Editorial Type:
Article
Article Type:
Research Article

A Method Using the Huber Function for Inversion of Tidal Open Boundary Conditions of the M2 Constituent in the Bohai and Yellow Seas

Yuchun Gao aFrontier Science Center for Deep Ocean Multispheres and Earth System, Ocean University of China, Qingdao, China
bPhysical Oceanography Laboratory, Ocean University of China, Qingdao, China
cQingdao National Laboratory for Marine Science and Technology, Qingdao, China

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Shengyi Jiao aFrontier Science Center for Deep Ocean Multispheres and Earth System, Ocean University of China, Qingdao, China
bPhysical Oceanography Laboratory, Ocean University of China, Qingdao, China
cQingdao National Laboratory for Marine Science and Technology, Qingdao, China

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Kai Fu dSchool of Mathematical Sciences, Ocean University of China, Qingdao, China

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Xueying Zeng dSchool of Mathematical Sciences, Ocean University of China, Qingdao, China

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Xianqing Lv aFrontier Science Center for Deep Ocean Multispheres and Earth System, Ocean University of China, Qingdao, China
bPhysical Oceanography Laboratory, Ocean University of China, Qingdao, China
cQingdao National Laboratory for Marine Science and Technology, Qingdao, China

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Online Publication:
19 Jan 2023
Print Publication:
01 Jan 2023
DOI:
https://doi.org/10.1175/JTECH-D-22-0030.1
Page(s):
113–127
Free access

Abstract

The adjoint assimilation method has been widely used in various ocean models, and a series of technical schemes have been developed at the same time. Open boundary conditions (OBCs) in the two-dimensional (2D) tidal model of the M2 tidal constituent in the Bohai and the Yellow Seas (BYS) were inverted successfully using the adjoint assimilation methods in previous studies. However, the cost function in the adjoint assimilation method usually used the L2 norm in the past, which is difficult to maintain the robustness of the method when there are outliers. Meanwhile, using the L1 norm with strong robustness will shield the outliers’ information fully. Therefore, we propose a new scheme that replaces the L2 norm with the Huber function to improve the robustness of the adjoint assimilation method and absorb the data’s useful information to some extent. This scheme was verified in the ideal experiments in which magnitudes of the misfit vector were significantly reduced and the quality control (QC) process was simplified consequently. In the practical experiments, the introduction of the Huber function improved the accuracy of inversion in the inshore area using mixed data containing tide gauges and satellite altimetry. With this scheme, the root-mean-square errors (RMSEs) between the estimation and the observed values at tide gauge stations were reduced from ∼8 cm with the original scheme to ∼6 cm. Testing the new scheme in more complex models and how it might be affected remains a topic for future study.

Significance Statement

The adjoint assimilation method has been effectively applied in various ocean models. The cost function in the adjoint assimilation is usually in the form of the L2 norm, which presents poor robustness. By using the Huber function instead of the L2 norm as the cost function, we proposed a new scheme that can perfectly handle the potential outliers in data and noticeably improve the robustness of the adjoint assimilation method. The new method was applied to the inversion of tidal open boundary conditions of the M2 constituent in the Bohai and the Yellow Seas. Both the ideal and practical experiments verified the effectiveness of the developed scheme.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kai Fu, kfu@ouc.edu.cn; Xueying Zeng, zxying@ouc.edu.cn

Abstract

The adjoint assimilation method has been widely used in various ocean models, and a series of technical schemes have been developed at the same time. Open boundary conditions (OBCs) in the two-dimensional (2D) tidal model of the M2 tidal constituent in the Bohai and the Yellow Seas (BYS) were inverted successfully using the adjoint assimilation methods in previous studies. However, the cost function in the adjoint assimilation method usually used the L2 norm in the past, which is difficult to maintain the robustness of the method when there are outliers. Meanwhile, using the L1 norm with strong robustness will shield the outliers’ information fully. Therefore, we propose a new scheme that replaces the L2 norm with the Huber function to improve the robustness of the adjoint assimilation method and absorb the data’s useful information to some extent. This scheme was verified in the ideal experiments in which magnitudes of the misfit vector were significantly reduced and the quality control (QC) process was simplified consequently. In the practical experiments, the introduction of the Huber function improved the accuracy of inversion in the inshore area using mixed data containing tide gauges and satellite altimetry. With this scheme, the root-mean-square errors (RMSEs) between the estimation and the observed values at tide gauge stations were reduced from ∼8 cm with the original scheme to ∼6 cm. Testing the new scheme in more complex models and how it might be affected remains a topic for future study.

Significance Statement

The adjoint assimilation method has been effectively applied in various ocean models. The cost function in the adjoint assimilation is usually in the form of the L2 norm, which presents poor robustness. By using the Huber function instead of the L2 norm as the cost function, we proposed a new scheme that can perfectly handle the potential outliers in data and noticeably improve the robustness of the adjoint assimilation method. The new method was applied to the inversion of tidal open boundary conditions of the M2 constituent in the Bohai and the Yellow Seas. Both the ideal and practical experiments verified the effectiveness of the developed scheme.

