Correction method for scale bias in GNSS-IR sea level retrieval

Xiaolei Wang 1✉ Emailchd_wxl@qq.com

Minfeng Song 1

Xiufeng He 1

1 School of Earth Sciences and Engineering Hohai University 211100 NanKing Jiangsu China

Xiaolei Wang, Minfeng Song, Xiufeng He

1 School of Earth Sciences and Engineering, Hohai University, NanKing, Jiangsu 211100, China

*Corresponding author: Xiaolei Wang (chd_wxl@qq.com)

Abstract

Since the principles and methods of Global Navigation Satellite System-Interferometric Reflectometry (GNSS-IR) sea level monitoring technology based on geodetic receivers were first proposed, the technology has undergone over a decade of development. Currently, this method typically utilizes the frequency of the SNR signal, converts it into the reflector height (RH), and further derives the sea level. Height variation error and tropospheric delay error are widely recognized as systematic errors. Both have corresponding mathematical models that can be used for error mitigation. Random errors and gross errors are generally handled or constrained through quality control or multi-GNSS combination methods. In 2024, a correction method based on the initial phase of the SNR arc was proposed, and related studies suggest it may exhibit superior performance in correcting errors. Beyond the aforementioned errors, many studies have observed a scale bias in inversion results that appears to be correlated with sea-level magnitude. This bias is generally believed to be related to tropospheric delay error, as the magnitude of the tropospheric delay is associated with RH magnitude. However, in many studies, the magnitude of the scale bias exceeds the magnitude of tropospheric delay model value. Through analysis, we infer that the scale bias in some studies is related to the assumption of uniform sea level changes within the time windows during multi-GNSS combination processing. This assumption introduces errors, leading to underestimation of tidal peaks and overestimation of tidal troughs. The primary objective of this paper is to correct scale biases in GNSS-IR sea level retrieval, including those exhibited by raw RHs and those observed in combined RHs. Considering that stations with larger tidal amplitudes tend to show more significant scale biases, GNSS data from two such stations, BRST and HKQT, were used. The results indicate that the phase-based error correction method is more effective than traditional correction method in reducing scale bias in raw RHs. Additionally, we propose a method to mitigate scale bias caused by the assumption of uniform changes. Results indicate that this method effectively corrects scale bias in the combined RHs.

Keywords:

GNSS-Interferometric Reflectometry

sea level

scale bias correction

tropospheric delay error

phase-based error correction method

multi-GNSS combination

1. Introduction

In 2000, Anderson (2000) proposed the idea of using Global Navigation Satellite System (GNSS) geodetic receivers for sea level monitoring. In 2013, Larson et al. (2013a) introduced a method for estimating reflector height (RH) based on the frequency of SNR, laying the foundation for the theory and methodology of GNSS-Interferometric Reflectometry (GNSS-IR) sea level retrieval. Height variation error and tropospheric delay error are widely recognized as systematic errors in GNSS-IR tide level inversion. Height variation error arises from the assumption of a static reflective surface without considering its undulations. By accounting for the RH variation rate (

$\:\dot{h}$

), Larson et al. (2013b) derived a new mathematical relationship between SNR frequency and RH, enabling the development of a correction model. The traditional height variation correction strategy involves a two-step iterative correction algorithm. The key to this algorithm is solving

$\:\dot{h}$

, typically obtained by curve fitting or tidal analysis of the raw RHs, followed by deriving

$\:\dot{h}$

from the fitted curve. The error is then computed based on

$\:\dot{h}$

and subtracted from the raw RHs (Löfgren et al., 2014; Larson et al., 2017). Tropospheric delay error refers to the impact of signal bending and delays caused by the troposphere on RH inversion values. A common correction strategy is to use classical tropospheric models and mapping functions to calculate the tropospheric delay error (Williams and Nievinski, 2017).

Multi-GNSS combination method can also be used to correct height variation errors. This approach simultaneously solves for RH and

$\:\dot{h}$

, incorporating the height variation model into the observation function (Roussel et al., 2015; Wang et al., 2019). Since the simultaneous solution theoretically provides more accurate

$\:\dot{h}$

values and is more robust to gross and random errors, this method often achieves better correction results when redundant observation data are available (e.g., with multi-constellation multi-frequency GNSS antennas). Common multi-GNSS combination methods include least squares, weighted least squares, and robust estimation (Roussel et al., 2015; Tabibi et al., 2017; Wang et al., 2019), which are fundamentally based on least squares or extended least squares techniques. While this method can improve accuracy using redundant data, it typically assumes uniform sea level changes within the processing time window. This assumption introduces error, leading to underestimated tidal peaks and overestimated tidal troughs. Moreover, the longer the window, the greater the impact.

In 2024, a new error correction method—the phase-based error correction method—was proposed. Theoretically, the initial phase should contain information about systematic errors, including height variation error and tropospheric delay error. Wei et al. (2024) experimentally verified a strong linear correlation between the initial phase of the SNR arc and sea-level retrieval error. They proposed a method to estimate retrieval errors using a linear function fitted from sea level data recorded by tide gauges, thereby improving sea level inversion accuracy. Wang et al. (2025) mathematically derived the relationship between the initial phase and the error, and resolved the major challenge of integer ambiguity, enabling the phase-based correction method to operate solely on GNSS data without external inputs. However, scale biases were observed in the combined RH results of Wang et al. (2025). They suggested in their discussion that these biases might be related to tropospheric delay error or the assumption of uniform change within the processing window.

