Factor Correction Analysis of Nodal Tides in Taiwan Waters

Abstract

Nodal tides, which follow an 18.6-year cycle, influence tidal variations at any given location in the ocean. Conventional nodal tide theory neglects land effects and topological change. Due to the complex seabed topography around Taiwan waters, the purpose of this paper is to use the long-term tidal data of six stations to discuss the effects of perigean and nodal tides on 20 constituents and to compare the results with previous theories. A modulation method is employed to fit the annual amplitude estimated by harmonic analysis (HA). The top four constituents of the fitted and theoretical values of nodal amplitude factor (AF) and phase factor (PF) are O₁, K₁, K₂, and Q₁. We find that perigean tides or second-order nodal tides considered in the fitting contribute to almost identical performance. The linear time change considered in the AF fitting has better fitting than the mean water level involved.

1. Introduction

Tides at a point on the Earth vary with tractive force, depending on the gravitational attraction from the Moon and the Sun and the centrifugal force resulting from the rotational motion of the Sun–Earth–Moon system [1]. The tide-producing force can be determined by the distance from the center of the two-body motion because the Moon’s orbit around the Earth and the Earth’s orbit around the Sun are close to an ellipse rather than a circle. The distances between the Sun and Earth and between the Earth and the Moon vary due to their respective orbits and produce tidal variability, such as semidiurnal, diurnal, fortnightly, monthly, and annual cycles [2,3,4]. Thomson [5] and Ferrel [6] independently developed HA for astronomical tidal constituents [7]. Doodson [3] classified tidal variability into different groups and listed the name and speed of 60 constituents for HA. Schureman [8] used the least-squares method to estimate HA coefficients and compiled an operational manual for tide prediction. Foreman and Henry [9] developed an HA using high and low tides. Parker [10] provided comprehensive introductions to tidal theory and HA and presented applications for tide-level prediction.

In developing the HA algorithm, Pawlowicz et al. [11] used the signal–noise ratio (SNR) threshold to select suitable coefficients and developed a T_Tide model that considered the influence of nodal tides. Codiga [12] developed the UTide model based on the maximum correlation between the basic development parameters of Foreman et al. [13] and T_Tide. The T_Tide and UTide models have been widely adopted to analyze the characteristics of tide levels at different locations. Leffler and Jay [14] used iterative weighting instead of the original least-squares method in HA. The tidal frequency in HA is assumed to be a known value related to the period of astronomical movement in the least-squares calculation. Mousavian and Hossainali [15] and Amiri-Simkooei et al. [16] proposed least-squares harmonic estimation and multivariate least-squares harmonic estimation to solve the problem of assuming that the frequency of each constituent is unknown. Forrester [17] provided a systematic overview of HA and authored the Canadian Tidal Manual. Australia’s Intergovernmental Committee on Surveying and Mapping (ICSM) has published a tide-level operation manual and developed a public program for TASK-2000 tide-level analysis and prediction [18].

The variation in the Moon’s declination follows an 18.61-year cycle, which can induce global changes in tidal amplitude. The variability in tidal features with a period of 18.6 years is called the nodal tide. Houston and Dean [19] accounted for the nodal tide and eliminated its effect on estimates of sea level trends and acceleration. The amplitude and spatial dependence of the nodal tide in mean sea level (MSL) records was discussed by Woodworth [20]. In addition to analyzing MSL records, alternative analyses of time series at different percentile water levels and analyses of monthly mean high waters (MMHWs) or monthly mean higher-high waters (MMHHWs) have been extensively used to characterize the 18.61-year lunar nodal cycle and estimate its contribution to high water levels [21,22,23,24,25,26].

The description of nodal modulation is reflected in the amplitude and phase of some tidal constituents, initially referred to as the ‘reduction of the harmonic constant’ [1,4]. The reduction in the harmonic constant results in nodal factors related to the amplitude and phase of each tidal component [3,27]. The most important constituent of tides is M₂, and there are numerous studies on the influence of nodal tides on M₂ [28,29,30,31,32,33,34]. The worldwide distribution of nodal variations for the major tidal constituents has been investigated [23,35].

This section reviews recent studies from the past decade that investigated the effects of nodal modulation on the amplitudes and phases of selected tidal constituents. Feng et al. [36] investigated the spatio-temporal changes in the five main tidal constituents along the coasts of China’s seas and in adjacent seas using 17 tide gauge records spanning the period 1954–2012. Godin [37] concluded that the main M₂ constituent exhibits significant amplitude variability demodulated by the nodal correction, and the constituents with the most effectively stabilized nodal correction are O₁, K₁, and K₂. The Q₁, M₂, OO₁, and J₁ constituents are at sites where the tide is predominantly linear and where third-order effects are minimal. Hein et al. [38] proposed a self-programmed, adapted HA to demonstrate differing degrees of amplification of compound tides modulated by nodal tides in shallow waters. Hagen et al. [39] showed that the amplitude factor is significantly overestimated for M₂ and N₂ and underestimated for K₁, K₂, and O₁, and that the calculated factor coincides well with deviations of less than 2.0° in shallow, friction-dominated parts of the North Sea. Friction causes the generation of shallow water tides, which influence the nodal modulation of lunar constituents and S₂ significantly. Pan et al. [40] used both tide gauges and satellite altimeter observations for AF and showed that the AFs of M₂ and N₂ significantly exceed theoretical values, while those of K₂, K₁, and O₁ are significantly lower than the theoretical values in the Gulf of Tonkin. Zong et al. [41] investigated the nodal modulation of the M₂ and N₂ constituents along the Norwegian coast using an enhanced HA that accounted for temporal variability in the amplitude and phase of nodal cycles.

