Two-Dimensional Legendre Polynomial Method for Internal Tide Signal Extraction

Abstract

This study employs the two-dimensional Legendre polynomial fitting (2-D LPF) method to fit M2 tidal harmonic constants from satellite altimetry data within the region of 53°E–131°E, 34°S–6°N, extracting internal tide signals acting on the sea surface. The M2 tidal harmonic constants are derived from the sea surface height (SSH) data of the TOPEX/Poseidon (T/P), Jason-1, Jason-2, and Jason-3 satellites via t-tide analysis. We validate the 2-D LPF method against the 300 km moving average (300 km smooth) method and the one-dimensional Legendre polynomial fitting (1-D LPF) method. Through cross-validation across 42 orbits, the optimal polynomial orders are determined to be seven for 1-D LPF, and eight and seven for the longitudinal and latitudinal directions in 2-D LPF, respectively. The 2-D LPF method demonstrated superior spatial continuity and smoothness of internal tide signals. Further single-orbit correlation analysis confirmed generally higher correlation with topographic and density perturbations (correlation coefficients: 0.502, 0.620, 0.245; 0.420, 0.273, −0.101), underscoring its accuracy. Overall, the 2-D LPF method can use all regional data points, overcoming the limitations of single-orbit approaches and proving its effectiveness in extracting internal tide signals acting on the sea surface.

1. Introduction

Internal tides, generated by the oscillation of internal waves within the ocean’s density stratification, are a crucial component of ocean dynamics. These tides are globally distributed and significantly influence the transport of energy and mass in the ocean [1]. Studies have shown that the energy of internal tides gradually attenuates as they propagate into the deep ocean, leading to significant energy accumulation in shallow seas and continental shelf areas, thereby impacting local ocean dynamic processes [2,3]. Internal tides play a key role in the global climate system. The energy dissipation and transformation processes of internal tides are crucial for regulating deep-sea heat fluxes and the internal temperature and salinity structure of the ocean [4,5]. For this reason, internal tides play a key role in the global climate system. By enhancing vertical mixing and nutrient transport, they directly influence ocean biological productivity and ecosystems [6,7].

Research on internal tides in the central Indian Ocean has gained increasing attention globally. The central Indian Ocean is characterized by unique topographic features, such as seamounts and ridges, which provide favorable conditions for the generation and propagation of internal tides [8,9]. Specifically, the Mascarene Plateau in the central Indian Ocean is a key region for internal tide generation. The generation and propagation of internal tides in this area is influenced by complex topography and water mass structures, leading to an irregular vertical distribution of the internal tide energy [10]. Additionally, climatic phenomena such as the Indian Ocean Dipole (IOD) and the El Niño–Southern Oscillation (ENSO) have significant effects on the spatiotemporal distribution of internal tides [11].

With the continuous development of satellite altimetry technology, using altimeter data for internal tide research has become an important method. The high-resolution and long-term sea surface height data can be precisely measured by altimeters [12]. These data can be used not only to enhance the accuracy and comprehensiveness of the research when combined with other observational methods, such as underwater acoustic detection and hydrological observation, but also to extract internal tide signals [13,14]. The manifestation of internal tides on the sea surface is primarily reflected by periodic changes in the sea surface height. Studies have shown that the amplitude of internal tide fluctuations is more pronounced in coastal regions and on continental shelves [15]. For instance, Ray and Mitchum (1996) used satellite altimeter data near Hawaii to reveal significant internal tide signals on the sea surface [16]. Additionally, satellite altimeters can monitor the propagation paths and energy attenuation of internal tides, providing critical observational data for understanding the dynamics of internal tides [17]. In the central Indian Ocean, regions such as the Mascarene Plateau exhibit more significant internal tide signals on the sea surface due to their unique topographic features. Zhao et al. (2021) conducted a detailed analysis of the seasonal variations and energy transfer of internal tides in this area using altimeter data, finding that the energy distribution of internal tides is highly complex and significantly influenced by topography and water mass structures [18]. These findings not only help reveal the generation and propagation mechanisms of internal tides in the central Indian Ocean but also provide valuable scientific insights into the region’s ocean dynamics. Furthermore, climatic phenomena such as the Indian Ocean Dipole (IOD) and the El Niño–Southern Oscillation (ENSO) significantly affect the signals of internal tides on the sea surface. Studies have shown that these climatic phenomena alter water mass structures and ocean currents, thus influencing the generation and propagation of internal tides [19]. This further demonstrates the importance of altimeter data in studying the relationship between internal tides and climate change.