© 2023 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Kai Fu, kfu@ouc.edu.cn; Xueying Zeng, zxying@ouc.edu.cn
Keywords: Tides; Inverse methods; Data assimilation; Optimization

1. Introduction

Data assimilation plays an important role in ocean models and forecasting works (Ghil and Malanotte-Rizzoli 1991; Stammer 2004) which could help improve the ability of model estimation and reduce estimation uncertainty by correcting state variables, parameters, boundary, and initial conditions (Cho et al. 2020). It also plays an important role in other fields such as geoscience (Stehly et al. 2009; Van Leeuwen 2010). When people realize that merely depending on the model, the modeling error may lead to a huge deviation between the predicted results and the true values, they begin to incorporate the observed data into the prediction (Lawless 2013). Among various data assimilation methods, the adjoint assimilation method is a well-known variation approach based on the variational principle and optimal control theory to inverse some important parameters of different oceanography models (Tziperman and Thacker 1989; Yu and O’Brien 1992; Ayoub 2006; Storto et al. 2018; He et al. 2022).

Researchers have studied and improved the adjoint assimilation method in different aspects in recent years, focusing on its cost and performance in various conditions and models. Altaf et al. (2013) proposed a variational data assimilation method based on a proper orthogonal decomposition which was tested in a groundwater subsurface contaminant model to reduce the computational cost. Zheng et al. (2016) proposed a new estimation method based on the incremental four-dimensional (4D) variational method (4D-Var) in a storm surge model. Matsuda and Miyatake (2021) presented a numerical method to obtain the exact gradient, which could be seen as a generalization of partitioned Runge–Kutta methods. For the shallow water equations usually used in oceanic research, Kevlahan et al. (2019) studied the convergence of one-dimensional (1D) shallow water equations with sparse observations, and a two-dimensional (2D) scenario applied for the optimization of a tsunami model was studied afterward (Khan and Kevlahan 2022). The method was also used to estimate the unknown parameters of the new agricultural subsurface drainage model with discharge observations and was improved in optimization (Chelil et al. 2022). Considerable prompts in different parts (numerical computation, optimization algorithms, etc.) of adjoint assimilation have been made for new demands in research. In this paper, a modification of the cost function in the 2D tidal wave model will be proposed for robust estimation.

The tidal wave model is one standard model concerning the application of adjoint methods in oceanography. Interest in the application of variational methods was raised for parameters estimation in tidal wave models several decades ago (Bennett and Mcintosh 1982; Das and Lardner 1992; Lardner et al. 1993; Chertok and Lardner 1996; Lyard 1999; Heemink et al. 2002), and their studies mainly depend on observations from tide gauges. Estimation of different parameters such as the initial condition, bottom friction coefficients, and open boundary conditions (OBCs) were explored for various equation formations and water bodies in different areas nowadays (Gao et al. 2015; Qian et al. 2021; Wang et al. 2021).

The OBCs are important parameters to determine the tidal evaluation, which is a difficulty in tidal models. On the long open boundary, where the water body studied adjoins an adjacent body of water such as the open ocean, significant uncertainties will be brought into the model by the OBCs (Lardner 1993). If there are errors in OBCs, they will propagate into the interior area and affect the model (Seiler 1993). Therefore, they are important control parameters in the tidal model. However, it is challenging to get accurate values relying on some empirical methods (Guo et al. 2012). But with the help of the adjoint assimilation method, it is easier to obtain more accurate OBCs. Researchers have done various work on the inversion of OBCs using the adjoint assimilation method. Lardner (1993) deduced the process of obtaining the adjoint equations and the open boundary correction relationship from the discrete forward tidal model. This method has been applied to open boundary inversion in the Bohai and the Yellow Seas (BYS) in the following research. Zhang et al. (2003) used the limited memory Broyden–Fletcher–Goldfarb–Shanno (L-BFGS) quasi-Newton method in large-scale optimization for the OBCs. Cao et al. (2012) used the Cressman interpolation to fit the coefficients of open boundary points in the independent point scheme, and Pan et al. (2017) used spline interpolation to get a more accurate and smooth inverted result. Jiang et al. (2018) used the adjoint assimilation method to invert the OBCs of the 3D internal tide model and improve the original independent point scheme.

In the inversion mentioned above, the cost function in the adjoint assimilation is usually in the form of the L2 norm, and the L2 norm has been widely used in most situations for its ease of use and obtained reasonable estimations in previous research. With the development of science and technology (especially the precise measurement of sea surface height using satellite altimetry since the 1990s), people can obtain a large amount of data through various observation means, and the scale of the data is also increasing rapidly which put forward higher requirements to identify, preprocess, and analyze data.

The handling of outliers is usually the first thing to think about in data processing. There are two common types of outliers. The first type of outlier, which is what we usually understand intuitively and generally represents the abnormal samples in observations, is defined as observations that deviate from the pattern set by most of the data. The second type of outlier is defined as observations whose error distributions have heavier tails than the traditional Gaussian distribution. In practical problems, data measurement is often affected by various natural and man-made factors. For example, in many atmospheric and oceanic forecasting works, outliers may appear due to various reasons (abnormal weather conditions, multitype data mixing, malfunctioning machines or some political emergencies, etc.), which leads to the problem that the observation error becomes larger or does not obey the Gaussian distribution, and causes the decrease of estimation precision of the adjoint assimilation method. The traditional data assimilation method using the L2 norm is statistically based on the assumption that the errors are independent and identically distributed with Gaussian. However, both types of outliers mentioned above will destroy this assumed distribution of the observation error. Therefore, the L2 norm cannot effectively handle the dataset containing more observations with relatively larger errors. Quality control (QC) of actual observed data is an essential part of any data assimilation process (Lorenc and Hammon 1988). In the process of data assimilation, the purpose of QC is to make sure that the correct observed data are used in the whole data assimilation process and that the incorrect values that deviate too much are discarded (Tavolato and Isaksen 2015).

