A Method Using the Huber Function for Inversion of Tidal Open Boundary Conditions of the M2 Constituent in the Bohai and Yellow Seas

Abstract

Abstract The adjoint assimilation method has been widely used in various ocean models, and a series of technical schemes have been developed at the same time. Open boundary conditions (OBCs) in the two-dimensional (2D) tidal model of the M2 tidal constituent in the Bohai and the Yellow Seas (BYS) were inverted successfully using the adjoint assimilation methods in previous studies. However, the cost function in the adjoint assimilation method usually used the L2 norm in the past, which is difficult to maintain the robustness of the method when there are outliers. Meanwhile, using the L1 norm with strong robustness will shield the outliers’ information fully. Therefore, we propose a new scheme that replaces the L2 norm with the Huber function to improve the robustness of the adjoint assimilation method and absorb the data’s useful information to some extent. This scheme was verified in the ideal experiments in which magnitudes of the misfit vector were significantly reduced and the quality control (QC) process was simplified consequently. In the practical experiments, the introduction of the Huber function improved the accuracy of inversion in the inshore area using mixed data containing tide gauges and satellite altimetry. With this scheme, the root-mean-square errors (RMSEs) between the estimation and the observed values at tide gauge stations were reduced from ∼8 cm with the original scheme to ∼6 cm. Testing the new scheme in more complex models and how it might be affected remains a topic for future study. Significance Statement The adjoint assimilation method has been effectively applied in various ocean models. The cost function in the adjoint assimilation is usually in the form of the L2 norm, which presents poor robustness. By using the Huber function instead of the L2 norm as the cost function, we proposed a new scheme that can perfectly handle the potential outliers in data and noticeably improve the robustness of the adjoint assimilation method. The new method was applied to the inversion of tidal open boundary conditions of the M2 constituent in the Bohai and the Yellow Seas. Both the ideal and practical experiments verified the effectiveness of the developed scheme.