As the research on internal tides deepens, methods for extracting internal tide signals using satellite altimeter data have significantly advanced, with the one-dimensional (1D) along-track extraction method being a commonly used approach. Richard D. Ray (1996) argues that the wavelengths of same-frequency internal tides are significantly shorter than those of barotropic tides, with very few exceeding 400 km [16]. Thus, high-pass filtering was applied to the T/P satellite altimeter data over the Hawaiian Sea, along-track, to filter out the wavelengths greater than 400 km associated with barotropic tides. The process revealed the spatial distribution characteristics of the M2 internal tides in this region [16]. Dushaw B. D. (2002) employed a 300 km sliding average method to filter the sea surface signals of internal tides over the Hawaiian Sea region [20]. An eighth-order polynomial fit was utilized to filter and extract internal tide signals from the T/P satellite altimeter data over the North Pacific Ocean region [21]. Despite the effectiveness of the one-dimensional (1D) along-track extraction method in isolating internal tide signals, several limitations hinder its broader application in complex oceanic environments. Firstly, the method requires analyzing each satellite orbit individually, which is labor-intensive and time-consuming. This impedes rapid and comprehensive analysis over large oceanic areas. Secondly, the accuracy of the signal extraction can be compromised for shorter orbits, especially in regions where the satellite coverage is irregular. This reduces the method’s overall representativeness and accuracy, limiting its utility in extensive and diverse oceanic studies. These factors underscore the need for alternative approaches that can provide a more efficient and precise analysis across varied oceanic conditions.

This study employs a simple and effective method, the two-dimensional Legendre polynomial (2-D LPF) method, to extract internal tide signals acting on the sea surface. We chose Legendre polynomials for fitting due to their numerous advantages: orthogonality (Legendre polynomials are orthogonal over the interval [−1, 1], which means that the inner product of the different order polynomials is zero within this interval. This orthogonality makes them extremely useful in function approximation and weighted integration, as it effectively reduces the interference between basis functions); robustness (Legendre polynomials exhibit good stability in numerical computations, minimizing the accumulation of numerical errors); and simplicity (Legendre polynomials have simple recurrence relations, allowing the efficient computation of higher-order polynomials) [22].

In this study, the 2-D LPF method is introduced in Section 2. Section 3 demonstrates the internal tide signals extracted using the 2-D LPF method compared to those obtained by the 1-D LPF method. Section 4 discusses the correlation between the 2-D LPF method and the 1-D LPF method with respect to topographic and density perturbations.

2. Materials and Methods

2.1. Study Region and Data

The study region is 53°E–95°E, 34°S–6°N. Figure 1 illustrates the topography of the study area overlaid with orbit track data. The colors represent different depths, with the color bar indicating depth variations from shallow waters (light blue) to deep ocean (dark blue) in meters. The black grid lines denote the satellite orbit paths. The study region features various complex topographic structures such as seamounts, ridges, and abyssal plains. By considering these topographic features, as tidal currents flow over seamounts and ridges, they cause disturbances in the water column, which can trigger internal tides.

The density structure, like the influence of topography, serves as crucial evidence for the presence of internal tides in the study region. The internal tides cause vertical and horizontal density variations within the water column as they generate or propagate, resulting in density perturbations. To further substantiate the presence of internal tides in the study region, we incorporate the analysis of density structure. We use the following formula to calculate density perturbations ${\rho }_{(x,y,t)}^{\prime }$:

$${\rho }_{(x,y,t)}^{\prime }={\int }_z^0({\rho }_{(x,y,z,t)}-{\int }_0^t{\rho }_{(x,y,z,t)}dt)dz$$

(1)

where ${\rho }_{(x,y,z,t)}$ is calculated based on temperature and salinity data. The temperature and salinity data were obtained from the HYCOM global daily snapshot 0Z 1/12 degree GOFS 3.1 analysis (GLBy0.08) Temperature/Salinity dataset. The HYCOM GLBy0.08 dataset features a spatial resolution of 1/12 degree (approximately 9 km) and a temporal resolution of daily snapshots at 00:00 UTC. This dataset is generated by the Hybrid Coordinate Ocean Model (HYCOM), part of the Global Ocean Forecast System (GOFS) version 3.1. The HYCOM model combines isopycnal, sigma, and z-level coordinates, accurately simulating the physical state of the ocean. GOFS 3.1 enhances the accuracy of ocean condition predictions and analyses by assimilating observational data [23]. The temperature and salinity data from 3 February to 17 February 2020 are selected to calculate density perturbations. The results for the 13th day are shown in Figure 2, representing the density perturbation results, with three orbits passing through areas with different density perturbation characteristics. Figure 1 and Figure 2 show the strong correlation between density perturbations and topography, indicating that they both play a significant role in the generation and propagation of internal tides.

Using the sea surface height (SSH) data from the TOPEX/Poseidon (T/P), Jason-1, Jason-2, and Jason-3 satellite altimeters, harmonic constants for 22,260 M2 tidal constituent are calculated through t-tide harmonic analysis [24]. The SSH information from satellite altimeter data involves a time resolution of 9.9156 days. This relatively long time resolution exceeds the semidiurnal and diurnal periods, resulting in various tidal constituents aliasing within the SSH data. To accurately extract the harmonic constants of the M2 tidal constituent, we utilize data from each point with a time series length of approximately 29 years (1993–2022), which far surpasses the minimum time series length (14.76 days) required to separate the M2 tide as the revised Rayleigh criterion [25]. Given such a long time series, we employ the t-tide harmonic analysis method to precisely extract the required harmonic constants of the M2 tidal constituent from other tidal constituents. Figure 3 presents the amplitude and phase lag information of the M2 tidal constituent. By examining the amplitude information in the left figure, it is evident that in areas with significant topographic variations and large density perturbations, the continuity of the amplitude is noticeably disrupted. This phenomenon indicates the presence of internal tide signals in these areas.

