A Method for Constructing an Empirical Model of Short-Term Offshore Ocean Tide Loading Displacement Based on PPP

Abstract

The ocean tide loading (OTL) can result in displacements of centimeters or even decimeters at nearshore stations. Global ocean tide models exhibit errors in nearshore regions, which limit the accuracy of maintaining the coordinates of these stations. GNSS positioning can obtain tidal load displacements in nearshore areas, but it often requires long-term observation data and cannot provide timely correction models for newly established reference stations. This paper proposes a method for an empirical correction model of short-term OTL displacements using GNSS observations, where the kinematic coordinate sequences are first obtained by multi-GNSS precise point positioning with ambiguity resolution (PPP-AR), and then the OTL corrections are obtained by window-sliding forecast based on random forest modeling. Through experiments conducted in the Hong Kong region, the empirical model with a window of 15 days is established by the proposed method. After applying the empirical model, root mean square errors of the residuals are reduced by 1.5 (30.6%), 3.7 (53.6%), and 3.7 mm (37.8%) in the East, North, and Up (ENU) components, respectively. When using the global ocean tide model FES2014, the RMSE values are reduced by 1.2 (24.5%), 0.3 (4.3%), and 3.7 mm (37.8%) in the ENU components, respectively. The empirical model shows better effects for the OTL displacement compared to FES2014, especially in the N component, with an improvement ratio of about 49.3%.

1. Introduction

The Sun, Moon, and other planets exert gravitational forces on the Earth, causing tides. Tides can be classified into solid tides, ocean tides, and polar tides. Among them, ocean tides cause the periodic rise and fall of the sea surface and the redistribution of seawater mass, leading to periodic deformations in the solid Earth. This phenomenon is known as ocean tide loading (OTL) [1]. In coastal areas, OTL can result in displacements at the measurement stations ranging from centimeters to decimeters [2]. Thus, the impact of OTL cannot be ignored in high-precision GNSS positioning research [3]. In current GNSS data processing, the main approach to eliminate the influence of OTL is by introducing ocean tide models. Examples of such models include Schwiderski [4], HAMTIDE11A.2011 [5], FES2014 [6], and EOT20 [7]. These models are primarily derived from continuous and high-precision satellite altimetry data. However, in coastal areas, the accuracy of global ocean tide models is limited due to the complex seafloor topography and unique coastline features. Satellite altimetry data cannot accurately capture the variations in sea surface height in these nearshore regions [8]. Shum et al. found that different global ocean tide models yield tidal heights that differ by only 2–3 cm in open ocean regions. However, in nearshore areas, the differences in tidal heights are much larger, with maximum discrepancies reaching decimeter levels [9]. Chung et al. compared the accuracy of five tidal models (Andersen 2006, EOT08a, GOT4.7, TPXO7.2, and NAO.99b) using continuous GPS data from ten stations along the Taiwan coast over a period of two years. The results showed that NAO.99b, released by the Japanese Astronomical Observatory, provided the best fit for the coastal waters around Taiwan. This is attributed to the inclusion of tide stations’ data from Japan and South Korea in the model [10]. In order to improve the applicability of ocean tide models in nearshore regions, it is necessary to employ alternative methods for refining the accuracy of these models.

With the advancement of geophysics and geodesy, space geodetic techniques have been gradually applied to tidal variation observations and obtaining high-precision OTL models. These techniques primarily include three methods: superconducting gravimeter (SG) observation technology, very long baseline interferometry (VLBI) technology, and global navigation satellite system (GNSS) positioning technology. Among these, GNSS technology has advantages such as a large number of monitoring stations, wide coverage, independence from time and weather limitations, low cost, and relatively high positioning accuracy [2]. In the field of GNSS, precise point positioning (PPP) [11], especially, achieves millimeter-level accuracy and does not eliminate the OTL displacement signal like real-time kinematic (RTK). Thus, it is widely applied in tidal observation and research [12,13,14,15,16,17,18,19,20,21,22,23].

There are two main methods for obtaining OTL displacements using PPP technology. The first method is the static method [12], which involves solving for 48 harmonic displacement parameters along with other unknown parameters in static PPP (also known as the harmonic coefficient estimation method). Schenewerk et al. calculated OTL displacement parameters for global IGS stations using the static method [13]; Allinson et al. analyzed 1000 days of continuous GPS data and found a poorer performance for K1 and K2 tidal waves when comparing OTL displacements with model predictions [14]; Urschl et al. validated global ocean tide models using GNSS estimates from 140 IGS stations, showing a good consistency with the FES95.2, FES99, and GOT00.2 models [15]; Wang et al. found that the combination of GPS, GLONASS, and Galileo provided better OTL displacement estimates than GPS alone [16]. The static method allows for the direct determination of absolute amplitudes and phases of tidal constituents. However, it may average out certain tidal information and make it difficult to observe certain constituents, such as the 1/3 diurnal tide. Additionally, in static precise point positioning, when obtaining the sequence of station coordinate changes, a large sampling interval can lead to significant errors in estimating OTL displacement parameters.