Nevertheless, the QC process is generally independent of the adjoint assimilation, which depends on empirical knowledge and is usually complicated. To simplify the QC process and incorporate it into the adjoint assimilation to avoid extra selection work, based on the theory of the adjoint assimilation, a more appropriate definition of the cost function is needed to improve the robustness of the L2 norm (i.e., the assumption of the Gaussian distribution needs to be changed). In the early variational QC algorithm, Anderson and Järvinen (1999) changed the probability density function (PDF) of the data error using the combination of a Gaussian distribution and a flat distribution over a bounded interval of values to alter the cost function in 4D-Var. In their approach, the probability density of very large observation errors falls back to Gaussian, which is the assumption that needs to be avoided in robust statistics (Rao et al. 2017). Concerning robustness, it occurs to people directly that we could replace the L2 norm with the L1 norm for its strong robustness. But the use of the L1 norm will ignore some useful information containing in the observed data during some extreme conditions.

To solve this problem, the Huber function was naturally introduced which will be expounded in detail in section 2. Roh et al. (2013) improved the robustness of the ensemble Kalman filter using a Huber type method. On the use of the Huber function in data assimilation, Tavolato and Isaksen (2015) gave a detailed expression of the application of the Huber function in the QC process in 4D-Var. Duan et al. (2017) used the Huber norm distribution in the QC process on the sea surface wind field to improve typhoon prediction. Later Rao et al. (2017) applied the L1 norm and the Huber function in the data assimilation to test a variety of data assimilation methods and verified their feasibility. Its effectiveness has been reflected in some simple examples, such as the shallow water wave model in which the Huber function shows a better result. Therefore, we could try to use the Huber function in the 2D tidal model and develop the corresponding adjoint assimilation method.

The purpose of this paper is to improve the robustness of the adjoint assimilation method in the 2D tidal model with a new scheme using the Huber function. Aiming at the idea of using the Huber function to reduce the influence of outliers on the adjoint assimilation method, we carry out ideal experiments to test its effectiveness and invert the OBCs of M2 constituent in the BYS. In practical conditions, to invert the open boundary, water level observation data are needed. Tide gauges and satellite altimetry are standard methods to measure sea surface height. Due to their characteristics, the two kinds of observation methods will produce mismatched data to some extent, which will be introduced at length in section 4. To bring the information contained in the tidal station observation data into the assimilation process and improve the accuracy of inshore simulation, based on the ideal experiments, we design practical experiments to reduce errors at tide gauge stations.

The remainder of the paper is organized as follows: Section 2 introduces the theoretical process of the adjoint assimilation, the Huber function, and the algorithm steps; section 3 introduces the experimental settings and the results of the ideal experiments which will show that the Huber function can screen outliers to some extent and keep robust under certain conditions. Section 4 presents practical problems caused by the difference between the two kinds of observed data (tidal gauge data and satellite altimeter data), and tries to use the Huber function to improve the accuracy of the inshore simulation. Section 5 is the summary of this paper.

2. Model and optimization method

In this part, a complete adjoint assimilation algorithm is established using the Huber function based on the 2D tidal wave model.

a. Forward model

The 2D depth-averaged tidal model considered in this paper is as follows:
∂ξ∂t+∂[(h+ξ)u]∂x+∂[(h+ξ)υ]∂y=0,
∂u∂t+u∂u∂x+υ∂u∂y−fυ+kuu2+υ2h+ξ−A(∂2u∂x2+∂2u∂y2)+g∂(ξ−ξ¯)∂x=0,
∂υ∂t+u∂υ∂x+υ∂υ∂y+fu+kυu2+υ2h+ξ−A(∂2υ∂x2+∂2υ∂y2)+g∂(ξ−ξ¯)∂y=0,
where x and y refer to the latitude and longitude coordinates, h(x, y) is the hydrostatic depth, ξ(x, y, t) is the change of the tide level relative to the hydrostatic depth, u(x, y, t) and υ(x, y, t) are velocity components in x and y directions, k is the bottom friction coefficient, A is the horizontal eddy viscosity coefficient, f is the Coriolis force parameter, g is the gravitational acceleration, and ξ¯ is the effect on the tide elevation caused by the tide-generating force. The area studied in this work covers the BYS and the rectangular coordinate system is used.
The OBCs are given as follows:
ξOpen(x,t)=a0(x)+∑i=1N[ai(x)cos⁡(ωit)+bi(x)sin⁡(ωit)],
where N is the number of tides with different frequency, {ωi: i = 1, …, N} are the angular frequencies of the tidal constituents considered along the open boundary, {ai, bi: i = 1, …, N} are the coefficients of the Fourier expansions of the OBCs of different tidal constituents.

