Estimation of Bottom Friction Coefficients Based on an Isopycnic-Coordinate Internal Tidal Model with Adjoint Method

Based on an isopycnic-coordinate internal tidal model with the adjoint method, three groups of ideal experiments are carried out in order to investigate the estimation of spatially varying bottom friction coefficients (BFCs). In group 1, five values of distance between independent points (DIP) are used to invert the BFCs with the distribution of conical surface. Results show that the BFCs can be inverted successfully with independent point scheme and the strategy with a DIP of 50 can yield the best results. In group 2, five values of interpolation radius (IR) are used to invert the BFCs with the distribution of conical surface. Results show that the strategy with an IR of 1.9 times of DIP can yield the best results. Based on the results of the first two groups, group 3 adopts the optimal DIP and IR to estimate 4 kinds of spatially varying BFCs. The results indicate that the isopycnic-coordinate internal tidal model with the adjoint method has a good ability to estimate the spatially varying BFCs; the inversion results of the BFCs with the distribution of revolution paraboloid are better than those with the distribution of conical surface.


Introduction
Internal tides are internal waves with tidal frequency and generated in a stratified ocean as the barotropic tidal currents interact with topography [1][2][3].They are ubiquitous in the oceans and play an important role in the dissipation of barotropic tides and in mixing the deep ocean [4][5][6][7][8].Generally speaking, internal tides are mainly studied by analytical and numerical modeling because the in situ observations are rare.Rattray Jr. [9], Baines [10,11], Gerkema [12], Garrett and Gerkema [13], Llewellyn Smith and Young [14], and Balmforth and Peacock [15] constructed analytical models and obtained the analytical solutions on ideal topographies so as to investigate the generation and propagation of internal tides.Although these analytical models aided us in understanding the tide-topography interaction process, they had to face some mathematical difficulties.Fortunately, numerical simulation has been able to provide us with an effective mean to study internal tides.Chuang and Wang [16], Holloway [17], Kang et al. [18], Niwa and Hibiya [19], Jan et al. [20], and Buijsman et al. [21] carried out many numerical studies on internal tides, which can be divided into two categories.One is the studies on generation mechanism and influence factors of internal tidal over an ideal topography, and the other is the studies on generation and propagation of internal tides in real oceans.
Determination of parameters is an important step in the numerical simulation of internal tides.As the most important parameter, the open boundary conditions have been studied very thoroughly [22][23][24][25][26][27].But usually the open boundary conditions can be treated as being distributed in a line.On the other hand, the bottom friction coefficients (BFCs) with spatial distribution are also important coefficients in the study of tides.So far, however, the relevant studies are still very few.Chevalier et al. [28] carried out a regional study on the tidal hydrodynamics of the Hudson Bay with a finite element model T-UGOm.In the study, they prescribed the importance of regionally varying friction coefficients in computing energy budget.With a rapid development in computing technology and the progress of ocean observation technology, the adjoint method has been widely used in BFCs estimation.Das and Lardner [29,30] estimated the spatially varying BFCs and depth based on a 2D tidal model with the adjoint method.Ullman and Wilson [31] estimated the temporal and spatial variability of BFCs by assimilation data of ADCP in the Hudson River using the adjoint method.Heemink et al. [32] used the adjoint method to optimize open boundary conditions, spatially varying depth, and BFCs based on a 3D tidal model.Lu and Zhang [33] studied the inversion problem of spatially varying BFCs based on a 2D tidal model with the adjoint method and simulated the M 2 tide in the East China Sea.Zhang and Lu [34] constructed a three-dimensional numerical barotropic tidal model with the adjoint method and studied the estimation of the open boundary conditions, BFCs, and the vertical eddy viscosity coefficients based on their model.Zhang et al. [35] optimized the BFCs in the Bohai and Yellow Seas based on a 2D tidal model using the adjoint method.Altaf et al. [36] used the adjoint method to estimate the depth values and the BFCs in a large-scale shallow sea model.However, these studies did not involve how to determine BFCs in an internal tidal model.This paper is to present a train of thought to solve this problem.
Three groups of ideal experiments are performed to investigate the estimation of spatially varying BFCs based on an isopycnic-coordinate internal tidal model with the adjoint method.Group 1 investigates the influence that the distance between independent points (DIP) exerts on the inversion results.Group 2 investigates the influence of interpolation radius (IR) on the inversion results.And group 3 investigates the inversion problem of BFCs with 4 prescribed distributions.This paper is organized as follows.The isopycniccoordinate internal tidal model with the adjoint method is briefly introduced in Section 2. Experimental design is presented in Section 3. Results are discussed in detail in Section 4. Finally, we make a summary and draw some conclusions in Section 5.