Taiwan is located on the western edge of the west Pacific Ocean, surrounded by seas. On the eastern coast, the Pacific has deeper and varied bathymetry; it reaches 4000 m in depth 50 km from the coast. The east–west orientation of the Okinawa Trench and associated topographic variations significantly influence tidal propagation in this region. On the western coast is the Taiwan Strait, where the shallow bathymetry is around 60 m on average. The tides on the Taiwan Strait are well explained by the diffraction effect due to Taiwan and the shoaling effect due to the rapid topography change in the continental shelf. The seafloor topography surrounding Taiwan is highly complex, which contributes to similar complex tidal characteristics. These observations have been documented in previous studies [42,43,44,45].

According to equilibrium tide theory [3], nodal tides modulate harmonic constants solely based on the relative positions of the Sun and the Moon, without considering the geographic location on Earth. However, due to the influence of the land, the actual tide differs from the equilibrium tide, particularly near the coast and areas where the depth of the water changes rapidly [43,44]. It is worth exploring whether reduction calculations of the amplitude and phase factors of 20 constituents, rather than some main constituents, is feasible for actual tide data.

Six tide stations along the coast of Taiwan, each with over 20 years of tidal data, were selected for analysis. HA was conducted on an annual basis to extract the amplitudes and phases of 20 tidal constituents. The data fitting was used to modulate each constituent’s long-term variation in amplitude and phase. The O₁ constituent was selected as a representative example across the six stations. In addition, the Penghu station was used as a case study to demonstrate the analysis procedure and evaluate the fitting accuracy of estimated versus predicted nodal corrections. The fitting performance of the 20 constituent nodal corrections was then evaluated for the six tidal stations to determine which constituents have a better fit. Finally, we specifically discuss the effect of the perigean tide and time change rate on fitting agreement.

2. Materials and Methods

2.1. Data

2.1.1. Location of Tide Gauges

Hourly observational data were obtained from the six tide gauges selected for this study. The locations of these stations are shown in Figure 1. In addition to the main consideration, that is, the length of the data, the location of the selected tide stations can also cover the different tidal characteristics of Taiwan. There are three tide stations on the east coast of Taiwan and three on the west coast. They are roughly distributed across south, north, and central Taiwan.

2.1.2. Tidal Data

These six stations are now equipped with acoustic gauges. All these stations are currently sampling at one hertz, which is one sample per second. However, 6 min sampling data are used in routine analysis. This study uses the available hourly tide. Tide observations could be very noisy and discontinuous with gaps and offsets. A data screening process is required. In this study, all tide data were calibrated, adjusted, and quality-checked by Lan et al. [46].

The full names, abbreviations, and data length and mean tidal range of the 6 tide gauges are listed in

Table 1. Full names, abbreviations, and data length and mean tidal range of 6 tide gauges.

Full Name	Abb.	Start Time of Data	End Time of Data	Data Length (Year)	Mean Tidal Range (cm)	Mean Spring Tide Range (cm)	Type of Tidal Regimes
Jhuwei	JW	1 January 1993	31 December 2018	26.01	287.8	439.4	Semidiurnal
Penghu	PH	1 January 1993	28 May 2019	26.42	226.0	347.4	Semidiurnal
Kaohsiung	KH	1 January 1993	31 December 2018	26.01	70.6	155.8	Mixed diurnal dominant
Suao	SA	21 March 1997	31 December 2018	21.79	121.6	237.2	Mixed semidiurnal dominant
Hualien	HL	1 November 1997	25 August 2019	21.83	120.2	223.1	Semidiurnal
Chenggong	CG	1 March 1993	2 September 2019	26.52	131.4	261.6	Semidiurnal

. In this study, English initials of Chinese names are used as abbreviated names for easier depiction and tabulation. These tidal data begin in 1993 or 1997 and end in 2018 or 2019. The shortest data length is 21.79 years, and the longest is 26.52 years. Based on the long-term tidal level statistics of all the stations under the Central Weather Administration, the average tidal range, spring tide range, and tidal type of these six stations are listed in the last three columns of Table 1. The tidal ranges of the three eastern stations are approximately 125 cm. However, the average tidal range of the three stations in the west is very significantly different. The spring tide range is significantly higher than the average tidal range: close to 1.5–2 times.

A simple form factor is used to classify tidal observations into four types of tidal regimes. The form factor is defined as F = (K₁ + O₁)/(M₂ + S₂), where K₁, O₁, M₂, and S₂ are the amplitudes of the four main tidal constituents. Depending on the F values, the four types of tidal regimes are classified as semidiurnal (F < 025), mixed semidiurnal dominant (F > 0.25 to F < 1.5), mixed diurnal dominant (F > 1.5 to F < 3), or diurnal (F > 3) [47]. The types of tidal regimes at the six stations are listed in the last column. Four stations, JW, PH, HL, and CG, are classified as the semidiurnal type. KH is of the mixed diurnal dominant type. SA is of the mixed diurnal semidominant type.

