## 1. Introduction

*F*is defined using a quadratic drag law through the coefficient

*k*:

*k*is usually set either as a constant or is used as a function of depth (Kang et al. 1998; Lee et al. 2002; Egbert et al. 2004; Sannino et al. 2004). But a more efficient and accurate way to estimate the BFC is through data assimilation using the adjoint method, as demonstrated in Wang et al. (2014).They made a comprehensive comparison of different schemes of BFCs, including a constant for the entire domain, different constants for different subdomains, depth-dependent BFCs, and BFCs inverted with the adjoint method; and concluded that the best simulation of the M

_{2}constituent was obtained with BFCs inverted by the adjoint method.

In data assimilation, an optimal initial model state or model parameter can be found to minimize the distance between the model output and observations. As one parameter of particular importance, the BFC has been estimated through data assimilation in many studies (Das and Lardner 1991, 1992; Lu and Zhang 2006). Because it is impractical and sometimes impossible to take the BFCs at all grid points as parameters (Ullman and Wilson 1998; Heemink et al. 2002), the independent points scheme is usually adopted, which means that BFCs at some selected points, or independent points, are treated as independent parameters and that values at other points are obtained by interpolation. This scheme reduces the number of control variables, serving as a great solution for the ill-posedness of the inversion problem (Zhang and Wang 2014).

*k*

_{i,j}is the BFC at the grid point (

*i*,

*j*),

*k*

_{l}is the

*l*th independent BFC, and

*i*,

*j*) and the

*l*th independent point, and

*R*is the influence radius set based on experience. Although the interpolation with the Cressman weight (CI) yields satisfying results in general (Lu and Zhang 2006), the reconstructed BFC field is unsmooth, sometimes showing unrealistic bull’s-eye patterns around the independent points. This motivates us to seek a more reasonable interpolation method to combine with the adjoint method. As one popular technique for data interpolation, the surface spline (also known as thin plate spline) has the advantage, among others, of producing smooth surfaces. It was first proposed by Harder and Desmarais (1972) for interpolating wing deflections and computing slopes for aeroelastic calculations. The surface spline and its variations have been applied successfully in oceanographic and meteorological contexts (Brankart and Brasseur 1996; Luo et al. 2008), where the spatial distribution of many parameters is smooth surface. Based on its proven great performance, the surface spline interpolation (SSI) is applied to the inversion of BFCs with an adjoint tidal model in this study.

The paper is organized as follows. The numerical model and the implementation of SSI in the model is introduced in section 2. Twin experiments and practical experiments are conducted in section 3 to assess the performance of SSI in the inversion of BFCs. The summary in section 4 completes the paper.

## 2. Numerical model and settings

### a. Forward model

*t*is time;

*h*is the undisturbed sea level;

*ζ*is the sea surface elevation above

*h*;

*u*and

*υ*are zonal and meridional velocity, respectively;

*f*is the Coriolis parameter;

*g*is the acceleration as a result of gravity;

*A*is the horizontal eddy viscosity coefficient; and

*k*is the BFC to be estimated in this study.

### b. Adjoint model

*J*:

*K*

_{ζ}is a constant, Σ is the set of observation locations, and

*λ*,

*μ*, and

*υ*are adjoint variables of

*ζ*,

*u*, and

*υ*, respectively. According to the typical theory of the Lagrangian multiplier method, the adjoint model can be constructed by the following equations:

### c. Optimization of independent BFCs

*J*using the optimization and adjoint methods. According to the typical steepest descent method,

*i*denotes the iteration step,

*k*

_{l}is the

*l*th parameter,

*p*is the negative of the gradient of

*J*in respect to

*k*

_{l}. It should be noted that in this study, the gradient is scaled, that is,

*N*is the number of independent parameters. The gradient can be obtained by solving the equation,

*l*th independent BFC is

*i*,

*j*) denotes grid points within the influence radius of the

*l*th independent BFC.

When CI is adopted, the coefficient of interpolation *W* is calculated according to Eq. (3). In this study, we combine SSI with the adjoint method and the SSI coefficients are derived as follows.

*N*points, the whole BFC field can be constructed with a variation of the surface spline. The BFC at grid points (

*i*,

*j*) is expressed as (Liu et al. 2002)

*r*

_{ij,l}is the distance between point (

*i*,

*j*) and the

*l*th independent point,

*R*is a prescribed radius, and

*r*

_{ij}is the distance between the

*i*th and

*j*th independent points. If we set

*l*th independent BFC at the point (

*i*,

*j*).

### d. Independent points scheme

The independent points are selected based on the gradient of the water depth. The basic idea is that as the bottom friction is closely related to water depth, more independent points will be needed to depict the distribution of BFCs if the water depth varies greatly or has a large gradient. Details on how to select independent points can be found in Lu and Zhang (2006). In this study, 14 independent points are selected and shown in Fig. 1.

