Fitting Cotidal Charts of Eight Major Tidal Components in the Bohai Sea, Yellow Sea Based on Chebyshev Polynomial Method

Abstract

High-precision tidal harmonic constants are necessary for studies involving tides. This study proposes a new method combined with the adjoint assimilation model and the Chebyshev polynomial fitting (CPF) method to obtain the tidal harmonic constants in the shallow-water region of the Bohai and Yellow Sea (BYS). Based on the CPF method, the full-field harmonic constants and reliable cotidal charts of the eight major constituents (M₂, S₂, K₁, O₁, N₂, K₂, P₁ and Q₁) were fitted from the X-TRACK products briefly and this method was effectively for coastal conditions. Compared with the observations of the X-TRACK products and tidal gauges, for the M₂ constituent, the TPXO9, Finite Element Solutions 2014 (FES2014), National Astronomical Observatory 99b (NAO.99b) and Empirical Ocean Tide 20 (EOT20) models yield the root-mean-square errors (RMSEs) of 18.50, 7.31, 18.73 and 13.32 cm, respectively, while the CPF method yields an RMSE of 10.74 cm. These results indicate that the CPF method could maintain high resolution and obtain accurate cotidal charts consistent with the simulations of the four models in shallow-water regions.

1. Introduction

Ocean tide is an important basic phenomenon in ocean physical mechanisms, which is involved in many other physical and ecological ocean mechanisms. Ocean tides originate from the gravitational attraction of the sun and moon, and there is about 3.7 TW of tidal dissipation in the ocean [1], which is an important energy source for inducing internal waves and intensifying turbulent mixing, in addition to its influence on the climate [2]. Ocean tides can influence many aspects of human life; the superposition of strong wind conditions and high tides during the landfall of cyclones can create a strong surge that poses a huge threat to the coastline and nearby habitat [3,4,5,6]. The tidal processes and dynamics associated with tides and tidal currents can lead to huge sediment transport and variations in sea surface temperature, leading to significant changes in coastal ecosystems [7,8,9,10]. To investigate high-frequency dynamical processes more accurately, efforts have been made in several studies to isolate tidal signals like storm surges, coastal extreme water levels, and ocean circulation [11,12,13,14]. Indeed, extraction of accurate harmonic constants is helpful for solving the scientific problems that involve tides [15,16].

Investigation of the tidal dynamics in shallow water is very important. With advances in observational methods, it has become possible to study tidal harmonic constants in shallow-water regions [17,18,19,20]. However, there are few satellite and tide gauge observation points in the Bohai and Yellow Sea (BYS), and it is necessary to obtain the full field harmonic constants even if the spatial resolution presents a challenge. Youn et al. [21] compared the TOPEX/POSEIDON (T/P) altimeter data and observations from 10 tide gauges near the satellite tracks in the East Asian marginal seas and calculated that the RMSEs were lower than 6.7 cm. Fang [22] used 10 years of T/P altimeter data to obtain empirical cotidal charts (1/12°) with limited satellite altimeter data in the BYS [23]. Low RMSEs were achieved in these studies in the whole area of the Bohai, Yellow and East Sea. However, the resolution of these studies can be improved, and the studies did not specifically calculate the tidal harmonic constants in shallow-water regions.

Generally, the complexities associated with tides in the deeper ocean are negligible. In contrast, the non-linear interactions of the tides and tidal currents with shallow bathymetry [24,25], coastal circulation and hydrography of the region lead to significant tidal variations within the shelf-slope regions [15,26]. The study region is an area of shallow water in BYS, which has a complicated tidal regime [22]. The study focuses on the eastern area of the study region, where the tidal amplitude is relatively large. These factors make it more difficult to obtain the tidal harmonic constants on the coast than in the open ocean, which motivates the development of a rapid and effective method for obtaining precise harmonic constants in this study region. In the present study, a new method based on CPF is proposed to calculate the full-field tidal harmonic constants in the shallow-water region of BYS, which provides a new direction for obtaining reliable harmonic constants in this region. The CPF was successfully applied in the area near Hawaii (the open ocean) and obtained precise cotidal charts of eight major constituents and minor tidal constituents [27,28]. This paper presents the application and results of the CPF method in the shallow-water region of BYS.

