## 1. Introduction

Tides are the earliest applications of the oceanography to human activities (Munk 2002). As the basic periodic motions of seawater, tides, caused by the tide-generating force, have a critical impact on the wind wave, ocean circulation, storm surge, and other ocean phenomenon. The purposeful and systematic theoretical research on such motions is still an important component part of marine science, especially for the continental marginal seas.

Addressing the main characteristics of different constituents with harmonic constants extracted from observations has been diligently sought after by many investigators since the establishment of equilibrium and dynamic theory of tides (Luther and Wunsch 1975; Fang et al. 2004). For the history of extracting harmonic constants, the tidal gauge station observations derived from highly sensitive force transducers play a significant role in the preliminary investigations. The cotidal charts have been reconstructed to address the characteristics of different constituents based on the tidal gauge station observations (Luther and Wunsch 1975; McCammon and Wunsch 1977; Fang 1986). However, an overwhelming majority of those tidal gauge stations are located at coastal area or island, and the deep ocean is poorly covered with the observations. With the development of the satellite altimeters, especially the launching of *Seasat-A*, *Geosat*, *Geosat Follow-On* (*GFO*), TOPEX/Poseidon (T/P), *ERS-1/2*, *Envisat*, *Jason-1/2*, and Hai Yang 2 (HY-2), the satellite altimeter observations are applied to investigate tides (Brown et al. 1981; Chelton and McCabe 1985; Nerem et al. 2010), and they are proved to be an effective approach for global tide research (Matsumoto et al. 2000; Egbert and Erofeeva 2002; Munk 2002; Fang et al. 2004; Albuquerque et al. 2018; Hamlington et al. 2019).

In terms of depicting the tidal characteristics, noticeable progress has been made since the combination of dynamic numerical model and observations (Schwiderski 1980; Parke and Hendershott 1980). The regional ocean model has a wide application and plays an indispensable role in tidal investigation for the continental marginal seas. During the modeling of tidal processes, the harmonic constants for each tidal constituent must be prescribed to provide the tidal dynamic information at the open boundary, which have a critical impact on the modeling results (Lardner et al. 1993; Cao et al. 2012; Guo et al. 2012; Jiang et al. 2018). So far, the following methods have been employed to deal with the open boundary conditions (OBCs). One is to interpolate the available observations near the open boundaries, and the other is to derive from larger-scale model with the help of nested-grid approach, such as the Oregon State University T/P global inverse solution (Egbert 1997; Egbert and Erofeeva 2002). However, the OBCs derived by these two methods require sufficiently accurate and adequate observations or a mass of computing resources, and artificial adjustment by experience is also needed to get satisfactory simulated results, which remains a great difficulty for the application of regional ocean model (Lardner et al. 1993; Pan et al. 2017). Direct interpolation of sparse observations in regional oceans used to be the top priority for the utilization of existing harmonic constants from observations, which is vital to analyze the spatial–temporal structure of tidal processes. In response to the difficulty faced by the regional ocean model, a suitable interpolation method, supported by observations from tidal gauge stations and satellite altimeters, is still deserved to be developed for exploiting the tidal dynamic information in the regional oceans.

According to whether the dynamic constrained factors are considered or not, the interpolation methods can be divided into two categories: the dynamically unconstrained interpolation method and the dynamically constrained interpolation method (DCIM). The DCIM integrates the statistical information from observations with dynamical constraints from dynamic numerical models, which indicates that the statistical signals and dynamical processes are considered as an integral whole. Therefore, the DCIM can significantly improve the precision of the final interpolated results and perform better than the dynamically unconstrained interpolation method (Hoteit et al. 2009; Yaremchuk and Sentchev 2013; Mao et al. 2018). Based on the adjoint assimilation method, the DCIM has been applied to different scenarios by interpolating the acoustic Doppler current profiler records, satellite altimeters, and in situ observations (Griffin and Thompson 1996; Lebedev et al. 2003; Mao et al. 2018). In this paper, the DCIM is developed and applied for the utilization of observations from tidal gauge stations and T/P altimeter in the Bohai, Yellow, and East China Seas (BYECS). Specifically, a substantial quantity of enhanced “observations,” generated from the interpolated results of the domain, are further interpolated with the sparse observations in the subdomain. It is the enhanced “observations” that play an important role in obtaining a better interpolated result.

