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The independent point scheme (IPS) is applied to inverting initial condition with the adjoint method for the ocean pollutant transport model in this work. As an improvement, the linear Cressman interpolation is removed and the surface spline interpolation is implemented in the IPS. A series of numerical experiments are carried out to test and compare the improved IPS. And experiment results show that through applying the improved IPS, what is further reduced is mean absolute errors between simulation results and observations. Moreover, the inverted distributions are more smooth, accurate and reasonable. In addition, the application of improved IPS also reduces the variables that need to be inverted and promotes the computational efficiency. By these numerical experiment results, it is demonstrated that the combination of improved IPS and adjoint method can be used for the inversion of initial conditions and parameters estimation more effectively and reliably.

Activities in coastal oceans such as marine fishing, marine shipping and inshore aquaculture are crucial for local economic development, but great damage is caused to marine ecosystems (Øyvind et al.,

Numerical ocean pollutant transport models have gradually become a powerful tool to study pollutant diffusion process (Arkhipov et al.,

In numerical studies of pollutant diffusion, researchers gradually realized that the initial conditions, for instance, the location and concentration of released pollutant, have an enormous influence on accurate prediction of pollutant diffusion. Kojima et al. (

Data assimilation is an effective method to make the best of limited observations, and has been widely used in meteorological and oceanographic predictions in recent years. With the development of data assimilation technology, an advanced data assimilation method which involves the adjoint technique is used to optimize and estimate many parameters of numerical models (Dimet and Talagrand,

In previous works, the Cressman interpolation (CI) was used in IPSs (Fan and Lv,

On the contrary, the surface spline interpolation (SSI) has a better performance in producing a smooth curved surface and it has a wide range of applications (Guo et al.,

This article is organized as follows. Section 2 describes the ocean pollutant transport model, the adjoint model and model setting. The selection of independent points and the introduction of the SSI are included in Section 3. Descriptions and results of numerical experiments are shown in Section 4. Discussion and summary are presented in Sections 5 and 6, respectively.

The governing equation of the ocean pollutant transport model is a three-dimensional advection diffusion equation, which is written as:
_{H}_{V}

In order to guarantee mass conservation, the open boundary condition is described as:

The adjoint method is a powerful tool for inverting initial condition. The basic idea of the adjoint method is very simple that the satisfying results can be obtained through optimizing control parameters including initial conditions, boundary conditions and empirical parameters. The cost function indicates the misfit between the simulated results and observations and it is defined by
_{C}_{C}_{C}

The Lagrangian function is constructed based on Lagrangian multiplier method:

In fact, Equation

In this article, the hydrodynamic background field that is applied to forcing the ocean pollutant transport model is calculated by the three-dimensional Regional Ocean Model System (ROMS; Shchepetkin and McWilliams, 2005). The computing area is the Bohai Sea shown in Fig. _{2}, S_{2}, K_{1}, O_{1}) obtained from the TOPEX/Poseidon Global Inverse Solution (TPXO7.2) (Egbert and Erofeeva,

Topography of the Bohai Sea (depth in meters) and routine monitoring stations (blue dots) in May, 2009. Dots represent the location of stations and the size of each dot indicates the relative concentration of total nitrogen (TN) in May, 2009.

Numerical experiments are performed to test the feasibility and reliability of the modified IPS when be applied to inversing the initial distribution of pollutant. The computing area is the Bohai Sea including the Liaodong Bay, the Bohai Bay, the Laizhou Bay and the central area. The integral time step length is 6 h and the total simulation time is 30 days.

The IPS is that the values of model parameters at some selected points, called independent points, are treated as independent variables and the values of model parameters at other points are obtained by interpolating the independent variables (Guo et al.,

Determining an IPS means the determination of independent points and interpolation method. For independent points, there are two ways to select: uniformly in study area or according to the situation of the study. These two ways were both applied and contrasted by Lu and Zhang (

The Bohai Sea is surrounded by land on three sides, and the Bohai Strait is the only channel on the east side with the Yellow Sea. So, the ability of water exchange of the Bohai Sea is weak (Wang et al.,

The observed pollutant concentration in Fig.

Locations of independent points. Dots represent independent points and stars represent estuaries.

CI is the main method in previous works and its formulation can be found in Li et al. (_{i,j}_{i, j, k}_{k}^{–1}, _{k}_{kk}

The TN concentration value at any other grid point can be obtained through interpolating values at independent points. Therefore, the optimization at all grid points can be achieved through optimizing values at independent points, which effectively reduces the number of variables that need to be inverted. Optimization of values at independent points can be realized through the following steps.

The gradient of cost function with respect to concentration values at independent points is calculated as:
_{kk}^{−6}.

Assumed distribution of TN concentration (mg/L) in the Bohai Sea in ideal experiments.

The number of iteration steps of assimilation has a great impact on the inversion results, which depends on many factors, such as the initial guess value and the magnitude of smoothness (Wang et al.,

The normalized cost function (NCF) and mean absolute error (MAE) are significant assessment criteria for the validity of adjoint method. Figure

Results of ideal experiments. (a) The Normalized cost functions (NCFs), (b) mean absolute errors between simulated values and ‘observation’ at routine monitoring stations (MAE1) and (c) mean absolute errors between the inverted values and prescribed ones at all grid points in the computing area (MAE2s). Note that, in all panels, the blue solid lines are results by the Cressman interpolation (CI) and the red dashed lines are by the surface spline interpolation (SSI).

