Different atmosphere-ocean general circulation models produce significantly different projections of climate change in response to increases in greenhouse gases and aerosol concentrations in the atmosphere. The main reasons for this disagreement are differences in the sensitivities of the models to external radiative forcing and differences in their rates of heat uptake by the deep ocean. In this study, these properties are constrained by comparing radiosonde-based observations of temperature trends in the free troposphere and lower stratosphere with corresponding simulations of a fast, flexible climate model, using techniques based on optimal fingerprinting. Parameter choices corresponding either to low sensitivity, or to high sensitivity combined with slow oceanic heat uptake are rejected. Nevertheless, a broad range of acceptable model characteristics remains, such that climate change projections from any single model should be treated as only one of a range of possibilities.