In this study, the data, the adjoint assimilation model and CPF methods are introduced in Section 2. A comparison between the CPF results and other tidal model results, as well as the eight cotidal charts in BYS, is presented in Section 3. Finally, the paper is summarized in Section 4.

2. Data and Methodology

2.1. Data

The study region is 34–40° N, 121.5–127.5° E (Figure 1), near the BYS (~water depth < 150 m). The harmonic constants (M₂, S₂, K₁, O₁, N₂, K₂, P₁ and Q₁) from the X-TRACK tidal products are adopted for further calculations. According to the long-time-series missions (Topex/Poseidon + Jason-1 + Jason-2), X-TRACK tidal constant products can be used to calculate the 73 tidal constituents using harmonic analysis. This product is a very important component of ocean observation systems with the aim of understanding ocean dynamics [29,30]. The X-TRACK along-track tidal constants are sufficiently accurate to estimate the results of the CPF method at the satellite observation points.

In this study, the harmonic constants calculated by the CPF are compared with the Finite Element Solutions 2014 (FES2014) model at a horizontal resolution of 1/16° [31], the EOT20 model at a horizontal resolution of 1/8° [32], the National Astronomical Observatory 99b (NAO.99b) model at a horizontal resolution of 1/2° [33], and the TPXO9 at a horizontal resolution of 1/6° [34,35]. Refer to the study of Xu, Wang, Wang, Lv and Chen [28] for more details on the NAO.99b, TPXO9 and FES2014 models, and refer to the study of Wang, Zhang, Xu, Wang and Lv [27] for more details on the EOT20 model. The harmonic constants of eight tidal gauges (Figure 1) were also compared with the harmonic constants of the CPF, TPXO9, FES2014, NAO.99b and EOT20 solutions.

2.2. The Adjoint Assimilation Model

In order to solve the problem of the sparse observation data available in this area, the two-dimensional tidal model and adjoint assimilation model are introduced to assimilate the X-TRACK satellite observation into the full-field harmonic constants at low resolution. Although the adjoint assimilation model is very reliable, and valid in many applications [36,37,38,39], it requires more computing resources for high-resolution tidal simulation. This study learns from the study of Zheng, Mao, Lv and Jiang [20], which used the method of nesting the adjoint assimilation model layer by layer to obtain high-resolution harmonic constants for a small area, thus saving computing resources. This study uses the adjoint assimilation model nested within the CPF method to maintain relatively high accuracy and reduce computing time.

The horizontal resolution of the model is 10′ × 10′, and the time step is 372.618 s (~1/120 of the period of the M₂ constituent). The tidal model is the same as that employed by Cao, Guo and Lü [39]. The set of parameters is employed with reference to Lu and Zhang [40]. The harmonic constants of the M₂ constituent calculated by the adjoint assimilation model are shown in Figure 1, as a demonstration of the results.

2.3. The CPF Method

For polynomial fitting problems, to improve the fitting accuracy, it is necessary to use higher-order polynomials. However, when the number of fitting polynomials is higher, the normal equations are often ill conditioned. The higher the order of the coefficient matrix of the normal equations, the more serious the ill conditioning. Therefore, the orthogonal polynomial fitting method is used to avoid the ill conditioning of normal equations [41]. On the other hand, when using orthogonal Chebyshev polynomials for data fitting, the main diagonal of the coefficient matrix is strictly dominant, which greatly improves the calculation efficiency.

The low-resolution harmonic constants in the full field calculated by the adjoint assimilation model and X-TRACK satellite observation are together fitted by the Chebyshev polynomial fitting (CPF) method. The CPF method has fast convergence properties and can effectively fit low-resolution X-TRACK satellite data to high-resolution grids (1/30°) [42].

