# Gap-free GNSS-R wind field reconstruction: A neural mapping scheme and initial validation

Hao Du, Ronan Fablet, Thi Thuy Nga Nguyen, Weiqiang Li, Estel Cardellach,

Bertrand Chapron

► To cite this version:

Hao Du, Ronan Fablet, Thi Thuy Nga Nguyen, Weiqiang Li, Estel Cardellach, et al., Gap-free GNSS-R wind field reconstruction: A neural mapping scheme and initial validation. Remote Sensing of Environment, 2026, 334, pp.115218. 10.1016/j.rse.2025.115218. hal-05502409

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# Gap-free GNSS-R wind field reconstruction: A neural mapping scheme and initial validation

Hao Du<sup>a, b, c</sup> ◐, Ronan Fablet<sup>d, f</sup>, Thi Thuy Nga Nguyen<sup>d, f ◐</sup>, Weiqiang Li<sup>a, b, \*</sup> ◐,
Estel Cardellach<sup>a, b</sup> ◐, Bertrand Chapron<sup>e, f ◐</sup>

## HIGHLIGHTS

* A GNSS-R wind speed mapping scheme based on 4DVarNet is proposed.

* High-resolution, gap-free GNSS-R ocean wind maps are generated.

* Reconstructed wind maps demonstrate strong agreement with ECMWF and AMSR2 data.

* Reconstructed winds effectively capture the evolution of Super Typhoon Surigae.

* The asymmetries in Tropical Storm Kompasu are observed.

## ARTICLE INFO

Edited by Dr. Menghua Wang

**Keywords:**

* Global navigation satellite system reflectometry (GNSS-R)
* Ocean wind speed
* Spatiotemporal evolution
* Neural mapping scheme
* Tropical cyclones

## ABSTRACT

Spaceborne Global Navigation Satellite System Reflectometry (GNSS-R) has been widely demonstrated as an effective method for ocean wind speed retrieval. This study explores the feasibility of using track-wise GNSS-R wind products to generate gap-free wind fields. A physics-informed neural mapping scheme, 4DVarNet, is adapted to reconstruct wind fields. Results indicate that the root mean square errors (RMSEs) of the 1-hour, 3-hour, and 6-hour 4DVarNet winds are 1.13 m/s, 1.16 m/s, and 1.24 m/s compared to European Center for Medium-Range Weather Forecast (ECMWF) ERA5 wind products, while 1.40 m/s, 1.41 m/s, and 1.48 m/s are referred to Advanced Microwave Scanning Radiometer-2 (AMSR2) all-weather winds. Spatial and temporal error analyses further confirm the robustness of 4DVarNet-derived winds, with daily RMSEs remaining below 1.6 m/s. Error decomposition reveals discrepancies between ECMWF and GNSS-R winds, which may support future recalibration of GNSS-R wind products or enhancements to ECMWF forecasts. A case study of Super Typhoon Surigae proves that 4DVarNet winds closely align with the International Best Track Archive for Climate Stewardship (IBTrACS) track data. The reconstructed winds detect the peak intensity temporally consistent with IBTrACS data, whereas ECMWF forecasts exhibit a two-day lag. Moreover, asymmetries in Tropical Storm Kompasu are observed, with the radius of maximum wind ($R_{max}$) over the Northeast quadrant 38% larger than that over the Northwest quadrant. Despite the absence of background wind field inputs, 4DVarNet effectively learns wind patterns from ECMWF data and integrates GNSS-R observations to generate gap-free wind mappings, exhibiting strong agreement with ECMWF wind fields. The reconstruction performance is degraded at high winds due to the underestimation of referenced ECMWF ERA5 winds and the small quantity of observations. This limitation could be alleviated through denser GNSS-R observations from multiple missions such as Fengyun-3, Tianmu-1, recently launched HydroGNSS, etc., and other training references with more high winds for improving the representation of 4DVarNet at high winds.

<footnote><sup>a</sup> Institute of Space Sciences (ICE, CSIC), Carrer de Can Magrans, Cerdanyola del Vallès, 08193, Barcelona, Spain</footnote>

<footnote><sup>b</sup> Institut d'Estudis Espacials de Catalunya (IEEC), Esteve Terradas 1, Campus UPC, Castelldefels, 08860, Barcelona, Spain</footnote>

<footnote><sup>c</sup> Faculty of Earth Sciences, University of Barcelona, Martí i Franqués, S/N, Distrito de Les Corts, 08028, Barcelona, Spain</footnote>

<footnote><sup>d</sup> UMR CNRS Lab-STICC, IMT Atlantique, Technopôle Brest-Iroise CS 83818, Brest, 29238, Bretagne, France</footnote>

<footnote><sup>e</sup> Laboratoire d'Oceanographie Physique et Spatiale (LOPS), Ifremer, ZI Pointe du Diable CS 10070, Brest, 29280, Bretagne, France</footnote>

<footnote><sup>f</sup> Inria, Odyssey team, 263 avenue Général Leclerc CS 74205, Rennes, 35042, Bretagne, France</footnote>

<footnote>* Corresponding author at: Earth Observation Group, Institute of Space Sciences, Barcelona 08193, Spain.
Email address: weiqiang@ice.csic.es (W. Li).</footnote>
# 1. Introduction

Ocean surface wind, one of the essential climate variables (ECVs) (Bojinski et al., 2014), profoundly affects air-sea interactions, influencing ocean circulation and Earth's climate. In recent years, high-resolution spatiotemporal wind field monitoring and forecasting have become a key research focus. Meanwhile, advancements in space science have driven unprecedented progress in remote sensing. Satellites capable of detecting ocean winds provide enormous observations, significantly enhancing state-of-the-art numerical weather prediction (NWP) products. These wind products are generally categorized into two types: point-wise and image-based. Observations from altimeters, e.g., Sentinel-3 (Donlon et al., 2012), are point-wise, offering spatial resolutions of a few kilometers. In contrast, scatterometers, including the Advanced Scatterometer (ASCAT) (Lindsley et al., 2016), and radiometers, such as the Advanced Microwave Scanning Radiometer-2 (AMSR2) (Hihara et al., 2015), provide wide-swath coverage with spatial resolutions of tens of kilometers. Additionally, synthetic aperture radar (SAR) satellites, e.g., Sentinel-1 (Torres et al., 2012), deliver wind field measurements with spatial resolutions on the order of tens of meters, making them particularly useful for detecting tropical cyclones (TCs). Surface Water and Ocean Topography (SWOT), launched in 2022, provides both swath and nadir measurements of surface water elevation across both oceans and freshwater bodies at high resolution (Vinogradova et al., 2025). However, these satellite observations typically have a long revisit period. Due to the high cost, deploying a sufficient number of satellites to reduce this revisit period remains challenging. On the other hand, to improve NWP, especially for extreme events such as TCs, wind measurements need to achieve high spatial and temporal resolution while maintaining all-weather capability.

As an emerging remote sensing technique, Global Navigation Satellite System Reflectometry (GNSS-R) was first proposed in 1993 (Martin-Neira et al., 1993). This method utilizes reflected L-band Global Navigation Satellite System (GNSS) signals to retrieve Earth's geophysical parameters, such as ocean wind speed, ocean ice, land water, etc. The effectiveness of spaceborne GNSS-R in ocean wind speed retrieval has been widely demonstrated (e.g., Zavorotny and Voronovich, 2000; Gleason, 2013; Zavorotny et al., 2014; Foti et al., 2015), owing to its advantages of low cost, low power consumption, short revisit time, and strong penetration capability. Numerous GNSS-R satellites have been launched or are planned for deployment, such as UK Technology Demonstration Satellite-1 (Unwin et al., 2016), Cyclone GNSS (CYGNSS) (Ruf et al., 2016), Fengyun-3 (Yang et al., 2022), Spire GNSS-R (Jales et al., 2020), HydroGNSS (Unwin et al., 2021), etc. For example, CYGNSS, an eight-satellite GNSS-R mission (seven available now) designed for TC detection, has provided billions of measurements with a mean revisit time of seven hours between 38°S and 38°N since March 2017. Its official level 2 (L2) wind products feature a spatial resolution of 25 km, with a root mean square error (RMSE) of approximately 1.4 m/s for winds below 20 m/s and an RMSE uncertainty of about 17% for winds exceeding 20 m/s (Ruf et al., 2016).

