Derivation of a Time-area Curve for an Elliptical Shaped Basin

In this study, the time-area curve for an elliptical shaped basin was analytically derived. The assumptions introduced to derive the time-area curve are as follows. First, an infinite number of raindrops reach the impervious surface of the elliptical shaped basin evenly in space. Second, the vertical axis of the ellipse is assumed to be the channel of the elliptical shaped basin. Third, the direction of the surface runoff is always perpendicular to the channel flow. Fourth, the flow velocity remains the same in any location within each channel and land surface, the ratio of their flow velocity is consistent everywhere within the basin. As a result, the time-area curve for the basin can be derived with the equivalent ellipse of the basin along with the proper characteristic velocity within each channel and land surface. Furthermore, by multiplying the flow velocity with this time-area curve, one can derive the inflow to the linear reservoir, of which the outflow becomes the Clark IUH.