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Adjoint method is used to assimilate pseudoobservations to simultaneously estimate the OEVP and the WSDC in an oceanic Ekman layer model. Five groups of experiments are designed to investigate the influences that the optimization algorithms, step-length, inverse integral time of the adjoint model, prescribed vertical distribution of eddy viscosity, and regularization parameter exert on the inversion results. Experimental results show that the best estimation results are obtained with the GD algorithm; the best estimation results are obtained when the step-length is equal to 1 in Group 2; in Group 3, 8 days of inverse integral time yields the best estimation results, and good assimilation efficiency is achieved by increasing iteration steps when the inverse integral time is reduced; in Group 4, the OEVP can be estimated for some specific distributions; however, when the VEVCs increase along with the depth at the bottom of water, the estimation results are relatively poor. For this problem, we use extrapolation method to deal with the VEVCs in layers in which the estimation results are poor; the regularization method with appropriate regularization parameter can indeed improve the experiment result to some extent. In all experiments in Groups 2-3, the WSDCs are inverted successfully within 100 iterations.

The development of ocean observation technology provides a wealth of fruits, especially the satellite data as well as acoustic doppler current profilers (ADCP) data. It is a mass of observations that make the application of data assimilation techniques to oceanography possible. In recent years, researchers have developed some data assimilation methods, such as successive corrections [

During the past few decades, many researches have showed that the adjoint method occupies an increasingly important position in data assimilation [

In this paper, the adjoint method in conjunction with an oceanic Ekman layer model is used to obtain the optimal estimates of the vertical distribution of eddy viscosity and the surface wind drag coefficients from pseudoobservations of wind velocities and current velocity profiles within the water columns. At present, the error of ADCP survey technique could be controlled within 5 meters. Therefore, the Ekman layer (the thickness is 100 meters in this paper) could be divided into 20 layers evenly.

To solve the problem above, good many feasible large-scale optimization algorithms can be found in [

The organization of this paper is as follows. The numerical model is given briefly in Section

A horizontally unbounded ocean surface layer with depth

A cost function which quantifies the discrepancy between the model results and the observations can be constructed as

Equations (

The aim is that the cost function is minimized subject to the initial boundary value problem (

As shown in Section

In the first place, the performances of GD, CG, and L-BFGS are compared via a group of ideal experiments in Section

As we all know, step-length affects the result involved with numerical optimization. If VEVC is estimated successfully when the inverse integral time of the adjoint model is reduced, the workload of calculation will be saved greatly. The actual oceanic eddy viscosity profile cannot be observed and it is unknown. If different prescribed distributions of VEVCs are estimated successfully, it can be inferred that the model based on the adjoint method has a good ability to estimate the VEVCs. Based on the above considerations, the following four groups of experiments are carried out in order to investigate the influences that the three optimization algorithms, step-length, inverse integral time of the adjoint model, and prescribed distribution of the VEVCs exert on the inversion results, respectively. In all four groups, the given value of the WSDC is set to be 0.0012. Therefore, the effect of

The performances of GD, CG, and L-BFGS algorithms are compared via a series of ideal experiments in this group. Because of simplicities of the WSDCs in this model, the experiments are performed with only VEVCs briefly. The inverse integral time of the adjoint model is 10 days. In the process of parameter optimization, we tried different iteration numbers. And the best result in experiments with different iteration numbers is adopted as the result of the algorithm. Clearly, the step-length is set to 1 as a matter of priority [

The prescribed and inverted VEVCs and variation curves of cost functions are shown in Figure

The RMSEs between VEVCs before and after assimilation in Group 1.

Algorithms | GD | CG | L-BFGS |
---|---|---|---|

Before assimilation (×10^{−2}) |
1.57 | 1.57 | 1.57 |

After assimilation (×10^{−4}) |
1.68 | 12 | 2010 |

(a) The prescribed and inverted VEVCs and (b) variation curves of cost functions in Group 1.

Figure

The prescribed and inverted VEVCs with L-BFGS.

Based on the results above, we choose the GD algorithm to estimate the VEVCs and WSDC in the following groups of experiments.