2.2. Methods

In the study of extracting internal tide signals, common methods include the 300 km moving average method and polynomial fitting method. The 300 km moving average method cannot accurately extract internal tide signals across the entire region due to the varying wavelengths of internal tides in the study region. On the other hand, the polynomial fitting method is not affected by changes in internal tide wavelengths, making it a more suitable method for internal tide signal extraction. For the polynomial fitting method, results represent the barotropic tidal values, and by filtering out the barotropic tidal components, the signal of the M2 internal tide acting on the sea surface can be effectively extracted.

The general form of the Legendre polynomial is

$$P_n(x)={\displaystyle \frac{1}{2^{n}n!}}{\displaystyle \frac{{\mathrm{d}}^{\mathrm{n}}}{{\mathrm{d}\mathrm{x}}^{\mathrm{n}}}}{(x^2-1)}^n,n=\mathrm{0,1},2,\cdots$$

(2)

The specific form of the Legendre polynomial is

$$\begin{array}{l}{P}_0(x)=1\\ P_1(x)=x\\ P_2(x)={\displaystyle \frac{1}{2}}\left(3x^2-1\right)\\ P_3(x)={\displaystyle \frac{1}{2}}\left(5x^3-3x\right)\\ \cdots \cdots \cdots \cdots \\ P_n(x)={\displaystyle \sum _{k=0}^{[{\displaystyle \frac{n}{2}}]}} {\displaystyle \frac{(-1{)}^k(2n-2k)!}{2^{n}k!(n-k)!(n-2k)!}}{x}^{n-2k},n=0,1,2,\cdots \end{array}$$

(3)

where n represents the order of the Legendre polynomial; it is a critical parameter influencing the polynomial fitting results. To avoid the complex calculation of fitting phase lag, the polynomial fitting method is used to fit the values of A = HcosG and B = HsinG, where H and G are the amplitude and phase lag in the harmonic constants, respectively. After fitting, H and G can be obtained by

$$\begin{array}{l}H=\sqrt{A^2+B^2},\\ G=arctan(B/A).\end{array}$$

(4)

For the Legendre polynomial fitting method, the choice of polynomial order significantly impacts the fitting results: a polynomial order that is too low leads to underfitting, failing to capture the complexity of the data, whereas an excessively high order results in overfitting, making the model sensitive to noise and reducing predictive performance. To determine the optimal polynomial order, we employ cross-validation. Specifically, we randomly divide the dataset into five average subsets and perform 5-fold cross-validation. In each fold, four subsets are used as the training set, while the remaining subset serves as the validation set. By repeating this process five times, each time using a different subset as the validation set, we are able to assess the Legendre polynomial fitting method’s performance at different polynomial orders. Finally, by comparing the root mean square errors (RMSes) in the validation sets across different polynomial orders, we select the optimal order to ensure the accuracy as well as prevent overfitting. The RMSes are calculated as follows:

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{e}={\left\{{\displaystyle \frac{1}{2}}\left({H_o}^2+{H_s}^2\right)-H_o{H}_s\mathrm{c}\mathrm{o}\mathrm{s}\left(G_o-G_s\right)\right\}}^{1/2}$$

(5)

where $H_o$ and $G_o$ are the harmonic constants of satellite altimeter data, and $H_s$ and $G_s$ are the processed harmonic constants calculated by polynomial fitting method.

2.2.1. One-Dimensional Legendre Polynomial Fitting (1-D LPF) Method

The one-dimensional Legendre polynomial fitting (1-D LPF) method is employed to individually analyze each of the 42 orbits within the study region along their respective orbital directions.

The formula for fitting using one-dimensional Legendre polynomials is:

$$D_n(x)=B_n{P}_n(x)$$

(6)

For the fitting along one orbit in the study region, n represents the order of the polynomial fit along the orbital direction and x denotes the position projected onto the orbital direction for each point along this orbit. $B_n$ is the coefficient of different orders Legendre polynomial, which can be calculated by the least squares method. $D_n(x)$ is the function value, which is the approximation of A or B at x.

Table 1. The RMSes (units: cm) of the cross-validation results with different orders of the one-dimensional Legendre polynomial fitting method for each orbit.

