Fusion-Based Regional ZTD Modeling Using ERA5 and GNSS via Residual Correction Kriging

Highlights

What are the main findings?

• RK ZTD applies Kriging interpolation to the GNSS–ERA5 residuals, rather than directly interpolating GNSS ZTD, to correct the ERA5 ZTD grids.
• In the Netherlands during 2023, RK ZTD achieves high accuracy, with an RMSE of 5.70 mm and a bias of 0.41 mm.
• RK ZTD outperforms both the original ERA5 ZTD and the GNSS-Kriging ZTD derived from GNSS ZTD alone, demonstrating the effectiveness of the residual-correction framework.

What are the implications of the main findings?

• The residual-correction strategy effectively mitigates the systematic overestimation in ERA5 ZTD.
• By correcting ERA5 ZTD while preserving its original grid structure, the method maintains spatial completeness for regional applications.
• The stable performance across different seasons and during Storm Ciarán suggests that the method is reliable for regional ZTD refinement under both typical and extreme weather conditions.

Abstract

Zenith Tropospheric Delay (ZTD) and its associated atmospheric water vapor information constitute essential environmental variables for Earth observation (EO)-based atmospheric monitoring and environmental variable retrieval. High-quality ZTD products are therefore of great importance for the post-processing, refinement, and reconstruction of atmospheric environmental variables at regional scales. Among existing observation techniques, Global Navigation Satellite System (GNSS) measurements provide high-precision ZTD estimates and have become an important means for retrieving tropospheric delay and water vapor. However, the sparse and uneven spatial distribution of GNSS stations limits their direct applicability for continuous environmental monitoring. Reanalysis-based products, such as ERA5 provided by the European Centre for Medium-Range Weather Forecasts (ECMWF), offer EO big data with excellent spatiotemporal continuity but suffer from pronounced systematic biases compared to precision GNSS retrievals, restricting their direct use in high-accuracy regional applications. To address these limitations, this study proposes a Residual Correction Kriging method for ZTD (RK ZTD) that integrates GNSS ZTD and ERA5 ZTD grids through a multi-source data fusion framework. High-precision GNSS ZTD is treated as reference data, and the differences between GNSS ZTD and ERA5 ZTD at modeling stations are defined as residuals to characterize the systematic bias in ERA5 ZTD grids. A Kriging interpolation algorithm is then employed to model the spatial distribution of these residuals and generate residual correction grids. By superimposing the interpolated residual grids onto the ERA5 ZTD grids, a refined and high-precision regional ZTD product is reconstructed. Experiments were conducted using observations collected in 2023 from 36 GNSS stations in the Netherlands, including 10 modeling stations and 26 independent validation stations, together with concurrent ERA5-derived ZTD grids. The results demonstrate that the proposed RK ZTD model provides spatially robust and high-precision ZTD products across the study region. The RK ZTD achieves a Root Mean Square Error (RMSE) of 5.70 mm, representing improvements of 58.4% and 35.4% compared with the original ERA5 ZTD (13.69 mm) and the GNSS-Kriging ZTD (8.82 mm), respectively. Moreover, the absolute bias is reduced to 0.41 mm, in contrast to 5.15 mm for the ERA5 ZTD, indicating that systematic biases are effectively mitigated. Spatial and seasonal analyses further confirm that the proposed method maintains stable performance across all seasons and significantly alleviates interpolation inaccuracies caused by sparse GNSS stations, even under extreme weather conditions such as Storm Ciarán, proving its value for advanced Earth environmental science applications.

1. Introduction

The Global Navigation Satellite System (GNSS) positioning technique offers centimeter- to millimeter-level positioning accuracy and has been widely applied in geodetic and Earth observation fields [1,2]. However, zenith tropospheric delay (ZTD) constitutes a major error source in GNSS positioning, significantly affecting both positioning accuracy and convergence time. ZTD is usually decomposed into zenith hydrostatic delay (ZHD) and zenith wet delay (ZWD) [3]. ZHD can be accurately modeled using classical models such as the Saastamoinen model, while ZWD accounts for only approximately 10% of ZTD but is strongly influenced by the spatiotemporal variability of atmospheric water vapor, exhibiting fluctuations that are difficult to capture and challenging for existing models to estimate precisely [4].

Given the inherent uncertainty in wet delay modeling, many scholars have explored empirical ZTD models based on meteorological parameters. Traditional methods include the Saastamoinen [5], Hopfield [6], and Black [7] models. Although these methods can provide a priori ZTD, most GNSS stations are not likely equipped with meteorological sensors, making it impossible to obtain measured meteorological parameters such as pressure, temperature, and water vapor pressure [8]. To overcome these limitations, the GNSS community has developed empirical models that do not rely on such meteorological parameters, such as the UNB3m [9], GPT3 [10], GSTDS [11], and TropGrid [12] models. These models fit harmonic functions to the periodic variations of meteorological parameters obtained from historical radiosonde and reanalysis data, allowing direct estimation of ZTD values using only variables like the day of the year. However, the accuracy of these models is limited, exhibiting significant spatiotemporal discrepancies in different regions, and they typically suffer from systematic biases of several centimeters, requiring further refinement [13].

Some studies have explored strategies that combine data from Numerical Weather Models (NWM) with empirical models [14]. For instance, Lu et al. utilized forecast data from the European Center for Medium-Range Weather Forecasts (ECMWF) to generate ZTD parameters, effectively shortening the PPP convergence time [15]. In 2019, ECMWF released the fifth-generation meteorological reanalysis (ERA5), which is based on ERA-Interim and offers significant improvements in both temporal and spatial resolution [16]. Although ERA5 provides spatial continuity, its model accuracy is still inferior to that of direct GNSS estimation, and obvious bias exists between the model values and GNSS ZTD, especially in localized regions with complex terrain [17].