This paper is organized as follows: The dynamical equations and DCIM are introduced in section 2. A practical experiment for the M_{2} constituent is carried out in section 3 to demonstrate the feasibility and effectiveness of the DCIM with enhanced “observations.” Finally, a summary is presented in section 4.

## 2. Model and method

In this section, the dynamical equations for tidal processes are first introduced, which are the dynamical constraints for the DCIM. Thereafter, the DCIM is developed to couple those dynamical processes and statistical signals.

### a. The dynamical equations

Consider a homogeneous and incompressible fluid with hydrostatic approximation, governing equations of the two-dimensional tidal model in the Cartesian coordinates can be written as

where *t* is the time, *x* and *y* are the eastward and northward coordinates of the horizontal Cartesian coordinate frame, *h* is the undisturbed depth, *ζ* is the free-surface elevation above the undisturbed level, *u* and *υ* are the eastward and northward components of depth mean velocity, respectively, *f* = 2Ω sin*θ* is the Coriolis parameter, Ω is the angular velocity of Earth’s rotation, *θ* is the latitude, *g* is the gravity acceleration, *k* is the bottom friction coefficient (BFC), *A* is the horizontal eddy viscosity coefficient, and Δ is the Laplace operator.

The initial conditions for the tidal model are: *ζ* = 0, *u* = 0, *υ* = 0, i.e., the initial free-surface elevation and horizontal velocity components are zero. The closed boundary conditions are zero flow normal to the coast: **u** ⋅ **n** = 0, where **u** represents the velocity vector and **n** represents the outward unit vector of the coast. The water elevations of the M_{2} constituent along the open boundaries are described as: *ζ* = *a* cos(*ωt*) + *b* sin(*ωt*), where *a* and *b* are the Fourier coefficients (FCs), and *ω* denotes the angular frequency of the M_{2} constituent (1.405 189 025 × 10^{−4} s^{−1}).

### b. The DCIM

The DCIM realizes the integration of the dynamical processes and statistical signals. More concretely, the dynamic numerical model, such as the tide model based on the above dynamical equations, is used to interpolate the observations under the dynamical constraints. However, the errors from the model parameters, boundary conditions, and truncation error may be introduced into the final interpolated results, which means that the final interpolated results may be inconsistent with effective observations. The DCIM integrates the adjoint assimilation method to optimize the interpolated results iteratively by adjusting the key model parameters until the stop criterion is satisfied. Based on the governing Eqs. (1)–(3) of the tidal model, its adjoint model can be constructed as follows. First, a cost function is defined to facilitate estimating the squared differences between the interpolated results and observations according to Lu and Zhang (2006), Guo et al. (2017), Wang et al. (2018a,b), and Mao et al. (2018):

where *K*_{ζ} is a constant (in this study, *K*_{ζ} = 1), Σ is the set of the observation locations, *ζ* and *n* means the sea level elevations at *n* time step. Then the Lagrangian function is defined as

where *λ*, *μ*, and *υ* denote the adjoint variables of *ζ*, *u*, and *υ*, respectively. For the sake of minimizing the cost function under the limitations of dynamical equations, and initial and boundary conditions, the first-order derivate of *L* with respect to the variables and parameters should be zero (Zhang and Lu 2010):

Equation (6) are essentially the two-dimensional tidal model [Eqs. (1)–(3)]. The adjoint equations can be derived from Eqs. (7):

The gradients of the cost function with respect to the FCs of OBCs can be calculated from Eqs. (8). To minimize the cost function, FCs of OBCs are modified using the typical steepest descent, while the BFC is spatially varying but remain unchanged and the horizontal eddy viscosity coefficient keeps constant during the DCIM. In addition, the Arakawa C grid (*ζ*, *u*, and *υ* locate at the center, eastern, and northern sides of the cell, respectively) is adopted as the finite difference scheme to establish the DCIM.

### c. The calculation process of the DCIM

The main calculation process of the DCIM is shown as follows:

Run the tidal model [Eqs. (1)–(3)] with the initial values of all the model parameters (FCs, BFC, and horizontal eddy viscosity coefficient) to obtain the interpolated results.

Calculate the cost function as Eq. (4). The difference of water elevations between the interpolated results from step 1 and observations serves as the external force of the DCIM. Values of adjoint variables are obtained through backward integration of the adjoint equations [Eqs. (9)–(11)].