The second one is the kinematic method [17], which involves extracting the amplitude and phase of tidal waves from high-sampling-rate coordinate time series computed through kinematic PPP. Khan and Tscherning introduced a kinematic method for solving OTL displacements using a 49-day GPS dataset in Alaska [18]; King et al. applied kinematic solutions to five years of GPS data in Antarctica, obtaining vertical OTL displacements for eight major tidal components [19]; Bos et al. used a kinematic approach to analyze four years of data from 20 European stations, showing that GNSS kinematic methods can enhance existing ocean tide models [20]; Abbaszadeh et al. processed 3–7 years of data from 49 global stations with kinematic precise point positioning, finding that GPS and GLONASS combined improve the K1 and K2 tidal component accuracy [21]. Peng et al. used dynamic precise point positioning to assess tidal loading effects on U.S. West Coast DInSAR interferograms, integrating global/regional ocean tidal models [22]. Matviichuk et al. estimated OTL displacements for Australia using 5.5 years of GPS and GLONASS data from 360 stations with a kinematic method [23]. For kinematic methods, the accuracy of PPP processing directly affects the precision of OTL displacements.

Traditional PPP techniques provide float solutions, where phase biases from the receiver and satellite hardware are absorbed into the ambiguities in PPP and cannot be eliminated during data processing [24]. Ambiguity fixing in PPP has been considered challenging. However, with further research, several methods for undifferenced ambiguity fixing have been proposed, including the integer clock model [25], decoupled clock model [26], uncalibrated phase delay (UPD) model [27], and phase clock/bias model [28]. As a result, precise point positioning with ambiguity resolution (PPP-AR) has been developed. Compared to PPP, AR can provide higher-precision fixed solutions and faster convergence during estimation [29]. Currently, machine learning has emerged as an important and effective modeling approach and is widely applied in fields such as remote sensing and computer vision. Numerous studies on random forests [30] have demonstrated that it is one of the machine-learning prediction models with a high accuracy, and it is less affected by outliers and noise [31]. Consequently, we may consider using PPP-AR to obtain high-precision OTL displacement time series for a region, and to construct an empirical model of OTL displacements using random forest regression, which can be used to predict OTL displacements for the region in the subsequent period.

Although the above achievements provide solid theoretical support for constructing OTL displacement models using PPP, obtaining complete and stable parameters typically requires several years of GNSS observation data. However, for newly established GNSS stations in coastal areas, the length of observation data needed for modeling greatly limits the effective construction of OTL displacement models. Therefore, this paper proposes a method to address this issue: utilizing short-term multi-GNSS data from coastal stations, based on PPP-AR and random forest regression, to construct a high-precision and simplified empirical model for regional OTL displacements. Through a study and discussion focused on the Hong Kong region, the effectiveness of this method is validated. It can be applied to correct the impact of OTL displacements on newly established GNSS stations in coastal areas. The structure of this paper is organized as follows: Section 2 introduces the proposed modeling method and provides theoretical details on specific steps. Section 3 describes the dataset used for validating the method. Section 4 presents the modeling process and results. Section 5 compares and discusses the partial tides of the empirical model and FES2014. Finally, the conclusions are summarized.

2. Methodology

In this section, the process of the proposed method is presented. The principles of AR and random forest regression are then introduced in detail, followed by a discussion on their feasibility in applying them to OTL displacement modeling.

The specific workflow for constructing the empirical model for regional OTL displacements is illustrated in Figure 1 and includes the following steps:

• Collect short-term GNSS data from multiple stations in a nearby coastal area;
• Perform PPP-AR to obtain high-precision coordinate sequences for the stations;
• Input the coordinate sequences into a random forest regression to fit the empirical model for OTL displacements in the region;
• Evaluate the accuracy of the empirical model.

2.1. Precise Point Positioning

PPP was initially proposed by Zumberge from the Jet Propulsion Laboratory (JPL) [11]. It is a post-processing GNSS positioning technique that uses observation data to correct various errors, allowing high-precision absolute position coordinates to be obtained with a single receiver.