Orbit	Order
	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
1	9.62	1.74	1.91	1.98	2.83	1.94	1.76	3.30	5.07	3.16	2.38	5.86	2.54	7.40	5.40
2	23.45	22.26	5.46	5.07	2.98	2.73	2.10	1.68	2.14	23.12	18.74	1.50	11.14	15.30	10.49
3	0.92	1.03	0.38	0.32	1.17	0.22	0.23	0.30	0.34	0.30	0.29	0.27	0.27	0.27	0.27
4	11.61	11.39	2.12	0.89	1.97	0.78	0.90	0.73	2.21	5.72	2.80	15.06	6.76	1.25	10.29
5	30.00	20.99	10.58	9.47	8.56	12.54	8.74	8.75	24.24	12.88	109.38	97.55	195.68	211.87	219.88
6	14.29	15.69	1.87	2.03	1.95	0.94	1.60	1.12	1.46	1.30	0.88	16.60	16.77	2.78	15.97
7	24.92	12.80	4.70	4.70	5.34	5.71	3.56	3.79	16.29	20.17	5.09	14.60	9.46	73.15	74.27
8	21.25	21.96	4.52	5.71	4.50	2.00	5.67	26.87	12.87	14.75	22.98	49.87	223.53	271.55	342.28
9	2.34	1.15	1.09	1.39	2.21	1.27	1.79	2.51	3.33	15.85	17.82	14.56	4.56	25.65	27.75
10	10.85	7.71	1.07	0.92	2.01	1.12	1.08	1.03	1.54	15.44	10.76	13.56	9.80	35.99	12.29
11	25.99	19.59	7.04	6.44	2.76	11.30	4.09	3.02	7.44	4.21	6.97	44.17	46.46	37.35	43.55
12	13.36	14.70	2.17	2.62	2.61	1.17	1.60	6.84	3.27	6.06	35.47	10.85	12.36	64.90	84.68
13	31.37	11.27	13.58	7.41	9.39	8.13	9.08	14.09	46.97	17.77	120.20	83.71	250.85	243.62	307.89
14	19.22	21.17	4.01	2.10	2.37	1.69	3.73	0.96	2.59	10.48	20.86	29.75	87.16	138.45	113.44
15	4.27	1.46	1.25	0.87	2.10	0.89	0.87	0.90	3.47	4.47	3.44	4.38	2.75	5.08	18.57
16	10.11	4.86	0.98	1.06	1.91	0.94	1.11	1.93	3.26	3.84	1.12	2.35	9.03	34.33	37.97
17	24.45	20.97	6.21	6.16	2.60	3.17	1.42	1.75	1.23	2.53	12.09	6.90	20.93	3.44	29.43
18	11.99	13.22	2.04	1.16	1.91	0.55	0.66	0.76	1.36	0.81	3.14	9.29	7.88	16.92	11.25
19	34.36	17.40	17.12	10.74	13.33	10.21	14.20	20.57	59.95	69.10	152.89	237.50	324.47	540.77	554.37
20	15.08	16.46	2.06	1.36	1.89	0.86	0.76	0.98	0.84	0.85	1.77	3.74	5.41	0.90	2.00
21	14.23	8.53	2.95	2.04	3.13	2.11	1.96	1.94	3.57	5.59	5.98	8.96	11.94	38.28	66.38
22	11.84	3.20	0.99	0.68	1.78	1.33	1.55	1.09	11.66	6.06	14.46	22.35	53.21	54.85	111.21
23	18.39	20.30	6.17	6.40	4.31	7.02	3.21	4.56	7.11	44.62	11.69	80.23	127.17	118.60	207.86
24	26.88	17.28	2.74	2.25	2.81	1.36	1.26	3.30	14.73	11.56	5.71	65.27	42.60	54.78	60.15
25	19.41	20.16	6.37	7.71	14.24	6.47	8.25	11.27	18.82	40.16	93.88	130.57	116.89	333.25	401.69
26	23.91	20.90	1.70	2.08	3.04	1.39	1.43	4.20	4.76	12.89	64.79	50.68	48.21	48.64	124.94
27	12.34	1.88	0.98	1.06	4.01	1.27	1.58	1.86	3.12	5.48	7.21	22.26	31.77	62.82	80.54
28	8.35	1.85	0.59	0.62	1.71	1.87	2.07	2.00	0.80	1.64	2.41	9.93	16.67	2.15	12.60
29	17.83	18.95	4.57	5.05	2.78	15.94	1.40	1.37	2.34	4.91	45.14	39.62	15.86	11.62	11.94
30	21.14	11.43	3.14	1.29	2.94	1.16	6.23	2.73	13.34	1.26	15.20	11.62	46.33	54.82	20.09
31	19.24	21.35	11.49	9.74	9.46	17.31	11.21	13.53	51.45	83.62	112.18	118.01	240.89	419.69	522.38
32	25.87	20.44	1.37	1.44	1.99	1.08	0.83	3.69	0.75	1.57	15.59	16.44	16.46	17.20	7.27
33	15.24	5.86	5.23	4.06	4.39	3.24	3.76	4.70	2.94	12.55	14.56	19.62	47.18	45.64	48.38
34	4.81	1.08	0.65	0.56	1.73	0.77	0.84	2.16	3.98	6.62	3.72	4.28	26.64	45.92	30.71
35	19.77	20.00	3.84	4.46	3.04	1.79	3.25	2.47	1.83	4.07	6.57	112.28	157.07	43.25	75.28
36	15.72	6.52	1.97	1.63	1.95	2.48	1.40	9.01	7.79	8.93	20.21	37.37	57.59	39.98	83.73
37	19.72	22.06	8.01	6.56	5.57	9.68	3.77	6.08	3.39	54.19	31.71	16.82	11.29	32.90	115.46
38	26.70	19.23	1.96	1.66	2.08	0.75	0.94	10.76	8.47	12.84	6.71	2.61	17.04	20.44	37.90
39	19.05	16.12	5.25	3.41	5.85	2.87	5.07	2.77	5.81	7.00	56.08	59.63	11.01	32.40	107.89
40	0.72	0.37	0.38	0.42	1.37	0.27	0.26	0.29	0.47	0.43	0.49	0.57	0.82	0.80	0.56
41	21.59	20.31	2.11	2.64	2.98	1.08	4.39	3.65	4.50	1.62	4.44	7.47	10.38	17.42	58.77
42	3.21	0.44	0.46	0.37	1.44	0.49	0.34	0.33	0.61	0.86	0.36	0.83	0.93	0.91	0.84
Average	16.80	12.76	3.88	3.30	3.69	3.57	3.09	4.66	8.86	13.36	25.91	35.70	56.09	76.84	97.82