### e. Settings

The model adopts the Arakawa C grid. The area of computation is the Bohai Sea, covering 37°–41°N, 117.5°–122.5°E with a space resolution of 1/6° × 1/6°. As we focus on the development of the method, only the M_{2} tide is simulated. Therefore, the time step is 1/120 of the period of the M_{2} tide. The bathymetry of the Bohai Sea, the positions of tidal gauge stations, and TOPEX/Poseidon (T/P) satellite tracks in this area are shown in Fig. 1.

## 3. Numerical experiments

### a. Twin experiments

Twin experiments (TEs) are conducted to evaluate the feasibility of SSI in inversion of BFCs. In TEs, harmonic constants of the M_{2} tide along the T/P tracks and at tidal gauge stations are obtained by the forward model that is run with a prescribed BFC field. These data are regarded as “observations.” Then run the forward model with an initial guess of BFCs and the difference between the simulated results and the observations along the T/P tracks serves as the external force for the adjoint model. Thereafter, run the adjoint model. With variables obtained from the forward and adjoint models, BFCs are optimized according to Eq. (8). Repeat the procedure with BFCs being optimized until certain criteria are met.

*E*computed according to

*x*is the tidal amplitude or phase, or BFC in the twin experiments because the “true” BFC field is known;

In TEs, four BFC fields are prescribed (Fig. 2): a paraboloid opening upward (TE1), a paraboloid opening downward (TE2), a bilinear surface (TE3), and a surface composed of four bilinear patches (TE4).

Four prescribed BFC fields in TEs: (a) a paraboloid opening upward (TE1), (b) a paraboloid opening downward (TE2), (c) a bilinear surface (TE3), and (d) a surface composed of 4 bilinear patches (TE4).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

Four prescribed BFC fields in TEs: (a) a paraboloid opening upward (TE1), (b) a paraboloid opening downward (TE2), (c) a bilinear surface (TE3), and (d) a surface composed of 4 bilinear patches (TE4).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

Four prescribed BFC fields in TEs: (a) a paraboloid opening upward (TE1), (b) a paraboloid opening downward (TE2), (c) a bilinear surface (TE3), and (d) a surface composed of 4 bilinear patches (TE4).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

The results of the four TEs are summarized in Table 1. The model skill in terms of the amplitude and phase of the M_{2} tide is 0.99 or 1 in all experiments, demonstrating the successful simulation of sea surface elevation. Therefore, we focus on the inversion of BFCs to compare the two interpolation methods.

Statistical analysis of the amplitude and phase of the M_{2} tide, as well as the BFCs in four TEs. Skill is the model skill defined in Eq. (18), RMS (10^{−4}) is the root-mean-square error, and *E* (%) is the relative average error.

In TE1, the skill of the model combined with SSI to invert BFCs is 0.97, showing a slight advantage over that with CI, which is 0.93. The advantage is also embodied in a smaller root-mean-square error and relative average error. Figure 3 shows that BFC fields inverted by the adjoint model combined with both methods are consistent with the prescribed field. However, the CI results display a bull’s-eye pattern, which is a common problem caused by the interpolation method. An analysis of the TE2 results lead to the same conclusions: In TE3 and TE4, CI outperforms SSI by a little, demonstrated by the model skill, the root-mean-square error, and the relative average error shown in Table 1. However, the bull’s-eye pattern still exists in the results obtained with CI (Fig. 4).

BFC fields inverted by the adjoint model combined with (a) SSI and (b) CI in TE1, in which the BFC field is prescribed as a paraboloid opening upward.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

BFC fields inverted by the adjoint model combined with (a) SSI and (b) CI in TE1, in which the BFC field is prescribed as a paraboloid opening upward.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

BFC fields inverted by the adjoint model combined with (a) SSI and (b) CI in TE1, in which the BFC field is prescribed as a paraboloid opening upward.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

As in Fig. 3, but for TE3 in which the BFC field is prescribed as a bilinear surface.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

As in Fig. 3, but for TE3 in which the BFC field is prescribed as a bilinear surface.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

As in Fig. 3, but for TE3 in which the BFC field is prescribed as a bilinear surface.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

### b. Practical experiments

Performance of the adjoint model combined with the two methods is also evaluated in a practical context. Harmonic constants of the M_{2} constituent extracted from T/P along-track data (from October 1992 to July 2002) are assimilated into the model. Tidal gauge data are used for evaluation of the simulated results. Statistical results are shown in Table 2, which demonstrates that the tidal simulation is successful with both interpolation methods. This statement is further confirmed by cotidal charts of the M_{2} constituent (Fig. 5), which show the accurate location of amphidromic points near Qinghuangdao and the Yellow River delta. The inverted BFC fields are shown in Fig. 6. On the whole the cotidal charts show similar patterns, that BFCs are larger in Laizhou Bay, Bohai Bay, and Liaodong Bay, while BFCs are smaller in deeper water. However, on closer inspection the BFC surface obtained with SSI is found to be smoother, while that obtained with CI has bumps and depressions around several independent points.