Given I₀ discrete points on the x-axis, $x_1,x_2,\cdots ,x_{I_0}$, the following Chebyshev polynomial is defined at these discrete points

$$\begin{array}{l}{\varphi }_0(x_i)=1\hfill \\ {\varphi }_1(x_i)=x_i-P_{1,0}{\varphi }_0(x_i)\hfill \\ {\varphi }_2(x_i)=x_i^2-P_{2,1}{\varphi }_1(x_i)-P_{2,0}{\varphi }_0(x_i)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=1,2,\cdots ,I_0\hfill \\ \cdots \hfill \\ {\varphi }_{\mathrm{k}}(x_i)=x_i^k-P_{k,k-1}{\varphi }_{\mathrm{k}-1}(x_i)-P_{k,k-2}{\varphi }_{\mathrm{k}-2}(x_i)-\cdots -P_{k,0}{\varphi }_0(x_i)\hfill \end{array}$$

(1)

where ${\varphi }_{\mathrm{k}}(x_i)$ is the basis function of the Chebyshev polynomial. The Chebyshev polynomials are orthogonal and satisfy

$${\displaystyle \sum _{i=1}^{I_0}{\varphi }_k(x_i)\cdot {\varphi }_l\left(x_i\right)=0},k\ne l$$

(2)

The coefficients in Equation (1) can be obtained by using the following Equation:

$$P_{k,s}=\frac{{\displaystyle \sum _{i=1}^{I_0}{x}_i^k\cdot {\varphi }_s(x_i)}}{{\displaystyle \sum _{i=1}^{I_0}{\varphi }_s^2(x_i)}}$$

(3)

If the function $Z\left(x\right)$ is defined on the same interval and its value at the lattice point $x_i$ is denoted as $Z(x_i)$, then the function $Z(x)$ can be expanded on these lattice points using the Chebyshev polynomial:

$$\tilde{Z}(x_i)={\displaystyle \sum _{k=0}^{K_0}{A}_k{\varphi }_k}(x_i)$$

(4)

where $\tilde{Z}(x_i)$ is the value of $Z\left(x_i\right)$ fitted using Equation (4), $K_0$ is the cutoff order of the polynomial taken, and $K_0\le I_0-1$. $A_k$ is the expansion coefficient, which can be obtained according to the orthogonal polynomial principle:

$$A_k=\frac{{\displaystyle \sum _{i=1}^{I_0}Z(x_i){\varphi }_k\left(x_i\right)}}{{\displaystyle \sum _{i=1}^{I_0}{\varphi }_k^2\left(x_i\right)}}$$

(5)

For the rectangular two-dimensional field $Z\left(x,y\right)$, assuming that there are N discrete points $\left(x_1,y_1\right),\left(x_2,y_2\right),\cdots ,\left(x_N,y_N\right)$ in the interval, the corresponding expansion formula is as follows:

$${\tilde{Z}}_i=\tilde{Z}\left(x_i,y_i\right)={\displaystyle \sum _{k=0}^{K_0}{\displaystyle \sum _{s=0}^{S_0}{A}_{k,s}{\varphi }_k(x_i){\psi }_s(y_i),}}i=1,2,\cdots ,N$$

(6)

where $K_0,S_0$ are polynomial cutoff orders taken in the x-direction and y-direction, respectively, $A_{k,s}$ is the expansion coefficient, ${\varphi }_k(x_i)$ is the k-order polynomial in the x-direction, ${\psi }_s(y_i)$ is the s-order polynomial in the y-direction, and ${\tilde{Z}}_i$ is the value of $Z_i$ obtained by fitting with Equation (6) The sum of error squares is defined as zero:

$$\epsilon \left(A_{0,0},A_{1,0},\cdots ,A_{k,s},\cdots ,A_{K_0,S_0}\right)={{\displaystyle \sum _{i=1}^N\left[Z_i-{\displaystyle \sum _{k=0}^{K_0}{\displaystyle \sum _{s=0}^{S_0}{A}_{k,s}{\varphi }_k(x_i){\psi }_s(y_i)}}\right]}}^2$$