presents the cross-validation results for each orbit from the first to the fifteenth order. The results indicate that for all orbit data, both lower and higher order Legendre polynomial fitting results yield higher RMSes. Considering these factors comprehensively, we have selected the seventh-order one-dimensional Legendre polynomial fitting method for extracting tidal signals within the study area.

2.2.2. Two-Dimensional Legendre Polynomial Fitting (2-D LPF) Method

The two-dimensional Legendre polynomial fitting (2-D LPF) method is employed to fit all the data in the whole study region. Compared to analyzing each orbit separately, this approach offers better overall coherence and does not have to consider the effect of too few data points on a particular orbit.

The formula for fitting using two-dimensional Legendre polynomials is:

$$\left\{\begin{array}{l}D(x,y)={\displaystyle \sum _{k=0}^m} {\displaystyle \sum _{s=0}^n} B_{k,s}{T}_k(x)P_s(y)\\ T_k(x)={\displaystyle \sum _{i=0}^{[{\displaystyle \frac{k}{2}}]}} {\displaystyle \frac{(-1{)}^i(2k-2i)!}{2^{n}i!(k-i)!(k-2i)!}}{x}^{k-2i}\\ P_s(y)={\displaystyle \sum _{i=0}^{[{\displaystyle \frac{s}{2}}]}} {\displaystyle \frac{(-1{)}^i(2s-2i)!}{2^{n}i!(s-i)!(s-2i)!}}{y}^{s-2i}\end{array}\right.$$

(7)

In the two-dimensional Legendre polynomial fitting method, $T_k(x)$ and $P_s(y)$ represent the Legendre polynomials along the x and y directions, respectively. The variables m and n denote the polynomial orders in the longitudinal and latitudinal directions. $B_{k,s}$ represents the polynomial coefficient with longitudinal order k and latitudinal order s, which can be determined using the least squares method. $D(x,y)$ is the approximate value at (x, y) obtained from the fitted coefficient $B_{k,s}$.

Similar to the one-dimensional Legendre polynomial fitting method, the two-dimensional Legendre polynomial fitting method also requires the determination of the optimal longitudinal and latitudinal orders m and n through five-fold cross-validation. However, unlike the one-dimensional polynomial fitting method, the two-dimensional polynomial fitting method can analyze all orbit data in the whole study region during cross-validation without needing to analyze each orbit separately.

Table 2. The RMSes (units: cm) of the cross-validation results with different longitudinal order and latitudinal order of the two-dimensional Legendre polynomial fitting method.

RMSe (cm)		Latitudinal Order
		2	3	4	5	6	7	8	9	10	11	12	13	14	15
longitudinal order	2	11.15	9.20	9.03	9.00	8.98	8.99	9.00	8.99	9.00	9.00	9.01	9.01	9.01	9.01
	3	6.63	2.97	2.31	2.23	2.16	2.16	2.16	2.16	2.17	2.17	2.18	2.18	2.19	2.19
	4	6.56	2.52	1.67	1.53	1.46	1.45	1.45	1.45	1.47	1.48	1.49	1.50	1.50	1.51
	5	6.62	2.42	1.50	1.31	1.24	1.24	1.24	1.24	1.25	1.26	1.28	1.29	1.30	1.30
	6	6.71	2.39	1.41	1.18	1.07	1.06	1.05	1.05	1.07	1.08	1.10	1.11	1.12	1.13
	7	6.73	2.40	1.39	1.15	1.02	1.00	0.99	0.99	0.99	1.00	1.01	1.02	1.04	1.04
	8	6.79	2.4	1.38	1.15	0.99	0.93	0.93	0.95	0.95	0.96	0.97	0.99	1.01	1.02
	9	6.85	2.42	1.38	1.17	1.01	0.99	1.01	1.06	1.10	1.13	1.13	1.15	1.19	1.20
	10	6.89	2.46	1.38	1.15	0.96	0.93	0.94	0.96	0.98	1.01	1.04	1.07	1.12	1.14
	11	7.00	2.51	1.38	1.12	0.94	0.94	0.95	0.92	0.96	1.04	1.07	1.08	1.12	1.16
	12	7.21	2.55	1.39	1.14	0.94	0.95	0.97	1.05	1.12	1.15	1.24	1.37	1.38	1.45
	13	7.35	2.57	1.41	1.16	0.97	0.93	0.94	1.02	0.98	0.99	1.17	1.59	1.56	1.74
	14	7.49	2.58	1.42	1.17	0.99	0.97	0.98	1.10	1.12	1.52	1.45	1.58	1.67	1.83
	15	7.57	2.61	1.44	1.17	1.00	0.95	0.99	1.24	1.48	1.61	2.54	3.37	3.58	3.60

shows the RMSes obtained from five-fold cross-validation using different orders for all data in the study region. As shown in Table 2, the smallest RMSe (0.93 cm) is achieved with a longitudinal order of 8 and a latitudinal order of 7. Therefore, we select the two-dimensional Legendre polynomial fitting method with a longitudinal order of 8 and a latitudinal order of 7 to extract internal tidal signals in the study region.