Spatial interpolation of regional tropospheric delays based on high-precision ZTD products from real-time or near-real-time GNSS stations is an effective approach. However, existing ZTD interpolation methods are subject to certain limitations. For example, the Polynomial Interpolation method can obtain ZTD estimates at any position in 3D space based on the least-squares principle, but its interpolation accuracy is limited to small areas [18]. The Spherical Harmonic Interpolation (SHI) method offers higher accuracy but is only applicable to interpolating variables that are not dependent on elevation [19]. Given the strong correlation between ZTD and elevation, the inherent limitations of the SHI method in handling variables with strong elevation dependency make it difficult to achieve ideal interpolation results. More importantly, the performance of these interpolation methods is highly dependent on the density and distribution of GNSS stations. However, as the deployment of ground-based stations is inevitably influenced by the evolution of urban spatial structure and socio-economic factors [20], the distribution is often spatially heterogeneous. Consequently, in regions with sparse or unevenly distributed stations, the interpolation accuracy decreases significantly, failing to meet the current demands for high-spatiotemporal-resolution ZTD modeling.

To overcome the limitations of traditional interpolation methods, such as the dependency on GNSS station distribution and the insufficient accuracy of ERA5 ZTD, fusing high-precision GNSS data with reanalysis data has become an effective technical approach [21,22]. This strategy leverages the excellent spatiotemporal continuity of ERA5 ZTD and the high accuracy of GNSS ZTD. Multiple studies have confirmed that a systematic bias exists between reanalysis data and high-precision GNSS ZTD. Lou et al. revealed a mean systematic bias of 0.36 cm when analyzing ERA-Interim and GNSS data; in their subsequent research, they utilized ERA-Interim data to establish an inverse scale height model, which was then used for fusing the GNSS network data [23]. Wang et al. explicitly pointed out that a systematic bias exists between ERA5 ZTD and GNSS ZTD and that this bias is more pronounced in regions with abundant water vapor [8]. Xia et al. constructed a real-time ZTD model for the China region by fusing GNSS observations and ERA5 data [22]. The key to this method was using ERA5 data to establish an elevation normalization factor grid model that accounted for annual, semiannual, and daily cycles and optimizing the Gaussian distance weighting function for the interpolation process. Their results demonstrated that, compared to GNSS post-processing results, the model’s Root Mean Square Error (RMSE) was better than 1.44 cm, and it could shorten the vertical convergence time of real-time PPP by more than 33%.

However, these data fusion strategies still present challenges in practical application. On one hand, whether in direct interpolation of GNSS ZTD or in the interpolation steps involved in fusion models, the accuracy and resolution are highly limited by the distribution density of the participating GNSS stations. In regions with sparse station coverage, the accuracy of interpolation models is far lower than in station-dense areas [24]. On the other hand, the systematic bias existing between ERA5 ZTD and high-precision GNSS ZTD, which possesses significant spatiotemporal variability, remains a critical factor for improving fusion model performance [25].

To address these challenges, this study proposes a Residual Correction Kriging method for ZTD (RK ZTD), which improves the interpolation accuracy in regions with sparse GNSS stations and corrects the systematic bias of ERA5 ZTD. The method first uses the spatially continuous ERA5 ZTD grid as an initial value. Then, it calculates the residual series by comparing high-precision GNSS ZTD from modeling stations. Subsequently, ordinary Kriging interpolation is applied to generate the residual correction grid, which is then superimposed onto the ERA5 ZTD grid to produce the fused ZTD product. This strategy aims to fully exploit the complementary advantages of GNSS and ERA5 data, constructing a high-precision regional ZTD model.

This manuscript is organized as follows. Section 1 is the Introduction, briefly describing the limitations of current ZTD modeling and interpolation methods. Section 2 introduces the methodologies for ZTD acquisition from GNSS and ERA5 and details the proposed RK ZTD model based on ERA5 residuals. Section 3 analyzes the inherent biases in ERA5 ZTD and outlines the validation scheme, including the GNSS-Kriging ZTD models used for comparison. Section 4 evaluates the performance of the proposed RK ZTD model by assessing its overall accuracy, spatial characteristics, and seasonal stability. Finally, Section 5 provides a summary of the key findings. This study aims to develop and validate a high-accuracy, high-resolution ZTD fusion product by modeling and correcting the systematic residuals in ERA5 data.

2. Methods

2.1. ZTD Retrieval from the GNSS

The ionosphere-free (IF) combination is a commonly used functional model in GNSS data processing. It utilizes dual-frequency or multi-frequency observations to construct the IF combination, which eliminates the first-order ionospheric delay in pseudorange and carrier phase measurements. Taking the first and second GNSS frequencies, $f_1$ and $f_2$, as an example, the ionosphere-free combination observations for pseudorange $P_{r,IF}^s$ and carrier phase $L_{r,IF}^s$ can be expressed as follows [26]:

$$\left\{\begin{array}{l}{P}_{r,IF}^s={\alpha }_{IF}\times P_{r,1}^s+{\beta }_{IF}\times P_{r,2}^s\\ L_{r,IF}^s={\alpha }_{IF}\times L_{r,1}^s+{\beta }_{IF}\times L_{r,2}^s\end{array}\right.$$

(1)

where ${\alpha }_{IF}$ and ${\beta }_{IF}$ are the IF combination coefficients, taken as ${\alpha }_{\mathrm{IF}}={\displaystyle \frac{f_1^2}{f_1^2-f_2^2}}$ and ${\beta }_{IF}=-{\displaystyle \frac{f_2^2}{f_1^2-f_2^2}}$, respectively. The dual-frequency IF combination observation equations can then be expressed as follows:

$$\left\{\begin{array}{l}{P}_{r,IF}^s={\rho }_r^s+c(t_r-t^s)+T_r^s+b_{r,IF}-b_{IF}^s+e_{r,IF}^s\\ L_{r,IF}^S={\rho }_r^s+c(t_r-t^s)+T_r^s+{\lambda }_{IF}(N_{r,IF}^s+B_{r,IF}-B_{IF}^s)+{\epsilon }_{r,IF}^s\end{array}\right.$$

(2)

where $b_{r,IF}$ and $b_{IF}^s$ are the combined pseudorange hardware delays at the receiver and satellite; $B_{r,IF}$, $B_{IF}^s$, and $N_{r,IF}^s$ are, respectively, the receiver and satellite phase delays and the IF ambiguity; $e_{r,IF}^s$ and ${\epsilon }_{r,IF}^s$ represent the observation noise on the pseudorange and carrier phase observations, respectively. The slant tropospheric delay $T_r^s$ can be expressed as follows [27]:

$$T_r^s\left(E,\alpha \right)=M{F}_h\left(E\right)\cdot ZHD+M{F}_w\left(E\right)\cdot ZWD+M{F}_g\left(E\right)\cdot \left[G_n\cos \left(\alpha \right)+G_e\sin \left(\alpha \right)\right]$$