For GNSS dual-frequency observations from station to satellite, the raw observation equation for original pseudorange and carrier-phase of the i-th frequency (i = 1,2) in the unit of length is

$$\left\{\begin{array}{c}{P}_{r,i}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})+{\displaystyle \frac{A}{f_i^2}}+d_{r,i}-d_i^s\hfill \\ L_{r,i}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})-{\displaystyle \frac{A}{f_i^2}}+{\lambda }_i{N}_{r,i}^s+b_{r,i}-b_i^s\hfill \end{array}\right.$$

(1)

where $P_{r,i}^s$ are pseudorange observations; $L_{r,i}^s$ are carrier-phase observations; ${\rho }_r^s$ is the sum of the station-satellite geometric distance and the slant troposphere delay; $c$ is the speed of light in vacuum; $\delta t_r$ and $\delta t^s$ are the receiver and satellite clock errors, respectively; $\delta t_{ISB}$ is the inter system bias (ISB) between satellite systems, using GPS time as the reference time scale; A denotes the impact of the first-order ionosphere delays; f₁ and f₂ are the frequencies of L₁ and L₂; ${\lambda }_1$ and ${\lambda }_2$ are the corresponding wavelength; $N_{r,1}^s$ and $N_{r,2}^s$ are integer ambiguities; $d_{r,i}$ and $d_i^s$ denote the pseudorange bias of the i-th frequency, which is caused by the hardware delay of the receiver and the satellite; and $b_{r,i}$ and $b_i^s$ denote the phase bias of the i-th frequency, which is caused by the hardware delay of the receiver and the satellite; for simplicity, high-order ionospheric delay, multipath effect, and random noise are omitted. The spatiotemporal reference is GPS time and IGS14.

For the first-order ionospheric delay in Equation (1), it can be eliminated by forming ionosphere-free linear combination of dual-frequency measurements. The ionosphere-free observation equation is then formed as:

$$\left\{\begin{array}{c}{P}_{r,0}^s=\alpha P_{r,1}^s-\beta P_{r,2}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})+d_{r,0}-d_0^s\hfill \\ L_{r,0}^s=\alpha L_{r,1}^s-\beta L_{r,2}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})+\alpha {\lambda }_1{N}_{r,1}^s-\beta {\lambda }_2{N}_{r,2}^s+b_{r,0}-b_0^s\hfill \end{array}\right.$$

(2)

where $\alpha =f_1^2/(f_1^2-f_2^2)$ and $\beta =f_2^2/(f_1^2-f_2^2)$; $d_0^s=\alpha d_1^s-\beta d_2^s$, $d_{r,0}=\alpha d_{r,1}-\beta d_{r,2}$, $b_{r,0}=\alpha b_{r,1}-\beta b_{r,2}$, and $b_0^s=\alpha b_1^s-\beta b_2^s$; and $P_{r,0}^s$ and $L_{r,0}^s$ are the ionosphere-free pseudorange observation and carrier-phase observation, respectively. Correspondingly, $d_{r,0}$ and $d_0^s$ are the ionosphere-free pseudorange bias at the receiver end and the satellite end, and $b_{r,0}$ and $b_0^s$ are the ionosphere-free combination phase bias at the receiver end and the satellite end, respectively. The ionosphere-free combination eliminates the first-order term of ionospheric delay and is the most commonly used observation model in PPP.

During the data processing of PPP, various errors in signal transmission, propagation, and reception need to be considered. These errors include satellite orbit errors, satellite clock errors, relativistic effects, ionospheric delay, tropospheric delay, receiver clock errors, and so on. Typically, models or parameter estimation methods are used to mitigate or eliminate these errors. Coastal stations are subject to tidal effects such as solid tides, ocean tides, and polar tides, which cause systematic variations in the station’s coordinate sequence throughout the day. According to the IERS2010 (International Earth Rotation Service) protocol, effective corrections for solid tides and polar tides can be applied using corresponding models. The correction accuracy has reached the level of 1 mm, meeting the requirements specified by the IERS standards [32]. The accuracy of global ocean tide models in the coastal regions is insufficient, which results in the inability to meet the precision requirements for OTL displacement corrections specified by the IERS standards [33]. Therefore, after subtracting the effects of solid tides and polar tides, PPP can be employed as an effective observational method to obtain the displacement series caused by OTL.

2.2. PPP with Ambiguity Resolution

For ionosphere-free phase observations $L_{r,0}^s$, fixing the ambiguities directly can be challenging due to their short wavelengths. This requires the use of the Melbourne–Wübbena combination to split them into wide-lane and narrow-lane components [34,35]:

$$\begin{array}{c}\begin{array}{cc}{L}_{r,m}^s& ={\lambda }_w\left({\displaystyle \frac{L_{r,1}^s}{{\lambda }_1}}-{\displaystyle \frac{L_{r,2}^s}{{\lambda }_2}}\right)-{\lambda }_n\left({\displaystyle \frac{P_{r,1}^s}{{\lambda }_1}}+{\displaystyle \frac{P_{r,2}^s}{{\lambda }_2}}\right)\end{array}\\ ={\lambda }_w\left(N_{r,w}^s+{\displaystyle \frac{b_{r,1}-b_1^s}{{\lambda }_1}}-{\displaystyle \frac{b_{r,2}-b_2^s}{{\lambda }_2}}\right)-{\lambda }_n\left({\displaystyle \frac{d_{r,1}-d_1^s}{{\lambda }_1}}+{\displaystyle \frac{d_{r,2}-d_2^s}{{\lambda }_2}}\right)\end{array}$$