2.2. Methods

2.2.1. Harmonic Analysis and Nodal Factors

The tides can be regarded as being composed of the constituents of each simple harmonic oscillation with different periods. HA aims to separate the harmonic constants of each tidal constituent, the constituent amplitude, and the angular velocity of each tidal constituent from the observation data [3]. We use 60 constituents referred to by Doodson [3]. The estimated tidal level, h(t), at time t, can be expressed as Equation (1), which includes mean sea level, z₀, periodic astronomical tides, and the residual component driven by forces other than the tractive astronomical force, e(t).

$$h\left(t\right)=z_0+{\displaystyle \sum _{j=1}^{60}{f}_j{H}_j}\cos \left({\omega }_{j}t-{(V_0+u(t))}_j-{\kappa }_j\right)+e(t)$$

(1)

where H_j and κ_j are the two unknown parameters, amplitude and phase; ω is the frequency of the chosen constituent j; and V₀ is a linear function of the time-dependent longitudes of the Moon and the Sun. The coefficients f_j and u_j are the AF and PF of constituent j. To avoid a lengthy introduction to HA, a brief description and the operational details of HA can be found in Appendix A.1.

Node factor is the change in the amplitude and phase of an original short-period tidal constituent reduced by the long period and low amplitude of the nodal tide with a period of about 18.6 years. Appendix A.2 introduces the derivation of AF and PF for the short-period tide reduction [1] and the correction formulas for AF and PF [3,4], which are used for the theoretical AF and PF calculated in this paper. There is also a formula for calculating V₀ [27].

2.2.2. Fitting of the Amplification and Phase Modulation

To assess the influence of nodal tides on individual tidal constituents, harmonic analysis should be applied to monthly or annual tidal records obtained from tide gauge stations. A time series of more than 19 years is required to capture the full nodal cycle. Long-term amplitude variation may include a linear trend and periodic modulation. In addition to the 18.61-year cycle of nodal tides, which affects the amplitude of the main constituents, the 8.85-year cycle of lunar perigee also causes a change in the amplitude of some constituents, especially for L2 and M1. The influence on high tidal levels has also been identified as a quasi-4.4-year cycle [22,23].

Feng et al. [36] investigated the long-term changes in the amplitude and phase of five main tidal constituents—O₁, K₁, M₂, N₂, and S₂—along the coasts of China using data fitting of the modulation of a single lunar constituent, as follows:

$$A\left(t\right)=b_0+b_{1}t+b_2\cos \left({\omega }_{n\left(p\right)}t\right)+b_3\sin \left({\omega }_{n\left(p\right)}t\right)$$

(2)

where b_i (i = 0, 1, 2, 3) are undetermined coefficients and ω_n(p) is the angular frequency of the nodal or perigean tide. ${\mathrm{b}}_0$ indicates the mean sea level during the data period, and $b_1$ shows the rate of SLR. The variation in the amplitude of only the N₂ constituent resulting from the perigean tide is examined, and those of the other four constituents from the nodal tide are considered.

It is necessary to fit the periodic variation in the annual amplitude of these tidal constituents in an expression considering both nodal and perigean tides. Therefore, Equation (2), for time-varying amplitudes, can be modified to be

$$A\left(t\right)=b_0+b_{1}t+b_2\cos \left({\omega }_{n}t\right)+b_3\sin \left({\omega }_{n}t\right)+b_4\cos \left({\omega }_{p}t\right)+b_5\sin \left({\omega }_{p}t\right)$$

(3)

where b_i (i = 0, 1, …, 5) are unknown coefficients and ω_n = 2π/T_n and ω_p = 2π/T_p are the angular frequencies of the nodal tide and the perigean tide, respectively. T_n = 18.61 years and T_p = 8.85 years. The value of b₀ indicates the mean value of the amplitude of this constituent, and b₁ shows the corresponding rate of increase in amplitude. The amplitude and phase lag of both nodal and perigean tides can be estimated by

$$A_n=\sqrt{b_2^2+b_3^2}; {\phi }_n={\tan }^{-1}{\displaystyle \frac{b_3 }{b_2}}$$

(4)

and

$$A_p=\sqrt{b_4^2+b_5^2}; {\phi }_p={\tan }^{-1}{\displaystyle \frac{b_5 }{b_4}}$$

(5)

The least-squares fitting is used to obtain the undetermined coefficients in Equation (3) and to have the amplitude and phase lag of the nodal and perigean tides by Equations (4) and (5).

The formula of phase factor according to

Table A1. Correction formulas for AF and PF [3,4].

Constituent	AF(f)	PF(u°)
Mm	1.000 − 0.130cosN + 0.0013cos2N	0
Mf	1.0429 + 0.4135cosN − 0.0040cos2N	−23.74sinN + 2.68sin2N − 0.38sin3N
O1, Q1, σ1, Q1, ρ1	1.0089 + 0.1871cosN − 0.0147cos2N + 0.0014cos3N	10.80sinN − 1.34sin2N + 0.19sin3N
K1	1.0060 + 0.1150cosN − 0.0088cos2N + 0.0006cos3N	−8.86sinN + 0.68sin2N − 0.07sin3N
J1, χ1, θ1	1.0129 + 0.1676cosN − 0.017cos2N + 0.0016cos3N	−12.94sinN-1.34sin2N − 0.19sin3N
OO1	1.1027 + 0.6504cosN + 0.0317cos2N −0.0014cos3N	−36.68sinN + 4.02sin2N − 0.57sin3N
M2, 2N2, μ2, N2, ν2	1.0004 + 0.0373cosN + 0.0002cos2N	−2.14sinN
K2	1.0241 + 0.2863cosN + 0.0083cos2N −0.0015cos3N	−17.74sinN + 0.68sin2N − 0.04sin3N
L2	f cosu = 1.00 − 0.2505cos2p − 0.1102cos(2p-N) − 0.0156cos(2p-2N) − 0.037cosN; f sinu = −0.2505sin2p − 0.1102sin(2p-N) − 0.0156sin(2p-2N) − 0.037sinN
M1	f cosu = 2cosp + 0.4cos(p-N); f sinu = sinp + 0.2sin(p-N)

is only the sine function of N, 2N, and 3N, but does not include a constant term. Since the phase factor directly affects the phase component, the influence of temporal trends and perigean tides is not considered in this context. The phase of Equation (1) contains the term V₀, and only the first two terms of the PF formula in Table A1 are considered to fit the measured values; the fitting expression is thus written as follows:

$$u\left(t\right)=b_0+b_2\cos \left({\omega }_{n}t\right)+b_3\sin \left({\omega }_{n}t\right)+b_4\cos \left(2{\omega }_{n}t\right)+b_5\sin \left(2{\omega }_{n}t\right)$$

