An Elevation-Coupled Multivariate Regression Model for GNSS-Based FY-4A Precipitable Water Vapor

Abstract

The measurement of atmospheric moisture content is essential for the monitoring of severe weather events and hydrological studies. This paper proposes a multivariate linear regression correction model that integrates elevation data with Global Navigation Satellite System (GNSS)-derived precipitable water vapor (PWV) to refine the water vapor content based on FY-4A satellite remote sensing data, thereby improving its accuracy. Taking Hong Kong as an experimental area, we investigated the correlation between GNSS PWV and FY-4A PWV, confirming the feasibility of utilizing GNSS PWV to calibrate FY-4A PWV. Subsequently, by examining the differences between the two PWV values, we found that the elevation of the stations affects the consistency of PWV measurement. Based on this finding, the elevation data are introduced to construct a multivariate linear regression correction model with a first-order polynomial. To evaluate the performance of the proposed model, a comparison with other correction models is made, including second-order polynomials and power functions. The results indicate that the elevation-integrated water vapor correction model improves the root mean square error (RMSE) by 27.4% and the MAE by 26.7%, and reduces the bias from 0.592 to nearly 0. Its accuracy surpasses that of second-order polynomial and power function models, demonstrating a considerable improvement in the precision of FY-4A.

1. Introduction

Water vapor monitoring is essential for weather forecasting and extreme event analysis [1,2,3,4]. Common retrieval methods include radiosondes, GNSS, radiometers, and satellite remote sensing [5], each with limitations: radiosondes have spatiotemporal constraints [6,7,8,9], while GNSS offers all-weather, high-precision, real-time capabilities but has sparse station coverage [10,11]. Satellite methods face cloud contamination, and radiosondes lack resolution, highlighting the need for multi-technology fusion approaches [12].

In response to the limitations of remote sensing technology in water vapor retrieval, such as data gaps, product biases, and the inability to simultaneously achieve high spatial and temporal resolutions, researchers have proposed various solutions [13,14,15]. Many scholars have used GNSS data to modify MODIS water vapor products or establish water vapor inversion models. Fang et al. proposed a method to correct MODIS water vapor with ground-based GPS PWV, generating regional continuous and high-precision PWV data. Liu et al. conducted a comparative study on the accuracy of regional and single-station models, and proved that the two models achieved comparable accuracy [16]. Zhang et al. developed a grid-based calibration model based on annual and semi-annual cyclical error fluctuations between MOD-NIR-PWV and ERA-PWV [17]. Wang et al. divided multiple regions in China based on environmental and climate factors, and established a MODIS correction model for each region. Experiments have shown that this method effectively improved the accuracy of MODIS water vapor data [18]. Lu et al. proposed a convolutional neural network (CNN)-based data fusion method that combines the advantages of MODIS and the techniques of the European Center for Medium-Range Weather Forecasts Reanalysis Version 5 to generate high-accuracy and high-resolution PWV data [19]. However, current mainstream research on water vapor remote sensing correction is predominantly concentrating on MODIS products, resulting in relatively limited studies on the FY satellite series.

FY-4A is a Chinese meteorological satellite dedicated to weather observation and meteorological data collection [20]. Incorporating the Advanced Geostationary Radiation Imager (AGRI), it provides layered atmospheric water vapor products [21,22]. In 2021, Long et al. [23] compared precipitation observation data from FY-4A AGRI and the Global Precipitation Measurement satellite (GPM IMERG), revealing that FY-4A AGRI exhibits smaller RMSE and higher probability of detection on an hourly and daily scale. However, the FY-4A satellite’s coarse spatial resolution compared with GNSS imposes inherent constraints on resolving localized water vapor variations, thereby limiting the accuracy of the correction. Current studies predominantly employ spatial interpolation for data matching, failing to effectively resolve the scale mismatch between satellite pixels and GNSS point observations.

