File:cost fig3.png

The normalized cost function J computed within inner-loops (curves) and JNL the cost function computed relative to the nonlinear model during outer-loops (open circles). The various cases shown are for a 1-day assimilation window: (10,3) and Ritz preconditioning using 2 Ritz vectors (black curve, black circles), and 10 Ritz vectors (red curve, red asterisks); and (10,3) and spectral preconditioning using 2 eigenvectors (blue line, blue circles).The horizontal dashed blue line represents the theoretical minimum value of the cost function given by half the number of assimilated observations.