Most studies on ocean wind applications focus on developing geophysical model functions (GMFs) to map the measured bistatic radar cross section (BRCS) $\sigma_0$ to wind speed. These approaches typically rely on either traditional empirical models (Clarizia et al., 2014; Clarizia and Ruf, 2016) or machine learning techniques (Asgarimehr et al., 2019; Guo et al., 2022; Du et al., 2024; Qiao and Huang, 2024). Post-processing CYGNSS data using neural networks has achieved an RMSE of about 1.0 m/s when compared to the European Center for Medium-Range Weather Forecast (ECMWF) ERA5 10-meter wind products (Du et al., 2024). Moreover, the expected benefits of assimilating CYGNSS wind retrievals were assessed by a variational analysis method (VAM) using simulated data (Annane et al., 2018). In 2020, Cardellach et al. assimilated uncalibrated observables into NWP wind fields to enhance high wind speed retrievals (Cardellach et al., 2020). Similarly, Huang et al. observed performance improvement when assimilating delay Doppler

maps (DDMs), one of the main GNSS-R observables, into ECMWF operational wind forecasts compared to scatterometer wind data (Huang et al., 2020, 2021). In 2022, Pu et al. analyzed the impact of assimilating CYGNSS L1 data into the Hurricane Weather Research and Forecasting (HWRF) model, demonstrating improved numerical simulations, and forecasts of hurricanes and precipitation (Pu et al., 2022). GNSS-R provides an unprecedented opportunity to improve ocean wind monitoring and holds significant potential for operational meteorological applications. To fully exploit its capabilities, the development of level 3 (L3) and level 4 (L4) GNSS-R wind products is essential. CYGNSS L3 gridded wind products are available on an hourly basis, however, with a quite low coverage rate for each map. Moreover, remaining calibration errors degrade the performance of GNSS-R wind products, hindering their meteorological applications. Recently, CYGNSS has published an L3 merged wind product based on a storm-centric coordinate system, with a temporal resolution of six hours and a spatial resolution of 0.1°×0.1° (Warnock et al., 2024). However, this dataset is limited to TC events and still contains observation gaps in each map.

Interpolation is a widely used technique for filling gaps between satellite observation tracks or swaths and converting gappy data into three-dimensional (3D) grids across time, latitude, and longitude. In 2008, Tenerelli et al. deduced the theoretical basis for propagating existing L-band spaceborne observations to adjacent space-time domains (Tenerelli et al., 2008). When generating gap-free L3 products, a trade-off typically exists between spatial coverage and temporal resolution. Among interpolation methods, data assimilation is extensively employed in operational meteorological products (Carrassi et al., 2018), including techniques such as optimal interpolation (OI), variational data assimilation (VAR), filter methods, etc. In ECMWF ERA5 reanalysis products, four-dimensional VAR (4DVAR) (Fisher et al., 2011) is applied to assimilate observations into the forecast model, demonstrating strong performance for wind speeds between 0–25 m/s. This method propagates the model state by assimilating observations within a predefined time window while maintaining physical consistency with the forecast model. However, the optimization process is computationally intensive and time-consuming due to the complexity of the underlying physical models.

To address this limitation, data-driven approaches have been explored, where space-time processes are reconstructed directly from data-driven representations (Brajard et al., 2020). While data-driven models show good overall performance, observations with a small amount in the total dataset, especially in some extreme events, e.g., TCs, can cause larger performance degradation compared to traditional model-driven methods. In this context, physics-informed learning has gained great attention due to its ability to integrate both physical prior knowledge and observational data (Raissi et al., 2019). These physics-informed models are designed based on realistic physical mechanisms and implemented using neural networks. Among state-of-the-art data assimilation methods, 4DVarNet is an end-to-end neural variational framework (Fablet et al., 2021), whose effectiveness and transferability have been validated in reconstructing sea surface height (Febvre et al., 2024), ocean currents (Fablet et al., 2024), and ocean color (Vient et al., 2021), etc. The model jointly trains both the representation of the dynamic prior and the gradient solver through a supervised learning strategy. This gradient solver operates based on the principle of weak-constrained 4DVAR (Fisher et al., 2011), thereby significantly improving the robustness of reconstructed gap-free fields compared to traditional interpolation methods.

In this paper, we reconstruct gap-free GNSS-R wind speed fields using 4DVarNet at 1-hour (1-h), 3-hour (3-h), and 6-hour (6-h) intervals with a spatial resolution of 0.25° × 0.25° as follows:

* First, track-wise corrected NOAA CYGNSS Version 1.2 (V1.2) products from January 2018 to September 2022 are gridded with the spatial resolution of 0.25°×0.25°, and 1-hour ECMWF ERA5 winds are
downsampled to 3-hour and 6-hour resolutions. Considering computational efficiency, we focus on the region between [0, 37°N) and [100°E, 160°E), where TCs are more active.

* Next, 4DVarNet models for ocean wind speed reconstruction are developed after hyperparameter optimization. The reconstructed wind fields are evaluated and compared to OI winds, using ECMWF and AMSR2 winds as references. Additionally, another 6-hour 4DVarNet model with the same configuration is trained to compute the interpolation error of 4DVarNet.

* Subsequently, we examine error characteristics to assess the stability of spatial and temporal performance.

* Finally, two practical case studies are conducted to further demonstrate the model's potential applications.

The rest of this paper is organized as follows: Section 2 introduces datasets and data processing methods. The details of 4DVarNet for ocean wind field reconstruction and post-processing are provided in Section 3. Section 4 presents the results and corresponding discussions. Further error analyses related to calibration and wind comparisons in TCs are explored in Section 5, and we finally conclude this work and provide an outlook in Section 6.

## 2. Datasets and data processing

In this work, NOAA CYGNSS V1.2 L2 wind speed data from January 2018 to September 2022 were selected. As indicated in Ruf et al. (2024), this product shows a smaller overall RMSE and better performance at high winds compared to the CYGNSS state-of-the-art Version 3.2 (V3.2) L2 Fully Developed Seas (FDS) wind product. The quality control (QC) criteria are detailed in Saïd et al. (2022). As one of the most used climate data, ECMWF ERA5 reanalysis products (Hersbach et al., 2018) were chosen as the reference for model development and evaluation due to their high spatiotemporal resolution and high accuracy. It provides gap-free, 1-hour, 0.25° global 10-meter wind speed data. However, its accuracy degrades quickly as wind speed increases, especially when exceeding 25 m/s. AMSR2 is a multi-purpose microwave imager that operates across multiple frequency bands and can measure various parameters, including sea surface temperature, precipitation, and wind speed. This capability enables it to provide all-weather wind speed products with an RMSE smaller than 2 m/s compared to numerical weather models (Alsweiss et al., 2021). Therefore, the latest AMSR2 L3 daily all-weather wind speed product was used as another reference. For initial validation, considering the heavy computational burden of the large 4DVarNet, we focused on a region in Southeast Asia, specifically [0, 37°N) and [100°E, 160°E), where high wind events occur more frequently.

Fig. 1 illustrates the data processing workflow. After quality control, to generate the 1-hour, 3-hour, and 6-hour gap-free ocean wind fields with a spatial resolution of 0.25°, L2 GNSS-R wind data were resampled onto all spatial grids at each time interval. The temporal resolutions were

<figure><img src="image_3.png" /><figcaption>Fig. 1. Overview of the data flow and the model development. GNSS-R L2 data were gridded in the spatiotemporal domain after QC. The dataset was divided into training, validation, and test sets according to temporal coverage. To analyze the interpolation error in 4DVarNet, an additional 4DVarNet was also developed for 6-hour data using ECMWF winds as input, which was masked (GT_mask) according to the GNSS-R observation (Obs). Performance analyses and discussions were conducted in the test set.</figcaption></figure>

1-hour, 3-hour, and 6-hour, and the spatial resolution was 0.25° × 0.25°. If multiple samples existed within a grid, the wind speed of the sample closest in time to the target interval was assigned as the grid value. The final size of gridded data is 148 × 240 for each time interval. For ECMWF winds, 3-hour and 6-hour datasets were generated by downsampling the original 1-hour data. For AMSR2 winds, there is a 0.125° shift in latitude and longitude bins compared to those of GNSS-R and ECMWF gridded winds. To correct this misalignment, bilinear interpolation was applied after temporally downsampling. Additionally, all land observations were filtered out for all three datasets. To train 4DVarNets, the data were divided into a training set (2018-2020), a validation set (2022), and a test set (2021), ensuring no temporal overlap. The training set was used for model development, while the validation set was applied to control the training iterations and tune hyperparameters. All performance analyses were conducted on the test set. Finally, the possible practical applications were discussed according to the error characteristics of reconstructed wind fields.

## 3. Methodology

In this section, we briefly introduce the principles of 4DVarNet and describe its implementation in ocean wind mapping, including patch generation, the prior dynamic module, the variational gradient solver, etc. Then, a triangular weighting strategy is applied to the reconstructed winds in temporally overlapped patches to produce the final smoothed gap-free wind field. Additionally, we define the evaluation metrics used to assess the reconstructed results.

### 3.1. 4DVarNet for ocean wind

The space-time reconstruction of the ocean wind field can be regarded as a data assimilation problem (Evensen, 2009). Typically, a state-space formulation can be established as

$$ \left\{ \begin{array}{l} \frac{\partial x(t)}{\partial t} = \mathcal{M}(x(t)) + \eta(t) \\ y_m(t) = \mathcal{H}_m(x(t)) + \epsilon_m(t) \quad \forall t \in [t_0, t_1, \dots, t_{N-1}], \forall m \in \Omega_t \end{array} \right. \quad (1) $$

where $x(t)$ is a time-dependent state within the selected space domain at $t$, $\mathcal{M}$ is the dynamical prior of $x$, $y_m$ is the observation data over observation domain $m$, $\mathcal{H}_m$ is an observation operator to map the state $x$ to observation $y_m$, and $\eta$ and $\epsilon$ are model and observation errors respectively.

