Incorporating multi-modal information such as vertical rate, aircraft performance parameters, and weather conditions (e.g., wind and temperature) to enrich the altitude-related input features. These additional modalities provide complementary cues about vertical dynamics that are not fully captured by position and velocity alone. By integrating such heterogeneous data, the model can achieve a more comprehensive understanding of altitude variations, thereby enhancing prediction accuracy.

Methods
The proposed WTFTP+ framework
In the flight process, the overall motion trends of the aircraft are typically determined by historical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a certain route, the longitude and latitude always evolve along the heading between two consecutive waypoints, indicating that the planned route (in the flight plans) can imply the overall flight trend. At the same time, local variations are introduced by weather conditions, aircraft performance parameters, and pilot-specific operations. In other words, a flight trajectory can be regarded as a combination of global trends and local details. While different flights in similar scenarios may follow similar overall trends, their local motion details often vary significantly, which limits the prediction accuracy of the conventional FTP approaches. Fortunately, the wavelet transform provides an intuitive solution to this challenge by decomposing time series into multi-scale components, thereby decoupling global trends from local details.

Thanks to the effectiveness of the time-frequency analysis, the proposed WTFTP+ framework inherits the core concept of the WTFTP framework, i.e., generating optimal wavelet coefficients to achieve the FTP. Based on the mentioned limitations in WTFTP, the resulting improvements lie in that the WTFTP+ framework directly generates wavelet coefficients of predicted multiple horizons, while WTFTP only performs single-horizon predictions and achieves the multi-horizon prediction in an iterative manner. To this end, an improved Transformer-style encoder-decoder neural architecture is designed to build the FTP model, primarily including four key modules: a trajectory encoder, a wavelet component prompt, a scale-aware wavelet components decoder (SAWCD), and multiple prediction heads. As illustrated in Fig. 6, the overall prediction paradigm is described as follows.

Let P_{N-M:N-1} ∈ R^{6×M} be a trajectory sequence observed over the last M timestamps. The trajectory encoder is designed to project the trajectory sequence into high-dimensional trajectory embeddings TE ∈ R^{d×M}:

TE = TrajectoryEncoder(P_{N-M:N-1}) (6)

where d represents the dimension of trajectory embeddings.

The objective of the SAWCD is to generate multi-scale wavelet components that imply the time-frequency features of predicted horizons. Specifically, the initial wavelet embedding (IWE) is firstly constructed by wavelet component prompt to serve as the input to the SAWCD, including initial low-frequency embedding (IL) and initial high-frequency embedding (IH). Since tail vector of trajectory (defined in this study as the trajectory attributes at the last historical timestep minus that at the previous timestep) is approximately aligned with the trajectory transition trend, it is utilized to construct the prompt of the low-frequency flight pattern, i.e., IL. Considering that it is hard to infer the local motions of predicted horizons from the past horizons, a couple of learnable parameters are trained from historical data to represent high-frequency wavelet embeddings, i.e., IH.

As shown in Fig. 6b, the IWE and TE are fed to the SAWCD to obtain optimal wavelet components (OWC), in which the CDA is introduced to fuse historical temporal features into multi-scale flight patterns, and time-frequency bridging mechanism is designed to promote inter-scale inherent correlating. It can be described as follows:

OWC = SAWCD(IWE, TE) (7)

The SAWCD is implemented by two parallel branches.
• Main branch: follows a Transformer-style architecture, where cross-domain interactions are performed under the assumption that time-domain trajectories lie in the approximate space at a certain scale and multi-scale features are distributed within its subspaces.
• Bypass branch: serves as a time-frequency bridging bypass: wavelet-domain features are first transformed into the time domain via the W2T block, then a self-attention mechanism is designed to capture intrinsic relationships within the time-domain sequences and build the correlations among multi-scale features. The processed features are transformed back into wavelet components through the T2W block to support subsequent time-frequency interactions.

In this work, the dual-branch of the SAWCD focuses on effectively capturing flight transition patterns from both the time and wavelet domains while preserving the orthogonality of wavelet components to facilitate accurate trajectory prediction.

To obtain the multi-horizon trajectory, a set of prediction heads are designed to project the corresponding scale of time-frequency representations into wavelet components. During the prediction procedure, a differential prediction strategy is utilized to avoid unnecessary prediction of the direct-current component, and the inverse discrete wavelet transformation of OWC is applied to obtain the differential sequence of predicted horizons. Finally, an inversely difference operation is employed to obtain trajectory attributes of the next K horizons. The prediction process can be described as:

^P_{N:N+K-1} = IDIFF(IDWT(OWC)) (8)

where IDWT(⋅) is the inverse discrete wavelet transform operation, and IDIFF(⋅) represents the inversely difference operation.

Note that the goal of the WTFTP+ framework is to predict the wavelet coefficients of the future trajectory sequence by considering the time-frequency features of the observations. During the training process, the mean squared error (MSE) loss is used to quantify the difference between the predicted and actual wavelet coefficients, thereby driving the optimization of the neural network parameters. The implementation details of each module are described in the following sections.

Trajectory encoder
The trajectory encoder is constructed by trajectory point embedding block and Transformer encoder blocks, which aims to extract the spatial-temporal dependencies of the observations. Specifically, to better learn the latent features from the observations, it is essential to map the trajectory points to high-dimensional embedding space to extract discriminative spatial features. Similar to the WTFTP framework, a simple yet effective trajectory point embedding block is employed to project the trajectory point into embedding space, which can be described as:

I = σ(P_{N-M:N-1} * W_{i1} + b_{i1}) * W_{i2} + b_{i2} (9)

where P_{N-M:N-1} ∈ R^{M×6} represents the observed trajectory sequence. W_{i1} ∈ R^{6×D//2}, and W_{i2} ∈ R^{D//2×D} are learnable transformation matrices. b_{i1} ∈ R^{D//2} and b_{i2} ∈ R^D are bias vectors. σ(⋅) represents the ReLU activation function. M is the number of observed trajectory points, and D is the feature space dimension. // denotes the floor division operator.

The WTFTP+ framework employs the Transformer encoder33 to extract the temporal dependencies among observed trajectory points. Since the self-attention mechanism is permutation-invariant, a positional encoding (PE) strategy is designed to establish the relative positional relationships of trajectory sequence. Therefore, the observed trajectory embeddings TE can be obtained as follows:

TE = TransformerEncoder(PE(I)) (10)

where TransformerEncoder(⋅) is multi-layer Transformer encoder. PE(⋅) applies sine and cosine functions to generate positional embeddings with dimension of D and add them into I.

Scale-aware wavelet component decoder
The SAWCD is stacked by several ITD layers, which incorporates time-frequency analysis capabilities by the proposed CDA and time-frequency bridging mechanism. Specifically, the initial wavelet embedding IWE is constructed to serve as the input of the SAWCD, which is further sequentially fed to the ITD layers into generate multi-scale wavelet component representations. As shown in Fig. 6b, the core architecture of the ITD layer follows the standard Transformer decoder layer with the following improvements:
• The scale-aware mechanism is incorporated into the self-attention and feed-forward network to preserve the orthogonality of inter-scale features, allowing the model to learn single-scale flight transition pattern during the training procedure.
• Since the multi-scale features are distributed in mutually orthogonal subspaces of the observed trajectory approximation space, the CDA is designed to incorporate embeddings of the observed trajectory sequence into the multi-scale features.
• Most importantly, a time-frequency bridging mechanism is proposed to extract inherent time-frequency correlations among multi-scale features while maintaining their orthogonality, which serves as the bridging bypass in the decoder.

Initial wavelet embedding. The initial wavelet embedding consists of two stacked parts: the initial low-frequency (IL) embeddings and the initial high-frequency (IH) embeddings. These embeddings are arranged in a descending order of time resolution, and the embeddings of the same resolution are sequentially grouped as illustrated in Supplementary Fig. 1. In general, low-frequency wavelet components represent the global trend of the trajectory sequence, while the high-frequency wavelet components imply the local motion details. In this work, the low-frequency flight pattern is transformed into a high-dimensional vector and replicated to generate the IL ∈ R^{t_L×D}, as shown below:

IL = REP(σ((P_{N-1} - P_{N-2})^T * W_{d1} + b_{d1}) * W_{d2} + b_{d2}, t_L) (11)

where W_{d1} ∈ R^{D×D//2} and W_{d2} ∈ R^{D//2×D} are learnable weight matrices. b_{d1} ∈ R^{D//2} and b_{d2} ∈ R^D are learnable bias vectors. t_L is the sequence length of low-frequency wavelet component in level-L wavelet analysis. The operation REP(a, b) replicate the vector a along the column dimension for b times.

For initial high-frequency components, considering the complicated predictability of future local motions under incomplete maneuvering intentions, a couple of learnable parameters are employed to model the distribution of fast dynamics from historical data. In addition, considering the orthogonality of inter-scale wavelet components, a learnable scale encoding (SE) strategy is proposed to enhance the scale discriminability of the initial embeddings. The PE strategy is also applied to enhance sequential discriminability along the temporal dimension. Mathematically, the initial wavelet embedding can be obtained by:

IWE = PE(Σ_{i=1}^L {IH_{i} [t_i : t_i, :]} + SE[j, :], IL + SE[j+1, :]) (12)

where IWE ∈ R^{t_A×D} serves as the input of the SAWCD, which corresponds to the wavelet components of the future flight trajectory. IH ∈ R^{(t_A-t_L)×D} is the learnable IH embedding. SE ∈ R^{(L+1)×D} is the learnable scale embedding. L is the level of wavelet analysis. {t_i}_{i∈[1,L]∩Z} stands for the sequence length of level-i wavelet component, while t_A = t_L + Σ_{i=1}^L t_i stands for that of stacked wavelet components in L levels of wavelet analysis. {t_i}_{i∈[1,L]∩Z} can be calculated below:

t_i = {H, i=0; ⌊t_{i-1}/2⌋+l, i=1, ..., L} (13)

where H is the predicted steps.

Scale-aware self-attention. To investigate sequential dependency of intra-scale flight patterns, in each ITD layer, the self-attention mechanism is applied to obtain enhanced wavelet embedding (EWE) derived from IWE. Therefore, a scale-aware self-attention (SA2) mechanism is designed to preserve temporal causality and prevent inter-scale feature leakage by employing a well-crafted mask matrix. The residual connection, followed by layer normalization (LN), is utilized to retain the learned semantics during the information propagation process. The inference process can be described as follows:

EWE = LN(SA2(IWE) + IWE) (14)

where EWE ∈ R^{t_A×D} is the enhanced wavelet embeddings. The attention weight W ∈ R^{t_A×t_A} is calculated by:

W = Softmax(Q * K^T / √{D/m} + Mask_{sa2}) (15)

Mask_{sa2}[i, j] = {0, i∈[t_A-t_L, t_A]∩Z, j∈[t_A-t_L, i]∩Z; -1, otherwise} (16)

where Q, K ∈ R^{t_A×(D/m)} are the query and key features of IWE. m is the number of heads. Mask_{sa2} ∈ R^{t_A×t_A} is the mask matrix.

Cross-domain attention. To learn the underlying flight transition patterns of the observation sequence, a CDA mechanism is utilized to incorporate the prior transition patterns from observed trajectory representation into enhanced wavelet embeddings. Since multi-scale features are generally distributed in the subspace of the time-domain approximation space, the CDA is expected to perform feature interaction between the trajectory encoder and the SAWCD, which learn wavelet-domain flight pattern from observed trajectory. Specifically, attention weights are obtained using: EWE serves as the queries, and the observed trajectory representation TE serves as both keys and values, as shown below:

FWE = LN(CDA(EWE, TE) + EWE) (17)

where FWE ∈ R^{t_A×D} is the output, i.e., fused wavelet embedding. Similarly, the residual connection, followed by LN, is utilized to retain semantics during the information propagation process.