(3)

where the calculation method for ZHD can adopt the Saastamoinen model [28]:

$$ZHD={\displaystyle \frac{0.002277\cdot P_S}{f(\phi ,h)}}$$

(4)

$$f(\phi ,h)=1-0.0026\cdot \cos (2\phi )-0.00028\cdot h$$

(5)

where $P_S$ is the atmospheric pressure, $\phi$ is the geodetic latitude, and $h$ is the ellipsoidal altitude; $E$ and $\alpha$ represent the satellite elevation angle and azimuth angle, respectively; $G_n$ is the horizontal gradient in the north–south direction, and $G_e$ is the horizontal gradient in the east–west direction; $M{F}_h\left(E\right)$ and $M{F}_w\left(E\right)$ represent the mapping functions for the hydrostatic and wet components at elevation angle $E$, respectively; and $M{F}_g\left(E\right)$ is the tropospheric horizontal gradient mapping function at elevation angle $E$. The Neill Mapping Functions (NMF) model, used for their calculation, consists of a dry mapping function and a wet mapping function. The dry mapping function is shown in Equation (6), and the wet mapping function is shown in Equation (7) [29]:

$$M{F}_h(E)={\displaystyle \frac{1+a_h/(1+b_h/(1+c_h))}{\sin E+\frac{a_h}{\sin E+\frac{b_h}{\sin E+c_h}}}}+\left[{\displaystyle \frac{1}{\sin E}}-{\displaystyle \frac{1+a_{ht}/(1+b_{ht}/(1+c_{ht}))}{\sin E+\frac{a_{ht}}{\sin E+\frac{b_{ht}}{\sin E+c_{ht}}}}}\right]\times {\displaystyle \frac{h}{1000}}$$

(6)

$$M{F}_w(E)={\displaystyle \frac{1+a_w/(1+b_w/(1+c_w))}{\sin E+\frac{a_w}{\sin E+\frac{b_w}{\sin E+c_w}}}}$$

(7)

where $h$ is the ellipsoidal altitude; $a_{ht}$, $b_{ht}$, and $c_{ht}$ take empirical values; and the calculation process for the dry component coefficients $a_h$, $b_h$, $c_h$ and the wet component coefficients $a_w$, ${\mathrm{b}}_w$, ${\mathrm{c}}_w$ is detailed in reference [29].

In this study, GNSS ZTD was retrieved using static Precise Point Positioning (PPP) processing. Compared with relative positioning, PPP enables each station to be processed independently, so that the derived ZTD values are less affected by inter-station dependence and can better reflect the spatial distribution characteristics of tropospheric delay over the study region. This property is particularly suitable for the subsequent residual modeling and spatial interpolation framework adopted in this study. The main PPP processing settings are summarized in

Table 1. Main PPP processing settings.

Item	Setting
Estimation interval	1 h
Elevation cutoff angle	7°
Precise orbit	WHU SP3 products
Precise clock	WHU CLK products
ERP correction	WUM ERP products
Ionosphere correction	IF combination
Troposphere correction	STO ZTD; PWC grad (720 min)
Mapping function	NMF
Ambiguity	Fixed
Antenna phase center correction	IGS20 ANTEX
Parameter estimation	LSQ
Tides correction	Solid Earth, pole, ocean loading

, including the 1 h ZTD estimation interval, a 7° elevation cutoff angle, precise orbit and clock products, ionosphere-free combination, stochastic estimation of ZTD, piecewise-constant gradient modeling, NMF mapping function, ambiguity resolution, and tide corrections.

2.2. ZTD Retrieval from ERA5 Products

The ERA5 atmospheric reanalysis dataset provides standard pressure level data at 37 layers for each grid point, with a spatial resolution of $0.25°\times 0.25°$ and a temporal resolution of 1 h. In this study, the pressure, temperature, specific humidity, and geopotential data from ERA5 during 2023 were used to calculate the corresponding ZTD. To derive the ERA5-based ZTD for a given GNSS station, vertical and horizontal interpolation or extrapolation is performed to obtain the meteorological parameters at the specific site. Before horizontal and vertical interpolation, the elevations of the ERA5 ZTD and GNSS ZTD need to be unified to the same reference system, since the ERA5 product provides geopotential (G) data based on pressure-level distributions. In practical applications, it is usually necessary to convert to geopotential height (GH). The relationship between geopotential and geopotential height is shown in Equation (8):

$$GH={\displaystyle \frac{G}{g}}$$

(8)

The resulting geopotential height is then converted to orthometric height; the conversion formula is expressed as Equation (9) [30]:

$$\left\{\begin{array}{l}OH(GH,\phi )={\displaystyle \frac{r(\phi )\cdot g_0\cdot GH}{g(\phi )\cdot r(\phi )-g_0\cdot GH}}\hfill \\ g(\phi )=9.780325\cdot {\displaystyle \frac{\sqrt{1+0.00193185\cdot {\sin }^2(\phi )}}{\sqrt{1-0.00669435\cdot {\sin }^2(\phi )}}}\hfill \\ r(\phi )={\displaystyle \frac{6378.137}{1.006803-0.006706\cdot {\sin }^2(\phi )}}\hfill \end{array}\right.$$

(9)

where $g_0$ is the standard gravitational acceleration; $g(\phi )$ is the gravitational acceleration at latitude $\phi$; and $r(\phi )$ is the effective Earth radius at latitude $\phi$.

The height system used for GNSS station data is ellipsoidal height (EH). The horizontal coordinates in this study adopt the European Terrestrial Reference System (ETRS89), which was aligned with the WGS84 framework used by the GNSS in 1989; the vertical datum utilizes the official Amsterdam Ordnance Datum (NAP) of the Netherlands, which provides normal heights based on mean sea level. The ellipsoidal height under the ETRS89 framework and the normal height under the NAP framework belong to different height systems. The conversion between them is realized via the geoid undulation, which is approximately 40 to 45 m in the study area and varies with geographic location. The basic relationship for converting orthometric height to ellipsoidal height using geoid undulation is shown in Equation (10) [31]:

$$h = OH + N_g$$

(10)

where $h$ is the ellipsoidal height, $OH$ is the orthometric height, and $N_g$ is the geoid undulation.