(3)

where ${\lambda }_w=c/(f_1-f_2),$ ${\lambda }_n=c/(f_1+f_2)$, and $N_{r,w}^s=N_{r,1}^s-N_{r,2}^s$; ${\lambda }_w$ and ${\lambda }_n$ are the wide-lane wavelength and narrow-lane wavelength, respectively; and $N_{r,w}^s$ is the ambiguity of wide lane. M–W combination eliminates ionospheric delay, and geometric distance from satellite to receiver, satellite clock, and receiver clock. It is only affected by multipath effect, measurement noise, and hardware delay. Because the wide-lane wavelength is up to 86cm, it is easy to determine its integer ambiguity; that is, the wide-lane ambiguity is solved directly using the rounding method [27]. The corresponding receiver phase deviation and satellite phase deviation are:

$$\left\{\begin{array}{c}{b}_{r,w}={\lambda }_w({\displaystyle \frac{b_{r,1}}{{\lambda }_1}}-{\displaystyle \frac{b_{r,2}}{{\lambda }_2}})-{\lambda }_n({\displaystyle \frac{d_{r,1}}{{\lambda }_1}}+{\displaystyle \frac{d_{r,2}}{{\lambda }_2}})\hfill \\ b_w^s={\lambda }_w({\displaystyle \frac{b_1^s}{{\lambda }_1}}-{\displaystyle \frac{b_2^s}{{\lambda }_2}})-{\lambda }_n({\displaystyle \frac{d_1^s}{{\lambda }_1}}+{\displaystyle \frac{d_2^s}{{\lambda }_2}})\hfill \end{array}\right.$$

(4)

After the ambiguity of wide lane is resolved through M–W combination, we can substitute $N_{r,2}^s=N_{r,1}^s-{\check{N}}_{r,w}^s$ into ionosphere-free combination Equation (2) which can then be transformed into

$$\left\{\begin{array}{c}{P}_{r,0}^s=\alpha P_{r,1}^s-\beta P_{r,2}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})+d_{r,0}-d_0^s\hfill \\ {\overline{L}}_{r,0}^s=L_{r,0}^s-\beta {\lambda }_2{\check{N}}_{r,w}^s={\rho }_r^s+c(\delta t_r-\delta t^s+\delta t_{ISB})+{\lambda }_n{N}_{r,1}^s+b_{r,0}-b_0^s\hfill \end{array}\right.$$

(5)

where ${\check{N}}_{r,w}^s$ denotes the resolved wide-lane ambiguity; the $N_{r,1}^s$ in this formula is also called narrow-lane ambiguity, obtained through a search using LAMBDA method [36]; and ${\overline{L}}_{r,0}^s$ is the ionosphere-free combined carrier-phase observation after correcting the wide-lane ambiguity.

In this study, the phase clock/bias model [28] was employed for precise point positioning with ambiguity resolution. It calculates the mean value of narrow-lane ambiguity in a single day and fixes it in subsequent data processing, then re-estimates the clock parameter, and absorbs the residual narrow-lane phase bias relative to the mean value of narrow-lane ambiguity into the clock error parameter. Therefore, the required integer ambiguity and its bias of narrow lane are the integer part and fractional part of the mean value of narrow-lane ambiguity, respectively. In the phase bias/clock model, the narrow-lane phase bias between the receiver and the satellite is:

$$\left\{\begin{array}{c}{b}_{r,n}=\Delta b_{i,0}-\Delta d_{i,0}\hfill \\ b_n^s=\Delta b_0^s-\Delta d_0^s\hfill \end{array}\right.$$

(6)

Accordingly, the user’s mathematical model for PPP-AR using the phase bias/clock model is as follows:

$$\left\{\begin{array}{c}{L}_{r,m}^s+{\widehat{b}}_w^r={\lambda }_w{N}_{r,w}^s+b_{r,w}\hfill \\ P_{s,0}^r+c{\widehat{t}}_F^s\approx {\rho }_r^s+c{t}_{r,F}\hfill \\ L_{r,0}^s+c{\widehat{t}}_F^s-\beta {\lambda }_2{\check{N}}_{r,w}^s+{\widehat{b}}_n^s={\rho }_r^s+c{t}_{r,F}+{\lambda }_n{N}_{r,1}^s+b_{r,n}\hfill \end{array}\right.$$

(7)

where ${\widehat{b}}_w^r$ and ${\widehat{b}}_n^r$ are the phase bias products of wide lane and narrow lane at the satellite end; and ${\widehat{t}}_F^s$ is the satellite clock product. The narrow-lane phase bias $b_{r,n}$ at the receiver end will be absorbed by the receiver clock $t_{r,F}$.