(6)

2.2.3. Performance Index

A statistical measure is needed to assess the approximation of model predictions to the real data points. The coefficient of determination, denoted R², is a common measure of the goodness of fit of a model [48]. R² is defined as follows:

$$R^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^N{\left[A_m\left(t_i\right)-A_c\left(t_i\right)\right]}^2 }{{{\displaystyle \sum }}_{i=1}^N{\left[A_m\left(t_i\right)-\overline{A_m}\right]}^2}}$$

(7)

where A_m and A_c are the estimated and computed amplitudes, respectively, at time t_i, and N is the total number of estimated amplitudes.

R² normally ranges from 0 to 1, indicating the proportion of the variation in the dependent variable that can be predicted from the independent variable. The most common interpretation of the coefficient of determination is how well the regression model fits the estimated data. Therefore, the higher the R², the closer the predicted value is to the measured value. In simple linear regression, R² indicates the squared correlation coefficient (CC) between two variables [49]. There are many different guidelines for interpreting the correlation coefficient because findings can vary substantially between study fields [49,50,51]. We choose the very strong relationship among the 5-level CC as the threshold value for screening appropriate constituents, which is CC > 0.8; that is, R² > 0.64.

When a standard error has been determined, 95% confidence intervals can be estimated using standard techniques. A shorter confidence interval is generally associated with more accurate results at the same confidence level. Pawlowicz et al. [11] alternatively used a signal-to-noise power ratio (SNR) to show the variability in each constituent in t_tide. The SNR is traditionally defined as the ratio of signal power to noise power, and it is commonly used in science and engineering to show the level of a signal to the level of background noise. Pawlowicz et al. [11] accepted the linear procedure of HA to be adequate when the SNR > 10, and it may still be acceptable for an SNR as low as 2 or 3.

The annual tidal amplitude is mainly affected by nodal tides, and there are slight variations due to local weather effects. The prediction by Equation (3) or (6) can be considered as a modulated signal, while the small deviations in amplitude from these signals are treated as noise. For the series of annual tidal amplitude, we follow the definition of the SNR by Sherman and Butler [52] to obtain

$$SNR={\displaystyle \frac{{{\displaystyle \sum }}_{i=1}^N{\left[A_c\left(t_i\right)-(b_0+b_1{t}_i\right]}^2 }{{{\displaystyle \sum }}_{i=1}^N{\left[A_m\left(t_i\right)-A_c\left(t_i\right)\right]}^2}}$$

(8)

Since the SNR only reflects the amplitude of the periodic signal, the amplitude used in the numerator of Equation (8) must be detrended prior to calculation.

3. Results

3.1. Example Description of Modulated and Calculated Nodal Factors

In order to illustrate the estimated AF from yearly tide data and the corresponding calculation for the 20 constituents in

Table 2. R² and RMSE of the fitted and calculated AFs related to the estimated ones for the O₁ constituent.

Station	AF					PF
	Fitted		Calculated		Fitted		Calculated
	R2	RMSE	R2	RMSE	R2	RMSE	R2	RMSE
JW	0.9938	0.0101	0.9708	0.0232	0.9959	0.4752	0.9165	2.1480
PH	0.9908	0.0115	0.9817	0.0220	0.9924	0.6654	0.9378	1.9258
KH	0.9849	0.0159	0.9725	0.0222	0.9777	1.1978	0.9296	2.2063
SA	0.9898	0.0137	0.9827	0.0183	0.9934	0.6136	0.9269	2.0518
HL	0.9814	0.0181	0.9723	0.0227	0.9877	0.8662	0.9268	2.1200
CG	0.9901	0.0128	0.9773	0.0201	0.9919	0.6857	0.9198	2.1571
Mean	0.9885	0.0137	0.9762	0.0214	0.9898	0.7506	0.9262	2.1015
Std	0.0045	0.0029	0.0051	0.0018	0.0065	0.2528	0.0075	0.0999

, we choose the O₁ constituent at the six stations as an example. The results are shown using six panels in Figure 2, in which the blue open circles, red dashed lines, and black solid lines indicate the estimated, fitted, and calculated AFs.

The values of R² and the root-mean-square error (RMSE) of the fitted and calculated AFs related to the estimated ones at the six stations are shown in the left panel of Table 2. RMSE is commonly used to measure the differences between estimated values and as a complementary performance indicator. The last two rows of Table 2 represent the average and standard deviation (Std) of R² of the six stations, respectively. Table 2 shows that the R² of the fitted AF at the six stations is higher than the calculated one. The mean and Std of the fitted R² are 0.9885 and 0.0045, respectively, while those of the calculated R² are 0.9762 and 0.0051. The RMSE of the AF in Table 2 shows that the fitted values range between 0.0101 and 0.0181, with a mean of 0.0137 and a Std of 0.0029. As for the RMSE of the calculated values, they range between 0.0183 and 0.0232, with an average of 0.0214 and a Std of 0.0018. The high R² and low RMSE of both the fitted and calculated AFs for the O₁ constituent indicate that the theoretical AFs closely match the estimated values. It also proves that the AF estimated via HA is accurate and can be expressed by the theoretical formula. Due to the extremely high R² values between the fitted and calculated AFs and the estimated values, the two curves in Figure 2 are nearly indistinguishable.