b. Huber function

The Huber function is a loss function used in robust regression to handle the potential outliers in data or other departures from model assumptions. We use the linear regression problem to introduce our motivation of using the Huber function for data assimilation. Linear regression is used to model the linear relationship between response and explanatory variables, it usually can be described as follows:
y=Xβ+ε,
where y is the vector of observed response variables, X is the matrix of explanatory variables, ε is the residual error and β is the parameters to be estimated.
The most commonly used method for estimating β is the least squares (LS) method that can be described as
β=arg⁡min⁡β12‖y−Xβ‖22.
The LS method can be seen as the maximum likelihood estimation (MLE) under the assumption that the residual error belongs to the Gaussian distribution. It works well for most cases because the residual error usually approximately obeys the Gaussian distribution. However, the LS method will break down once there exist outliers in data, which may appear in extreme conditions. The outliers will lead the residual error to be close to the Laplace distribution whose distribution function is fat tailed and has a higher peak than the Gaussian distribution (Kotz et al. 2001). The MLE under a Laplace distribution for residual error is the least absolute deviation (LAD) regression, which uses the L1 norm to decrease the sensitivity to outliers.
Using the L1 norm is equivalent to completely ignoring all outliers and hence has strong robustness to shield the large residual errors. However, the L1 norm inevitably has several drawbacks. First, outliers may still contain some information, the best way is to set appropriate small weights for them rather than ignore them. Second, the data that are not corrupted by the outliers still have residual error modeled by the Gaussian distribution, and the L1 norm could not fit them well. Third, the L1 norm is nondifferentiable which may lead to large fluctuations in results and affect the smoothness of the solution in numerical calculation. This motivates us to use the Huber function as the fitting term for regression. The 1D Huber function has the form of
LH(ε)={12ε2,if|ε|<δ,δ|ε|−12δ2,else, 
where δ is threshold value. Figure 1a shows the shapes of the above-mentioned three estimators in 1D situation. Figure 1b shows the shapes of the Huber function which is adjustable with different threshold values. Moreover, the derivative of LH is
dLHdε={ε,if|ε|<δ,δsign(ε),else,
which clearly shows how the Huber function works.
Fig. 1.
Fig. 1.

The comparison between different kinds of norms and different threshold values for the Huber function. (a) The function values of the L1, L2 norm, and the Huber function; (b) the function values of different threshold values for the Huber function; (c) the derivative values of different threshold values for the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Combining with Fig. 1b, the Huber function operates with such a mechanism: when the error is limited within a certain range, the Huber function works like the L2 norm and the derivative is as same as that of the L2 norm; otherwise, it will present the quality of the L1 norm and the gradient behaves as a constant δ (Fig. 1c). As can be seen, the Huber function combines the advantages of the L2 and the L1 estimators by being convex and smooth when close to the minimum and less steep for extreme values. Specifically, the Huber function uses the L2 norm (the residuals with small deviation are considered to obey the Gaussian distribution) when the data points are close to the mean value of the dataset, while it uses the L1 norm (the residuals with large deviation are considered to obey the Laplace distribution) when data points are far from the mean value of the dataset. The scale of this transition can be controlled by δ. Therefore, in the process of the problem in this paper, using the Huber function instead of the L2 norm as the cost function for adjoint assimilation can improve the robustness of the adjoint assimilation method and recover its effectiveness in extreme cases.

c. Adjoint assimilation

Variational data assimilation can be regarded as a method derived from the Bayesian theory, which assumes that the state variables and observation errors of the system obey the Gaussian distribution. In this case, computing the maximum a posterior (MAP) of the system state variables can be transformed into a linear (or nonlinear) LS problem. Freitag (2020) gave a detailed derivation of the above process using the Gaussian distribution and the original data assimilation problem can be formulated to a constrained optimization problem.