2.3. Random Forest Regression

In recent years, machine-learning techniques have demonstrated excellent modeling performance in various fields, especially in dealing with complex nonlinear relationships and large-scale data. However, the application of these advanced techniques in the GNSS domain is still relatively limited. The problem of predicting OTL displacements is essentially a regression problem, where the goal is to predict future displacement changes based on historical time-series data. Random forest models excel in handling regression problems, particularly when dealing with time-series data [30]. By integrating multiple decision trees, they can capture complex patterns and temporal dependencies in the data, thereby achieving more accurate predictions. The basic principle is illustrated in Figure 2.

The random forest regression model also has numerous advantages, making it highly promising for OTL displacement modeling. Firstly, it can handle high-dimensional data and effectively identify important features, which is crucial for complex OTL displacement data. Secondly, random forest regression has good resistance to overfitting, maintaining high prediction accuracy even with a small sample size. Additionally, the model shows strong robustness to missing data and outliers, making it particularly suitable for GNSS data processing in practical applications. Thus, in this experiment, PPP-AR is used to obtain the OTL displacement time series in the Hong Kong area, and random forest regression is employed to construct a short-term empirical model for OTL displacement. This model can be utilized to predict OTL displacement for subsequent periods, providing a more accurate OTL displacement correction model for new stations in the region.

3. Dataset

To validate the effectiveness of the proposed method, this study takes Hong Kong, China as a research case and calculates the impact of OTL on the region. An empirical model for OTL displacements in Hong Kong is constructed. Hong Kong is located on the southern coast of China, bordered by Shenzhen City, Guangdong Province to the north, facing Macau across the sea to the west, and facing the South China Sea to the south. Its geographical coordinates range from approximately 22°08′N to 22°35′N latitude and 113°49′E to 114°31′E longitude. The coastlines in the Hong Kong region are complex, making it more susceptible to the effects of OTL.

Since 2000, Hong Kong has been constructing a Continuously Operating Reference System (CORS). Currently, there are 18 CORS stations in operation, including nine mountain-top stations: HKKT, HKLT, HKMW, HKNP, HKOH, HKSL, HKSS, HKST, and HKWS, marked on Figure 3; as well as nine rooftop stations: T430, HKFN, HKQT, HKCL, HKSC, HKPC, HKTK, HKKS, and HKLM. Considering factors such as visibility, environment, stability, and sources of interference, this study utilizes the multi-GNSS observation data from nine mountain-top stations in the Hong Kong region for a continuous period of 60 days, from the day of the year (DOY) 001 to 060 in 2021.

4. Model Construction and Evaluation

4.1. PPP-AR Solutions

This paper uses the PRIDE PPP-AR v3.0 software to perform PPP-AR solutions, with the parameter settings for the solutions shown in

Table 1. Parameters and processing strategies for PPP-AR solutions.

Parameters	Processing Strategies
Satellite system	GPS: L1/L2; Galileo: E1/E5a; GLONASS: R1/R2
Interval	30 s
Weighting	Elevation-dependent data weighting
Observation cut-off	10°
Satellite orbits	SP3 products by Wuhan University [38]
Satellite clocks	CLK products by Wuhan University [38]
ERP parameters	ERP products by Wuhan University [38]
Undifferenced ambiguity resolution	BIA products by Wuhan University [28] Ambiguity fixing for the GPS and Galileo Float solution for the GLONASS
PCV and PCO	ATX files by IGS [39]
Ionospheric delay	Iono-Free LC
Tropospheric delay	Saastamoinen model [40]
Reference frame	IGS14
Geocenter motion	IERS Conventions 2010 [41]
Solid Earth tides and pole tides	IERS Conventions 2010

. The final output was the daily PPP fixed solutions for each station. It is worth noting that the correction for OTL was not applied during the solution process. Due to the fact that BDS-3 had only recently become operational in 2021, some frequencies of the BDS-3 from the CORS stations were missing. Thus, BDS could not be included in this experiment. Additionally, other mass loadings, such as atmospheric load, hydrological load, and non-tidal oceanic load, have been shown by research to have a smaller impact [37]. Consequently, these factors are neglected in this experiment.