Time-frequency bridging. Although the CDA effectively extracts informative flight transition patterns from the observed trajectory, due to the orthogonality of these wavelet components, it is still hard to capture inherent dependency between global trend and local motion patterns. In this work, a time-frequency bridging mechanism is proposed to extract the intrinsic correlations across multi-scale wavelet components to preserve the wavelet orthogonality, which is expected to fully leverage the time-frequency analysis capability of wavelet transform in the FTP task. Note that time-frequency bridging mechanism is performed in parallel with the CDA procedure as a bypass operation on EWE, which achieves an effective and efficient feature learning. Specifically, the bridging bypass first transforms multi-scale components of enhanced wavelet embeddings into a unified time domain by a trainable W2T block. Then time-domain self-attention is designed to establish the interaction bridge for inter-scale features and capture significant dependency among multi-scale flight patterns. To facilitate subsequent procedure, a trainable T2W block is proposed to regain time-frequency representations to support the whole time-frequency analysis for the FTP task.

Let low- and high-frequency features in wavelet analysis at the i-th level be Lo_i, Hi_i ∈ R^{t_i×D}, respectively. Taking L=3 levels of wavelet analysis as an example, the time-frequency bridging process is illustrated in Supplementary Fig. 5 by cascading W2T block, time-domain self-attention, and T2W block. The W2T procedure can be described as follows:

Lo_{i-1} = σ(ALT(Lo_i, Hi_i)^T * W_{lw2t} + b_{lw2t})^T, i=1,2,...,L (18)

Lo_L = EWE[t_A-t_L : t_A, :] (19)

Hi_i = EWE[t_A-t_L-t_i : t_A-t_L, :], i=1,2,...,L (20)

where W_{lw2t} ∈ R^{2t_i×t_{i-1}} is learnable inverse wavelet transform matrix at the i-th level of the W2T procedure, and b_{lw2t} ∈ R^{t_{i-1}} serves as learnable bias. ALT(Lo, Hi) alternate the rows of Lo and Hi along the time dimension. Notably, the learnable inverse wavelet transform matrix consists of: i) a sparse matrix constructed from wavelet filter parameters (as described in Supplementary Information, Section “The wavelet-to-time domain procedure”), and ii) learnable parameters to pad the sparse elements in i). Thus, the W2T procedure is able to achieve the inverse wavelet transform-like operations, in which the design of the learnable parameters facilitates adaptive adjustment of the bridging bypass and promotes the interaction of inter-scale features.

Through iterative process of Eq. (18), low- and high-frequency features at the same level are jointly reconstructed into the lower-frequency. As the level i decreases in the W2T procedure, the time resolution of Lo_i goes higher, which are ultimately transformed into Lo_0 in time domain. Then, the self-attention mechanism is employed to perform feature interactions by extracting inherent correlations within Lo_0, as shown below:

^Lo_0 = LN(TSA(Lo_0) + Lo_0) (21)

where ^Lo_0 represents the output time-domain features. TSA(⋅) represents time-domain self-attention. In addition, the residual connection followed by LN is utilized to retain semantics during information propagation.

As an inverse operation, the T2W procedure aims to regain time-frequency representations to facilitate subsequent inference, which in practice transforms ^Lo_0 into multi-scale wavelet features, as shown below:

^Lo_i = LN(σ(^Lo_{i-1}^T * W_{lt2wl} + b_{lt2wl})^T + Lo_i), i=1,2,...,L (22)

^Hi_i = LN(σ(^Lo_{i-1}^T * W_{lt2wh} + b_{lt2wh})^T + Hi_i), i=1,2,...,L (23)

BWE = {^Hi_i, ^Lo_L}_{i∈[1,L]∩Z} (24)

where ^Lo_i, ^Hi_i ∈ R^{t_i×D} represent low- and high-frequency features in T2W procedure at the i-th level. W_{lt2wl}, W_{lt2wh} ∈ R^{t_{i-1}×t_i} are learnable low- and high-frequency wavelet transform matrices at the i-th level of the T2W procedure, and b_{lt2wl}, b_{lt2wh} ∈ R^{t_i} serve as learnable low- and high-frequency bias vectors. The learnable low- and high-frequency wavelet transform matrices are constructed in the similar way with the learnable inverse wavelet transform matrices (as described in Supplementary Information, Section “The time-to-wavelet domain procedure”), and also consist of two parts, the wavelet filter parameters and the learnable parameters. After the time-frequency bridging, ^Lo_L serves as the low-frequency component, while {^Hi_i}_{i∈[1,L]∩Z} serve as the multi-scale high-frequency components. Finally, the bridged wavelet embeddings BWE ∈ R^{t_A×D} is obtained by concatenating the ^Lo_L and {^Hi_i}_{i∈[1,L]∩Z}. The T2W procedure is the inverse process of the W2T procedure. As the level i increases, the time-domain feature ^Lo_0 is decomposed back into wavelet domain with frequency resolution gradually going higher.

Scale-aware feed-forward network. In each ITD layer, the FWE and BWE are aggregated and further extracted in the scale-aware feed-forward networks (SAFFNs) by a shared architectures with unique parameters, preserving the orthogonality of inter-scale features and yielding complete wavelet embeddings CWE ∈ R^{t_A×D}. The SAFFN can be expressed as:

CWE = [^Q_L, ^Q_{L-1}, ..., ^Q_0] (25)

^Q_L = LN(SAFFN(Σ_{i=1}^L MWE[t_i, :]) + Σ_{i=1}^L MWE[t_i, :]) (26)

^Q_{L-j} = LN(SAFFN(Σ_{i=1}^L MWE[t_i : t_{L-j+1} + t_i, :]) + Σ_{i=1}^L MWE[t_i : t_{L-j+1} + t_i, :]), j=1,2,...,L (27)

MWE = FWE + BWE (28)

where MWE ∈ R^{t_A×D} is the sum of FWE and BWE. ^Q_L ∈ R^{t_L×D} is the low-frequency components of the CWE, while {^Q_i}_{i∈[1,L]∩Z} ∈ R^{t_{L-i+1}×D} are the multi-scale high-frequency components. {SAFFN(⋅)}_{i∈[1,L]∩Z} represents the scale-aware feedforward network for Hi, while SAFFN(⋅) for Lo_L. Similarly, the residual connection followed by LN is also utilized to retain semantics during information propagation.

Prediction and post-processing
Due to the multi-scale and orthogonality properties of wavelet components, multiple prediction heads are required to map complete wavelet embeddings of the last ITD layer to obtain wavelet components of certain predicted horizons. Each prediction head is implemented by multi-layer perceptrons (MLP) (i.e., two linear layers with an ReLU activation function). The prediction process can be described as follows:

^Q_i = PredictionHead_i(Q), i=0,1,2,...,L (29)

where {PredictionHead_i(⋅)}_{i∈[1,L]∩Z} represents the i-th prediction head for generating ^Q_i, i.e., the desired WTC according to definition of wavelet component in our previous work22.

Subsequently, to achieve the FTP task with trajectory attributes of predicted horizons, the IDWT operation is applied to transform predicted wavelet components back to the time domain, further reconstructing the predicted future trajectory sequence, as in our previous work22. Mathematically, the prediction for the j-th attribute ^P'_{N:N+H-1}[j-1, :] is shown as:

^P'_{N:N+H-1}[j-1, :] = IDIFF(TRIM(IDWT({^Q_i[j-1, :]}_{i∈[0,L]∩Z}))), j=1,2,...,6 (30)

where TRIM(⋅) is applied to crop out the redundant segments at the end of the trajectory sequence due to the signal extension, i.e., the first H elements along the time dimension retained. IDIFF(⋅) represents the inversely difference operation to avoid unnecessary prediction for the direct-current component.

Loss function
To facilitate the learning of multi-scale flight patterns, the MSE loss function performed on wavelet components is proposed to update the learnable parameters of the WTFTP+ framework, following the similar strategy to the WTFTP framework. Furthermore, an L2 regularization penalty is applied to optimize the learnable parameters within the W2T and T2W transform matrices. This regularization constrains the deviation from the standard wavelet filters, thereby preserving the effectiveness of the time-frequency bridging process. The loss function is formulated as follows:

L = (epoch/total_epoch) * (L_{w2t} + L_{t2w}) + L_{wavelet} (31)

where L_{w2t}, L_{t2w} are one part of the loss function and the L2 regularization for the W2T and T2W transform matrices, respectively. epoch and total_epoch are the current and total epochs during the training procedure, whereby a proper ratio allows the learnable parameters to have a broader optimization exploration space in the early stages of training, while gradually converging toward the wavelet filter as training progresses. The details of L_{w2t}, L_{t2w} are described as follows:

L_{w2t} = β * Σ_{i=1}^l Σ_{j=1}^L (||W_{w2t, i,j}||_F^2 + ||b_{w2t, i,j}||_2^2) (32)

L_{t2w} = β * Σ_{i=1}^l Σ_{j=1}^L (||W_{t2w, i,j}||_F^2 + ||W_{t2w, i,j}||_F^2 + ||b_{t2w, i,j}||_2^2 + ||b_{t2w, i,j}||_2^2) (33)

where β is the weight for both of the L2 regularizations. L is the level of wavelet analysis. l is the number of decoder layers. W_{w2t, i,j}, b_{w2t, i,j} are the W2T transform matrix and bias vector at the j-th level of wavelet analysis in the i-th layer of the ITD. W_{t2w, i,j}, b_{t2w, i,j} are the T2W transform matrix and bias vector at the j-th level of wavelet analysis in the i-th layer of the ITD for the low-frequency, while W_{t2w, i,j}, b_{t2w, i,j} for the high-frequency. ||⋅|| is the Euclidean norm. Since the spectral norm is difficult to calculate for transform matrices in time-frequency bridging, the Frobenius norm, ||⋅||_F, is utilized to optimize the upper bound of the spectral norm. The MSE loss of the wavelet component, i.e., L_{wavelet}, is calculated by:

L_{wavelet} = 1/(L-k) * d * Σ_{i=1}^{L-k} Σ_{j=1}^d (c_{i,j}^k - ^c_{i,j}^k)^2 (34)

