We seek to determine if a small number of measurements of upper ocean temperature and currents can be used to make estimates of the drag coefficient that have a smaller range of uncertainty than previously found. We adopt a numerical approach using forward models of the ocean's response to a tropical cyclone, whereby the probability density function of drag coefficient values as a function of wind speed that results from adding realistic levels of noise to the simulated ocean response variables is sought. Allowing the drag coefficient two parameters of freedom, namely the values at 35 and at 45 m/s, we found that the uncertainty in the optimal value is about 20% for levels of instrument noise up to 1 K for a misfit function based on temperature, or 1.0 m/s for a misfit function based on 15 m velocity components. This is within tolerable limits considering the spread of measurement-based drag coefficient estimates. The results are robust for several different instrument arrays; the noise levels do not decrease by much for arrays with more than 40 sensors when the sensor positions are random. Our results suggest that for an ideal case, having a small number of sensors (20-40) in a data assimilation problem would provide sufficient accuracy in the estimated drag coefficient.