Model Introduction
In this paper, the isopycnic-coordinate internal tidal model with the adjoint method is the same as that constructed by Chen et al. [26], which contains a forward model and an adjoint one.The forward model is used to simulate internal tides and the adjoint one is used to optimize parameters using the adjoint method.The reasonability and feasibility of the model have been tested by Chen et al. [26].And the formulation will not be detailed in this paper.In the following part, the derivation of BFCs adjustment and the independent point scheme are shown in detail.

BFCs Adjustment.
According to the derivation of Chen et al. [26], we have the following first derivative of Lagrangian function with respect to BFC: where where   is the potential density in the th layer, where  is the step length,   , and  +1 , are the priori and adjusted values of BFC in the th iteration, respectively, and (/ , ) () is the gradient of cost function with respect to the BFC, whose L2-norm is ‖(/ , ) () ‖.In order to guarantee that the cost function declines steadily, formula (3) could be modified as follows:

Independent Point Scheme.
Considering that the amount of available observations is limited in practical applications, the independent point scheme, which is easy to implement mathematically and reasonable physically, is employed in parameter optimization.A proper independent point scheme can improve the accuracy of parameter estimation, so as to improve the accuracy of simulation results [37].
The basic idea of optimizing BFCs with independent point scheme is described as follows.Some grids in computational area are selected as independent points.Values of BFCs at independent points are obtained by using adjoint method and those at other grids are obtained by interpolation, which can be described as follows:  where  , is the value of BFC at independent point (, ),  , is the value at other grids (, ), and  ,,, is the weight coefficient, which adopts the form of Cressman [38]: where  ,,, is the distance between (, ) and (, ) and  is the interpolation radius.Then the gradient of cost function with respect to  , is computed as where / , is the gradient of cost function with respect to  , , which is calculated by (2).

Experimental Design
The calculation region (from 116 ∘ E to 124 ∘ E and from 18 ∘ E to23 ∘ E) with an ideal topography is shown in Figure 1 Based on the model, three groups of ideal experiments (i.e., groups 1 to 3) are carried out to investigate the estimation of the spatially varying BFCs.In all groups, values of the prescribed BFC range from 1 × 10 −3 m 2 /s to 3 × 10 −3 m 2 /s.In groups 1 and 2, BFCs with the distribution of conical surface are prescribed and shown in Figure 2. In group 3, BFCs with 4 distributions are prescribed and shown in Figure 3.
In group 1, five strategies are implemented to investigate the influence that the DIP exerts on the inversion results.The DIPs are 30  , 40  , 50  , 60  , and 70  , and the IR are 60  , 80  , 100  , 120  , and 140  corresponding to 5 strategies, respectively.That is to say, the IR is 2 times of DIP in this group.
In group 2, the optimal DIP in group 1 is employed, and 5 strategies are implemented to investigate the influence of IR on the inversion results.The interpolation radii corresponding to 5 strategies are 1.7, 1.8, 1.9, 2.0, and 2.1 times of DIP, respectively.
In group 3, the optimal DIP and IR obtained based on the results of the previous experiments are employed to invert the BFCs with 4 prescribed distributions.The prescribed distributions are shown in Figure 3, in which panels (a) and (b) are two conical surfaces, while panels (c) and (d) are two paraboloids of revolution.