The primary goal of this paper is to correct scale biases in GNSS-IR sea level retrieval, including those observed in raw RHs and in combined RHs. Height variation error correction, tropospheric delay error correction and phase-based error correction methods were used and compared. The principles of these three correction methods are presented in Section 2. Furthermore, considering that errors due to the assumption of uniform change within the processing window depend on window length, we propose a method to correct such errors by comparing and modeling multi-GNSS combined results across windows of varying lengths. The corresponding methodology is detailed in Section 3. Considering that stations with larger tidal amplitudes tend to show more significant scale biases, GNSS data from two such stations, BRST and HKQT, were used. The results are analyzed in Section 4.

2. Theory

According to the coherent superposition theory of direct and reflected signals, the detrended SNR

$\:\delta\:\text{S}\text{N}\text{R}$

exhibits a gradually decaying cosine function with respect to

$\:\text{s}\text{i}\text{n}e$

, where

$\:e$

representing the elevation (Nievinski and Larson 2014a,b):

$\:\delta\:\text{S}\text{N}\text{R}=\text{A}\text{m}\text{p}\bullet\:{\text{e}}^{-{k}^{2}{d}^{2}{\text{s}\text{i}\text{n}}^{2}e}\bullet\:\text{c}\text{o}\text{s}(\frac{2\pi\:\text{s}\text{i}\text{n}e}{\lambda\:}f+{\varphi\:}_{0})$

where

$\:f$

is the oscillation frequency,

$\:d$

the decay factor,

$\:\text{A}\text{m}\text{p}$

is the amplitude,

$\:{\varphi\:}_{0}$

is the initial phase,

$\:\:k=\frac{2\pi\:}{\lambda\:}$

$\:\lambda\:$

is the signal wavelength.

In the absence of various errors, the extra path traveled by the reflected signal compared to the direct signal is

$\:D=2h\text{s}\text{i}\text{n}e$

, where

$\:h$

is the RH indicating the height between surface and antenna. Assuming

$\:h$

and

$\:e$

remain constant during the corresponding period, the phase difference between the direct signal and the reflected signal based on

$\:D$

is calculated and then its derivative of with respect to

$\:\text{s}\text{i}\text{n}e$

is:

$\:2\pi\:f=\frac{2\pi\:}{\lambda\:}\frac{\delta\:D}{\text{s}\text{i}\text{n}e}=\frac{4\pi\:}{\lambda\:}\stackrel{\sim}{h}$

where

$\:\stackrel{\sim}{h}$

is the RH with assumption that

$\:h$

and

$\:e$

remain constant. By performing spectral analysis of

$\:\delta\:\text{S}\text{N}\text{R}$

using the Lomb-Scargle Periodogram (LSP; Lomb 1976; Scargle 1982), the frequency

$\:f$

can be obtained. According to this formula, the

$\:\stackrel{\sim}{h}$

is given by

$\:\stackrel{\sim}{h}=\left(\lambda\:f\right)/2$

2.1. Height variation error

When considering the variation rates of

$\:h$

and

$\:e$

$\:\dot{h}=\frac{d\text{h}}{dt}$

and

$\:\dot{e}=\frac{de}{dt}$

, Eq. (2) can be derived as:

$\:2\pi\:f=\frac{2\pi\:}{\lambda\:}\frac{\delta\:D}{\text{s}\text{i}\text{n}e}=\frac{4\pi\:}{\lambda\:}\frac{\text{t}\text{a}\text{n}\left(e\right)}{\dot{e}}\dot{h}+\frac{4\pi\:}{\lambda\:}h\:$

From Equations (2) and (3), the relationship between

$\:\stackrel{\sim}{h}$

and

$\:h$

is:

$\:\stackrel{\sim}{h}=h+\frac{\text{t}\text{a}\text{n}\left(e\right)}{\dot{e}}\dot{h}=h+{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

where

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}=\frac{\text{t}\text{a}\text{n}\left(e\right)}{\dot{e}}\dot{h}$

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

is the height variation error, i.e., the correction model of the height variation error.

2.2. Tropospheric delay error

Tropospheric refraction in the troposphere is a common source of error in signal propagation. The reflected signal has a greater tropospheric delay than the direct signal caused by additional path. Consequently, it will lead a larger calculated RH than the actual RH. Williams and Nievinski (2017) improved a traditional method to correct this error by integrating the Global Temperature and Pressure (GPT2w) model with the VMF1 mapping function (Böhm et al., 2004, 2006) to correct for tropospheric delays. The tropospheric delay

$\:{\tau\:}_{\text{t}\text{r}\text{o}\text{p}}$

can be modeled as:

$\:{\tau\:}_{\text{t}\text{r}\text{o}\text{p}}=2\varDelta\:{{\uptau\:}}_{\text{h}\text{y}\text{d}}^{z}\times\:{m}_{\text{h}\text{y}\text{d}}\left(e\right)+2\varDelta\:{{\uptau\:}}_{\text{w}\text{e}\text{t}}^{z}\times\:{m}_{\text{w}\text{e}\text{t}}\left(e\right)$

where the superscript

$\:z$

denotes the zenith delay, while the subscript

$\:\text{h}\text{y}\text{d}$

and

$\:\text{w}\text{e}\text{t}$

represents the hydrostatic and wet components.