where h_{L-k} is the length of WTC_k 22, satisfying h_{L-k} = t_{L-k+1} except h_L = t_L. d=6 is the number of trajectory attributes. ^c_{i,j}^k is the value of the i-th predicted wavelet coefficient in the j-th attribute of WTC after applying the DWT operation to the future trajectory differential sequence, while c_{i,j}^k is the ground truth.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper introduces an enhanced WTFTP framework that demonstrates robust performance in flight trajectory prediction.\nThe manuscript presents a well-structured study with sound methodology. However, addressing the following points would\nfurther strengthen its rigor, clarity, and impact prior to publication:\n1.The current \"Introduction\" section provides an overview of existing methods but lacks a critical analysis of their limitations.\nThe authors should explicitly identify the unresolved issues in current approaches. They could better emphasize how their\nproposed method addresses these problems, thereby highlighting its innovativeness.\n2.The problem definition and introduction suggest that filter-based approaches (e.g., particle filters, Kalman filters)\ndemonstrate applicability for flight trajectory prediction (FTP) tasks. To strengthen the comparative analysis, the authors\nshould incorporate experimental comparisons with these baseline methods. Furthermore, a quantitative evaluation of the\nhyperparameter selection process for the proposed enhanced WTFTP framework would significantly enhance the\nmethodological rigor of the study.\n3.From Table 1, it can be seen that WTFTP+ does not have an advantage in the RMSE of ALT prediction. Additionally,\nFlightBERT++ has an advantage in the MAPE (%) for long-term prediction. Please provide additional analysis and\nexplanation.", "reviewer_2": "This paper presents an enhanced version of the previously proposed WTFTP framework, named WTFTP+, aiming to\nimprove multi-horizon flight trajectory prediction by mitigating cumulative errors of stepwise inference and enhancing\nwavelet component interactions. The proposed architecture introduces a direct encoder-decoder-based prediction paradigm\nand a novel time-frequency bridging mechanism. The work is well-structured and tackles an urgent problem in real-world air\ntraffic control systems. Furthermore, the release of the source code will benefit both academic and industrial communities.\nHowever, there remain several technical and presentation concerns that should be addressed before publication.\nMajor Comments:\n1.The paper frequently mentions that wavelet components lie in “orthogonal spaces.” However, the exact meaning of this\nclaim and the origin of orthogonality are not fully explained. Is this based on the properties of the wavelet basis used, or is it\nempirically observed in the learned representations? Please clarify.\n2.While WTFTP+ aims to improve prediction accuracy, its encoder-decoder architecture and bridging mechanisms increase\nmodel complexity. Since trajectory prediction in air traffic control is a time-sensitive task, it is critical to evaluate whether\nWTFTP+ meets real-time inference requirements. Please provide a quantitative comparison of model parameters and\ninference time between baselines and WTFTP+.\n3.The learnable transformation matrix used in the time-frequency bridging module lacks interpretability. It would be\nbeneficial to provide an intuitive explanation of this matrix. Additionally, the contribution of this component should be further\nvalidated through ablation experiments.\n4.The current implementation uses trajectory tail vector as a proxy for short-term motion trend (i.e., initial low-frequency\nwavelet embedding). Since ATC instructions often represent long-term macroscopic intentions, have the authors considered\nfusing such intention vectors into the initial wavelet embeddings? This might enhance semantic guidance in multi-horizon\nforecasting.\n5.The proposed ITD mechanism converts features from the wavelet domain to the time domain and then back to the wavelet\ndomain before generating predictions. Have the authors explored the possibility of skipping the T2W process and directly\npredicting in the time domain after fusion?\n6.The proposed method performs slightly worse than some baselines in terms of MAE and RMSE for the altitude prediction\ntask. While this is discussed in the Section Discussion, the experimental analysis should also be included in the quantitative\nanalysis section, such as error accumulation or sensitivity to vertical motion features.\nMinor Comments:\n1.Fig. 1: Replace “Temporal domain” with “Time domain” for consistency.\n2.Fig. 5: The extensive use of blue tones reduces color contrast; consider improving visual distinctiveness.\n3.Acronyms: Many abbreviations are defined multiple times (e.g., FTP, ATC). Acronyms should only be defined upon first\nusage.\n4.Notation Consistency: Ensure consistent formatting for terms like \"pseudo-observations\" vs. \"pseudo observations\",\n\"WTC\" vs. \"WTCs\".\n5.Grammar/Typo Improvements:\nSection 2.1: “In general, the short-term FTP tasks can generally be…” -> “...can be…”\nSection 2.2: “...the region of interest is...” -> “...the regions of interest are...”\nSection 2.3: “long short term memory” -> “long short-term memory”\nSection 2.5: Remove “are detailed reported” -> “…reported…”\nPage 6: “which supports our motivation in this work and provide...” -> “... provides...”\nPage 7: “By combing...” -> “By combining...”\nPage 12: Clarify “MRE” -> possibly intended to be “RMSE”?\nPage 9: “expected for” -> “except for”\nPage 10: “the level-3 exhibit” -> “exhibits”\nPage 15: “underdo” -> likely intended “undergo”\n6.Page 7: FAA report statistics are slightly misrepresented. It should read: “Takeoff/climb and approach/descent phases\naccount for 21% and 46% of total accidents, respectively,” rather than presenting them as absolute accident rates.", "reviewer_3": "The paper under review proposes a wavelet-based method for an efficient flight trajectory prediction in the multi-horizon\nsetting. The work appears promising, as it employs frequency-domain analysis in the flight trajectory prediction task, in\ncontrast to data-driven methods. The proposed method can be viewed as an extension of the existing Wavelet Transform-\nbased Flight Trajectory Prediction (WTFTP) framework to circumvent its limitations, in that WTFTP suffers from a\nperformance bottleneck in multi-horizon prediction due to its inherent single-horizon inference paradigm. This novel\nframework (WTFTP+) is implemented with a generalized encoder-decoder architecture, in which the encoder is employed to\nextract the abstract features from the observed trajectory sequence, while the decoder is applied to generate the optimal\nwavelet coefficients for the predicted horizons. In order to maintain orthogonality of the multi-scale wavelet components, in\nthe decoder, the orthogonal wavelet components are first bridged into the time domain for promotive interaction, and\nsubsequently bridged back into the wavelet-transformed domain to obtain enhanced features of inherent flight patterns. The\nvalidity of the WTFTP+ method is demonstrated via several experiments on a real-world flight trajectory dataset, whereas\nsome baseline comparisons are incorporated into the experimental analysis to provide a comprehensive evaluation of the\nmethod.\nIn conclusion, I find the paper very well-written and the results quite interesting from an empirical perspective. Though a\nreasonable discussion on the theoretical underpinnings of the method can be found in the paper, an extensive theoretical\nperformance analysis of the WTFTP+ method is not provided. Moreover, the choice of wavelet function, decomposition\nlevels, and LSTM hyperparameters may significantly impact results. In this regard, a more systematic sensitivity analysis\nwould be beneficial. Since the model performs well on U.S. flight data, it is unclear how well it would generalise to other\nairspaces or data sources without retraining. Overall, the WTFTP+ method is somewhat more explainable in contrast to the\ndata-driven frameworks.", "reviewer_4": "The paper proposes an enhanced flight trajectory prediction framework (WTFTP+) that leverages time-frequency wavelet\ntransforms and a direct multi-horizon prediction paradigm. This model builds upon the authors' earlier work by introducing a\ntime-frequency bridging mechanism designed to enable inter-scale feature interaction without violating wavelet component\northogonality.\nOverall, the approach is well-explained and supported by comprehensive experimentation. However, I have the following\nmajor and minor comments:\nMajor Comment\nWhile the incorporation of wavelet transforms into trajectory prediction is interesting, the core contribution appears\nincremental. The WTFTP+ framework primarily refines existing idea, such as direct multi-horizon prediction and inter-scale\nattention, rather than offering fundamentally new algorithmic or theoretical insights. As such, the contributions fall short of the\nnovelty and depth typically expected for a journal publication.\nMinor Comments\nThe introduction references technical concepts like \"wavelet components\" and \"orthogonality of approximation subspace\"\nwithout sufficient context. Providing a brief explanation or definition upon first mention would improve readability for a\nbroader audience.\nWhile the evaluation is thorough, the abstract would benefit from including specific performance outcomes using key metrics\n(e.g., prediction accuracy) to better highlight the method’s advantages from the outset.\nThe caption for Figure 3 is unclear.\nThe conclusion notes that the model underperforms in altitude prediction. It would strengthen the paper if the authors\nproposed more concrete next steps to address this limitation beyond general remarks."}, "author_response": "Revision Summary\nDear Editors and Reviewers,\nThanks for your efforts in reviewing our work. In this revision, we have carefully reviewed the\ncomments and have revised the manuscript accordingly. A brief description is provided here to\ncapture the core revisions in this version.\n To provide a more comprehensive comparison with classical FTP methods, a Kalman\nfilter-based FTP baseline has been added to the manuscript with corresponding\nexperimental analysis and visualizations.\n To better evaluate the effectiveness and robustness of the proposed framework, detailed\nablation studies have been conducted and added to the manuscript, including variations\nin network depth, embedding dimensions, and wavelet basis functions.\n Alternative strategies for integrating ATC instructions have been explored to evaluate the\nscalability of the WTFTP+ framework under instruction-driven flight scenarios.\n Comparisons of computational efficiency and model parameters have been provided to\nthe manuscript to clarify the trade-off between performance and complexity.\n An additional dataset from U.S. airspace was constructed and used for retraining to\nvalidate the generalization capability of the proposed WTFTP+ framework across different\noperational environments.\n More theoretical details are provided in Section Method and Supplementary Information\nto elaborate on our motivation and enhance clarity.\n The manuscript has been thoroughly double-checked and proofread to avoid grammatical\nand typographical errors. In addition, the figures have also been refined to improve clarity\nand readability.\nThe point-by-point response to the comments are provided as below.\nHope this revision can address all your concerns.\nResponse to Reviewer #1\nThe paper introduces an enhanced WTFTP framework that demonstrates robust performance\nin flight trajectory prediction. The manuscript presents a well-structured study with sound\nmethodology. However, addressing the following points would further strengthen its rigor, clarity,\nand impact prior to publication:\nComment #1: The current \"Introduction\" section provides an overview of existing methods\nbut lacks a critical analysis of their limitations. The authors should explicitly identify the\nunresolved issues in current approaches. They could better emphasize how their proposed\nmethod addresses these problems, thereby highlighting its innovativeness.\nAuthor response: We sincerely appreciate the your insightful comments, which have been\ninstrumental in improving the clarity and depth of our manuscript.\nIn the revised version, the “Introduction” section has been updated to include a more explicit\nand critical analysis of the limitations in existing methods. In addition, we have also emphasized\nhow the proposed WTFTP+ framework addresses these limitations by leveraging time–\nfrequency modeling and multi-horizon prediction paradigm. The detailed modifications are\nlisted as below.\nHope the following revisions can directly address your concerns and more clearly highlight the\nnovelty of the proposed WTFTP+ framework.\nRevision:\nAlthough these methods achieved the desired performance improvements, they still struggle to\ncapture transition patterns due to the high dependency on dynamic models and are prone to\nerror accumulation over extended future horizons, thereby limiting their multi-horizon prediction\ncapability. With the development of artificial intelligence techniques [17, 18], data-driven\nmethods have emerged as primary approaches for FTP modeling, which learns complex\ntransition patterns and temporal-spatial dependencies by fitting large-scale historical trajectory\ndata [19], and finally enable more accurate and adaptive trajectory predictions in various\nconditions [20, 21, 22, 23, 24]. Nevertheless, most existing data-driven FTP methods primarily\nfocus on advancing temporal–spatial modeling techniques, while often overlooking the\nexploration of the frequency domain. The inability to model time-frequency patterns inevitably\nimpairs performance in relatively high maneuver scenarios. (Pages 1-2, the 2nd paragraph of\nSection 1, Main Text)\nIn this paradigm, the WTFTP+ framework not only can fully exploit the advantages of the time–\nfrequency modeling technique but also is able to mitigate the error accumulation problem in\nmulti-horizon prediction scenarios. (Page 2, the 4th paragraph of Section 1, Main Text)\nComment #2: The problem definition and introduction suggest that filter-based approaches\n(e.g., particle filters, Kalman filters) demonstrate applicability for flight trajectory prediction (FTP)\ntasks. To strengthen the comparative analysis, the authors should incorporate experimental\ncomparisons with these baseline methods. Furthermore, a quantitative evaluation of the\nhyperparameter selection process for the proposed enhanced WTFTP framework would\nsignificantly enhance the methodological rigor of the study.\nAuthor response: Thanks for your insightful comments. It is very important to help us\nimprove the quality of the manuscript.\nIn this revision, the following modifications have been provided in the main manuscript and\nSupplementary information.\n1. The Kalman filter method has been incorporated as a comparative baseline in the main\nmanuscript, and corresponding analysis and discussions have been added to the Results\nsection.\n2. Following your insightful suggestion, a detailed presentation of the hyperparameter\nselection process has been provided in the Supplementary Information Section 7.2,\nincluding the number of Transformer layers in the trajectory encoder and SAWCD, as well\nas the feature dimensions of trajectory embeddings. Due to space limitations in the main\ntext, these results have been included in the Supplementary Information. According to the\njournal policy, the supplementary material will be made publicly available upon acceptance\nof the manuscript.\nHope these revisions can address your concerns.\nRevision:\nIn this work, a total of 7 competitive baselines with diverse technical frameworks are selected\nto validate the effectiveness of the proposed WTFTP+ framework, as listed below:\n Kalman filter: A classical recursive state estimation algorithm that predicts and corrects\nflight states by integrating historical observations with system dynamics models [15]. (Page\n5, Section 2.3, Main Text)\nBased on the inference style of the multi-horizon prediction, the baselines are further\ncategorized into iterative prediction models (Kalman filter, CNN-LSTM, FlightBERT, WTFTP)\nand direct prediction models (Attn-LSTM, TDTransformer, and FlightBERT++). (Page 5, Section\n2.3, Main Text)\nIn addition, the Kalman filter-based FTP method only obtain inferior performance, especially\nfacing longer prediction horizon scenarios, because they rely solely on pseudo observations\nwithout real collected measurement updates during the iterative prediction process. (Page 7, the\n3rd paragraph of Section 2.5, Main Text)\nSimilar to the quantitative analysis, the Kalman filter-based method exhibits larger error\naccumulation at longer horizons without real-time observations. In contrast, for more complex\nflight scenarios, such as the coupled descent and right-turn case during the approach phase\nshown in Fig. 2c and d, it can be seen that the WTFTP+ framework significantly outperforms\nother baselines in the visualization results. Except for the CNN-LSTM, Kalman filter, and\nWTFTP affected by error accumulation, other baseline models can only capture long-term and\nturn-right trends but fail to accurately predict the latitude-longitude transition during the descent\nphase. (Page 9, the 2nd paragraph of Section 2.7, Main Text)\n7.2. Hyperparameters (Pages 9-11, Supplementary Information)\nThis section presents ablation studies on the hyperparameters of the WTFTP+ framework,\nwhich mainly focus on the dimension of trajectory embeddings and the number of Transformer\nlayers in the trajectory encoder and the SAWCD module.