Then, the reconstruction problem can be expressed as the minimization of the 4DVAR cost function within the time domain

$$ U_{\Phi}(\mathbf{x}, \mathbf{y}, \mathbf{\Omega}) = \sum_{m} \lambda_{m} \|\mathbf{y}_{m} - \mathcal{H}_{m}(\mathbf{x})\|^{2} + \gamma \|\mathbf{x} - \mathcal{H}_{\mathcal{M}}(\mathbf{x})\|^{2} \quad (2) $$

where $\mathbf{x}$ represents the states to be estimated within $[t_0, t_1, \dots, t_{N-1}]$, $\mathcal{H}_m(\mathbf{x})$ denotes the time integration of the dynamic model $\mathcal{M}$, and $\lambda_m$ and $\gamma$ are weight coefficients for observation and prior terms, respectively. 4DVarNet is a weak-constrained neural scheme, acknowledging that the prior dynamic model is not perfect (Fablet et al., 2021). Consequently, $\mathcal{H}_m(\mathbf{x})$ in Eq. (2) is reformulated as a one-step-forward flow operator that propagates state $\mathbf{x}$ from $t_i$ to $t_{i+1}$. Further details on the physics-informed 4DVarNet optimization process can be found in Fablet et al. (2021, 2024). Here we focus on the fundamental principles of 4DVarNet for ocean wind field reconstruction.

For GNSS-R ocean wind field reconstruction, the mapping scheme is shown in Fig. 2, including a prior model $\Phi$ and a gradient-based solver. The variational cost for L2 GNSS-R winds in Eq. (2) can be simplified as

$$ U_{\Phi}(\mathbf{x}, \mathbf{y}) = \|\mathbf{y} - \mathbf{x}\|_{\Omega}^{2} + \|\mathbf{x} - \Phi(\mathbf{x})\|^{2} \quad (3) $$

where $\Omega$ is the spatial sampling mask of observation, and $\mathbf{y}$ is L2 GNSS-R wind speed observation. If $\mathbf{y}$ consists of L1 GNSS-R observations, the forward operator $\mathcal{H}$ (a physical or empirical model) should be applied to $\mathbf{x}$. For the interpolation problem, there is no background field. Therefore, a prediction model $\Phi$ will also be developed to predict the next state.
<figure><img src="image_4.png" /><figcaption>Fig. 2. The end-to-end 4DVarNet scheme for GNSS-R ocean wind field reconstruction. It consists of a gradient solver and a bilinear AE. The neural scheme then minimizes the 4DVAR cost function, as defined in Eq. (3). Gap-free ECMWF winds serve as the ground truth. The state estimation from the Kth LSTM cell is the final reconstructed result.</figcaption></figure>

To minimize this cost function, an iterative gradient solver should be established. In 4DVarNet, this process is implemented by the Long Short-Term Memory (LSTM) network (Hochreiter and Schmidhuber, 1997), which exhibits exceptional capability in capturing the temporal evolution patterns in time-series data

$$ \begin{cases} \mathbf{h}(k+1) = \mathcal{R}[\nabla_x U(\mathbf{x}^k, \mathbf{y}), \mathbf{h}(k), \mathbf{c}(k)] \\ \mathbf{x}^{k+1} = \mathbf{x}^k - \mathcal{L}(\mathbf{h}(k+1)) \end{cases} \quad (4) $$

where $\mathcal{R}$ is a two-dimensional LSTM cell, $\mathbf{h}(k)$ and $\mathbf{c}(k)$ are the hidden state and cell state at iteration $k$ respectively, $\nabla_x U$ is the gradient of the 4DVAR cost (Eq. eq (3)) w.r.t the state $\mathbf{x}$, and $\mathcal{L}$ is a mapping operator

$$ \mathcal{L}(\mathbf{h}(k+1)) = \frac{1}{k+1} \mathbf{h}(k+1) + \alpha \frac{k+1}{K} \nabla_x U(\mathbf{x}^k, \mathbf{y}) \quad (5) $$

Inspired by Meta-learning (Andrychowicz et al., 2016), $\nabla_x U$ is incorporated into the operator with a weight coefficient $\alpha$, where $K$ is the total number of iterations (i.e., LSTM cells). Automatic differentiation tools in PyTorch (Paszke et al., 2017) are used to compute gradients for updating the state. Each LSTM cell is equivalent to one iteration of the gradient descent process. The variational cost at each iteration is calculated according to Eq. (3). A bilinear autoencoder (AE) is selected as the prior model $\Phi$, enabling the embedding of non-linear intrinsic features in dynamical systems (Fablet et al., 2018). As shown in Fig. 2, the concatenation of the output of Layer Conv3_1 and the linear element-wise product of Layer Conv3_2 itself is linearly mapped to the model wind field. Each hidden layer contains 128 neurons.

More details about 4DVarNet can be found in Fablet et al. (2021, 2024), Febvre et al. (2024) and the project can be found at https://github.com/CIA-Oceanix/4dvarnet-starter.

## 3.2. Training setting

We use a supervised strategy to jointly train the autoencoder prior model and LSTM gradient solver within the end-to-end 4DVarNet

framework. As expressed in Eq. (6), the loss function is defined as the weighted sum of four components

$$ \mathcal{J}(\hat{\mathbf{x}}, \mathbf{x}_{\text{true}}) = \sum_{n} \left[ w_1 \|\hat{\mathbf{x}}_n - \mathbf{x}_{\text{true},n}\|^2 + w_2 \|\nabla_{\hat{\mathbf{x}}_n} - \nabla_{\mathbf{x}_{\text{true},n}}\|^2 \right. \\ \left. + w_3 \|\hat{\mathbf{x}}_n - \Phi(\hat{\mathbf{x}}_n)\|^2 + w_4 \|\mathbf{x}_{\text{true},n} - \Phi(\mathbf{x}_{\text{true},n})\|^2 \right] \quad (6) $$

where $\hat{\mathbf{x}}$ and $\mathbf{x}_{\text{true}}$ are reconstructed winds and reference winds, respectively, $\nabla$ is the gradient operator (Sobel filter is used), and $w_1, w_2, w_3, w_4$ are weight coefficients. The last two components are used to constrain the prior dynamic model $\Phi$ with the reconstructed result and reference ECMWF wind as the input, which can improve the robustness of $\Phi$ compared to using the gappy input with a low coverage rate. The set of weights is optimized to balance the actual scale of the four error components. More details about this strategy can be found in (Fablet et al. (2021)).

The fixed-interval gridded GNSS-R wind data are divided into 3D patches with continuous time slices, i.e., wind maps. The stride between two patches is one map, meaning that each patch starts one map after the previous one, resulting in substantial overlap between consecutive patches. To handle overlapping outputs, a temporal triangular weighting strategy was applied to obtain the final reconstructed wind field, which will be explained later. The Adam optimizer with a $1 \times 10^{-4}$ learning rate and cosine annealing was used with a total of 150 epochs. The model with the best performance on the validation set was selected. Configuration differences for the 1-hour, 3-hour, and 6-hour datasets are presented in Table 1. The total number of parameters in 4DVarNet is about 1.8 million, making it a computationally efficient model compared to traditional physical interpolation models. Model training was conducted using four NVIDIA A100 80-GB GPUs.

## 3.3. Weighting strategy

As mentioned above, both the input and output are structured as a patch, representing a time window that includes multiple wind maps.
<table><caption>**Table 1**
The configurations of 4DVarNets for 1-hour, 3-hour, and 6-hour data.</caption><thead><tr><th>Settings</th><th>1-hour</th><th>3-hour</th><th>6-hour</th></tr></thead><tbody><tr><td>Train set</td><td>2019–2020</td><td>2018–2020</td><td>2018–2020</td></tr><tr><td>Patch size</td><td>[24, 148, 240]</td><td>[16, 148, 240]</td><td>[12, 148, 240]</td></tr><tr><td>Batch size</td><td>12</td><td>16</td><td>16</td></tr><tr><td>LSTM cells</td><td>10</td><td>10</td><td>12</td></tr><tr><td>Weights in J</td><td>[50, 1000, 1, 10]</td><td>[50, 1000, 1, 0]</td><td>[50, 4000, 100, 50]</td></tr><tr><td>α</td><td></td><td>0.02</td><td></td></tr><tr><td>Hidden dimension in LSTM cell</td><td></td><td>128</td><td></td></tr><tr><td>Hidden dimension in AE</td><td></td><td>128</td><td></td></tr></tbody></table>

For example, in the case of 6-hour data, the time window consists of sixteen time steps, corresponding to four days of wind data. Apart from the first and last eight wind maps, each wind map has sixteen reconstructed results, distributed across sixteen consecutive patches. To ensure a robust and smooth reconstructed wind field, a triangular weighting strategy is applied to the overlapping patches.

For convenience, we illustrate an example using 6-hour data to determine the reconstructed result of the 16th ($t_{16}$) map in the first patch as Fig. 3. Each wind map has sixteen reconstructed results distributed across sixteen consecutive patches. To ensure a smooth reconstruction, larger weights should be assigned to results closer to the center of the patch. At patch $p_8$, the 16th wind map is positioned at the center. Therefore, the triangular weight is assigned as

$$W(p_i) = 1 - \left| \frac{1 + 2\delta t - p_8}{p_8} \right| \qquad (7)$$

where $\delta t = p_i - p_8$, $p_i \in [p_1, p_2, \dots, p_{16}]$, and $p_i$ is the index of the patch. Then, the final reconstructed wind map can be calculated as

$$\hat{\mathbf{x}}_{\text{rec}} = \frac{\sum_{i}^{16} W(p_i) \hat{\mathbf{x}}_i}{\sum_{i}^{16} W(p_i)} \qquad (8)$$

In fact, a wind map is two-dimensional, and all pixels within the map are assigned the same weight based on the time offset, as defined in Eq. (8).

