## 1. Introduction

As a basic type of movement of ocean water, tides and tidal currents are of great importance to research on storm surge, ocean circulation, and water mass (Munk 1997). Tides play an important role in marginal seas. Research has been conducted on the Bohai and Yellow Seas—typical marginal seas adjacent to China—revealing fundamental characteristics of tides and tidal currents, especially the four principal constituents (Fang 1986; Kang et al. 1998; Lefevre et al. 2000).

Treatment of open boundary conditions (OBCs), one of the most important control parameters in the tidal model, remains a great challenge for numerical simulation of tides and tidal currents (Lardner et al. 1993). Since the 1990s, the adjoint method has been widely implemented in the inversion of OBCs. Lardner et al. (1993) achieved effective control of the OBCs in a depth-averaged numerical tidal model by data assimilation. Seiler (1993) successfully estimated the OBCs for a quasigeostrophic ocean model using the adjoint method. Heemink et al. (2002) improved the predictive capabilities of a 3D shallow sea model by estimating OBCs, the friction parameter, the viscosity parameter, and the depth values with the adjoint method. Zhang et al. (2002) implemented the adjoint data assimilation technique for estimation of lateral tidal OBCs.

However, in these studies all points along the open boundary were taken as control variables, which were often beset by the ill posedness of the inverse problem (Smedstad and O’Brien 1991; Chertok and Lardner 1996; Navon 1998). Another problem was that cotidal charts obtained with all points serving as control variables were not accurate in regions near the open boundary. As is shown in Fig. 1, the coamplitude and cophase lines are in disorder near the open boundary, though the characteristics of the M_{2} tide is generally consistent with reality.

For this problem, the independent points (IPs) scheme serves as an effective solution, in which OBCs at some selected IPs were taken as control variables and those at other points along the open boundary were calculated by interpolating values at IPs. Zhang and Lu (2008) established a three-dimensional numerical barotropic tidal model and inverted the Fourier coefficients of OBCs, the bottom friction coefficients, and the vertical eddy viscosity profiles with the IP scheme. The influence of initial guesses, model errors, and data number on inversion was also discussed. As an extension, Zhang and Lu (2010) explored the inversion of OBCs in detail with the same model. Cao et al. (2012) inverted two-dimensional tidal OBCs in the Bohai and Yellow Seas (BYS), showing that using two IPs yielded the best simulated results. Guo et al. (2012) suggested that the scheme with three IPs and an interpolation radius of 1° worked best for inverting OBCs in the Bohai Sea. Chen et al. (2014) applied the IP scheme to the estimation of OBCs for a numerical internal tidal model. In the IP scheme used above, values at the IPs were linearly interpolated, which resulted in unsmooth open boundary curves. As an improvement, we proposed a new IP scheme, in which the Cressman interpolation was replaced by the spline interpolation, to obtain smooth open boundary curves.

This paper is organized as follows. Section 2 shows the derivation of the correction of Fourier coefficients at IPs along the open boundary when the spline interpolation is implemented. Twin experiments and practical experiments are conducted in sections 3 and 4, respectively. Conclusions are presented in section 5.

## 2. Correction of the Fourier coefficients at IPs

### a. Independent points scheme

*j*th point along the open boundary is conducted as follows:where

In the above mentioned equations,

Equation (4) shows that the gradient of the cost function with respect to control variables at IPs is composed of two parts: one corresponds to IPs themselves and the other corresponds to other points that will be assigned to IPs according to the weighting coefficients.

### b. Interpolation methods

Interpolation methods such as kriging and inverse distance weighting (IDW) have been widely used in many areas (Goodin et al. 1979; Franke 1982; Willmott et al. 1985; Biau et al. 1999; Largueche 2006). They can be expressed by Eq. (3).

*R*is the interpolation radius and

*j*th point and the

*jj*th point.

Equation (4) shows that the gradient of the cost function with respect to nonindependent points is assigned to IPs according to the weighting coefficients. It requires that the weighting coefficients are determined merely by the positions of points; that is, *j* and *jj*. Therefore, some novel interpolation methods like information diffusion interpolation cannot be used in the IP scheme.

The Cressman interpolation was traditionally used in IPs scheme and the interpolation radius was set as the distance between neighboring IPs. In this case, the interpolation result is a polygonal line. To obtain the smooth interpolated curve, we adopt the spline method. Spline interpolation is a form of interpolation whose interpolant uses a special type of piecewise polynomial called a spline. The spline interpolation is widely used and often better than polynomial interpolation because the interpolation error can be made very small (Knott 1999). Wijnands et al. (2016) established a model of near-surface wind speeds in tropical cyclones based on the spline interpolation. A comparison with an earlier linear wind model showed that the spline model produced more accurate wind estimates. Yaghoobi et al. (2017) described a robust, accurate, and efficient scheme based on the spline interpolation to solve a class of nonlinear variable-order fractional equations with time delay. Liang et al. (2017) proposed a method combining the spline interpolation and the harmonic analysis of time series (HANTS) algorithm to reconstruct high-quality NDVI time series. The proposed method generated more accurate results than the traditional statistical method.