Experimental Results and Discussions
In the process of parameter optimization, different iteration numbers are tried in group 1, and 100 iterations are a good choice based on the following considerations: (1) the misfit between "observations" and the simulated results is very small and approximately constant when it declines to a certain value after 100 iterations; (2) the cost functions are almost no longer falling after 100 iterations; (3) the calculation amount is acceptable for the 100 iterations.In the other two groups, the iteration number is also taken as 100.  4 and the mean absolute errors (MAEs) between them are shown in Figure 5. From Figure 4 we can find that all the inverted distributions show good agreements with the prescribed one, indicating that, by using independent point scheme, the BFCs with the distribution of conical surface can be inverted successfully.The MAE in BFCs before assimilation is 3.67 × 10 −4 .After assimilation, all the MAEs are decreased especially for Strategy 3, of which the value reaches the minimum in this group.Statistics of the relative errors (REs) of inversion results and average vector differences (AVDs) between pseudoobservations and simulated results are listed in Tables 1 and  2, respectively.At most grids, the REs are less than 10%, while only at very few grids, the values are more than 20%, suggesting the feasibility of the model.The AVD in interface elevation before assimilation is 7.17 × 10 −4 .From Table 2 we can find that, after assimilation, the AVDs are decreased dramatically compared with the value before assimilation.
The iteration histories of the cost functions and their gradient norms with respect to BFCs (all values are normalized  by their own initial values) are shown in Figure 6.The cost functions and their gradient norms are declining dramatically during the assimilation process.After 100 iterations, the cost functions and their gradient norms are reduced by more than 2 orders of magnitude and by more than 1 order of magnitude compared with their initial values, respectively, implying the efficiency of the gradient descent method.Based on the above analysis, the following conclusions can be summarized: the BFCs with the distribution of conical surface can be inverted successfully based on the isopycniccoordinate internal tidal model with the adjoint method; The use of a proper DIP can indeed effectively improve the precision of parameter estimation, and the DIP of 50  can lead to the best inversion results.

Results and Discussions of Group 2.
According to the conclusion in Section 4.1, the DIP in this group is set to be 50  .And 5 strategies are implemented to investigate the influence of IR on the inversion results.The interpolation radii in 5 strategies are 1.7, 1.8, 1.9, 2.0, and 2.1 times of DIP, respectively.The prescribed and inverted distributions of BFCs are shown in Figure 7 and the MAEs between them are shown in Figure 8. From Figure 7 we can find that all the inverted distributions show good agreements with the prescribed one.The MAE in BFCs before assimilation is 3.67 × 10 −4 .After assimilation, all the MAEs are decreased especially for Strategy 3, of which the value reaches the minimum.
In addition, numerical experimental results show that, after 100 iterations, the cost functions and their gradient norms are reduced by more than 2 orders of magnitude and by more than 1 order of magnitude compared with their initial values, respectively.
Based on the above analysis, the use of a proper IR can indeed effectively improve the precision of parameter estimation, and the IR of 1.9 times of DIP can lead to the best inversion results.
For the BFCs with the distribution of conical surface, the DIP is set to be 50  and the IR is set to be 1.9 times of DIP which leads to the best inversion results.

Results and Discussions of Group 3.
Based on the results of the first two groups of experiments, the DIP is set to be 50  and the IR is set to be 1.9 times of DIP in this group.Four   experiments are carried out to invert 4 distributions of BFCs.The inverted distributions of BFCs are shown in Figure 9.
From Figure 9 we can find that all the inverted distributions show good agreements with the prescribed ones, indicating that all 4 prescribed distributions of BFCs can be inverted successfully.
The MAEs in BFCs before and after assimilation are listed in Table 3, which shows that the MAEs after assimilation are decreased compared with the values before assimilation; the MAEs after assimilation with the distributions of two conical surfaces exceed those with the distributions of two revolution paraboloids.The iteration histories of the cost functions and their gradient norms (all values are normalized by their own initial values) are shown in Figure 10, which shows that the cost functions and their gradient norms are declining dramatically during the assimilation process; after 100 iterations, the cost functions and their gradient norms are reduced by more than 2 orders of magnitude and by more than 1 order of magnitude compared with their initial values, respectively; the descent of the two indexes (i.e., the cost functions and their gradient norms) for the two revolution paraboloids are superior to those for the two conical surfaces.
Based on the above analysis, the isopycnic-coordinate internal tidal model with the adjoint method has a good ability in estimating the spatially varying BFCs, and the inversion results of the BFCs with the distribution of revolution paraboloid are better than those with the distribution of conical surface.We think that the prescribed BFCs with the distribution of conical surface change more remarkably than those with the distribution of revolution paraboloid; The prescribed BFCs with the distribution of conical surface have a cuspidal point, while those with the distribution of revolution paraboloid do not.