### 3.4. Metrics

The performance of reconstructed winds is quantitatively evaluated using RMSE, mean absolute error (MAE), bias, and relative error (RE). The observation term is two-dimensional over latitude and longitude. All metrics were calculated for each wind map and then averaged temporally.

The RMSE is a quadratic scoring rule used to measure the magnitude of the residuals between reconstructed maps and corresponding reference maps

$$\text{RMSE} = \frac{1}{N_t} \sum_{t=1}^{N_t} \sqrt{\frac{1}{N_{\text{lat}}} N_{\text{lon}}} \sum_{i=1}^{N_{\text{lat}} \cdot N_{\text{lon}}} |\hat{\mathbf{x}}(t) - \mathbf{x}_{\text{true}}(t)|^2 \qquad (9)$$

where $N_{\text{lat}} = 148$, $N_{\text{lon}} = 240$, and $N_t$ depends on the temporal coverage in performance analyses. The RMSE will give a relatively high weight to large errors. In contrast, the MAE is a linear scoring rule, assigning equal weights to all errors

$$\text{MAE} = \frac{1}{N_t} \sum_{t=1}^{N_t} \frac{1}{N_{\text{lat}} N_{\text{lon}}} \sum_{i=1}^{N_{\text{lat}} \cdot N_{\text{lon}}} |\hat{\mathbf{x}}(t) - \mathbf{x}_{\text{true}}(t)| \qquad (10)$$

To analyze the systematic difference, the bias is also defined as

$$\text{bias} = \frac{1}{N_t} \sum_{t=1}^{N_t} \frac{1}{N_{\text{lat}} N_{\text{lon}}} \sum_{i=1}^{N_{\text{lat}} \cdot N_{\text{lon}}} [\mathbf{x}_{\text{true}}(t) - \hat{\mathbf{x}}(t)] \qquad (11)$$

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** The triangular weighting strategy for the 6-hour 4DVarNet. When the map is positioned at the center of the patch, more temporal evolution information from adjacent maps can be utilized, thereby with higher reconstruction accuracy. Larger weights are assigned to the reconstructed results of $t_{16}$ for $p_8$ and $p_9$. As the time interval between a patch and $p_8$ increases, the assigned weights gradually decrease.</figcaption></figure>
<table><caption>**Table 2**
Errors of 1-hour, 3-hour, and 6-hour reconstructed winds of 4DVarNet and OI models with ECMWF winds as reference.</caption><thead><tr><th>Model</th><th>Interval</th><th>RMSE (m/s)</th><th>MAE (m/s)</th><th>Bias (m/s)</th><th>RE (%)</th></tr></thead><tbody><tr><td rowspan="3">OI</td><td>1-hour</td><td>1.57</td><td>1.09</td><td>-0.304</td><td>28.96</td></tr><tr><td>3-hour</td><td>1.55</td><td>1.09</td><td>-0.280</td><td>28.92</td></tr><tr><td>6-hour</td><td>1.58</td><td>1.11</td><td>-0.313</td><td>29.59</td></tr><tr><td rowspan="3">4DVarNet</td><td>1-hour</td><td>1.13</td><td><strong>0.83</strong></td><td>-0.021</td><td><strong>24.94</strong></td></tr><tr><td>3-hour</td><td>1.16</td><td>0.85</td><td><strong>-0.002</strong></td><td>25.12</td></tr><tr><td>6-hour</td><td>1.24</td><td>0.90</td><td>-0.065</td><td>26.03</td></tr></tbody></table>

Errors tend to be larger at higher wind speeds compared to lower ones. Therefore, a scale-independent metric provides a more intuitive performance assessment. The RE is defined as the ratio of the absolute error to the ground truth

$$ RE = \frac{1}{N_f} \sum_{t=1}^{N_f} \frac{1}{N_{lat} N_{lon}} \sum_{N_{lat}, N_{lon}} \frac{|\hat{x}(t) - \mathbf{x}_{\text{true}}(t)|}{\mathbf{x}_{\text{true}}(t) + \varepsilon} \times 100\% \quad (12) $$

The Pearson correlation coefficient was computed overall for the performance analysis at different wind ranges

$$ R = \frac{\sum_{i=1}^{N} (\hat{x}^{i} - \bar{x})(\mathbf{x}_{\text{true}}^{i} - \bar{\mathbf{x}}_{\text{true}})}{\sqrt{\sum_{i=1}^{N} (\hat{x}^{i} - \bar{x})^2} \sqrt{\sum_{i=1}^{N} (\mathbf{x}_{\text{true}}^{i} - \bar{\mathbf{x}}_{\text{true}})^2}} \quad (13) $$

where *N* is the total number of wind speed grids with a valid value, and $\bar{x}$ and $\bar{x}_{\text{true}}$ are the expectations of $\hat{x}$ and $\mathbf{x}_{\text{true}}$, respectively.

## 4. Results

This section presents the reconstructed results from 4DVarNet along with a comprehensive error analysis. The performance of the reconstructed 4DVarNet winds is analyzed using ECMWF winds, as well as the interpolation error from 4DVarNet itself. The robustness of their spatial and temporal performance is also evaluated. Additionally, AMSR2 wind speed is used as a third reference to further evaluate the performance of 4DVarNet winds.

### 4.1. Performance compared to ECMWF winds

Table 2 compares the performance of OI and 4DVarNet across all time intervals. Due to computational constraints, the observation cube was divided into patches with the size of 10×37×40 for coarsening observations in OI implementation, ensuring efficiency and reliability when calculating the weighting matrices. OI serves as a baseline method, yielding a mean RMSE of approximately 1.6 m/s compared to ECMWF winds. 4DVarNet achieves superior reconstruction performance, with improvements of 28.0%, 25.2%, and 21.5%, where the 1-hour results exhibit the highest accuracy. This indicates that 4DVarNet effectively captures dynamic evolution features when provided with higher temporal resolution input. Conversely, OI winds show no significant performance variation across different time intervals due to the same configuration of the reference lengths. Additionally, systematic biases exist between OI winds and ECMWF winds due to the incorporation of HWRF winds when developing the GMF for wind speed retrieval in this GNSS-R product. In 4DVarNet winds, these biases are nearly eliminated, benefiting from the data-driven neural scheme.

Furthermore, the performance of 4DVarNet winds is analyzed for different wind speed ranges: low (0-12 m/s), medium (12-20 m/s), and high (> 20 m/s) as summarized in Table 3. The RMSE and bias rise obviously with increasing wind speeds, especially for wind speeds larger than 20 m/s. At 0-12 m/s and 12-20 m/s, 1-hour results show the largest correlation coefficient *R*, indicating better performance. However, 3-hour reconstructed winds have a smaller RMSE and bias, as well as a larger *R* for ECMWF wind speed larger than 20 m/s. It demonstrates that

<table><caption>**Table 3**
Errors of 1-hour, 3-hour, and 6-hour reconstructed winds of 4DVarNet for low (0-12 m/s), medium (12-20 m/s), and high winds (> 20 m/s) with ECMWF winds as reference.</caption><thead><tr><th>Range (m/s)</th><th>Interval</th><th>Counts</th><th>RMSE (m/s)</th><th>MAE (m/s)</th><th>Bias (m/s)</th><th>R</th></tr></thead><tbody><tr><td rowspan="3">0-12</td><td>1-hour</td><td>222,379,727</td><td>1.09</td><td>0.80</td><td>-0.075</td><td>0.914</td></tr><tr><td>3-hour</td><td>74,073,695</td><td>1.12</td><td>0.82</td><td>-0.057</td><td>0.909</td></tr><tr><td>6-hour</td><td>37,056,135</td><td>1.20</td><td>0.87</td><td>-0.117</td><td>0.898</td></tr><tr><td rowspan="3">12-20</td><td>1-hour</td><td>9,874,165</td><td>1.82</td><td>1.37</td><td>1.139</td><td>0.628</td></tr><tr><td>3-hour</td><td>3,343,555</td><td>1.88</td><td>1.42</td><td>1.128</td><td>0.604</td></tr><tr><td>6-hour</td><td>1,652,521</td><td>1.98</td><td>1.48</td><td>1.041</td><td>0.554</td></tr><tr><td rowspan="3">>20</td><td>1-hour</td><td>131,916</td><td>5.41</td><td>4.67</td><td>4.581</td><td>0.408</td></tr><tr><td>3-hour</td><td>44,686</td><td>5.30</td><td>4.61</td><td>4.458</td><td>0.431</td></tr><tr><td>6-hour</td><td>22,312</td><td>5.51</td><td>4.73</td><td>4.489</td><td>0.369</td></tr></tbody></table>

the 3-hour resolution could be a good choice, balancing the coverage rate of each input GNSS-R wind map and temporal resolution to capture dynamic features. It can be found that the performance above 20 m/s is degraded significantly. Due to the extremely uneven distribution of wind speeds and low coverage rate, the current reconstructed results based on 4DVarNet indicate the overfitting phenomenon to some extent, which is common for deep learning. It could be resolved by fine-tuning the model using more high-wind inputs.