Therefore, this paper delved into the strategy of collaborative water vapor detection by integrating GNSS and FY-4A. Subsequently, a multiple linear regression water vapor correction model that incorporates elevation data was constructed. This model capitalizes on the continuous high-precision monitoring capabilities of GNSS and the high-coverage advantages of FY-4A. This study provides a methodological breakthrough to resolve the inherent “high-coverage versus low-accuracy” dilemma in geostationary satellite-based water vapor products. As a result, it offers valuable data support for extreme weather warnings and quantitative forecasting.

2. Methods

2.1. GNSS PWV

Precise Point Positioning (PPP) represents a high-precision positioning approach based on pseudo-range and carrier-phase measurements, which could achieve millimeter-level positioning accuracy [24,25,26]. The GNSS PWV estimates are derived using satellite signals with elevation angles of 10°. For instance, at 3 km altitude, this corresponds to a horizontal scale of 17 km from the receiver. This paper utilizes the PPP method to calculate water vapor. First, the real-time undifferenced observation PPP technique is employed to estimate the zenith tropospheric delay (ZTD) value. The meteorological data are input into the Saastamoinen model (Equation (2)) to compute the zenith hydrostatic delay (ZHD) value. After that, based on Equation (1), the wet delay (ZWD) is derived [27]. Then, the PWV of GNSS can be calculated by using Equation (3) [28].

$$ZTD=ZHD+ZWD,$$

(1)

$$ZHD=\left(2.2768\pm 0.0024\right)\times {\displaystyle \frac{P_s}{f\left(\phi ,h_s\right)}},$$

(2)

$$PWV=\Pi \cdot (ZTD-ZHD)$$

(3)

In Equations (1)–(3), $P_s$ denotes surface pressure, measured in hPa, at the GNSS receiver’s location; $f(\phi ,h_s)=1-0.00266\times cos2\phi -0.00028\times h_s$, $h_s$ is the elevation of the ground point (m), where $\phi$ denotes the station latitude. $\Pi$ is the water vapor conversion factor, and this paper employs Chen Yongqi’s model to derive the atmospheric weighted mean temperature (T_m) [29].

2.2. FY-4A PWV

The specific process for extracting FY-4A PWV involves converting the longitude and latitude coordinates of stations into row and column numbers in the water vapor product, and then reading the PWV values of corresponding pixels, and can be expressed as Equations (4)–(9):

(1) Before calculating PWV of FY-4A, the geographic latitude and longitude should be transformed into the geocentric latitude and longitude as Equations (4) and (5), respectively:

$${\lambda }_e=lon,$$

(4)

$${\varphi }_e=\mathit{arctan}({\displaystyle \frac{e{b}^2}{e{a}^2}}\times \mathit{tan}(lat)),$$

(5)

where ${\lambda }_e$ is the geocentric longitude ($lon$), which is the longitude of a point as seen from the center of the Earth, and ${\varphi }_e$ is the geocentric latitude, which is the latitude ($lat$) of a point as seen from the center of the Earth. $e_a=6378.137$ km is the semi-major axis of the Earth, and $e_b=6356.7523$ km is the semi-minor axis of the Earth.

(2) The projection coordinates of the point on the Earth’s surface are calculated using Equations (6) and (7):

$$x=\mathit{arctan}(-{\displaystyle \frac{r_2}{r_1}})\times {\displaystyle \frac{180°}{\pi }},$$

(6)

$$y=\mathit{arcsin}(-{\displaystyle \frac{r_3}{r_n}})\times {\displaystyle \frac{180°}{\pi }},$$