\nFor trajectory embeddings, in addition to the original configuration of 128 dimensions,\nexperiments are conducted with 64 and 256 dimensions. As shown in Supplementary Table 4,\nthe results indicate that 128 dimensions provide the best balance between representation\ncapability and computational efficiency. It is believed that the proposed model (with 64-\ndimensional embedding) is insufficient to capture the complex spatial–temporal patterns of\nflight trajectories, which finally results in under-representation and degraded prediction\naccuracy. Conversely, increasing the dimension to 256 might introduce redundant features that\ndo not substantially improve predictive performance while significantly increasing the\ncomputational efficiency of the model. The above experimental results suggest that 128\ndimensions are the most suitable selection to extract trajectory representations in the WTFTP+\nframework.\nFor the number of Transformer layers, we further evaluate two additional configurations beyond\nthe original setting (4 stacked Transformer layers in Trajectory encoder and 2 stacked\nTransformer layers for SAWCD): (i) 2 stacked Transformer layers in Trajectory encoder and 1\nTransformer layer for SAWCD, and (ii) 4 stacked Transformer layers in Trajectory encoder and\n4 stacked Transformer layers for SAWCD. Experimental results reveal the following two\nimportant insights: first, reducing the depth of the model slightly (i.e., setting (i)) leads to\ndegraded performance, likely due to overfitting and the disruption of the balance between\nfeature abstraction and interaction depth. Second, further improving the depth of the model to\nsetting (ii) obtains comparable performance to the original configuration, which indicates deeper\nlayers can maintain but not substantially enhance the model accuracy.\nOverall, these results demonstrate that the adopted hyperparameter settings represent a\nrational and empirically validated design choice for the WTFTP+ framework.\nComment #3: From Table 1, it can be seen that WTFTP+ does not have an advantage in the\nRMSE of ALT prediction. Additionally, FlightBERT++ has an advantage in the MAPE (%) for\nlong-term prediction. Please provide additional analysis and explanation.\nAuthor response: Thanks for your insightful comments. It is very important to help us\nimprove the quality of the manuscript.\nAs described in Discussion section, the transition patterns derived from wavelet components in\nthe altitude attribute differ significantly from those in the longitude and latitude attributes. This\ncan be attributed to the fact that altitude typically exhibits noticeable changes only during\nclimbing or descending maneuvers, whereas longitude and latitude undergo continuous\nchanges throughout flight operations. Consequently, the unified time–frequency modeling\napplied across longitude, latitude, and altitude attributes may limit the predictive robustness for\nthe altitude attribute, leading to relatively higher RMSE in altitude prediction.\nRegarding the MAPE (%) for long-term horizons, the results in Table 1 show that WTFTP+\nachieves consistently lower absolute errors and spatial deviations across most horizons,\nindicating its ability to capture fine-grained trajectory variations more effectively. Nevertheless,\nFlightBERT++ achieves slightly better MAPE at the 15-step horizon, which can be attributed to\nits tendency to focus on long-term trend modeling. In contrast, WTFTP+ demonstrates a\nbalanced capability in both short- and long-term prediction, which aligns with our motivation of\nleveraging wavelet analysis to capture both local details and global trends.\nRevision:\nWTFTP+ exhibits slightly higher RMSE in the altitude dimension, primarily due to the limited\nadaptability of unified time–frequency modeling to the distinct variation patterns of altitude\ncompared with longitude and latitude (see Discussion). In addition, FlightBERT++ achieves\nmarginally better MAPE at the 15-step horizon, which can be attributed to its tendency to focus\non long-term trend modeling. In contrast, WTFTP+ consistently achieves lower absolute errors\nand spatial deviations across most horizons and demonstrates a balanced capability in both\nshort- and long-term prediction, which aligns with our motivation of leveraging wavelet analysis\nto capture both local details and global trends. (Page 7, the 4th paragraph of Section 2.5, Main Text)\nResponse to Reviewer #2\nThis paper presents an enhanced version of the previously proposed WTFTP framework,\nnamed WTFTP+, aiming to improve multi-horizon flight trajectory prediction by mitigating\ncumulative errors of stepwise inference and enhancing wavelet component interactions. The\nproposed architecture introduces a direct encoder-decoder-based prediction paradigm and a\nnovel time-frequency bridging mechanism. The work is well-structured and tackles an urgent\nproblem in real-world air traffic control systems. Furthermore, the release of the source code\nwill benefit both academic and industrial communities. However, there remain several technical\nand presentation concerns that should be addressed before publication.\nMajor Comments:\nComment #1: The paper frequently mentions that wavelet components lie in “orthogonal\nspaces.” However, the exact meaning of this claim and the origin of orthogonality are not fully\nexplained. Is this based on the properties of the wavelet basis used, or is it empirically observed\nin the learned representations? Please clarify.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nIt should be noted in this paper that the proposed method is based on the orthogonal wavelet.\nTightly supported orthogonal wavelets achieve superior temporal localization capabilities,\nenabling more precise characterization of trajectory features during maneuvering phases.\nFurthermore, the detail spaces generated by wavelet functions are mutually orthogonal across\ndifferent scales and orthogonal to the approximation space at the same scale. Based on these\nproperties, this paper employs orthogonal Haar wavelets for trajectory prediction, generating\nmulti-scale wavelet components lied in mutually orthogonal subspaces. A more detailed\ndescription of the orthogonal space formulation under multi-resolution analysis is provided in\nSupplementary Information Section 3.\nRevision:\nIn our previous work [22], a wavelet transform-based flight trajectory prediction (WTFTP)\nframework is proposed to tackle the single-horizon FTP tasks, which leverages time-frequency\nanalysis to capture global flight trends and local motion details represented by wavelet\ncomponents (WTCs) 1. (Page 2, the 3rd paragraph of Section 1, Main Text)\nConsidering the orthogonality of approximation subspace of wavelet components, each\ndecoder performs an independent prediction process without desired feature interaction 2.\n(Page 2, the 3rd paragraph of Section 1, Main Text)\n1The descriptions of the time-frequency analysis and WTCs can be found in Supplementary Information\nSection 3. (Page 2, footnote, Main Text)\n2A detailed description of the orthogonal space formulation under multi-scale analysis is provided in\nSupplementary Information Section 3. (Page 2, footnote, Main Text)\n3. Multi-resolution analysis (Pages 2-4, Section 3, Supplementary Information)\nMulti-resolution analysis (MRA) theory [1], also known as multi-scale analysis, provides a\ncoarse-to-fine feature representation of signals while simplifying complex integration\ncalculations, as the essence of discrete wavelet transforms. In MRA, signals undergo a series\nof filtering and downsampling operations to yield feature sequences with physical significance\nat different scales. The general procedure of MRA involves decomposing a signal into two\nfundamental components: low-frequency subbands and high-frequency subbands. The former,\nobtained through low-pass filtering, constitutes the approximation coefficients; the latter,\ngenerated by high-pass filtering, represents the detail coefficients. To achieve finer frequency\nresolution, the low-frequency subbands undergo further decomposition. As decomposition\nlevels progress, MRA delivers multi-scale time-frequency representations of the signal.\nMRA describes a sequence of closed subspaces {V} , which are nested like V ⊂ V. V is\nj j∈Z j−1 j j\nthe approximation space spanned by translations and scalings of the scale function φ(x),\ndefined as:\nwhere j is the scale factor, and k is the translation factor. {φ } is standard orthogonal\nj,k j∈Z,k∈Z\nbasis of V. The larger the scale (the larger j), the higher the resolution and the more detail V\nj j\ncontains, satisfying V → L2(R) when j → +∞. Since V is a subspace of V, they share\nj j−1 j\nsubstantial common information. Due to the finer scale of V, some detail information in which\nj\ncan not be projected onto V . According to linear algebra theory, every subspace possesses\nj−1\na unique orthogonal complement. Therefore, denoting the orthogonal complement of V in V\nj−1 j\nas W , i.e., the detail space of scale j − 1, V can be described as:\nj−1 j\nwhere ⊕ denotes direct sum and the spaces are orthogonal. W is generated by translations\nj−1\nand scalings of the wavelet function ψ(x), defined as:\nwhere j is the scale factor, and k is the translation factor. Thus, the signal’s ”overall profile”\n(captured by V) and ”newly revealed details” (captured by W) constitute two entirely distinct\nj j\nand independent types of information. The approximation spaces can be iteratively\ndecomposed into coarser approximation spaces and their orthogonal complement spaces (as\nshown in Supplementary Fig. 2). By projecting the signal onto these spaces, we obtain its multi-\nscale time-frequency information.\nIn practice, with the signal projection onto the approximation space at scale j, it can be\ndecomposed using wavelet filters to obtain projections onto the approximation space and detail\nspace at scale j−1. These are referred to as the low-frequency component and high-frequency\ncomponent, respectively. By repeatedly decomposing the lowfrequency component at each\nlevel, we can obtain the high-frequency components at corresponding scales and the low-\nfrequency component at the smallest scale. Collectively, these are termed wavelet components,\nresiding in the subspace of the approximation space at scale j and mutually orthogonal to each\nother.\nIn the flight process, the overall motion trends of the aircraft are typically determined by\nhistorical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a\ncertain route shown in Supplementary Fig. 3, the trajectory always evolves toward the\ndesignated waypoint (in the flight plans), indicating that the planned route can imply the overall\nflight trend. At the same time, local variations are introduced by weather conditions, aircraft\nperformance parameters, and pilot-specific operations. In other words, a flight trajectory can be\nregarded as a combination of global trends and local details. While different flights in similar\nscenarios may follow similar overall trends, their local motion details often vary significantly,\nwhich limits the prediction accuracy of the conventional FTP approaches.\nThe wavelet transform provides an intuitive solution to this challenge by decomposing time\nseries into multiscale components, thereby decoupling local details from global trends. This\ntime–frequency analysis forms the theoretical foundation of both WTFTP and WTFTP+.\nOrthogonal wavelets with compact support facilitate the identification of maneuvers such as\nturns, climbs, or descents at different scales. Moreover, local motion details can influence global\npatterns, implying inherent dependencies across multi-scale wavelet components. WTFTP+\nbuilds upon this foundation by introducing scale-aware wavelet component decoder and time–\nfrequency bridging mechanisms, which enable effective interaction among multi-scale wavelet\ncomponents without breaking orthogonality. These technical improvements allow the model to\njointly capture global motion trends and fine-grained local dynamics, thereby addressing\nperformance bottlenecks inherent in conventional FTP approaches.\nComment #2: While WTFTP+ aims to improve prediction accuracy, its encoder-decoder\narchitecture and bridging mechanisms increase model complexity. Since trajectory prediction in\nair traffic control is a time-sensitive task, it is critical to evaluate whether WTFTP+ meets real-\ntime inference requirements. Please provide a quantitative comparison of model parameters\nand inference time between baselines and WTFTP+.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nIn this revision, to further evaluate the practical applicability of the proposed WTFTP+\nframework, a new subsection, i.e., Computational performance, has been included in the\nResults section. The detailed modifications are presented as below.\nHope this revison can address your concerns.\nRevision:\n2.8 Computational performance (Pages 9-11, Main Text)\nTo further evaluate the practical applicability of the proposed WTFTP+ framework, Fig. 3\npresents the model parameters and computational efficiency of WTFTP+ and selected data-\ndriven baseline models. The vertical axis is log-scaled to improve readability and\ndistinguishability. To ensure the comparison fairness, the batch size and prediction horizons is\nfixed to 1 and 15 for all models, respectively.\nAlthough the Kalman filter-based FTP method achieves superior MTC performance (0.64ms),\nits prediction error rapidly propagates to reduce the performance as the prediction horizon\nincreases (as shown in Table 1), which demonstrates its inapplicability for long-horizon\nprediction scenarios. Except for FlightBERT and FlightBERT++, the number of model\nparameters of all models is maintained within the order of millions. Although FlightBERT++\nadopts a non-autoregressive prediction mechanism and obtains a faster inference speed, it still\nrequires additional computational overhead for the binary encoding process in practice. Prior\nworks has reported that the average visual reaction time of air traffic controllers can reach up\nto 170ms [32]. Therefore, all FTP models except FlightBERT can meet real-time operational\nrequirements. The slower performance of FlightBERT is primarily due to its base transformation\nof trajectory attributes, which further burdens computational overhead during iterative multi-\nhorizon prediction. Compared with the WTFTP framework, WTFTP+ removes the multi-decoder\nstructure but increases the latent feature dimensionality and incorporates a Transformer-style\nbackbone, thereby resulting in larger trainable parameters than that of the WTFTP framework.\nFortunately, the proposed WTFTP+ follows a direct multi-horizon prediction paradigm,\nachieving 15-step predictions within 24ms even under four-level wavelet decomposition. In\nsummary, considering both the model parameters and MTC measurement, WTFTP+ achieves\na favorable balance between model complexity and computational efficiency, while maintaining\nsuperior prediction accuracy and robustness across multi-horizon prediction scenarios. The\nresults indicate that the WTFTP+ is a promising solution for the real-world operational ATC\nsystems, further supporting real-time downstream applications, such as conflict detection and\ndecision-making.\nComment #3: The learnable transformation matrix used in the time-frequency bridging\nmodule lacks interpretability. It would be beneficial to provide an intuitive explanation of this\nmatrix. Additionally, the contribution of this component should be further validated through\nablation experiments.\nAuthor response: Thanks for your correction, it is very important to help us improve the\nquality of our manuscript.\nIn this revision, an ablation study has been added to quantitatively assess the contribution of\nthe learnable parameters in the time–frequency bridging module. Specifically, the learnable\ntransformation matrices in the wavelet-to-time (W2T) and time-to-wavelet (T2W) processes\nwere removed to conduct the corresponding experiments.\nThe results demonstrate that the learnable matrices play a positive role in enhancing the\nadaptability of the time–frequency bridging module. They allow the model to capture\ncorrelations among multi-scale wavelet representations, guiding the module to optimize the\ninformation exchanges according to the statistical properties of the trajectory data.\nFurthermore, even after removing the learnable matrices, the proposed WTFTP+ framework\nstill exhibits competitive performance. This result also verifies the inherent robustness of the\nWTFTP+ architecture, while also confirming that the learnable matrices can provide expected\nperformance improvements rather than being a critical dependency.