(7)

According to the principle of least squares, the partial derivative of each variable is obtained:

$$\frac{\partial \epsilon }{\partial A_{m,n}}=2{\displaystyle \sum _{i=1}^N\left[Z_i-{\displaystyle \sum _{k=0}^{K_0}{\displaystyle \sum _{s=0}^{S_0}{A}_{k,s}{\varphi }_k(x_i){\psi }_s(y_i)}}\right]{\varphi }_m(x_i){\psi }_n(y_i)=0}$$

(8)

The definition of an orthogonal polynomial is as follows:

$$\begin{array}{l}{\displaystyle \sum _{i=1}^N{\varphi }_k(x_i){\varphi }_m(y_i)=0,k\ne m},\hfill \\ {\displaystyle \sum _{i=1}^N{\psi }_s}(y_i){\psi }_n(y_i)=0,s\ne n,\hfill \end{array}$$

(9)

According to Equations (8) and (9), the expansion coefficient is as follows:

$$A_{k,s}=\frac{{\displaystyle \sum _{i=1}^{N}Z\left(x_i,y_i\right){\varphi }_k\left(x_i\right){\psi }_s\left(y_i\right)}}{{\displaystyle \sum _{i=1}^N{\varphi }_k^2\left(x_i\right){\psi }_s^2\left(y_i\right)}}$$

(10)

The harmonic constants calculated by the adjoint assimilation model and the harmonic constants of the X-TRACK satellite are together fitted by the CPF method, in order to obtain the high-resolution harmonic constants in the full field. More details on data, methodology and error assessment can be found in Part I (Xu et al., 2021).

In the meantime, to validate the accuracy of the results fitted by the CPF method, the mean absolute errors (MAEs) of amplitude and phase lag are defined as:

$$\Delta H=\frac{1}{N}{\displaystyle \sum _{n=1}^N\left|H_n-H^{*}{}_n\right|},\Delta G=\frac{1}{N}{\displaystyle \sum _{n=1}^N\left|G_n-G^{*}{}_n\right|},$$

(11)

where $H_n$ and $G_n$ represent the amplitude and phase lag of satellite altimeter or tidal models at the nth point, $H^{*}{}_n$ and $G^{*}{}_n$ are the amplitude and phase lag calculated by the CPF method at the same point, and N is the total number of points.

The RMSE can reflect the combined effects of the $\Delta H$ and $\Delta G$. The RMSE is defined as:

$$RMSE={\left\{\frac{1}{2}(H_o{}^2+H_s{}^2)-H_o{H}_s\cos (G_o-G_s)\right\}}^{1/2},$$

(12)

where subscripts o and s represent the actual and reconstructed groups of harmonic constants, respectively.

The CPF method has two important parameters: horizontal fitting order and vertical fitting order ($K_0,S_0$). The optimal fitting orders, which can make the accuracy higher and the result distribution more reasonable, are generally different in different sea areas.

From the previous experience of other researchers, too high an order can cause over-fitting [28]. To avoid over-fitting, the optimal order is determined using the cross-validation method. The X-TRACK data are divided into ten parts to provide 10 experiments. In the ten different experiments, one part of the data is chosen as the evaluation data, and the other nine parts are fitted using the CPF method. In the ten different experiments, the orders of horizontal and vertical fitting are 2~6, respectively, and a total of 25 groups of experiments are set up. In each experiment, the results of the CPF method using the nine parts of the data are compared with both the observations from the tidal gauges and the one part of the data being employed as evaluation data, and the root-mean-square errors (RMSEs) of 25 groups are calculated. For the 25 groups in each of the ten different experiments, there are ten groups of identical horizontal and vertical orders from ten experiments. The RMSEs of ten groups of identical orders are averaged, so that the group of certain orders with the smallest RMSE can be chosen, and the optimal orders can be determined. The optimal orders (shown in

Table 1. The optimal orders of the Chebyshev polynomials for the eight major constituents.