(7)

where $r_1=\mathrm{h}-r_e\times \mathit{cos}({\varphi }_e)\times \mathit{cos}({\lambda }_e-{\lambda }_D)$; $r_2=-r_e\times \mathit{cos}({\varphi }_e)\times \mathit{sin}({\lambda }_e-{\lambda }_D)$; $r_3=r_e\times \mathit{sin}({\varphi }_e)$; $r_e={\displaystyle \frac{e_b}{\sqrt{1-\frac{{e_a}^2-{e_b}^2}{{e_a}^2}}\times {\mathit{cos}}^2\left({\varphi }_e\right)}}$; $r_n=\sqrt{r_1^2+r_2^2+r_3^2}$. The parameters $e_a$ and $e_b$ have the same meaning as in Equation (5). $r_1$, $r_2$, and $r_3$ are intermediate variables used to convert the geographic latitude and longitude into nominal row and column numbers. The parameter h = 42,164 km signifies the Earth-to-satellite center distance, and ${\lambda }_D$ = 104.7 is the longitude of the satellite’s sub-satellite point. $r_n$ represents a function of the Earth’s effective radius and geocentric latitude.

(3) The station’s geographic coordinates can be converted into row and column numbers in the water vapor product using Equations (8) and (9):

$$c=COFF+x\times 2^{-16}\times CFAC,$$

(8)

$$l=LOFF+y\times 2^{-16}\times LFAC,$$

(9)

where c denotes the column index, l indicates the row index, COFF stands for the column offset, CFAC refers to the column scaling coefficient, LOFF is the row offset, and LFAC signifies the row scaling coefficient. These values vary depending on the resolution of the FY-4A products. After determining the corresponding pixel position (row/column indices) through coordinate transformation, the PWV values stored in the layered precipitable water (LPW) product are extracted from the calculated grid location.

2.3. Radiosonde PWV

PWV is computed by vertically integrating the specific humidity (q) through the atmospheric column from the ground level to the top of the radiosonde observation, and it can be expressed as Equation (10):

$$V_{RS}={\displaystyle \frac{1}{{\rho }_{w}g}}{\int }_{P_{surf}}^{P_{top}}q dP$$

(10)

${\rho }_w$ denotes the density of liquid water, with units of kg/m³; q represents the mass of water vapor per unit mass of moist air, with units of kg/kg; g is the gravitational acceleration, with units of m/s²; $P_{surf}$ indicates the surface pressure; $P_{top}$ refers to the top pressure of the sounding profile.

Building on the first-order polynomial, this paper introduces elevation data and proposes the construction of a multivariate linear regression correction model for the FY-4A PWV. This model integrates elevation and GNSS water vapor data, and it can be expressed as Equation (11):

$$V_{\mathit{GNSS}}=a_0\times V_{\mathit{FY}4A}+a_1\times H+b$$

(11)

$V_{\mathit{GNSS}}$ represents GNSS PWV; $V_{\mathit{FY}4A}$ represents FY-4A PWV; H denotes the elevation of the corresponding GNSS station. The results will be compared with those derived from traditional linear correction models [30]. It should be noted that, to illustrate the quality of the GNSS water vapor retrieval results, we selected four commonly used accuracy metrics for the PWV, namely bias, mean absolute error (MAE), RMSE, and coefficient of determination (R²), to evaluate the retrieval results.

3. Data Processing

3.1. Experiment Location

Hong Kong is located along the coast of the South China Sea and is influenced by a maritime subtropical monsoon climate. At the same time, the GNSS Continuously Operating Reference Stations (CORS) in Hong Kong are relatively dense, making this region suitable for GNSS atmospheric water vapor data retrieval. The spatial representativeness of GNSS PWV (17 km radius at 3 km altitude) aligns well with Hong Kong’s CORS network density (average station spacing < 20 km). Among the 19 GNSS stations in Hong Kong, only 11 are equipped with meteorological sensors to obtain in situ meteorological parameters. In this paper, in order to ensure that the PWV retrieved from each station possesses a certain degree of regional representativeness, 11 GNSS stations were selected for the experiment. The locations are shown in Figure 1.

