Sound speed profiles in high spatiotemporal resolution using multigrid three-dimensional variational method: a coastal experiment off northern Shandong Peninsula

It is essential to acquire sound speed profiles (SSPs) in high-precision spatiotemporal resolution for undersea acoustic activities. However, conventional observation methods cannot obtain high-resolution SSPs. Besides, SSPs are complex and changeable in time and space, especially in coastal areas. We proposed a new space-time multigrid three-dimensional variational method with weak constraint term (referred to as STC-MG3DVar) to construct high-precision spatiotemporal resolution SSPs in coastal areas, in which sound velocity is defined as the analytical variable, and the Chen-Millero sound velocity empirical formula is introduced as a weak constraint term into the cost function of the STC-MG3DVar. The spatiotemporal correlation of sound velocity observations is taken into account in the STC-MG3DVar method, and the multi-scale information of sound velocity observations from long waves to short waves can be successively extracted. The weak constraint term can optimize sound velocity by the physical relationship between sound velocity and temperature-salinity to obtain more reasonable and accurate SSPs. To verify the accuracy of the STC-MG3DVar, SSPs observations and CTD observations (temperature observations, salinity observations) are obtained from field experiments in the northern coastal area of the Shandong Peninsula. The average root mean square error (RMSE) of the STC-MG3DVar-constructed SSPs is 0.132 m/s, and the STC-MG3DVar method can improve the SSPs construction accuracy over the space-time multigrid 3DVar without weak constraint term (ST-MG3DVar) by 10.14% and over the spatial multigrid 3DVar with weak constraint term (SC-MG3DVar) by 44.19%. With the advantage of the constraint term and the spatiotemporal correlation information, the proposed STC-MG3DVar method works better than the ST-MG3DVar and the SC-MG3DVar in constructing high-precision spatiotemporal resolution SSPs.

     

An application of the A-4DEnVar to coupled parameter optimization

In variational methods, coupled parameter optimization (CPO) often needs a long minimization time window (MTW) to fully incorporate observational information, but the optimal MTW somehow depends on the model nonlinearity. The analytical four-dimensional ensemble-variational (A-4DEnVar) considers model nonlinearity well and avoids adjoint model. It can theoretically be applied to CPO. To verify the feasibility and the ability of the A-4DEnVar in CPO, “twin” experiments based on A-4DEnVar CPO are conducted for the first time with the comparison of four-dimensional variational (4D-Var). Two algorithms use the same background error covariance matrix and optimization algorithm to control variates. The experiments are based on a simple coupled ocean-atmosphere model, in which the atmospheric part is the highly nonlinear Lorenz-63 model, and the oceanic part is a slab ocean model. The results show that both A-4DEnVar and 4D-Var can effectively reduce the error of state variables through CPO. Besides, two methods produce almost the same results in most cases when the MTW is less than 560 time steps. The results are similar when the MTW is larger than 560 time steps and less than 880 time steps. The largest MTW of 4D-Var and A-4DEnVar are 1 200 time steps. Moreover, A-4DEnVar is not sensitive to ensemble size when the MTW is less than 720 time steps. A-4DEnVar obtains satisfactory results in the case of highly nonlinear model and long MTW, suggesting that it has the potential to be widely applied to realistic CPO.