A Note on the Uniqueness Problem of Variational Adjustment Approach to Four-Dimensional Data Assimilation

Abstract

A model for nonlinear resonant Rossby waves is utilized to obtain analytical solutions. "Identical twin" approach is adopted, in which one specific solution is chosen as the control which serves as pseudo-observation. Then initial condition is varied and thus specified various solutions are compared with the control. The cost function is computed and plot vs. the initial condition. Multiple minima are revealed in the cost function. In the context of variational adjustment approach to four-dimensional data assimilation, which is usually conducted with some minimization scheme based on the gradient information, this multiple-minimum phenomenon in cost function implies nonuniqueness, i. e., the final convergent solution depends upon the first guess for the initial condition. Such nonuniqueness problem becomes more serious due to the increasing complexity of the cost function as the assimilation period gets longer. The relation to nonlinear energy exchange is discussed.