3.2. Data Introduction

To obtain the water vapor sequence calculated by GNSS, this paper derives GNSS-based PWV observation data from 11 eligible stations in the Hong Kong CORS network, with the standard RINEX format data (1 s/5 s/30 s intervals, multi-rate sampling) of June 2023 obtained from the University of Hong Kong. The temporal coverage of all other datasets is consistent with that of the GNSS data. For radiosonde data, those obtained from the University of Wyoming’s Upper Air Database (http://weather.uwyo.edu (accessed on 1 June 2024)) during the same period serve as the validation benchmark for the GNSS PWV accuracy assessment. The remote sensing water vapor product used in this paper is the layered water vapor product (LPW) provided by the FY-4A satellite. The LPW employed in this research measures water vapor in units of g/kg, is convertible to cm, and provides a temporal resolution of 15 min with a spatial resolution of 4 km. For this experiment, the FY-4A products only provide planar coordinates of pixel points (25 June 2023) (as shown in Figure 2). Though the elevation data for pixels at GNSS station locations can be directly obtained from the station measurements, elevation data for pixels at other locations must be extracted from the Digital Terrain Model, which can be downloaded from the Hong Kong Lands Department’s official website: (https://www.landsd.gov.hk/sc/spatial-data/open-data/kf_dtm.html (accessed on 1 June 2024)). The geospatial referencing adopts the local coordinate system “Hong Kong 1980 Grid” and vertical datum “Hong Kong Principal Datum”. It displays the terrain in a 5 m × 5 m grid (including non-ground information such as elevated roads and bridges). The model’s elevation range was from −24 to 957 m, with an elevation accuracy of ±5 m. If the land was covered by vegetation, the terrain was shown at the height of the vegetation.

3.3. Data Preprocessing

The data preprocessing in this study is primarily conducted in two steps: (1) GNSS water vapor retrieval above the station and (2) FY-4A PWV extraction and analysis.

The experiment employed the open-source software RTKlib (v2.4.3) for PPP. The data of the HKSC station were used, and the corresponding processing approach is outlined as follows: ionospheric errors are mitigated using the dual-frequency ionosphere-free combination model. Satellite orbit and clock errors were corrected using precise ephemeris and clock products from IGS (https://igs.com (accessed on 1 July 2024)). The processing parameters include a 10° elevation mask and 5 min temporal resolution for ZTD estimation. The sampling interval was set to 5 min. The GNSS PWV series was calculated, as well as the radiosonde PWV. Since the HKSC station is the closest CORS station to the King’s Park radiosonde station among all available CORS stations in Hong Kong, this study selected GNSS data from June 1 to 30 at this station to retrieve water vapor data, which were then compared with radiosonde-derived water vapor data. The variation trend of PWV is shown in Figure 3.

As can be seen from Figure 3, for the HKSC site, comparative analysis demonstrates strong trend consistency between GNSS-derived and radiosonde-based PWV estimates, with a correlation value of 0.923, indicating a clear correlation between the GNSS PWV and radiosonde-derived PWV. Additionally, the RMSE is 1.876 mm and the MAE is 1.576 mm, both of which meet the practical accuracy requirement of 2 mm [30]. The strong agreement between GNSS and radiosonde PWV further confirms the reliability of GNSS as an independent reference dataset for PWV validation and calibration. Unlike radiosondes (typically launched twice daily), GNSS provides real-time, uninterrupted PWV estimates regardless of cloud cover or precipitation, ensuring consistent calibration potential.

The PWV data from FY-4A at the locations of the radiosonde stations were also extracted. Specifically, the FY-4A PWV 30 min before and after the specified time was selected. Then, the extracted FY-4A PWV data were compared with the radiosonde PWV. Figure 4 shows the comparison of the trends of the FY-4A and radiosonde PWV.

Based on Figure 4, a strong correlation was observed between FY-4A and radiosonde PWV measurements, with a correlation coefficient of 0.922. Accuracy assessment of FY-4A data from 41 samples revealed statistical metrics, including 4.417 mm RMSE and 3.870 mm MAE. Nevertheless, systematic underestimation was identified in FY-4A data, showing a negative bias of 3.767 mm compared with radiosonde observations. The accuracy assessment revealed that FY-4A PWV measurements exhibited clearly higher uncertainty than GNSS-derived estimates [31]. This also proved that it is reasonable and feasible to optimize the FY-4A PWV data by leveraging GNSS PWV data. With widely distributed ground stations, GNSS can provide localized PWV corrections that account for regional atmospheric variations, which is particularly valuable for improving FY-4A’s performance in complex terrains.