From Fig.

The blue solid lines represent inverted results by the CI and the red dashed lines are by the SSI in Fig.

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The comparison of distribution inverted by two interpolation methods are depicted in Fig.

Results of ideal experiments. (a, b) The inverted initial distributions, (c, d) the absolute errors between the inverted initial distribution and the assumed shown in Fig.

Through ideal experiments, the effectiveness and advantage of the SSI over the CI are verified by lower MAEs and better similarity of inverted initial distribution to the prescribed.

In ideal experiments, ‘observations’ are obtained through strict mathematical calculation, meaning that there is no error in ‘observations’. But we all know that the practical in situ observations contain noises definitely. Thus, as sensitivity tests, artificial errors are added into ‘observations’, which are 10%, 50% and 80% of the accurate ‘observations’, respectively. Those noisy ‘observations’ are used instead of the error free ones, and then repeat above ideal experiments.

Using two interpolation methods, initial distribution inverted by assimilating ‘observations’ containing errors are shown in Fig.

Inverted initial distributions of sensitivity experiments. (a, b) Inverted by assimilating ‘observations’ containing 10% errors, (c, d) containing 50% errors, (e, f) contains 80% errors. Note that, (a, c, e) are inverted by the CI and (b, d, f) are by the SSI.

In sensitivity experiments, the SSI has a better performance in terms of the inverted TN concentration distribution and MAE than the CI when assimilates inaccurate ‘observations’, which demonstrates the reliability of the SSI and it is more suitable for the application in practical experiments.

The advantage of the SSI is presented through the comparisons with the CI in ideal and sensitivity experiments, and then this interpolation method is implemented with both the real position and values of routine monitoring stations depicted in Fig.

The variations of NCF (Fig.

Results of practical experiments. (a) The NCFs, (b) the MAE1s, (c) the initial distribution inverted by the CI, (d) the initial distribution inverted by the SSI.

In view of the inversion distributions by two interpolation methods in three bays, the comparisons highlight the differences in these places, and the MAE1 in each bay is calculated and listed in Table

The BHB, LDB, and LZB represent the MAE1 in the Bohai Bay, the Liaodong Bay and the Laizhou Bay after 50 iterations, respectively.

In practical experiments, the comparisons of SSI and CI are displayed through the inverted distributions and MAEs in three bays. The more reasonable inverted distribution and the reduced error indicate that the application of SSI causes meaningful and comprehensive improvements.

For acquiring a good initial condition, an appropriate interpolation is the crucial factor, and the application of SSI further improves the accuracy of the inverted initial condition. In fact, the advantage of SSI has been identified by many previous works. The comparison with 4-nearest neighbours method was tested by Perrin et al. (

The spline is a special function that defined piecewise by polynomials, so it is good at smoothing in nature. Pan et al. (

Comparison of the interpolation results by the CI and SSI. (a) The prescribed surface, (b) the interpolation result by the CI, (c) the interpolation result by the SSI.

In addition, the inversion results are also influenced by the number of independent points besides the interpolation method. This factor has been discussed by Wang et al. (

As a result, the application of SSI provides a more smoothness, accuracy and reasonableness inversion result, which also reduces the quantity of independent points and variables that need to be inverted, and improves the computational efficiency of the adjoint model.

More attentions have been paid to the diffusion of pollutant due to serious threats from human activities to marine ecosystem, and a better initial condition is needed for accurate prediction. The ocean pollutant transport model combined with IPS is a power tool to acquire an accurate initial condition. Based on the IPS, the influence of the interpolation method on the accuracy of initial condition must be considered. The CI is mainly used in IPS in pervious works, but its inverted result has inferior smoothness and large errors. On the contrary, the SSI could improve the smoothness of result and it has been used in many studies. As a result, the CI and the SSI are applied to inverting and contrasting the initial condition in the ocean pollutant transport model with the adjoint method.

In ideal experiments, MAE1 and MAE2 by the SSI are lower than those by the CI. Moreover, the initial distribution inverted by the SSI is also closer to the prescribed one. Those results indicate the effectiveness and advantage of SSI over CI. In sensitivity experiments, the MAEs by SSI are always lower than by CI no matter what the artificial error is, also, the distributions inverted by SSI are more similar to the prescribed. Results of the comparisons demonstrate the reliability of SSI. In practical experiments, MAE1s by the SSI are all lower than those by the CI in three bays, and more importantly, the initial distribution inverted by the SSI is more reasonable.

Results of numerical experiments certify the effectivity, reliability and reasonability of the SSI combined with IPS and adjoint method to invert the initial condition for the ocean pollutant transport model. It is worth to do more works based on the SSI.

In this work, the study area is the Bohai Sea, which is the water west of the Bohai Strait (117.5°E–122.5°E, 37°N–41°N). Section 2.3.2 describes the Bohai Sea in more detail and it is shown in Fig.

We gratefully acknowledge the support from Haidong Pan on the program of spline interpolation.

No potential conflict of interest was reported by the authors.