\nDue to space limitations in the main text, these results have been included in the\nSupplementary Information. According to the journal policy, the supplementary material will be\nmade publicly available upon acceptance of the manuscript.\nThe detailed modifications are listed below.\nHope this revision can address your concerns.\nRevision:\n7.1 Learnable transformation matrix (Page 8, Supplementary information)\nTo evaluate the contribution of the learnable T2W and W2T procedures, the learnable\ntransformation matrices in both blocks are replaced by fixed operations to construct the\nWTFTP+ framework. The experimental results are presented in Supplementary Table 3. It can\nbe seen from the results that introducing learnable parameters into the T2W and W2T\nprocedure obtains better performance in terms of the proposed metrics. The learnable\nparameters allow the model to adaptively refine the mapping among multi-scale wavelet\nrepresentations, rather than relying solely on fixed mathematical transformations. The resulting\nflexibility enables the model to capture task-specific time–frequency dependencies and\nempirical correlation in flight patterns that may not be well represented by standard wavelet\nbasis functions. In addition, the learnable transformation enhances the ability of the wavelet\ndomain features to align with trajectory dynamics, thereby improving the model’s sensitivity to\nboth local fluctuations and long-term motion patterns. Moreover, even after removing the\nlearnable parameters, WTFTP+ exhibits only a slight degradation in long-horizon prediction\nperformance. This result highlights the effectiveness of the proposed time–frequency bridging\nmechanism, which ensures stable information flow between time and frequency domains.\nNevertheless, the performance improvements observed with the learnable transformation\nconfirm its effectiveness in enhancing adaptability and improving predictive accuracy.\nComment #4: The current implementation uses trajectory tail vector as a proxy for short-term\nmotion trend (i.e., initial low-frequency wavelet embedding). Since ATC instructions often\nrepresent long-term macroscopic intentions, have the authors considered fusing such intention\nvectors into the initial wavelet embeddings? This might enhance semantic guidance in multi-\nhorizon forecasting.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nFollowing your suggestion, we arranged the instruction embeddings before the initial low-\nfrequency wavelet embeddings as prior prompt to fine-tune the model. However,\nexperimental results indicate that integrating the instruction embeddings with the wavelet\nembeddings fail to achieve better performance than the original fusion strategy. A possible\nexplanation is that the predicted wavelet components are derived from the initial components\nthrough multiple layers of nonlinear transformation (i.e., scale-aware wavelet component\ndecoder). The instruction embedding that extracted from textual ATC instructions via a BERT\nmodel, already contains high-level semantic representations and thus does not require\nadditional learning of wavelet- and time-domain information within the scale-aware decoder. In\ncontrast, both the instruction and trajectory embeddings can be regarded as prior knowledge,\nwhich can be more effectively aligned and fused with latent wavelet component representations\nvia a cross-domain attention mechanism in the decoder. This approach facilitates the learning\nability of the most relevant pripor information while mitigating interference from redundant\nfeatures.\nDue to space limitations in the main text, these results have been included in the\nSupplementary Information. According to the journal policy, the supplementary material will be\nmade publicly available upon acceptance of the manuscript.\nThe detailed modifications are listed below.\nHope this revision can address your concerns.\nRevision:\n5. Integrating ATC instruction into WTFTP+ framework (Page 7, Supplementary information)\nAs described in the main text, the WTFTP+ framework demonstrates strong extensibility and\ncan serve as a foundational model to incorporate ATC instructions, thereby enhancing its\nperformance in instruction-driven maneuver phases. This section presents detailed\nexperimental results as well as an exploration of alternative fusion strategies to integrate ATC\ninstructions. In general, ATC instructions convey long-horizon macroscopic intentions, while\nthe initial low-frequency wavelet embedding is applied to represent short-term flight trends.\nBased on this observation, a straightforward integration strategy is to arrange the instruction\nembeddings before the initial low-frequency wavelet embeddings and use them as prior\nprompts for fine-tuning the multi-modal FTP model.\nThe experimental results are presented in Supplementary Table 1. It is indicated that this fusion\nstrategy fails to improve short-horizon prediction and in some cases slightly degrades the\nperformance. However, it achieves certain performance improvement at longer horizons\ncompared to the original WTFTP+ framework (without instruction). The results suggest that the\nintroduction of instruction embeddings may interfere with the learning of initial wavelet\nembeddings, thereby weakening their ability to capture short-term motion dynamics. At the\nsame time, the consideration of high-level intent information (integrating ATC instruction)\nprovides informative guidance for longer-horizon trajectory prediction. Moreover, experimental\nresults also indicate that integrating the instruction embeddings with the wavelet embeddings\nfails to achieve better performance than the original fusion strategy (see Main Text). A possible\nexplanation is that the predicted wavelet components are derived from the initial components\nthrough multiple layers of nonlinear transformation (i.e., scale-aware wavelet component\ndecoder). The instruction embedding that was extracted from textual ATC instructions via a\nBERT model, already contains high-level semantic representations and thus does not require\nadditional learning of wavelet- and time-domain information within the scale-aware decoder. In\ncontrast, both the instruction and trajectory embeddings can be regarded as prior knowledge,\nwhich can be more effectively aligned and fused with latent wavelet component representations\nvia a cross-domain attention mechanism in the decoder. This approach facilitates learning of\nthe most relevant prior information while mitigating interference from redundant features.\nOverall, the above experimental results demonstrate the importance of carefully designing\nfusion strategies when integrating ATC instructions.\nComment #5: The proposed ITD mechanism converts features from the wavelet domain to\nthe time domain and then back to the wavelet domain before generating predictions. Have the\nauthors explored the possibility of skipping the T2W process and directly predicting in the time\ndomain after fusion?\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nIn fact, WTFTP+ still inherits the core concept of WTFTP, i.e., predicting optimal wavelet\ncoefficients rather than directly forecasting raw trajectory attributes. During model training,\nreal wavelet coefficients are used as the supervision, which enables the model to learn multi-scale\ntime–frequency representations of flight dynamics.\nTo address the limitations of the WTFTP framework, in this paper, a time-frequency bridging\nmechanism is proposed to extract inherent dependencies within flight patterns across different\ntime-frequency scales. Specifically, wavelet-domain features are first transformed into the time\ndomain (via the W2T block) to allow correlation extraction in a unified representation space.\nSubsequently, a transformation is applied to map these features back into the wavelet domain\n(via the T2W block), where multi-scale analysis can be explicitly leveraged to generate\npredictions.\nIn this contexts, skipping the T2W process and directly predicting in the time domain would\ndeviate from this design principle, since the model would no longer generate wavelet\ncoefficients as its prediction objective. Under such a setting, the prediction task would be\nreduced to producing raw trajectory attributes without explicit supervision from real wavelet\ncoefficients. This would prevent the framework from effectively modeling the intrinsic time–\nfrequency structure of flight trajectories, thereby limiting its ability to capture multi-scale flight\npatterns.\nHope the above explanation can address your concerns.\nComment #6: The proposed method performs slightly worse than some baselines in terms\nof MAE and RMSE for the altitude prediction task. While this is discussed in the Section\nDiscussion, the experimental analysis should also be included in the quantitative analysis\nsection, such as error accumulation or sensitivity to vertical motion features.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nAs described in Discussion section, the transition patterns derived from wavelet components in\nthe altitude attribute differ significantly from those in the longitude and latitude attributes. This\ncan be attributed to the fact that altitude typically exhibits noticeable changes only during\nclimbing or descending maneuvers, whereas longitude and latitude undergo continuous\nchanges throughout flight operations. Consequently, the unified time–frequency modeling\napplied across longitude, latitude, and altitude attributes may limit the predictive accuracy for\nthe altitude attribute, leading to a higher RMSE in altitude prediction.\nIn this revision, the corresponding analysis have been added to the quantitative analysis section.\nThe detailed modifications can be found in below.\nHope this revision can address your concerns.\nRevision:\nWTFTP+ exhibits slightly higher RMSE in the altitude dimension, primarily due to the limited\nadaptability of unified time–frequency modeling to the distinct variation patterns of altitude\ncompared with longitude and latitude (see Discussion). In addition, FlightBERT++ achieves\nmarginally better MAPE at the 15-step horizon, which can be attributed to its tendency to focus\non long-term trend modeling. In contrast, WTFTP+ consistently achieves lower absolute errors\nand spatial deviations across most horizons and demonstrates a balanced capability in both\nshort- and long-term prediction, which aligns with our motivation of leveraging wavelet analysis\nto capture both local details and global trends. (Page 7, the 4th paragraph of Section 2.5, Main Text)\nMinor Comments:\nComment #7: 1.Fig. 1: Replace “Temporal domain” with “Time domain” for consistency.\nAuthor response: Thanks for your corrections. The Fig. 1 have been updated to ensure the\nconsistency of the terms.\nHope this revision can address your concerns.\nRevision:\nComment #8: Fig. 5: The extensive use of blue tones reduces color contrast; consider\nimproving visual distinctiveness.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the readability of our manuscript.\nIn this revision, we have revised the Fig. 5 (Fig. 6 in this version) to enhance the readability.\nHope this revision can address your concerns.\nRevision:\nComment #9: Acronyms: Many abbreviations are defined multiple times (e.g., FTP, ATC).\nAcronyms should only be defined upon first usage.\nAuthor response: We sincerely appreciate your careful reading and helpful comment.\nIn this revision, redundant acronym definitions (e.g., FTP, ATC) have been removed, and each\nabbreviation is now defined only at its first occurrence in the manuscript.\nHope this revision can address your concerns.\nComment #10: Notation Consistency: Ensure consistent formatting for terms like \"pseudo-\nobservations\" vs. \"pseudo observations\", \"WTC\" vs. \"WTCs\".\nAuthor response: We sincerely appreciate your insightful comment. In this revision, all\ninconsistent notations have been standardized throughout the manuscript. For example,\n“pseudo-observations” is now used consistently instead of “pseudo observations”, and “WTCs”\nis adopted uniformly when referring to multiple instances of WTC.\nHope this revision can address your concerns.\nComment #11: Grammar/Typo Improvements:\nSection 2.1: “In general, the short-term FTP tasks can generally be…” -> “...can be…”\nSection 2.2: “...the region of interest is...” -> “...the regions of interest are...”\nSection 2.3: “long short term memory” -> “long short-term memory”\nSection 2.5: Remove “are detailed reported” -> “…reported…”\nPage 6: “which supports our motivation in this work and provide...” -> “... provides...”\nPage 7: “By combing...” -> “By combining...”\nPage 12: Clarify “MRE” -> possibly intended to be “RMSE”?\nPage 9: “expected for” -> “except for”\nPage 10: “the level-3 exhibit” -> “exhibits”\nPage 15: “underdo” -> likely intended “undergo”\nAuthor response: We sincerely appreciate your careful reading and constructive feedback.\nThe grammatical issues in the manuscript have been corrected as suggested. Due to the large\nnumber of minor modifications, detailed changes can be found directly in the revised manuscript.\nHope this revision can address your concerns.\nComment #12: Page 7: FAA report statistics are slightly misrepresented. It should read:\n“Takeoff/climb and approach/descent phases account for 21% and 46% of total accidents,\nrespectively,” rather than presenting them as absolute accident rates.\nAuthor response: We sincerely appreciate your careful reading and constructive feedback.\nThe FAA report statistics have been revised to accurately state: “Takeoff/climb and\napproach/descent phases account for 21% and 46% of total accidents, respectively,” in\naccordance with the source. This correction ensures that the figures are presented as\nproportions of total accidents rather than as absolute accident rates.\nHope this revision can address your concerns.\nRevision:\nFurthermore, according to the Global Flight Accident Report [31] issued by the Federal Aviation\nAdministration, takeoff/climb and approach/descent phases account for 21% and 46% of total\naccidents, respectively, whereas the cruise phase accounts for only 9%. (Page 8, the 1st\nparagraph of Section 2.6, Main Text)\nResponse to Reviewer #3\nThe paper under review proposes a wavelet-based method for an efficient flight trajectory\nprediction in the multi-horizon setting. The work appears promising, as it employs frequency-\ndomain analysis in the flight trajectory prediction task, in contrast to data-driven methods. The\nproposed method can be viewed as an extension of the existing Wavelet Transform-based\nFlight Trajectory Prediction (WTFTP) framework to circumvent its limitations, in that WTFTP\nsuffers from a performance bottleneck in multi-horizon prediction due to its inherent single-\nhorizon inference paradigm. This novel framework (WTFTP+) is implemented with a\ngeneralized encoder-decoder architecture, in which the encoder is employed to extract the\nabstract features from the observed trajectory sequence, while the decoder is applied to\ngenerate the optimal wavelet coefficients for the predicted horizons. In order to maintain\northogonality of the multi-scale wavelet components, in the decoder, the orthogonal wavelet\ncomponents are first bridged into the time domain for promotive interaction, and subsequently\nbridged back into the wavelet-transformed domain to obtain enhanced features of inherent flight\npatterns. The validity of the WTFTP+ method is demonstrated via several experiments on a\nreal-world flight trajectory dataset, whereas some baseline comparisons are incorporated into\nthe experimental analysis to provide a comprehensive evaluation of the method.\nIn conclusion, I find the paper very well-written and the results quite interesting from an\nempirical perspective.\nComment #1: Though a reasonable discussion on the theoretical underpinnings of the\nmethod can be found in the paper, an extensive theoretical performance analysis of the\nWTFTP+ method is not provided.\nAuthor response: Thanks for your insightful comments, which helps us further improve the\nquality and clarity of the manuscript.\nIn the flight process, the overall motion trends of the aircraft are typically determined by\nhistorical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a\ncertain route, the longitude and latitude always evolve along the heading between two\nconsecutive waypoints, indicating that the planned route (in the flight plans) can imply the\noverall flight trend. At the same time, local variations are introduced by weather conditions,\naircraft performance parameters, and pilot-specific operations. In other words, a flight trajectory\ncan be regarded as a combination of global trends and local details. While different flights in\nsimilar scenarios may follow similar overall trends, their local motion details often vary\nsignificantly, which limits the prediction accuracy of the conventional FTP approaches.\nThe wavelet transform provides an intuitive solution to this challenge by decomposing time\nseries into multi-scale components, thereby decoupling global trends from local details. This\ntime–frequency decomposition forms the theoretical foundation of both WTFTP and WTFTP+.