Based on an isopycniccoordinate internal tidal model with the adjoint method, the inversion of spatially varying vertical eddy viscosity coefficient (VEVC) is studied in two groups of numerical experiments. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Results demonstrate that the VEVCs can be inverted successfully with IPSs and the model has the best performance with the optimal IPSs. Using the optimal IPSs obtained in Group One, the inversions of VEVCs on two different Gaussian bottom topographies are carried out in Group Two. In addition, performances of two optimization methods of which one is the limitedmemory BroydenFletcherGoldfarbShanno (LBFGS) method and the other is a simplified gradient descent method (GDMS) are also investigated. Results of the experiments indicate that this adjoint model is capable to invert the VEVC with spatially distribution, no matter which optimization method is taken. The LBFGS method has a better performance in terms of the convergence rate and the inversion results. In general, the LBFGS method is a more effective and efficient optimization method than the GDMS.
Internal tide, which is the internal wave of tidal frequency, is a ubiquitous phenomenon in the oceans. Rattray [
Determination of the vertical eddy viscosity coefficient (VEVC), which describes the vertical mixing in the ocean, plays an important role in the study of energy exchange and material transportation. The VEVC is regularly regarded as a constant in numerical models. Schemes to determine the VEVC mainly include the Prandtl mixinglength hypothesis model, the
Satellite remote sensing technology and other related technologies provide us with a large number of data. Thus, it is one of the most important missions in physical oceanography to make use of the data efficiently and precisely as well as to combine the observation data with present numerical models. Indeed, data assimilation with the adjoint method provides an effective access to these missions. The use of the adjoint method in marine science can be traced back to 1980s. The adjoint model is capable of optimizing control parameters in numerical simulation. Bennett and McIntosh [
There are two main objectives of this paper. One is to study the inversion of the VEVC with an internal tidal model and the adjoint method. According to the introductions above, a lot of studies have been carried out to investigate the inversion of the control parameters of internal tide such as the open boundary condition [
Two groups of numerical experiments are carried out to study the inversion of spatially varying VEVCs based on an isopycniccoordinate internal tidal model with the adjoint method. In Group One, the influences of independent point schemes (IPSs) exerting on parameter inversion are discussed. Group Two investigates the inversions of VEVCs on two different Gaussian bottom topographies and the performances of two optimization methods which are the GDMS and the LBFGS methods.
This paper is organized as follows. Section
An isopycniccoordinate internal tidal model with adjoint assimilation method is employed in this paper. There are two parts in the internal tide model. One is forward model with the governing equations and the other is adjoint model with the adjoint equations. The two models are used to simulate the internal tide and to optimize the control variables, respectively. Chen et al. [
According to the equations and derivations of Chen et al. [
Accurately programming the adjoint in such problems as the present one is quite tricky and experience has shown that it is essential to check the accuracy of the adjoint computation before proceeding with the minimization runs [
Here,
In order to test the accuracy of the adjoint method, two experiments are carried out in which two different types of
Figure
Variation of
The available observation data may not be sufficient and control parameters to be determined may be excessive in practice. That may cause illposedness of the inversion problem. Richardson and Panchang [
In this paper, the IPS is used to optimize the control parameter. The basic idea of IPS is as follows: some grids (e.g.,
According to Section
The values of VEVC at the independent grids can be calculated inside the model and values at other grids are gained through interpolation using (
There have been many largescale optimization methods to solve the minimization problem [
Generally speaking, numerical methods to solve the minimization problems have the similar iterative formula as follows:
The GDM is a simple and feasible method to define the search direction as follows:
In the GDMS, the step length
LBFGS is an optimization algorithm in the family of quasiNewton methods that approximates the BFGS algorithm using a limited amount of computer memory. This method is first described in the work of Nocedal [
It requires the search direction to be
Note that
All the experiments in this paper are implemented in an ideal regional area from 116°E to 124.17°E and from 18°N to 23.17°N with in mind the practical sea area located around the Luzon strait. The horizontal resolution is
Eastern and western boundaries:
North and south boundaries:
Similar as Chen et al. [
The T/P altimeter data is widely spread throughout the ocean and it can be used to invert VEVC. In this work, we pick 89 calculating points based on the distribution features of T/P altimeter observation as the observation points (Figure
Planform of topography (e.g., topography A) and locations of the observations (white dots).
Two kinds of topographies are tested in this paper and they are generated based on the two formulas in (
Topographies A and B. Note that the abscissa and ordinate axes are labeled with zonal index and meridional index, respectively.
For each experiment, the optimization of the VEVC can be implemented with the following steps.
The prescribed VEVC is given and the forward model is run. The whole simulation time is 20 period of the
Initial value of the control parameter (VEVC) is given and forward model is run to get the simulated results of all the state variables such as current velocity and water elevation. The value of cost function
Difference between the simulated elevation and the “pseudoobservation” plays as the external force of the adjoint model. Via backward integrating the adjoint equations in a period of the
Using formula (
Update the unknown control variables with a certain optimization method.
If the stopping criterion of iteration is reached, bring the iteration to an end and return the optimized parameter. Otherwise, update all the parameters and go back to Step
In the experiments of this paper, all initial values of VEVC are set to 0.005 and the total number of iterations is allowed to be 100 at most. The chosen convergence criterion is that the last two values of the cost function are sufficiently close, which is defined by
Two groups of numerical experiments are carried out: the influence of IPSs on the inversion of VEVC is studied in Group One; in Group Two the ability of this internal tide model to invert different kinds of VEVC with spatial distribution is examined. Two kinds of spatial distribution of VEVC are prescribed and given in Figure
Planform of two prescribed spatial distributions of VEVC.
In Group One, nine experiments are carried out to discuss the influence of IPS on the inversion of VEVC. The distance between independent points (IP) ranges from 20′ (length of 2 grids) to 100′ (length of 10 grids) and details are listed in Table
Settings of independent point schemes in Group One.
IPS  1  2  3  4  5  6  7  8  9 