Statistical analysis of the amplitude and phase of the M_{2} tide.

Cotidal charts for the M_{2} constituent in the Bohai Sea obtained with the adjoint model combined with (a) SSI and (b) CI. Shown are coamplitude lines (m, dashed lines) and cophase lines (°, solid lines).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

Cotidal charts for the M_{2} constituent in the Bohai Sea obtained with the adjoint model combined with (a) SSI and (b) CI. Shown are coamplitude lines (m, dashed lines) and cophase lines (°, solid lines).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

Cotidal charts for the M_{2} constituent in the Bohai Sea obtained with the adjoint model combined with (a) SSI and (b) CI. Shown are coamplitude lines (m, dashed lines) and cophase lines (°, solid lines).

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

BFC fields inverted with the adjoit method combined with (a) SSI and (b) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

BFC fields inverted with the adjoit method combined with (a) SSI and (b) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

BFC fields inverted with the adjoit method combined with (a) SSI and (b) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

### c. Discussion

BFCs are introduced for parameterization of the bottom friction. However, the real BFC field is difficult to measure directly for evaluation of inversion results. Here arises the problem of choosing a more appropriate interpolation method to obtain a BFC field closer to the real one.

A phenomenon found in the abovementioned experiments is that the CI results usually show unrealistic extrema around independent points. Although we would like to reduce the number of independent parameters while capturing features of the BFCs field (Lu and Zhang 2006; Zhang and Wang 2014), the distribution of independent points selected according to water depth is irregular. To ensure that BFCs at any grid point can be obtained by interpolation, the CI influence radius should be larger than a certain value. Consequently, the bull’s-eye pattern, especially around dense independent points, is hard to eliminate. In this respect, SSI is more appropriate for the inversion of BFCs.

Although the real BFCs are difficult to measure directly, previous studies show that they are associated with water depth and bottom roughness (Mofjeld 1988; Ullman and Wilson 1998; Li et al. 2004). The fact is obvious that the distribution of water depth and bottom roughness in the Bohai Sea (Li et al. 2014) is nonlinear along any direction. Thus, the real distribution of BFCs, as a function of water depth, bottom roughness, and other parameters, is more likely to be a nonlinear surface. Twin experiments demonstrate that the adjoint model combined with SSI performs better in the inversion of nonlinear surfaces (TE1 and TE2), while that combined with CI performs better in the inversion of bilinear surfaces (TE3 and TE4). This is confirmed by interpolating the bathymetry of the Bohai Sea. We use the same independent points as in previous experiments to reconstruct the bathymetry of the Bohai Sea by the two interpolation methods. Figure 7 shows that the bathymetry interpolated with the surface spline is closer to the real one. The skills of SSI and CI are 0.89 and 0.83, respectively, indicating the superiority of SSI. Therefore, we believe that the BFC field inverted by the model combined with SSI is closer to the real field.

(a) Bathymetry of the Bohai Sea (m) and those obtained with (b) SSI and (c) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

(a) Bathymetry of the Bohai Sea (m) and those obtained with (b) SSI and (c) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

(a) Bathymetry of the Bohai Sea (m) and those obtained with (b) SSI and (c) CI.

Citation: Journal of Atmospheric and Oceanic Technology 34, 9; 10.1175/JTECH-D-17-0012.1

## 4. Summary

The adjoint method is an efficient way to determine BFCs, but too many control variables often lead to ill-posedness of the inverse problem. The independent points strategy serves as a satisfactory solution, where the BFC field is constructed by interpolating the values of a group of independent points under the assumption of spatially varying BFCs. This study explores the feasibility of SSI as an alternative to the traditionally used linear interpolation. The adjoint model combined with SSI is evaluated in the simulation of the M_{2} tide in the Bohai Sea by optimizing the BFCs. In twin experiments, SSI outperforms CI in the inversion of the nonlinear distribution of BFCs. In addition, the surface of BFCs reconstructed by SSI is smoother, while BFCs reconstructed by CI has unexpected extrema. In practical experiments, the BFCs are optimized and the M_{2} tide is simulated by assimilating T/P data. The accurate cotidal charts indicate the feasibility of SSI in the inversion of BFC in practical application.

## Acknowledgments

This research was supported by the National Natural Science Foundation of China through Grant 41606006, the National Key Research and Development Plan through Grant 2016YFC1402304, the Key Research and Development Plan of Shandong Province through Grant 2016ZDJS09A02, the National Natural Science Foundation of China through Grant 41506015, and the Natural Science Foundation of Zhejiang Province through Grant LY16D060001.

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