Similar to the above AF analysis, the results of the PF are shown in Figure 3 and the right panel of Table 2. The mean and Std of the fitted R² are 0.9898 degrees and 0.0065 degrees, respectively, while those of the calculated R² are 0.9262 and 0.0075. The RMSE of the PF in Table 2 shows that the fitted values range between 0.4752 and 1.1978 degrees, with a mean of 0.7506 degrees and a Std of 0.2528 degrees. As for the RMSE of the calculated values, they range between 1.9258 and 2.2063 degrees, with an average of 2.1015 and a Std of 0.0999. Similar to what we have concluded for the AFs, the R² of the fitted PF is also higher than that of the calculated one.

The R² and RMSE results show that the fitted AF and PF are slightly more consistent with the estimated values by HA than the calculated values. This is reasonable and acceptable because the calculated value does not take into account the actual influencing factors such as depth and weather changes, while the fitted value is for fitting the temporal change in the estimated amplitudes.

Comparing Figure 2 and Figure 3, we can see that there is a phase difference between the peak values of AF and PF. The mean phase difference for these six stations is ${88.71}^{\circ }$, with a Std of ${7.28}^{\circ }$. The expression of AF in Table 2 is a function of cos(N), and PF is a function of sin(N). The cosine and sine functions have a phase difference of ${90}^{\circ }$. It can be seen that the method used here can correctly describe the variation in the AF and PF of the O₁ constituent of the nodal tide. In addition, Figure 2 and Figure 3 also show that the highly consistent values of AFs and PFs are among the six sites. This result confirms the theoretical value in Table A1, indicating that AF and PF are only related to the node time and have nothing to do with the location. The slight changes in AF and PF may be explained by the influence of Taiwan’s topography.

As for the AF and PF of the other constituents, we choose the PH station with the highest R² in Table 2 as an example to illustrate the analyzed results. We use the measures of R² and the SNR to evaluate the fitted AF for each constituent. The results are shown in Figure 4, where blue open circles represent R² values and red crosses indicate SNR values. The two dashed lines represent the respective thresholds of 0.64 for R² and 2 for the SNR. The SNR of the O₁ constituent is beyond the upper limit of Figure 4, so a cross is not presented. The evaluation shows that, if a constituent meets the R² threshold, it also satisfies the SNR threshold. There are nine constituents that satisfied the test at the same time, which are O₁, K₁, J₁, OO₁, Q₁, M₂, K₂, L₂, and N₂.

As for the PF fitting of PH station, its R² and SNR are shown in Figure 5, in which two dashed lines are the thresholds. Figure 5 shows that seven constituents pass the R² threshold, but all constituents pass the SNR threshold. If the SNR threshold is raised to six, the constituents passing this threshold will be the same as those passing the R² threshold. The seven constituents that pass the R² test are O₁, K₁, OO₁, Q₁, M₂, K₂, and µ₂. Among them, the first six (excluding µ₂) are also included in the nine constituents with the highest R² values in Figure 4. SNR changes are sensitive to signal interference and noise intensity, so the commonly used threshold value, 2, may not effectively differentiate the fitted PF.

Next, we explore the R² of the AF and PF values calculated using the formulas in Table 2 and the estimated values from the data, as shown in Figure 6. It can be seen from Figure 6 that the six constituents that pass the AF threshold are O₁, K₁, OO₁, K₂, M₁, and Q₁, and those that pass the PF threshold are O₁, K₁, OO₁, M₂, K₂, Q₁, and µ₂. The two R² values of the AF and PF in Figure 6 pass the threshold at the same time, including O₁, K₁, OO₁, K₂, and Q₁. The R² value of the PF for the M₁ constituent is 0.063, and the R² value of the AF for the µ₂ constituent is 0.044. This means that both the estimated PF of the M₁ constituent and the estimated AF of the µ₂ constituent from the tide data cannot be well described by the theoretical formula. Although the AF of the M₂ constituent does not pass the threshold, its R² is 0.594, which is slightly lower than the threshold. The result indicates that its AF formula can still describe the measured AF.

3.2. Performance Statistics of the Six Stations

3.2.1. AF Performance

Based on the results and discussion of the PH station in the previous subsection, the fitting accuracy of the nodal tides at the other five stations is considered in this subsection. The value of R² is between 0 and 1, so the variance in the R² value is limited. However, the variation in the SNR is large. The variance in the SNR of the six stations may be too large to be represented graphically. Therefore, the goodness of fit for the AF and PF can be better estimated by R² than by the SNR. The error bar plot of R² of the AF is shown in Figure 7, in which the open circle is the mean and the error bar is defined as the range of one standard deviation (Std) from the mean. The mean R² with the Std between the fitted or calculated and the estimated AF of each constituent for the six stations exceeding 0.64 is listed in

Table 3. Mean R² and Std between the fitted or calculated and estimated AFs and PFs.