After the ERA5 ZTD at the nearby surrounding grid locations is vertically corrected to the GNSS station height, the data then need to be interpolated horizontally. The height-corrected ERA5 ZTD values from the nearby grid locations are interpolated to the GNSS station location, thereby yielding the ERA5 ZTD at the GNSS station. This study adopts the Inverse Distance Weighting (IDW) method, for which the functional model is as follows [32]:

$$Z_{\mathrm{site}}={\displaystyle \sum _{i=1}^n{\lambda }_i}{Z}_{\mathrm{grid},i}$$

(11)

where $Z_{\mathrm{site}}$ is the estimated meteorological elements at the specific station; $Z_{\mathrm{grid},i}$ are the sample values, and $n$ denotes the number of grid points.

$${\lambda }_i={\displaystyle \frac{1/d_i^x}{{\displaystyle \sum _{i=1}^{n}1}/d_i^x}}$$

(12)

where $d_i^x$ is the distance from the $i$-th grid point to the station; ${\lambda }_i$ is the weight of the $i$-th grid point; and $x$ is the power, generally taken as 1.3.

Air temperature, pressure, and specific humidity are the basis for retrieving ERA5 ZTD. Methods for calculating ZTD from ERA5 data typically include the integration method and the Saastamoinen model. In this study, the integration method is adopted to calculate the ZTD values at the nearby grid points surrounding each GNSS station, which are then horizontally interpolated to the GNSS station location. The ERA5 dataset provides 37 standard pressure levels for each grid point. The pressure, geopotential height, temperature, and specific humidity from each layer are required to calculate the atmospheric refractive index for the corresponding layers [33]:

$$N=k_1{\displaystyle \frac{p_d}{T}}+k_2{\displaystyle \frac{e}{T}}+k_3{\displaystyle \frac{e}{T^2}}$$

(13)

where $p_d$ is the dry air pressure, $e$ is the water vapor pressure, and $T$ represents the temperature; $k_1$ is 77.6 K/hpa, $k_2$ is 70.4 K/hpa, and $k_3$ is 373,900 K²/hpa.

Subsequently, the atmospheric refractive index is integrated layer by layer to compute the ZTD for different initial altitudes.

$$ZTD={10}^{-6}{{\displaystyle \int Ndh=10}}^{-6}\times {\displaystyle \frac{{\displaystyle \sum _{j=1}^{n-1}\left(N_{j+1}+N_j\right)\times \left(h_{j+1}+h_j\right)}}{2}}$$

(14)

where $n$ represents the total number of reanalysis data layers above the station; $h_j$ and $h_{j+1}$ are the heights of the $j$-th and $j+1$-th layer, respectively; and $N_j$ and $N_{j+1}$ denote the atmospheric refractive indices at the $j$-th and $j+1$-th layer, respectively.

2.3. Residual Correction Kriging Method for ZTD

Reanalysis products can provide spatially continuous ZTD grids, but systematic differences often exist between ERA5 ZTD and ground-based GNSS ZTD. To reduce these discrepancies, a residual-correction model based on Kriging interpolation was developed in this study. The basic idea is to use GNSS-derived ZTD as the reference, extract the systematic residuals between GNSS ZTD and ERA5 ZTD, and then model the spatial distribution of these residuals. The interpolated residual field is finally added back to the original ERA5 ZTD grid to generate a refined ZTD product with improved accuracy and preserved spatial continuity.

The implementation of RK ZTD is carried out as follows. First, GNSS ZTD and ERA5 ZTD are matched at the GNSS station locations, and the residuals between the two datasets are calculated. Second, the residuals at the selected modeling stations are used to characterize the spatial bias pattern of ERA5 ZTD over the study area. Third, Kriging interpolation is applied to these residuals to generate a continuous residual correction surface. Finally, the interpolated residual correction is superimposed onto the ERA5 ZTD grids to obtain the RK ZTD product. The corresponding flowchart of the RK ZTD model is shown in Figure 1.

3. Results

3.1. Data Sources

The study area for this research comprises the terrestrial region of the Netherlands (50–54°N, 3–8°E). The country is located in Western Europe, adjacent to the North Sea, and its most significant geographical feature is its low-lying terrain, with a considerable portion of its territory below mean sea level. This unique geographical environment exposes it directly to the warm, moist air currents from the North Atlantic, resulting in a typical temperate maritime climate. Consequently, the atmospheric water vapor content over the Netherlands is perennially high, and its spatiotemporal distribution exhibits high complexity and dynamic variability. Accurately capturing and simulating the tropospheric delay variations within this complex moisture environment presents a key practical challenge.

The data sources for this experiment include the ERA5 products for the study area during 2023 and observations from 36 continuously operating GNSS reference stations. The ERA5 meteorological parameters were obtained at an hourly temporal resolution. Correspondingly, the ERA5 ZTD grids were computed at 1 h intervals. GNSS ZTD was generated using batch processing with a 1 h ZTD estimation interval to match the ERA5 temporal resolution. Consequently, both ERA5 ZTD and GNSS ZTD used in this study share a consistent 1 h temporal resolution, providing a reliable temporal foundation for residual modeling and subsequent Kriging interpolation.

The variables utilized in this research to calculate the ERA5-derived ZTD include geopotential height, specific humidity, pressure, and temperature. As presented in Figure 2, to enable model construction and independent validation, the 36 reference stations were divided into 10 modeling stations and 26 validation stations. The modeling stations were selected to ensure a homogeneous spatial distribution and their data were used to construct the ZTD residual grid. To further describe the spatial characteristics of the GNSS network, the nearest-neighbor distances among the 36 stations were calculated. The average nearest-neighbor distance is approximately 21.6 km, which is comparable to the ERA5 grid spacing of 0.25 degrees, corresponding to a spatial scale of about 20 to 30 km in the study region. This indicates that the GNSS stations provide relatively dense but discrete observations, while the ERA5 product supplies spatially continuous fields at a similar scale. Conversely, the validation stations were completely excluded from the modeling process; they served exclusively for the objective external assessment of the final product’s accuracy. The geographical distribution and grouping of the stations are shown in Figure 2, where the modeling and validation stations are represented by red and black triangles.