The purpose of Group 2 is to investigate the influence that the step-length exerts on the inversion results. The inverse integral time of the adjoint model is 10 days. In the process of parameter optimization, we tried 1000 iterations. Clearly, the step-length is set to 1 as a matter of priority [

strategy 1:

strategy 2:

strategy 3:

strategy 4:

where

The RMSEs between VEVCs before and after assimilation are listed in Table

The RMSEs between VEVCs before and after assimilation in Group 2.

Strategies | 1 | 2 | 3 | 4 |
---|---|---|---|---|

Before assimilation (×10^{−2}) |
1.57 | 1.57 | 1.57 | 1.57 |

After assimilation (×10^{−4}) |
1.68 | 4.11 | 2.21 | 1.79 |

The prescribed and inverted VEVCs in Group 2.

The variation curves of cost functions (

The modeled (black dotted) and “observed” (black solid) current velocities and their differences (blue solid) at (a) 25 m and (b) 75 m after 1000 iterations in Group 2 with strategy 1.

The prescribed and inverted values of WSDC in Group 2 with 4 strategies.

Figure ^{−4} in magnitude. At other depths of water, the simulated velocities are also highly consistent with the observation values. The WSDCs are able to be successfully inverted after 95, 51, 37, and 47 iterations with four strategies, respectively.

The results show that the oceanic Ekman layer model with adjoint method could estimate OEVP and WSDC successfully; the step-length has an effect on inversion results: the best assimilation efficiency can be obtained with strategy 1; that is, the step-length which is equal to 1 obtains the best results in Group 2.

In this section, the influence that the inverse integral time of the adjoint model exerts on the inversion is investigated. Strategy 1 described in Group 2 is used for the step-length. The iteration number is set to be 500 mainly based on the following considerations: (1) the misfit between observation values and the simulated results is very small and approximately constant when it declines to a certain value after 500 iterations; (2) the cost functions are almost no longer falling after 500 iterations. A series of inverse integral times are tested, that is, 10 days, 9 days, 8 days, 7 days, 6 days, 5 days, 4 days, 3 days, 2 days, and 1 day.

Figure

The prescribed and inverted VEVCs in Group 3.

The RMSE in VEVCs before assimilation is

The RMSEs in OEVCs after assimilation in Group 3.

The result indicates that the inversion effects can be associated with the inverse integral time of the adjoint model, and the best assimilation efficiency is obtained with 8 days of inverse integral time.

In order to demonstrate the capability of the oceanic Ekman layer model with adjoint method in estimating the OEVP, different distributions of the VEVCs are designed in this section. Strategy 1 described in Group 2 is used to determine the step-length, and the inverse integral time of the adjoint model is 8 days. One conclusion drawn from experiments of Group 3 is that preferable results can be obtained by increasing iteration steps appropriately. Therefore, in this group, different iteration steps are adopted according to the distributions of the OEVCs. Six different distributions are designed as follows:

distribution 1:

distribution 2:

distribution 3:

distribution 4:

distribution 5:

distribution 6:

We obtained reasonable inversion results after 1000 iterations for distribution 1. For some complicated distributions, increasing iteration steps appropriately can also yield good inversion result. Taking distributions 3, 5, and 6, for examples, the iteration steps need to reach 20000. For other complicated distributions, such as distributions 2 and 4, however, the inversion results are poor at the bottom of water even though the iteration number is very large. That is, when the VEVCs increase along with the depth at the bottom of water, the estimation results are relatively poor. The RMSEs between VEVCs before and after assimilation are listed in Table

The RMSEs between VEVCs before and after assimilation in Group 4.

Distributions | 1 | 2 | 3 | 4 | 5 | 6 |
---|---|---|---|---|---|---|

Before assimilation (×10^{−2}) |
2.27 | 2.27 | 2.20 | 1.37 | 2.21 | 1.33 |

After assimilation (×10^{−4}) |
1.95 | 51.47 | 2.47 | 55.63 | 7.32 | 2.63 |

The prescribed and inverted VEVCs in Group 4. (a)–(f) denote the prescribed (black line) and inverted (red line) VEVCs for six distributions, respectively.