3.4. Model Development

In this paper, we selected GNSS PWV data from the stations equipped with meteorological instruments and compared it with the extracted FY-4A PWV data to conduct a correlation analysis [32,33]. The linear relationship between the two datasets was evaluated using the Pearson correlation coefficient (R). A scatter plot was generated with the two datasets from all stations. The accuracy assessment is presented in

Table 1. Accuracy evaluation of FY-4A and GNSS PWV data at all stations.

GNSS Station Name	R	RMSE (mm)	Bias (mm)	Sample Size	Elevation (m)
HKKT	0.844	3.809	−2.321	1474	34.576
HKLT	0.829	3.081	0.068	1764	125.922
HKNP	0.773	6.523	5.526	1809	350.672
HKOH	0.877	3.291	1.933	1364	166.401
HKPC	0.832	4.119	−2.762	1791	18.130
HKSC	0.860	3.794	−2.538	1399	20.239
HKSL	0.807	3.223	−0.144	1712	95.297
HKSS	0.876	3.806	−2.611	1511	38.713
HKST	0.871	3.470	2.062	1339	258.704
HKWS	0.876	3.205	−1.703	1369	63.791
T430	0.847	5.708	−4.870	1586	41.323

.

The results presented in Table 1 show that, except for a slightly lower correlation between the GNSS water vapor and FY-4A water vapor above station HKNP, the correlations at the remaining stations are all above 0.8, indicating a strong overall correlation between the FY-4A PWV and GNSS PWV. Most RMSE values fall within the range of 3–4 mm, with negative bias values, indicating that the FY-4A PWV data underestimate the GNSS PWV measurements in the Hong Kong region. Stations such as HKNP, HKOH, and HKST share common characteristics of having high elevations and exhibiting positive bias values, whereas stations like HKPC, HKSC, and HKSS are characterized by low elevations and negative bias values. It is seen that the elevation of the stations influences the consistency between the FY-4A PWV and GNSS PWV. In the construction of the correction model, elevation data can be incorporated to correct the FY-4A PWV, ensuring more consistency between the FY-4A-derived and GNSS-retrieved PWV measurements.

This paper constructed an elevation-integrated multivariate linear regression correction model for FY-4A PWV based on GNSS water vapor data, as shown in Equation (11). Of the sample data, 70% were randomly selected as the training set, while the remaining 30% served as the validation set. By combining the least squares method, the model parameters are calculated as follows: a₀ = 0.860, a₁ = −0.026, b = 11.312, and R² = 0.747. Due to the varying elevations of each station, this paper also used scatter plots to visually represent the fitting performance of the correction model on the training data, as shown in Figure 5.

Figure 5 contains multiple fitting lines formed by connected scatter points, representing different linear relationships between the FY-4A PWV and GNSS PWV across various elevation zones. It can be seen that the multivariate linear regression model incorporating elevation achieves an R² of 0.747 for the training dataset. The original FY-4A PWV data points predominantly fall below the 1:1 reference line, indicating a systematic underestimation of PWV by FY-4A compared to GNSS observations. After correction using the multivariate linear model, the data points show significantly better alignment with the 1:1 reference line, demonstrating the model’s effectiveness in reducing the systematic bias in FY-4A PWV retrievals. This result indicates that, compared with both GNSS and FY-4A retrieval methods, the model shows enhanced accuracy in quantifying atmospheric water vapor.

By applying the validation dataset to conventional regional water vapor correction models (including linear polynomial, quadratic polynomial, exponential function, logarithmic function, and power function models) as well as the elevation-integrated water vapor correction model, the corrected RMSE, MAE, bias, and residual sequence are obtained and shown in

Table 2. Changes in overall accuracy of validation data.