To assess performance across different wind conditions, comparisons between ECMWF winds and input GNSS-R winds, OI winds, and 4DVarNet winds are illustrated in Fig. 4. NOAA GNSS-R winds show good consistency with ECMWF winds, although slightly larger in the 10-20 m/s range. OI largely preserves the pattern of input winds, as indicated by the similar slopes of best-fit lines and correlation coefficients. However, 4DVarNet winds show a stronger correlation with ECMWF winds, primarily due to improvements in the 5-10 m/s range. As shown in Fig. 5, samples within this range account for approximately 57% of the total dataset. 4DVarNet tends to estimate wind speeds in this range with higher probability, leading to an underestimation of winds exceeding 15 m/s across all three datasets. This issue is a known challenge in deep learning, often referred to as deep imbalanced regression (Yang et al., 2021). Consistent with the findings from Fig. 4, the distributions of OI winds closely resemble those of the input winds. Interestingly, results indicate that the underestimation of 4DVarNet winds for winds larger than 25 m/s is alleviated. The use of a coarser temporal resolution increases the number of observations per wind map, providing more effective information for reconstructing the wind field. Additionally, a larger weight $w_2$ of the gradient loss for the 6-hour configuration can improve the model performance in cases with large wind gradients, which typically happen during TCs. This finding suggests that the performance of 4DVarNet is strongly influenced by the coverage rate of each input map, allowing it to reconstruct wind fields by leveraging observations rather than relying solely on the pattern of ground truth data. Moreover, 4DVarNet tends to preserve more of the ECMWF wind patterns in the 1-hour data due to lower observation coverage. Despite using only gappy GNSS-R winds as input, the learning-based 4DVarNet effectively mimics the assimilation of these winds into NWP model winds through its end-to-end scheme.

Fig. 6 presents the RMSE and bias curves of reconstructed winds with ECMWF winds as the reference. Below 21 m/s, 4DVarNet winds exhibit consistently smaller RMSE compared to OI winds. As discussed earlier, OI outperforms 4DVarNet for winds exceeding 21 m/s, inheriting the good performance of GNSS-R winds. However, the implementation of 4DVarNet for GNSS-R gap-filling is a supervised scheme, introducing the wind patterns of ECMWF ERA5 winds, which are normally underestimated above 20 m/s. Overfitting can further degrade performance as discussed above. Another notable result is that among 4DVarNet winds, the 3-hour dataset yields the best results. In the 1-hour dataset, observation locations are more accurate with minimal displacement but
<figure><img src="image_6.png" /><figcaption>Fig. 4. Density plots comparing ECMWF winds with gappy GNSS-R winds, OI winds, and 4DVarNet winds for 1-hour, 3-hour, and 6-hour datasets. The dashed lines represent the 1:1 reference lines, while the solid lines indicate the best-fit linear regression lines.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>Fig. 5. The probability distribution of GNSS-R, ECMWF, OI, and 4DVarNet winds.</figcaption></figure>

have limited coverage. Conversely, in the 6-hour dataset, observation coverage increases, but the relative movement of observations is more significant. The 3-hour dataset appears to provide an optimal tradeoff between relative movement and coverage rate. This phenomenon under-scores the importance of selecting the appropriate temporal resolution in L3/L4 processing, also balancing accuracy and computational efficiency. Future work could incorporate the movement of GNSS-R wind observations through a time-difference matrix. Additionally, the increasing error at higher wind speeds is not only due to the uneven data distribution but also to the rapid degradation of ECMWF wind performance at

high wind speeds (Gandoin and Garza, 2024). To further evaluate the reconstructed results, AMSR2 winds are used as an additional reference.

To intuitively compare the reconstructed results, Fig. 7 shows an example of reconstructed wind fields alongside the corresponding gappy GNSS-R winds and ECMWF winds. First, it is evident that the coverage range of GNSS-R winds varies. Fig. 8 summarizes the coverage rate after removing land pixels. Coverage rate is the percentage of grids containing CYGNSS observations. For the 1-hour dataset, fewer than 12% of pixels contain observations, whereas the 3-hour and 6-hour datasets have coverage rates of approximately 24% and 36%, respectively. As observed
<figure><img src="image_8.png" /><figcaption>Fig. 6. RMSE (left) and bias (right) curves of OI and 4DVarNet winds versus different wind speeds compared to ECMWF winds.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. One sample on October 8, 2021, at 12:00:00, of 1-hour, 3-hour, and 6-hour reconstructed wind fields by OI and 4DVarNet, as well as corresponding gappy NOAA winds and ECMWF winds. Bias maps of OI and 4DVarNet winds are also computed with reference to ECMWF wind maps. It should be noted that 4DVarNet winds are generated using gappy maps within the time window, not just the one shown here.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Distribution of GNSS-R observation coverage rates at 1-hour, 3-hour, and 6-hour intervals. Coverage rate is defined as the percentage of grids containing CYGNSS observations, with land areas excluded. For example, a coverage rate of 3% corresponds to a GNSS-R observation map in which 0–6% of grids are observed.</figcaption></figure>

in Fig. 7, 4DVarNet produces smoother wind fields compared to OI. With fewer pixels for the 1-hour dataset, the reconstructed field more closely resembles ECMWF wind fields in both magnitude and gradient, whereas there are some isolated clusters in OI results according to the bias maps. 4DVarNet can dig out large-scale wind field characteristics, benefiting from its end-to-end neural scheme. However, there are some misalignments in the orientations of reconstructed cyclones. Limited by the current CYGNSS observations, tracks are still sparse and anisotropic due to the fixed orbit inclination angle. More observations, such as 22-satellite Tianmu-1, can provide denser input mapping to further refine the reconstructed wind patterns (Liu et al., 2025). In this framework, the prior model learns the dynamic features, while the gradient solver optimizes the overall solution by minimizing the 4DVAR cost function. Moreover, 4DVarNet reconstructs one year of 1-hour wind fields within 10 minutes, more than ten times faster than OI owing to efficient matrix multiplications.

As the number of observations increases in the 3-hour and 6-hour datasets, some 'track-wise' errors become apparent in the 4DVarNet wind fields. As previously discussed, GNSS-R winds are derived from a different wind reference, leading to greater divergence from ECMWF winds as wind speeds increase. Furthermore, observation displacement becomes more pronounced in the 6-hour dataset, amplifying positioning errors during reconstruction, while ECMWF winds maintain a 1-hour resolution across all time intervals. This finding could contribute to recalibrating GNSS-R wind products, and we will discuss it in Section 4.2.

## 4.2. Decomposition of reconstruction errors

As shown in Fig. 9, the biases of 4DVarNet winds are nearly independent of the input GNSS-R for wind speeds below 20 m/s, particularly in the 1-hour dataset. Furthermore, as wind speed increases, mean
<figure><img src="image_11.png" /><figcaption>Fig. 9. Comparison between the bias of reconstructed wind fields and GNSS-R input winds.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Fig. 10. Spatially averaged RMSE and bias of OI and 4DVarNet winds and averaged ECMWF winds for 1-hour, 3-hour, and 6-hour data.</figcaption></figure>

4DVarNet winds tend to be larger than ECMWF winds, which is inherited from the improvement of NOAA V1.2 wind data at high wind speeds. For the 3-hour dataset, the bias becomes increasingly positive for wind speeds exceeding 20 m/s. According to Table 1, the prior cost component of the reference winds is set to zero in Eq. (6). During training, the autoencoder prior model may adopt a more conservative approach, producing wind estimates closer to the mean value of the inputs, thereby increasing the positive bias at higher wind speeds. For the 6-hour dataset, pixels with a larger coverage rate assign more weight to observations within the 4DVarNet framework, leading to a more negative bias at high wind speeds.

We further evaluate the spatial distribution of errors. Fig. 10 illustrates the gridded average RMSE and bias of OI and 4DVarNet winds, along with the average ECMWF wind speed. It can be observed that 4DVarNet winds exhibit smaller errors compared to OI winds, especially in the region between [35°N, 37°N). Notably, the 1-hour reconstructed wind fields show the smallest spatial errors, which can be attributed to their higher temporal resolution. According to the design of CYGNSS satellites, the orbit transitions from ascending to descending over this region, resulting in temperature variations and increased calibration errors. Additionally, the availability of surrounding spatial correlation information is limited at the edges of wind maps. As a result, OI exhibits significantly degraded performance in this region, as well as in coastal regions. In the L2 processing of CYGNSS data, measurements within 25 km of the coastline are filtered out. Furthermore, mean ECMWF wind speeds in these regions are slightly higher than those observed elsewhere, contributing to greater uncertainty according to Fig. 6. These errors are largely mitigated in 4DVarNet winds, particularly in the 1-hour dataset, which benefits from the learning-based and data-driven scheme. The autoencoder prior model effectively captures

spatial patterns, while the gradient solver models temporal evolution. Nevertheless, positive biases persist in coastal regions due to the absence of data across all maps.

Next, we perform a time-series analysis of daily averaged RMSEs and biases. To identify trends, a 7-day rolling average is applied, as shown in Fig. 11. The 1-hour 4DVarNet winds have a consistently lower RMSE compared to the other datasets, with a maximum RMSE of less than 1.6 m/s. A negative bias is observed in the 6-hour 4DVarNet winds, attributed to the increasing number of GNSS-R observations and relative movements, as previously discussed. Additionally, due to the CYGNSS orbit period, errors exhibit a periodic increase corresponding to variations in coverage rate. The period length is similar across all three datasets due to a comparable number of observed pixels within the 7-day window, whereas the RMSE magnitude is higher in the 6-hour dataset.