To obtain the coordinate sequence for each station, the obtained solution results in spatial cartesian coordinates (X/Y/Z) need to be transformed into station-centered coordinates (east/north/up). The reference coordinates for each station during the conversion are determined using daily static PPP. Additionally, the coordinate sequences for the same station over the continuous 60-day period need to be concatenated. Figure 4 illustrates the coordinate sequences for each station during the first seven days (DOY 001-007) of the year 2021.

From Figure 4, it can be seen that, despite the differences in the locations and elevations of the nine stations, the coordinate variations of all stations during the seven days are relatively similar. This indicates that, in a small nearshore area like Hong Kong, the OTL has a consistent impact on all GNSS stations, regardless of their elevation. To further confirm this point, this study analyzed the correlation coefficients of the displacement sequences of these nine stations for a continuous period of 60 days, as shown in Figure 5. It can be observed that the correlation coefficients of the east, north, and vertical components of the nine stations are all above 0.7, with most stations exhibiting even higher correlation coefficients around 0.9. This indicates that the impact of the tidal load on GNSS stations in the Hong Kong area is generally consistent.

The consistency of the displacement time series obtained from various stations using AR is highest in the N direction, with an average correlation coefficient of up to 98%. In the E direction, only the correlation coefficient between the HKSL and HKWS stations is relatively low (79%), but the average correlation coefficient is 95%. In the U direction, the correlation coefficients between the HKSL and HKWS, and between the HKSL and HKNP stations are 72% and 77%, respectively, while the correlation coefficients between other stations are above 81%, resulting in an overall average correlation coefficient of 93%. Based on the above analysis, despite differences in latitude, longitude, and elevation among the stations, the deformation series influenced by tidal loading within a range of 50 km exhibits a good consistency in the ENU direction. To extract more reliable tidal loading displacement information, this study calculates the average displacement series separately in the ENU direction for each station and then performs empirical modeling. The specific ENU sequences are shown in Figure 6. It is evident that the vertical component of OTL displacements in Hong Kong is significantly larger than the horizontal components. Although it includes tidal wave signals with multiple periods, their variations are synchronous, reaching peak and trough values almost simultaneously.

4.2. Model Construction

To begin with, it is necessary to determine the length of the empirical model. This can be achieved by calculating the correlation coefficients of displacement series with different durations. Specifically, assuming the possible period of the displacement series is n days (1 ≤ n ≤ 30), the correlation coefficients of the coordinate sequences at each station over a span of n days in the ENU directions are computed. Then, the average of these correlation coefficients is calculated. Additionally, for groups with a high average correlation coefficient (greater than 0.6), the standard deviation of the correlation coefficients is also computed. Finally, the “n” corresponding to the maximum mean correlation coefficient and the smallest standard deviation is identified as the length of the model. The calculation results are depicted in Figure 7, where the height of the bars represents the mean correlation coefficient, the red bars indicate a mean correlation coefficient greater than 0.6, and the error bars represent the standard deviation of the correlation coefficients. The results show that, when “n = 15”, the mean correlation coefficient is maximized and the standard deviation is minimized. Thus, the length of the model is set to 15 days.

Due to the presence of longer period signals such as monthly and seasonal variations [42] in ocean tides, an OTL displacement empirical model with only a 15-day length would not be able to address this issue. Therefore, it is necessary to continuously update the empirical model to maintain its best performance and minimize errors. The model update process is illustrated in Figure 8. Specifically, the GNSS data during DOY 001-015 can be used for modeling to predict and correct OTL displacements for DOY 016-Y030. After 15 days, the model is updated again using GNSS data during DOY 016-030 to model, predict, and correct tidal loading displacement for DOY 031-045, and so on. This process is repeated continuously.

Using the data from DOY 001-030 as an example, the PPP-AR results provide the OTL displacement time series E, N, and U for various stations in Hong Kong, which include the time t (0 ≤ t ≤ 30 days, interval of 30 s) and the coordinate values E, N, and U. The tidal load displacement series from DOY 001-015 are used as training data: with time as the feature x (0 ≤ x ≤ 15 days), and E, N, and U as labels, the Fe(x), Fn(x), and Fu(x) models are trained, respectively. These models are then utilized to predict OTL displacement series for DOY 016-030, based on the assumption that E(t) = Fe(t − 15 days), N(t) = Fn(t − 15 days), and U(t) = Fu(t − 15 days). Fifteen days later, new GNSS data are employed to construct updated models, which are then used to predict the subsequent segment of OTL displacements.

For this experiment, the displacement series during DOY 001-015 was used as the training data to construct the empirical model. The displacement series during DOY 016-030 was then used as the validation data to assess the accuracy of the model. Figure 9 shows the modeling results. Overall, the empirical model constructed using random forest regression shows a good training accuracy for the ENU components, with all of the R² values above 0.80. Among them, the model has the highest training accuracy for the north component and the lowest training accuracy for the up component, with R² values of 0.89 and 0.85, respectively. This is primarily related to the largest magnitude of OTL displacement in the up direction.