Tide	M2	S2	N2	K2	K1	O1	P1	Q1
m	6	5	6	6	6	6	5	6
n	6	6	6	4	6	6	5	6

) are assessed quantitatively by the mean RMSE of the ten experiments.

3. Experiments and Results

3.1. Comparison with the Results from The X-TRACK Satellite Altimeters

In order to compare the results of the CPF method and the four tidal models at the satellite observation points, the RMSEs and MAEs at the eight tidal gauges were averaged in space.

Table 2. The RMSEs and MAEs of the harmonic constants of the results calculated by the CPF method, the TPXO9, FES2014, NAO.99b and the EOT20 models for the eight major constituents, compared with the X-TRACK coastal satellite observations.

RMSE (cm)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	3.81	1.36	1.46	1.29	1.38	0.77	0.84	0.69
TPXO9	5.70	2.30	2.14	1.56	1.09	1.39	0.83	0.67
FES2014	0.52	0.67	0.95	0.36	0.34	0.41	0.66	0.36
NAO.99b	4.35	2.44	1.77	1.60	1.53	1.02	0.90	0.58
EOT20	1.34	0.80	1.11	0.56	0.40	0.48	0.67	0.34
$\Delta H$ (cm)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	4.27	1.25	1.33	1.38	1.52	0.75	0.74	0.42
TPXO9	4.22	1.64	0.93	0.67	0.81	1.59	0.67	0.79
fFES2014	0.55	0.59	0.66	0.25	0.29	0.37	0.38	0.30
NAO.99b	5.01	2.70	1.10	1.97	1.64	1.09	0.69	0.70
EOT20	1.21	0.64	0.60	0.48	0.36	0.46	0.40	0.26
$\Delta G$ (deg)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	1.62	2.23	4.08	3.08	3.12	4.38	9.60	12.48
TPXO9	3.36	3.71	11.23	6.98	3.49	5.16	10.23	9.15
FES2014	0.22	0.98	5.39	1.76	1.01	2.23	10.22	7.12
NAO.99b	1.36	2.95	6.69	2.87	3.63	5.04	11.28	8.05
EOT20	0.62	1.14	6.42	2.05	0.95	2.66	9.21	6.18

shows the spatially averaged MAEs and RMSEs of the CPF method, the TPXO9, FES2014, NAO.99b and EOT20 models, compared with the X-TRACK satellite observations.

For semidiurnal tides (the M₂, S₂, N₂ and K₂ constituents), the CPF method and the four models have approximately equal RMSEs. For the M₂, S₂, K₂ constituents, the RMSEs of the CPF method (3.81, 1.36 and 0.77 cm) are smaller than those of the TPXO9 (5.70, 2.30 and 1.39 cm) and NAO.99b (4.35, 2.44 and 1.02 cm) models. For the N₂ constituent, the RMSE of the CPF method (1.38 cm) is smaller than RMSE of the NAO.99b model (1.53 cm). For diurnal tides, the results of the CPF method also have a qualified performance. For the K₁ and O₁ constituents, the RMSEs of the CPF method are smaller than those of the TPXO9 and NAO.99b models. For the P₁ constituent, the RMSEs of the CPF method (0.84 cm) and the TPXO9 model (0.83 cm) are smaller than those of the NAO.99b model (0.90 cm). For the Q₁ constituent, the RMSEs of the CPF method (0.69 cm) and the TPXO9 model (0.67 cm) are similar to that of the FES2014 (0.36 cm), NAO.99b (0.58 cm) and EOT20 (0.34 cm) models. The mean RMSE of the CPF method is 1.45 cm, and those of the TPXO9, TPXO9, FES2014, NAO.99b and EOT20 models are 1.96, 0.53, 1.77 and 0.71 cm, respectively.