The adjoint assimilation can be used to optimize the initial field, various model parameters, and boundary conditions. Based on variational analysis and optimal control theory (Asch et al. 2016), adjoint assimilation takes the model as the control condition and the “distance” between the observed data and the computational results as the “drive,” calculates the gradient of the cost function concerning parameters by solving the adjoint equations, and obtains the values of the objective parameters. From the optimal control theory, two important procedures are to obtain the adjoint equations and the gradient of the cost function with respect to the objective parameters (Le Dimet and Talagrand 1986). The cost function of adjoint assimilation is often used to measure the difference between the observed value and the calculated value. It usually takes the following form:
F(ξ)=12ρ[ϕ(ξ)−y],
where ξ is the computed state variable, y is the observed value, ϕ is a map from the state variable space to the observation space, ρ represents the distance function. Considering that the traditional adjoint assimilation method usually takes ρ as the form of the L2 norm which is not robust, we choose the Huber function as a substitute to improve the robustness of the method. According to the background of the problem and the forward model in section 2a, let the cost function be defined as Eq. (7),
F(ξ,a,b)=∑i∑jKi,j∑t∈TLH[ξ(xi,yj,t)−ξi,j*(t)]+12∑lKl{[a(xl)−a*(xl)]2+[b(xl)−b*(xl)]2},
where (xi, yj) is the position that the observations are measured; Ki,j is the weight at (xi, yj), which may depend on the accuracy of observed value partially (Sasaki 1958); ξi,j*(t) denotes the measured value; t is the temporal variable and it belongs a subset T, which contains the whole time interval; l denotes the lth measured position of the open boundary; Kl is the weight at xl; a∗ and b∗ are the prior Fourier expansion coefficients at open boundary points; and LH is the Huber function introduced in section 2b. The problem could be written as
{arg⁡min⁡ξ,a,bF(ξ,a,b),Eqs. (1a)–(1c).
The Lagrange function of (P) could be written as
L(ξ,a,b,λ,μ,ν)=F(ξ,a,b)+∫t∫Ω(λE+μF+νG)dΩdt,
where λ, μ, and ν are the Lagrange multipliers, E, F, and G are the left terms of Eqs. (1a)–(1c), respectively, and Ω is the calculated domain. By taking the variation of the Lagrange function and performing integration by parts, we can obtain the adjoint equations as follows:
∂λ∂t+u∂λ∂x+υ∂λ∂y+kμuu2+υ2(h+ξ)2+kνυu2+υ2(h+ξ)2+g∂μ∂x+g∂ν∂y=∂LH∂ξ,
∂μ∂t−[f+kuυ(h+ξ)u2+υ2]ν−μ∂u∂x−ν∂υ∂x+∂(μu)∂x+∂(μυ)∂y     +(h+ξ)∂λ∂x+A(∂2μ∂2x+∂2μ∂2y)−k(2u2+υ2)(h+ξ)u2+υ2μ=0,
∂ν∂t+[f−kuυ(h+ξ)u2+υ2]μ−μ∂u∂y−ν∂υ∂y+∂(νu)∂x+∂(νυ)∂y     +(h+ξ)∂λ∂y+A(∂2ν∂2x+∂2ν∂2y)−k(u2+2υ2)(h+ξ)u2+υ2ν=0,
where ∂LH/∂ξ is the derivative using the Huber function which is different from that of frequently used L2 norm. Its expression is shown in Eq. (9) which is similar to Eq. (5), and this is the only difference between the new method and the original one in computational process. The Huber function changes the adjoint assimilation method exactly by affecting this part of the adjoint equations:
(∂LH∂ξ)i,j={Ki,j(ξi,j−ξi,j*),if|ξi,j−ξi,j*|<δ,Ki,jδsign(ξi,j−ξi,j*),else.

d. Optimization and discretization

Suppose the observed data ξ* have been obtained; a and b are the guessed initial values. According to the discussion above, we propose the following improved optimization algorithm to solve (P) in ideal experiments:

  • Step 1: Solve the forward model in positive direction using the guessed a and b to get the predicted solution ξ.

  • Step 2: Calculate the cost function value using ξ and ξ*. If the stopping criterion is satisfied or the iteration step exceeds the maximum iteration number, break the iteration and output a and b; otherwise, continue to step 3.

  • Step 3: Solve the adjoint Eqs. (8a)–(8c) in reverse direction to obtain the gradients of the cost function with respect to a and b.

  • Step 4: Update a and b with proper optimization algorithm, set the updated a and b as the new guessed values, and then turn back to step 1.

For the numerical solution of the forward model and adjoint equations, we use the Arakawa C grid for the studied area. The tidal height is taken at the center of the grid and the velocities are taken at the edge of the grid. We use uniform grid for the time interval [0, T] with step size Δt. Accordingly, the original continuous adjoint equations are transformed into the discrete form. The used finite difference schemes are similar to Lu and Zhang (2006).

As for the optimization algorithm, the gradient descent (GD) method and the Broyden–Fletcher–Goldfarb–Shanno (BFGS) method are tested. The way of updating a and b is the same as that of previous studies (Lardner 1993; He et al. 2004; Pan et al. 2017). In our studied region, the M2 constituent is stronger than other constituents, so only the M2 constituent is considered in our experiment, and the formula is simplified accordingly. Moreover, for better calculation results, independent points scheme is used in which the other points except the chosen independent points are obtained by the spline interpolation. The specific formation of spline interpolation is
K(aI−aI*)+∑l=1Mfl,I[∑j=1JTljcos⁡(ωjΔt)]=0,K(bI−bI*)+∑l=1Mfl,I[∑j=1JTljsin⁡(ωjΔt)]=0,
in which the coefficients are the same as Pan et al. (2017); M is the number of the independent points, fl,I is the coefficient of the spline interpolation, and Tlj depends on the position which has the formation as follows:
{Tlj=−gμml,nlj/Δx,(ml,nl) is on the left of the computing domain,Tlj=−gμml−1,nlj/Δx,(ml,nl) is on the right of the computing domain,Tlj=−gνml,nlj/Δy,(ml,nl) is  below the computing domain,Tlj=−gνml,nl−1j/Δy,(ml,nl) is  above the computing domain.

3. Numerical experiments

In this section, for the purpose of comparing the robustness of the two schemes, we designed ideal experiments which are carried out to get the inversion of the OBCs of the M2 tidal constituent in the BYS. Ideal experiments in two cases are conducted to compare the L2 norm and the Huber function with outliers. To ensure the reliability of the experiments, sensitivity experiments with varying numbers and magnitude of outliers are carried out.

a. Experimental settings

The process could refer to the algorithm discussed in section 2. The computation area of the experimental simulation area shown in Fig. 2 covers 34°–41°N, 117°30′–126°40′E with a resolution of 10′ × 10′.