$\:\varDelta\:{{\uptau\:}}_{}^{z}={{\uptau\:}}_{}^{z}\left(-h\right)-{{\uptau\:}}_{}^{z}\left(0\right)$

represents the zenith delay difference across antenna and reflecting surface.

$\:m$

is the mapping function.

$\:{{\uptau\:}}_{\text{h}\text{y}\text{d}}^{z}$

and

$\:{{\uptau\:}}_{\text{w}\text{e}\text{t}}^{z}$

can be estimated from GPT2w.

$\:{m}_{\text{h}\text{y}\text{d}}$

and

$\:{m}_{\text{w}\text{e}\text{t}}$

can be estimated from VMF1. The tropospheric delay error

$\:{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}$

in RH caused by

$\:{\tau\:}_{\text{t}\text{r}\text{o}\text{p}}$

can be derived from Equations (2) and (5) (Williams and Nievinski, 2017):

$\:{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}=\frac{1}{2}\frac{{\tau\:}_{\text{t}\text{r}\text{o}\text{p}}}{\sigma\:\text{s}\text{i}\text{n}e}$

Traditional correction methods combine the corrections for height variation error and tropospheric delay error:

$\:h=\stackrel{\sim}{h}-{{\Delta\:}h}_{\text{v}\text{a}\text{r}\text{i}}-{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}=\stackrel{\sim}{h}-{{\Delta\:}h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

where

$\:{{\Delta\:}h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}={{\Delta\:}h}_{\text{v}\text{a}\text{r}\text{i}}+{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}$

2.3. Error related to SNR initial phase

As mentioned above, various errors exist in

$\:\stackrel{\sim}{h}$

. The sum of these errors can be represented as

$\:{\Delta\:}h$

. Substituting

$\:h=\stackrel{\sim}{h}-{\Delta\:}h$

into Eq. (1) yields:

$\:\delta\:\text{S}\text{N}\text{R}=\text{A}\text{m}\text{p}\bullet\:{\text{e}}^{-{k}^{2}{d}^{2}{\text{s}\text{i}\text{n}}^{2}e}\bullet\:\text{cos}\left(\frac{4\pi\:\text{s}\text{i}\text{n}e}{\lambda\:}\stackrel{\sim}{h}-\frac{4\pi\:\text{s}\text{i}\text{n}e}{\lambda\:}{\Delta\:}h+{\varphi\:}_{0}\right)$

Using least squares estimation (Larson et al. 2008; Chew et al. 2013; Huang et al. 2024) or optimization methods (Wang et al., 2025), the initial phase

$\:{\varphi\:}_{0}$

in Eq. (8) can be estimated, with the fitted value denoted as

$\:{\varphi\:}_{\text{f}\text{i}\text{t}}$

. Considering the integer ambiguity during fitting, the following relationship can be established:

$\:{\varphi\:}_{\text{f}\text{i}\text{t}}+2\pi\:{N}_{\varphi\:}=-\frac{4\pi\:\text{s}\text{i}\text{n}e}{\lambda\:}{\Delta\:}h+{\varphi\:}_{0}$

where

$\:{N}_{\varphi\:}$

represents the integer ambiguity of the phase. The method for determining

$\:{N}_{\varphi\:}$

and the phase-based error correction approach are detailed in Wang et al. (2025) and will not be elaborated here. Based on the estimated initial phase

$\:{\varphi\:}_{\text{f}\text{i}\text{t}}$

, the corrected value

$\:{\Delta\:}h$

can be calculated as:

$\:{{\Delta\:}h}_{\varphi\:}=\frac{\lambda\:}{4\pi\:\text{s}\text{i}\text{n}e}\left({\varphi\:}_{0}-2\pi\:{N}_{\varphi\:}-{\varphi\:}_{\text{f}\text{i}\text{t}}\right)$

where

$\:{{\Delta\:}h}_{\varphi\:}$

specifically refers to the phase-based correction value, and

$\:h=\stackrel{\sim}{h}-{{\Delta\:}h}_{\varphi\:}$

Theoretically,

$\:{{\Delta\:}h}_{\varphi\:}$

encompasses systematic errors, including the height variation error and the tropospheric delay error, as well as other potential systematic error sources not yet recognized.

3. Method

3.1. Multi-GNSS combination Method

A sliding window is used to segment the

$\:\stackrel{\sim}{h}$

sequence. Based on Equations (4) and (6), after traditional height variation and tropospheric delay corrections, the following equation can be derived:

$\:{\stackrel{\sim}{h}}_{i,j,l}-{{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}}_{i,j,l}={h}_{i,j,l}+\frac{\text{t}\text{a}\text{n}{e}_{i,j,l}}{{\dot{e}}_{i,j,l}}\dot{h}$

where the subscripts

$\:i,j,l$

represent that the parameter belongs to signal

$\:j$

and trajectory

$\:l$

in the

$\:i$

th window. Assume that within the

$\:i$

th window, the change in RH is uniform, with a rate of

$\:{\dot{h}}_{i}$

. In other words, it is assumed that the trajectory of RH within the window is linear (shorted for window linearization), the following equation holds:

$\:{h}_{i,j,l}=\left({t}_{i,j,l}-{t}_{i}\right){\dot{h}}_{i}+{h}_{i}$

Using Equations (12) and (13), the system of equations within the window can be written as:

$\:\left[\begin{array}{c}\dots\:\\\:\begin{array}{c}{\stackrel{\sim}{h}}_{i,j,l-1}-{{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}}_{i,j,l-1}\\\:{\stackrel{\sim}{h}}_{i,j,l}-{{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}}_{i,j,l}\\\:{\stackrel{\sim}{h}}_{i,j,l+1}-{{{\Delta\:}h}_{\text{t}\text{r}\text{o}\text{p}}}_{i,j,l+1}\end{array}\\\:\dots\:\end{array}\right]=\left[\begin{array}{c}\dots\:\\\:\begin{array}{cc}\frac{\text{t}\text{a}\text{n}{e}_{i,j,l-1}}{{\dot{e}}_{i,j,l-1}}+({t}_{i,j,l-1}-{t}_{i})&\:1\\\:\frac{\text{t}\text{a}\text{n}{e}_{i,j,l}}{{\dot{e}}_{i,j,l}}+({t}_{i,j,l}-{t}_{i})&\:1\\\:\frac{\text{t}\text{a}\text{n}{e}_{i,j,l+1}}{{\dot{e}}_{i,j,l+1}}+({t}_{i,j,l+1}-{t}_{i})&\:1\end{array}\\\:\dots\:\end{array}\right]\left[\begin{array}{c}{\dot{h}}_{i}\\\:{h}_{i}\end{array}\right]$

Robust estimation methods are used to solve the matrix Eq. (14), which effectively handles outliers in the RH retrieval sequence (Wang et al., 2019, 2021). If instead of using the traditional correction method, a phase-based error model is used for correction, Eq. (14) can be written as:

$\:\left[\begin{array}{c}\dots\:\\\:\begin{array}{c}{\stackrel{\sim}{h}}_{i,j,l-1}-{{{\Delta\:}h}_{\varphi\:}}_{i,j,l-1}\\\:{\stackrel{\sim}{h}}_{i,j,l}-{{{\Delta\:}h}_{\varphi\:}}_{i,j,l}\\\:{\stackrel{\sim}{h}}_{i,j,l+1}-{{{\Delta\:}h}_{\varphi\:}}_{i,j,l+1}\end{array}\\\:\dots\:\end{array}\right]=\left[\begin{array}{c}\dots\:\\\:\begin{array}{cc}({t}_{i,j,l-1}-{t}_{i})&\:1\\\:({t}_{i,j,l}-{t}_{i})&\:1\\\:({t}_{i,j,l+1}-{t}_{i})&\:1\end{array}\\\:\dots\:\end{array}\right]\left[\begin{array}{c}{\dot{h}}_{i}\\\:{h}_{i}\end{array}\right]$

Equation (15) also uses robust estimation to solve the system of equations within each window.

3.2. Method to correct scale bias caused by window linearization

Equations (14) and (15) are both based on the assumption of uniform RH variation within the window. However, this assumption clearly does not align with the actual tide variations. Theoretically, this assumption will inevitably introduce an error, leading to the underestimation of tidal peaks and overestimation of tidal troughs, which manifests as scale bias. Moreover, the longer the window, the greater the impact. To theoretically avoid this scale error, a short time window should be used. However, in practical applications, more redundant data leads to more accurate and reliable estimates. Considering that short time windows exhibit smaller scale biases but fewer inversions and lower precision, while long time windows have larger scale biases but more inversions and better precision, this section proposes a scale bias correction method for long time window combination values based on short time window combination values. The steps for this method are as follows:

1. Use long time windows to separate

$\:\stackrel{\sim}{h}$

and calculate the combination system and obtain the estimate

$\:{h}_{i,\text{l}\text{o}\text{n}\text{g}}$

2. Use a short time window to separate

$\:\stackrel{\sim}{h}$

and calculate the system and obtain the estimate

$\:{h}_{i,\text{s}\text{h}\text{o}\text{r}\text{t}}$

3. Interpolate

$\:{h}_{i,\text{l}\text{o}\text{n}\text{g}}$

based on the corresponding times of

$\:{h}_{i,\text{s}\text{h}\text{o}\text{r}\text{t}}$

, and fit a linear relationship (

$\:\text{y}=\text{a}\text{x}+\text{b}$

) between them.