Calculate the gradients of the cost function with respect to the FCs [Eq. (8)] and then adjust FCs using the typical steepest descent.

Repeat steps 1–3 until the number of iteration steps is exactly 100, as the difference between observations and interpolated results will decrease slowly after 100 steps.

## 3. Practical experiment

A practical experiment is conducted in this section. The main content of the experiment is to fully describe the character of the M_{2} constituent in the investigation area with the DCIM using observations from tidal gauge stations and T/P altimeter. More details of the experiment are shown as follows.

### a. Model settings

Three subdomains (I, II, and III), chiefly located in the BYECS, the continental shelf sea of the East China Sea and the Zhejiang–Fujian coastal area, are defined to serve as the investigation areas for the experiment. The 1-min-resolution topography provided by the Laboratory for Coastal and Ocean Dynamics Studies, Sung Kyun Kwan University, Korea (Choi 1999), and locations of open boundaries for each subdomain are shown in Fig. 1a. The Coriolis parameter *f* takes the local value. Limited by the Courant–Friedrich–Levy (CFL) condition, the time step for the experiment is set as 62.10 s, which is 1/720 of the M_{2} tidal cycle. More details about the settings of the experiment can be found in Table 1.

Specific settings for each subdomain of the practical experiment.

### b. Observations

Based on harmonic analysis, harmonic constants of the M_{2} constituent are extracted from the sea surface height data observed by tidal gauge stations and T/P altimeter and further used in the DCIM. As mentioned by Talke and Jay (2020), tidal properties are changing for natural and anthropogenic factors, such as astronomical forcing, wetland reclamation, and channel dredging, especially for shallow water and coastal areas. Therefore, a total of 37 observations from tidal gauge stations and 419 from the T/P altimeter with at least 25 km offshore (haversine distance formulation) and 10 m depth, are chosen as the effective observations. The effective observations are sparsely distributed along the coastline and tracks of T/P altimeter (Fig. 1b).

### c. Validation of final interpolated results

In this study, the effective observations of harmonic constants from tidal gauge stations are not interpolated in the DCIM and only used as independent effective observations to assess the accuracy of final interpolated results for each subdomain. To describe the interpolation effects quantitatively, the root-mean-square (RMS) error between the final interpolated results and the independent effective observations from tidal gauge stations are calculated for each subdomain as the following formulation (King and Padman 2005; Stammer et al. 2014; Seifi et al. 2019):

where *N* is the number of the tidal gauge stations for each subdomain, *A*_{i} and *P*_{i} are the tidal amplitudes and phase lags of the M_{2} constituent for the *i*th point, and the superscripts “obs” and “interp” mean the observations from the tidal gauge stations and final interpolated results using the DCIM, respectively. Smaller RMS error means better model performance of recovering the known data.

### d. Process of the practical experiment

In the practical experiment, the basic settings of the DCIM are listed in Table 1. The platform for the practical experiment is a desktop workstation with an Intel FORTRAN compiler. The main hardware for interpolation is an Intel Core i7-8700 Processor with a random access memory (RAM; 16 Gb).

The bathymetry and open boundaries for subdomain I are shown in Fig. 2a. A total of 419 effective observations from the T/P altimeter are interpolated with the DCIM, and 37 independent effective observations from tidal gauge stations are used to validate the performance of the DCIM in subdomain I (Fig. 2b). It takes about 5.5 min and 211 Mb of RAM to run the DCIM on the platform mentioned above, which mainly depends on the size of the computer grid. During the iteration process, the normalized cost function (defined as the ratio of the cost function and its initial value) is used to describe the decline of the cost function, which is significantly reduced and tends to be stable with the increase of iteration step (Fig. 2c). After 100 iteration steps, the cost function can reach 0.21% of its initial values or less. The RMS error between the final interpolated results from the T/P altimeter and the independent effective observations from tidal gauge stations is about 5.00 cm (Table 2), implying that the final interpolated results have a good agreement with the majority of the observations in subdomain I. Note that the RMS error for most other methods and global models is greater than 5.24 cm (Stammer et al. 2014). In addition, the regional tidal solution named as China Seas and Indonesia 2016 (Egbert and Erofeeva 2002), which covers subdomain I with a higher resolution (2 min), is also used to obtain the harmonic constants of the M_{2} constituent for each tidal gauge station in subdomain I. The RMS error between the results from China Seas and Indonesia 2016 with the independent effective observations from the tidal gauge stations is about 14.18 cm. A possible cause for the discrepancy is the different source of bathymetry data. From the perspective of the RMS error, the DCIM has a better performance than most of previous efforts. The final interpolated results of the M_{2} constituent are shown in the form of cotidal chart in Fig. 2d. The major tidal characteristic in subdomain I is the four rotating systems with amphidromic points (located in the Liaodong Bay, the Yellow River delta, the North Yellow Sea, and the South Yellow Sea). In addition, the M_{2} cotidal chart shown in Fig. 2d coincides fairly well with those in Fang et al. (2004) and Lu and Zhang (2006) in the BYECS.