Tide	AF				Tide	PF
	Fitted		Calculated			Fitted		Calculated
	Mean	Std	Mean	Std		Mean	Std	Mean	Std
O1	0.9885	0.0045	0.9762	0.0051	O1	0.990	0.006	0.942	0.002
K1	0.9772	0.0177	0.9602	0.0222	K1	0.978	0.013	0.926	0.007
K2	0.9575	0.0315	0.9323	0.0382	K2	0.917	0.101	0.895	0.795
Q1	0.9459	0.0119	0.9087	0.0137	Q1	0.881	0.091	0.795	0.134
M2	0.8391	0.1132	0.6450	0.1233	J1	0.722	0.076	0.551	0.246
OO1	0.8166	0.1276	0.7744	0.1184	-	-	-	-	-
N2	0.7111	0.1126	-	-	-	-	-	-	-
J1	0.6542	0.1359	-	-	-	-	-	-	-

.

Comparing Figure 7a,b, it is obvious that, except for the M₁ constituent, the R² of the other constituents in Figure 7a is higher than that in Figure 7b. Since Equation (3) considers the perigean tides, the AF that fits the estimated values should be more suitable than the theoretical formula in Table 2, which only considers nodal tides. Figure 7a and Table 3 show that there are seven constituents where the mean R² exceeds 0.64, and four of them exceed 0.9, which are O₁, K₁, K₂, and Q₁. The corresponding Std for O₁, K₁, K₂, and Q₁ is less than 0.032. The other four constituents are M₂, OO₁, N₂, and J₁, with a Std ranging from 0.107 to 0.135.

Figure 7b and Table 3 show only six constituents with a mean R² of the calculated AF exceeding 0.64, and the ranking of the constituents with the top four mean R² values is in the same order as that of the fitted AF. The corresponding Std remains below 0.04. The mean R² of OO₁ is 0.774, which is higher than that of M₂, which is 0.645. This ranking is the reverse of the one estimated in the fitted AF results.

3.2.2. PF Performance

Figure 8 shows the error bar plot for the R² of PF. The mean and standard deviation of the ranked constituents that exceed the threshold are listed in the right panel of Table 3. The R² of the fitted PF is higher than the R² of the theoretically calculated PF. Five constituents exceed the R² threshold, which are O₁, K₁, K₂, Q₁, and J₁. The mean R² of the top three is over 0.9. The mean and Std of the R² for the Q₁ and J₁ constituents are 0.881, 0.722 and 0.091, 0.076, respectively. The ranking of constituents by the PF’s R² is the same as that of the first four AFs, but that of the fifth one is different: see J₁ vs. M₂.

3.2.3. Amplitude Ratio of Perigean Tide to Nodal Tide

In order to understand the influence of perigean tide on the tidal amplitudes year by year, we show the error bar plot of the ratio of A_p to A_n in Figure 9.

Figure 9 shows that the average amplitude ratios of the 14 tides are all lower than one, and the others exceed the upper limit of the vertical axis. Six constituents with the lowest amplitude ratios—O₁, K₁, K₂, Q₁, M₂, and OO₁—have an average value lower than 0.2 and a small standard deviation of less than 0.06. From the above model performance and discussion, we can draw a specific conclusion: five constituents in the waters near Taiwan have tides obviously affected by nodal tides: O₁, K₁, K₂, Q₁, and M₂. The annual amplitude of each station has a very small upward trend with minimal spatial variation. Two constituents, M₂ and OO₁, are still significantly affected by nodal tides. Except for these six constituents, the annual amplitudes of the other constituents affected by nodal tides cannot show periodic changes, and it is difficult to draw reliable conclusions.

3.2.4. Time Increase Rate of AF

From the definition of the SNR in Equation (8), it can be seen that its numerator only represents the amplitude of the periodic oscillation of a signal, excluding the trend and average value, which are represented by b₁ and b₀, respectively. When the trend is smaller, the signal’s cycling amplitude increases. Therefore, when b₁ is smaller, the cycling amplitude is larger, i.e., the SNR is larger. The denominator of Equation (8) indicates the difference between the predicted and estimated signals, that is, the strength of noises. A high SNR may result from weak noises, indicating that the signal is close to pure periodic oscillation.

Figure 10 shows the error bar plot of the relative rate of increase: b₁/b₀. The relative linear trend is considered primarily because the average amplitude of each constituent at the six stations is different, i.e., to unify the trend. It is shown in Figure 10 that the mean relative rate of increase for the O₁, K₁, K₂, and Q₁ constituents is close to zero and their Std is small, especially for O₁ and K₁. This result suggests that the annual amplitudes of O₁, K₁, K₂, and Q₁ at the six stations are influenced by nodal tides and exhibit regular periodic variation, but with a minimal long-term rising trend. The annual amplitude changes in the other constituents either demonstrate an obviously rising trend or are very different from each other. When this phenomenon becomes more apparent, the predicted annual amplitude changes will lead to poorer fitting accuracy.

3.2.5. Nodal Correction of S₂

Doodson’s theory [3] specifically pointed out that the S2 constituent is not affected by nodal tides. In this subsection, we discuss whether this conclusion is changed by Taiwan’s complex terrain environment. Figure 11 presents the estimated amplitudes of perigean and nodal tides for the S₂ constituent. These values are expressed relative to b₀, which represents the mean amplitude. The b₀ value, that is, the mean annual amplitude, at the six stations ranges between 6.90 cm and 34.64 cm. Its mean value is 20.4 cm, and the Std is 8.94 cm.

Figure 11 shows that both the relative amplitudes of perigean and nodal tides for the S₂ constituent at the six stations are quite low, at about 0.01. Its mean value of perigean tide is 0.0077 and the standard deviation is 0.0026. For the nodal tide, the mean and Std are 0.0109 and 0.0031, respectively. The maximum value is 0.015, which occurs at the CG station. From this analysis, it can be confirmed that nodal tides do not affect the S₂ constituent, even under Taiwan’s complex submarine topography. This result is consistent with the theoretical analysis and further validates the reliability of this analysis method.