4.3. Accuracy Assessment

The accuracy of the OTL displacement model was quantitatively evaluated using the root mean square error (RMSE) of the residuals, as shown in Equation (8). The empirical model for OTL displacement and the global model was used to correct the OTL at the stations in the Hong Kong region. The RMSE of the coordinate sequences after correction was compared, and it can be calculated as follows:

$$RMSE=\sqrt{{\displaystyle \frac{1}{N}}\sum _{i=1}^N{\left(x_i-f_i\right)}^2}$$

(8)

where N represents the number of epochs; i represents the epoch number; f_i represents the OTL displacement components provided by the model; and x_i represents the E/N/U component of the validation data.

The accuracy of the empirical model is evaluated using validation data, compared with global ocean tide models including FES2004, FES2014, and TPXO9, as shown in Figure 10. The first column represents the validation data, while the remaining columns correspond to the residuals of the OTL displacements in Hong Kong after correction using the empirical model constructed in this study, FES2004, FES2014, and TPXO9, respectively.

In the Hong Kong region, the three global ocean tide models FES2004, FES2014, and TPXO9 (in Figure 10c–e) have comparable corrections of the OTL displacements. They effectively corrected the OTL displacements in the E and U components. In the case of FES2014, the RMSE of the E- and U-directions were reduced to 3.7 mm (24.5%) and 6.1 mm (37.8%), respectively, However, the correction for the N component was very small, with an improvement in the RMSE of only 4.3%. For the empirical model proposed in this study, it not only reduces the RMSE to 3.4 mm (30.6%) and 6.1 mm (37.8%) in the E- and U-directions, respectively, but also significantly reduces the RMSE from 6.9 mm to 3.2 mm in the N-direction, with a correction rate of 53.6%. The above comparison shows that the established empirical model exhibits a superior correction in Hong Kong compared to the two global models.

Although the empirical modelling proposed in this paper is able to capture the OTL signals within days, the empirical model construction uses 15 days of GNSS data, resulting in its inability to capture the OTL signals at the monthly scale and above, and, thus, the empirical model will inevitably lead to a degradation of the OTL correction as the forecast arc length increases. To verify the accuracy of the empirical OTL model for different forecast arcs, the RMSEs for 15-day and 30-day forecasts were counted in this work, and the specific results are shown in

Table 2. Differences in accuracy of training and testing models using different time intervals.

Training Data	Validation Data	RMSE (mm)
		East	North	Up
DOY 001-015	DOY 016-030	3.4	3.2	6.1
	DOY 031-045	3.8	3.9	7.4
	DOY 046-060	5.1	5.0	8.5
DOY 016-030	DOY 031-045	3.4	3.2	6.1
	DOY 046-060	4.0	4.3	7.5
DOY 031-045	DOY 046-060	3.8	3.8	6.9

.

When training the empirical model using data during DOY 001-015 and validating the model accuracy using data during DOY 016-030, 031-045, and 046-060, it can be noticed that the ability of the empirical model to correct the OTL displacements gradually decreases. In the E-direction, the correction ratio decreases from 30.6% to 26.9%, and, eventually, there is no correction at all. In the N direction, the correction ratio decreases from 53.6% to 45.8% and further to 18.0%. In the U-direction, the correction ratio decreases from 37.8% to 32.7% and finally to 17.5%. A similar pattern was observed for the model trained using 016-030 data. It can be seen that the closer the training and validation data periods are (e.g., training data: 001-015 days, and validation data: 016-030 days), the more accurate the model is and the better it is at correcting the OTL displacement. Comparing the results in Table 2, it can be seen that, if the empirical model is not updated for a long period (more than 30 days), its ability to correct the OTL displacements will be weaker than that of the global ocean tidal models.

To solve the problem of performance degradation of the constructed empirical model due to the long forecast arc length, this paper recursively realizes the model update of the newly established tracking station by using a fixed window form, and the sliding of the window is shown in the previous Figure 8. For instance, the model is constructed using GNSS data from the first half of April to predict the OTL displacements for the second half of April; after 15 days, the new empirical model is constructed using GNSS data from the second half of April to predict the OTL displacements for the period from 1 May to 15 May. Comparing the results in Table 2, it can be seen that the ability of the empirical model to correct OTL displacements can be maintained at the same level as that of Figure 10 when the empirical model is continuously updated over a long period.