Fig. 2.
Fig. 2.

The studied area in the experiments. The small black dots present the positions of satellite altimeter observations and the hollow circles present the open boundary points.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

As shown in Fig. 2, the open boundary is a horizontal line located around 34°N. The Coriolis parameter takes the local value. The time interval is set as 1/120 of one M2 tidal cycle, which is 372.618 s. The maximum number of iteration steps is set to 100. According to the “observed values” which are obtained from the forward model using the given OBCs, we test different norms when using a and b in two cases, and measure their properties with the mean square errors (MSEs) and the mean absolute errors (MAEs) of H and φ. In case one, we set a(x) = 0.4 sin(πx/18), x ∈ [0, 36], while in case two we set a(x) = 0.4 cos(πx/18), x ∈ [0, 36]. And b(x) ≡ 0 in both cases. The initial guess of a and b are set as the same.

The GD method and the BFGS method are also tested. The results (25% outliers with 20 cm incremental water level) are shown in Fig. 3. To show the error of H and φ, the root-mean-square errors (RMSEs) between the estimation and the observed values are introduced as Eq. (10),
RMSE=12K∑i=1K(α2+β2)1/2,
where α=Hiobscos⁡(φiobs)−Hiestcos⁡(φiest), β=Hiobssin(φiobs)−Hiestsin(φiest), “obs” means the observed values, “est” means the estimated values obtained from the computation process, and K is the total number of the points (King and Padman 2005; Stammer et al. 2014). According to the decrease of the cost function and the RMSEs, the decrease speed with the BFGS method is faster than the GD method. Moreover, the result that the BFGS method ultimately falls to is no worse than that of the GD method. Therefore, in the ideal experiment, the BFGS method is used for updating the parameters.
Fig. 3.
Fig. 3.

The decrease of the log10 of relative value of the cost function and the RMSEs using the Huber function. (a),(b) Case 1; (c),(d) case 2.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

To optimize the effect of the Huber function, a test is performed for selecting threshold values when there are outliers. We display one situation of tests (25% outliers with 20 cm incremental water level). For both cases, since the MSEs and the MAEs of H and φ reduce to a similar extent when δ is around 0.01 or lower (seen in Figs. 4 and 5), we choose a relatively bigger δ which equals 0.01 for a faster rate of descent. Generally, it is accessible that δ is located in a reasonable range. Moreover, the step size α in the following part decreases with the number of iterations increases.

Fig. 4.
Fig. 4.

Error histogram of amplitude and argument in case 1. (a) MSEs and MAEs of the amplitude H; (b) MSEs and MAEs of the argument φ.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Fig. 5.
Fig. 5.

Error histogram of amplitude and argument in case 2. (a) MSEs and MAEs of the amplitude H; (b) MSEs and MAEs of the argument φ.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

b. Experimental results

In the experiment, the effectiveness of the L2 norm and the Huber function are compared when there are outliers. The proportion of outliers and incremental water levels are changed at all time points for different groups. Since different cost functions could not be compared together, to observe the evolution quantitatively during the iteration, the decreases of RMSEs are used for comparison. The last iteration result of RMSEs is used to show the inversion results, and the results of sensitivity experiments are presented in Table 1.

Table 1

RMSEs (cm) of the two cases in sensitivity experiments.
Table 1

The RMSEs in Table 1 demonstrate that the accuracy of the inversion is affected by the proportion and magnitude of the outliers. For both the L2 norm and the Huber function, the RMSEs increase roughly as the proportion and the magnitude increase. The Huber function shows its superiority and presents its stability. On the one hand, when using the Huber function, the RMSEs are kept on the order of 10−3 while the L2 norm mostly has RMSEs on the order of 10−2. The variation range of RMSEs with the proportion and magnitude of the outliers also differs between the L2 norm and the Huber function. For example, in case one, the RMSE is increased to around 7 times the original when using the L2 norm with 50 cm outliers as the proportion of outliers increases from 5% to 25%. However, it is just increased to around 1.5 times the original when using the Huber function. Similar results are also shown in other test groups. Therefore, the Huber function is more stable in the face of outliers than the L2 norm. Moreover, to carefully compare and present the difference between the two methods, the detailed test results with the proportion of 25% outliers and 20 cm incremental water level are shown as follows.

It is easy to see that the Huber function has enhanced the robustness of the adjoint method and avoided the effect of outliers. As shown in Figs. 6 and 7, when there are outliers, the curves representing the Fourier expansion coefficients of the OBCs computed using the L2 norm exhibit a certain degree of deformation, while the results computed using the Huber function keep a better shape.

Fig. 6.
Fig. 6.

The inversion result of Fourier expansion coefficient a in case one (25% outliers with 20 cm incremental water level). (a) The L2 norm; (b) the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Fig. 7.
Fig. 7.

The inversion result of Fourier coefficient b in case two (25% outliers with 20 cm incremental water level). (a) The L2 norm; (b) the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

The equation
MV=12{[Hobscos⁡(φobs)−Hestcos⁡(φest)]2+[Hobssin⁡(φobs)−Hestsin⁡(φest)]2}1/2
shows the formation of magnitudes of the misfit vector (MV). MVs are shown in Figs. 8 and 9, which indicate a notable difference: the magnitudes of the new scheme using the Huber function are smaller than the original one. For the Huber function, the misfit vector in the studied area barely exceeds 0.01, which is much smaller than that of the L2 norm.
Fig. 8.
Fig. 8.