The estimation results are relatively poor when the VEVC increases along with the depth at the bottom of water (i.e., distributions 2 and 4). We design new strategy to solve the problem. Because there are only a few layers at the bottom of water (called the bottom layers in this paper) where the results are poor, the extrapolation method can be used to calculate the VEVCs. In this paper, linear extrapolation is applied because of its simplicity. More specifically, the VEVCs in bottom layers are calculated by linear extrapolation using the values of VEVCs in subbottom layers above on the bottom layers. The RMSEs between VEVCs before and after extrapolation for distributions 2 and 4 are listed in Table

The RMSEs between OEVCs before and after extrapolation.

Distributions | 2 | 4 |
---|---|---|

Before extrapolation (×10^{−4}) |
51.47 | 55.63 |

After extrapolation (×10^{−4}) |
5.21 | 6.95 |

The prescribed and inverted VEVCs. (a)-(b) denote the prescribed (black line) and inverted (red line) OEVCs for distributions 2 and 4, respectively.

The results indicate that the prescribed distributions at the bottom of water are inverted successfully with extrapolation method. In practical application, we can learn from the new strategy to deal with the VEVCs in a few layers at the bottom of water. From the experimental results in this group, it is inferred that the oceanic Ekman layer model with adjoint method is likely to have a good capability of estimating the spatially varying VEVCs. This brings hope for us to apply the model to a real case.

One classical approach that is often used to solve the ill-posed inverse problems is regularization method. Tikhonov regularization which is pioneered by Tikhonov [

The smoothing functional is constructed as follows:

In this section, the regularization parameters are set to 10^{2}, 10, 10^{0} (i.e., 1.0), and 0, respectively. It should be noted that there is no Tikhonov stabilizer when

(a) The prescribed and inverted VEVCs and (b) variation curves of cost functions in Group 5.

(a) RMSEs and (b) correlation coefficients between the inverted and prescribed VEVCs in Group 5.

As we can see in Figure

According to the results of ideal experiments, we can draw the following conclusions: using the adjoint method to assimilate ocean observation values into an oceanic Ekman layer model, the distribution of the VEVCs and the WSDC could be inverted at the same time.

In Group 1, the best estimation results are obtained with the GD algorithm, and even the CG and L-BFGS get a better convergence rate. The GD algorithm, which is controllable and easy to implement, should be regarded seriously as a choice. In Group 2, four strategies of step-length are tested. For all strategies, the inversion results of the VEVCs are satisfactory, and it is proved that when the step-length is equal to 1, it improves the assimilation efficiency. In Group 3, the inverse integral time of the adjoint model is tested. The results indicate that the inversion effects are associated with the inverse integral time and 8 days of inverse integral time leads to the best results in experiments of Group 4; the good assimilation efficiency is achieved by increasing iteration steps if the inverse integral time is short. In Group 4, the inversion of six distributions of the VEVCs is examined. The results indicate that some complicated distribution can be inverted by increasing the assimilation steps. We also conclude that, at the bottom of water, if the VEVC decreases along with the depth, the inversion results are satisfactory, while if the VEVC increases along with the depth, the results are relatively poor. To solve this problem, we try to use extrapolation method to calculate the VEVCs in layers where the inversion results are unsatisfactory. The experimental results indicate that the attempt is successful. In a practical application, we can learn from the attempt to deal with VEVCs in a few layers at the bottom of water. The regularization method with appropriate regularization parameter can indeed improve the experiment result to some extent in this inverse problem. The WSDCs in all experiments are able to be inverted successfully within 100 iterations.

The factors tested in this paper are closely combined with inversion efficiency. The work inspires us with an effective way to determine the OEVP and the WSDC in the numerical simulation of Ekman wind-driven currents.

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 State Ministry of Science and Technology of China through Grant 2013AA122803, the Natural Science Foundation of Shandong Province of China through Grant ZR2014DM017, and the National Natural Science Foundation of China through Grants 41321004, 41371496, and 41206001.