For the eight major constituents, the RMSEs of the four models and the CPF method compared with the X-TRACK satellite observations are shown in Figure 2. For the eight major constituents, the RMSEs of the CPF method at the points of satellite observation are similar to those of the four models, indicating that the fitted harmonic constants are consistent with satellite observations at all points and spatial-averaged. In general, the results fitted by the CPF method at the position of the X-TRACK observation can achieve the accuracy to be applicated in other scientific research.

3.2. Comparison with the Harmonic Constants from the Tidal Gauges

Because of the influence of complex terrain in the study region, and the quite short distance between the tidal gauges and the shore, the RMSEs and MAEs between the simulated results and the observations of the coastal tidal gauges are greater than those of the X-TRACK satellite. In order to validate the results of the CPF method, the TPXO9, FES2014, NAO.99b and EOT20 models, the harmonic constants of those models located at the tidal gauges are obtained by linear interpolation from grid points existing data.

Table 3. The RMSEs and MAEs of the harmonic constants of the results calculated by the CPF method, the TPXO9, FES2014, NAO.99b and the EOT20 models for the eight major constituents, compared with the tidal gauges observations.

RMSE (cm)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	10.74	6.17	2.46	1.74	2.66	1.88	0.97	1.29
TPXO9	18.50	8.94	2.61	1.02	3.16	2.68	1.13	0.74
FES2014	7.31	4.02	1.37	1.33	1.69	1.36	0.58	0.57
NAO.99b	18.73	8.33	3.38	3.08	4.22	2.46	1.51	0.85
EOT20	13.32	6.98	1.86	1.91	2.97	2.04	0.69	0.58
$\Delta H$ (cm)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	8.60	3.71	0.84	0.75	2.19	1.46	0.28	0.50
TPXO9	9.59	5.08	1.55	0.32	1.73	1.56	0.92	0.43
FES2014	2.98	2.96	0.86	0.67	0.91	0.63	0.19	0.10
NAO	15.82	7.75	3.68	2.81	3.10	2.07	1.71	0.64
EOT20	9.28	4.80	0.72	0.70	1.99	1.40	0.30	0.30
$\Delta G$(deg)	M2	S2	K1	O1	N2	K2	P1	Q1
CPF	5.31	9.59	6.34	5.85	5.22	9.98	8.43	22.40
TPXO9	10.33	15.18	5.95	3.67	8.23	16.39	6.22	11.18
FES2014	4.14	5.68	3.37	4.59	4.08	7.86	5.07	11.13
NAO	7.92	8.40	3.94	6.95	9.99	8.47	6.29	13.38
EOT20	6.91	10.01	4.70	6.75	7.84	11.73	5.65	9.25

shows the spatially averaged MAEs and RMSEs of the harmonic constants between the tidal gauge observations and the CPF method, and the MAEs and RMSEs of the harmonic constants between the tidal gauge observations and the four tidal models.