\nOrthogonal wavelets with compact support facilitate the identification of maneuvers such as\nturns, climbs, or descents at different scales. Moreover, local motion details can influence global\npatterns, implying inherent dependencies across multi-scale wavelet components.\nWTFTP+ builds upon this foundation by introducing scale-aware wavelet components decoder\nand time–frequency bridging mechanisms, which enable effective interaction among multi-\nscale wavelet components without breaking orthogonality. These mechanisms allow the model\nto jointly capture global motion trends and fine-grained local dynamics, thereby addressing\nperformance bottlenecks inherent in conventional FTP approaches.\nIn this revision, the above description have been updated to the Section method to elaborate\non our motivation and serves as the theoretical foundation of the proposed WTFTP and\nWTFTP+ framework. In addition, more theoretical analysis of the WTFTP+ have been provided\nin the Supplementary Information Section 3.\nThe detailed modifications are listed below. Hope this revision can address your concerns.\nRevision:\nIn the flight process, the overall motion trends of the aircraft are typically determined by\nhistorical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a\ncertain route, the longitude and latitude always evolve along the heading between two\nconsecutive waypoints, indicating that the planned route (in the flight plans) can imply the\noverall flight trend. At the same time, local variations are introduced by weather conditions,\naircraft performance parameters, and pilot-specific operations. In other words, a flight trajectory\ncan be regarded as a combination of global trends and local details. While different flights in\nsimilar scenarios may follow similar overall trends, their local motion details often vary\nsignificantly, which limits the prediction accuracy of the conventional FTP approaches.\nFortunately, the wavelet transform provides an intuitive solution to this challenge by\ndecomposing time series into multi-scale components, thereby decoupling global trends from\nlocal details. (Page 16, the 1st paragraph of Section 4.1, Main Text)\n3. Multi-resolution analysis (Pages 2-4, Section 3, Supplementary Information)\nMulti-resolution analysis (MRA) theory [1], also known as multi-scale analysis, provides a\ncoarse-to-fine feature representation of signals while simplifying complex integration\ncalculations, as the essence of discrete wavelet transforms. In MRA, signals undergo a series\nof filtering and downsampling operations to yield feature sequences with physical significance\nat different scales. The general procedure of MRA involves decomposing a signal into two\nfundamental components: low-frequency subbands and high-frequency subbands. The former,\nobtained through low-pass filtering, constitutes the approximation coefficients; the latter,\ngenerated by high-pass filtering, represents the detail coefficients. To achieve finer frequency\nresolution, the low-frequency subbands undergo further decomposition. As decomposition\nlevels progress, MRA delivers multi-scale time-frequency representations of the signal.\nMRA describes a sequence of closed subspaces {V} , which are nested like V ⊂ V. V is\nj j∈Z j−1 j j\nthe approximation space spanned by translations and scalings of the scale function φ(x),\ndefined as:\nwhere j is the scale factor, and k is the translation factor. {φ } is standard orthogonal\nj,k j∈Z,k∈Z\nbasis of V. The larger the scale (the larger j), the higher the resolution and the more detail V\nj j\ncontains, satisfying V → L2(R) when j → +∞. Since V is a subspace of V, they share\nj j−1 j\nsubstantial common information. Due to the finer scale of V, some detail information in which\nj\ncan not be projected onto V . According to linear algebra theory, every subspace possesses\nj−1\na unique orthogonal complement. Therefore, denoting the orthogonal complement of V in V\nj−1 j\nas W , i.e., the detail space of scale j − 1, V can be described as:\nj−1 j\nwhere ⊕ denotes direct sum and the spaces are orthogonal. W is generated by translations\nj−1\nand scalings of the wavelet function ψ(x), defined as:\nwhere j is the scale factor, and k is the translation factor. Thus, the signal’s ”overall profile”\n(captured by V) and ”newly revealed details” (captured by W) constitute two entirely distinct\nj j\nand independent types of information. The approximation spaces can be iteratively\ndecomposed into coarser approximation spaces and their orthogonal complement spaces (as\nshown in Supplementary Fig. 2). By projecting the signal onto these spaces, we obtain its multi-\nscale time-frequency information.\nIn practice, with the signal projection onto the approximation space at scale j, it can be\ndecomposed using wavelet filters to obtain projections onto the approximation space and detail\nspace at scale j−1. These are referred to as the low-frequency component and high-frequency\ncomponent, respectively. By repeatedly decomposing the lowfrequency component at each\nlevel, we can obtain the high-frequency components at corresponding scales and the low-\nfrequency component at the smallest scale. Collectively, these are termed wavelet components,\nresiding in the subspace of the approximation space at scale j and mutually orthogonal to each\nother.\nIn the flight process, the overall motion trends of the aircraft are typically determined by\nhistorical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a\ncertain route shown in Supplementary Fig. 3, the trajectory always evolves toward the\ndesignated waypoint (in the flight plans), indicating that the planned route can imply the overall\nflight trend. At the same time, local variations are introduced by weather conditions, aircraft\nperformance parameters, and pilot-specific operations. In other words, a flight trajectory can be\nregarded as a combination of global trends and local details. While different flights in similar\nscenarios may follow similar overall trends, their local motion details often vary significantly,\nwhich limits the prediction accuracy of the conventional FTP approaches.\nThe wavelet transform provides an intuitive solution to this challenge by decomposing time\nseries into multiscale components, thereby decoupling local details from global trends. This\ntime–frequency analysis forms the theoretical foundation of both WTFTP and WTFTP+.\nOrthogonal wavelets with compact support facilitate the identification of maneuvers such as\nturns, climbs, or descents at different scales. Moreover, local motion details can influence global\npatterns, implying inherent dependencies across multi-scale wavelet components. WTFTP+\nbuilds upon this foundation by introducing scale-aware wavelet component decoder and time–\nfrequency bridging mechanisms, which enable effective interaction among multi-scale wavelet\ncomponents without breaking orthogonality. These technical improvements allow the model to\njointly capture global motion trends and fine-grained local dynamics, thereby addressing\nperformance bottlenecks inherent in conventional FTP approaches.\nComment #2: Moreover, the choice of wavelet function, decomposition levels, and LSTM\nhyperparameters may significantly impact results. In this regard, a more systematic sensitivity\nanalysis would be beneficial.\nAuthor response: Thanks for your insightful comments, which are very important for\nimproving both the quality and readability of our manuscript.\n1. In the previous version, we selected the Haar wavelet as the basis function and analyzed\nperformance under different decomposition levels (Section 2.10.1. Levels of the wavelet\nanalysis). Following your suggestions, in this revision, we additionally implement the\nWTFTP+ with the Symlets (sym2) and Coiflets (coif1) wavelet basis functions, and\nconducted evaluations across decomposition levels 1–4.\n2. Compared with WTFTP, the WTFTP+ model adopts a Transformer-style architecture for\nboth encoder and decoder, rather than LSTM-based networks. The Transformer\narchitecture enables better parallelism, facilitates long-range dependency modeling, and\nimproves the capability to capture multi-scale temporal patterns. As you pointed out,\nhyperparameters in neural networks may significantly affect results. In the WTFTP+ \nframework, the hyperparameters were determined through extensive empirical tuning\nduring the preliminary investigations, but the details were not provided in the original\nsubmission due to space limitations. In this revision, we have added ablation details\nregarding the choice of trajectory embedding dimensions and the number of Transformer\nlayers in both the encoder and decoder.\nDue to space limitations in the main text, the above experiments have been included in the\nSupplementary Information. According to the journal policy, the supplementary material will be\nmade publicly available upon acceptance of the manuscript.\nThe detailed modifications are listed below.\nHope this revision can address your concerns.\nRevision:\n7.3. Wavelet functions and decomposition levels (Pages 9-11, Supplementary Information)\nIn this section, the impacts of different wavelet basis functions and decomposition levels are\ninvestigated to validate the effectiveness of the selected Haar wavelet basis function.\nSpecifically, the Symlets (sym2) and Coiflets (coif1) are selected as alternatives to the Haar\nwavelet used in the original WTFTP+ framework. As shown in Supplementary Table 5, the\nWTFTP+ framework still achieves the desired FTP performance even with other wavelet basis\nfunctions, which demonstrates the robustness of the proposed WTFTP+ framework. However,\nthe performance exhibited a slight degradation when the WTFTP+ was implemented with\nSymlets and Coiflets functions. Considering that the predicted trajectory length is shorter than\nthat of common time series, the wavelet basis with a relatively short length of compact support\nsuffices to achieve short-term prediction. The Haar wavelet, as the simplest basis function with\na length of support being only 1, enables superior sensitivity in capturing abrupt changes and\nlocal discontinuities in trajectory data. In contrast, despite Symlets with 2 orders and Coiflets\nwith 1 order offering smoother frequency responses, their extended support means a single\nwavelet coefficient may incorporate information from multiple time points, making it impossible\nto pinpoint the exact instant at which a maneuver occurred. This “tail effect” or “temporal\nambiguity” weakens the significance of local critical information in short-term forecasts, thereby\ndegrading prediction performance. Heisenberg's uncertainty principle dictates that a signal\ncannot achieve arbitrarily high resolution simultaneously in both the time domain and the\nfrequency domain. Frequency resolution can be enhanced through multi-level wavelet analysis.\nTherefore, when selecting a wavelet basis, greater emphasis should be placed on temporal\nlocality in short-term forecasts. The tightly supported Haar wavelet with the shortest support\nemerges as the optimal choice within the WTFTP+ framework, leading to superior prediction\nperformance.\n7.2. Hyperparameters (Pages 9-10, Supplementary Information)\nThis section presents ablation studies on the hyperparameters of the WTFTP+ framework,\nwhich mainly focus on the dimension of trajectory embeddings and the number of Transformer\nlayers in the trajectory encoder and the SAWCD module.\nFor trajectory embeddings, in addition to the original configuration of 128 dimensions,\nexperiments are conducted with 64 and 256 dimensions. As shown in Supplementary Table 3,\nthe results indicate that 128 dimensions provide the best balance between representation\ncapability and computational efficiency. It is believed that the proposed model (with 64-\ndimensional embedding) is insufficient to capture the complex spatial–temporal patterns of\nflight trajectories, which finally results in under-representation and degraded prediction\naccuracy. Conversely, increasing the dimension to 256 might introduce redundant features that\ndo not substantially improve predictive performance while significantly increasing the\ncomputational efficiency of the model. The above experimental results suggest that 128\ndimensions are the most suitable selection to extract trajectory representations in the WTFTP+\nframework.\nFor the number of Transformer layers, we further evaluate two additional configurations beyond\nthe original setting (4 stacked Transformer layers in Trajectory encoder and 2 stacked\nTransformer layers for SAWCD): (i) 2 stacked Transformer layers in Trajectory encoder and 1\nTransformer layer for SAWCD, and (ii) 4 stacked Transformer layers in Trajectory encoder and\n4 stacked Transformer layers for SAWCD. Experimental results reveal the following two\nimportant insights: first, reducing the depth of the model slightly (i.e., setting (i)) leads to\ndegraded performance, likely due to overfitting and the disruption of the balance between\nfeature abstraction and interaction depth. Second, further improving the depth of the model to\nsetting (ii) obtains comparable performance to the original configuration, which indicates deeper\nlayers can maintain but not substantially enhance the model accuracy.\nOverall, these results demonstrate that the adopted hyperparameter settings represent a\nrational and empirically validated design choice for the WTFTP+ framework.\nComment #3: Since the model performs well on U.S. flight data, it is unclear how well it would\ngeneralise to other airspaces or data sources without retraining. Overall, the WTFTP+ method\nis somewhat more explainable in contrast to the data-driven frameworks.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality and readability of our manuscript.\nFirst, all experiments in the manuscript were conducted on flight trajectory data collected from\nthe industrial ATC system in China. As described in Section 2.2 Dataset and evaluation metric,\nthe dataset covers the period from February 19 to February 27, 2021, with a sampling interval\nof 20 seconds. The spatial region of interest are [94.616°, 113.689°] in longitude, [19.305°,\n37.275°] in latitude, and [0, 12,500] meters in altitude.\nSecond, it should be noted that both WTFTP and WTFTP+ are data-driven FTP approaches.\nTheir core idea is to predict multi-scale wavelet components of trajectory attributes to capture\nglobal motion trends and local movement details, thereby achieving more accurate and robust\npredictions. To learn the transition patterns of flight trajectory in the time–frequency domain and\nto predict multi-scale wavelet coefficients, the neural network–based prediction models are\ndesigned and trained through a supervised learning paradigm using a real-world flight trajectory\ndataset. In this way, the proposed FTP framework can not only leverage the strengths of\nwavelet transformation in modeling time–frequency features, but also fully exploits the\npredictive power of data-driven methods, ultimately improving both prediction accuracy and\ninterpretability.\nIn data-driven FTP methods, the normalization algorithms (e.g., min–max scaling and z-score\nnormalization) are often required to map the input variables (trajectory attributes) into a fixed\nrange or standardized distribution. This process not only accelerates network convergence and\nstabilizes training, but also prevents features with large numeric ranges from dominating the\nlearning process. However, because the value ranges of trajectory attributes vary across\ndifferent airspaces, a certain optimized model trained on one dataset may struggle to generalize\nto another dataset or scenarios without retraining.\nTo further evaluate the generalization capability of WTFTP+, we constructed an additional\ndataset using trajectory data collected from the U.S. airspace. Both WTFTP+ and the best\nbaseline (FlightBERT++) were retrained on this new dataset. Experimental results show that\nWTFTP+ consistently outperforms FlightBERT++ across all evaluation metrics. This\ndemonstrates that the proposed WTFTP+ framework can effectively adapt to different\nairspaces after retraining, and its performance advantage is maintained even under changes\nin traffic patterns, route structures, and operational environments.\nDue to space limitations in the main text, the above experiments have been included in the\nSupplementary Information. According to the journal policy, the supplementary material will be\nmade publicly available upon acceptance of the manuscript.\nThe detailed modifications are listed below.\nHope this revision can address your concerns.