The advantage of the spline interpolation over the Cressman and kriging interpolations is shown in the next experiment. Select five uniformly distributed points—A, B, C, D, and E—as IPs, which are left endpoint, maximum value point, inflection point, minimum value point, and right endpoint, respectively, of *R* is 9. Figure 2 shows that the curve obtained with the spline interpolation is smooth and very close to the original curve. The curve obtained with the Cressman interpolation displays a similar pattern to the original one, though it is not smooth. The kriging interpolation, which was conducted with the geostatistical MATLAB (mGstat) toolbox (Hansen 2004), with parameters set as follows: 1) A semivariogram model is specified as the spherical semivariogram model, with a range of nine and a sill of one. 2) A universal kriging method is selected. The curve obtained with the kriging interpolation shows greater difference from the original curve than the spline and Cressman interpolations. Moreover, it is hard to apply the kriging method to the IP scheme due to its complexity; therefore, it is excluded from the following experiments.

### c. Derivation of the correction of the Fourier coefficients at IPs

*l*th point, and

is on the left of the computing domain; is on the right of the computing domain; is below the computing domain; and is above the computing domain,

*i*is omitted for clarity. Fourier coefficients at IPs are denoted as

*L*is the number of IPs, and values at other points—

*l*th IP for the

*k*th point along the open boundary.

Equation (11) is a more specific expression of Eq. (5) in this model and so is the relationship between Eqs. (9) and (1), as well as Eqs. (10) and (3).

When we use the spline interpolation, the weighting coefficients can be calculated as follows.

*k*th point in the interval

*l*th IP can be expressed as

## 3. Twin experiments

### a. Model settings

To evaluate the feasibility of the model and the improved IP scheme, twin experiments are conducted. The model adopts the Arakawa C grid, where the water level is defined at the center and the two velocity components are defined to be perpendicular to the edges of the cell. The computation area covers 34°–41°N, 117°30′–126°40′E with a horizontal resolution of 1/6° × 1/6° and the real bathymetry of the BYS is implemented. The Coriolis parameter takes the local value; the bottom friction coefficient is set to be 0.002; and the time step interval is set to be 372.618 s, 1/120 of a M_{2} tidal cycle. The points along the TOPEX/Poseidon (T/P) ground tracks serve as observation sites (Fig. 3). Only the M_{2} constituent is simulated because we focus on the assessment of the scheme.

The twin experiments’ calculation procedure is designed as follows: 1) Run the forward model with the prescribed OBCs and the simulated water levels at the observation sites are taken as “observations” in the following steps. Here, only the positions of T/P satellite tracks are used and the observations are not real, but the simulated results of the forward model based on the prescribed OBCs. 2) Assign the initial values of the Fourier coefficients (taken as zero here) to the OBCs to run the forward model. 3) The misfits between the simulated water levels and the observations serve as a driven force for the adjoint model. Backward integration of the adjoint equations enables the calculation of the gradient of the Fourier coefficients. 4) Using Eqs. (10) and (11), we can update the Fourier coefficients. 5) Assign the updated Fourier coefficients to the OBCs to run the forward model. Repeat steps 3–5; the Fourier coefficients will be optimized and the difference between the simulated values and the observations will decrease continuously. The inversion terminates when a certain criterion is met.

In the twin experiments 1–4 (TE1–4), four prescribed distributions of Fourier coefficient *a* (Fig. 5) are inverted, respectively, while *b* is set to be zero. For the inversion of each distribution, we adopt five IPs; the SD method; and one of the two interpolation methods, that is, either the Cressman or the spline interpolation.

### b. Results and discussion

The cost functions normalized by their values at the first iteration step are shown in Fig. 4. After 100 iterations, the cost functions decrease to 1% of their initial values. The cost functions corresponding to the spline interpolation decrease more rapidly to smaller values than those corresponding to the Cressman interpolation, implying the advantage of the improved IP scheme. Figure 5 displays the prescribed and inverted OBCs. Although satisfactory results can be obtained with both IP schemes, the open boundary curves inverted with the spline interpolation are much smoother. The mean absolute errors (MAEs) between the prescribed and inverted OBCs are quantified in Table 1, showing that the errors in the experiments adopting the spline interpolation are much smaller. The difference between the observations and the simulated values are presented in Table 2. The magnitude of the misfit vector is smaller than 1.45 cm in all the experiments, demonstrating the feasibility of the model and the IP scheme. The change in the interpolation method from the Cressman method to the spline method results in a decrease of errors, especially in the first two groups of the experiments.

MAE of the inversion of OBCs in the twin experiments (cm).

MAE of the inversion of observations in the twin experiments.

### c. Twin experiments around Hawaii

To further assess the method proposed in this study, we conduct a twin experiment for the inversion of OBCs of the M_{2} tide around Hawaii. The computation area covers 18°–24°N, 154°–164°W, and the real bathymetry is adopted. Four open boundaries and T/P tracks are shown in Fig. 6. The rest of the experiment design is the same as before. Figure 7 and Table 3 show that the curves along the open boundaries obtained with the spline method are smoother and closer to the prescribed one.

