**8. Conclusions**

The algorithm of DDM in the problem of variational DA is considered. The theoretical study of the problem has been carried out, including the study of the unique and dense solvability of the inverse problem. The iterative algorithm is proposed and the theorem concerning its convergence has been formulated. To illustrate the theoretical results the numerical experiments for the linearized shallow water equations have been carried out. The results of the numerical experiments show that the iterative algorithm converges with the parameter *τk* described in this paper. The DDM does not significantly affect the modelling results. The result of the simulation with variational DA method fits the observational data. However, the additional noise in the observational data yields a noise in the solution results. Experiments show that the effect of a noise may be mitigated by the regularisation. It is worth to be noted that the regularisation parameter and the iteration parameter should be chosen depending on a problem to be solved. These parameters affect the accuracy, convergence and rate of convergence.

In this paper we focus on the theoretical study of the inverse problem for the linearized system of shallow water equations. To illustrate the theoretical results the simplified example was considered. To estimate the approach, its quality and the possibility of application to realistic problems, more experimental results should be provided. We are going to continue the study and test the algorithm in the regional ocean models (e.g., [22,31]). In this case the algorithm will be implemented at each time step to the problem corresponding to step 3-c of the splitting method (mentioned in Section 1). The general approach outlined here may be extended to other regional ocean models. However, for each special problem, boundary conditions on the inner and outer liquid boundaries should be formulated depending on chosen simplifications and the corresponding theoretical study should be carried out. It is worth noting that the methodology based on the theory of optimal control and adjoint equations (used in this paper) may be applied to nonlinear problems. In addition, it may become possible to use meshes of different scales in subdomains. Moreover, the general approach may be improved to become suitable for modelling multiphysics and multiscale coastal ocean processes.

**Author Contributions:** V.A. stated the problem and suggested the methodology; N.L. and T.S. investigated the problem, proposed an algorithm, carried out the numerical experiments; V.A., N.L. and T.S. analysed the experiments.

**Funding:** The work was supported by the Russian Science Foundation (project 19–71–20035, studies in Sections 2 and 7) and by the Russian Foundation for Basic Research (project 18-31-00096 mol\_a, studies in Sections 3–6).

**Conflicts of Interest:** The authors declare no conflict of interest.
