An overview and illustration of sensitivity of forecasts with respect to initial conditions

Sensitivity of forecasts with respect to initial conditions is an intermediate step in studies dealing with the impact of features in the initial conditions, or of observations in a data assimilation system, on the quality of the forecast. The sensitivity diagnostics however are obtained as approximations, and various formulations have been proposed over recent years. Since the sensitivity is assessed by computing a distance measure, the corresponding cost function can be estimated by a Taylor expansion. More recently, quadrature formulations have also been proposed. In this paper, these two approaches are re-assessed and attention is paid to specific assumptions made in those developments: the sometimes conflicting use of assumptions of linearity, the way non-linear aspects are taken into account and the relative formal merits of each technique. The presence, omission or absence of the second-order model derivative is also addressed. Eventually, a proposal for extending sensitivity diagnostics in a non-linear iterative context is made. The various diagnostics are illustrated and discussed in the framework of the Lorenz 96 toy model, for which both the first- and second-order model derivatives have been implemented. Several initial condition states are obtained by the means of a four-dimensional variational assimilation (4D-Var) algorithm. The sensitivity diagnostics are presented with respect to the observed dynamical model regime as verification time, beyond the assimilation window, is increased: respectively for a quasi-linear, a weakly non-linear and an early-phase, starting non-linear regime. The numerical results suggest that quadrature-based approximations of cost function reduction are more robust than Taylor-expansion-based ones. Furthermore, the quadrature-based diagnostics do not require the usual omission or simplification of high-order derivatives. A straightforward extension to an iterative context also is possible and illustrated.