MAE of the inversion of OBCs in the twin experiments around Hawaii (cm).

### d. Influence of optimization algorithm on inversion

As one of the popular algorithms used in the minimization of the inverse problem, the limited-memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) method (Nocedal 1980; Liu and Nocedal 1989) is known for its efficiency. In this section, the experiment design is the same as before except for the optimization algorithm, which was changed to the L-BFGS method. The decrease process of the cost functions is shown in Fig. 8. In the experiments with the spline interpolation, the cost function corresponding to the L-BFGS method decreases at a faster speed while that corresponding to the SD method decreases to a smaller value. But for the Cressman interpolation, a smaller value is obtained with the L-BFGS method and the cost functions do not decrease monotonically. Prescribed and inverted open boundary curves are shown in Fig. 9. It can be found that the optimization algorithm has little influence on the inverted OBC when the spline interpolation is implemented. However, for the Cressman interpolation, the results obtained with the L-BFGS method are more accurate.

## 4. Practical experiments

In the practical experiments, the M_{2} tide in the BYS is simulated by assimilating T/P altimetry data. The TOPEX/Poseidon data were provided by PO.DAAC of JPL. The sampling interval of the T/P satellite is 9.9156 days. Harmonic constants of the tidal constituents at the observation sites are extracted by harmonic analysis conducted on the along-track sea surface height anomalies spanning October 1992–July 2002 (the 2nd to 363rd cycles). In the inversion of the OBCs, five IPs are selected and four combinations of different interpolation methods (the Cressman and Spline methods) and optimization algorithms (the SD and the L-BFGS methods) are used.

The normalized cost functions are shown in Fig. 10. After 100 iterations, all of them decrease to 1% of their initial values. When the L-BFGS algorithm is adopted, the cost function decreases more rapidly but not monotonically.

Figure 11 displays the inverted open boundary curves. It can be found that the results differ from each other greatly. The OBCs inverted with the spline method are smoother. However, considering the magnitude of the misfit vector (Table 4), we regard all these inverted OBCs as acceptable. The misfits between the simulated water levels and the observations are presented in Table 4. For all combinations of the interpolation methods and the optimization algorithms, the MAEs in amplitude and phase between the simulation and T/P data are smaller than 4.3 cm and 3.4°, respectively, and those between the simulation and the tidal gauge data are smaller than 5.4 cm and 4.7°, implying successful simulation. The spline interpolation and the SD algorithm display a better performance than their counterparts. Figure 12 shows tidal gauge positions and their serial numbers in the BYS. Figure 13 displays the magnitude of the misfit vector between the simulation and each tidal gauge for the combinations of one of the two interpolation methods and the SD method. It can be found that there are significant differences between the results of the two interpolation methods at the tidal gauges (1–4) located in the eastern part of the Yellow Sea. The change in the interpolation method from the Cressman method to the spline method results in a significant decrease of the misfits at the tidal gauges (1–4).

MAE of the inversion of observations in the practical experiments.

The cotidal charts for the M_{2} constituent in the BYS obtained with the combination of one of the two interpolation methods and the SD method are shown in Fig. 14. Both of the cotidal charts show the following characteristics: there are two amphidromic points in the Bohai Sea, which are located near Qinhuangdao, China, and the Yellow River delta; the other two amphidromic points located in the Yellow Sea, specifically, to the north of Chengshantou, China, and in the vicinity of Jiangsu, China. The magnitudes of the misfit vector of results obtained with the combination of the SD method and the two interpolation methods are shown in Fig. 15. It can be found that there are significant differences near the open boundary and the eastern part of the Yellow Sea, which is consistent with Fig. 13. Cotidal charts obtained with the L-BFGS algorithm show similar patterns (not shown).

## 5. Conclusions

In this paper we developed an improved IP scheme: values at selected IPs are taken as control variables and those at other points are obtained by the spline interpolation instead of the traditionally used Cressman interpolation. The improved scheme was calibrated with a series of twin experiments. The results showed that OBCs were successfully inverted by assimilating “observations,” and those obtained with the spline interpolation were more accurate and smoother. In practical experiments, OBCs were inverted successfully by assimilating observations with all combinations of the different interpolation methods and optimization algorithms. The spline method showed better performance than the Cressman method. The cotidal charts of the M_{2} constituent showed similar characteristics.

The open boundary curves inverted with the spline method are smoother and are therefore more consistent with reality. In this study the bottom friction coefficients are taken as a known constant. It is worthwhile to apply the improved scheme to the inversion of bottom friction coefficients.

We deeply thank the reviewers and the editor for their constructive criticism of an earlier version of the manuscript. Partial support for this research was provided by the Natural Science Foundation of Shandong Province of China through Grant ZR2014DM017, the National Natural Science Foundation of China through Grants 41606006 and 41371496, the Natural Science Foundation of Zhejiang Province through Grant LY15D060001, the State Ministry of Science and Technology of China through Grant 2013AA09A502, the National Science and Technology Support Program through Grant 2013BAK05B04, and the Ministry of Education’s 111 Project through Grant B07036. The TOPEX/Poseidon data were provided by PO.DAAC of JPL.

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