PDF of N_ap calculated with the filters U in Fig. 3. The orange lines are determined with the simulations and the orange shaded region is the standard deviation from 48 sub-patches. The black dashed lines correspond to the results from the new model, and for comparison the red dashed line in the upper left panel is from the old model. Lower panels: residuals Δp(N_ap) of all lines with respect to the simulations.