Model	RMSE (mm)	MAE (mm)	Bias (mm)
No correction	4.182	3.303	−0.592
Linear	4.051	3.172	−0.019
Quadratic	4.055	3.176	−0.214
Exponential	4.230	3.298	−1.112
Logarithmic	4.050	3.173	−0.043
Power function	4.052	3.175	0.087
Multiple linear regression	3.036	2.420	−0.022

. The GNSS PWV data from the monitoring stations were compared with FY-4A PWV data both before and after model correction, resulting in an RMSE comparison plot, as shown in Figure 6.

By comparing the accuracy indicators before and after correction in Table 2, it is demonstrated that after using the water vapor correction model that integrated elevation data, the overall RMSE increased by 27.4%, the MAE improved by 26.7%, and the bias was reduced from 0.592 to close to 0. Figure 6 shows that the FY-4A PWV corrections demonstrated obvious improvements at stations HKNP and T430, though all 11 monitoring stations exhibited systematic improvements in PWV values after correction. The RMSE of the water vapor data at other monitoring station locations was also improved to below 3.0 mm.

The residual values were calculated as the differences between the GNSS PWV measurements and the corresponding FY-4A PWV values, both before and after model correction. The residual distributions before and after correction of the validation dataset are plotted as scatter plots, as shown in Figure 7.

Figure 7 indicates that the range of residuals was narrowed from −10 to 10 mm before the correction to −5 to 10 mm after the correction. In addition, the proportion of the samples, with residuals within the range of −5 to 5 mm increasing from 76.7% to 90.1%, shows an obvious improvement in accuracy.

4. Discussion

The elevation-coupled water vapor correction model proposed in this study clearly enhances the precision of the FY-4A PWV. After the application of corrections, the RMSE is reduced to 3.036 mm, the MAE is reduced to 2.420 mm, and the bias is narrowed from 0.592 mm to −0.022 mm. Additionally, the proportion of samples with residuals within the range of −5 to 5 mm increases from 76.7% to 90.1%, indicating a clear improvement in the accuracy. This suggests that the model effectively captures the relationship between the GNSS PWV and FY-4A PWV and clearly enhances the accuracy of the FY-4A PWV. The introduction of elevation data clearly enhances the performance of the correction model, which aligns with findings in the literature regarding the impact of elevation on the water vapor content. In this study, it was found that incorporating elevation data improves the model’s R² from 0.55 to 0.747 and reduces the RMSE from 4.2 mm to 3.036 mm. This finding indicates that elevation plays an important role in the disparities between the FY-4A PWV and GNSS PWV. The GNSS-corrected FY-4A PWV enables nowcasting of heavy rainfall and fog formation in complex terrain. Based on this, future research can be directed towards a more in-depth exploration of how the accuracy of elevation data affects water vapor correction.

5. Conclusions

This research selected Hong Kong as the experimental location. First, through the correlation analysis of GNSS and FY-4A PWV datasets, an elevation-based correction model for GNSS/FY-4A PWV was developed. Building on existing studies, the proposed model shows its ability to enhance the accuracy of FY-4A’s LPW products. As a result, it can effectively improve the satellite-derived PWV measurements. The key findings are summarized as follows. (1) FY-4A PWV and GNSS PWV exhibited a strong correlation. When using general linear correction models, the minimum RMSE of the PWV correction was 4.050 mm, with an overall RMSE of around 4.2 mm and an R² of approximately 0.55. This indicates that individual linear models have limited effectiveness in correcting the sample data. Additionally, it is also noted that abnormally high RMSE values in some regions might be associated with the elevation of the GNSS stations in those regions. (2) By adopting a first-order polynomial and incorporating elevation data, a multivariate linear regression correction model that integrates elevation was constructed. The model achieved an R² of 0.747 and reduced the RMSE to 3.036 mm, demonstrating an improved correction performance. This study did not explore whether the constructed water vapor correction model would have different accuracies in other regions because of varying climates and terrains, which could be a topic for future research.