For subdomain I, (a) the bathymetry and open boundaries (red stars). (b) The positions of tidal gauge stations (red dots) and tracks of T/P altimeter (black squares). (c) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis). (d) The cotidal charts for the M_{2} constituent. Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

For subdomain I, (a) the bathymetry and open boundaries (red stars). (b) The positions of tidal gauge stations (red dots) and tracks of T/P altimeter (black squares). (c) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis). (d) The cotidal charts for the M_{2} constituent. Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

For subdomain I, (a) the bathymetry and open boundaries (red stars). (b) The positions of tidal gauge stations (red dots) and tracks of T/P altimeter (black squares). (c) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis). (d) The cotidal charts for the M_{2} constituent. Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

Number of the observations and RMS error for each subdomain.

In subdomain II (Fig. 3a), there are 235 effective observations from T/P altimeter and 8 independent effective observations from tidal gauge stations. At first, the 235 effective observations from T/P altimeter are the only source data to be interpolated in the DCIM. The cost function reaches about 0.26% of its initial value and its descent process tends to be stable after nearly 100 iteration steps (Fig. 3c). The RMS error between the final interpolated results from the T/P altimeter and the independent effective observations from 8 tidal gauge stations is about 7.27 cm (Table 2), which is a little larger than that in subdomain I. In other words, the M_{2} constituent obtained through DCIM in subdomain II may be not insufficient precise. A possible reason is the limited effective observations from T/P altimeter interpolated in the DCIM in subdomain II. To obtain the M_{2} constituent in subdomain II more accurately, 1122 enhanced “observations” from the final interpolated results of the subdomain I, of which the locations are at least 25 km offshore and depths are larger than 10 m, are employed to work together with 235 effective observations from the T/P altimeter in the DCIM (Fig. 3b). After 100 iteration steps, the cost function reaches about 0.10% of its initial value and its descent process also becomes stable (Fig. 3e). The RMS error between the final interpolated results and the independent effective observations from 8 tidal gauge stations is about 4.75 cm (Table 2), which is significantly decreased with the introduction of the 1122 enhanced “observations.” This means that the main characteristic of the M_{2} constituent in subdomain II presented in Fig. 3f is more precise than in Fig. 3d. The farther away from the coastline, the smaller the tidal amplitude and phase, and the coamplitude line gradually parallels to the coastline. Combining all the results in subdomain II, it can be concluded that the enhanced “observations” play a crucial role in improving the accuracy of DCIM. Additionally, the DCIM spends about 4.5 min and 180 Mb of RAM to get the final interpolated results.

(a),(b) As in Fig. 2, but for subdomain II. (c),(e) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis) without and with the enhanced “observations.” (d),(f) The cotidal charts for the M_{2} constituent without and with the enhanced “observations.”; Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

(a),(b) As in Fig. 2, but for subdomain II. (c),(e) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis) without and with the enhanced “observations.” (d),(f) The cotidal charts for the M_{2} constituent without and with the enhanced “observations.”; Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

(a),(b) As in Fig. 2, but for subdomain II. (c),(e) The descent process with time steps (horizontal axis) for the normalized cost function in log form (vertical axis) without and with the enhanced “observations.” (d),(f) The cotidal charts for the M_{2} constituent without and with the enhanced “observations.”; Black dashed line denotes the coamplitude line (m). Black solid line denotes the cophase line (°).