3.2. Analysis of ERA5 ZTD Bias Characteristics

To quantitatively assess the consistency and deviation between the ERA5 ZTD and the GNSS ZTD within the study area, we calculated the annual mean bias, defined as the difference between GNSS ZTD and ERA5 ZTD for the 10 modeling stations. The statistical results are presented in

Table 2. Statistics of the bias between GNSS ZTD and ERA5 ZTD at the 10 modeling stations.

Station	Bias (mm)	Station	Bias (mm)
AMST	−6.4	EHVN	−4.6
APEL	−8.8	SASG	−4.3
DHEL	−4.3	STAV	−3.5
DLF1	−4.8	TERS	−4.6
EBRG	−5.3	WSRA	−7.9

.

As evidenced in Table 2, the bias values for all 10 modeling stations are consistently negative. The ubiquity of these negative values indicates a prevalent systematic discrepancy across the study region, suggesting that the ERA5 ZTD tends to systematically overestimate the ZTD values compared to the GNSS ZTD. Specifically, the values range from −8.8 mm at the APEL station to −3.5 mm at the STAV station.

To further demonstrate the temporal characteristics of this prevalent systematic bias, this study selected the EBRG station for a detailed time series analysis. This station was chosen for its statistical representativeness, as its RMSE and Bias values closely approximate the average levels of all validation stations.

Figure 3 visually illustrates the time series comparison between the GNSS ZTD and the ERA5 ZTD at the representative EBRG station. As shown in the upper panel of Figure 3, the ERA5 ZTD demonstrates a high degree of consistency in its overall annual trend with the GNSS ZTD, which is used as the reference value. Distinct seasonal patterns are evident in both datasets, with peaks occurring in summer and autumn.

However, a clear and systematic deviation exists between those two time series. Over the study period, the ERA5 ZTD values are systematically higher than those of the GNSS ZTD. The residual sequence in the bottom panel of Figure 3 further quantifies this difference. This residual sequence remains below the zero line for most of the year, consistent with the negative bias shown in Table 2, confirming a prevalent systematic overestimation in the ERA5 ZTD within this region.

Furthermore, this residual sequence is not a constant offset but rather exhibits complex temporal fluctuations. Particularly during the summer, which is characterized by more active water vapor, and during partial extreme weather events, the residual amplitude and short-term oscillations are markedly amplified. This fully demonstrates that the systematic bias between the ERA5 ZTD and the GNSS ZTD is a time-varying and complex variable closely related to meteorological conditions, rather than a simple constant error. Therefore, Figure 3 clearly reveals the necessity of constructing a refined and spatiotemporal residual correction model, which serves as the core basis for the subsequent work in this study.

3.3. Performance of the RK ZTD

To quantify the accuracy improvement gained from incorporating ERA5 grid ZTD and to provide an objective performance assessment of the proposed RK ZTD, the GNSS-Kriging ZTD model was applied, which is generated entirely from GNSS observations using Kriging interpolation and is intended to simulate the ZTD modeling accuracy achievable without relying on external meteorological products.

The procedure for this GNSS-Kriging ZTD model is as follows. First, it employs the identical set of 10 modeling stations used by the RK ZTD model. The high-precision GNSS ZTD time series from these stations are used directly as the input samples for interpolation. Subsequently, Ordinary Kriging is applied to spatially interpolate ZTD values directly to the validation station locations. The resulting ZTD estimates are derived exclusively from the interpolation of discrete GNSS station data, independent of any external model data.

To quantitatively assess the overall performance of the proposed Kriging-based residual correction model, this study conducted a year-long, comprehensive accuracy evaluation in comparison with two models, ERA5 ZTD and GNSS-Kriging ZTD, at 26 validation stations. Figure 4 clearly illustrates the scatter density distributions comparing the estimates from the three models with the GNSS ZTD reference values.

As shown in Figure 4, the ERA5 ZTD product exhibits the lowest accuracy. Its scatter points show significant dispersion and a 5.2 mm systematic bias, with an RMSE of 13.7 mm and an R² of only 0.9319. The coefficient of determination R² serves as a crucial metric to quantify the goodness of fit, reflecting the proportion of the variance in the reference GNSS ZTD that is predictable from the model. A higher R² value signifies a stronger capability of the model to capture the complex temporal variations and dynamic trends of the actual tropospheric delay. In comparison, the GNSS-Kriging ZTD model, constructed solely by interpolating from sparse modeling stations, demonstrates a significant improvement in accuracy. Its data points are more concentrated, with the RMSE decreasing to 8.8 mm and the R² increasing to 0.9728, which proves the fundamental advantage of using high-precision GNSS observations as a data source.

The proposed RK ZTD demonstrates the optimal performance. Its scatter distribution is highly concentrated, exhibiting the minimal dispersion among others. Statistical results show that the RMSE of RK ZTD is only 5.7 mm, representing an accuracy improvement of 58.4% and 35.4% compared to the original ERA5 and GNSS-Kriging models, respectively. Simultaneously, its absolute Bias is only 0.4 mm, indicating the systematic bias is nearly eliminated, and its R² reaches 0.9865, showing extremely high consistency with the GNSS reference values. This demonstrates that the RK ZTD correction model proposed in this study substantially improves the accuracy of ERA5 ZTD.

To further compare the accuracy performance of the proposed RK ZTD with the ERA5 ZTD and GNSS-Kriging ZTD, Figure 5 presents the probability density functions of the Bias values for the 26 validation stations. As illustrated, the ERA5 ZTD exhibits a distinct positive systematic deviation. Its distribution is mainly concentrated in the positive range of 2 to 9 mm, with the peak shifted significantly to the right around 5.5 mm and reaching a probability of over 30%. This quantitative shift confirms a consistent overestimation of ZTD by the ERA5 model across the region. The results are presented in Figure 5.

Conversely, the GNSS-Kriging model, while derived from observations, displays the poorest stability. Its distribution curve is notably broad and flat, spanning a wide interval from approximately −12 mm to 4 mm. The peak density is relatively low and shifts negatively to around −6 mm. These characteristics indicate significant station-to-station fluctuations and large error dispersion caused by interpolation uncertainties from the sparse modeling stations.