Summary and Conclusions
Based on the isopycnic-coordinate internal tidal model with the adjoint method, three groups of ideal experiments are carried out in order to investigate the estimation of spatially varying BFCs.
In group 1, five strategies are implemented to investigate the influence that the DIP exerts on the inversion results.The numerical experimental results show that by using independent point scheme, the BFCs with the distribution of conical surface can be inverted successfully; the MAEs in BFCs after assimilation are decreased compared with the value before assimilation, and especially, the MAE obtained with the strategy with a DIP of 50  reaches the minimum; The cost functions and their gradient norms with respect to BFCs are declining dramatically during the assimilation process.The numerical experimental results indicate that the strategy with a DIP of 50  leads to the best inversion results.
In Group 2, the DIP is set to be 50  , and 5 strategies are implemented to investigate the influence of IR on the inversion results.The numerical experimental results show that the MAEs after assimilation are decreased compared with the value before assimilation, and, especially, the MAE obtained with the strategy with an IR of 1.9 times of DIP reaches the minimum; after assimilation, the cost functions and their gradient norms are all reduced dramatically.The numerical experimental results indicate that the strategy with an IR of 1.9 times of DIP leads to the best inversion results.
In group 3, the DIP is set to be 50  and the IR is set to be 1.9 times of DIP based on the results of the first two groups of experiments.Four experiments are carried out to invert 4 distributions of BFCs, respectively.The numerical experimental results show that all 4 prescribed distributions of BFCs are inverted successfully; the MAEs after assimilation are decreased compared with the value before assimilation, and furthermore the MAEs with the distributions of two conical surfaces exceed those with the distributions of two revolution paraboloids; after 100 iterations, the cost functions and their gradient norms are all reduced dramatically, and the cost functions and their gradient norms with the two conical surfaces are superior to those with the two revolution paraboloids.The numerical experimental results indicate that the isopycnic-coordinate internal tidal model with the adjoint method has a good ability in estimating the spatially varying BFCs, and the inversion results of BFCs with the distribution of revolution paraboloid are better than those with the distribution of conical surface.
The ideal experiments successfully invert the spatially varying BFCs based on the isopycnic-coordinate internal tidal model with the adjoint method.The optimal independent point scheme can be used as a reference for numerical simulation of internal tides in the South China Sea.The work in this paper provides a way for parameter optimization in study of internal tide.For the spatially varying coefficients of the interface friction and the horizontal eddy viscosity, we can borrow this way to determine the values in internal tide simulation.

Figure 1 :
Figure 1: Plan view of calculation region (black dots are observation locations).

Figure 2 :
Figure 2: The prescribed BFCs with distribution of conical surface.

Figure 4 :
Figure 4: The prescribed and inverted distributions of BFCs obtained with 5 strategies.

Figure 7 :
Figure 7: The prescribed and inverted distributions of BFCs obtained with 5 strategies.

Figure 10 :
Figure10: The iteration histories of the cost functions and their gradient norms ( 0 and  0 are the values of cost function and its gradient norm before assimilation, resp.).
is Lagrangian function and  , is the value of BFC at grid (, ).The gradient of cost function (i.e., ) with respect to  , can be deduced by

Table 1 :
Statistics of the REs of inversion results obtained with 5 strategies.Results and Discussions of Group 1.The DIPs in 5 strategies in group 1 are 30  , 40  , 50  , 60  , and 70  , respectively.The prescribed and inverted distributions of BFCs are shown in Figure

Table 2 :
The AVDs in interface elevation obtained with 5 strategies (AA indicates after assimilation, BA indicates before assimilation).

Table 3 :
The MAEs in BFCs before assimilation (BA) and after assimilation (AA).