5. Discussion

To further compare the empirical model and FES2014, it would be beneficial to consider their differences in tidal components. According to the principles of classical harmonic analysis of tides, the tidal phenomenon can be decomposed into the superposition of multiple tidal waves [43], as shown in Equation (9). Typically, only eight constituent tides are considered, including four semi-diurnal tides (M2, S2, N2, and K2) and four diurnal tides (K1, O1, P1, and Q1). Among these eight major constituent tides, M2, S2, K1, and O1 have the largest amplitudes and are referred to as the four major constituents [44].

$$Z\left(t\right)=S_0+\sum _{j=1}^J \left(H_j\cos \left({\sigma }_{j}t-g_j\right)\right)$$

(9)

where Z(t) represents the observed tide at time; and σ_j, H_j, and g_j represent the angular frequency, amplitude, and phase of the j-th constituent tide, respectively. S₀ represents the mean sea level.

The S_TIDE toolbox is used to decompose the empirical model into the superposition of multiple tidal waves [44]. The amplitudes and phases of these tidal waves can be obtained and compared with the tidal parameters of the FES2014 model at the HKSL station (data sourced from the classical OTL website: http://holt.oso.chalmers.se/loading/index-aside-2404271219.html, accessed on 11 March 2023), as shown in

Table 3. Comparison of tidal wave parameters at HKSL.

Direction	Tidal Constituent	Empirical Model		FES2014
		Amplitude (mm)	Phase (°)	Amplitude (mm)	Phase (°)
East	M2	3.56	127.7	2.23	129.5
	S2	1.08	137.2	1.00	137.0
	K1	2.22	−86.9	1.69	−76.1
	O1	1.43	−98.6	1.36	−98.8
North	M2	4.03	139.3	1.47	23.6
	S2	1.49	−70.4	0.52	58.1
	K1	8.47	−79.8	2.58	177.7
	O1	2.49	149.1	2.22	142.3
Up	M2	6.03	−167.5	6.80	−167.9
	S2	2.28	−150.1	1.78	−145.1
	K1	6.52	−16.5	6.78	−17.3
	O1	6. 95	−44.8	7.32	−49.6

. Due to the limited length of the empirical model (15 days), it is not possible to discern all tidal waves according to the Rayleigh criterion [45]. Thus, only the parameters of the four major constituents are compared.

The empirical model and the FES2014 model show consistent amplitudes and phase parameters for the E and U components of tidal constituents, but exhibit significant differences in the N component. Considering the correction effect of the models on OTL displacements, the distribution pattern of the data is reasonable, which demonstrates the effectiveness of the proposed method. It is worth noting that, for the Hong Kong region, FES2014 performs well in correcting OTL displacements compared to some global ocean tide models, such as FES2004, TPXO9, and so on. Thus, the empirical model constructed in this study shows a small deviation from FES2014 in the east and up directions. However, it can be anticipated that, for regions with more intricate coastlines and complex coastal topography, the empirical model developed using the method proposed in this study will exhibit more significant correction capabilities for OTL displacements.

6. Conclusions

This study proposed a method for the development of a short-term regional OTL displacement empirical model, utilizing PPP-AR and random forest regression techniques applied to multi-GNSS data. The proposed model is capable of predicting OTL displacements and enhancing the precision of GNSS precise positioning. However, it is important to note that the model’s construction is based on a relatively brief dataset and does not account for long-period signals inherent in tidal variations. Consequently, to maintain prediction accuracy, the model necessitates continuous updates.

Through experimental investigations conducted in the Hong Kong region, it has been observed that the empirical model exhibits superior precision. Specifically, when the empirical model is applied for correction, the reductions in RMSE of the residuals are 1.5 mm (30.6%) for the east component, 3.7 mm (53.6%) for the north component, and 3.7 mm (37.8%) for the up component. When FES2014 is employed for correction, the reductions in RMSE values of the residuals are 1.2mm (24.5%) for the east component, 0.3 mm (4.3%) for the north component, and 3.7 mm (37.8%) for the up component. Notably, the empirical model demonstrates a significantly enhanced correction performance for OTL displacements in the north component, with an improvement ratio of approximately 49.3% relative to the FES2014.

From the perspective of data sources, the modeling approach in this study does not rely on any prior ocean tide models as input or constraints. Instead, it solely relies on the kinematic positioning results obtained through GNSS PPP-AR. The improved model performance is attributed to the increased number of redundant observations from multi-GNSS satellites and the enhanced accuracy and stability provided by AR fixed solutions. In the future, multi-GNSS PPP-AR, as an independent and high-precision observation technique, may contribute to regional and global tidal modeling.

To bolster the reliability of the proposed method, it is advisable to conduct experimental verifications across a broader range of regions. Furthermore, the method is tailored for newly established GNSS stations with relatively short observation periods. Once the station accumulates sufficient observation data, harmonic analysis methods can be employed to develop enduring OTL displacement models, which remain valid over extended periods.