4. Discussion

4.1. Fitting of Second-Order Nodal Correction

This fitting method simultaneously accounts for the influence of both nodal and perigean tides on the amplitude of certain constituents. However, the AF correction formula in Table 2 includes only terms related to the third-order nodal tide. In order to compare the fitting consistency between Equation (3) and the AF formula in Table 2, the last two terms in Equation (3) have been changed to second-order nodal tides, which are $b_4\cos \left(2{\omega }_{n}t\right)$ and $b_5\sin \left(2{\omega }_{n}t\right).$ To facilitate the evaluation of the fitting performance of the two methods, the former is called NPT, and the latter is called N2T.

Because R² only describes how well the regression model fits the estimated data, it does not indicate the amount of calculation error. The mean R² and RMSE of eight constituents for the six stations and their mean and Std are shown in the two panels of

Table 4. Mean R² and RMSE between the fitted or calculated and estimated AF for eight constituents using three fitting formulas: NPT, N2T, and N2C.

Tide	R2			RMSE
	NPT	N2T	N2C	NPT	N2T	N2C
O1	0.9885	0.9886	0.9880	0.0137	0.0136	0.0139
K1	0.9772	0.9767	0.9733	0.0114	0.0116	0.0127
K2	0.9575	0.9572	0.9549	0.0408	0.0409	0.0420
Q1	0.9459	0.9440	0.9415	0.0305	0.0309	0.0316
M2	0.8391	0.8387	0.8157	0.0125	0.0126	0.1929
OO1	0.8166	0.8167	0.7400	0.1917	0.1921	0.0161
N2	0.7111	0.7246	0.6730	0.0219	0.0213	0.0232
J1	0.6542	0.6496	0.6311	0.0979	0.0982	0.1006
Mean	0.8613	0.8620	0.8397	0.0526	0.0527	0.0541
Std	0.1276	0.1262	0.1441	0.0630	0.0632	0.0630

.

The ranking of the top eight constituents whose R² values exceed the threshold is identical for NPT and N2T. Moreover, the R² and RMSE obtained from both methods are nearly identical, differing only at the fourth decimal place. The mean R² for NPT and N2T is 0.8613 and 0.8620, respectively, while the mean RMSE of NPT and N2T is 0.0526 and 0.0527, respectively. The Stds of NPT and N2T are very close. It is determined that NPT and N2T have an extremely high ability to fit the amplitude variation in each tidal constituent in time.

The term $\cos \left(2{\omega }_{n}t\right)$ indicates that the period of this component is T_n/2 = 9.31 years, which is close to the period of T_p = 8.85 years. Perhaps, the two periods are close, and the data length is only about 20 years, which leads to such analysis results. In the future, longer data or data from different locations will be needed to evaluate which method is more appropriate.

4.2. Effect of Time Change Rate of AF on Fitting Analysis

The time-varying term of the AF is also considered in Equation (3). With 20 years of data length, it is worth exploring whether this analysis method needs to consider the time change rate of the AF. If the linear time change term, $b_{1}t$, is removed, we refer to this fitting as N2C. The results of the N2C evaluation are listed in the last column of each panel of Table 4. It is clear that the R² of N2T is slightly higher than that of N2C for all constituents, and the RMSE of N2T is slightly lower than that of N2C. The mean R² and RMSE of N2C are 0.8397 and 0.0541, respectively. Its Stds are 0.1441 and 0.0630. N2C has an R² lower than N2T by about 0.02. As a result, it is still necessary to consider the time changes in AF in this issue.

4.3. Comparison with Godin’s Results

Godin [37] used HA to calculate the amplitude and phase of some constituents from four tide stations over a 20-year period spanning the 1960s to the 1980s. The four stations were Manzanillo in Mexico, Tofino in Canada, Quebec within the Saint Lawrence River, and Baltra in Galapagos. The estimated annual changes in amplitude and phase were then visually compared with the theoretical values to show their consistency. Our tide data are from three tide gauges in the Taiwan Strait, all facing the Pacific Ocean. The former may be affected by tidal shallowing and diffraction. We used R² or the SNR to objectively evaluate the consistency between the estimated value and the theoretical value.

Godin’s important conclusions are as follows: (1) The nodal corrections are effective for K₁, O₁, and K₂. (2) The constituents Q₁, J₁, OO₁, and M₁ are stabilized at sites where the tide is predominately linear and where the third-order effects are minimal. (3) The effectiveness of the corrections for μ₂ and ν₂ is not apparent. (4) It is not prudent to apply any a priori corrections to 2N₂, N₂, or L₂. (5) S₂ may at times be modulated by its nonlinear interaction with M₂. (6) M₂ may be demodulated by nodal corrections where the local tide is linear. The amplitude corrections for M₂ become excessive when nonlinear effects become preponderant, although the phase corrections continue to help.

The estimated and calculated AFs are shown in Figure 7b, and the PFs are shown in Figure 8b. Table 3 shows that the top eight highest R² values are for O₁, K₁, K₂, Q₁, OO₁, M₂, M₁, and J₁. The ranking of the effectiveness of nodal correction on the amplitude of constituents is the same as that of Godin [37], except for M₂. The case where the R² is lower than 0.4 in Figure 7b and Figure 8b shows that the calculated AF and PF are consistent with their estimated counterparts. The constituents described in Godin’s [37] third and fourth conclusions belong to this category.