4. Based on the linear relationship, calculate the scale correction value

$\:{h}_{i,\text{l}\text{o}\text{n}\text{g},\:\text{s}\text{l}\text{o}\text{p}\text{e}\_\text{c}\text{o}\text{r}\text{r}}$

for

$\:{h}_{i,\text{l}\text{o}\text{n}\text{g}}$

$\:{h}_{i,\text{l}\text{o}\text{n}\text{g},\:\text{s}\text{l}\text{o}\text{p}\text{e}\_\text{c}\text{o}\text{r}\text{r}}=\text{a}{\times\:h}_{i,\text{l}\text{o}\text{n}\text{g}}+b$

Result

4.1. Site description

Considering that stations with larger tidal amplitudes tend to exhibit more pronounced scale biases, GNSS data from two such stations, BRST and HKQT, were utilized in this study. BRST is situated in Brest harbor on France's west coast (48.4°N, 4.5°W), where sea-level changes can reach up to 8.0 m. A Trimble NETR9 GNSS receiver with a Trimble TRM57971.00 antenna is positioned approximately 17 m above sea level. The station supports multi-GNSS signals, including GPS (L1, L2, L5), GLONASS (G1, G2), Galileo (E1, E5A, E5B, E6), and BDS (B1, B2, B3). 1 Hz GNSS data during 2022 DOY 032 to 058 were used. A tide gauge managed by Réseaux de Référence des Observations Marégraphiques (REFMAR) is located less than 500 m from BRST. According to Fig. 1 (right), the usable ranges are 5°–12° elevation range for the azimuth range of 130°–165° and 13°–20° elevation range for the azimuth range of 165°–330°.

Station HKQT is located in Quarry Bay, Hong Kong (114.21°E, 22.29°N). It is equipped with a Trimble NetR9 receiver and a Trimble 59800.0 antenna. Observations at HKQT include multi-system signals such as GPS (L1C/A, L2P, L2C, L5), GLONASS (G1C, G1P, G2C, G2P), Galileo (E1, E5, E7, E8), and BDS (B1, B2). The sampling frequency is 5 seconds. In this study, SNR arcs with an elevation angle range of 5°–15° were selected for data inversion, with the azimuth angle range limited to -60°–105°. The data used were collected from 2022 DOY 001 to 200. Additionally, an IOC (Intergovernmental Oceanographic Commission) tide gauge located 2 m from the HKQT station provides measured tide-level data.

Fig. 1

Screenshot of first Fresnel zones with elevation angles of 5°, 10° and 20° projected on a Google Earth image for BRST with RH of 17 m (right) and HKQT with RH of 7 m (bottom). Plotted using the software provided by Roesler and Larson (2018).

4.2. Scale bias in RHs and the correction effectiveness of the three methods

Due to the surrounding environmental conditions at the BRST station, the reflection region is divided into two parts: 5°–12° elevation angle for the 130°–165° azimuth range and 13°–20° elevation angle for the 165°–330° azimuth range. As shown in Wang et al. (2025), the retrieval results for these two regions differ. Therefore, we analyze these two regions sequentially. First, we analyze the sea level retrieval results for the SNR arcs in the region with better performance: 5°–12° elevation angle for the 130°–165° azimuth range. Figure 2 presents the sea level retrieval values after applying height variation correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

), tropospheric delay correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

) and phase-based error correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\varphi\:}$

), along with the corresponding Van de Casteele diagram. Table 1 provides the Root Mean Square Error (RMSE) and Correlation Coefficient (CORR) values for the corresponding sea level retrievals.

Fig. 2

Sea level retrieval results for BRST used SNR arcs in 5°–12° elevation angle for the 130°–165° azimuth range. (Top): From top to bottom, subplots show the sea level retrievals after applying height variation correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

), tropospheric delay correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

) and phase-based error correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\varphi\:}$

) respectively. The black line represents the measured sea levels. (Bottom): From left to right, the Van de Casteele diagrams correspond to sea level retrievals after height variation correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

), tropospheric delay correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

) and phase-based error correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\varphi\:}$

) respectively.

Table 1

RMSE and CORR between the retrievals in Fig. 2 and measured sea levels.
Sea level retrievals with correction of			$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$		$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$		$\:{\varDelta\:h}_{\varphi\:}$
System	Signal	Number (pc/day)	RMSE (cm)	RMSE (cm)	CORR (%)	RMSE (cm)	RMSE (cm)	CORR (%)
GPS	L1	9.54	13.05	99.73	12.43	99.73	8.11	99.92
	L2	8.88	13.32	99.63	13.15	99.63	7.43	99.91
	L5	7.81	8.70	99.86	8.67	99.86	5.72	99.95
GLONASS	G1	5.62	10.93	99.93	9.86	99.93	5.57	99.97
GLONASS	G2	5.77	7.06	99.91	7.32	99.91	5.97	99.95
Galileo	E1	5.54	11.19	99.83	10.64	99.82	7.30	99.93
	E5A	5.69	9.22	99.84	9.32	99.84	9.67	99.85
	E5B	5.69	8.79	99.87	8.64	99.87	8.23	99.93
	E6	5.69	8.81	99.86	9.02	99.86	6.49	99.93
BDS	B1	8.23	8.21	99.87	8.27	99.87	5.64	99.94
	B2	8.08	9.36	99.87	9.07	99.87	7.87	99.90
	B3	1.08	14.34	99.57	14.05	99.57	8.80	99.89

According to Fig. 2, it can be seen that for multiple signals’ SNR arcs in 5°–12° elevation angle for the 130°–165° azimuth range at BRST, the retrievals after applying the three error correction strategies show good correlation with the measured sea levels. Figure 2 (bottom left) and Fig. 2 (bottom center) both exhibit a “leftward skew”. After applying the tropospheric delay correction, Fig. 2 (bottom center) does not show an observable correction of the skewed trend seen in Fig. 2 (bottom left). This suggests that the scale correction effect based on the traditional tropospheric delay correction method may not be as effective as expected. To rule out any potential issues with the program code, we reproduced Fig. 2 from Williams and Nievinski (2017) using our program and found identical results, eliminating the possibility of errors in the theory or code. This confirms that the traditional tropospheric delay correction method has a weak scale bias correction effect for this station in our experiment.