There are three components in the error of 4DVarNet winds as Eq. (14)

$$ \sigma_{\text{rec/ECMWF}}^2 = \sigma_{\text{interp}}^2 + \sigma_{\text{GNSS-R}}^2 - \sigma_{\text{ECMWF}}^2 \quad (14) $$

where $\sigma_{\text{rec/ECMWF}}^2$ represents the mean square error (MSE) between reconstructed winds and ECMWF winds, $\sigma_{\text{interp}}^2$ denotes the interpolation error, and $\sigma_{\text{GNSS-R}}^2$ and $\sigma_{\text{ECMWF}}^2$ are the actual errors of GNSS-R winds and ECMWF winds with respect to ground truth, respectively. To determine $\sigma_{\text{interp}}^2$, we developed another 4DVarNet model using 6-hour ECMWF winds as input, masked by GNSS-R observations. $\sigma_{\text{rec/ECMWF}}^2$ has been computed in the 6-hour 4DVarNet model. Thus, their difference indicates the discrepancy between $\sigma_{\text{GNSS-R}}^2$ and $\sigma_{\text{ECMWF}}^2$.

As shown in Fig. 12, the MSE of 4DVarNet winds (GT_mask) is consistently lower than that of 4DVarNet winds (Obs), indicating that GNSS-R winds have a larger variance than ECMWF winds. The Pearson
<figure><img src="image_13.png" /><figcaption>Fig. 11. Daily averaged RMSE and bias for 1-hour, 3-hour, and 6-hour data. Rolling averaging is applied with a 7-day window.</figcaption></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /><figcaption>Fig. 12. Daily averaged MSEs of 4DVarNet winds with the input of GNSS-R winds (Obs) and ECMWF winds with the same track mask (GT_mask), as well as their differences (top), and the normalized counts of daily observations (bottom). Both models are trained using the 6-hour dataset. A 3-day rolling average was applied. The correlation coefficient between the count and the MSE of GT_mask winds is −0.849, between the count and the MSE of Obs winds is −0.485, and between the count and their difference is 0.553.</figcaption></figure>

correlation coefficient between the MSE of 4DVarNet winds (GT_mask) and the pixel counts is −0.849, confirming that interpolation error is highly correlated with input sparsity. However, when using GNSS-R observations as input, the correlation decreases to −0.485, as σ²<sub>GNSS-R</sub> − σ²<sub>ECMWF</sub> is influenced by additional factors beyond pixel count. The mean value of MSE differences is about 0.5 m/s, showing a moderate correlation (0.553) with pixel count, which can be attributed to systematic biases between GNSS-R and ECMWF winds. Certain noticeable errors could be used to recalibrate GNSS-R wind products. Simultaneously, with the continuous improvement of GNSS-R observations and ocean wind retrievals, it can also contribute to analyzing errors in ECMWF winds, particularly in TC cases. We will discuss these cases in Section 5.

### 4.3. Validation with a third reference: AMSR2

To further validate reconstructed wind fields, AMSR2 wind speed products are also used as a reference, and the performances of 1-hour, 3-hour, and 6-hour OI and 4DVarNet winds are summarized in Table 4.

<caption>**Table 4**</caption>

Errors of 1-hour, 3-hour, and 6-hour reconstructed winds of 4DVarNet and OI models as well as gappy GNSS-R input NOAA V1.2 winds and ECMWF winds with AMSR2 winds as reference.

<table><caption>**Table 5**</caption><thead><tr><th>Interval</th><th>Wind</th><th>RMSE (m/s)</th><th>MAE (m/s)</th><th>Bias (m/s)</th><th>RE (%)</th></tr></thead><tbody><tr><td rowspan="4">1-hour</td><td>NOAA</td><td>1.30</td><td>0.97</td><td>0.089</td><td>24.12</td></tr><tr><td>OI</td><td>1.62</td><td>1.17</td><td>0.103</td><td>30.18</td></tr><tr><td>4DVarNet</td><td>1.40</td><td>1.00</td><td>0.379</td><td>32.27</td></tr><tr><td>ECMWF</td><td>1.42</td><td>1.01</td><td>0.411</td><td>27.27</td></tr><tr><td rowspan="4">3-hour</td><td>NOAA</td><td>1.32</td><td>0.939</td><td>0.096</td><td>29.58</td></tr><tr><td>OI</td><td>1.59</td><td>1.10</td><td>0.093</td><td>28.27</td></tr><tr><td>4DVarNet</td><td>1.41</td><td>0.97</td><td>0.361</td><td>28.79</td></tr><tr><td>ECMWF</td><td>1.43</td><td>0.98</td><td>0.399</td><td>25.45</td></tr><tr><td rowspan="4">6-hour</td><td>NOAA</td><td>1.38</td><td>0.97</td><td>0.067</td><td>26.78</td></tr><tr><td>OI</td><td>1.63</td><td>1.12</td><td>0.087</td><td>27.79</td></tr><tr><td>4DVarNet</td><td>1.48</td><td>1.01</td><td>0.300</td><td>27.98</td></tr><tr><td>ECMWF</td><td>1.55</td><td>1.06</td><td>0.417</td><td>26.70</td></tr></tbody></table>

Errors of 1-hour, 3-hour, and 6-hour reconstructed winds of 4DVarNet and ECMWF winds for 0–17 m/s and > 17 m/s with AMSR2 winds as reference. Wind speed range is determined using ECMWF ERA5 winds.

<table><thead><tr><th>Range (m/s)</th><th>Interval</th><th>RMSE (m/s)</th><th>MAE (m/s)</th><th>Bias (m/s)</th><th>R</th></tr></thead><tbody><tr><td rowspan="3">4DVarNet</td><td>0-17</td><td>1-hour</td><td>1.39</td><td>0.93</td><td>0.366</td><td>0.909</td></tr><tr><td></td><td>3-hour</td><td>1.40</td><td>0.95</td><td>0.350</td><td>0.904</td></tr><tr><td></td><td>6-hour</td><td>1.47</td><td>1.00</td><td>0.288</td><td>0.896</td></tr><tr><td rowspan="3">> 17</td><td>1-hour</td><td>6.58</td><td>4.98</td><td>4.83</td><td>0.532</td></tr><tr><td>3-hour</td><td>6.13</td><td>4.69</td><td>4.46</td><td>0.504</td></tr><tr><td>6-hour</td><td>6.82</td><td>5.07</td><td>4.76</td><td>0.425</td></tr><tr><td rowspan="4">ECMWF</td><td>0-17</td><td>1-hour</td><td>1.44</td><td>0.96</td><td>0.407</td><td>0.904</td></tr><tr><td></td><td>3-hour</td><td>1.44</td><td>0.96</td><td>0.396</td><td>0.902</td></tr><tr><td></td><td>6-hour</td><td>1.55</td><td>1.06</td><td>0.413</td><td>0.890</td></tr><tr><td></td><td>1-hour</td><td>4.46</td><td>2.88</td><td>1.80</td><td>0.670</td></tr><tr><td rowspan="3">> 17</td><td>3-hour</td><td>4.23</td><td>2.79</td><td>1.64</td><td>0.628</td></tr><tr><td></td><td>6-hour</td><td>4.73</td><td>3.05</td><td>1.94</td><td>0.635</td></tr></tbody></table>

Similar to the results in Table 2, the RMSEs of 4DVarNet winds are reduced by 13.6%, 11.3%, and 9.2%, respectively, compared to OI winds, although OI winds have a lower bias. It can be observed that L2 GNSS-R winds show lower deviation from AMSR2 winds compared to ECMWF winds. OI generates wind fields purely based on GNSS-R winds, whereas 4DVarNet ‘assimilates’ GNSS-R winds into ECMWF model winds. Depending on the reference dataset, 4DVarNet winds show an error pattern more similar to that of ECMWF winds. However, the overall error of 4DVarNet winds is slightly lower than that of ECMWF winds. Table 5 summarizes the errors of 4DVarNet and ECMWF winds at 0–17 m/s and > 17 m/s with reference to AMSR2 winds. The range is determined according to ECMWF winds, as AMSR2 winds tend to be overestimated in the 5–15 m/s range (Hihara et al., 2015). Below 17 m/s, the RMSE and bias of 4DVarNet winds are 3.5% and 10.0%
<figure><img src="image_16.png" /><figcaption>Fig. 13. Density plots comparing AMSR2 winds with gappy GNSS-R winds, 4DVarNet winds, and ECMWF winds for 1-hour, 3-hour, and 6-hour datasets. The dashed lines represent the 1:1 reference lines, while the solid lines indicate the best-fit linear regression lines.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>Fig. 14. RMSE (left) and bias (right) curves of OI, 4DVarNet, and ECMWF winds versus different wind speeds compared to AMSR2 winds.</figcaption></figure>

lower than those of ECMWF winds for 1-hour data. The correlation co-
efficient is about 0.9, indicating a strong linear relationship between
4DVarNet winds and AMSR2 winds. For winds above 17 m/s, however,
the performance of 4DVarNet winds degrades more obviously compared
to ECMWF winds, which is mainly attributed to overfitting as discussed
above.