3. Results

3.1. Sea Surface Signal of the M2 Internal Tide Extracted by 1-D LPF Method

By applying the 1-D LPF method to the satellite altimeter harmonic constant data of the M2 tidal constituent for each orbit, we obtained the values of the barotropic tides and derived a residual that corresponds to the difference with the satellite altimeter data. This residual is interpreted as the internal tide signal plus background noise. The signal, shown in Figure 4, displays the distribution of amplitude changes (ΔH), phase lag changes (ΔG), and root mean square errors (RMSes) in ascending and descending orbit data, which changes represents the original information minus the processed information.

As shown in Figure 4, the range of ΔH in the ascending orbit data is between −1.5 cm and 1.5 cm, and besides a few obvious signals, most areas show small amplitude changes, concentrated between −0.5 cm and 0.5 cm, which is the effective noise level. Both the ascending and descending orbit data demonstrate significant spatial distribution characteristics of internal tide signals. Specifically, in the ascending orbit data, the region between 56°E and 64°E and 4°S and 20°S exhibits noticeable amplitude changes, indicating the presence of internal tides. The descending orbit data show a similar distribution pattern of amplitude changes, with a range between −2 cm and 2 cm. Significant height changes are observed in the region between 54°E and 72°E and 2°S and 24°S. These areas of height changes indicate the consistency of internal tide signals across different orbits, particularly in the western and northern parts of the study region.

Moreover, in the ascending orbit data, the range of ΔG is between −4 degrees and 4 degrees, and significant phase lag changes occur in the area between 56°E and 64°E and 0°S and 24°S, suggesting stronger internal tide signals in this area. In the descending orbit data, the range of phase lag changes is between −5 degrees and 5 degrees, and significant phase lag changes are observed in the region between 57°E and 62°E and 4°S and 24°S. These areas of phase changes are also more consistent with the spatial distribution of amplitude changes, further confirming the consistency and distribution characteristics of the internal tide signals across different orbits.

Additionally, the RMSes in both the ascending and descending orbits show regions of higher values, particularly aligning with areas of significant amplitude and phase lag changes. In the ascending orbit data, RMSe values range up to 3 cm, with noticeable increases in the areas between 56°E and 64°E and 4°S and 20°S. In the descending orbit data, RMSe values similarly range up to 3.5 cm, with significant areas of higher RMSes observed between 56°E and 64°E and 4°S and 20°S. These elevated RMSe values corroborate the regions of internal tide activity, highlighting the reliability of the detected signals.

3.2. Sea Surface Signal of the M2 Internal Tide Extracted by 2-D LPF Method

In order to reduce the restrictions of the 1-D LPF method application, the 2-D LPF method is employed to fit the satellite altimeter harmonic constant data of the M2 tidal constituent, aiming to extract the internal tidal signals acting on the sea surface. To further demonstrate the effectiveness of the 2-D LPF method, we also compared it to the current dominant 300 km moving average (300 km smooth) method. Figure 5 displays the ΔH, ΔG, and RMSes extracted using the 2-D LPF method (right part) compared to the results obtained using the 300 km smooth method (left part) and the 1-D LPF method (middle part).

For the amplitude change, ΔH extracted using the 2-D LPF method ranges from −2 cm to 2 cm, showing more detailed and continuous spatial distribution characteristics. Throughout the study region, especially in the area between 53°E and 68°E and 0°S and 28°S, the amplitude change is significant and continuous. Compared to the 300 km smooth method and the 1-D LPF method, the 2-D LPF method can more integrally reflect amplitude changes, avoiding the issues of scattered and discontinuous signals. Additionally, the 2-D LPF method exhibits clearer amplitude change patterns in the region between 64°E and 80°E and 8°S and 16°S, indicating higher spatial resolution in capturing the internal tidal signals.

In the phase lag change extracted using the 2-D LPF method, the ΔG ranges from −5 degrees to 5 degrees, displaying more complex and continuous spatial distribution characteristics. In the area between 53°E and 66°E and 0°S and 24°S, the phase lag change is more consistent and detailed, eliminating the discontinuity problem brought by the 300 km smooth method and the 1-D LPF method. Furthermore, the 2-D LPF method shows more obvious and integral phase lag changes in the area between 6°N and 3°S, demonstrating the intricate features of the internal tidal signals. This higher spatial resolution and overall coherence make the 2-D LPF method significantly advantageous in revealing the propagation paths and impact ranges of internal tidal signals.