Number of IPs  360  160  96  60  40  35  24  24  15 
Distance between IPs (′)  20  30  40  50  60  70  80  90  100 
In Group Two, four numerical experiments (NEs) are carried out which are numbered as NE1~NE4, respectively. Each experiment is implemented with GDMS and the LBFGS methods. Information of topographies and prescribed VEVCs for all NEs is listed in Table
Topographies and distributions of VEVC in Group Two.
Experiment  Topography  Distribution 

NE1  A  a 
NE2  A  b 
NE3  B  a 
NE4  B  b 
All the experiments in Group One and Group Two are carried out following Steps
Figure
MAEs versus IP distance in Group One. The abscissas indicates distance between adjacent IPs (unit: ′) while the ordinate indicates MAE of inversion results. The solid lines are values of different experiments and the dashed lines indicate the minimum values of two solid lines, respectively.
As is shown in Figure
With the respective optimal IPSs and the iteration process in Section
Planform of inversion results in Group Two.
NE1(I)
NE1(II)
NE2(I)
NE2(II)
NE3(I)
NE3(II)
NE4(I)
NE4(II)
Comparison of the inversion results with the prescribed VEVCs indicates that all the given spatial distributions of VEVC are successfully inverted after 100 iteration steps. The main features of all distributions can be recovered very well. Surfaces of the inverted VEVC with the LBFGS are much smoother than those with the GDMS. Compared against the inversion results with the GDMS (left panels), patterns with the LBFGS method (right panels) are closer to the prescribed VEVC. More statistic data will be presented in the next paragraphs.
MAEs of the four numerical experiments after assimilation are calculated and listed in Table
Inversion errors of VEVC in Group Two (unit: m^{2}/s).
Method  Experiment  

NE1  NE2  NE3  NE4  
GDMS(I) 




LBFGS(II) 




To compare the effectiveness of the two methods to invert the VEVC, we make statistics on the percentages of the grids at which the MAEs are less than
Effectiveness analyses of inversions in Group Two.
Method  Experiment  

NE1  NE2  NE3  NE4  
GDMS(I)  42.53%  42.24%  40.99%  40.91% 
LBFGS(II)  79.76%  79.76%  79.76%  79.76% 
Combining the inversion patterns, the inversion errors of the VEVC, and the effectiveness analyses, conclusions can be drawn that the LBFGS has a better performance in reducing the inversion errors.
Finally we come to the optimization history for all the experiments carried out in Group Two. The variations of the cost function normalized by its initial value, that of the
Optimization history for experiments of Group Two, about (a) the cost function normalized by its initial value
Note that all experiments with the LBFGS method reach the convergence criterion and stop after 4 iterations, which indicates that the computation time for the LBFGS method is one twentyfifth of that for the GDMS. Figure
It is also clear in Figure
Based on an isopycniccoordinate internal tidal model, the inversion of VEVC is studied in this paper. A series of numerical experiments are carried out to examine the influence factors on the inversion of VEVCs in four aspects: independent point schemes (IPS), topography, the spatial distribution of VEVC, and the optimization methods. For each experiment, the cost function, the
The IPS is introduced and discussed in Group One. All the VEVCs can be inverted successfully with IPS. MAE is regarded as the comparison criterion of the result. After comparing the 9 experiments, the correctness of the IPS is confirmed and the optimal IPSs are selected for the GDMS and the LBFGS methods, respectively.
Based on the optimal IPSs in Group One, two kinds of VEVC distributions are successfully inverted with this adjoint model on two kinds of topography in Group Two. MAEs after optimization are at the level of 10^{−4} (10^{−5}) for the GDMS (LBFGS), which is one (two) order(s) of magnitude lower than the initial value. All the cost functions and their gradient norms with respect to the VEV lead satisfactory declines no matter which optimization method is taken. Compared with the GDMS, the LBFGS method has a remarkably better performance, not only in terms of the convergence rate but also in terms of the final inversion results. The computation time for the LBFGS method is much shorter than that for the GDMS. To sum up, the LBFGS method is a more effective and efficient method than the GDMS in terms of the inversion of the VEVC. Nevertheless, the GDMS is more convenient and controllable so it should not be ignored and should be taken seriously as a choice for the inversion of the VEVC with spatially distribution.
The success of numerical experiments lays a solid foundation for the practical experiments and encourages us to carry out experiments in practical sea area with measured data and the real T/P altimeter data.
Let us start with the governing equations in Chen et al. [
Layer 1 (surface layer)
Layer
Layer
The variables and background of the governing equations have been introduced in Chen’s [
The cost function is defined as
Finally, according to the typical theory of Lagrangian multiplier method, we have the following firstorder derivate of Lagrangian function with respect to the control parameter
The authors declare that there is no conflict of interests regarding the publication of this paper.
Partial support for this research was provided by the National Natural Science Foundation of China through Grant 41371496, the State Ministry of Science and Technology of China through Grant 2013AA122803, and the Fundamental Research Funds for the Central Universities 201262007 and 201362033.