On the other hand, Fig. 2 (bottom right) shows a clear correction of the leftward skew observed in Fig. 2 (bottom left), even exhibiting a slight “rightward skew”. Figure 2 (bottom right) not only demonstrates the reduction of scale error by the phase-based correction method but also shows a more “slim” Van de Casteele diagram, indicating better retrieval accuracy. This improvement in precision is more clearly reflected in Table 1. The RMSE after

$\:{\varDelta\:h}_{\varphi\:}$

correction is approximately 30% smaller than the other two correction methods. The RMSE of the retrieval values after

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

correction is generally smaller than that after

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

correction, indicating that the traditional tropospheric delay correction method still provides some degree of correction.

Next, we analyze the sea level retrievals for the SNR arcs in the 13°–20° elevation angle for the 165°–330° azimuth range. Same steps were repeated.

Fig. 3

Sea level retrieval results for BRST used SNR arcs in 13°–20° elevation angle for the 165°–330° azimuth range. The interpretation of subplots is the same as in Fig. 2.

Table 2

RMSE and CORR between the retrievals in Fig. 3 and measured sea levels.
Sea level retrievals with correction of			$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$		$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$		$\:{\varDelta\:h}_{\varphi\:}$
System	Signal	Number (pc/day)	RMSE (cm)	RMSE (cm)	CORR (%)	RMSE (cm)	RMSE (cm)	CORR (%)
GPS	L1	31.15	23.97	99.00	21.99	99.01	25.86	98.81
	L2	31.08	21.02	99.24	19.32	99.26	22.97	99.03
	L5	22.38	22.52	99.02	21.78	99.01	25.01	98.76
GLONASS	R1	22.62	25.84	98.91	23.60	98.92	27.36	98.73
GLONASS	R2	21.92	24.02	99.14	21.30	99.14	26.13	98.84
Galileo	E1	17.92	22.33	99.15	20.73	99.14	24.92	98.92
	E5A	21.42	20.19	99.26	18.90	99.25	22.67	98.99
	E5B	19.77	20.22	99.33	18.41	99.32	23.55	98.98
	E6	21.88	20.70	99.35	18.39	99.34	23.19	99.02
BDS	B1	28.08	20.78	99.33	18.62	99.33	24.11	98.98
	B2	26.73	25.00	98.85	24.16	98.85	27.47	98.60
	B3	3.77	20.70	99.54	18.92	99.54	24.31	99.23

Compared to Fig. 2, the bottom subplots in Fig. 3 show that the results in this range have greater dispersion and lower accuracy. Considering the use of higher elevation angle range, such results are understandable. Although Fig. 3 (bottom) exhibits higher dispersion, similar conclusions to those in Fig. 2 can still be observed. In Fig. 3 (bottom center), the scale bias correction effect for Fig. 3 (bottom left) is not significant. However, Fig. 3 (bottom right) shows a clear scale correction effect. It is worth noting that there are two distinct clusters in the areas marked by the red box and near the zero value in Fig. 3. The clustering in the red-boxed area of Fig. 3 is caused by incorrect integer ambiguity determination. Table 2 also exhibits the same issue: the RMSE of the retrieval values after

$\:{\varDelta\:h}_{\varphi\:}$

correction is even higher. This is consistent with the findings of Wang et al. (2025). In the SNR arcs of this range, the higher elevation range may result in poor SNR quality, difficulty in fixing the initial phase, more gross errors, and other influences. These factors cause a bias between the errors estimated from

$\:\frac{\text{tan}\left(e\right)}{\dot{e}}\dot{h}$

and the true errors, leading to incorrect calculation of

$\:{N}_{\varphi\:}$

. Considering this, Wang et al. (2025) did not use data from this range to combine retrievals of multiple GNSS signals using Eq. (15). However, in this study, we used the data from Fig. 3 (bottom right) for multi-GNSS combination. Given the redundancy of observations in both Fig. 2 (bottom right) and Fig. 3 (bottom right), robust estimation should allow for the identification and removal or reduction of outliers in the red-boxed area of Fig. 3.

Last, we analyze the results for HKQT. Similar to the BRST station, we calculated and plotted the sea level retrievals after applying the three correction methods. During this process, we found that the retrievals of different signals exhibited significant inter-frequency offsets. As shown in Fig. 4 (bottom left), after applying height variation correction, the Van de Casteele diagram for different signals showed clear center offsets. We calculated the mean RH for each signal and found that the offset was related to the frequency. This type of inter-frequency offset has been observed in previous studies and at other stations (Santamaría-Gómez and Watson, 2016; Wang et al., 2021). Since this phenomenon does not occur at all stations but is seen at some, it suggests that the inter-frequency offset is likely related to the receiver’s or antenna’s hardware and software. The offset of RH for each signal, compared to the L1 signal, is presented in Table 3. We subtracted the inter-frequency offset from RHs and then applied the three correction methods (Fig. 4).

