  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.