MVs in case one (25% outliers with 20 cm incremental water level) (unit: cm). (a) The L2 norm; (b) the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Fig. 9.
Fig. 9.

MVs in case two (25% outliers with 20 cm incremental water level) (unit: cm). (a) The L2 norm; (b) the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

As shown in Table 2, the MSEs of H and φ could decrease to around 12% and 6% of the MSEs using the L2 norm in case one, respectively, while for case two, the MSEs of H and φ could decrease to around 9% and 26%, respectively. The degrees of decline in H are similar in two cases, while those of φ have some difference. Therefore, in terms of the degree of improvement, the role of the Huber function is also affected by the structure of Fourier expansion coefficients.

Table 2

MSEs and MAEs of the H and φ using different norms with outliers (25% outliers with 20 cm incremental water level).
Table 2

The decreases of the cost function are shown in Figs. 10a and 11a, indicating they reduce to 1.4% and 0.7% of the initial value. From the results shown in Figs. 10b and 11b, the reduction rates of the RMSEs are similar between the L2 norm and the Huber function, while for the extent of the reduction, the Huber function presents its superiority and the RMSE of the Huber function is lower than that of the L2 norm. In addition, the time costs in each optimization cycle of both methods are considered, as shown in Figs. 10c and 11c, presenting decreases of RMSEs with time. The computational time of both methods is the same. When there are outliers, under the same computation time, the degree of decline using the Huber function is greater than using the L2 norm. And the time when they begin to converge is different in different cases.

Fig. 10.
Fig. 10.

Case 1: (a) the decrease of the log10 of relative value of cost function (the Huber function); (b) the decrease of the RMSEs; (c) the decrease of the RMSEs with time.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Fig. 11.
Fig. 11.

Case 2: (a) the decrease of the log10 of relative value of cost function (the Huber function); (b) the decrease of the RMSEs; (c) the decrease of the RMSEs with time.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

4. Practical experiments

Tide gauges and satellite altimetry are standard methods to measure sea surface height. During the practical application, it is found that they have their own advantages and disadvantages. Tide gauge is the major technique for sea level observation and can be dated back to the 1770s (Cipollini et al. 2017). Its precision is constantly improving with the development of high technology (Míguez et al. 2008). Tide gauges are grounded to the land, and provide long-term sea level observation data that are usually used for validating satellite altimetry performance (Birol et al. 2016; Aldarias et al. 2020; Peng and Deng 2020). However, tide gauges may have discontinuous observations and are restricted by their uneven spatial distribution. Also, they are unsuitable for conditions in the deep sea and could be affected by land vertical movement (Jevrejeva et al. 2014; Wöppelmann and Marcos 2016; Cipollini et al. 2017; Khairuddin et al. 2019). Simultaneously, they are irregularly distributed on the continental margins in the BYS.

Satellite altimetry has been providing a precise global measurement of sea surface height since the early 1990s (Adebisi et al. 2021). Satellite altimetry can be used as an alternative method to partly make up for the shortages of tide gauges. It can provide accurate and continuous datasets with global coverage and medium spatiotemporal resolution (Cazenave and Llovel 2010). However, satellite altimetry always has difficulties in the coastal zone due to land contamination for various factors (Anzenhofer et al. 1999; Chelton 2001; Marcos et al. 2019) and incorrect geophysical correction models (Desportes et al. 2007). Satellite altimeter data in the coastal zone are always manually discarded depending on different local conditions since they are regarded as unreliable there. Many advances in coastal altimetry are made to weaken this influence; efforts are still needed to improve the accuracy of altimetry in the coastal zone yet through some analysis of data (Cipollini et al. 2017). Although many researchers have made a lot of use and evaluation of these two observation methods, there is still a lack of enough tests and attention to their integrated use (Adebisi et al. 2021).

To sum up, the satellite altimeter data along the coast have lower precision than in the basin center, which explains the main source of the outliers. The use of the Huber function could provide a way to reduce the influence of inshore satellite altimeter data with larger errors in the adjoint assimilation. Tidal gauge data are not only used as observations along the coast but also as a measurement of the inversion results. To improve the accuracy in the coastal region, we try to absorb information from tidal gauge data in the inshore areas since they are more reliable along the coast.

X-Track is a postprocessing algorithm to recover altimetry data over marginal seas (Birol et al. 2016). According to the above background, in the studied region, we use T/P–Jason satellite altimeter data processed by X-Track and tidal gauge data together as observed values to invert the open boundary. The T/P–Jason satellite altimeter data contain T/P altimetry data and Jason-1 to Jason-3 altimetry data. They have the same orbits and the same sampling intervals, which is 9.915 642 days.