Fig. 4

Sea level retrieval results for HKQT used SNR arcs in 5°–15° elevation angle for − 60°–06° azimuth range. (Top): The same as in Fig. 2. (Bottom): From left to right, the Van de Casteele diagrams correspond to sea level retrievals after height variation correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

) without inter-frequency offset correction, and after height variation correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

), tropospheric delay correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

) and phase-based error correction (

$\:\stackrel{\sim}{h}-{\varDelta\:h}_{\varphi\:}$

) with inter-frequency offset correction.

Table 3

RMSE and CORR between the retrievals in Fig. 4 and measured sea levels.
Sea level retrievals with correction of				$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$		$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$		$\:{\varDelta\:h}_{\varphi\:}$
System	Signal	Offset (m)	Number (pc/day)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)
GPS	L1C/A	0	25.78	17.48	93.76	18.11	93.75	16.47	94.82
	L2P	0.10	22.74	11.65	97.28	11.93	97.28	9.02	98.26
	L2C	0.12	18.55	10.01	97.86	9.51	97.85	7.34	98.86
	L5	0.14	12.81	9.48	97.95	9.08	97.93	7.66	98.78
GLONASS	G1C/A	-0.03	20.51	16.56	94.66	17.33	94.66	15.55	95.29
	G1P	-0.03	20.93	14.58	96.95	14.13	96.94	12.27	97.21
	G2C/A	0.12	20.18	15.25	95.41	14.47	95.41	13.05	96.10
	G2P	0.12	22.40	11.09	97.64	10.83	97.63	10.00	97.94
Galileo	E1	0	10.35	17.26	94.76	17.76	94.71	15.84	95.48
	E5	0.12	10.41	13.61	96.44	12.97	96.40	12.89	96.91
	E7	0.12	10.39	14.46	96.20	13.67	96.16	12.90	96.92
	E8	0.12	10.40	14.84	96.79	13.26	96.74	13.44	97.02
BeiDou	B2	0.12	5.35	16.14	96.35	18.01	96.33	16.28	97.27
BeiDou	B3	0.12	0.89	8.28	98.93	8.04	98.92	5.76	99.42

Based on Fig. 4, it can be seen that for multiple signals at the HKQT station, the retrievals after applying the three error correction strategies show good correlation with the measured sea levels. Figure 4 (second subplot from the left) and Fig. 4 (second subplot from the bottom right) both show slight leftward skew in the scale bias, while Fig. 4 (bottom right) does not show any noticeable skew. Similar to Fig. 2, Fig. 4 does not display a significant correction of scale bias by the traditional tropospheric delay correction method, but the phase-based correction method demonstrates a clear improvement in scale bias, as well as a more “slim” Van de Casteele diagram. The results in Table 3 are consistent with those in Table 1. The RMSE of the retrievals after

$\:{\varDelta\:h}_{\varphi\:}$

correction is approximately 20% smaller than the other two correction methods. The RMSE of the retrievals after

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i},\text{t}\text{r}\text{o}\text{p}}$

correction is generally smaller than that after

$\:{\varDelta\:h}_{\text{v}\text{a}\text{r}\text{i}}$

correction, indicating that the traditional tropospheric delay correction method still provides some degree of correction.

4.3. Scale bias in combined RHs and the correction effectiveness of long-and-short window method

In this section, we focus on analyzing the scale bias caused by window linearization during multi-GNSS combination and the correction effectiveness of the long-and-short window method described in Section 3.2 on this bias. First, we compute the combined RHs over longer time windows using both the traditional correction method and the phase-based correction method, as represented by Equations (14) and (15). According to the number of valid SNR arcs listed in Tables 1–3, BRST has approximately 350 usable arcs per day, while HKQT has about 210. To ensure sufficient redundancy within the window (around 10–30 arcs), the longer time window is set to 2 hours for BRST and 3 hours for HKQT, with a step size of 10 minutes for both. The inversion results of the combined RHs are shown in the top and middle subplots of Figs. 5 and 6. It can be observed that, for both stations, the estimated results of Eq. (15) yield more concentrated Van de Casteele diagrams and better precision than these of (14). Moreover, the upper and middle subplots of Figs. 5 and 6 both exhibit a “leftward skew”. This indicates that the sea level retrievals from long-window combined RHs underestimate the actual tidal peaks and overestimate the actual tidal troughs. Theoretically, this issue arises from the assumption of window linearization.

Fig. 6

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Further, the scale bias is corrected using the method described in Section 3.2. The core of this correction lies in using shorter time windows since smaller windows reduce the influence of the linearization assumption. In this experiment, the short time window is set to 20 minutes for BRST and 30 minutes for HKQT, with the step remaining at 10 minutes. The retrieval results from short-window combined RHs are shown in the bottom subplots of Figs. 5 and 6.

Fig. 5

Sea level retrievals of BRST using three combination strategies (left) and their corresponding Van de Casteele diagrams (right). From top to bottom, the results correspond to Eq. (14) with a 2-hour time window, Eq. (15) with a 2-hour time window, and Eq. (15) with a 20-minute time window.

Fig. 6

Sea level retrievals of HKQT using three combination strategies (left) and their corresponding Van de Casteele diagrams (right). From top to bottom, the results correspond to Eq. (14) with a 3-hour time window, Eq. (15) with a 3-hour time window, and Eq. (15) with a 30-minute time window.