For the root mean square error (RMSe), the higher RMSe values mean that more internal tide signals are extracted, and Figure 4 indicates that the 2-D LPF method shows an improvement over the 300 km smooth method and the 1-D LPF method. The RMSe values from the 2-D LPF method are generally higher in the study region, indicating a more accurate extraction of the internal tide signals. Specifically, in the areas between 53°E and 60°E and 8°S and 16°S, which corresponds to areas of internal tide propagation, the 2-D LPF method exhibits higher RMSe values, demonstrating its ability to extract the signals effectively. Similarly, in the areas between 70°E and 74°E and 6°N and 8°S, the 2-D LPF method maintains higher RMSe values, highlighting its superior spatial resolution and overall coherence.

The previous sections have provided the spatial distribution of topography and density perturbation within the study area, which is crucial for understanding the generation and propagation of internal tidal signals. By comparing the internal tidal signals extracted using the 2-D LPF method, the 300 km smooth method, and the 1-D LPF method with the spatial distribution of topography and density perturbation, it is evident that the signals extracted by the 2-D LPF method align more closely with the underlying topography and density perturbation. In the area between 72° E and 80° E, the variations in the internal tidal signals extracted by the 2-D LPF method coincide remarkably with the topographic undulations and density perturbation, further proving the accuracy and reliability of the 2-D LPF method in capturing internal tidal signals. In contrast, due to their limitations, the 300 km smooth method and the 1-D LPF method struggle to accurately extract signals against the backdrop of complex topography and density perturbation.

To quantitatively compare the internal tidal signals extracted by the 300 km smooth method, the 1-D LPF method, and the 2-D LPF method, we performed normal distribution analyses on the internal tidal signals obtained by all three methods. Additionally, we counted the number of signals within different ranges, as illustrated in Figure 6. To incorporate the combined characteristics of amplitude and phase lag signals of the internal tide acting on the sea surface within the analysis of three distinct orbits, we employed RMSes to represent the internal tide signal comprehensively.

In Figure 6a, it is evident that the 2-D LPF method exhibits a lower probability density near 0 cm and higher probability densities at larger intensity ranges (e.g., beyond ±1.5 cm). This indicates that the signals extracted by the 2-D LPF method have a more dispersed intensity distribution with a higher proportion of strong signals.

Figure 6b further illustrates the counts of signals within different ranges. It is apparent that the 2-D LPF method extracts more signals than the 300 km smooth method and the 1-D LPF method. For instance, in the 0.5 cm to 1 cm range, the number of signals extracted by the 2-D LPF method is almost two times that of the 300 km smooth method and one and a half times that of the 1-D LPF method. Additionally, in the rest of the ranges, the number of signals extracted by the 2-D LPF method is also significantly higher than that of the 300 km smooth method and the 1-D LPF method. These results suggest that the 2-D LPF method is more effective in capturing the internal tide signals, thereby demonstrating superior signal extraction capabilities compared to the 300 km smooth method and the 1-D LPF method.

In order to demonstrate the effectiveness of the 2-D LPF method in extracting the internal tide signals acting on the sea surface, we chose the central basin of the South China Sea (110°E–120°E, 8°N–18°N) as another study region for further research. For detailed information, please see the Supplementary Materials.

The 2-D LPF method, when processing data from the entire study region, can consider all the orbit data, improving analysis efficiency and result reliability. The internal tidal signals extracted by this method have more continuous and smooth spatial distributions, avoiding the limitations brought by analyzing individual orbits. Overall, the 2-D LPF method can better extract the signal from the M2 internal tide response to the sea surface than the 300 km smooth method and the 1-D LPF method.

4. Discussion

Given that the generation and propagation of internal tides are intricately related to topography and density structure, this section aims to more accurately assess the effectiveness of 300 km smooth method, 1-D LPF method, and 2-D LPF method in extracting internal tide signals. To this end, three distinct orbits were selected for analyzing the correlation between internal tide signals extracted by the different methods and the density perturbations and topography beneath these orbits. The positions of the three orbits are illustrated in Figure 2. The RMSes, topographies, and density perturbations obtained by the different methods for three orbits are illustrated in Figure 7, where the blue lines stand for the results obtained using the 300 km smooth method, the black lines denote the results obtained using the 1-D LPF method, and the red lines represent the results from the 2-D LPF method. The correlation coefficients between the RMSes derived from the 300 km smooth method, the 1-D LPF method, the 2-D LPF method, and the topography and density perturbations are shown in

Table 3. The correlation coefficients between the RMSes derived from 300 km smooth method, 1-D LPF method, 2-D LPF method, and the topography and density perturbations.

r–topo	Orbit 1	Orbit 2	Orbit 3
300 km smooth	0.498	0.601	0.229
1-D LPF	0.465	0.593	0.158
2-D LPF	0.502	0.620	0.245
r–Δρ	Orbit 1	Orbit 2	Orbit 3
300 km smooth	0.403	0.339	−0.032
1-D LPF	0.386	0.262	−0.031
2-D LPF	0.420	0.273	−0.101

.

