To elucidate the underlying assumption behind four existing gradient wind (GW) models in relation to a translating tropical cyclone, we theoretically derived all the GW models solely from the storm-relative (SR) momentum equations in a unified manner. Then, we characterized the five GW models, including a new model, by deriving the mathematical form of the residuals of each GW model from the SR momentum equations, and evaluated the quantitative characteristics of the residuals with a numerical example. The numerical results indicated that the GW model used by Georgiou et al. (1983) exhibited the largest residuals among the five models, whereas the GW model used by Kepert (2001) exhibited virtually zero residuals as theoretically expected if the geostrophic balance and the synoptic-scale environmental pressure superposition can be assumed. The performance of each GW model was evaluated by root mean square errors (RMSEs) with reference to the horizontal wind velocities observed by wind profilers at high altitudes during the passage of Typhoon Jebi 2018 toward and across Japan. The RMSE results are found consistent with the residual characteristics theoretically deduced and numerically evaluated, indicating the excellence of the Kepert model in performance and adequacy of the related assumptions.