In contrast, the proposed RK ZTD demonstrates the optimal performance with the most robust error control. Its probability distribution is highly concentrated within a narrow range of ±2.5 mm. The curve is sharply peaked, reaching a probability of approximately 34% while centering almost perfectly on the zero line. This confirms that the RK model effectively eliminates the systematic bias of the ERA5 ZTD while significantly reducing the error dispersion observed in the GNSS-Kriging ZTD. Consequently, the proposed method achieves high-precision, unbiased ZTD estimation across all validation stations, verifying the effectiveness of the residual correction strategy.

To further analyze the performance of the RK ZTD model at different geographical locations, this section calculates the per-station RMSE for the 26 validation stations. Their spatial distribution is presented in Figure 6. The figure simultaneously presents the RMSE distribution of the GNSS-Kriging model for comparison, and both subplots utilize a unified color scale to facilitate direct comparison.

A comparison of the overall color tones in Figure 6 reveals that the RMSE values of the RK ZTD model in panel a of Figure 6 are significantly lower than those of the GNSS-Kriging model in panel b of Figure 6 at nearly all stations, demonstrating the universal superiority of the proposed residual Kriging strategy. Specifically, the RMSE distribution of RK ZTD is more uniform across the stations, with values primarily concentrated between 4 and 8 mm. In contrast, the GNSS-Kriging model exhibits not only higher RMSE values but also greater spatial variability, with errors exceeding 12 mm at some central-eastern stations.

These spatial distribution results indicate that the introduction of ERA5 grid data effectively mitigates the decrease in interpolation accuracy caused by the uneven distribution of GNSS stations. This significantly enhances the spatial consistency and robustness of the ZTD product across the entire study region.

To visually quantify the accuracy gains achieved by the proposed RK ZTD model relative to the comparative methods, Figure 7 presents the spatial distributions of the RMSE improvement rates against both ERA5 ZTD and GNSS-Kriging ZTD.

Panel a of Figure 7 reveals that the improvement of RK ZTD over the original ERA5 ZTD is extremely significant. At all 26 validation stations, the accuracy improvement rates are positive, with the majority of stations exceeding 40% and some even surpassing 60%. This demonstrates that the residual correction strategy proposed in this paper possesses a universal and highly efficient capability for correcting the systematic bias of the original ERA5 ZTD.

The comparison in panel b of Figure 7 more clearly reveals the accuracy gain brought by fusing ERA5 data. Compared to the GNSS-Kriging model, which relies solely on the interpolation of 10 modeling stations, RK ZTD likewise exhibits an RMSE reduction at nearly all stations. Notably, the regions where the GNSS-Kriging model performed worst are shown in panel b of Figure 7, specifically the southwestern coastal stations between 51°N and 52°N and the northeastern stations north of 52.5°N, all with RMSEs exceeding 10 mm, which corresponds precisely to the regions where the RK ZTD model demonstrates the most significant improvement rates in panel b of Figure 7. This spatial correlation provides strong evidence that the proposed residual correction method can effectively utilize the continuous spatial information provided by the ERA5 grid data to precisely compensate for the interpolation shortfalls caused by the uneven distribution of GNSS stations, particularly in regions where modeling stations are sparse.

To examine the performance stability of the models under different seasonal conditions, this section presents the RMSE distributions for the three models across the four seasons. The results are presented as box plots in Figure 8 and summarized numerically in

Table 3. Statistical summary of seasonal RMSE (mm) for the RK ZTD, GNSS-Kriging ZTD, and ERA5 ZTD.

Season	RMSE (mm)
	RK ZTD	GNSS-Kriging ZTD	ERA5 ZTD
Spring	4.5	7.9	9.7
Summer	6.8	10.0	11.8
Autumn	6.6	9.3	19.9
Winter	4.4	7.8	10.9

.

Two primary features can be observed in Figure 8. First, the accuracy of all models exhibits distinct seasonal characteristics, with RMSE values in summer generally being higher than in spring and winter. This aligns with the physical characteristics of the region, which experiences more active water vapor dynamics during the summer. Second, the proposed RK ZTD consistently demonstrates the highest accuracy. As detailed in Table 3, the RK ZTD maintains the lowest RMSE throughout the year, ranging from 4.4 mm in winter to 6.8 mm in summer, significantly outperforming both the GNSS-Kriging ZTD and the ERA5 ZTD.

Specifically, during summer, the season with the most active water vapor, the RK ZTD effectively suppresses the error growth, keeping the RMSE at a low level of 6.8 mm, whereas the ERA5 ZTD reaches 11.8 mm. Concurrently, as shown in Figure 8, the interquartile range of RK ZTD remains narrow in all seasons, indicating high consistency in performance across the different validation stations. It is particularly noteworthy that in the autumn statistics shown in Table 3, the RMSE of the ERA5 ZTD surges abnormally to 19.9 mm, the worst performance among all seasons. In contrast, the RK ZTD remains stable with an RMSE of 6.6 mm, correcting the error by nearly 67%. This anomaly in autumn is closely related to an extreme weather event during the study period.

However, a significant anomaly is observed in the autumn statistics. As shown in Table 3, the RMSE of the ERA5 ZTD surges abruptly to 19.9 mm, which is the worst performance among all seasons and far exceeds the typical error range. In sharp contrast, the RK ZTD remains stable with an RMSE of 6.6 mm, effectively correcting this large deviation. This unusual degradation in ERA5 accuracy suggests the presence of specific meteorological factors or extreme events during this period that the reanalysis data failed to capture accurately. To investigate the physical mechanism behind this autumn anomaly, a detailed analysis of a specific extreme weather event occurring in November will be presented in the following section.

To further investigate the spatial distribution characteristics of the seasonal differences observed in Figure 8, Figure 9 presents the per-station RMSE spatial distribution results for the three models across the four seasons. This figure provides richer spatial dimension information for the box plots in Figure 8 and validates the seasonal performance of the models.