\nRevision:\n6. Generalizing to other airspace (Pages 8-9, Supplementary Information)\nTo validate the generalization capability of the proposed WTFTP+ framework, an additional\ndataset was collected from U.S. airspace, comprising 13473 trajectories. The region of interest\nspans [−93.82°, −66.00°] in longitude, [24.00°, 49.99°] in latitude, and [0, 11898] meters\nin altitude. The dataset was randomly split into training, validation, and testing subsets with a\nratio of 8:1:1. For a fair comparison, both WTFTP+ and the strongest baseline model,\nFlightBERT++, were retrained on this dataset. The experimental results are reported in\nSupplementary Table 2. It can be seen from the results that the WTFTP+ also achieves the\ndesired performance in this new dataset. In particular, WTFTP+ consistently yields lower errors\nin longitude and latitude prediction, which demonstrates its enhanced ability to capture\nhorizontal maneuver dynamics. Although altitude prediction remains a challenging task for both\nmodels, WTFTP+ demonstrates greater robustness and reliability over extended prediction\nhorizons. Overall, these results confirm that the WTFTP+ framework generalizes effectively to\nunseen airspace conditions and maintains superiority over the state-of-the-art baselines. It is\nindicated that the proposed technical improvements not only enhance predictive accuracy in\nthe original dataset but also have strong adaptability across diverse operational environments.\nResponse to Reviewer #4\nThe paper proposes an enhanced flight trajectory prediction framework (WTFTP+) that\nleverages time-frequency wavelet transforms and a direct multi-horizon prediction paradigm.\nThis model builds upon the authors' earlier work by introducing a time-frequency bridging\nmechanism designed to enable inter-scale feature interaction without violating wavelet\ncomponent orthogonality.\nOverall, the approach is well-explained and supported by comprehensive experimentation.\nHowever, I have the following major and minor comments:\nMajor Comment：\nComment #1: While the incorporation of wavelet transforms into trajectory prediction is\ninteresting, the core contribution appears incremental. The WTFTP+ framework primarily\nrefines existing idea, such as direct multi-horizon prediction and inter-scale attention, rather\nthan offering fundamentally new algorithmic or theoretical insights. As such, the contributions\nfall short of the novelty and depth typically expected for a journal publication.\nAuthor response: Thanks for your insightful comments, which help us to better clarify the\nnovelty and contributions of our work.\nAlthough the WTFTP+ framework retains the core idea (time-frequency modelling) from, it\naddresses several fundamental limitations that have not been effectively solved by existing FTP\napproaches, including both time-domain and frequency-domain methods.\nIn the original WTFTP, multi-horizon predictions were obtained through an iterative paradigm,\nwhich inevitably leads to error accumulation, especially for high-frequency wavelet components\nto capture local motion details. Moreover, the independent decoding of each wavelet scale,\nbased on the strict mathematical orthogonality assumption, with limited considerations on the\ninherent inter-scale dependencies that usually occur in real flight scenarios.\nThe proposed WTFTP+ introduces three key innovations to overcome these bottlenecks:\n1. Direct multi-horizon prediction paradigm: WTFTP+ replaces the iterative strategy with\na generalized direct-prediction encoder–decoder framework. This significantly mitigates\ncumulative errors over extended horizons and enables more reliable performance in real-\nworld ATC operations where long-term forecasts are critical. In addition, compared to the\nWTFTP framework, a Transformer-style architecture is designed to implement the neural\nnetwork of the WTFTP+, which enables more effective sequential modeling for multi-\nhorizon FTP tasks.\n2. Scale-aware wavelet component decoder (SAWCD): Instead of separate decoders for\neach scale, SAWCD integrates the learning process to simultaneously model intra-scale\nflight patterns and capture inter-scale correlations,while preserving the orthogonality.\n3. Time–frequency bridging mechanism: To enhance inherent dependencies within flight\npatterns across different time-frequency scales and retain orthogonality and enable inter-\nscale interaction, WTFTP+ proposes a bidirectional bridging between wavelet and time\ndomains via learnable W2T and T2W procedures. This mechanism allows the model to\nexploit intrinsic dependencies that are invisible when each wavelet scale is treated in\nisolation.\nExtensive experiments and ablation studies confirm that WTFTP+ achieves substantial and\nconsistent performance improvements over both the original WTFTP and competitive baselines,\nwith up to 40% relative reduction in mean deviation error at the 5-minute horizon. Meanwhile,\nWTFTP+ not only addresses the key limitation of WTFTP in long-horizon prediction, but also\nprovides the flexibility to incorporate ATC instructions, thereby enhancing its ability to model\ninstruction-driven maneuvers. Furthermore, in this revision, additional experiments on a new\ndataset (U.S. airspace) further verify the strong generalization capability of the proposed\nframework.\nBy jointly tackling error accumulation and cross-scale dependency modeling, i.e., two key\nissues in existing time–frequency FTP research. WTFTP+ expands the applicability of wavelet-\nbased methods from single-horizon, component-wise prediction to robust multi-horizon\nforecasting in operationally complex ATC environments. Therefore, it is believed that the\ncontributions of WTFTP+ are not incremental refinements, but rather represent a meaningful\nstep toward integrating time–frequency analysis with modern sequence modeling for the flight\ntrajectory prediction task.\nHope the above explanation can address your concerns.\nMinor Comments\nComment #2: The introduction references technical concepts like \"wavelet components\" and\n\"orthogonality of approximation subspace\" without sufficient context. Providing a brief\nexplanation or definition upon first mention would improve readability for a broader audience.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nConsidering that the definitions of these concepts are both fundamental and technically\ninvolved, a more detailed explanation has been provided in Supplementary Information Section\n3. In addition, cross-references and footnotes have been added at the first occurrence of these\nterms in the main text to improve the readability.\nHope this revision can address your concerns.\nRevision:\nIn our previous work [22], a wavelet transform-based flight trajectory prediction (WTFTP)\nframework is proposed to tackle the single-horizon FTP tasks, which leverages time-frequency\nanalysis to capture global flight trends and local motion details represented by wavelet\ncomponents (WTCs) 1. (Page 2, the 3rd paragraph of Section 1, Main Text)\nConsidering the orthogonality of approximation subspace of wavelet components, each\ndecoder performs an independent prediction process without desired feature interaction 2.\n(Page 2, the 3rd paragraph of Section 1, Main Text)\n1The descriptions of the time-frequency analysis and WTCs can be found in Supplementary Information\nSection 3. (Page 2, footnote, Main Text)\n2A detailed description of the orthogonal space formulation under multi-scale analysis is provided in\nSupplementary Information Section 3. (Page 2, footnote, Main Text)\n3. Multi-resolution analysis (Pages 2-4, Section 3, Supplementary Information)\nMulti-resolution analysis (MRA) theory [1], also known as multi-scale analysis, provides a\ncoarse-to-fine feature representation of signals while simplifying complex integration\ncalculations, as the essence of discrete wavelet transforms. In MRA, signals undergo a series\nof filtering and downsampling operations to yield feature sequences with physical significance\nat different scales. The general procedure of MRA involves decomposing a signal into two\nfundamental components: low-frequency subbands and high-frequency subbands. The former,\nobtained through low-pass filtering, constitutes the approximation coefficients; the latter,\ngenerated by high-pass filtering, represents the detail coefficients. To achieve finer frequency\nresolution, the low-frequency subbands undergo further decomposition. As decomposition\nlevels progress, MRA delivers multi-scale time-frequency representations of the signal.\nMRA describes a sequence of closed subspaces {V} , which are nested like V ⊂ V. V is\nj j∈Z j−1 j j\nthe approximation space spanned by translations and scalings of the scale function φ(x),\ndefined as:\nwhere j is the scale factor, and k is the translation factor. {φ } is standard orthogonal\nj,k j∈Z,k∈Z\nbasis of V. The larger the scale (the larger j), the higher the resolution and the more detail V\nj j\ncontains, satisfying V → L2(R) when j → +∞. Since V is a subspace of V, they share\nj j−1 j\nsubstantial common information. Due to the finer scale of V, some detail information in which\nj\ncan not be projected onto V . According to linear algebra theory, every subspace possesses\nj−1\na unique orthogonal complement. Therefore, denoting the orthogonal complement of V in V\nj−1 j\nas W , i.e., the detail space of scale j − 1, V can be described as:\nj−1 j\nwhere ⊕ denotes direct sum and the spaces are orthogonal. W is generated by translations\nj−1\nand scalings of the wavelet function ψ(x), defined as:\nwhere j is the scale factor, and k is the translation factor. Thus, the signal’s ”overall profile”\n(captured by V) and ”newly revealed details” (captured by W) constitute two entirely distinct\nj j\nand independent types of information. The approximation spaces can be iteratively\ndecomposed into coarser approximation spaces and their orthogonal complement spaces (as\nshown in Supplementary Fig. 2). By projecting the signal onto these spaces, we obtain its multi-\nscale time-frequency information.\nIn practice, with the signal projection onto the approximation space at scale j, it can be\ndecomposed using wavelet filters to obtain projections onto the approximation space and detail\nspace at scale j−1. These are referred to as the low-frequency component and high-frequency\ncomponent, respectively. By repeatedly decomposing the lowfrequency component at each\nlevel, we can obtain the high-frequency components at corresponding scales and the low-\nfrequency component at the smallest scale. Collectively, these are termed wavelet components,\nresiding in the subspace of the approximation space at scale j and mutually orthogonal to each\nother.\nIn the flight process, the overall motion trends of the aircraft are typically determined by\nhistorical transition patterns, flight plans, and real-time ATC instructions. To be specific, for a\ncertain route shown in Supplementary Fig. 3, the trajectory always evolves toward the\ndesignated waypoint (in the flight plans), indicating that the planned route can imply the overall\nflight trend. At the same time, local variations are introduced by weather conditions, aircraft\nperformance parameters, and pilot-specific operations. In other words, a flight trajectory can be\nregarded as a combination of global trends and local details. While different flights in similar\nscenarios may follow similar overall trends, their local motion details often vary significantly,\nwhich limits the prediction accuracy of the conventional FTP approaches.\nThe wavelet transform provides an intuitive solution to this challenge by decomposing time\nseries into multiscale components, thereby decoupling local details from global trends. This\ntime–frequency analysis forms the theoretical foundation of both WTFTP and WTFTP+.\nOrthogonal wavelets with compact support facilitate the identification of maneuvers such as\nturns, climbs, or descents at different scales. Moreover, local motion details can influence global\npatterns, implying inherent dependencies across multi-scale wavelet components. WTFTP+\nbuilds upon this foundation by introducing scale-aware wavelet component decoder and time–\nfrequency bridging mechanisms, which enable effective interaction among multi-scale wavelet\ncomponents without breaking orthogonality. These technical improvements allow the model to\njointly capture global motion trends and fine-grained local dynamics, thereby addressing\nperformance bottlenecks inherent in conventional FTP approaches.\nComment #3: While the evaluation is thorough, the abstract would benefit from including\nspecific performance outcomes using key metrics (e.g., prediction accuracy) to better highlight\nthe method’s advantages from the outset.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nFollowing your valuable suggestion, we have revised the abstract to explicitly report key\nexperimental results. In particular, we now highlight that WTFTP+ achieves more than 40%\nreduction in mean deviation error at the 5-minute prediction horizon compared with WTFTP,\nwhile retaining superior single-horizon performance.\nIt is believed that this revision makes the advantages of our method clearer from the outset and\nstrengthens the contribution of the work. Hope this revision can address your concerns.\nRevision:\nExperiments on real-world datasets demonstrate that the WTFTP+ not only retains the superior\nsingle-horizon prediction performance of WTFTP but also significantly improves multi-horizon\nprediction accuracy, achieving over 40% mean deviation error reduction at the 5-minute horizon\ncompared with WTFTP. (Page 1, Abstract, Main Text)\nComment #4: The caption for Figure 3 is unclear.\nAuthor response: Thanks for your insightful comments, it is very important to help us\nimprove the quality of our manuscript.\nThe caption has been modified as: a-b The ground truth and predicted trajectory along the\nlongitude and the latitude, respectively. c-j Trends and details illustrated by multi-scale WTCs\nin the WTFTP+ framework, which are obtained by the IDWT procedure of the involved WTCs\n(The involved one is indicated in parentheses of y-axis label. Not involved ones will be replaced\nby zeros). x-axis represents time steps. k The selected trajectory in the 3D earth space. Since\nthe Kalman filter-based FTP method suffers from significant error accumulations, its results are\nnot displayed for better visualization.\nIt is noted that , the corresponding Figure 3 in the original manuscript is renumbered as Figure\n4 in the revision.\nHope this revision can address your concerns.\nRevision:\nComment #5: The conclusion notes that the model underperforms in altitude prediction. It\nwould strengthen the paper if the authors proposed more concrete next steps to address this\nlimitation beyond general remarks.\nAuthor response: We sincerely thank the reviewer for pointing out that our discussion on\naltitude prediction limitations was not sufficiently concrete.\nFollowing your suggestion, we have revised the conclusion to provide more specific future\ndirections. In particular, two possible improvements are further discussed in the manuscript: (i)\ndesigning attribute-independent modeling strategies to decouple vertical and horizontal\ndynamics, (ii) incorporating additional multi-modal information such as vertical rate, aircraft\nperformance parameters, and weather conditions to enrich altitude-related features.\nIt is believed that the abovementioned potential improvements can provide a clear and feasible\nsolution to enhance the predictive capability of WTFTP+ for altitude, while maintaining its\nadvantages in multi-horizon prediction for longitude and latitude.\nThe detailed modifications are provided as follows. Hope this revision can address your\nconcerns.\nRevision:\nIn future work, the following potential directions can be further explored to address this limitation:\n1) Designing attribute-independent modeling strategies that decouple vertical dynamics from\nhorizontal motion, thereby capturing the sparse yet abrupt transition patterns of altitude more\neffectively. Meanwhile, it is essential to design an appropriate interaction mechanism to capture\nthe inherent dependencies between vertical and horizontal dynamics. A channel-attention\nmechanism may then be employed to capture inter-channel correlations, which alleviates the\nlimitations of unified modeling across attributes with distinct dynamic characteristics while\npreserving their potential dependencies.\n2) Incorporating multi-modal information such as vertical rate, aircraft performance parameters,\nand weather conditions (e.g., wind and temperature) to enrich the altitude-related input features.\nThese additional modalities provide complementary cues about vertical dynamics that are not\nfully captured by position and velocity alone. By integrating such heterogeneous data, the\nmodel can achieve a more comprehensive understanding of altitude variations, thereby\nenhancing prediction accuracy. (Page 16, the 2nd paragraph of Section Discussion, Main Text)"}, {"round": 2, "reviewer_comments": {"reviewer_1": "The reviewer appreciates the revision from all the authors. I am glad to recommend this work for publication.", "reviewer_2": "The paper has been well revised and I have no further suggestions.", "reviewer_3": "I have carefully evaluated the second revised version of the manuscript entitled \"Multi-horizon flight trajectory prediction\nenabled by time-frequency wavelet transform”. It is pertinent to mention that the authors have addressed all the major\nconcerns raised by the anonymous reviewers in earlier rounds of review and have made substantial improvements in both\npresentation and technical accuracy. I am quite satisfied with the current revision and have no further substantive comments.\nI recommend the manuscript for acceptance in its present form."}, "author_response": ""}]