For semidiurnal tides, the harmonic constants calculated by the CPF method are significantly better than those of the TPXO9, NAO.99b and EOT20 models. For the M₂ constituents, the RMSE of the CPF method (10.74 cm) is 19–43% lower than that of the TPXO9 (18.50 cm), NAO.99b (18.73 cm) and EOT20 (13.32 cm) models. For the S₂ constituents, the RMSE of the CPF method (6.17 cm) is 13–31% lower than that of the TPXO9 (8.94 cm), NAO.99b (8.33 cm) and EOT20 (6.98 cm) models. For the N₂ constituent, the RMSE of the CPF method (2.66 cm) is 10–37% lower than that of the TPXO9 (3.16 cm), NAO.99b (4.22 cm) and EOT20 (2.97 cm) models. For the K₂ constituent, the RMSE of the CPF method (1.88 cm) is 8–30% lower than that of the TPXO9 (2.68 cm), NAO.99b (2.46 cm) and EOT20 (2.04 cm) models. For diurnal tides, the results of CPF are approximately equal to the other four models. For the K₁ constituent, the RMSE of the CPF method (2.46 cm) is 6% and 27% lower than that of the TPXO9 (2.61 cm) and NAO.99b (3.38 cm) models, respectively. For the O₁ constituent, the RMSE of CPF (1.74 cm) is 9% and 44% lower than that of the EOT20 (1.91 cm) and NAO.99b (3.08 cm) models, respectively. For the P₁ constituent, the RMSE of the CPF method (0.97 cm) is 14% and 36% lower than that of the TPXO9 (1.13 cm) and NAO.99b (1.51 cm) models, respectively. For the Q₁ constituent, the CPF method has no better performance. In this study, region, the amplitude of the Q₁ constituent is much lower than that of the other seven major constituents, which makes the simulation of the harmonic constants more difficult using the adjoint assimilation model. Especially in the coastal region, the $\Delta G$ is harder to reduce, which leads to bigger RMSEs compared with the four models after the next step of fitting. At the same time, the K₁ and O₁ constituents with bigger amplitude (than the P₁ and Q₁ constituents) have small $\Delta G$ compared with both of that derived from the X-TRACK products and the tidal gauge observations.

The mean RMSE of the CPF method is 3.49 cm, and those of the TPXO9, FES2014, NAO.99b and EOT20 models are 4.85, 2.28, 5.32 and 3.79 cm, respectively. The CPF method calculates enough accurate harmonic constants near the tidal gauges, which illustrates that the CPF has better performance for eight major constituents in the area closer to the land.

For the eight major constituents, the RMSEs of the four models and the CPF method compared with the observations of the eight tidal gauges are shown in Figure 3. For the eight major constituents, the RMSE of the CPF method at any tidal gauge is smaller or a little larger than that of the four models. Therefore, not only the spatial-averaged RMSE of CPF is relatively ahead of the four models, but also approximate equal to the four models at each tidal gauge. These results indicate that the fitted harmonic constants are consistent with observations.

The harmonic constants of the four models and the CPF method compared with the observations of the eight tidal gauges are shown in Figure 4. The correlation coefficients of the harmonic constants between the CPF method and the eight tidal gauges are also calculated in Figure 4. For the eight major tidal constituents, the harmonic constants of the CPF method show well consistent with those derived from the tidal gauges. Except for the phase lags of the Q₁ constituent, the other correlation coefficients are bigger than 0.92. The high correlation indicates that the harmonic constants calculated by the CPF method have a good accuracy near the coast. For the Q₁ constituent, the correlation coefficients of amplitude and phase lags are 0.93 and 0.75, respectively, which indicates that parts of gauges have bigger $\Delta G$ (like Usuyong, Mokpo, Eocheongdo and Yeonggwang in Figure 3). This further explain that the RMSE of the Q₁ (1.29 cm) constituent mainly comes from the bigger $\Delta G$.

3.3. Cotidal Charts Obtained by the CPF Method and the Four Models

The cotidal charts of the M₂, S₂, K₁, O₁, N₂, K₂, P₁ and Q₁ constituents are given in Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12. It can be observed that the amplitudes of the eight constituents are bigger in the eastern and northern of the study region and generally small in the center of the study region. Because of the influence of the complex terrain, the area on the east coast near 37° N has the biggest amplitude in the study region. In this area, the amplitudes of the M₂ constituent can reach about 270 cm. The amphidromic points of diurnal tides from the CPF method and the four models are not in the study region. The amphidromic points of semidiurnal tides from the CPF method and the four models are shown in Figure 5, Figure 6, Figure 9 and Figure 10. The amphidromic points of semidiurnal tides appear in the region of 37–38° N, and the amphidromic points of diurnal tides appear in the southern of the 35° N. Residual analysis carried out in the EOT20 model used the results of the FES2014 model, so that the amphidromic points of both models have small differences and even are identical (Figure 6). Additionally, the phase contours of TPXO9, NAO.99b and EOT20 are discontinuous and the phase contours of NAO.99b are not connected to the amphidromic points. The phase contours of the CPF method are continuous and connected to the amphidromic points.