A column-wise comparison clearly shows that, regardless of the season, the RMSE of the proposed RK ZTD model is significantly lower than that of ERA5 ZTD and GNSS-Kriging ZTD at all stations. Based on the legend, RK ZTD presents as dark blue or blue-green at the vast majority of stations in all four seasons, indicating an RMSE below 10 mm. Especially during summer and autumn, when water vapor is most active and the errors of ERA5 ZTD and GNSS-Kriging ZTD are generally high, RK ZTD still effectively controls the RMSE at a low level, demonstrating superior performance.

A row-wise comparison allows for the analysis of each model’s seasonal performance and spatial dependency. The error of ERA5 ZTD is worst in autumn, and its high-error stations show clear spatial clustering, mainly distributed in the central-eastern inland region of the Netherlands, which is highly consistent with the statistical results from the box plots in Figure 8. The GNSS-Kriging model also exhibits a significant accuracy decrease in summer and autumn, but its spatial characteristics differ from ERA5, with its high-error areas appearing more dispersed along the coast and in the southwest.

In contrast, the performance of the RK ZTD model is the most outstanding. It not only maintains the lowest RMSE level in all seasons but also features the most uniform spatial distribution, avoiding the obvious high-error “clusters” exhibited by the other two models. This result again confirms that the RK ZTD model proposed in this study possesses high accuracy and high spatial robustness under different seasonal and complex meteorological conditions.

3.4. Impact of Storm Ciarán on ZTD Accuracy

To investigate the physical mechanism behind the significant accuracy degradation of the ERA5 ZTD observed in November, this study conducted a detailed meteorological analysis of the extreme weather event, Storm Ciarán, which impacted the study area from 1 to 4 November. Figure 10 illustrates the temporal variations of four key meteorological parameters including relative humidity, temperature, atmospheric pressure, and wind speed at the three representative stations from 26 October to 9 November.

As shown in panel c of Figure 10, the pressure variation serves as the most direct indicator of the trajectory of the storm. Prior to 31 October, the pressure remained stable above 990 hPa. However, commencing on 1 November, a rapid and precipitous drop was recorded across all stations, hitting a deep trough on 2 November. The Deelen station (52°N, 6°E) recorded a minimum pressure of approximately 966 hPa, while the coastal Rotterdam station (52°N, 4.5°E) dropped even lower. This sharp V-shaped pressure curve signifies the rapid passage of a deep low-pressure system. Following the departure of the storm, pressure rebounded sharply from 3 November, recovering to normal levels above 1000 hPa by 7 November.

The wind dynamics exhibited a strong correlation with the pressure drop as depicted in panel d of Figure 10. Wind speeds remained relatively low before the event but surged continuously starting 31 October. On 2 November, coinciding with the pressure minimum, wind speeds peaked across all stations. Notably, observations at the Rotterdam station were consistently higher than those at the inland stations, with a peak speed exceeding 50 km/h, reflecting the severe impact of the storm on coastal areas. Additionally, the passage of the storm induced significant fluctuations in hydrothermal conditions. Panel a of Figure 10 shows that relative humidity experienced violent oscillations during the storm, dropping sharply to nearly 60% on 3 November after the rainband passed, indicating a rapid air mass exchange. Similarly, temperature exhibited irregular diurnal patterns during the event, as shown in panel b of Figure 10, further confirming the atmospheric instability.

These extreme meteorological conditions provide a physical explanation for the large systematic bias and RMSE observed in the ERA5 ZTD during this period. The Zenith Hydrostatic Delay is strictly proportional to surface pressure. The drastic pressure drop of over 30 hPa within 48 h caused a massive variation in the ZHD component. ERA5, as a reanalysis model with limited temporal resolution and spatial grid averaging, often fails to perfectly synchronize with such rapid local pressure plunges, leading to significant residuals in the hydrostatic component. Furthermore, the violent fluctuations in wind and humidity imply intense turbulent water vapor transport. Under such dynamic conditions, the moisture distribution becomes highly heterogeneous. The ERA5 model tends to smooth out these local extremes, whereas the GNSS stations sensitively capture the real-time and high-frequency water vapor variations.

Consequently, the inability of the ERA5 background field to accurately replicate these extreme physical changes resulted in the abnormal bias observed in the autumn statistics. In contrast, the RK ZTD model successfully corrected these background errors by incorporating high-precision GNSS observations, which inherently contain this real-time meteorological information. This demonstrates the superior robustness of the proposed method even under such extreme weather conditions.

To further visually analyze the impact of the extreme weather event on the tropospheric delay, Figure 11 depicts the time series of GNSS ZTD for the 10 modeling stations alongside the background validation stations from 26 October to 9 November. In this visualization, the time series of the 10 modeling stations are highlighted with distinct colored lines, whereas the data from the 26 validation stations are plotted in grey to form a background envelope. The distinct phases of the storm approach and departure are marked with vertical dashed lines. In addition, the mean ERA5 ZTD derived from the ERA5 ZTD grids at the 10 modeling station locations is plotted as a dashed blue curve to facilitate a direct comparison with the GNSS ZTD time series during the storm period. This specific plotting strategy serves to validate the spatial representativeness of the modeling network. By superimposing the trajectories of the modeling stations onto the background variability range defined by the validation stations, Figure 11 demonstrates that the variation patterns of the modeling stations are highly consistent with the overall regional trend. This confirms that the selected 10 modeling stations effectively capture the dominant atmospheric dynamics and ZTD evolution characteristics of the entire study area, even under the volatile conditions of an extreme weather event.

As illustrated in Figure 11, the GNSS ZTD time series exhibited a highly consistent variation pattern across all stations, indicating that the region was controlled by a large-scale weather system. Prior to the arrival of the storm, the GNSS ZTD values fluctuated at a relatively high level between 2360 mm and 2430 mm. However, coinciding with the approach of Storm Ciarán on 2 November, the ZTD values at all stations experienced a precipitous decline. Within a short period, the ZTD plummeted from approximately 2430 mm to a trough of around 2300 mm. This sharp V-shaped drop in the ZTD sequence is physically consistent with the drastic reduction in atmospheric pressure observed during the storm passage, as the Zenith Hydrostatic Delay component of ZTD is strictly proportional to the surface pressure. The dashed blue curve in Figure 11 further shows that the ERA5 ZTD is generally higher than the GNSS-derived ZTD throughout this period, which is consistent with the systematic ERA5 overestimation identified at the EBRG station in Figure 3 and the corresponding negative residuals. During the storm peak, the offset between the ERA5 mean curve and the GNSS station envelope becomes more evident at certain hours, indicating that the ERA5 ZTD grids do not fully reproduce the intensity and rapid evolution of the extreme event captured by the GNSS observations.