S₂ is one of the four main tidal components, so its amplitude is relatively large. Therefore, the annual amplitude of the analysis has some slight changes, or there is a nodal tide cycle due to the nonlinear effect of M₂. However, the amplitude of this change is only about 1% of the average amplitude of S₂, as shown in Section 3.2.5, which is consistent with nodal tide theory.

4.4. Land Effect on AF and PF of Four Main Constituents

Under the assumption that water is uniformly distributed on the Earth’s surface and that there is no influence from land, theoretically, the global AF and PF are only related to time and have nothing to do with location [3,4]. The seafloor topography surrounding Taiwan is highly complex, and this is especially evident from a series of complex topographic features in the Taiwan Strait, between the fixed boundaries of Taiwan and China. It is worth studying whether complex topography can contribute to the spatial variation in nodal tides. However, there is currently no research exploring this topic. We can only use the numerical calculation results of Jan et al. [44] to infer the possible reasons why the nodal tide is affected by the topography (O₁, K₁, M₂, and S₂).

Jan et al. [44] calculated the amplitudes and phases of five constituents at 11 tide gauge stations across the Taiwan Strait using the fine-resolution Taiwan Strait Nowcast System (TSNOW). We plot the results of four main constituents—O₁, K₁, M₂, and S₂—at five tide gauge stations in Taiwan, as shown in Figure 12. The Tamsuei (TS) station is located in the north of JW, while Hsinchu (HC) is in the south. The Taichung (TC) station is located between JW and PH. The Budai (BD) station is located between PH and KH.

Jan et al. [44] showed that the semidiurnal M₂ tide, propagating southward, is Kelvin wave-like, which results from Coriolis force and land-side boundaries along the mainland coast, and its amplitude decreases offshore. On the other hand, the semidiurnal tide is a standing wave along the Taiwan coast where the tidal phase is nearly constant from HC to BD. The largest amplitude of M₂ is about 1.73 m near TC. The tidal amplitude drops to about 0.99 m at TS and to about 0.18 m at KH. The spatial variation in S₂ is similar to that in M₂, but the magnitude is quite small, about 1/3 of the M2 amplitude.

The K₁ tide propagates southward in the strait with little variation in amplitude and nearly constant speed, which manifests as a linear increase in phase. The spatial variation in O₁ is similar to that in K₁. The results of Jan et al. [44] indicate that the influence of the topography and land boundaries of the Taiwan Strait on the amplitude and phase of M₂ and S₂ is greater than that of O₁ and K₁.

We can infer that, when the waves incident on the Taiwan Strait from the open sea are slightly different each year, the amplitude and phase changes in M₂ and S₂ will be greater than those of O₁ and K₁. Therefore, the long-term amplitude and phase changes in M₂ and S₂ are not only affected by the node tide, but also by the topography. This explanation may provide the reason the AF and PF of O₁ and K₁ obtained in this paper are in better agreement with the theoretical values than those of M₂ and S₂. Whether the nodal tide is affected by the topography along the coast of Taiwan still needs to be confirmed by future studies.

5. Conclusions

Varying tides are very important for human life on the coast and ocean navigation. In addition to understanding the characteristics of semidiurnal, diurnal, fortnightly, monthly, and annual cycles on the Earth’s surface through HA, we use nodal tides with a cycle of 18.6 years, which affect tidal characteristics. The purpose of this paper is to use the six stations’ long-term tidal data to discuss the effects of perigean and nodal tides on 20 constituents in Taiwan waters with complex seabed topography and compare the results with previous theories. We modified the modulation method [36] to fit the estimated annual amplitude using HA. Two model performance indicators were used to assess the degree of agreement between each constituent’s amplitude factor and phase factor and the theoretical value. First, we used the O₁ constituent as an example to explain in detail the analysis of nodal tides at the six stations and then analyzed the overall results of all the constituents to draw some conclusions.

Comparing the theoretical and estimated amplitude and phase factors of the 20 constituents, O₁, K₁, K₂, and Q₁ have the highest four R² and SNRs. The results show that the annual amplitudes of the four constituents in the Taiwan waters are obviously affected by nodal tides and can be described theoretically. From the model evaluation results, it can be seen that the annual amplitudes of the M₂, OO₁, and J₁ constituents described by theoretical values are acceptable. The estimated annual amplitudes of the other 14 constituents deviate greatly from the theoretical values. The relative estimated amplitudes of perigean and nodal tides for the S₂ constituent at the six stations are determined to be quite low, at about 0.01, which is very close to the theoretical zero value. In particular, it is determined that the estimated annual amplitude of the constituents that are in good agreement with the theoretical value has almost no rising trend, and the ratio of the perigean amplitude to the nodal amplitude is lower than 0.2. In the future, these results must be theoretically verified.

Considering the addition of perigean tides or second-order nodal tides, the agreement between the fitted tidal amplitudes is quite consistent. Considering the linear time change leads to better fitting results than not considering it. A detailed comparison of the results of this study with those of Godin [37] is included in the Discussion Section. In theory [3,4], the nodal factors are only related to time, not location. The results of this paper are limited to Taiwanese waters. In the future, some global tide data, such as GESLA [53] or PSMSL [54], should be used to examine the effectiveness of nodal factors for each constitution.

The lowest astronomical tide (LAT) calculation is commonly used to define chart data. The LAT needs to be calculated from the estimated data of a tide station or from the simulated tidal levels of an ocean tide model using HA. This study shows that tidal amplitudes vary from year to year. The harmonic parameters of HA may be different when using tide data from different years. The different harmonic parameters may be combined to produce varying LATs. Future research should aim to select the appropriate AF and PF of nodal tides in LAT calculations to obtain a more stable and correct LAT.