Handling errors in four-dimensional variational data assimilation by balancing the degrees of freedom and the model constraints: A new approach

For many years, strongly and weakly constrained approaches were the only options to deal with errors in four-dimensional variational data assimilation (4DVar), with the aim of balancing the degrees of freedom and model constraints. Strong model constraints were imposed to reduce the degrees of freedom encountered when optimizing the strongly constrained 4DVar problem, and it was assumed that the models were perfect. The weakly constrained approach sought to distinguish initial errors from model errors, and to correct them separately using weak model constraints. Our proposed i4DVar* method exploits the hidden mechanism that corrects initial and model errors simultaneously in the strongly constrained 4DVar. The i4DVar* method divides the assimilation window into several sub-windows, each of which has a unique integral and flow-dependent correction term to simultaneously handle the initial and model errors over a relatively short period. To overcome the high degrees of freedom of the weakly constrained 4DVar, for the first time we use ensemble simulations not only to solve the 4DVar optimization problem, but also to formulate this method. Thus, the i4DVar* problem is solvable even if there are many degrees of freedom. We experimentally show that i4DVar* provides superior performance with much lower computational costs than existing methods, and is simple to implement.