In the bottom subplots of the Van de Casteele diagrams in Figs. 5 and 6, although there are larger outliers compared to the upper two subplots, no observable scale bias is present. This indicates that using short windows for multi-GNSS combination effectively avoids the influence of window linearization. Based on the results of Figs. 5 and 6, the retrieval values used the long and short windows are further temporally matched, and a linear relationship is fitted, as shown in the left subplots of Figs. 7 and 8. The fitted linear equations are

$\:y=1.0295x-0.4971$

for BRST and

$\:y=1.0704x-0.4602$

for HKQT. Using these equations, the long-window retrievals are corrected, and the corrected retrievals and corresponding Van de Casteele diagrams are plotted, as shown in the middle and right subplots of Figs. 7 and 8. The RMSE and CORR of the retrievals in Figs. 5–8 are summarized in Table 4.

Fig. 7

(Left) scatter plot and linear relationship of retrieval values of long and short windows for BRST. (Middle) retrievals from the long time window corrected based on the linear relationship in the left subplot and (Right) the corresponding Van de Casteele diagram.

Fig. 8

(Left) scatter plot and linear relationship of retrieval values of long and short windows for HKQT. (Middle) retrievals from the long time window corrected based on the linear relationship in the left subplot and (Right) the corresponding Van de Casteele diagram.

Table 4

RMSE and CORR of retrievals corresponding to the four combination methods described in this section
Site	Combination method using	Eq. (14) Figure 5/6 (Top)		Eq. (15) Figure 5/6 (Middle)		Eq. (15) Figure 5/6 (Bottom)		Eq. (15) with scale correction Figure 7/8 (Middle)
BRST	Window length	2 hour		2 hour		20 min		2 hour
	Data Analysis	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)
	Data Analysis	10.33	99.85	8.91	99.92	12.96	99.66	6.54	99.92
HKQT	Window length	3 hour		3 hour		30 min		3 hour
	Data Analysis	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)	RMSE (cm)	CORR (%)
	Data Analysis	9.18	98.66	6.57	99.43	14.76	95.87	5.40	99.44

In the right subplots of Figs. 7 and 8, the scale-corrected combined retrievals show a significant improvement in the “leftward skew” trend compared to Figs. 5 and 6. According to Table 4, it is evident that the scale-corrected retrievals achieve an approximate 20% improvement in accuracy compared to those without the correction. The results from Figs. 5–8 and Table 3 further demonstrate that short time windows exhibit smaller scale biases but result in fewer inversions and lower accuracy, while long time windows have larger scale biases but yield more inversions with higher accuracy. The long-and-short window correction method proposed in Section 3.2 takes into account the advantages and disadvantages of both window types. This experiment confirms that this method is effective and reliable.

5. Conclusion and discussion

This paper primarily analyzes and corrects the scale biases in GNSS-IR sea level retrievals, including the scale bias exhibited by the raw RH of individual signals and those observed in the RH after multi-signal combination. For the former, the experiments in Section 4.2 demonstrate that the phase-based error correction method effectively rectifies the scale bias in the raw RHs. As shown in Figs. 3 and 4, this correction significantly reduces the leftward skew in the Van de Casteele diagram. For the latter, the paper proposes a long-and-short window correction method. The experiments in Section 4.3 confirm that this method can effectively mitigate the scale bias in the combined RHs. Notably, this method does not rely on external data to correct the scale bias and achieves a significant improvement in accuracy.

From Figs. 5, 6, and Table 3, it is evident that short time windows exhibit smaller scale biases but result in fewer inversions with lower accuracy. Here, the reduction in the number of inversions output by short time windows is further quantified. For the BRST station, the output from short time windows is 17% less than that of long time windows, while for the HKQT station, the reduction is 89%. However, as shown in Figs. 7 and 8, this substantial reduction in quantity or increase in gross errors does not hinder linear fitting. Therefore, there is no need to worry about the reduced output or lower accuracy of inversions from short time windows.

The advantages of the phase-based correction method over other methods are emphasized here. As described in Section 2.3, in principle,

$\:{{\Delta\:}h}_{\varphi\:}$

includes systematic errors such as height variation errors and tropospheric delay errors, as well as other unidentified sources. The experiments in Section 4.2 show that this method’s ability to suppress errors is significantly better than traditional methods. It is hoped that researchers in the GNSS-IR field will conduct further studies. We will also continue to refine this approach to enhance the theory and methodology of error correction.

Data availability

We thank the International GNSS Service (IGS; http://www.igs.org) for providing GNSS data for BRST, and REFMAR (http://refmar.shom.fr/) for supplying tide gauge data from Brest. GNSS data for HKQT are available from the Hong Kong GNSS Satellite Positioning Reference Station Network (https://www.geodetic.gov.hk/). Tide gauge data near HKQT are provided by the IOC (http://www.ioc-sealevelmonitoring.org).

Funding sources

This work is supported by National Nature Science Foundation of China (42474048, 41804005).

Conflict of interest:

The authors declare that they have no conflict of interest.

Author Contribution

X.W. drafted the manuscript and conducted the majority of the experiments. X.H. provided the experimental platform. M.S. performed a portion of the experimental work.

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