Decision letter
Comments on the manuscript NCOMMS-25-44002
Multi-horizon flight trajectory prediction enabled by time-frequency
wavelet transform
The paper under review proposes a wavelet-based method for an efficient flight trajectory
prediction in the multi-horizon setting. The work appears promising, as it employs
frequency-domain analysis in the flight trajectory prediction task, in contrast to data-driven
methods. The proposed method can be viewed as an extension of the existing Wavelet
Transform-based Flight Trajectory Prediction (WTFTP) framework to circumvent its
limitations, in that WTFTP suffers from a performance bottleneck in multi-horizon prediction
due to its inherent single-horizon inference paradigm. This novel framework (WTFTP+) is
implemented with a generalized encoder-decoder architecture, in which the encoder is
employed to extract the abstract features from the observed trajectory sequence, while the
decoder is applied to generate the optimal wavelet coefficients for the predicted horizons.
In order to maintain orthogonality of the multi-scale wavelet components, in the decoder,
the orthogonal wavelet components are first bridged into the time domain for promotive
interaction, and subsequently bridged back into the wavelet-transformed domain to obtain
enhanced features of inherent flight patterns. The validity of the WTFTP+ method is
demonstrated via several experiments on a real-world flight trajectory dataset, whereas
some baseline comparisons are incorporated into the experimental analysis to provide a
comprehensive evaluation of the method.
In conclusion, I find the paper very well-written and the results quite interesting from
an empirical perspective. Though a reasonable discussion on the theoretical underpinnings
of the method can be found in the paper, an extensive theoretical performance analysis of
the WTFTP+ method is not provided. Moreover, the choice of wavelet function,
decomposition levels, and LSTM hyperparameters may significantly impact results. In this
regard, a more systematic sensitivity analysis would be beneficial. Since the model performs
well on U.S. flight data, it is unclear how well it would generalise to other airspaces or data
sources without retraining. Overall, the WTFTP+ method is somewhat more explainable in
contrast to the data-driven frameworks.
I recommend a minor revision before the article is accepted for publication in Nature
Communications.