 Solution

Treat the quoted solar spectrum as irradiance normal to the beam, approximately constant over the narrow band. The incident band irradiance is $F_{\perp }=19\mathrm{W}{\mathrm{m}}^{-2}$ and the horizontal value is $F_0=F_{\perp }\sin 30^{\circ }=9.5\mathrm{W}{\mathrm{m}}^{-2}$. The cloud's vertical optical depth is $25(0.4)=10$; including the clear atmosphere gives ${\tau }_v=10.07$. The slant optical depth is ${\tau }_s={\tau }_v/\sin 30^{\circ }=20.14$. Thus the Beer-Lambert law givesThe removed fraction is $1-t_{\mathrm{dir}}=0.9999999982$, almost the full incident horizontal $9.5\mathrm{W}{\mathrm{m}}^{-2}$. Beam-normal surface irradiance is twice the horizontal value. These are direct-beam quantities, not the total diffuse-plus-direct surface flux. If the supplied irradiance were already horizontal, the projection factor would instead be omitted.