In Figure 5, Figure 6, Figure 9 and Figure 10, the positions of amphidromic points of the M₂, S₂, N₂ and K₂ constituents from the TPXO9, FES2014, NAO.99b and EOT20 models are around the amphidromic points of the CPF method, while simultaneously not being far from the amphidromic points of the other four models, indicating that the amphidromic points of the CPF method are sufficiently precise. For the M₂, S₂, N₂ and K₂ constituents, the distances between the amphidromic points derived from the CPF method and the FES2014 model are less than ${10}^{\prime }$. Because the FES2014 model has the smallest RMSEs, the amphidromic points of the CPF method and the FES2014 model may be the most reasonable.

For each constituent, the cotidal chart derived from the CPF method is similar to those of the four models in the full field, which illustrates that the harmonic constants fitted by the CPF method can be not only precise in terms of the points of the X-TRACK satellite observation and the tidal gauges, but also sufficiently precise in the full field. In Figure 5, Figure 6, Figure 7, Figure 8, Figure 9, Figure 10, Figure 11 and Figure 12, the spatial resolution of cotidal charts obtained by the CPF method is superior to that of the TPXO9 and NAO.99b models. The adjoint assimilation model nested the CPF method, as a brief method, not only can calculate the precise full-field harmonic constants effectively, but also has a high resolution.

4. Conclusions

A brief and efficient method is adopted to calculate the harmonic constants of the eight major tidal constituents (the M₂, S₂, K₁, O₁, N₂, K₂, P₁ and Q₁) with small RMSEs and MAEs using the X-TRACK satellite observation only.

The accurate assimilation method is an important basis for the further optimization of the harmonic constants. The adjoint assimilation model is used to provide low-resolution harmonic constants in the full field to fit by the CPF method, which effectively solves the problem of fewer observational data, and provides the possibility of obtaining better fitting results.

In the study region employed for this paper (121.5–127.5° E, 34–40° N), the optimal fitting order in the horizontal and vertical of each constituent is chosen by the cross-validation method. A series of experiments indicate that the CPF method can effectively fit the low-resolution X-TRACK satellite observation and digital analog data to a high-resolution grid. The harmonic constants of the CPF method, the TPXO9, FES2014, NAO.99b and EOT20 models are compared with the X-TRACK satellite observation and the tidal gauges observation. Compared with the X-TRACK satellite observation, the results of the CPF method yield a mean RMSE of 1.45 cm, that of the TPXO9, TPXO9, FES2014, NAO.99b and EOT20 models yield mean RMSEs of 1.96, 0.53, 1.77 and 0.71 cm, respectively. The results of the CPF method have a qualified performance for fitting the eight major tides. Compared with the tidal gauges observation, the results of the CPF method yield a mean RMSE of 3.49 cm, that of the TPXO9, TPXO9, FES2014, NAO.99b and EOT20 models yield mean RMSEs of 4.85, 2.28, 5.32 and 3.79 cm, respectively. Although the observation of tidal gauges does not fit within the scope of this study, the harmonic constants are highly correlated with those of the tidal gauges near the coast, which shows the superiority of the CPF method.

In addition, the position of amphidromic points of the CPF method and the four models are also compared. The comparisons indicate that the harmonic constants calculated by the CPF method are accurate for obtaining reliable cotidal charts of the eight major constituents and to be applicated in the other research. In conclusion, the results obtained using the CPF method maintain high precision, and the cotidal charts derived from it are very close to that of the TPXO9, FES2014, NAO.99b and EOT20 models, indicating that the CPF method is an effective and reasonable method for obtaining high-resolution harmonic constants.