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

Finally, the practical experiment focuses on subdomain III (Fig. 4a). A total of 162 effective observations from T/P altimeter without any enhanced “observations” are first interpolated with the DCIM as the basic experiment of subdomain III. Besides, a total of 1843 observations, including 162 effective observations from T/P altimeter mentioned above and 1681 enhanced “observations” selected from the final interpolated results of subdomain II, are also interpolated with the DCIM as a contrast experiment. The 7 independent effective observations from tidal gauge stations are used to evaluate the accuracy of the final interpolated results (Fig. 4b). After 100 iteration steps, the normalized cost function in the contrast experiments is 0.19%, which is smaller than that (0.25%) in the basic experiments. At the same time, the descent process of cost function in the contrast experiment (Fig. 4e) is more stable than that in the basic experiments (Fig. 4c). The RMS errors in the contrast and basic experiments are 4.63 and 4.92 cm (Table 2), respectively, which illustrate that the enhanced “observations” are effective to remedy the lack of observations to derive a better final interpolated result. Figures 4f and 4d are the cotidal charts for the M_{2} constituent in subdomain III derived by the DCIM with or without the enhanced “observations,” respectively. With the enhanced “observations,” Fig. 4f can fully depict the characteristic of amplitudes and phase lags of the M_{2} constituent in the Zhejiang–Fujian coastal area. It is worth mentioning that the time and RAM consumed in subdomain III are close to 16 min and 569 Mb, as the grid size is larger than those in the subdomains I and II, but it is still easy for any mainstream workstations.

As in Fig. 3, but for subdomain III.

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

As in Fig. 3, but for subdomain III.

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

As in Fig. 3, but for subdomain III.

Citation: Journal of Atmospheric and Oceanic Technology 37, 7; 10.1175/JTECH-D-19-0212.1

Viewed from the entire process of the experiment, a substantial quantity of enhanced “observations” play a significant role in the interpolation for the sparse observations in the subdomain. For the final interpolated results of each subdomain, the normalized cost function is about 0.20% and tends to be stable after 100 iteration steps, and the RMS error between the final interpolated results from the T/P altimeter as well as enhanced “observations” and independent effective observations from tidal gauge stations is smaller than 5.00 cm. This indicates that the final interpolated results are consistent with the observations in each subdomain. Meanwhile, the cotidal charts can depict the main tidal characteristics of the M_{2} constituent in the BYECS, the continental shelf sea of the East China Sea and the Zhejiang–Fujian coastal area, respectively, which also demonstrate the feasibility and effectiveness of the DCIM to obtain high-accuracy regional oceanic cotidal charts.

## 4. Summary

In this paper, the DCIM is introduced and applied to interpolate the harmonic constants extracted from the observations of the M_{2} constituent in the BYECS. Specifically, a substantial quantity of enhanced “observations,” generated from the interpolated results of the domain, also play a significant role in the interpolation for the sparse observations in the subdomain. The tidal model provides the dynamical constraints to interpolate the observations, and the adjoint assimilation method, supported by the rigorous mathematical foundations, provides the iterative optimization by adjusting the key model parameters in the DCIM. The cost function is significantly reduced and tends to be stable. The RMS error between the final interpolated results from the T/P altimeter as well as enhanced “observations” and the independent effective observations from tidal gauge stations is smaller than 5.00 cm. In addition, the cotidal maps reconstructed by the final interpolated results accurately address the main characteristics in the BYECS, the continental shelf sea of the East China Sea and the Zhejiang–Fujian coastal area, respectively. Therefore, the interpolated results are reasonable and demonstrate that the DCIM is feasible and effective to obtain high-accuracy cotidal charts for regional ocean.

In conclusion, there is no doubt that sufficient amount and rational utilization of observations play an essential role in the investigation of natural phenomena in the ocean. Supported by the mass observations, the DCIM is an effective method to improve the utilization of observations and it is worthwhile to extend this method into other adjacent fields.

## Acknowledgments

We sincerely thank the anonymous reviewers and editor for their insightful comments and suggestions to improve this manuscript. We also would like to thank Dr. Anzhou Cao (Ocean College, Zhejiang University, Zhoushan, China) for his assistance. This work is supported by the National Natural Science Foundation of China (Grant U1806214) and the National Key Research and Development Plan of China (Grants 2016YFC1402705 and 2016YFC1402304).

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