Data Assimilation Based On Level Set And Optimal Transport Applied To Oceanic Pollution Forecast:Theory And Algorithm

In recent years,with the rapid development of marine exploration,pollution acci-dents such as oil spill have occurred frequently,causing serious damage to the ecological environment.It is of great significance to improve the ability of marine pollution fore-casting.The forecast of petroleum and other pollutants on the sea surface are mostly simulated by the forward model.Limited by initial and boundary conditions,param-eters,and numerical discretization,long-term tracking of pollutants will produce large errors.Currently,variational data assimilation(VDA)is an efficient method of retrieving the state of dynamical equations,which combines the mathematical information provided by equations with the physical information contained in observation in an optimal way.Considering the particularity and difficulty of the oil spill tracking problem,in this pa-per,the level-set equation is adopted as the pollution transport model,and some efficient approaches are proposed for assimilating the information of diffusion scope in the obser-vation.Therefore,the initial contour could be well optimized.In addition,to improve the performance of ocean currents reconstruction,we propose an adaptive sparsity-promoting image data assimilation via dictionary learning method,in which more physical structures of the flows are concerned.Real test cases are carried out to demonstrate the viability of the proposed methods.The details are as follows:Firstly,due to the particularity of the remote sensing observation of oil pollutant,it is hard to obtain relatively accurate concentration.In this paper,we introduce the level-set method into the framework of VDA.A contour-fitting cost function is established for combining the level-set model with the information about diffusion scope in the observa-tion.In addition,the gradient of the new cost function with respect to the initial contour is derived by the adjoint approach.Numerical results of real test cases demonstrate that the contour information in the observation can be assimilated by the proposed method,and the performance on complex topological changes generated during the evolution of oil slicks can be improved.Then,considering the position errors contained in the observation of oil pollutant,topological VDA based on optimal transport theory is introduced.A quadratic Wasser-stein metric is used to characterize the difference between the contours from the observa-tion and the simulation by the level-set model.It can avoid the double penalty effect pro-duced by the l~2-norm-based method.Furthermore,the gradient of the new cost function with respect to the initial contour is derived based on the Kantorovich potential.Numer-ical results of real test cases show that the proposed method can overcome non-Gaussian error from the position and shape of the background,and reduce the impact caused by data missing on the result of assimilation.Finally,regarding the ill-posedness of the sea surface currents reconstruction,we propose a sparsity constraint 4D-Var via dictionary learning,and the gradient of the Gateaux differential term in the new objective function with respect to initial velocity is derived by the adjoint method.The proposed method combines the advantages of the vorticity characterizing the physical feature of the flow with the adaptive sparse transform with multi-scale and multi-directional characteristics,therefore the optimal currents will contain more oceanic features.The l~1+l~2 mixed norm non-smooth optimization problem is solved by the split Bregman algorithm.Numerical results of real test cases show that the speed of currents construction can be accelerated,and the artifacts can be also reduced by the proposed method.