Fig. 13 compares AMSR2 winds with input GNSS-R winds, 4DVarNet winds, and ECMWF winds. More points are centered around the 1:1 line for GNSS-R winds, while some points in ECMWF winds are underestimated. This characteristic is inherited by 4DVarNet winds, where the uneven distribution further degrades performance for wind speeds larger than 20 m/s, as also observed in Fig. 14. 4DVarNet winds show
<figure><img src="image_18.png" /><figcaption>Fig. 15. 4DVarNet wind maps with a 7° × 7° region during Super Typhoon Surigae. The gray line is the best track data from IBTrACS, where the center of the TC is marked out in each figure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

lower RMSE between 2 m/s and 11 m/s, but the error increases rapidly for wind speeds exceeding 20 m/s, although the 1-hour winds perform the best. Conversely, OI winds demonstrate the lowest RMSE and bias above 20 m/s, better than ECMWF winds. As previously explained, OI retains the error patterns of GNSS-R winds. To address this issue, fine-tuning (Hu et al., 2022) can be applied to the trained 4DVarNet model using datasets with more high-wind observations. Another approach is to use wind speed references with more high-wind observations, such as Marine Atmosphere eXtreme Satellite Synergy (MAXSS) products, including multiple mission data (Portabella et al., 2022).

In conclusion, the reconstructed gap-free 4DVarNet wind fields align well with ECMWF and AMSR2 wind fields, despite the GNSS-R observation coverage rate being less than 12% for the 1-hour dataset. The bias remains nearly independent of the input GNSS-R wind speed, and the performance of 4DVarNet winds is stable both spatially and temporally.

## 5. Discussions

In this section, we select the results between April 16, 2021, and April 24, 2021, during Super Typhoon Surigae, which is the strongest April

typhoon ever observed in the western Pacific. The peak intensity of 10-minute sustained winds reached 220 km/h, i.e., about 61.1 m/s (Knapp et al., 2010). It formed south of Micronesia, passed by Philippine waters, and dissipated in the open North Pacific Ocean. The International Best Track Archive for Climate Stewardship (IBTrACS) product (Knapp et al., 2010) is used in this analysis. Additionally, the radius ($R_{max}$) of maximum wind ($V_{max}$) in 4DVarNet winds during the early stage of Tropical Storm Kompasu is analyzed to explore its capability of detecting the asymmetry.

### 5.1. Performance in super typhoon surigae

Fig. 15 presents the reconstructed wind fields using the 1-hour dataset. Similar plots of input GNSS-R, OI, and ECMWF winds are provided in the supplementary file. Although with a lower spatial resolution, the overall patterns and magnitudes of 4DVarNet winds are consistent with those of ECMWF winds as shown in Fig. S3. Moreover, the location of maximum reconstructed wind speeds in each map closely aligns with the TC center from IBTrACS data. According to Fig. S1, results with relatively larger misalignments are primarily due to sparse
<figure><img src="image_19.png" /><figcaption>**Fig. 16.** The time series of normalized maximum wind speeds from input GNSS-R, 4DVarNet, ECMWF, AMSR2, and IBTrACS winds.</figcaption></figure>

input observations, such as on 2021-04-20T12:00:00 and 2021-04-22T12:00:00, etc. On April 18, the reconstructed wind field reached its peak intensity of approximately 32 m/s. Additionally, the cyclone area with high wind speeds expanded, suggesting the onset of an eyewall replacement cycle. Subsequently, on April 19, the maximum wind speed decreased while the cyclone maintained its size as shown in Fig. 15, corresponding to the replacement of the inner eyewall by the outer eyewall. Due to the upwelling of cooler waters beneath the slow-moving typhoon (Zhang and Han, 2024) and the entrainment of dry air, Surigae gradually weakened and eventually dissipated. However, the track-wise wind pattern can still be observed in 4DVarNet winds, inherited from the input CYGNSS winds.

AMSR2 provides higher maximum wind speeds (about 70 m/s), with magnitudes closely matching IBTrACS data, except for two points with lower wind speeds due to AMSR2 swaths being located more than 2° from the TC center. The magnitude of the reconstructed winds is lower than IBTrACS maximum sustained wind speed. This discrepancy arises because IBTrACS data for Surigae is based on 10-minute averaged wind speed (Knapp et al., 2010), whereas 4DVarNet winds have an actual spatial resolution coarser than 25 km, representing measurements larger than a 10-minute averaging period. Additionally, differences in spatial resolution further contribute to this discrepancy. To analyze the tendency of Surigae, the normalized maximum wind speed of different wind data is shown in Fig. 16. Sharp increases in 4DVarNet winds correspond to GNSS-R observed high winds. Despite being constrained by the coverage rate, 4DVarNet efficiently maps GNSS-R observations through spatial and temporal correlations. Overall, 4DVarNet retrieves the development of Surigae, detecting the peak intensity on April 18, which is consistent with IBTrACS and AMSR2 data. However, wind speed from ECMWF reaches its maximum on April 20, a two-day delay compared to IBTrACS and AMSR2 data. Similar to OI winds, some oscillations occur between April 21 and April 24, which could be caused by the periodic overpass of CYGNSS tracks to the storm area as shown in Fig. S1. Additionally, in NOAA CYGNSS V1.2 products, CDF matching is applied to boost ECMWF winds using HWRF winds in the high wind regime, and HWRF winds are used as the reference to generate the model $σ_0$ whenever they are available (Saïd et al., 2022). Wind speeds between 20–30 m/s fall within a transition range from fully developed seas (FDS) to young seas with limited fetch (YSLF), where GNSS-R wind speed retrievals exhibit a different error pattern. As shown in Fig. 6, the RMSE tends to be flat from 19 m/s, showing an improved performance compared to other GNSS-R FDS wind products. For maps where observation tracks intersect the TC, reconstructed wind speeds increase. Conversely, for maps with fewer observed pixels, 4DVarNet assigns greater weight to learned ECMWF wind patterns, leading to lower wind speed estimates.

These results introduce a new approach for TC detection, with potential applications in estimating TC structures, such as the radius of maximum winds. Regarding the wind magnitude, future work will focus on datasets with sufficiently high-wind reference maps designed for TC cases, as well as developing a multimodal 4DVarNet incorporating both GNSS-R observations and NWP wind fields.

## 5.2. Gradient feature of reconstructed wind fields

Beyond the wind magnitude, gradient features are more obvious in TCs due to intense wind shear in the eyewall. Fig. 17 presents a case study of Surigae. Additionally, to analyze track-wise error, the 6-hour results, which include more observation tracks per map, are examined here. The maximum wind speed recorded in IBTrACS data is 54.0 m/s, while ECMWF winds report a lower value of 29 m/s. The reconstructed wind fields from 4DVarNet (GT_mask) show a wind pattern similar to the reference, albeit with reduced spatial resolution and lower magnitude in the eyewall. This limitation is inherent to the performance of the 4DVarNet scheme. According to the conclusions from Fig. 12, $σ_{interp}^2$ is highly related to the coverage rate. Future improvements can be achieved by expanding 4DVarNet and incorporating additional GNSS-R observations from other missions. For 4DVarNet (Obs) winds, the maximum magnitude increases by 73.8%, and the counterclockwise feature of the TC can be observed. However, due to relative movements between observation tracks, the reconstructed field appears less smooth, especially in the cyclone region, where ECMWF winds deviate more significantly from GNSS-R observations. OI (Obs) winds seem smoother than 4DVarNet winds and recover more high wind speeds, benefiting from the isotropic Gaussian assumption. However, non-physical wind patterns over the region larger than 129°E can be observed due to the limited interpolation window and no consideration of the gradient information.

To intuitively analyze this error, gradient maps are plotted in Fig. 18. Overall, 4DVarNet effectively reconstructs wind gradients by introducing the gradient loss component. However, underestimation still exists, primarily due to the highly uneven distribution of wind speed gradients, as shown in Fig. 18 (right). For 4DVarNet (Obs) winds, gradient magnitudes are improved, and the eye of Surigae is clearly detected and consistent with the IBTrACS predicted typhoon center. From Fig. 18 (middle), some track-wise gradient patterns correspond to input GNSS-R observation tracks. However, these error patterns disappear in 4DVarNet (GT_mask) winds. This remaining error can be attributed to the selection of reference winds during training models. As discussed in Section 4.2, it highlights discrepancies between ECMWF winds and NOAA CYGNSS v1.2 wind products. At low-to-medium wind speeds, these track-wise errors could be outliers, which may aid in recalibrating GNSS-R wind products. Above all, relative movements within the time interval should be compensated by some techniques, such as time-weighted averaging, physics-based oversampling (Sun et al., 2018), etc. These improvements could be particularly beneficial for the CYGNSS Science Data Record (SDR) wind products, addressing calibration issues such as inter-satellite biases and radiometric interference.

