Calculation of a velocity distribution from particle trajectory end-points.

The longitudinal component of the velocity of a particle at or near a glacier surface is considered, its position as a function of time being termed its trajectory. Functional relationships are derived for obtaining the trajectory from the spatial distribution of velocity and for obtaining the velocity distribution from the trajectory. It is established that the trajectory end-points impose only an integral condition on the velocity distribution, and that no individual point on the velocity distribution can be determined if only the end-points are known. An example is given of a deduced velocity distribution that is consistent with (although not uniquely determined by) the end-points of several trajectories on the lower reach of Columbia Glacier, Alaska. It is shown that constructing a velocity distribution by assigning the average trajectory velocity to the trajectory mid-point can be subject to errors of several per cent for velocity distribution features that are typical of actual glaciers. The error in this method is determined, and closed-form expressions for the trajectory are obtained, for linear velocity distributions and for two classes of second-degree distributions. The class of functions is identified to which the velocity distribution must belong for this error to be zero.