Orbit 1 spans 53°E, passing through 0°S, 8°S, 16°S, and down to 34°S. This orbit crosses several areas with significant topographic variation and density perturbations, particularly between 3°S and 5°S, 16°S and 18°S. The RMSe graph reveals that the values obtained from the 2-D LPF method are significantly higher than those from the 300 km smooth method and the 1-D LPF method at 5°S and 16°S. The topographic profile along this orbit illustrates dramatic changes in the seafloor, with several seamounts and deep valleys. These variations are mirrored in the RMSe graph, indicating that the topography is a key factor in generating an internal tide. The density perturbation graph for orbit 1 shows strong disturbances in areas with significant topographic features, with these perturbations being more evident in the results of the 2-D LPF method. The correlation coefficients for orbit 1 with the topography are 0.498 for the 300 km smooth method, 0.465 for the 1-D LPF method, and 0.502 for the 2-D LPF method, and with density perturbations of 0.403 for the 300 km smooth method, 0.386 for the 1-D LPF method, and 0.420 for the 2-D LPF method, further confirming the superiority of the 2-D LPF method in capturing internal tide signals.

Orbit 2 extends from 68°E, passing through 6°N, 8°S, 16°S, and down to 34°S. The topographic variation and density perturbations along this orbit are mainly concentrated between 5°S and 7°S. The RMSe graph shows that the values from the 2-D LPF method are higher than those from the 300 km smooth method and the 1-D LPF method between 6°S and 7°S. This orbit intersects with a prominent seamount, as indicated in the topographic profile. The density perturbation graph for orbit 2 also shows significant disturbances in areas with topographic changes, particularly around 6°S, with the 2-D LPF method results standing out more prominently. The correlation coefficients for orbit 2 with the topography are 0.601 for the 300 km smooth method, 0.593 for the 1-D LPF method, and 0.620 for the 2-D LPF method, and with density perturbations of 0.339 for the 300 km smooth method, 0.262 for the 1-D LPF method, and 0.273 for the 2-D LPF method.

Orbit 3 starts at 60°E and passes through 6°N, 8°S, 16°S, and down to 34°S. The topographic variation and density perturbations along this orbit are more scattered, weaker than those of orbit 1 and orbit 2. The area through which the orbit passes is more similar to an area of internal tidal propagation. The RMSe graph demonstrates that, overall, the results obtained from the 2-D LPF method are higher. The topographic profile and density perturbation show less variation relative to orbit 1 and orbit 2. However, in terms of correlation coefficients, as with orbit 1 and orbit 2, the correlation coefficients obtained from the 2-D LPF method had increased compared to the correlation coefficients obtained from the 300 km smooth method and the 1-D LPF method. The correlation coefficients for orbit 3 with topography were 0.229 for the 300 km smooth method, 0.158 for the 1-D LPF method, and 0.245 for the 2-D LPF method, and with density perturbations of −0.032 for the 300 km smooth method, −0.031 for the 1-D LPF method, and −0.101 for the 2-D LPF method.

In conclusion, results from the 2-D LPF method show generally higher correlation with both topographic features and density perturbations than those from the 300 km smooth method and the 1-D LPF method. This demonstrates that the 2-D LPF method is superior to the 300 km smooth method and the 1-D LPF method in capturing internal tide signals, regardless of generation or propagation.

5. Conclusions

In this study, the two-dimensional Legendre polynomial fitting (2-D LPF) method is employed to fit the M2 tidal harmonic constants from satellite altimetry data within the region of 53°E–131°E, 34°S–6°N. By decomposing the barotropic component and baroclinic components from the M2 tidal harmonic constants, we extract the internal tide signals acting on the sea surface. The M2 tidal harmonic constants are obtained from the sea surface height (SSH) data of the TOPEX/Poseidon (T/P), Jason-1, Jason-2, and Jason-3 satellites through t-tide harmonic analysis.

To validate the effectiveness of the 2-D LPF method, we apply the 300 km moving average (300 km smooth) method and the one-dimensional Legendre polynomial fitting (1-D LPF) method to fit and extract internal tide signals from 42 orbits within the study region, and we compare these results with those obtained from the 2-D LPF method. Since the fitting results of polynomial methods are influenced by the choice of polynomial order, we conduct cross-validation for each orbit in the study region using the 1-D LPF method, selecting the optimal order as seven. Concurrently, we perform cross-validation for all data points in the study region using the 2-D LPF method, selecting the optimal orders as a longitudinal order of eight and latitudinal order of seven.

The comparison reveals that the internal tide signals extracted by the 2-D LPF method exhibit more spatial continuity and smoothness, and the signal characteristics are more pronounced. To further demonstrate the accuracy of the internal tide signals extracted by the 2-D LPF method, we select three orbits passing through different topographic and density perturbation features for single-orbit correlation analysis. The results indicate that the correlation coefficients between the internal tide signals extracted by the 2-D LPF method and those calculated from topographic and density perturbations (0.502, 0.620, 0.245; 0.420, 0.273, −0.101) are generally higher than those obtained by the 300 km smooth method (0.498, 0.601, 0.229; 0.403, 0.339, −0.032) and the 1-D LPF method (0.465, 0.593, 0.158; 0.386, 0.262, −0.031), further proving the accuracy of the 2-D LPF method in extracting internal tide signals.

Overall, the 2-D LPF method can utilize all the data points within the study region for internal tide signal extraction, avoiding the drawbacks of single-orbit signal extraction and proving that the 2-D LPF method can effectively extract the internal tide signals acting on the sea surface.