## 5.3. Asymmetries in tropical storm kompasu

Tropical Storm Kompasu, which formed on October 7, 2021, and dissipated on October 14, 2021, is characterized by its large size and pronounced asymmetry. Although its maximum sustained wind speed was only about 30 m/s, the storm caused extensive damage due to its broad footprint. As shown in Fig. 19, the asymmetry is obvious on October 8 and 9. The wind speeds over the Northeast (NE) and Southwest (SW) quadrants are higher and evolve faster than those over the Southeast (SE) and Northwest (NW) quadrants. Moreover, the eye-like structure of Kompasu is clearly detected, whose location and range are consistent with those in ECMWF ERA5 winds as shown in Fig. S4. Limited by current spatial resolution, it is difficult to detect fine-scale structures of the storm and its counterclockwise rotation.
Typhoon SURIGAE - Wind Speed Comparison
2021-04-18T12:00:00 UTC

<figure><img src="image_20.png" /><figcaption>**Fig. 17.** One sample on April 18, 2021, at 12:00:00, of the 6-hour input GNSS-R winds, 4DVarNet (GT_mask) winds, 4DVarNet (Obs) winds, OI (Obs) winds, ECMWF winds, and AMSR2 winds. The maximum wind speeds are about 45 m/s, 23 m/s, 41 m/s, 41 m/s, 29 m/s, and 70 m/s, respectively.</figcaption></figure>

Typhoon SURIGAE - Wind Gradient Comparison
2021-04-18T12:00:00 UTC

<figure><img src="image_21.png" /><figcaption>**Fig. 18.** One sample on April 18, 2021, at 12:00:00, of the gradient of 4DVarNet (GT_mask) winds (left), 4DVarNet (Obs) winds (middle), and ECMWF winds (right). In the left and middle figures, track-wise errors can be observed, especially related to the input wind speed. The color of points denotes wind speed similar to Fig. 17. For convenience, the gradient magnitudes of ECMWF winds are constrained to the range of [0, 4], while they can reach up to 9.3 m²/s² within cyclone regions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
<figure><img src="image_22.png" /><figcaption>**Fig. 19.** 4DVarNet wind maps during Tropical Storm Kompasu from October 8 to October 9, 2021. The gray line is the best track data from IBTrACS, where the center of the TC is marked out in each figure. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

To quantitatively analyze the asymmetry of Kompasu, 4DVarNet winds are mapped to a storm-centric coordinate (radius and azimuth) using IBTrACS data as the center. For example, Fig. 20 shows the detected $V_{max}$ and $R_{max}$ on 2021–10-09T06:00:00, respectively. Among the four quadrants, $V_{max}$ and $R_{max}$ are detected in the NE quadrant with 17 m/s and 587.5 km. $R_{max}$ is 362.5 km in the NW quadrant, 38% shorter than the NE $R_{max}$. Overall, $R_{max}$ of the NE and SW quadrants is larger than that of the NW and SE quadrants. According to IBTrACS track data in Fig. 19, Kompasu had a northwestward motion, fighting easterly wind shear. This asymmetry prevents the consolidation of a tight eyewall, yielding a slow intensification. The findings further demonstrate the capability of using these gap-free GNSS-R ocean wind maps for monitoring tropical storms or typhoons, especially when the radius of the footprint is larger than 100 km.

## 6. Conclusions

Spaceborne GNSS-R has provided unprecedented Earth observations with high temporal resolution and quasi-global coverage, showing great

potential to improve ocean wind speed retrieval and forecasting. To obtain gap-free ocean wind fields from observations, interpolation methods such as OI are commonly used. However, the computational efficiency constrains the spatiotemporal range of reconstruction. In this study, we utilized a neural data mapping scheme, 4DVarNet, to reconstruct gap-free ocean wind fields using NOAA CYGNSS L2 wind speed products. Wind maps were reconstructed at 1-hour, 3-hour, and 6-hour intervals, achieving overall RMSEs of 1.13 m/s, 1.16 m/s, and 1.24 m/s, respectively, compared to ECMWF ERA5 wind speed products, despite measurement coverage rates of less than 36%. Compared to AMSR2 all-weather winds, the RMSEs of 4DVarNet winds were 1.40 m/s, 1.41 m/s, and 1.48 m/s, showing a similar error pattern to ECMWF winds. For both reference datasets, 4DVarNet outperforms OI at low-to-medium winds, producing reconstructed wind fields that better conform to physical wind patterns observed in ECMWF winds. Additionally, results confirm that 4DVarNet winds maintain stable spatial and temporal performance. Through the time-series analysis, we found that the interpolation error of 4DVarNet is correlated with the coverage rate of GNSS-R winds. Furthermore, differences in error patterns between GNSS-R and ECMWF
<figure><img src="image_23.png" /><figcaption>**Fig. 20.** Radius and azimuth analysis of maximum 4DVarNet winds under a storm-centric coordinate on 2021-10-09T06:00:00. The azimuth 0 represents the North. The resolutions of radius and azimuth are 25 km and 30°, respectively. The left two plots show the detected $R_{max}$ in the polar coordinate and the Cartesian coordinate, where different colors indicate. The right four plots summarize the wind speeds against radii over the NE, SE, SW, and NW quadrants. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

winds suggest potential applications for recalibrating GNSS-R wind products or improving ECMWF wind estimates.

However, determined by the inherent limitations of deep learning, 4DVarNet showed degraded performance for wind speeds exceeding 20 m/s, where GNSS-R winds outperformed ECMWF winds when compared to AMSR2 winds. Nevertheless, a case study of Super Typhoon Surigae demonstrates that its evolution was effectively captured in the reconstructed 4DVarNet winds, which remained consistent with IBTrACS data. The time of reaching peak intensity was successfully detected, whereas ECMWF winds reached their maximum two days later. The maximum wind speeds of OI winds were typically higher than those of 4DVarNet, likely due to the unsupervised nature of OI and the better performance of GNSS-R winds at high wind speeds. For the 1-hour dataset, the low coverage rate led to fluctuations in OI winds, which were greatly reduced by 4DVarNet, benefiting from its larger patch size and end-to-end learning strategy. Moreover, 4DVarNet winds successfully detected the asymmetries in Tropical Storm Kompasu with $R_{max}$ of 587.5 km over the NE quadrant on 2021-10-09T06:00:00.

Ultimately, this study explores possible applications of GNSS-R wind observations and demonstrates the feasibility of reconstructing global ocean wind fields. Future work will focus on compensating for the relative movements of GNSS-R observations within the time interval and developing a multimodal 4DVarNet to incorporate GNSS-R winds into NWP winds on a global scale with stronger GPU resources. More validation in different regions will be performed to further demonstrate the generalization. Additionally, datasets dedicated to high-wind cases will be developed to fine-tune 4DVarNet and improve the accuracy of reconstructed magnitudes and gradients. Increasing spaceborne GNSS-R missions will provide denser observation maps to improve the reconstruction performance, especially for the more complicated high-wind cases, where more available inputs are necessary to mimic the space-time evolution. Moreover, the spatial resolution of spaceborne GNSS-R observations can be improved to enhance the sensitivity at high winds by using a wide-band antenna and some advanced processing, such as stare processing.

### CRediT authorship contribution statement

**Hao Du:** Writing – review & editing, Writing – original draft, Visualization, Validation, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization.
**Ronan Fablet:** Writing – review & editing, Writing –

original draft, Supervision, Software, Methodology, Investigation, Funding acquisition, Formal analysis, Conceptualization.
**Thi Thuy Nga Nguyen:** Writing – review & editing, Writing – original draft, Supervision, Software, Methodology, Investigation, Formal analysis, Conceptualization.
**Weiqiang Li:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Funding acquisition, Formal analysis, Conceptualization.
**Estel Cardellach:** Writing – review & editing, Writing – original draft, Supervision, Methodology, Funding acquisition, Conceptualization.
**Bertrand Chapron:** Writing – original draft, Supervision, Methodology, Formal analysis, Conceptualization, Writing – review & editing.

### Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

### Acknowledgments

This work is partially supported by Grants PID2021-126436OB-C22 and the Spanish program Unidad de Excelencia María de Maeztu CEX2020-001058-M, all funded by MCIN/AEI/10.13039/501100011033 with contributions by “European Union Next Generation EU/ PRTR”, and is also supported by Grant 20215AT007 funded by Spanish National Research Council. This work of Hao Du is supported in part by the China Scholarship Council (CSC) through a State Scholarship Fund (No.202206270011). This work is also partially funded by the French ANR, as part of the OceaniX (ANR-19-CHIA-0016); it is also supported by CNES (French Space Agency); and EU Horizon Europe project EDITO Model Lab (Grant 101093293). It benefits from resources from CPER AIDA GPU cluster supported by the Regional Council of Brittany and FEDER. This work is also supported by ESA Marine Atmosphere eXtreme Satellite Synergy (MAXSS) 4000132954/20/I-NB. The authors are grateful to the public NOAA CYGNSS Level 2 Science Wind Speed 25-km Product Version 1.2 (available at doi: https://doi.org/10.5067/CYGNN-22512), ECMWF ERA5 reanalysis wind products (available at doi: https://doi.org/10.24381/cds.adbb2d47), AMSR2 L3 daily all-weather wind products (available at doi: https://doi.org/10.56236/RSS-bq), and International Best Track Archive for Climate Stewardship (IBTrACS) tropical cyclone datasets (available at doi: https://doi.org/10.25921/82ty-9e16). W. Li and E.
Cardellach are members of the CYGNSS mission's extended science team.

## Appendix A. Supplementary data

Supplementary data for this article can be found online at doi:10.1016/j.rse.2025.115218.

## Data availability

Data will be made available on request.

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