Similar to the ideal experiments, the L2 norm and the Huber function are used to inverse the OBCs, respectively, with the observed values containing satellite altimeter data and tidal gauge data. We hope that the Huber function could absorb some information from tidal gauge data and help improve the inversion accuracy in the coastal region. The studied area and the position of the open boundary are the same as the ideal experiments. The initial guess of Fourier expansion coefficients a and b are set as 0; meanwhile, the initial tidal field is also set as 0. The bottom friction coefficient is not changed during the whole iterated process. In the ideal experiments, it has been verified that the BFGS method has a faster descent speed. Although the GD method has a slow convergence speed, it is stable and not easy to blow up. Given the stability, the GD method and the BFGS method are combined to complete the inversion of the OBCs. Specifically, the BFGS method is used in the first few steps and the GD method is then used in the subsequent steps. The threshold value of the Huber function depends on the dataset, and it is set to a lower value for the satellite altimeter data than for the tidal gauge data.

Figure 12 shows the inverted results of the Fourier expansion coefficients a and b obtained by the L2 norm and the Huber function, which are basically the same. The cotidal charts for the M2 constituent in the studied area are shown in Fig. 13. They basically reflect the same characteristics, that is, there are two amphidromic points in the Bohai Sea that are near Qinhuangdao, China, and the Yellow River delta while another two are sited in the Yellow Sea that one is at the north of Chengshantou, China, and the other is near JiangSu, China.

Fig. 12.
Fig. 12.

The inverted OBCs in practical experiments. (a) The Fourier expansion coefficient a. (b) The Fourier expansion coefficient b.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

Fig. 13.
Fig. 13.

The cotidal charts using (a) the L2 norm and (b) the Huber function.

Citation: Journal of Atmospheric and Oceanic Technology 40, 1; 10.1175/JTECH-D-22-0030.1

To make the experiment more complete, we also carry out the experiments using the L1 norm. On the premise that other parameters are completely consistent, the results of the three norms are shown in Table 3. The L1 norm produces an amplitude with larger errors and other indexes are similar to those of the L2 norm and the Huber function. For the tidal gauge data, the error can be improved by using the Huber function (seen in Table 3) compared with previous studies (Pan et al. 2017, Table 4). In the practical experiment, the RMSE at the tide gauge stations can be greatly reduced from ∼8 to ∼6 cm. Taking into account the magnitude difference between H and φ, the mean relative errors (MREs) are also calculated and indicate the same conclusion.

Table 3

MAE, MSE, MRE at tide gauges in the practical experiments and RMSE.
Table 3

In conclusion, the Huber function can improve the accuracy of adjoint assimilation at tide gauge stations to some extent and reduce the error of tide level simulation in the coastal region on the basis of ensuring that the overall error is at a low level.

5. Conclusions

Adjoint assimilation plays an important role in the inversion of the open boundary. In this paper, we have inherited lots of previous research results on the inversion of the open boundary and propose a new scheme that uses the Huber function to improve the original form (the L2 norm) of the cost function.

During the ideal experiments, we artificially add outliers to invert the open boundary of the M2 tidal constituent of the 2D tidal model of the BYS by using the adjoint assimilation method. When there are some outliers, the OBCs are still inverted successfully using the Huber function, which is tested in the ideal experiments. The quantitative results show that the MVs deduced by the Huber function are notably smaller than that deduced by the L2 norm, and the MSEs and the MAEs have a significant reduction compared to the original method. The results can verify that for the adjoint assimilation method, in some cases, the Huber function can avoid the influence of seriously deviated outliers and maintain robustness.

According to this result, considering the mismatch between the two types of data commonly used in the numerical simulation of 2D tidal wave models, the satellite altimeter data are better than the tidal gauge data in the deep sea, but the accuracy of satellite altimeter data is lower in the coastal region for various reasons. To improve the accuracy of the inversion in the inshore area in practical experiments, we used mixed data. The goal of controlling the overall error at a low level and absorbing the information of tidal gauge data is achieved by using the Huber function. In section 4, the RMSE at the tide gauge stations can be significantly reduced from ∼8 cm with the original scheme to ∼6 cm with the new scheme, and it is better than previous experimental results. Therefore, this scheme is helpful in improving the accuracy of numerical simulation of the 2D tidal wave model near the shore.

In addition, there are many alternative or improved methods for many steps in the experiment, including the optimization algorithm and the difference scheme in the optimization process. How to choose appropriate ways for different specific problems deserves discussion. Likewise, under the different choices above, whether the effectiveness of the Huber function will be affected and why they are worth negotiating. Ultimately, the selection of the threshold value of the Huber function also needs to be verified by experiments. For the two types of outliers in the 2D tidal wave model, the Huber function could produce a better effect on the biased observations and filter out the outliers hidden in the normal observations without independent empirical work; thus, it can simplify the QC process. In view of the wide application of the adjoint assimilation method in the field of meteorology and oceanography and the imperfection of current observation process technology, this improvement is of great practical significance.

Acknowledgments.

This work is supported by the Natural Science Foundation of Shandong Province of China ZR2020MD056 and ZR2021MA005, the National Natural Science Foundation of China through Grants 42076011, 11601497, U1806214, and 11871444, and Fundamental Research Funds for the Central Universities 202264006. The X-TRACK products are provided by the Center of Topography of the Ocean and Hydrosphere (CTOH; French observation service dedicated to satellite altimetry studies).

Data availability statement.

The regional along-track sea level anomaly data for T/P–Jason altimeters are provided by AVISO at https://www.aviso.altimetry.fr/en/data/products/auxiliary-products/coastal-tide-xtrack.html.

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    • Export Citation

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    • Export Citation

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    • Export Citation

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    • Export Citation

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