We report observations of the concentrations of hydrogen radicals, OH and HO2, in the middle and upper troposphere. Combined with simultaneous measurements of NO, O3, CO, H2O, CH4, non-methane hydrocarbons, and the ultraviolet and visible radiation field, the data set provides a direct demonstration of the processes that produce ozone in this region of the atmosphere. Comparison of the measured concentrations of OH and HO2 with calculations based on their production from water vapor, ozone, and methane, demonstrate that these sources are insufficient to explain the observed radical concentrations in the upper troposphere. We suggest that the photolysis of carbonyl and peroxide compounds transported from the lower troposphere to this region can control the concentrations of OH and HO2, and thus the production rate of ozone. Photochemical production of ozone, determined from the measured concentration of HO2 and NO, exceeds 1 part per billion by volume each day in most of the upper tropospheric air masses encountered. This production rate is faster than previously thought, and implies that anthropogenic activities that add NO to the upper troposphere, such as biomass burning and aviation, will lead to production of more ozone than expected.