This extreme atmospheric variability provides the fundamental explanation for the large systematic bias and RMSE observed in the ERA5 ZTD during this specific period. The reanalysis data, limited by their spatial and temporal resolution, often fail to fully capture the extreme depth of the low-pressure center and the rapid rate of the pressure drop. Since the hydrostatic delay constitutes the majority of the total delay, even a small discrepancy in the modeled surface pressure by ERA5 leads to a significant systematic error in the calculated ZTD. Furthermore, the ZTD time series in Figure 11 exhibits intensified high-frequency oscillations during the storm duration from 2 November to 4 November. These fluctuations reflect the intense turbulent transport of water vapor and the unstable atmospheric stratification. The ERA5 model tends to smooth out these local and rapid variations, whereas the GNSS observations sensitively record these real-time atmospheric dynamics. Consequently, the inability of the ERA5 background field to accurately replicate these extreme pressure drops and moisture anomalies resulted in the significant deviation from the GNSS truth values. This analysis confirms that the high-precision GNSS ZTD serves as a crucial reference for correcting numerical weather prediction models during extreme weather events.

4. Discussion

The performance of the RK ZTD model is not simply reflected by improved statistical metrics but is primarily attributable to the complementary integration of reanalysis data and ground-based GNSS ZTD products. ERA5 provides a spatially continuous grid that captures the large-scale structure and temporal evolution of tropospheric variability. However, systematic biases may arise from model parameterization and data assimilation uncertainties. The residual correction strategy proposed in this study effectively constrains the ERA5 ZTD grid using GNSS-derived ZTD, thereby reducing the systematic offset commonly associated with numerical weather prediction outputs. Related refinement strategies have been explored in other remote sensing applications, where prior information is used to improve the local accuracy of an initial prediction [34]. The relatively uniform error distribution observed across the study area further indicates that the proposed framework alleviates the spatial limitations inherent in conventional interpolation techniques, such as Kriging. In areas with sparse station coverage, purely distance-based interpolation methods may suffer from reduced stability and degraded accuracy. By contrast, the ERA5 grid provides a spatially continuous reference, allowing the residual correction to remain robust even in regions located far from reference stations. It should be noted that this study focuses on the performance of RK ZTD under a sparse modeling-station configuration, and denser GNSS station deployments are expected to further strengthen the residual constraints and potentially improve the correction performance.

A critical finding concerns the model’s resilience to volatile weather events. As shown in Figure 8, a pronounced anomaly occurs in autumn, where the RMSE of the ERA5 ZTD surges to 19.9 mm. This degradation is corroborated by widespread high-error clusters in Figure 9, directly linked to the passage of Storm Ciarán. Meteorological analysis, as presented in Figure 10, reveals that the storm induced a rapid pressure drop and wind speed surge, resulting in a synchronized V-shaped decline in the ZTD time series, as illustrated in Figure 11. This aligns with Lian et al. [35], who found that ERA5 ZTD often exhibits substantial biases during intense cyclonic systems, with RMS errors significantly larger than those derived from GNSS ZTD. In contrast, the positive systematic deviation observed in ERA5 ZTD during the storm is mainly attributable to the well-documented wet bias of numerical weather prediction models. As noted by He et al. [36] and Li et al. [37], NWP models often struggle to respond to rapid pressure drops and tend to overestimate atmospheric water vapor content and cloud liquid water path within active convective systems. This behavior leads to an inflated wet delay component in reanalysis products, which cannot be corrected in real time. By incorporating GNSS-derived residuals, the RK ZTD model effectively bridges this gap, capturing high-frequency atmospheric variations that are not adequately represented in ERA5 and maintaining a stable RMSE of 6.6 mm even under such highly volatile conditions.

Although the RK ZTD model demonstrates robust performance in the present study area, which is characterized by relatively flat terrain, its broader applicability to regions with complex topography warrants further investigation. Tropospheric delay exhibits a strong dependence on elevation, and neglecting the non-linear vertical stratification of ZTD may introduce systematic errors when applying two-dimensional interpolation schemes in mountainous regions. While elevation differences exert only a minor influence in the Netherlands, extending the proposed framework to areas with significant relief will require explicit vertical adjustment. The incorporation of refined height-dependent correction models, such as those based on cubic polynomial fitting, may help reconcile ZTD discrepancies across different elevations. Future research should therefore focus on integrating vertical normalization strategies prior to interpolation to improve the physical consistency and general applicability of the proposed approach.

5. Conclusions

This study constructed an RK ZTD model to reconstruct high-precision regional ZTD grids under sparse GNSS station conditions. The proposed strategy fuses the ERA5 grid with GNSS-derived ZTD products. By interpolating residuals via Ordinary Kriging, the method directly corrects systematic biases in the reanalysis data and suppresses the spatial distortion inherent in sparse networks. This integration yields a residual grid with superior spatial stationarity compared to raw ZTD data.

The model achieves an annual RMSE of 5.7 mm, representing accuracy improvements of 58.4% over the original ERA5 ZTD and 35.4% over the traditional GNSS-Kriging model. Furthermore, the residual correction reduces the absolute systematic bias from 5.2 mm in the ERA5 grid to a negligible 0.4 mm, ensuring uniform accuracy across the study area regardless of station distance.

Crucially, the model maintains robustness under extreme meteorological conditions. During Storm Ciarán, where the standalone ERA5 product failed to capture rapid pressure drops due to “wet bias,” the RK ZTD model accurately tracked high-frequency atmospheric dynamics. By incorporating real-time GNSS-derived residuals, the model maintained an RMSE of 6.6 mm even under volatile conditions, effectively compensating for reanalysis deficiencies.

In summary, the RK ZTD model generates spatiotemporally continuous ZTD grids resilient to seasonal and extreme weather variations. This framework provides a reliable external constraint for PPP and a precise calibration tool for meteorological and hydrological monitoring.